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Using Item Parameter Predictions for Reducing Calibration Sample Requirements--A Case Study Based on a High-Stakes Admission Test

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Title: Using Item Parameter Predictions for Reducing Calibration Sample Requirements--A Case Study Based on a High-Stakes Admission Test
Language: English
Authors: Esther Ulitzsch, Dmitry Belov, Oliver Lüdtke, Alexander Robitzsch
Source: Journal of Educational Measurement. 2026 63(1).
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 52
Publication Date: 2026
Document Type: Journal Articles
Reports - Research
Descriptors: High Stakes Tests, Test Items, Difficulty Level, Computation, Bayesian Statistics, Maximum Likelihood Statistics, Sample Size, Accuracy, Prediction
DOI: 10.1111/jedm.12426
ISSN: 0022-0655
1745-3984
Abstract: In item difficulty modeling (IDM), item parameters are predicted from the items' linguistic features, aiming to ultimately render item calibration redundant. Current IDM applications, however, commonly do not yield the required prediction accuracy. To immediately exploit even somewhat inaccurate IDM predictions, we blend IDM with established Bayesian estimation techniques. We propose a two-step approach where (1) IDM predictions are obtained and (2) employed to construct informative prior distributions. We evaluate the approach in a case study on small-sample calibration of the 3PL in a high-stakes test. First, concerning implementation, we find computationally efficient penalized maximum likelihood estimation to be comparable to the best-performing MCMC-based approach. Second, we investigate sample size reductions achievable with state-of-the-art IDM predictions, finding negligible gains compared to merely considering the historical distribution of parameters. Third, we evaluate the prediction accuracy required for a targeted sample size reduction by gradually increasing simulated IDM prediction accuracies. We find that required accuracies can counterbalance each other, allowing calibration sample size to be reduced when either high-quality item difficulty predictions or good predictions of item discriminations and pseudo-guessing parameters are available. We argue that these evaluations provide new, portable IDM benchmarks quantifying performance in terms of achievable sample size reductions.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1501461
Database: ERIC
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  Value: <anid>AN0192629996;mea01mar.26;2026Apr01.06:22;v2.2.500</anid> <title id="AN0192629996-1">Using Item Parameter Predictions for Reducing Calibration Sample Requirements—A Case Study Based on a High‐Stakes Admission Test </title> <sbt id="AN0192629996-2">Introduction</sbt> <p>In item difficulty modeling (IDM), item parameters are predicted from the items' linguistic features, aiming to ultimately render item calibration redundant. Current IDM applications, however, commonly do not yield the required prediction accuracy. To immediately exploit even somewhat inaccurate IDM predictions, we blend IDM with established Bayesian estimation techniques. We propose a two‐step approach where (<reflink idref="bib1" id="ref1">1</reflink>) IDM predictions are obtained and (<reflink idref="bib2" id="ref2">2</reflink>) employed to construct informative prior distributions. We evaluate the approach in a case study on small‐sample calibration of the 3PL in a high‐stakes test. First, concerning implementation, we find computationally efficient penalized maximum likelihood estimation to be comparable to the best‐performing MCMC‐based approach. Second, we investigate sample size reductions achievable with state‐of‐the‐art IDM predictions, finding negligible gains compared to merely considering the historical distribution of parameters. Third, we evaluate the prediction accuracy required for a targeted sample size reduction by gradually increasing simulated IDM prediction accuracies. We find that required accuracies can counterbalance each other, allowing calibration sample size to be reduced when either high‐quality item difficulty predictions or good predictions of item discriminations and pseudo‐guessing parameters are available. We argue that these evaluations provide new, portable IDM benchmarks quantifying performance in terms of achievable sample size reductions.</p> <p>To be able to retire items that have been overexposed and may, therefore, threaten test security, high‐stakes operational assessments rely on constant replenishment of their item pools. Calibration of newly developed items with customary marginal maximum likelihood (MML) estimation, however, requires large sample sizes to yield accurate item parameters, especially when item response theory (IRT) models with more than one item parameter such as the two‐ (2PL) and three‐parameter logistic (3PL) model are employed (Hulin et al., [<reflink idref="bib27" id="ref3">27</reflink>]). This renders calibration costly. To reduce calibration costs and risks of item exposure, many high‐stakes licensure, certification, and admission programs explored the potentials of item difficulty modeling (IDM) as a means to obtain item parameters <emph>without</emph> item response data. In the IDM framework, machine learning models are trained to predict item parameters with features commonly derived from the items' text with natural language processing (NLP) techniques, targeting predictions that are accurate enough to be employed as fixed item parameters and ultimately render item calibration redundant (e.g., Yaneva et al., [<reflink idref="bib56" id="ref4">56</reflink>]; Belov, [<reflink idref="bib5" id="ref5">5</reflink>]; Xue et al., [<reflink idref="bib55" id="ref6">55</reflink>]; Benedetto et al., [<reflink idref="bib9" id="ref7">9</reflink>]; Settles et al., [<reflink idref="bib47" id="ref8">47</reflink>]; Štěpánek et al., [<reflink idref="bib49" id="ref9">49</reflink>]; Qiu et al., [<reflink idref="bib44" id="ref10">44</reflink>]; Belov et al., [<reflink idref="bib8" id="ref11">8</reflink>]). While these applications have achieved impressive results, it is unclear whether the aim of almost perfect accuracy required for substituting calibrated for predicted item parameters can be achieved in the foreseeable future, especially when items are more complex, but arguably also more ecologically valid.</p> <p>To make even somewhat inaccurate IDM item parameter predictions immediately exploitable, we aim to blend this comparably young stream of research with established Bayesian techniques for incorporating collateral information into the estimation procedure. To this end, we suggest a two‐step approach where, in Step 1, IDM predictions are obtained and, in Step 2, employed to construct informative prior distributions for Bayesian estimation of small‐sample IRT models. Taking such an approach allows to immediately leverage even imperfect IDM predictions and to gradually decrease calibration sample sizes as IDM research evolves and improves these predictions. Using a high‐stakes admission test as a case study, the aim of the present study is (a) to derive guidelines for selecting a Bayesian estimator for Step 2 of the proposed approach, (b) to explore the potential of the proposed two‐step approach with current state‐of‐the‐art IDM item parameter predictions and (c) to derive benchmarks for the quality of IDM predictions needed to decrease calibration sample size requirements by a targeted amount.</p> <p>In what follows, we first briefly review customary MML estimation of IRT models. Then, the specification of IRT models with informative priors in the Bayesian approach is discussed and different Bayesian estimators that can be used to estimate IRT model parameters are described. We review current IDM approaches to obtain item parameter predictions based on features derived from the items' text using NLP, and delineate how IDM predictions can be used to construct informative prior distributions. We then evaluate the suggested two‐step approach in three simulation studies, using IDM item parameter predictions for the high‐stakes test at hand in a case study. Study I compares different candidate Bayesian estimators to derive guidelines for implementing the proposed approach. Study II explores the gains in accuracy of parameter estimates and reduction in calibration sample size requirements that can be achieved by using current state‐of‐the‐art IDM predictions for informative prior construction, employing both default prior settings as well as priors informed by the distribution of item parameters in the item pool as benchmarks. As current state‐of‐the‐art IDM predictions are not sufficiently accurate to achieve marked gains over prior distributions based on item pool information, Study III explores the accuracy level of item parameter predictions needed to achieve such gains. From these three simulation studies, we derive recommendations for implementing the proposed two‐step approach and discuss how it can be used to derive new benchmarks to gauge the utility of IDM predictions' accuracy levels.</p> <hd id="AN0192629996-3">Estimation of Item Response Theory Models</hd> <p>In the present study, we consider the 3PL, which is the analysis model employed in the considered high‐stakes admission test. The 3PL assumes the following item response function (IRF) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$p_j(\theta)$</annotation></semantics></math> </ephtml> , indicating the probability of a correct item response conditional on ability <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>θ</mi><annotation encoding="application/x-tex">$\theta$</annotation></semantics></math> </ephtml> : 1 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><msub><mi>c</mi><mi>j</mi></msub><mo linebreak="goodbreak">+</mo><mrow><mo>(</mo><mn>1</mn><mo linebreak="goodbreak">−</mo><msub><mi>c</mi><mi>j</mi></msub><mo>)</mo></mrow><mfrac><mrow><mi>exp</mi><mo>(</mo><mi>D</mi><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>θ</mi><mo>−</mo><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo>(</mo><mi>D</mi><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>θ</mi><mo>−</mo><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></mfrac><mo>,</mo><mspace width="1em" /><mtext>with</mtext><mspace width="0.33em" /><mi>θ</mi><mo>∼</mo><mi mathvariant="script">N</mi><mrow><mo>(</mo><msub><mi>μ</mi><mi>θ</mi></msub><mo>,</mo><msub><mi>σ</mi><mi>θ</mi></msub><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} p_j(\theta) = c_j + (1-c_j)\frac{\exp (Da_j(\theta -b_j))}{1+\exp (Da_j(\theta -b_j))}, \quad \text{with } \theta \sim \mathcal {N}(\mu _\theta,\sigma _\theta), \end{equation}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>a</mi><mi>j</mi></msub><annotation encoding="application/x-tex">$a_j$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>b</mi><mi>j</mi></msub><annotation encoding="application/x-tex">$b_j$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mi>j</mi></msub><annotation encoding="application/x-tex">$c_j$</annotation></semantics></math> </ephtml> give the item discrimination, difficulty, and pseudo‐guessing parameter for item <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>j</mi><annotation encoding="application/x-tex">$j$</annotation></semantics></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">$j=1,\ldots,J$</annotation></semantics></math> </ephtml> ). Mirroring operational practice, in the present study, the scale parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>D</mi><annotation encoding="application/x-tex">$D$</annotation></semantics></math> </ephtml> is set to 1.702 so that the units of the model will approximate those of normal ogive IRT models. For model identification, the mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>μ</mi><mi>θ</mi></msub><annotation encoding="application/x-tex">$\mu _\theta$</annotation></semantics></math> </ephtml> and standard deviation <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>σ</mi><mi>θ</mi></msub><annotation encoding="application/x-tex">$\sigma _\theta$</annotation></semantics></math> </ephtml> of the normal distribution assumed for ability are set to 0 and 1, respectively.</p> <hd id="AN0192629996-4">Marginal Maximum Likelihood Estimation</hd> <p>MML estimation is the standard method of choice for estimating IRT model parameters from the response matrix <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">X</mi><annotation encoding="application/x-tex">$\bm{X}$</annotation></semantics></math> </ephtml> , containing scored item responses <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">$x_{ij} \in \lbrace 0,1\rbrace$</annotation></semantics></math> </ephtml> for persons <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>i</mi><annotation encoding="application/x-tex">$i$</annotation></semantics></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">$i=1,\ldots,N$</annotation></semantics></math> </ephtml> ) and items <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>j</mi><annotation encoding="application/x-tex">$j$</annotation></semantics></math> </ephtml> . For simplicity, we assume that the response matrix is complete. For customary IRT models, the likelihood function is given by 2 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">X</mi><mo>|</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>J</mi></munderover><mi>p</mi><mrow><mo>(</mo><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mi mathvariant="bold-italic">ξ</mi><mi>j</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \mathcal {L}(\bm{X}|\bm{\theta },\bm{\xi }) = \prod _{i=1}^{N} \prod _{j=1}^{J} p(x_{ij}| \bm{\xi }_{j}, \theta _{i}), \end{equation}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold-italic">ξ</mi><mi>j</mi></msub><annotation encoding="application/x-tex">$\bm{\xi }_j$</annotation></semantics></math> </ephtml> is a vector containing the item parameters; in the present study <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ξ</mi><mi>j</mi></msub><mo>=</mo><mrow><mo>(</mo><msub><mi>a</mi><mi>j</mi></msub><mo>,</mo><msub><mi>b</mi><mi>j</mi></msub><mo>,</mo><msub><mi>c</mi><mi>j</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\bm{\xi }_j=(a_j,b_j,c_j)$</annotation></semantics></math> </ephtml> . In MML estimation of 3PL item parameters, latent variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>θ</mi><annotation encoding="application/x-tex">$\theta$</annotation></semantics></math> </ephtml> are integrated out over the standard normal distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">$F(\theta)$</annotation></semantics></math> </ephtml> assumed for ability, yielding the following log likelihood 3 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>log</mi><mfenced separators="" open="[" close="]"><mo>∫</mo><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>J</mi></munderover><mi>p</mi><mrow><mo>(</mo><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mi mathvariant="bold-italic">ξ</mi><mi>j</mi></msub><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mi mathvariant="normal">d</mi><mi>F</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} l(\bm{\xi }) = \sum _{i=1}^{N} \log {\left[ \int \prod _{j=1}^{J} p(x_{ij}| \bm{\xi }_{j}, \theta) \mathrm{d}F(\theta)\right]}. \end{equation}$$</annotation></semantics></math> </ephtml> Then, the MML estimate is given by: 4 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>ξ</mi><mo>̂</mo></mover><mtext>MML</mtext></msub><mo linebreak="badbreak">=</mo><munder><mrow><mi>arg</mi><mspace width="0.16em" /><mi>max</mi></mrow><mi>ξ</mi></munder><mspace width="0.16em" /><mi>l</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \hat{\xi }_{\text{MML}} = \underset{\xi }{\mathrm{arg\,max}}\, l(\bm{\xi }). \end{equation}$$</annotation></semantics></math> </ephtml></p> <p>When applied to large data sets, MML provides accurate parameter estimates even for IRT models with more than one item parameter. For the 3PL, simulation results from a rich body of studies suggest that—if tests are sufficiently long—MML yields accurate parameter estimates for sample sizes of approximately at least 1,000 examinees (Hulin et al., [<reflink idref="bib27" id="ref12">27</reflink>]; Ree & Jensen, [<reflink idref="bib46" id="ref13">46</reflink>]; Akour & Al‐Omari, [<reflink idref="bib1" id="ref14">1</reflink>]; Tang et al., [<reflink idref="bib52" id="ref15">52</reflink>]; Swaminathan & Gifford, [<reflink idref="bib50" id="ref16">50</reflink>]).[<reflink idref="bib1" id="ref17">1</reflink>] When the number of available data points is small, however, there is an increased risk of the likelihood being too flat, giving rise to convergence issues as well as biased and unstable parameter estimates (see Belov et al., [<reflink idref="bib7" id="ref18">7</reflink>], for illustrations). This renders MML impractical for small‐sample item calibration.</p> <hd id="AN0192629996-5">Bayesian Estimation with Informative Priors</hd> <p>Bayesian estimation, possibly with informative prior distributions, is a well‐studied solution for small‐sample latent variable modeling (see, e.g., Ulitzsch et al., [<reflink idref="bib54" id="ref19">54</reflink>]; Lüdtke et al., [<reflink idref="bib34" id="ref20">34</reflink>]; Smid et al., [<reflink idref="bib48" id="ref21">48</reflink>]; Chen et al., [<reflink idref="bib13" id="ref22">13</reflink>]; Lee & Song, [<reflink idref="bib33" id="ref23">33</reflink>]; Lee, [<reflink idref="bib32" id="ref24">32</reflink>]). In the Bayesian estimation approach, prior distributions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">π</mi></mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">$\bm{\pi }(\cdot)$</annotation></semantics></math> </ephtml> are assigned to the to‐be‐estimated model parameters—that is, to ability <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">θ</mi></mrow><annotation encoding="application/x-tex">$\bm{\theta }$</annotation></semantics></math> </ephtml> and item parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><annotation encoding="application/x-tex">$\bm{\xi }$</annotation></semantics></math> </ephtml> . These prior distributions capture researchers' beliefs on the plausible parameter space, and, thus, may provide additional guidance in parameter estimation. Parameter estimates are then obtained by summarizing the parameters' posterior distribution, which combines information from the observed data and the prior distributions, and is proportional to the likelihood times the prior: 5 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>|</mo><mi mathvariant="bold-italic">X</mi><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><mfrac><mrow><mi mathvariant="script">L</mi><mo>(</mo><mi mathvariant="bold-italic">X</mi><mo>|</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mrow><mo>∫</mo><mspace width="0.16em" /><mo>∫</mo><mi mathvariant="script">L</mi><mo>(</mo><mi mathvariant="bold-italic">X</mi><mo>|</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo><mi mathvariant="normal">d</mi><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mi mathvariant="normal">d</mi><mrow><mi mathvariant="bold-italic">ξ</mi></mrow></mrow></mfrac><mo linebreak="goodbreak">=</mo><mi>C</mi><mi mathvariant="script">L</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">X</mi><mo>|</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo></mrow><mi>π</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo></mrow><mi>π</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} p(\bm{\xi },\bm{\theta }|\bm{X}) = \frac{\mathcal {L}(\bm{X}|\bm{\xi },\bm{\theta })\pi (\bm{\theta })\pi (\bm{\xi })}{\int \,\int \mathcal {L}(\bm{X}|\bm{\xi },\bm{\theta })\pi (\bm{\theta })\pi (\bm{\xi }) \mathrm{d}\bm{\theta }\mathrm{d}\bm{\xi }} = C \mathcal {L}(\bm{X}|\bm{\xi },\bm{\theta })\pi (\bm{\theta })\pi (\bm{\xi }), \end{equation}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mo>=</mo><mn>1</mn><mo>/</mo><mo>∫</mo><mspace width="0.16em" /><mo>∫</mo><mi mathvariant="script">L</mi><mo>(</mo><mi mathvariant="bold-italic">X</mi><mo>|</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo><mi mathvariant="normal">d</mi><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mi mathvariant="normal">d</mi><mrow><mi mathvariant="bold-italic">ξ</mi></mrow></mrow><annotation encoding="application/x-tex">$C = 1/\int \,\int \mathcal {L}(\bm{X}|\bm{\xi },\bm{\theta })\pi (\bm{\theta })\pi (\bm{\xi }) \mathrm{d}\bm{\theta }\mathrm{d}\bm{\xi }$</annotation></semantics></math> </ephtml> is a normalizing constant.</p> <hd id="AN0192629996-6">Obtaining Bayesian Point Estimates</hd> <p>We now briefly review approaches for evaluating and summarizing the posterior distribution. A more detailed discussion is provided in, for example, Gelman et al. ([<reflink idref="bib20" id="ref25">20</reflink>]), Choi et al. ([<reflink idref="bib14" id="ref26">14</reflink>]), and Lüdtke et al. ([<reflink idref="bib34" id="ref27">34</reflink>]).</p> <hd id="AN0192629996-7">Summarizing the marginal posterior distribution</hd> <p>Bayesian point estimates are commonly obtained by summarizing the center—typically, the mean (expected a posteriori; EAP) or mode (maximum a posteriori, MAP)—of the marginal posterior distribution of the parameter of interest (e.g., a specific item difficulty parameter). Note that both the EAP and the MAP are functionals of the joint posterior distribution and involve high‐dimensional integration to obtain the marginal posterior distribution, as does the normalizing constant <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>C</mi><annotation encoding="application/x-tex">$C$</annotation></semantics></math> </ephtml> . To compute complicated or intractable marginal posterior densities and posterior summaries, Markov chain Monte Carlo (MCMC) algorithms such as the Gibbs sampler (as implemented in JAGS) or the No‐U‐turn sampler (as implemented in Stan) can be used. These approximate marginal posterior distributions by iteratively sampling from conditional distributions.</p> <hd id="AN0192629996-8">Summarizing the joint posterior distribution</hd> <p>Alternatively, the mode of the <emph>joint</emph> posterior can be obtained as a Bayesian point estimate using penalized maximum likelihood (PML) estimation. For PML estimation of IRT models, this can be achieved by introducing the prior distribution imposed on item parameters as a penalty term in MML estimation (see, e.g., Mislevy, [<reflink idref="bib38" id="ref28">38</reflink>]), maximizing 6 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>ξ</mi><mo>̂</mo></mover><mtext>PML</mtext></msub><mo linebreak="badbreak">=</mo><munder><mrow><mi>arg</mi><mspace width="0.16em" /><mi>max</mi></mrow><mi>ξ</mi></munder><mrow><mo>[</mo><mi>l</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mo linebreak="goodbreak">+</mo><mi>log</mi><mi>π</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \hat{\xi }_{\text{PML}} = \underset{\xi }{\mathrm{arg\,max}}[ l(\bm{\xi })+ \log \pi (\bm{\xi })], \end{equation}$$</annotation></semantics></math> </ephtml> commonly by means of (deterministic) numerical optimization algorithms such as (Quasi‐)Newton methods.</p> <p>For IRT models with diffuse priors, PML estimates have been found to be comparable to EAP estimates in comparably large samples. In smaller samples (less than 300 for the graded response model, Kieftenbeld & Natesan, [<reflink idref="bib29" id="ref29">29</reflink>], 500 per group for a multiple‐group 2PL model, Azevedo et al, 2012), however, EAP estimates exhibit a slight advantage over PML estimates in terms of the Root Mean Square Error (RMSE), which combines information on bias and variability of obtained point estimates.</p> <hd id="AN0192629996-9">Prior construction</hd> <p>In the present study, the prior distribution employed for ability parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">θ</mi></mrow><annotation encoding="application/x-tex">$\bm{\theta }$</annotation></semantics></math> </ephtml> is the standard normal used for model identification, that is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>)</mo><mo>=</mo><mi mathvariant="script">N</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">$\pi (\bm{\theta }) = \mathcal {N}(0,1)$</annotation></semantics></math> </ephtml> . For item parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><annotation encoding="application/x-tex">$\bm{\xi }$</annotation></semantics></math> </ephtml> , diffuse prior distributions can be employed. In this case, the results produced by the Bayesian approach can be expected to be similar to those obtained by (marginal) maximum likelihood estimation as the posterior distribution depends predominantly on the likelihood (Gelman et al., [<reflink idref="bib20" id="ref30">20</reflink>]). However, the Bayesian approach also offers the opportunity to provide additional guidance during the estimation procedure by imposing informative prior distributions on the to‐be‐estimated parameters. This can be especially advantageous when the number of data points is small and the likelihood does not contain sufficient information to obtain stable parameter estimates (see Ulitzsch et al., [<reflink idref="bib54" id="ref31">54</reflink>]; Smid et al., [<reflink idref="bib48" id="ref32">48</reflink>]; Lee, [<reflink idref="bib32" id="ref33">32</reflink>]; Lüdtke et al., [<reflink idref="bib34" id="ref34">34</reflink>]; Zitzmann et al., [<reflink idref="bib57" id="ref35">57</reflink>], for neighboring research on confirmatory factor analysis models).</p> <p>A fairly straightforward method is to construct weakly informative prior distributions based on considerations regarding the plausible parameter space; typically by imposing separate univariate prior distributions on item discrimination, difficulty, and pseudo‐guessing parameters (e.g., Mislevy, [<reflink idref="bib38" id="ref36">38</reflink>]; Paek et al., [<reflink idref="bib42" id="ref37">42</reflink>]). Likewise, weakly informative prior distributions can be constructed on the basis of the item parameter distribution in the to‐be‐replenished item pool.[<reflink idref="bib2" id="ref38">2</reflink>]</p> <p>For reducing 3PL calibration sample requirements with collateral information, informative prior distributions have mainly been imposed on item difficulty parameters only, either using expert judgments on item difficulty (Swaminathan et al., [<reflink idref="bib51" id="ref39">51</reflink>]) or simple item surface features (e.g., number of words or item type; Keller, [<reflink idref="bib28" id="ref40">28</reflink>]; Matteucci et al., [<reflink idref="bib36" id="ref41">36</reflink>]). Only recently, expert elicitation techniques for the pseudo‐guessing parameter were introduced that allow incorporating expert judgments into the prior construction for pseudo‐guessing parameters (Ames & Smith, [<reflink idref="bib2" id="ref42">2</reflink>]). While incorporating collateral information from expert judgments and simple item surface features has been shown to yield some gains in accuracy of small‐sample IRT parameter estimates (Keller, [<reflink idref="bib28" id="ref43">28</reflink>]; Matteucci et al., [<reflink idref="bib36" id="ref44">36</reflink>]; Swaminathan et al., [<reflink idref="bib51" id="ref45">51</reflink>]), these were oftentimes too minor to allow for substantive reductions of calibration sample sizes. Hence, to achieve such reductions, stronger, higher‐quality prior information is needed than expert judgments and simple item surface features can provide.</p> <hd id="AN0192629996-10">Item Difficulty Modeling: Predicting Item Parameters from Item Text Information</hd> <p>IDM approaches for obtaining IRT parameters from the items' text rather than observed item response data leverage the fact that many operational tests have large item pools containing the items on the one hand and the corresponding item parameters on the other. From the machine learning perspective taken in IDM, the item pool can be understood as a labeled data set, with the item parameters being the labels and the item text providing the raw data from which features can be extracted. This can, for example, be achieved by drawing on techniques from NLP to extract linguistic features of the items. Other approaches use word embeddings (i.e., representations of each word as a real‐valued vector) such as BERT (Bidirectional Encoder Representations from Transformers). Next, a machine learning model can be trained on the item pool to "learn" the (possibly nonlinear and complex) associations between the extracted features and item parameters. To this end, researchers typically draw on common machine learning algorithms, possibly comparing different alternatives, such as random forests, support vector machines, ridge regression, or neural networks (see, e.g., Benedetto et al., [<reflink idref="bib9" id="ref46">9</reflink>]; Yaneva et al., [<reflink idref="bib56" id="ref47">56</reflink>]; Štěpánek et al., [<reflink idref="bib49" id="ref48">49</reflink>]; Qiu et al., [<reflink idref="bib44" id="ref49">44</reflink>]; Pandarova et al., [<reflink idref="bib43" id="ref50">43</reflink>]; Belov et al., [<reflink idref="bib8" id="ref51">8</reflink>]; Huang et al., [<reflink idref="bib26" id="ref52">26</reflink>]). Note that in contrast to explanatory IRT models (e.g. Fischer, [<reflink idref="bib18" id="ref53">18</reflink>]; Embretson, [<reflink idref="bib17" id="ref54">17</reflink>]) aiming to understand the relationship between item features (e.g., format or topic area) and item parameters, in IDM, the relationship between item features and parameters is not of interest from a subject‐matter perspective but is merely exploited to obtain predictions that are as accurate as possible. Hence, IDM does not require the obtained textual features to be interpretable. Newly written items can be perceived as unlabeled data for which the labels (i.e., item parameters) are predicted using the trained machine learning model. Note that this approach rests on the assumption that items are drawn from the same population or "universe" of items.</p> <p>The majority of IDM studies assumes a Rasch model and, as such, focuses only on item difficulty (see Belov, [<reflink idref="bib5" id="ref55">5</reflink>]; Benedetto et al., [<reflink idref="bib9" id="ref56">9</reflink>]; Belov et al., [<reflink idref="bib8" id="ref57">8</reflink>], for exceptions), which is arguably the parameter that is easiest to predict. For simple tasks such as cloze‐type items (i.e., items where a part of the text is masked and the examinee is asked to fill in the masked portion of text), IDM applications obtained impressive results, reporting estimates that are highly correlated (.90) with those obtained from MML with large samples (Pandarova et al., [<reflink idref="bib43" id="ref58">43</reflink>]). For more complex tasks such as logical reasoning, reading comprehension, or highly specialized knowledge tests, for example, for medical licensure, recent applications outperformed difficulty estimates obtained from expert judgments (e.g., Štěpánek et al., [<reflink idref="bib49" id="ref59">49</reflink>]; Yaneva et al., [<reflink idref="bib56" id="ref60">56</reflink>]) and produced item parameter predictions that may be of use in the item development phase to feedback rough difficulty estimates to item developers. However, predictions were still far from being accurate enough to substitute item parameter calibration (Yaneva et al., [<reflink idref="bib56" id="ref61">56</reflink>]; Belov, [<reflink idref="bib5" id="ref62">5</reflink>]; Štěpánek et al., [<reflink idref="bib49" id="ref63">49</reflink>]; Benedetto et al., [<reflink idref="bib9" id="ref64">9</reflink>]; Huang et al., [<reflink idref="bib26" id="ref65">26</reflink>]; Qiu et al., [<reflink idref="bib44" id="ref66">44</reflink>]; Beinborn et al., [<reflink idref="bib4" id="ref67">4</reflink>]; Belov et al., [<reflink idref="bib8" id="ref68">8</reflink>]). Yaneva et al. ([<reflink idref="bib56" id="ref69">56</reflink>]), for instance, reported a correlation between true and predicted proportions correct for a medical licensure exam of.32 for their best‐performing machine learning model. For methods employed by Huang et al. ([<reflink idref="bib26" id="ref70">26</reflink>]) and Settles et al. ([<reflink idref="bib47" id="ref71">47</reflink>]) (as implemented and reevaluated by McCarthy et al., [<reflink idref="bib37" id="ref72">37</reflink>]), the highest correlation between true and predicted proportions correct achieved in the context of language assessment was just below.45.[<reflink idref="bib3" id="ref73">3</reflink>]</p> <hd id="AN0192629996-11">The Present Study</hd> <p>In the present study, we blend IDM with the Bayesian estimation approach in a two‐step approach, obtaining IDM predictions in Step 1, and using them for informative prior construction for small‐sample Bayesian calibration in Step 2. We evaluate and illustrate this approach in a case study based on a high‐stakes admission test and state‐of‐the‐art item parameter predictions achieved in Belov et al. ([<reflink idref="bib8" id="ref74">8</reflink>]). In what follows, we first briefly describe the considered high‐stakes test and its item parameter predictions achieved in Belov et al. ([<reflink idref="bib8" id="ref75">8</reflink>]). We then outline how informative priors can be constructed on the basis of IDM predictions and information on their quality. Finally, we study the approach in three simulation studies. Study I serves to provide guidelines on the implementation of the approach by evaluating its performance using different Bayesian estimators—the mode of the joint posterior distribution that is obtained from PML as well as the mean (EAP) and mode (MAP) of the marginal posterior distribution that are calculated using MCMC methods. Study II explores by how much calibration sample sizes for the considered high‐stakes admission test can be reduced when IDM predictions with accuracies as obtained in Belov et al. ([<reflink idref="bib8" id="ref76">8</reflink>]) are employed for constructing informative prior distributions. Study III serves to derive guidelines on the accuracy of predictions required to achieve a targeted reduction in calibration sample size without losses in accuracy.</p> <hd id="AN0192629996-12">Item Parameter Predictions for a High‐Stakes Admission Test</hd> <p>In the high‐stakes admission test under consideration, each item is based on short (3–4 sentences), ordinary‐language arguments, and it is assessed whether examinees can understand and critique the reasoning contained in the arguments. Figure 1 depicts the distribution of log‐transformed item discrimination <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> , difficulty <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> , and logit‐transformed pseudo‐guessing parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> in the 1,742‐item pool of the test.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0001.jpg" title="1 Densities and bivariate contour plots of log discrimination a$a$, difficulty b$b$, and logit‐transformed pseudo‐guessing parameters c$c$ in the item pool of the considered high‐stakes admission test." /> </p> <p></p> <p>Belov et al. ([<reflink idref="bib8" id="ref77">8</reflink>]) studied 3PL item parameter prediction for the test's items. They modified the approach developed by Belov ([<reflink idref="bib5" id="ref78">5</reflink>]), where a neural network was used to map an item's features into its discrete item characteristic curve (ICC). Belov et al. ([<reflink idref="bib8" id="ref79">8</reflink>]) analyzed two sets of features used by Belov ([<reflink idref="bib5" id="ref80">5</reflink>]) for predicting 3PL parameters and identified the most usable ones. The first set included seven features established by test developers: six categorical features and an item rank, which is an estimate of item difficulty from {1, 2, 3, 4}, with 1 as easiest and 4 as hardest (referred to as expert judgment). The second set contained automatically generated features with readability measures (Chall & Dale, [<reflink idref="bib12" id="ref81">12</reflink>]) and text features built from embeddings for the item stimulus, question, answer, and distractors.[<reflink idref="bib4" id="ref82">4</reflink>]</p> <p>The total number of features was 68. Finally, Belov et al. ([<reflink idref="bib8" id="ref83">8</reflink>]) performed a random search in the space of hyperparameters controlling the neural network structure and the convergence of its training algorithm in order to identify sets of hyperparameters providing the best predictive performance in a 10‐fold cross‐validation. They found two best sets of hyperparameters resulting in Model 1 and Model 2. Here, we use Model 1, as it provided the most accurate predictions in their final validation study.[<reflink idref="bib5" id="ref84">5</reflink>]</p> <p>The quality of obtained predictions in terms of achieved <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>R</mi><mn>2</mn></msup><annotation encoding="application/x-tex">$R^2$</annotation></semantics></math> </ephtml> values is summarized in Table 1. Neither textual features nor expert judgments were informative of discrimination and pseudo‐guessing parameters. Both, however, revealed substantive relationships with item difficulties. Belov et al. ([<reflink idref="bib8" id="ref85">8</reflink>]) did not achieve difficulty predictions more accurate than expert judgments when using textual features alone. Jointly considering expert judgments and textual features, however, yielded a marked increase in prediction accuracy. In the present case study, we evaluate the utility of the best‐performing item parameter predictions based on the combination of textual features and expert judgments as informative prior information.</p> <p>1 Table R2$R^2$ Achieved by Belov et al. (2022 July) in Predicting Log‐Transformed Item Discriminations a$a$, Item Difficulties b$b$, and Logit‐Transformed Pseudo‐Guessing Parameters c$c$ with Different Sources of Information</p> <p> <ephtml> <table><thead><tr><th /><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msup><mi>a</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$a^*$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msup><mi>c</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$c^*$</annotation></semantics></math></p></th></tr></thead><tbody><tr><td>Experts</td><td>.01</td><td>.19</td><td>.03</td></tr><tr><td>Text</td><td>.03</td><td>.11</td><td>.02</td></tr><tr><td>Text + Experts</td><td>.02</td><td>.23</td><td>.02</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mo>∗</mo></msup><mo>=</mo><mi>log</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$a^*=\log (a)$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>c</mi><mo>∗</mo></msup><mo>=</mo><mi>log</mi><mfenced separators="" open="(" close=")"><mfrac><mi>c</mi><mrow><mn>1</mn><mo>−</mo><mi>c</mi></mrow></mfrac></mfenced></mrow><annotation encoding="application/x-tex">$c^*=\log \left(\frac{c}{1-c}\right)$</annotation></semantics></math> </ephtml> . Experts: Features based on expert judgments. Text: Automatically generated features based on item text.</p> <hd id="AN0192629996-14">Constructing Priors from Item Parameter Predictions</hd> <p>For informative prior construction, we work with transformed discrimination and pseudo‐guessing parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>a</mi><mi>j</mi><mo>∗</mo></msubsup><mo>=</mo><mi>log</mi><mrow><mo>(</mo><msub><mi>a</mi><mi>j</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$a_j^*=\log (a_j)$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>c</mi><mi>j</mi><mo>∗</mo></msubsup><mo>=</mo><mi>log</mi><mfenced separators="" open="(" close=")"><mfrac><msub><mi>c</mi><mi>j</mi></msub><mrow><mn>1</mn><mo>−</mo><msub><mi>c</mi><mi>j</mi></msub></mrow></mfrac></mfenced></mrow><annotation encoding="application/x-tex">$c_j^*=\log \left(\frac{c_j}{1-c_j}\right)$</annotation></semantics></math> </ephtml> (see Mislevy, [<reflink idref="bib38" id="ref86">38</reflink>]), which, with an unrestricted range for all parameters, facilitates imposing multivariate normal prior distributions. To translate IDM predictions alongside estimates of their quality into informative prior distributions, we make use of three pieces of information: (a) the IDM predictions themselves, (b) the (presumed) quality of these predictions in the form of their residual standard deviations in the training data set, and (c) the relationship among item parameters in the pool. More specifically, we use transformed IDM predictions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi mathvariant="bold-italic">ξ</mi><mo>̂</mo></mover><mrow><mi>j</mi><mo>,</mo><mtext>IDM</mtext></mrow></msub><mo>=</mo><mrow><mo>(</mo><msubsup><mover accent="true"><mi>a</mi><mo>̂</mo></mover><mrow><mi>j</mi><mo>,</mo><mtext>IDM</mtext></mrow><mo>∗</mo></msubsup><mo>,</mo><msub><mover accent="true"><mi>b</mi><mo>̂</mo></mover><mrow><mi>j</mi><mo>,</mo><mtext>IDM</mtext></mrow></msub><mo>,</mo><msubsup><mover accent="true"><mi>c</mi><mo>̂</mo></mover><mrow><mi>j</mi><mo>,</mo><mtext>IDM</mtext></mrow><mo>∗</mo></msubsup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\hat{\bm{\xi }}_{j,\text{IDM}} = (\hat{a}^*_{j,\text{IDM}},\hat{b}_{j,\text{IDM}},\hat{c}^*_{j,\text{IDM}})$</annotation></semantics></math> </ephtml> for each item <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>j</mi><annotation encoding="application/x-tex">$j$</annotation></semantics></math> </ephtml> as item‐specific prior means and the variance‐covariance matrix of residuals in the training data set (i.e., the item pool) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold">Σ</mi><mrow><mi>ε</mi><mo>,</mo><mtext>pool</mtext></mrow></msub><annotation encoding="application/x-tex">$\bm{\Sigma }_{\epsilon,\text{pool}}$</annotation></semantics></math> </ephtml> as the variance‐covariance matrix for the multivariate normal prior imposed on item parameters, that is 7 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">ξ</mi><mi>j</mi></msub><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><msub><mi mathvariant="script">N</mi><mn>3</mn></msub><mrow><mo>(</mo><msub><mover accent="true"><mi mathvariant="bold-italic">ξ</mi><mo>̂</mo></mover><mrow><mi>j</mi><mo>,</mo><mtext>IDM</mtext></mrow></msub><mo>,</mo><msub><mi mathvariant="bold">Σ</mi><mrow><mi>ε</mi><mo>,</mo><mtext>pool</mtext></mrow></msub><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \pi (\bm{\xi }_j) = \mathcal {N}_3(\hat{\bm{\xi }}_{j,\text{IDM}},\bm{\Sigma }_{\epsilon,\text{pool}}). \end{equation}$$</annotation></semantics></math> </ephtml></p> <hd id="AN0192629996-15">Simulation Studies</hd> <p></p> <hd id="AN0192629996-16">Data Generation</hd> <p>In all three simulation studies, we evaluated item parameter estimation for 50 items sampled from the test's pool (see Figure 1 for a depiction of the pool's item parameter distribution). To make sure the sample accurately represents the item pool, we used a Monte‐Carlo test assembly algorithm (Belov & Armstrong, [<reflink idref="bib6" id="ref87">6</reflink>]). For the item sample assembly, we constrained the item parameter means, standard deviations, and correlations as well as the correlation of IDM predictions with the true values to resemble those in the item pool. Using the sampled items and generating person abilities from a standard normal distribution, per condition, we generated 200 data sets according to Equation 1, varying the sample size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>N</mi><annotation encoding="application/x-tex">$N$</annotation></semantics></math> </ephtml> (50; 100; 200; 500; 1,000). Data generation and all subsequent analyses were conducted with R version 4.3.3 (R Core Team, [<reflink idref="bib45" id="ref88">45</reflink>]).</p> <hd id="AN0192629996-17">Study I</hd> <p>In Study I, we compared the mode of the joint posterior distribution from PML estimation as well as the EAP and MAP obtained using MCMC methods as possible Bayesian estimators for Step 2 of the proposed two‐step approach.</p> <hd id="AN0192629996-18">Data analysis</hd> <p>In Study I, each data set was analyzed with informative priors constructed according to Equation 7 based on the predictions achieved by Belov et al. ([<reflink idref="bib8" id="ref89">8</reflink>]) using textual features and expert judgments. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>R</mi><mn>2</mn></msup><annotation encoding="application/x-tex">$R^2$</annotation></semantics></math> </ephtml> values are reported in Table 1. Standard deviations and correlations of residuals in the training data set (i.e., the item pool) are given in the right half of Table 2. Recall that these inform the variance‐covariance matrix of the multivariate normal prior imposed on item parameters.</p> <p>2 Table Standard Deviations and Correlations Employed for Multivariate Normal Prior Construction for Log‐Transformed Item Discriminations a$a$, Item Difficulties b$b$, and Logit‐Transformed Pseudo‐Guessing Parameters c$c$ under Different Prior Settings</p> <p> <ephtml> <table><thead><tr><th /><th align="center">Pool</th><th align="center">Text + Experts</th></tr><tr><th /><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msup><mi>a</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$a^*$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msup><mi>c</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$c^*$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msub><mi mathvariant="bold-italic">μ</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">$\bm{\mu }_{\text{pool}}$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msup><mi>a</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$a^*$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msup><mi>c</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$c^*$</annotation></semantics></math></p></th></tr></thead><tbody><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><mo>∗</mo><annotation encoding="application/x-tex">$a^*$</annotation></semantics></math></p></td><td>.32</td><td /><td /><td>−.26</td><td>.32</td><td /><td /></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p></td><td>.27</td><td>1.12</td><td /><td>.06</td><td>.23</td><td>.99</td><td /></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><mo>∗</mo><annotation encoding="application/x-tex">$c^*$</annotation></semantics></math></p></td><td>.42</td><td>.38</td><td>1.50</td><td>−2.08</td><td>.40</td><td>.33</td><td>1.49</td></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mo>∗</mo></msup><mo>=</mo><mi>log</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$a^*=\log (a)$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>c</mi><mo>∗</mo></msup><mo>=</mo><mi>log</mi><mfenced separators="" open="(" close=")"><mfrac><mi>c</mi><mrow><mn>1</mn><mo>−</mo><mi>c</mi></mrow></mfrac></mfenced></mrow><annotation encoding="application/x-tex">$c^*=\log \left(\frac{c}{1-c}\right)$</annotation></semantics></math> </ephtml> . <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold-italic">μ</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">$\bm{\mu }_{\text{pool}}$</annotation></semantics></math> </ephtml> gives the item pool mean of transformed item parameters. The diagonal displays standard deviations and the off‐diagonal correlations of transformed item parameters (left), respectively their residuals (right).</p> <p>Bayesian estimation was conducted using Stan version 2.19 (Carpenter et al., [<reflink idref="bib10" id="ref90">10</reflink>]) employing the rstan package version 2.19.3 (Guo et al., [<reflink idref="bib24" id="ref91">24</reflink>]). Note that Stan supports both MCMC‐based and PML estimation.</p> <p>For MCMC‐based estimation, we ran two MCMC chains with 1,500 iterations each, with the first half being employed as warm‐up. For sampling from the posterior distributions, Stan employs the No‐U‐Turn sampler (Hoffman & Gelman, [<reflink idref="bib25" id="ref92">25</reflink>]), an adaptive form of Hamiltonian Monte Carlo sampling (Neal, [<reflink idref="bib40" id="ref93">40</reflink>]). The sampling procedure was assessed on the basis of potential scale reduction factor (PSRF) values, with PSRF values below 1.10 for all parameters being considered as satisfactory (Gelman & Rubin, [<reflink idref="bib21" id="ref94">21</reflink>]; Gelman & Shirley, [<reflink idref="bib22" id="ref95">22</reflink>]).</p> <p>For PML estimation, we approximated the log likelihood by using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>=</mo><mn>41</mn></mrow><annotation encoding="application/x-tex">$Q=41$</annotation></semantics></math> </ephtml> equally spaced quadrature points <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>θ</mi><mi>q</mi></msub><annotation encoding="application/x-tex">$\theta _q$</annotation></semantics></math> </ephtml> between −5 and 5 8 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>log</mi><mfenced separators="" open="[" close="]"><munderover><mo>∑</mo><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow><mi>Q</mi></munderover><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>J</mi></munderover><mi>p</mi><mrow><mo>(</mo><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mi mathvariant="bold-italic">ξ</mi><mi>j</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>q</mi></msub><mo>)</mo></mrow><msub><mi>w</mi><mi>q</mi></msub></mfenced><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} l(\bm{\xi }) = \sum _{i=1}^{N} \log {\left[\sum _{q=1}^{Q} \prod _{j=1}^{J} p(x_{ij}| \bm{\xi }_{j}, \theta _q) w_q\right]}, \end{equation}$$</annotation></semantics></math> </ephtml> with normalized probability density weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>w</mi><mi>q</mi></msub><annotation encoding="application/x-tex">$w_q$</annotation></semantics></math> </ephtml> obtained from the standard normal distributions' probability density function assumed for ability. Recall that in PML estimation, priors are considered by adding the logarithm of the prior distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mi>π</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$\log \pi (\bm{\xi })$</annotation></semantics></math> </ephtml> as a penalty term to the to‐be‐maximized log likelihood function (see Equation 6). For PML estimation, Stan employs the L‐BFGS optimizer—a quasi‐Newton optimization algorithm—as its default.</p> <p>Note that log‐transformed discrimination and logit‐transformed pseudo‐guessing parameters were estimated. Their back‐transformed counterparts were calculated as derived parameters.</p> <hd id="AN0192629996-19">Evaluation criteria</hd> <p>Arguably, for evaluating the performance of estimators in small samples, both bias and variability need to be taken into account (Gelman et al., [<reflink idref="bib20" id="ref96">20</reflink>]; Zitzmann et al., [<reflink idref="bib57" id="ref97">57</reflink>]). Hence, to evaluate performance, we focused on the quality of point estimates in terms of the RMSE, which aggregates information on bias and variability into a single measure and is defined as9 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>RMSE</mtext><mrow><mo>(</mo><mover accent="true"><mi>γ</mi><mo>̂</mo></mover><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><msqrt><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></msubsup><msup><mrow><mo>(</mo><msub><mover accent="true"><mi>γ</mi><mo>̂</mo></mover><mi>r</mi></msub><mo>−</mo><mi>γ</mi><mo>)</mo></mrow><mn>2</mn></msup></mrow><mi>R</mi></mfrac></msqrt><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \text{RMSE}(\hat{\gamma }) = \sqrt {\frac{\sum _{r=1}^{R}(\hat{\gamma }_r-\gamma)^2}{R}}, \end{equation}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mover accent="true"><mi>γ</mi><mo>̂</mo></mover><mi>r</mi></msub><annotation encoding="application/x-tex">$\hat{\gamma }_r$</annotation></semantics></math> </ephtml> is the sample estimate for the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>r</mi><annotation encoding="application/x-tex">$r$</annotation></semantics></math> </ephtml> th replication ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mi>R</mi></mrow><annotation encoding="application/x-tex">$r=1,\ldots R$</annotation></semantics></math> </ephtml> ), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>γ</mi><annotation encoding="application/x-tex">$\gamma$</annotation></semantics></math> </ephtml> is the population value, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">$R$</annotation></semantics></math> </ephtml> is the number of replications.</p> <p>In addition, for completeness, we obtained the bias for all types of considered point estimates, which is reported in Appendices 1 and 3.</p> <hd id="AN0192629996-20">Item parameter estimates</hd> <p>For each item parameter type, we averaged RMSE values across all 50 sample items.</p> <hd id="AN0192629996-21">Item characteristic curves</hd> <p>To investigate estimation accuracy of ICCs implied by the obtained item parameter estimates, we investigated RMSE values for model‐implied probabilities correct (see Equation 1) averaged over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mn>30</mn></mrow><annotation encoding="application/x-tex">$S=30$</annotation></semantics></math> </ephtml> equally spaced supporting points over different ability intervals—again, averaged across all 50 sampled items. We evaluated four ability intervals. First, to investigate the overall recovery of ICCs, we obtained RMSE values over the ability interval [–5; 5]. Second, we investigated the recovery of ICCs over intervals of low ([–5; −1]), medium ([–1; 1]), and high ability ([1; 5]) in terms of their RMSE, that is 10 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>RMSE</mtext><mrow><mo>(</mo><mover accent="true"><mtext>ICC</mtext><mo>̂</mo></mover><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><msqrt><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>J</mi></msubsup><msubsup><mo>∑</mo><mrow><mi>S</mi><mo>=</mo><mn>1</mn></mrow><mi>S</mi></msubsup><msup><mfenced separators="" open="[" close="]"><msub><mover accent="true"><mi>p</mi><mo>̂</mo></mover><mrow><mi>j</mi><mi>r</mi></mrow></msub><mrow><mo>(</mo><msub><mi>θ</mi><mi>s</mi></msub><mo>)</mo></mrow><mo>−</mo><msub><mi>p</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>θ</mi><mi>s</mi></msub><mo>)</mo></mrow></mfenced><mn>2</mn></msup></mrow><mrow><mi>R</mi><mi>J</mi><mi>S</mi></mrow></mfrac></msqrt><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \text{RMSE}(\hat{\text{ICC}}) = \sqrt {\frac{\sum _{r=1}^{R}\sum _{j=1}^J\sum _{S=1}^S{\left[\hat{p}_{jr}(\theta _s)-p_j(\theta _s)\right]}^2}{RJS}}. \end{equation}$$</annotation></semantics></math> </ephtml></p> <p>We considered different ability intervals because, in the context of high‐stakes admissions and licensing exams, it may be more important to have precise ICCs for ability intervals with diagnostic relevance (e.g., those around important cut‐points) and to enable accurate fine‐grained differentiation among high‐ability examinees, while precision may be less critical for segments of the ability continuum far below the cut‐score required for, say, passing the exam.[<reflink idref="bib6" id="ref98">6</reflink>]</p> <p>Figure 2 illustrates that the precision of ICCs can be quite different for different ability segments, depicting ICCs implied by hypothetical true and two different sets of estimated item parameters. Note that in both figures, estimated item parameters markedly differ from their true counterparts. In Figure 2a, for the lower segment of the ability continuum, probabilities correct implied by estimated and true item parameters indeed differ. For higher ability segments, however, the estimated ICC closely aligns with the true one, even though item parameters do not coincide. Hence, this hypothetical item would likely facilitate obtaining trustworthy estimates for examinees with high but not for those with low ability. The opposite pattern emerges for Figure 2b, depicting probabilities correct implied by estimated parameters that markedly differ from their true counterparts in the medium and upper but not in the lower segments of the ability continuum.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0002.jpg" title="2 True and estimated item characteristic curves for true item parameters of a=.90$a=.90$, b=−.25$b=-.25$, and c=.15$c=.15$ and different sets of estimated item parameters." /> </p> <p></p> <hd id="AN0192629996-23">Ability estimates for selected examinees</hd> <p>In a similar vein, we also investigated the RMSE for EAP estimates for 5 selected examinees with abilities of −2, −1, 0, 1, and 2. To this end, for each replication, we generated item responses for these examinees to the 50 sampled items. Using the R package mstR (Magis et al., [<reflink idref="bib35" id="ref99">35</reflink>]), we then obtained EAP ability estimates, fixing item parameters to the estimates obtained from different estimators.</p> <hd id="AN0192629996-24">Results</hd> <p></p> <hd id="AN0192629996-25">Item parameter estimates</hd> <p>Figure 3 depicts RMSE values for item parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> obtained with the different Bayesian estimators. As was to be expected, estimation accuracy of all Bayesian estimators increased with increasing sample size. For all parameter types, the EAP estimate exhibited the lowest RMSE values. For item discriminations <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> , the MAP estimate yielded RMSE values comparable to the EAP estimate, while the mode of the joint posterior distribution obtained with PML yielded somewhat higher RMSE values across all sample sizes. Likewise, for item difficulties <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> , the mode of the joint posterior distribution obtained with PML yielded the highest RMSE values, followed by the MAP estimate. For pseudo‐guessing parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> , the mode of the joint posterior distribution obtained with PML yielded lower RMSE values than the MAP estimate in small samples up to 500. In large samples, both estimators exhibited comparable performance, but did not reach the accuracy levels provided by the EAP estimate.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0003.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0003.jpg" title="3 RMSE values for item parameters a$a$, b$b$, and c$c$ obtained with different Bayesian estimators for the two‐step approach plotted against sample size N$N$. EAP: mean of the marginal posterior distribution obtained with MCMC; MAP: mode of the marginal posterior distribution obtained from MCMC; PML: mode of the joint posterior distribution obtained with penalized maximum likelihood. Note that y$y$ ‐axs differ in scale. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <hd id="AN0192629996-27">Item characteristic curves</hd> <p>A different pattern emerged for RMSE values for ICCs, depicted in Figure 4. Here, again, estimation accuracy of all Bayesian estimators increased with increasing sample size. Across all considered segments of the ability continuum, the EAP estimate exhibited the lowest RMSE values. However, the mode of the joint posterior distribution obtained with PML exhibited RMSE values comparable to the best‐performing EAP estimate. The MAP estimate performed comparably to the EAP estimate only for the middle and upper ability segments, but exhibited higher RMSE values for the lower ability segment and across the entire ability interval [–5; 5].</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0004.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0004.jpg" title="4 RMSE values for item characteristic curves over different segments of the ability continuum obtained with different Bayesian estimators for the two‐step approach plotted against sample size N$N$. ICC all: [–5; 5]; ICC lower: [–5; −1]; ICC middle: [–1; 1]; ICC upper: [1; 5]. EAP: mean of the marginal posterior distribution obtained with MCMC; MAP: mode of the marginal posterior distribution obtained from MCMC; PML: mode of the joint posterior distribution obtained with penalized maximum likelihood. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <hd id="AN0192629996-29">Ability estimates for selected examinees</hd> <p>Figure 5 depicts RMSE values of EAP ability estimates for five selected examinees with item parameters fixed to values obtained with the different Bayesian estimators. We observed only minor differences in accuracy of ability estimates obtained with item parameters fixed to values obtained with the different Bayesian estimators for medium low ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta =-1$</annotation></semantics></math> </ephtml> ) to high ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta =2$</annotation></semantics></math> </ephtml> ) ability levels. Mirroring results for ICCs, for low ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta =-2$</annotation></semantics></math> </ephtml> ) ability levels, item parameters fixed to MAP estimates yielded RMSE values that were higher than those using item parameters obtained with the other two Bayesian estimators, especially when the sample size for estimating item parameters was small.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0005.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0005.jpg" title="5 RMSE values for EAP ability estimates for five selected examinees with item parameters fixed to values obtained from different Bayesian estimators for the two‐step approach under different sample size N$N$. EAP: mean of the marginal posterior distribution obtained with MCMC; MAP: mode of the marginal posterior distribution obtained from MCMC; PML: mode of the joint posterior distribution obtained with penalized maximum likelihood. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <p>Bias for all types of considered point estimates is reported in Appendix 1.</p> <hd id="AN0192629996-31">Conclusions from Study I</hd> <p>We found the EAP obtained with MCMC to yield the most accurate item parameter estimates and ICCs under informative multivariate normal prior settings. Nevertheless, the mode of the joint posterior distribution obtained with PML exhibited ICC accuracy levels that closely resembled the EAP estimate. Likewise, its fixed item parameters achieved accuracy of ability estimates closely resembling those using EAP item parameter estimates. Since the aim of licensure, certification, and admission programs is to draw accurate inferences on examinee ability levels and item parameter estimation is a mere means to that end, we argue that accurate recovery of ICCs and examinee ability is the more important criterion to evaluate an estimator's utility. Given PML's performance in Study I combined with the fact that it is computationally much less demanding than MCMC and markedly better aligns with the MML set‐up implemented in operational practice, we, therefore, recommend to implement the proposed two‐step approach using PML and will draw on PML estimation in Studies II and III, and provide a minimal example for its implementation in R and Stan in the OSF repository accompanying this study.</p> <hd id="AN0192629996-32">Study II</hd> <p>The aim of Study II was to contextualize the estimation accuracy of PML estimates of our two‐step approach obtained in Study I. Because prior construction in the proposed two‐step approach considers both IDM predictions and the relationship among parameters in the pool, we denote the resultant PML estimate in the following with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> . Study II considered three baselines for comparison: (a) PML with default prior settings (subsequently denoted with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> ), mirroring operational practice; (b) PML with informative prior settings based on the distribution of item parameters in the item pool (subsequently denoted with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> ), allowing to isolate the gains from considering state‐of‐the‐art IDM predictions from the effect of considering prior knowledge on the historical distribution of item parameters; and (c) the accuracy of the IDM predictions themselves, allowing to evaluate the gains from considering item response data in addition to item features. Note that the employed IDM predictions were essentially uninformative of item discrimination and pseudo‐guessing parameters. Nevertheless, it may well be that the incorporated information on item difficulty may also aid the estimation of the remaining two item parameters.</p> <hd id="AN0192629996-33">Data analysis</hd> <p>For default prior settings, we adhered to the default priors implemented in BILOG (Mislevy & Bock, [<reflink idref="bib39" id="ref100">39</reflink>]), employing diffuse normal priors for log item discriminations and difficulties, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mrow><mo>(</mo><msub><mi>a</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>∼</mo><mi mathvariant="script">N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0.5</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\log ({a}_{j})\sim \mathcal{N}(0,0.5)$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>j</mi></msub><mo>∼</mo><mi mathvariant="script">N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$b_j \sim \mathcal {N}(0,2)$</annotation></semantics></math> </ephtml> , as well as a beta prior for pseudo‐guessing parameters with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>j</mi></msub><mo>∼</mo><mtext>Beta</mtext><mrow><mo>(</mo><mn>20</mn><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>m</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>,</mo><mn>20</mn><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>m</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$c_j \sim \text{Beta}(20(1/m) + 1, 20(1-1/m) + 1)$</annotation></semantics></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>m</mi><annotation encoding="application/x-tex">$m$</annotation></semantics></math> </ephtml> denotes the number of response options. In the considered test, all items have five response options; hence <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">$m=5$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>j</mi></msub><mo>∼</mo><mtext>Beta</mtext><mrow><mo>(</mo><mn>5</mn><mo>,</mo><mn>17</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$c_j \sim \text{Beta}(5, 17)$</annotation></semantics></math> </ephtml> . For informative prior settings, the multivariate normal prior imposed on transformed item parameters was based on the item pool mean vector <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold-italic">μ</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">$\bm{\mu }_{\text{pool}}$</annotation></semantics></math> </ephtml> and variance‐covariance matrix <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold">Σ</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">$\bm{\Sigma }_{\text{pool}}$</annotation></semantics></math> </ephtml> , that is11 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">ξ</mi><mi>j</mi></msub><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><msub><mi mathvariant="script">N</mi><mn>3</mn></msub><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">μ</mi><mtext>pool</mtext></msub><mo>,</mo><msub><mi mathvariant="bold">Σ</mi><mtext>pool</mtext></msub><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \pi (\bm{\xi }_j) = \mathcal {N}_3(\bm{\mu }_{\text{pool}},\bm{\Sigma }_{\text{pool}}). \end{equation}$$</annotation></semantics></math> </ephtml> Values used for this multivariate normal prior are given in the left half of Table 2.</p> <hd id="AN0192629996-34">Evaluation criteria</hd> <p>We considered the same evaluation criteria as in Study I. In addition to RMSE values of item parameter estimates with different prior settings, ICCs, and ability estimates obtained using PML estimates as fixed item parameters, we also obtained the corresponding RMSE values for IDM predictions. Recall that these are obtained without item response data and, hence, do not vary across sample sizes.</p> <hd id="AN0192629996-35">Results</hd> <p></p> <hd id="AN0192629996-36">Item parameter estimates</hd> <p>Figure 6 depicts RMSE values for item parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> obtained with PML under different prior settings as well as for the IDM predictions. Again, as was to be expected, for all prior settings, estimation accuracy increased with increasing sample size. For item discriminations <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> yielded the lowest RMSE values, closely followed by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>PM</mi><msub><mi>L</mi><mtext>IDM+pool</mtext></msub></mrow><annotation encoding="application/x-tex">$\textit{PM}{L}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> . In small samples up to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\mathrm{PML}}_{\text{default}}$</annotation></semantics></math> </ephtml> yielded RMSE values that were markedly higher than both the two informative prior settings as well as the IDM predictions, and exhibited performance resembling the baseline prior settings only in large samples of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> . For item difficulties <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> , estimated item parameters exhibited markedly lower RMSE values than the IDM baseline across all sample sizes and prior settings. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> yielded comparable RMSE values. Even though Belov et al. ([<reflink idref="bib8" id="ref101">8</reflink>]) achieved fair IDM predictions for item difficulties, the accuracy of these predictions was not sufficient to entail increases in estimation accuracy when used for informative prior construction. In fact, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> exhibited RMSE values that were markedly higher than those obtained under the two baseline prior settings. For pseudo‐guessing parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> performed best in small samples up to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> . In larger samples, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> performed comparable to default prior settings. Throughout all sample sizes considered, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> yielded RMSE values that were higher than those of the IDM predictions used for prior construction.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0006.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0006.jpg" title="6 RMSE values for item parameters a$a$, b$b$, and c$c$ obtained with different prior settings as well as item difficulty modeling (IDM) plotted against sample size N$N$. PMLdefault${\rm PML}_{\text{default}}$: default priors as implemented in BILOG; PMLpool${\rm PML}_{\text{pool}}$: multivariate normal prior based on the distribution of item parameters in the item pool; PMLIDM+pool${\rm PML}_{\text{IDM+pool}}$: priors informed by IDM predictions. Note that y$y$ ‐axes differ in scale. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <hd id="AN0192629996-38">Item characteristic curves</hd> <p>Again, a different pattern emerged for ICC RMSE values, depicted in Figure 7. ICCs based on estimated item parameters exhibited markedly lower RMSE values than the IDM baseline for all ability segments across all sample sizes, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> yielded RMSE values closely resembling each other. For the entire ability continuum over [–5; 5] as well as the lower ability segment, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> yielded ICC accuracy levels comparable to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> . For the upper and, to a lesser extent, the middle ability segment, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> exhibited slightly higher RMSE values than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> .</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0007.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0007.jpg" title="7 RMSE values for item characteristic curves over different segments of the ability continuum obtained with different prior settings as well as item difficulty modeling (IDM) plotted against sample size plotted against sample size N$N$. ICC all: [–5; 5]; ICC lower: [–5; −1]; ICC middle: [–1; 1]; ICC upper: [1; 5]. PMLdefault${\rm PML}_{\text{default}}$: default priors as implemented in BILOG; PMLpool${\rm PML}_{\text{pool}}$: multivariate normal prior based on the distribution of item parameters in the item pool; PMLIDM+pool${\rm PML}_{\text{IDM+pool}}$: priors informed by IDM predictions. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <hd id="AN0192629996-40">Ability estimates for selected examinees</hd> <p>Figure 8 depicts RMSE values of EAP ability estimates for the five selected examinees with item parameters fixed to values obtained with different prior settings as well as using IDM. Mirroring ICC results, we observed only minor differences in accuracy of ability estimates achieved with item parameters fixed to values obtained under different prior settings for low ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta =-2$</annotation></semantics></math> </ephtml> ) to medium high ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta =1$</annotation></semantics></math> </ephtml> ) ability levels. For high ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta =2$</annotation></semantics></math> </ephtml> ) ability levels, item parameters fixed to estimates obtained with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> yielded RMSE values that were higher than those obtained with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> , especially when the sample size for estimating item parameters was small. In general, accuracy of ability estimates achieved with item parameters fixed to values obtained with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{pool}}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>IDM+pool</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{IDM+pool}}$</annotation></semantics></math> </ephtml> were almost indistinguishable. Interestingly, differences between IDM predictions and estimated item parameters were especially pronounced for extreme ability levels, that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta =-2$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta =2$</annotation></semantics></math> </ephtml> .</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0008.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0008.jpg" title="8 RMSE values for EAP ability estimates for five selected examinees with item parameters fixed to values obtained with different prior settings as well as item difficulty modeling (IDM) plotted against sample size plotted against sample size N$N$. PMLdefault${\rm PML}_{\text{default}}$: default priors as implemented in BILOG; PMLpool${\rm PML}_{\text{pool}}$: multivariate normal prior based on the distribution of item parameters in the item pool; PMLIDM+pool${\rm PML}_{\text{IDM+pool}}$: priors informed by IDM predictions. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <p>Again, we provide the bias for all types of considered point estimates in Appendix 1.</p> <hd id="AN0192629996-42">Conclusions from Study II</hd> <p>Considering the historical distribution of parameters in the item pool resulted in improved small‐sample accuracy of ICCs and ability estimates for upper ability segments. Considering state‐of‐the‐art IDM predictions for prior construction, however, did not yield additional gains in accuracy and even yielded less stable item parameters compared to considering the historical distribution of parameters in the item pool.</p> <hd id="AN0192629996-43">Study III</hd> <p>Recall that the employed state‐of‐the‐art IDM predictions evaluated in Study II provided fair predictions of item difficulties ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>=</mo><mo>.</mo><mn>23</mn></mrow><annotation encoding="application/x-tex">$R^2_b=.23$</annotation></semantics></math> </ephtml> ), but were uninformative about item discriminations and pseudo‐guessing parameters. We found these predictions to be insufficient for substantive reductions in calibration sample size requirements with the proposed two‐step approach. To evaluate the quality of item parameter predictions needed to achieve substantive reductions, we gradually increased the quality of simulated IDM predictions. To this end, we fully crossed <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>R</mi><mn>2</mn></msup><annotation encoding="application/x-tex">$R^2$</annotation></semantics></math> </ephtml> values for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>a</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$a^*$</annotation></semantics></math> </ephtml> (0 to.60 in steps of.20), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> (.20 to.80 in steps of.20), and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>c</mi><mo>∗</mo></msup><annotation encoding="application/x-tex">$c^*$</annotation></semantics></math> </ephtml> (0 to.60 in steps of.20). For each replication of the simulation study, item parameter predictions for the 50 sampled items satisfying the targeted <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>R</mi><mn>2</mn></msup><annotation encoding="application/x-tex">$R^2$</annotation></semantics></math> </ephtml> values were generated using the R package faux (DeBruine, [<reflink idref="bib15" id="ref102">15</reflink>]). We obtained PML estimates under the resultant <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>64</mn></mrow><annotation encoding="application/x-tex">$4\times 4\times 4=64$</annotation></semantics></math> </ephtml> prior settings with the same set‐up as in Study I, but focusing only on sample sizes up to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> (i.e., 50; 100; 200; 500).</p> <hd id="AN0192629996-44">Evaluation Criteria</hd> <p>Again, we investigated RMSE values for obtained item parameters, ICCs for different segments of the latent ability continuum, and ability estimates for selected examinees. When doing so, for each sample size, we evaluated whether or not priors informed by simulated IDM predictions yielded RMSE values falling below their counterparts obtained with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> for a sample size of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> . We considered RMSE values greater than 1.20 times the RMSE values of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline as worse, RMSE values within.80 times and 1.20 times the RMSE values of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline as comparable, and RMSE values smaller than.80 times the RMSE values of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline as improved estimation accuracy, albeit a smaller calibration sample size. If estimation accuracy was at least comparable, we concluded that the simulated quality of IDM predictions would be sufficient to reduce calibration sample requirements to the respective sample size. Note that we required <emph>all</emph> parameters of the same type (i.e., all item parameters, ICCs across different segments, and ability estimates for different examinees) to be at least comparable to the the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline to reach this conclusion.</p> <hd id="AN0192629996-45">Results</hd> <p>Due to the vastness of results (64 prior settings, four sample size conditions, and three types of outcome measures), we discuss results for Study III exemplarily. When doing so, we focus on two selected sample sizes ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> ) to illustrate how simulations using the proposed two‐step approach can be used to derive benchmarks for IDM prediction quality in terms of achievable reductions in calibration sample size requirements.</p> <p>For ICC estimation accuracy across all ability segments, we observed comparable or even improved estimation accuracy only for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> . Note that compared to the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline, this would correspond to halving the calibration sample size without losses in ICC estimation accuracy. Therefore, in the following, for item parameters and ICCs, only results for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> are discussed in greater detail. For ability estimates for all selected examinees—with item parameters obtained from the proposed two‐step approach—we were able to obtain estimates comparable to the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline already with a sample size as low as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math> </ephtml> under a variety of simulated IDM prediction quality combinations. Therefore, in the following, for ability estimates of selected examinees, only results for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math> </ephtml> are discussed in greater detail. Results for the remaining sample sizes are given in Appendix 2. Bias is reported in Appendix 3.</p> <hd id="AN0192629996-46">Item parameter estimates</hd> <p>Results for item parameter estimates are displayed in Table 3. For a calibration sample size of as low as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math> </ephtml> , simulated combinations of IDM predictions with highly accurate predictions for item difficulty <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>=</mo><mo>.</mo><mn>80</mn></mrow><annotation encoding="application/x-tex">$R^2_b=.80$</annotation></semantics></math> </ephtml> ) or a combination of good predictions for item difficulty <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> and pseudo‐guessing parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_b\ge.60$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_c\ge.60$</annotation></semantics></math> </ephtml> ) yielded RMSE values for all types of item parameters that were at least comparable (values displayed in bold) to the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline (values displayed in the table header).</p> <p>3 Table RMSE Values for Item Parameters a$a$, b$b$, and c$c$ Obtained with Varying IDM Prediction Accuracy under N=100$N=100$ and N=500$N=500$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (.17)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (.30)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> (.10)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (.17)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (.30)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> (.10)</th></tr></thead><tbody><tr><td>.00</td><td>.20</td><td>.00</td><td>.200</td><td>.494</td><td>.143</td><td>.163</td><td>.367</td><td>.117</td></tr><tr><td>.00</td><td>.20</td><td>.20</td><td>.194</td><td>.463</td><td>.127</td><td>.157</td><td>.353</td><td>.106</td></tr><tr><td>.00</td><td>.20</td><td>.40</td><td>.189</td><td>.433</td><td>.113</td><td>.152</td><td>.325</td><td>.095</td></tr><tr><td>.00</td><td>.20</td><td>.60</td><td>.185</td><td>.403</td><td>.099</td><td>.146</td><td>.289</td><td>.082</td></tr><tr><td>.00</td><td>.40</td><td>.00</td><td>.199</td><td>.495</td><td>.146</td><td>.163</td><td>.385</td><td>.122</td></tr><tr><td>.00</td><td>.40</td><td>.20</td><td>.192</td><td>.442</td><td>.125</td><td>.155</td><td>.344</td><td>.106</td></tr><tr><td>.00</td><td>.40</td><td>.40</td><td>.187</td><td>.418</td><td>.111</td><td>.150</td><td>.315</td><td>.092</td></tr><tr><td>.00</td><td>.40</td><td>.60</td><td>.182</td><td>.387</td><td>.095</td><td>.143</td><td>.281</td><td>.079</td></tr><tr><td>.00</td><td>.60</td><td>.00</td><td>.192</td><td>.449</td><td>.137</td><td>.159</td><td>.371</td><td>.121</td></tr><tr><td>.00</td><td>.60</td><td>.20</td><td>.186</td><td>.408</td><td>.119</td><td>.153</td><td>.325</td><td>.101</td></tr><tr><td>.00</td><td>.60</td><td>.40</td><td>.183</td><td>.389</td><td>.107</td><td>.146</td><td>.286</td><td>.087</td></tr><tr><td>.00</td><td>.60</td><td>.60</td><td>.179</td><td>.359</td><td>.090</td><td>.141</td><td>.257</td><td>.075</td></tr><tr><td>.00</td><td>.80</td><td>.00</td><td>.178</td><td>.359</td><td>.119</td><td>.146</td><td>.291</td><td>.102</td></tr><tr><td>.00</td><td>.80</td><td>.20</td><td>.177</td><td>.337</td><td>.106</td><td>.143</td><td>.256</td><td>.086</td></tr><tr><td>.00</td><td>.80</td><td>.40</td><td>.175</td><td>.325</td><td>.097</td><td>.140</td><td>.240</td><td>.078</td></tr><tr><td>.00</td><td>.80</td><td>.60</td><td>.175</td><td>.313</td><td>.086</td><td>.137</td><td>.224</td><td>.069</td></tr><tr><td>.20</td><td>.20</td><td>.00</td><td>.184</td><td>.509</td><td>.145</td><td>.154</td><td>.407</td><td>.122</td></tr><tr><td>.20</td><td>.20</td><td>.20</td><td>.179</td><td>.464</td><td>.128</td><td>.149</td><td>.360</td><td>.108</td></tr><tr><td>.20</td><td>.20</td><td>.40</td><td>.175</td><td>.428</td><td>.113</td><td>.144</td><td>.325</td><td>.094</td></tr><tr><td>.20</td><td>.20</td><td>.60</td><td>.170</td><td>.394</td><td>.096</td><td>.137</td><td>.293</td><td>.082</td></tr><tr><td>.20</td><td>.40</td><td>.00</td><td>.182</td><td>.494</td><td>.145</td><td>.154</td><td>.407</td><td>.126</td></tr><tr><td>.20</td><td>.40</td><td>.20</td><td>.179</td><td>.439</td><td>.124</td><td>.147</td><td>.352</td><td>.107</td></tr><tr><td>.20</td><td>.40</td><td>.40</td><td>.174</td><td>.410</td><td>.110</td><td>.142</td><td>.317</td><td>.092</td></tr><tr><td>.20</td><td>.40</td><td>.60</td><td>.171</td><td>.380</td><td>.094</td><td>.138</td><td>.289</td><td>.081</td></tr><tr><td>.20</td><td>.60</td><td>.00</td><td>.177</td><td>.442</td><td>.137</td><td>.150</td><td>.372</td><td>.122</td></tr><tr><td>.20</td><td>.60</td><td>.20</td><td>.173</td><td>.399</td><td>.117</td><td>.144</td><td>.322</td><td>.101</td></tr><tr><td>.20</td><td>.60</td><td>.40</td><td>.170</td><td>.376</td><td>.105</td><td>.138</td><td>.286</td><td>.087</td></tr><tr><td>.20</td><td>.60</td><td>.60</td><td>.168</td><td>.353</td><td>.090</td><td>.135</td><td>.253</td><td>.075</td></tr><tr><td>.20</td><td>.80</td><td>.00</td><td>.168</td><td>.351</td><td>.120</td><td>.139</td><td>.283</td><td>.102</td></tr><tr><td>.20</td><td>.80</td><td>.20</td><td>.168</td><td>.336</td><td>.106</td><td>.136</td><td>.260</td><td>.088</td></tr><tr><td>.20</td><td>.80</td><td>.40</td><td>.166</td><td>.322</td><td>.096</td><td>.134</td><td>.237</td><td>.076</td></tr><tr><td>.20</td><td>.80</td><td>.60</td><td>.165</td><td>.308</td><td>.085</td><td>.131</td><td>.222</td><td>.069</td></tr><tr><td>.40</td><td>.20</td><td>.00</td><td>.165</td><td>.502</td><td>.142</td><td>.142</td><td>.403</td><td>.121</td></tr><tr><td>.40</td><td>.20</td><td>.20</td><td>.162</td><td>.456</td><td>.128</td><td>.138</td><td>.357</td><td>.107</td></tr><tr><td>.40</td><td>.20</td><td>.40</td><td>.158</td><td>.416</td><td>.111</td><td>.133</td><td>.322</td><td>.093</td></tr><tr><td>.40</td><td>.20</td><td>.60</td><td>.157</td><td>.383</td><td>.096</td><td>.128</td><td>.279</td><td>.079</td></tr><tr><td>.40</td><td>.40</td><td>.00</td><td>.163</td><td>.474</td><td>.141</td><td>.140</td><td>.396</td><td>.124</td></tr><tr><td>.40</td><td>.40</td><td>.20</td><td>.159</td><td>.423</td><td>.122</td><td>.136</td><td>.338</td><td>.104</td></tr><tr><td>.40</td><td>.40</td><td>.40</td><td>.157</td><td>.392</td><td>.108</td><td>.132</td><td>.304</td><td>.090</td></tr><tr><td>.40</td><td>.40</td><td>.60</td><td>.154</td><td>.367</td><td>.092</td><td>.128</td><td>.269</td><td>.077</td></tr><tr><td>.40</td><td>.60</td><td>.00</td><td>.159</td><td>.431</td><td>.138</td><td>.137</td><td>.360</td><td>.121</td></tr><tr><td>.40</td><td>.60</td><td>.20</td><td>.157</td><td>.395</td><td>.117</td><td>.133</td><td>.311</td><td>.099</td></tr><tr><td>.40</td><td>.60</td><td>.40</td><td>.155</td><td>.367</td><td>.103</td><td>.129</td><td>.278</td><td>.084</td></tr><tr><td>.40</td><td>.60</td><td>.60</td><td>.153</td><td>.350</td><td>.090</td><td>.125</td><td>.253</td><td>.074</td></tr><tr><td>.40</td><td>.80</td><td>.00</td><td>.153</td><td>.350</td><td>.120</td><td>.131</td><td>.281</td><td>.102</td></tr><tr><td>.40</td><td>.80</td><td>.20</td><td>.153</td><td>.331</td><td>.107</td><td>.128</td><td>.256</td><td>.087</td></tr><tr><td>.40</td><td>.80</td><td>.40</td><td>.152</td><td>.321</td><td>.097</td><td>.125</td><td>.240</td><td>.078</td></tr><tr><td>.40</td><td>.80</td><td>.60</td><td>.150</td><td>.307</td><td>.085</td><td>.124</td><td>.220</td><td>.068</td></tr><tr><td>.60</td><td>.20</td><td>.00</td><td>.138</td><td>.481</td><td>.138</td><td>.123</td><td>.373</td><td>.114</td></tr><tr><td>.60</td><td>.20</td><td>.20</td><td>.137</td><td>.441</td><td>.123</td><td>.120</td><td>.333</td><td>.101</td></tr><tr><td>.60</td><td>.20</td><td>.40</td><td>.135</td><td>.400</td><td>.108</td><td>.118</td><td>.300</td><td>.089</td></tr><tr><td>.60</td><td>.20</td><td>.60</td><td>.134</td><td>.371</td><td>.093</td><td>.115</td><td>.262</td><td>.076</td></tr><tr><td>.60</td><td>.40</td><td>.00</td><td>.139</td><td>.459</td><td>.140</td><td>.122</td><td>.362</td><td>.117</td></tr><tr><td>.60</td><td>.40</td><td>.20</td><td>.136</td><td>.416</td><td>.121</td><td>.119</td><td>.320</td><td>.100</td></tr><tr><td>.60</td><td>.40</td><td>.40</td><td>.134</td><td>.385</td><td>.105</td><td>.117</td><td>.285</td><td>.086</td></tr><tr><td>.60</td><td>.40</td><td>.60</td><td>.132</td><td>.357</td><td>.092</td><td>.115</td><td>.259</td><td>.076</td></tr><tr><td>.60</td><td>.60</td><td>.00</td><td>.136</td><td>.421</td><td>.136</td><td>.121</td><td>.338</td><td>.116</td></tr><tr><td>.60</td><td>.60</td><td>.20</td><td>.135</td><td>.387</td><td>.115</td><td>.118</td><td>.302</td><td>.097</td></tr><tr><td>.60</td><td>.60</td><td>.40</td><td>.134</td><td>.359</td><td>.101</td><td>.116</td><td>.267</td><td>.083</td></tr><tr><td>.60</td><td>.60</td><td>.60</td><td>.132</td><td>.340</td><td>.090</td><td>.113</td><td>.244</td><td>.073</td></tr><tr><td>.60</td><td>.80</td><td>.00</td><td>.134</td><td>.340</td><td>.120</td><td>.116</td><td>.270</td><td>.100</td></tr><tr><td>.60</td><td>.80</td><td>.20</td><td>.134</td><td>.326</td><td>.107</td><td>.116</td><td>.249</td><td>.086</td></tr><tr><td>.60</td><td>.80</td><td>.40</td><td>.132</td><td>.309</td><td>.094</td><td>.115</td><td>.231</td><td>.075</td></tr><tr><td>.60</td><td>.80</td><td>.60</td><td>.130</td><td>.300</td><td>.085</td><td>.111</td><td>.214</td><td>.067</td></tr></tbody></table> </ephtml> </p> <p>3 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>For a calibration sample size of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> , all simulated combinations of IDM predictions with nonzero <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> for pseudo‐guessing parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> yielded RMSE values for all types of item parameters that were at least comparable (values displayed in bold) or even superior (values with gray background) to the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline. For conditions with uninformative predictions for pseudo‐guessing parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$R^2_c=0$</annotation></semantics></math> </ephtml> ), highly accurate predictions for item difficulty <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>=</mo><mo>.</mo><mn>80</mn></mrow><annotation encoding="application/x-tex">$R^2_b=.80$</annotation></semantics></math> </ephtml> ) or a combination of good predictions for item discrimination <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> and difficulty <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.40$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_b\ge.60$</annotation></semantics></math> </ephtml> , or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.60$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">$R^2_b\ge.40$</annotation></semantics></math> </ephtml> ) yielded comparable or improved estimation accuracy for all types of item parameters.</p> <hd id="AN0192629996-47">Item characteristic curves</hd> <p>As evidenced in Table 4, the item parameter estimation accuracy levels achieved with many of the IDM prediction quality combinations were not sufficient to yield mirroring ICC estimation accuracy across all ability segments considered. For <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math> </ephtml> , none of the simulated IDM prediction quality combinations was sufficient to reach the accuracy of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline for all considered ability segments. For <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> , we observed that item parameter prediction accuracies required to half the calibration sample size without losses (or even with improvements) in ICC estimation accuracy across all ability segments could counter‐balance each other. Comparable or even improved ICC estimation accuracy across all ability segments could be achieved with highly accurate predictions of item difficulties but poor predictions for the remaining two parameter types ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>20</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.20$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>=</mo><mo>.</mo><mn>80</mn></mrow><annotation encoding="application/x-tex">$R^2_b=.80$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>00</mn></mrow><annotation encoding="application/x-tex">$R^2_c\ge.00$</annotation></semantics></math> </ephtml> ), good predictions for item discriminations and pseudo‐guessing parameters but somewhat lower prediction accuracy for item difficulties ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.60$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>20</mn></mrow><annotation encoding="application/x-tex">$R^2_b\ge.20$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">$R^2_c\ge.40$</annotation></semantics></math> </ephtml> ), or good predictions for all types of item parameters ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.40$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">$R^2_b\ge.40$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_c\ge.60$</annotation></semantics></math> </ephtml> ).</p> <p>4 Table RMSE Values for ICCs over Different Ability Segments Obtained with Varying IDM Prediction Accuracy under N=100$N=100$ and N=500$N=500$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math></p></th></tr><tr><th /><th /><th /><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th>all</th><th>lower</th><th>middle</th><th>upper</th><th>all</th><th>lower</th><th>middle</th><th>upper</th></tr><tr><th /><th /><th /><th>(.04)</th><th>(.05)</th><th>(.03)</th><th>(.04)</th><th>(.04)</th><th>(.05)</th><th>(.03)</th><th>(.04)</th></tr></thead><tbody><tr><td>.00</td><td>.20</td><td>.00</td><td>.080</td><td>.108</td><td>.055</td><td>.051</td><td>.053</td><td>.068</td><td>.033</td><td>.041</td></tr><tr><td>.00</td><td>.20</td><td>.20</td><td>.080</td><td>.107</td><td>.055</td><td>.051</td><td>.053</td><td>.067</td><td>.033</td><td>.041</td></tr><tr><td>.00</td><td>.20</td><td>.40</td><td>.079</td><td>.106</td><td>.055</td><td>.051</td><td>.052</td><td>.066</td><td>.032</td><td>.041</td></tr><tr><td>.00</td><td>.20</td><td>.60</td><td>.078</td><td>.104</td><td>.055</td><td>.051</td><td>.051</td><td>.065</td><td>.032</td><td>.041</td></tr><tr><td>.00</td><td>.40</td><td>.00</td><td>.079</td><td>.106</td><td>.054</td><td>.050</td><td>.053</td><td>.067</td><td>.033</td><td>.041</td></tr><tr><td>.00</td><td>.40</td><td>.20</td><td>.079</td><td>.106</td><td>.055</td><td>.050</td><td>.053</td><td>.067</td><td>.033</td><td>.041</td></tr><tr><td>.00</td><td>.40</td><td>.40</td><td>.079</td><td>.106</td><td>.055</td><td>.050</td><td>.052</td><td>.066</td><td>.032</td><td>.041</td></tr><tr><td>.00</td><td>.40</td><td>.60</td><td>.078</td><td>.104</td><td>.054</td><td>.050</td><td>.051</td><td>.065</td><td>.032</td><td>.040</td></tr><tr><td>.00</td><td>.60</td><td>.00</td><td>.077</td><td>.103</td><td>.053</td><td>.050</td><td>.052</td><td>.066</td><td>.033</td><td>.041</td></tr><tr><td>.00</td><td>.60</td><td>.20</td><td>.078</td><td>.105</td><td>.053</td><td>.050</td><td>.052</td><td>.066</td><td>.032</td><td>.041</td></tr><tr><td>.00</td><td>.60</td><td>.40</td><td>.078</td><td>.104</td><td>.053</td><td>.049</td><td>.052</td><td>.066</td><td>.032</td><td>.040</td></tr><tr><td>.00</td><td>.60</td><td>.60</td><td>.076</td><td>.102</td><td>.053</td><td>.049</td><td>.051</td><td>.065</td><td>.031</td><td>.040</td></tr><tr><td>.00</td><td>.80</td><td>.00</td><td>.073</td><td>.096</td><td>.050</td><td>.049</td><td>.050</td><td>.063</td><td>.031</td><td>.040</td></tr><tr><td>.00</td><td>.80</td><td>.20</td><td>.073</td><td>.097</td><td>.050</td><td>.049</td><td>.050</td><td>.063</td><td>.030</td><td>.040</td></tr><tr><td>.00</td><td>.80</td><td>.40</td><td>.073</td><td>.098</td><td>.050</td><td>.049</td><td>.050</td><td>.063</td><td>.030</td><td>.040</td></tr><tr><td>.00</td><td>.80</td><td>.60</td><td>.073</td><td>.096</td><td>.050</td><td>.048</td><td>.049</td><td>.062</td><td>.030</td><td>.040</td></tr><tr><td>.20</td><td>.20</td><td>.00</td><td>.078</td><td>.105</td><td>.055</td><td>.049</td><td>.052</td><td>.067</td><td>.034</td><td>.040</td></tr><tr><td>.20</td><td>.20</td><td>.20</td><td>.078</td><td>.104</td><td>.054</td><td>.049</td><td>.052</td><td>.066</td><td>.033</td><td>.040</td></tr><tr><td>.20</td><td>.20</td><td>.40</td><td>.077</td><td>.103</td><td>.054</td><td>.049</td><td>.051</td><td>.066</td><td>.032</td><td>.039</td></tr><tr><td>.20</td><td>.20</td><td>.60</td><td>.075</td><td>.099</td><td>.054</td><td>.048</td><td>.050</td><td>.064</td><td>.032</td><td>.038</td></tr><tr><td>.20</td><td>.40</td><td>.00</td><td>.077</td><td>.103</td><td>.054</td><td>.048</td><td>.052</td><td>.067</td><td>.034</td><td>.039</td></tr><tr><td>.20</td><td>.40</td><td>.20</td><td>.076</td><td>.102</td><td>.054</td><td>.049</td><td>.051</td><td>.066</td><td>.033</td><td>.039</td></tr><tr><td>.20</td><td>.40</td><td>.40</td><td>.076</td><td>.102</td><td>.054</td><td>.049</td><td>.051</td><td>.065</td><td>.032</td><td>.039</td></tr><tr><td>.20</td><td>.40</td><td>.60</td><td>.075</td><td>.099</td><td>.053</td><td>.048</td><td>.050</td><td>.064</td><td>.032</td><td>.039</td></tr><tr><td>.20</td><td>.60</td><td>.00</td><td>.075</td><td>.100</td><td>.052</td><td>.048</td><td>.051</td><td>.066</td><td>.033</td><td>.039</td></tr><tr><td>.20</td><td>.60</td><td>.20</td><td>.075</td><td>.101</td><td>.052</td><td>.048</td><td>.051</td><td>.065</td><td>.032</td><td>.039</td></tr><tr><td>.20</td><td>.60</td><td>.40</td><td>.075</td><td>.100</td><td>.052</td><td>.047</td><td>.050</td><td>.065</td><td>.031</td><td>.038</td></tr><tr><td>.20</td><td>.60</td><td>.60</td><td>.073</td><td>.098</td><td>.052</td><td>.048</td><td>.050</td><td>.064</td><td>.031</td><td>.038</td></tr><tr><td>.20</td><td>.80</td><td>.00</td><td>.070</td><td>.093</td><td>.049</td><td>.047</td><td>.049</td><td>.062</td><td>.030</td><td>.038</td></tr><tr><td>.20</td><td>.80</td><td>.20</td><td>.071</td><td>.095</td><td>.049</td><td>.047</td><td>.049</td><td>.062</td><td>.030</td><td>.038</td></tr><tr><td>.20</td><td>.80</td><td>.40</td><td>.071</td><td>.095</td><td>.049</td><td>.047</td><td>.048</td><td>.062</td><td>.030</td><td>.038</td></tr><tr><td>.20</td><td>.80</td><td>.60</td><td>.070</td><td>.092</td><td>.049</td><td>.046</td><td>.048</td><td>.061</td><td>.029</td><td>.038</td></tr><tr><td>.40</td><td>.20</td><td>.00</td><td>.075</td><td>.101</td><td>.054</td><td>.047</td><td>.051</td><td>.066</td><td>.034</td><td>.037</td></tr><tr><td>.40</td><td>.20</td><td>.20</td><td>.075</td><td>.100</td><td>.053</td><td>.047</td><td>.050</td><td>.065</td><td>.032</td><td>.037</td></tr><tr><td>.40</td><td>.20</td><td>.40</td><td>.073</td><td>.098</td><td>.053</td><td>.046</td><td>.049</td><td>.064</td><td>.032</td><td>.037</td></tr><tr><td>.40</td><td>.20</td><td>.60</td><td>.071</td><td>.094</td><td>.052</td><td>.047</td><td>.048</td><td>.062</td><td>.031</td><td>.037</td></tr><tr><td>.40</td><td>.40</td><td>.00</td><td>.074</td><td>.100</td><td>.053</td><td>.047</td><td>.051</td><td>.066</td><td>.033</td><td>.037</td></tr><tr><td>.40</td><td>.40</td><td>.20</td><td>.074</td><td>.098</td><td>.052</td><td>.047</td><td>.050</td><td>.064</td><td>.032</td><td>.037</td></tr><tr><td>.40</td><td>.40</td><td>.40</td><td>.073</td><td>.097</td><td>.052</td><td>.046</td><td>.049</td><td>.064</td><td>.031</td><td>.036</td></tr><tr><td>.40</td><td>.40</td><td>.60</td><td>.070</td><td>.093</td><td>.052</td><td>.046</td><td>.048</td><td>.062</td><td>.030</td><td>.036</td></tr><tr><td>.40</td><td>.60</td><td>.00</td><td>.073</td><td>.097</td><td>.051</td><td>.046</td><td>.050</td><td>.065</td><td>.032</td><td>.036</td></tr><tr><td>.40</td><td>.60</td><td>.20</td><td>.072</td><td>.096</td><td>.051</td><td>.046</td><td>.049</td><td>.064</td><td>.031</td><td>.036</td></tr><tr><td>.40</td><td>.60</td><td>.40</td><td>.071</td><td>.095</td><td>.051</td><td>.045</td><td>.048</td><td>.063</td><td>.031</td><td>.036</td></tr><tr><td>.40</td><td>.60</td><td>.60</td><td>.070</td><td>.093</td><td>.050</td><td>.045</td><td>.048</td><td>.062</td><td>.030</td><td>.036</td></tr><tr><td>.40</td><td>.80</td><td>.00</td><td>.068</td><td>.090</td><td>.048</td><td>.045</td><td>.047</td><td>.061</td><td>.029</td><td>.036</td></tr><tr><td>.40</td><td>.80</td><td>.20</td><td>.068</td><td>.091</td><td>.048</td><td>.045</td><td>.047</td><td>.060</td><td>.029</td><td>.036</td></tr><tr><td>.40</td><td>.80</td><td>.40</td><td>.069</td><td>.091</td><td>.048</td><td>.045</td><td>.047</td><td>.061</td><td>.029</td><td>.035</td></tr><tr><td>.40</td><td>.80</td><td>.60</td><td>.066</td><td>.087</td><td>.048</td><td>.044</td><td>.046</td><td>.059</td><td>.029</td><td>.036</td></tr><tr><td>.60</td><td>.20</td><td>.00</td><td>.073</td><td>.098</td><td>.053</td><td>.045</td><td>.049</td><td>.064</td><td>.033</td><td>.034</td></tr><tr><td>.60</td><td>.20</td><td>.20</td><td>.071</td><td>.095</td><td>.052</td><td>.044</td><td>.048</td><td>.063</td><td>.031</td><td>.034</td></tr><tr><td>.60</td><td>.20</td><td>.40</td><td>.069</td><td>.092</td><td>.051</td><td>.043</td><td>.047</td><td>.062</td><td>.030</td><td>.033</td></tr><tr><td>.60</td><td>.20</td><td>.60</td><td>.066</td><td>.088</td><td>.051</td><td>.043</td><td>.045</td><td>.059</td><td>.029</td><td>.033</td></tr><tr><td>.60</td><td>.40</td><td>.00</td><td>.071</td><td>.096</td><td>.051</td><td>.044</td><td>.048</td><td>.063</td><td>.032</td><td>.034</td></tr><tr><td>.60</td><td>.40</td><td>.20</td><td>.070</td><td>.094</td><td>.051</td><td>.043</td><td>.047</td><td>.062</td><td>.031</td><td>.033</td></tr><tr><td>.60</td><td>.40</td><td>.40</td><td>.068</td><td>.090</td><td>.050</td><td>.042</td><td>.046</td><td>.061</td><td>.030</td><td>.033</td></tr><tr><td>.60</td><td>.40</td><td>.60</td><td>.066</td><td>.088</td><td>.050</td><td>.043</td><td>.046</td><td>.060</td><td>.029</td><td>.033</td></tr><tr><td>.60</td><td>.60</td><td>.00</td><td>.069</td><td>.093</td><td>.050</td><td>.043</td><td>.047</td><td>.062</td><td>.031</td><td>.033</td></tr><tr><td>.60</td><td>.60</td><td>.20</td><td>.068</td><td>.091</td><td>.049</td><td>.042</td><td>.046</td><td>.061</td><td>.030</td><td>.033</td></tr><tr><td>.60</td><td>.60</td><td>.40</td><td>.066</td><td>.088</td><td>.049</td><td>.042</td><td>.046</td><td>.060</td><td>.029</td><td>.033</td></tr><tr><td>.60</td><td>.60</td><td>.60</td><td>.065</td><td>.086</td><td>.049</td><td>.042</td><td>.045</td><td>.059</td><td>.029</td><td>.033</td></tr><tr><td>.60</td><td>.80</td><td>.00</td><td>.065</td><td>.087</td><td>.047</td><td>.041</td><td>.045</td><td>.059</td><td>.028</td><td>.033</td></tr><tr><td>.60</td><td>.80</td><td>.20</td><td>.065</td><td>.086</td><td>.046</td><td>.041</td><td>.045</td><td>.059</td><td>.028</td><td>.032</td></tr><tr><td>.60</td><td>.80</td><td>.40</td><td>.064</td><td>.085</td><td>.047</td><td>.041</td><td>.045</td><td>.058</td><td>.028</td><td>.033</td></tr><tr><td>.60</td><td>.80</td><td>.60</td><td>.062</td><td>.082</td><td>.046</td><td>.041</td><td>.044</td><td>.057</td><td>.027</td><td>.032</td></tr></tbody></table> </ephtml> </p> <p>4 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <hd id="AN0192629996-48">Ability estimates for selected examinees</hd> <p>For ability estimates for selected examinees (see Table 5), we, again, found that item parameter prediction accuracies required to achieve substantial reductions in calibration sample sizes without losses in the accuracy of ability estimates for the five selected examinees (displayed in bold) could counter‐balance each other: Comparable ability estimation accuracy for all considered levels could be achieved with highly accurate predictions of item difficulties but poor predictions for the remaining two parameter types ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>00</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.00$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>=</mo><mo>.</mo><mn>80</mn></mrow><annotation encoding="application/x-tex">$R^2_b=.80$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>00</mn></mrow><annotation encoding="application/x-tex">$R^2_c\ge.00$</annotation></semantics></math> </ephtml> ), fair predictions for item discriminations but good predictions for item difficulties and pseudo‐guessing parameters ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>20</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.20$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_b\ge.60$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>=</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">${R}_{c}^{2}=.60$</annotation></semantics></math> </ephtml> ), or fair prediction accuracy for item discriminations and good accuracy for item difficulties ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">$R^2_a\ge.40$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$R^2_b\ge.60$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><mo>≥</mo><mo>.</mo><mn>00</mn></mrow><annotation encoding="application/x-tex">$R^2_c\ge.00$</annotation></semantics></math> </ephtml> ) for a calibration sample size as low as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math> </ephtml> . For a calibration sample size of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math> </ephtml> , ability estimation accuracy was comparable to the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>PML</mi><mtext>default</mtext></msub><annotation encoding="application/x-tex">${\rm PML}_{\text{default}}$</annotation></semantics></math> </ephtml> baseline across all simulated combinations of IDM predictions; that is, essentially, accurate ability estimates could already be obtained by merely considering the distribution of item parameters in the pool and no accurate IDM predictions were required for halving the calibration sample size.</p> <p>5 Table RMSE Values for EAP Ability Estimates for Five Selected Examinees with Item Parameters Fixed to Values Obtained with Prior Settings of Varying IDM Prediction Accuracy under N=100$N=100$ and N=500$N=500$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th></tr><tr><th /><th /><th /><th>(.49)</th><th>(.33)</th><th>(.29)</th><th>(.29)</th><th>(.41)</th><th>(.49)</th><th>(.33)</th><th>(.29)</th><th>(.29)</th><th>(.41)</th></tr></thead><tbody><tr><td>.00</td><td>.20</td><td>.00</td><td>.634</td><td>.333</td><td>.284</td><td>.302</td><td>.412</td><td>.534</td><td>.317</td><td>.288</td><td>.298</td><td>.400</td></tr><tr><td>.00</td><td>.20</td><td>.20</td><td>.627</td><td>.332</td><td>.282</td><td>.307</td><td>.418</td><td>.532</td><td>.316</td><td>.288</td><td>.302</td><td>.399</td></tr><tr><td>.00</td><td>.20</td><td>.40</td><td>.622</td><td>.333</td><td>.282</td><td>.307</td><td>.418</td><td>.536</td><td>.318</td><td>.289</td><td>.301</td><td>.398</td></tr><tr><td>.00</td><td>.20</td><td>.60</td><td>.611</td><td>.332</td><td>.286</td><td>.308</td><td>.423</td><td>.536</td><td>.316</td><td>.290</td><td>.301</td><td>.398</td></tr><tr><td>.00</td><td>.40</td><td>.00</td><td>.612</td><td>.329</td><td>.285</td><td>.304</td><td>.416</td><td>.530</td><td>.317</td><td>.288</td><td>.300</td><td>.400</td></tr><tr><td>.00</td><td>.40</td><td>.20</td><td>.619</td><td>.331</td><td>.281</td><td>.307</td><td>.416</td><td>.531</td><td>.316</td><td>.287</td><td>.301</td><td>.400</td></tr><tr><td>.00</td><td>.40</td><td>.40</td><td>.612</td><td>.332</td><td>.281</td><td>.308</td><td>.418</td><td>.534</td><td>.319</td><td>.288</td><td>.301</td><td>.397</td></tr><tr><td>.00</td><td>.40</td><td>.60</td><td>.611</td><td>.332</td><td>.282</td><td>.309</td><td>.419</td><td>.531</td><td>.320</td><td>.287</td><td>.303</td><td>.395</td></tr><tr><td>.00</td><td>.60</td><td>.00</td><td>.592</td><td>.322</td><td>.283</td><td>.312</td><td>.418</td><td>.528</td><td>.316</td><td>.287</td><td>.303</td><td>.401</td></tr><tr><td>.00</td><td>.60</td><td>.20</td><td>.612</td><td>.332</td><td>.284</td><td>.311</td><td>.415</td><td>.533</td><td>.318</td><td>.288</td><td>.303</td><td>.399</td></tr><tr><td>.00</td><td>.60</td><td>.40</td><td>.601</td><td>.321</td><td>.279</td><td>.309</td><td>.414</td><td>.534</td><td>.313</td><td>.288</td><td>.303</td><td>.398</td></tr><tr><td>.00</td><td>.60</td><td>.60</td><td>.603</td><td>.325</td><td>.282</td><td>.312</td><td>.417</td><td>.531</td><td>.314</td><td>.287</td><td>.304</td><td>.396</td></tr><tr><td>.00</td><td>.80</td><td>.00</td><td>.557</td><td>.322</td><td>.286</td><td>.314</td><td>.410</td><td>.518</td><td>.318</td><td>.288</td><td>.304</td><td>.396</td></tr><tr><td>.00</td><td>.80</td><td>.20</td><td>.568</td><td>.320</td><td>.285</td><td>.314</td><td>.409</td><td>.527</td><td>.314</td><td>.287</td><td>.305</td><td>.393</td></tr><tr><td>.00</td><td>.80</td><td>.40</td><td>.576</td><td>.327</td><td>.286</td><td>.313</td><td>.406</td><td>.530</td><td>.318</td><td>.288</td><td>.305</td><td>.392</td></tr><tr><td>.00</td><td>.80</td><td>.60</td><td>.577</td><td>.319</td><td>.286</td><td>.309</td><td>.403</td><td>.525</td><td>.313</td><td>.287</td><td>.303</td><td>.390</td></tr><tr><td>.20</td><td>.20</td><td>.00</td><td>.619</td><td>.327</td><td>.288</td><td>.300</td><td>.411</td><td>.533</td><td>.318</td><td>.289</td><td>.298</td><td>.396</td></tr><tr><td>.20</td><td>.20</td><td>.20</td><td>.624</td><td>.332</td><td>.280</td><td>.303</td><td>.414</td><td>.539</td><td>.317</td><td>.287</td><td>.300</td><td>.396</td></tr><tr><td>.20</td><td>.20</td><td>.40</td><td>.617</td><td>.330</td><td>.285</td><td>.304</td><td>.416</td><td>.529</td><td>.317</td><td>.290</td><td>.301</td><td>.394</td></tr><tr><td>.20</td><td>.20</td><td>.60</td><td>.603</td><td>.332</td><td>.282</td><td>.306</td><td>.420</td><td>.532</td><td>.318</td><td>.288</td><td>.302</td><td>.397</td></tr><tr><td>.20</td><td>.40</td><td>.00</td><td>.610</td><td>.334</td><td>.287</td><td>.301</td><td>.411</td><td>.525</td><td>.320</td><td>.290</td><td>.299</td><td>.395</td></tr><tr><td>.20</td><td>.40</td><td>.20</td><td>.604</td><td>.326</td><td>.285</td><td>.311</td><td>.413</td><td>.527</td><td>.315</td><td>.289</td><td>.303</td><td>.395</td></tr><tr><td>.20</td><td>.40</td><td>.40</td><td>.607</td><td>.324</td><td>.285</td><td>.306</td><td>.413</td><td>.529</td><td>.316</td><td>.288</td><td>.302</td><td>.394</td></tr><tr><td>.20</td><td>.40</td><td>.60</td><td>.603</td><td>.325</td><td>.278</td><td>.310</td><td>.418</td><td>.531</td><td>.315</td><td>.286</td><td>.305</td><td>.395</td></tr><tr><td>.20</td><td>.60</td><td>.00</td><td>.587</td><td>.322</td><td>.288</td><td>.309</td><td>.420</td><td>.520</td><td>.317</td><td>.289</td><td>.303</td><td>.401</td></tr><tr><td>.20</td><td>.60</td><td>.20</td><td>.601</td><td>.325</td><td>.284</td><td>.304</td><td>.412</td><td>.529</td><td>.315</td><td>.288</td><td>.301</td><td>.394</td></tr><tr><td>.20</td><td>.60</td><td>.40</td><td>.600</td><td>.326</td><td>.285</td><td>.303</td><td>.409</td><td>.529</td><td>.318</td><td>.289</td><td>.301</td><td>.394</td></tr><tr><td>.20</td><td>.60</td><td>.60</td><td>.590</td><td>.325</td><td>.284</td><td>.304</td><td>.411</td><td>.529</td><td>.315</td><td>.289</td><td>.302</td><td>.393</td></tr><tr><td>.20</td><td>.80</td><td>.00</td><td>.547</td><td>.325</td><td>.289</td><td>.312</td><td>.411</td><td>.507</td><td>.319</td><td>.288</td><td>.303</td><td>.395</td></tr><tr><td>.20</td><td>.80</td><td>.20</td><td>.565</td><td>.320</td><td>.284</td><td>.309</td><td>.410</td><td>.517</td><td>.317</td><td>.287</td><td>.303</td><td>.396</td></tr><tr><td>.20</td><td>.80</td><td>.40</td><td>.566</td><td>.332</td><td>.281</td><td>.308</td><td>.401</td><td>.523</td><td>.320</td><td>.288</td><td>.302</td><td>.390</td></tr><tr><td>.20</td><td>.80</td><td>.60</td><td>.571</td><td>.323</td><td>.287</td><td>.305</td><td>.401</td><td>.526</td><td>.315</td><td>.289</td><td>.301</td><td>.388</td></tr><tr><td>.40</td><td>.20</td><td>.00</td><td>.610</td><td>.326</td><td>.285</td><td>.301</td><td>.407</td><td>.524</td><td>.321</td><td>.291</td><td>.297</td><td>.392</td></tr><tr><td>.40</td><td>.20</td><td>.20</td><td>.612</td><td>.326</td><td>.287</td><td>.302</td><td>.407</td><td>.528</td><td>.315</td><td>.289</td><td>.299</td><td>.390</td></tr><tr><td>.40</td><td>.20</td><td>.40</td><td>.603</td><td>.320</td><td>.286</td><td>.303</td><td>.405</td><td>.526</td><td>.314</td><td>.290</td><td>.302</td><td>.389</td></tr><tr><td>.40</td><td>.20</td><td>.60</td><td>.594</td><td>.324</td><td>.283</td><td>.306</td><td>.413</td><td>.522</td><td>.315</td><td>.289</td><td>.301</td><td>.393</td></tr><tr><td>.40</td><td>.40</td><td>.00</td><td>.598</td><td>.326</td><td>.286</td><td>.301</td><td>.408</td><td>.525</td><td>.317</td><td>.289</td><td>.299</td><td>.392</td></tr><tr><td>.40</td><td>.40</td><td>.20</td><td>.602</td><td>.329</td><td>.284</td><td>.301</td><td>.410</td><td>.529</td><td>.316</td><td>.289</td><td>.301</td><td>.392</td></tr><tr><td>.40</td><td>.40</td><td>.40</td><td>.594</td><td>.318</td><td>.287</td><td>.300</td><td>.409</td><td>.518</td><td>.313</td><td>.291</td><td>.298</td><td>.390</td></tr><tr><td>.40</td><td>.40</td><td>.60</td><td>.597</td><td>.325</td><td>.283</td><td>.299</td><td>.412</td><td>.529</td><td>.317</td><td>.289</td><td>.301</td><td>.391</td></tr><tr><td>.40</td><td>.60</td><td>.00</td><td>.576</td><td>.327</td><td>.283</td><td>.304</td><td>.404</td><td>.520</td><td>.321</td><td>.288</td><td>.301</td><td>.390</td></tr><tr><td>.40</td><td>.60</td><td>.20</td><td>.589</td><td>.326</td><td>.284</td><td>.301</td><td>.404</td><td>.520</td><td>.318</td><td>.289</td><td>.300</td><td>.388</td></tr><tr><td>.40</td><td>.60</td><td>.40</td><td>.587</td><td>.318</td><td>.285</td><td>.302</td><td>.406</td><td>.523</td><td>.316</td><td>.289</td><td>.301</td><td>.392</td></tr><tr><td>.40</td><td>.60</td><td>.60</td><td>.578</td><td>.328</td><td>.282</td><td>.306</td><td>.409</td><td>.523</td><td>.315</td><td>.287</td><td>.304</td><td>.391</td></tr><tr><td>.40</td><td>.80</td><td>.00</td><td>.542</td><td>.322</td><td>.290</td><td>.307</td><td>.409</td><td>.510</td><td>.318</td><td>.291</td><td>.302</td><td>.393</td></tr><tr><td>.40</td><td>.80</td><td>.20</td><td>.555</td><td>.323</td><td>.288</td><td>.307</td><td>.406</td><td>.515</td><td>.322</td><td>.287</td><td>.302</td><td>.393</td></tr><tr><td>.40</td><td>.80</td><td>.40</td><td>.570</td><td>.322</td><td>.290</td><td>.307</td><td>.403</td><td>.518</td><td>.317</td><td>.290</td><td>.304</td><td>.389</td></tr><tr><td>.40</td><td>.80</td><td>.60</td><td>.561</td><td>.319</td><td>.286</td><td>.304</td><td>.397</td><td>.521</td><td>.315</td><td>.288</td><td>.301</td><td>.386</td></tr><tr><td>.60</td><td>.20</td><td>.00</td><td>.604</td><td>.327</td><td>.287</td><td>.298</td><td>.401</td><td>.521</td><td>.320</td><td>.290</td><td>.299</td><td>.384</td></tr><tr><td>.60</td><td>.20</td><td>.20</td><td>.603</td><td>.325</td><td>.283</td><td>.300</td><td>.406</td><td>.524</td><td>.321</td><td>.289</td><td>.299</td><td>.387</td></tr><tr><td>.60</td><td>.20</td><td>.40</td><td>.591</td><td>.323</td><td>.291</td><td>.300</td><td>.405</td><td>.522</td><td>.311</td><td>.293</td><td>.300</td><td>.385</td></tr><tr><td>.60</td><td>.20</td><td>.60</td><td>.581</td><td>.315</td><td>.285</td><td>.300</td><td>.411</td><td>.517</td><td>.313</td><td>.290</td><td>.299</td><td>.390</td></tr><tr><td>.60</td><td>.40</td><td>.00</td><td>.587</td><td>.327</td><td>.285</td><td>.296</td><td>.401</td><td>.515</td><td>.323</td><td>.288</td><td>.297</td><td>.387</td></tr><tr><td>.60</td><td>.40</td><td>.20</td><td>.589</td><td>.321</td><td>.286</td><td>.294</td><td>.408</td><td>.518</td><td>.317</td><td>.291</td><td>.295</td><td>.391</td></tr><tr><td>.60</td><td>.40</td><td>.40</td><td>.591</td><td>.325</td><td>.284</td><td>.296</td><td>.403</td><td>.523</td><td>.317</td><td>.288</td><td>.298</td><td>.387</td></tr><tr><td>.60</td><td>.40</td><td>.60</td><td>.588</td><td>.322</td><td>.286</td><td>.297</td><td>.405</td><td>.525</td><td>.316</td><td>.290</td><td>.299</td><td>.388</td></tr><tr><td>.60</td><td>.60</td><td>.00</td><td>.565</td><td>.323</td><td>.289</td><td>.298</td><td>.401</td><td>.506</td><td>.319</td><td>.291</td><td>.298</td><td>.388</td></tr><tr><td>.60</td><td>.60</td><td>.20</td><td>.574</td><td>.322</td><td>.286</td><td>.298</td><td>.396</td><td>.517</td><td>.318</td><td>.289</td><td>.298</td><td>.385</td></tr><tr><td>.60</td><td>.60</td><td>.40</td><td>.584</td><td>.327</td><td>.283</td><td>.300</td><td>.402</td><td>.521</td><td>.321</td><td>.288</td><td>.299</td><td>.385</td></tr><tr><td>.60</td><td>.60</td><td>.60</td><td>.581</td><td>.325</td><td>.285</td><td>.297</td><td>.401</td><td>.523</td><td>.319</td><td>.290</td><td>.300</td><td>.387</td></tr><tr><td>.60</td><td>.80</td><td>.00</td><td>.535</td><td>.322</td><td>.290</td><td>.302</td><td>.403</td><td>.503</td><td>.320</td><td>.290</td><td>.299</td><td>.390</td></tr><tr><td>.60</td><td>.80</td><td>.20</td><td>.560</td><td>.319</td><td>.284</td><td>.298</td><td>.399</td><td>.514</td><td>.314</td><td>.288</td><td>.299</td><td>.388</td></tr><tr><td>.60</td><td>.80</td><td>.40</td><td>.565</td><td>.320</td><td>.293</td><td>.293</td><td>.397</td><td>.517</td><td>.316</td><td>.291</td><td>.297</td><td>.386</td></tr><tr><td>.60</td><td>.80</td><td>.60</td><td>.566</td><td>.315</td><td>.282</td><td>.295</td><td>.393</td><td>.522</td><td>.311</td><td>.287</td><td>.298</td><td>.383</td></tr></tbody></table> </ephtml> </p> <p>5 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <hd id="AN0192629996-49">Conclusions from Study III</hd> <p>Study III illustrated how the proposed two‐step approach can be used to inform benchmarks for IDM prediction quality in terms of achievable reductions in calibration sample size requirements. We found that in order to achieve substantive reductions in calibration sample size requirements without sacrificing accuracy, not all item parameters require good prediction quality. Rather, required prediction qualities can counterbalance each other such that with the proposed two‐step approach, calibration sample sizes can be reduced by both high‐quality item difficulty predictions or good predictions of item discriminations and pseudo‐guessing parameters. Further, we found achievable sample size reductions to vary for different outcome measures. When the primary aim is to obtain accurate ability estimates regardless of whether or not item parameters or ICCs are accurately estimated, calibration sample sizes can be much more reduced with the same quality of IDM item parameter predictions than when obtaining accurate ICCs is of importance.</p> <hd id="AN0192629996-50">Discussion</hd> <p>In the present study, we blended two approaches to reducing calibration sample size requirements: IDM, aiming to reduce calibration samples to zero by predicting item parameters from items' textual features and the incorporation of collateral information into Bayesian IRT estimation procedures. The resultant approach is a two‐step approach where, in Step 1, IDM predictions are obtained and, in Step 2, employed to construct informative prior distributions for Bayesian estimation of small‐sample IRT models. We illustrated and evaluated the approach in three simulation studies, informed by a case study using an item pool and IDM predictions of a high‐stakes admission test.</p> <p>Study I served to determine guidelines on implementation by comparing the small‐sample performance of three Bayesian estimators with informative prior distributions—the EAP and MAP obtained with MCMC as well as the mode of the joint posterior distribution obtained with PML. Corroborating results for other IRT models and diffuse prior settings (Kieftenbeld & Natesan, [<reflink idref="bib29" id="ref103">29</reflink>]; Azevedo et al., [<reflink idref="bib3" id="ref104">3</reflink>]), we found the EAP to exhibit an advantage over the MAP and PML estimates in the accuracy of item parameter estimates. Nevertheless, the item parameter estimates provided by PML yielded ICCs and, accordingly, ability estimates across different levels with accuracy closely resembling the EAP estimate. These results align with previous research pointing out that even unstable item parameter estimates may provide rather stable response functions (Ogasawara, [<reflink idref="bib41" id="ref105">41</reflink>]). Given that PML and its fixed item parameters are able to provide ICCs and ability estimates closely resembling the best‐performing EAP in terms of accuracy while aligning much closer with operational practice, we recommend implementing the proposed two‐step approach drawing on PML estimation.</p> <p>In Study II, we evaluated by how much calibration sample sizes for the considered high‐stakes admission test can be reduced when state‐of‐the‐art IDM predictions—yielding fair predictions of difficulties, but being uninformative about discriminations and pseudo‐guessing parameters—are employed for constructing informative prior distributions. Aligning with previous research on weakly informative collateral information (Keller, [<reflink idref="bib28" id="ref106">28</reflink>]; Matteucci et al., [<reflink idref="bib36" id="ref107">36</reflink>]; Swaminathan et al., [<reflink idref="bib51" id="ref108">51</reflink>]), we found negligible gains, especially in comparison with considering the historical distribution of item parameters in the pool. That is, for the type of items considered in the present study, current state‐of‐the‐art IDM predictions were not accurate enough to allow for substantive reductions of calibration sample size requirements with the proposed approach.</p> <p>In Study III, we evaluated the levels of IDM prediction accuracy required to achieve a targeted reduction in calibration sample size without losses in accuracy. Most importantly, we found that it is not necessary for all item parameters to have high prediction quality to achieve substantive reductions in calibration sample size. This is why we recommend to evaluate the quality of IDM predictions in an ensemble, as even weak predictions for one parameter type may aid calibration sample size reductions when they are counter‐balanced by good predictions for another. Further, we believe that Study III illustrated how the proposed approach can be used to provide benchmarks for evaluating the quality of IDM predictions—that, in our opinion, the field is lacking—and quantifying them in terms of achievable calibration sample size reductions when used for informative prior constructions. In contrast to current practice, which oftentimes evaluates the extent to which the best‐performing machine learning model used for IDM outperformed its (possibly poorly performing) competitors, these new benchmarks are portable across studies, facilitating to contextualize achieved IDM predictions. Note that the specific combinations of IDM prediction quality required to achieve calibration sample size reductions in our case study may be study‐specific and may not generalize to other settings. For IDM studies, we, therefore, recommend evaluating the quality of IDM predictions by drawing on simulations mirroring the conditions at hand, rather than using the results of Study III as "look‐up tables." Using the present study as a template, obtained IDM results could then be complemented with explorations of achievable reductions in calibration sample size requirements compared to a baseline mirroring operational practice if the obtained IDM predictions were used in the proposed two‐step approach.</p> <p>We point out that IDM prediction qualities required for achieving a substantive reduction of the calibration sample size (say, for halving it) exceed current state‐of‐the‐art predictions for complex tasks (as achieved in, e.g., Beinborn et al., [<reflink idref="bib4" id="ref109">4</reflink>]; Yaneva et al., [<reflink idref="bib56" id="ref110">56</reflink>]; Qiu et al., [<reflink idref="bib44" id="ref111">44</reflink>]; Belov et al., [<reflink idref="bib8" id="ref112">8</reflink>]). Hence, while we presented an approach that allows immediately exploiting even somewhat inaccurate IDM predictions, further research advancing IDM prediction accuracy is still needed.</p> <p>While investigating our approach with a case study allowed evaluations of its performance in real‐life settings, this set‐up also limits the generalizability of our findings. For instance, it may well be that the deviations of multivariate normality of item parameter distributions in the considered item pool (see Figure 1) may have impacted the performance of both informative prior settings, which assume item parameters, respectively, their IDM residuals, to be multivariate normally distributed. Future studies may, therefore, investigate replicability of our findings in other operational settings.</p> <p>Finally, we point out that there are current alternatives to the incorporation of collateral information into Bayesian IRT estimation procedure for small‐sample IRT estimation, and that the field has recently seen innovative and promising small‐sample IRT solutions. For instance, when to‐be‐calibrated items are administered jointly with items from the pool, fixed item parameter calibration can be employed to stabilize small‐sample IRT estimates (König et al., [<reflink idref="bib30" id="ref113">30</reflink>]). Further, small‐sample IRT estimation has been shown to immensely profit from careful construction of diffuse hierarchical priors (König et al., [<reflink idref="bib31" id="ref114">31</reflink>]). Another alternative is just‐emerging likelihood‐free estimation (Belov et al., [<reflink idref="bib7" id="ref115">7</reflink>]) that, rather than enriching the likelihood with prior information substitutes reliance on the likelihood with a neural network "learning" the relationship between response patterns and item parameters. Likewise, the latent‐variable‐free IRT Bayesian covariance structure model recently proposed by Fox ([<reflink idref="bib19" id="ref116">19</reflink>]) has exhibited impressive performance for the two‐parameter logistic model in small samples and its expansion to the 3PL can be hoped to achieve comparable results. Future research may compare these alternative small‐sample IRT solutions against the proposed approach.</p> <hd id="AN0192629996-51">Acknowledgments</hd> <p>Online materials for this article can be found in the OSF and are available via the following link: https://osf.io/2ju9z/. This work was partially supported by the Research Council of Norway through its Centres of Excellence scheme, project number 33160.</p> <hd id="AN0192629996-52">1 Appendix Study I and II: Bias of Item Parameter Estimates, Item Characteristic Curves, and...</hd> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0009.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0009.jpg" title="A1 Bias of item parameters a$a$, b$b$, and c$c$ obtained with different Bayesian estimators for the two‐step approach plotted against sample size N$N$. The light gray dashed horizontal line marks unbiased estimation. EAP: mean of the marginal posterior distribution obtained with MCMC; MAP: mode of the marginal posterior distribution obtained from MCMC; PML: mode of the joint posterior distribution obtained with penalized maximum likelihood. Note that y$y$ ‐axes differ in scale. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0010.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0010.jpg" title="A2 Bias of item characteristic curves over different segments of the ability continuum obtained with different Bayesian estimators for the two‐step approach plotted against sample size N$N$. Note that for obtaining the bias of item characteristic curves, absolute differences between true and estimated model‐implied probabilities correct at each supporting point are considered, such that bias cannot be negative. The light gray dashed horizontal line marks unbiased estimation. ICC all: [–5; 5]; ICC lower: [–5; −1]; ICC middle: [–1; 1]; ICC upper: [1; 5]. EAP: mean of the marginal posterior distribution obtained with MCMC; MAP: mode of the marginal posterior distribution obtained from MCMC; PML: mode of the joint posterior distribution obtained with penalized maximum likelihood. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0011.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0011.jpg" title="A3 Bias of EAP ability estimates for five selected examinees with item parameters fixed to values obtained from different Bayesian estimators for the two‐step approach under different sample size N$N$. The light gray dashed horizontal line marks unbiased estimation. EAP: mean of the marginal posterior distribution obtained with MCMC; MAP: mode of the marginal posterior distribution obtained from MCMC; PML: mode of the joint posterior distribution obtained with penalized maximum likelihood. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0012.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0012.jpg" title="A4 Bias of item parameters a$a$, b$b$, and c$c$ obtained with different prior settings as well as item difficulty modeling (IDM) plotted against sample size N$N$. The light gray dashed horizontal line marks unbiased es PMLdefault${\rm PML}_{\text{default}}$: default priors as implemented in BILOG; PMLpool${\rm PML}_{\text{pool}}$: multivariate normal prior based on the distribution of item parameters in the item pool; PMLIDM+pool${\rm PML}_{\text{IDM+pool}}$: priors informed by IDM predictions. Note that y$y$ ‐axes differ in scale. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0013.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0013.jpg" title="A5 Bias of item characteristic curves over different segments of the ability continuum obtained with different prior settings as well as item difficulty modeling (IDM) plotted against sample size plotted against sample size N$N$. Note that for obtaining the bias of item characteristic curves, absolute differences between true and estimated model‐implied probabilities correct at each supporting point are considered, such that bias cannot be negative. The light gray dashed horizontal line marks unbiased estimation. ICC all: [–5; 5]; ICC lower: [–5; −1]; ICC middle: [–1; 1]; ICC upper: [1; 5]. PMLdefault${\rm PML}_{\text{default}}$: default priors as implemented in BILOG; PMLpool${\rm PML}_{\text{pool}}$: multivariate normal prior based on the distribution of item parameters in the item pool; PMLIDM+pool${\rm PML}_{\text{IDM+pool}}$: priors informed by IDM predictions. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12426-fig-0014.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12426-fig-0014.jpg" title="A6 Bias of EAP ability estimates for five selected examinees with item parameters fixed to values obtained with different prior settings as well as item difficulty modeling (IDM) plotted against sample size plotted against sample size N$N$. The light gray dashed horizontal line marks unbiased estimation. PMLdefault${\rm PML}_{\text{default}}$: default priors as implemented in BILOG; PMLpool${\rm PML}_{\text{pool}}$: multivariate normal prior based on the distribution of item parameters in the item pool; PMLIDM+pool${\rm PML}_{\text{IDM+pool}}$: priors informed by IDM predictions. The x$x$‐axis ticks mark the studied sample sizes, with labels provided for the smallest (50) and largest (1,000) studied sample sizes." /> </p> <p></p> <hd id="AN0192629996-59">2 Appendix Study III: RMSE Values for N=50$N=50$ and N=200$N=200$</hd> <p>B1 Table RMSE Values for Item Parameters a$a$, b$b$, and c$c$ Obtained with Varying IDM Prediction Accuracy under N=50$N=50$ and N=200$N=200$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">$N=50$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>200</mn></mrow><annotation encoding="application/x-tex">$N=200$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (0.17)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (0.30)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> (0.10)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (0.17)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (0.30)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> (0.10)</th></tr></thead><tbody><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.208</p></td><td><p>.558</p></td><td><p>.151</p></td><td><p><bold>.189</bold></p></td><td><p>.444</p></td><td><p>.135</p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.201</p></td><td><p>.518</p></td><td><p>.134</p></td><td><p><bold>.183</bold></p></td><td><p>.420</p></td><td><p><bold>.121</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p><bold>.197</bold></p></td><td><p>.484</p></td><td><p><bold>.119</bold></p></td><td><p><bold>.176</bold></p></td><td><p>.382</p></td><td><p><bold>.107</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p><bold>.194</bold></p></td><td><p>.453</p></td><td><p><bold>.102</bold></p></td><td><p><bold>.170</bold></p></td><td><p><bold>.352</bold></p></td><td><p><bold>.093</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.205</p></td><td><p>.543</p></td><td><p>.151</p></td><td><p><bold>.188</bold></p></td><td><p>.445</p></td><td><p>.138</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.199</p></td><td><p>.497</p></td><td><p>.133</p></td><td><p><bold>.180</bold></p></td><td><p>.401</p></td><td><p><bold>.118</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p><bold>.195</bold></p></td><td><p>.467</p></td><td><p><bold>.115</bold></p></td><td><p><bold>.174</bold></p></td><td><p><bold>.365</bold></p></td><td><p><bold>.104</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p><bold>.193</bold></p></td><td><p>.438</p></td><td><p><bold>.100</bold></p></td><td><p><bold>.167</bold></p></td><td><p><bold>.336</bold></p></td><td><p><bold>.089</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.199</p></td><td><p>.491</p></td><td><p>.146</p></td><td><p><bold>.182</bold></p></td><td><p>.417</p></td><td><p>.133</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p><bold>.195</bold></p></td><td><p>.455</p></td><td><p>.126</p></td><td><p><bold>.176</bold></p></td><td><p>.368</p></td><td><p><bold>.111</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p><bold>.192</bold></p></td><td><p>.430</p></td><td><p><bold>.113</bold></p></td><td><p><bold>.170</bold></p></td><td><p><bold>.339</bold></p></td><td><p><bold>.099</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p><bold>.190</bold></p></td><td><p>.412</p></td><td><p><bold>.097</bold></p></td><td><p><bold>.164</bold></p></td><td><p><bold>.313</bold></p></td><td><p><bold>.085</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p><bold>.187</bold></p></td><td><p>.390</p></td><td><p>.129</p></td><td><p><bold>.168</bold></p></td><td><p><bold>.329</bold></p></td><td><p><bold>.112</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p><bold>.186</bold></p></td><td><p>.370</p></td><td><p><bold>.115</bold></p></td><td><p><bold>.165</bold></p></td><td><p><bold>.307</bold></p></td><td><p><bold>.098</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p><bold>.186</bold></p></td><td><p><bold>.361</bold></p></td><td><p><bold>.103</bold></p></td><td><p><bold>.162</bold></p></td><td><p><bold>.287</bold></p></td><td><p><bold>.089</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p><bold>.185</bold></p></td><td><p><bold>.349</bold></p></td><td><p><bold>.094</bold></p></td><td><p><bold>.160</bold></p></td><td><p><bold>.273</bold></p></td><td><p><bold>.079</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p><bold>.189</bold></p></td><td><p>.557</p></td><td><p>.151</p></td><td><p><bold>.176</bold></p></td><td><p>.457</p></td><td><p>.136</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p><bold>.185</bold></p></td><td><p>.510</p></td><td><p>.134</p></td><td><p><bold>.169</bold></p></td><td><p>.407</p></td><td><p><bold>.119</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p><bold>.181</bold></p></td><td><p>.480</p></td><td><p><bold>.118</bold></p></td><td><p><bold>.166</bold></p></td><td><p>.378</p></td><td><p><bold>.105</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p><bold>.178</bold></p></td><td><p>.447</p></td><td><p><bold>.100</bold></p></td><td><p><bold>.159</bold></p></td><td><p><bold>.339</bold></p></td><td><p><bold>.090</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p><bold>.187</bold></p></td><td><p>.537</p></td><td><p>.151</p></td><td><p><bold>.174</bold></p></td><td><p>.459</p></td><td><p>.141</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p><bold>.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127</bold></p></td><td><p><bold>.273</bold></p></td><td><p><bold>.086</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p><bold>.134</bold></p></td><td><p><bold>.338</bold></p></td><td><p><bold>.093</bold></p></td><td><p><bold>.124</bold></p></td><td><p><bold>.258</bold></p></td><td><p><bold>.077</bold></p></td></tr></tbody></table> </ephtml> </p> <p>6 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>B2 Table RMSE Values for ICCs over Different Ability Segments Obtained with Varying IDM Prediction Accuracy under N=50$N=50$ and N=200$N=200$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">$N=50$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>200</mn></mrow><annotation encoding="application/x-tex">$N=200$</annotation></semantics></math></p></th></tr><tr><th /><th /><th /><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th>all</th><th>lower</th><th>middle</th><th>upper</th><th>all</th><th>lower</th><th>middle</th><th>upper</th></tr><tr><th /><th /><th /><th>(0.04)</th><th>(0.05)</th><th>(0.03)</th><th>(0.04)</th><th>(0.04)</th><th>(0.05)</th><th>(0.03)</th><th>(0.04)</th></tr></thead><tbody><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.091</p></td><td><p>.122</p></td><td><p>.069</p></td><td><p>.056</p></td><td><p>.069</p></td><td><p>.092</p></td><td><p>.044</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.090</p></td><td><p>.120</p></td><td><p>.068</p></td><td><p>.056</p></td><td><p>.068</p></td><td><p>.091</p></td><td><p>.044</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.090</p></td><td><p>.119</p></td><td><p>.069</p></td><td><p>.056</p></td><td><p>.068</p></td><td><p>.090</p></td><td><p>.044</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.088</p></td><td><p>.116</p></td><td><p>.068</p></td><td><p>.056</p></td><td><p>.066</p></td><td><p>.088</p></td><td><p>.044</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.089</p></td><td><p>.119</p></td><td><p>.067</p></td><td><p>.055</p></td><td><p>.068</p></td><td><p>.090</p></td><td><p>.044</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.089</p></td><td><p>.119</p></td><td><p>.067</p></td><td><p>.055</p></td><td><p>.067</p></td><td><p>.089</p></td><td><p>.044</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.089</p></td><td><p>.118</p></td><td><p>.068</p></td><td><p>.055</p></td><td><p>.067</p></td><td><p>.089</p></td><td><p>.044</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.088</p></td><td><p>.116</p></td><td><p>.067</p></td><td><p>.055</p></td><td><p>.066</p></td><td><p>.088</p></td><td><p>.043</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.087</p></td><td><p>.116</p></td><td><p>.065</p></td><td><p>.055</p></td><td><p>.066</p></td><td><p>.088</p></td><td><p>.043</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.087</p></td><td><p>.117</p></td><td><p>.065</p></td><td><p>.055</p></td><td><p>.067</p></td><td><p>.088</p></td><td><p>.043</p></td><td><p><bold>.046</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.087</p></td><td><p>.116</p></td><td><p>.065</p></td><td><p>.055</p></td><td><p>.067</p></td><td><p>.089</p></td><td><p>.043</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.085</p></td><td><p>.113</p></td><td><p>.065</p></td><td><p>.054</p></td><td><p>.066</p></td><td><p>.087</p></td><td><p>.042</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.082</p></td><td><p>.109</p></td><td><p>.061</p></td><td><p>.053</p></td><td><p>.063</p></td><td><p>.082</p></td><td><p>.040</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.082</p></td><td><p>.108</p></td><td><p>.061</p></td><td><p>.053</p></td><td><p>.063</p></td><td><p>.083</p></td><td><p>.040</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.081</p></td><td><p>.107</p></td><td><p>.061</p></td><td><p>.053</p></td><td><p>.063</p></td><td><p>.083</p></td><td><p>.040</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.081</p></td><td><p>.107</p></td><td><p>.060</p></td><td><p>.053</p></td><td><p>.063</p></td><td><p>.083</p></td><td><p>.040</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.089</p></td><td><p>.118</p></td><td><p>.068</p></td><td><p>.055</p></td><td><p>.067</p></td><td><p>.089</p></td><td><p>.044</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.088</p></td><td><p>.117</p></td><td><p>.068</p></td><td><p>.055</p></td><td><p>.067</p></td><td><p>.089</p></td><td><p>.043</p></td><td><p><bold>.045</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.086</p></td><td><p>.114</p></td><td><p>.067</p></td><td><p>.054</p></td><td><p>.065</p></td><td><p>.087</p></td><td><p>.043</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.084</p></td><td><p>.110</p></td><td><p>.067</p></td><td><p>.055</p></td><td><p>.064</p></td><td><p>.085</p></td><td><p>.043</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.087</p></td><td><p>.116</p></td><td><p>.066</p></td><td><p>.054</p></td><td><p>.066</p></td><td><p>.088</p></td><td><p>.044</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.086</p></td><td><p>.115</p></td><td><p>.066</p></td><td><p>.054</p></td><td><p>.066</p></td><td><p>.087</p></td><td><p>.043</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.086</p></td><td><p>.114</p></td><td><p>.066</p></td><td><p>.054</p></td><td><p>.065</p></td><td><p>.087</p></td><td><p>.043</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.084</p></td><td><p>.111</p></td><td><p>.066</p></td><td><p>.054</p></td><td><p>.064</p></td><td><p>.085</p></td><td><p>.042</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.085</p></td><td><p>.113</p></td><td><p>.064</p></td><td><p>.054</p></td><td><p>.065</p></td><td><p>.086</p></td><td><p>.042</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.085</p></td><td><p>.113</p></td><td><p>.064</p></td><td><p>.053</p></td><td><p>.065</p></td><td><p>.086</p></td><td><p>.042</p></td><td><p><bold>.044</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.084</p></td><td><p>.112</p></td><td><p>.064</p></td><td><p>.052</p></td><td><p>.064</p></td><td><p>.085</p></td><td><p>.042</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.082</p></td><td><p>.109</p></td><td><p>.064</p></td><td><p>.053</p></td><td><p>.064</p></td><td><p>.085</p></td><td><p>.042</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.080</p></td><td><p>.105</p></td><td><p>.060</p></td><td><p>.051</p></td><td><p>.061</p></td><td><p>.080</p></td><td><p>.039</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.080</p></td><td><p>.106</p></td><td><p>.060</p></td><td><p>.051</p></td><td><p>.062</p></td><td><p>.081</p></td><td><p>.040</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.079</p></td><td><p>.104</p></td><td><p>.060</p></td><td><p>.051</p></td><td><p>.061</p></td><td><p>.081</p></td><td><p>.040</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.077</p></td><td><p>.102</p></td><td><p>.059</p></td><td><p><bold>.051</bold></p></td><td><p>.061</p></td><td><p>.080</p></td><td><p>.040</p></td><td><p><bold>.042</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.086</p></td><td><p>.115</p></td><td><p>.067</p></td><td><p>.053</p></td><td><p>.064</p></td><td><p>.086</p></td><td><p>.043</p></td><td><p><bold>.042</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.084</p></td><td><p>.112</p></td><td><p>.066</p></td><td><p>.052</p></td><td><p>.064</p></td><td><p>.085</p></td><td><p>.043</p></td><td><p><bold>.042</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.083</p></td><td><p>.109</p></td><td><p>.066</p></td><td><p>.053</p></td><td><p>.062</p></td><td><p>.083</p></td><td><p>.042</p></td><td><p><bold>.042</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.080</p></td><td><p>.105</p></td><td><p>.066</p></td><td><p>.053</p></td><td><p>.062</p></td><td><p>.081</p></td><td><p>.042</p></td><td><p><bold>.042</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.085</p></td><td><p>.113</p></td><td><p>.065</p></td><td><p>.053</p></td><td><p>.064</p></td><td><p>.085</p></td><td><p>.043</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.083</p></td><td><p>.111</p></td><td><p>.065</p></td><td><p>.053</p></td><td><p>.063</p></td><td><p>.084</p></td><td><p>.042</p></td><td><p><bold>.043</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.082</p></td><td><p>.109</p></td><td><p>.065</p></td><td><p>.052</p></td><td><p>.063</p></td><td><p>.084</p></td><td><p>.042</p></td><td><p><bold>.041</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.080</p></td><td><p>.104</p></td><td><p>.065</p></td><td><p>.052</p></td><td><p>.061</p></td><td><p>.080</p></td><td><p>.041</p></td><td><p><bold>.042</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.082</p></td><td><p>.110</p></td><td><p>.063</p></td><td><p>.051</p></td><td><p>.062</p></td><td><p>.083</p></td><td><p>.041</p></td><td><p><bold>.042</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.082</p></td><td><p>.108</p></td><td><p>.063</p></td><td><p>.052</p></td><td><p>.062</p></td><td><p>.083</p></td><td><p>.041</p></td><td><p><bold>.041</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.080</p></td><td><p>.106</p></td><td><p>.063</p></td><td><p>.051</p></td><td><p>.061</p></td><td><p>.081</p></td><td><p>.041</p></td><td><p><bold>.041</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.078</p></td><td><p>.102</p></td><td><p>.062</p></td><td><p><bold>.050</bold></p></td><td><p>.060</p></td><td><p>.080</p></td><td><p>.040</p></td><td><p><bold>.041</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.077</p></td><td><p>.102</p></td><td><p>.059</p></td><td><p><bold>.049</bold></p></td><td><p>.059</p></td><td><p>.078</p></td><td><p>.039</p></td><td><p><bold>.041</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.077</p></td><td><p>.101</p></td><td><p>.059</p></td><td><p><bold>.050</bold></p></td><td><p>.059</p></td><td><p>.078</p></td><td><p>.039</p></td><td><p><bold>.041</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.077</p></td><td><p>.102</p></td><td><p>.059</p></td><td><p><bold>.049</bold></p></td><td><p>.060</p></td><td><p>.079</p></td><td><p>.039</p></td><td><p><bold>.041</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.074</p></td><td><p>.097</p></td><td><p>.059</p></td><td><p><bold>.049</bold></p></td><td><p>.058</p></td><td><p>.076</p></td><td><p>.038</p></td><td><p><bold>.040</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.084</p></td><td><p>.112</p></td><td><p>.066</p></td><td><p><bold>.051</bold></p></td><td><p>.062</p></td><td><p>.083</p></td><td><p>.042</p></td><td><p><bold>.039</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.082</p></td><td><p>.108</p></td><td><p>.065</p></td><td><p><bold>.050</bold></p></td><td><p>.060</p></td><td><p>.081</p></td><td><p>.041</p></td><td><p><bold>.039</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.079</p></td><td><p>.104</p></td><td><p>.065</p></td><td><p><bold>.049</bold></p></td><td><p>.059</p></td><td><p>.079</p></td><td><p>.040</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.076</p></td><td><p>.098</p></td><td><p>.064</p></td><td><p><bold>.049</bold></p></td><td><p>.057</p></td><td><p>.076</p></td><td><p>.040</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.082</p></td><td><p>.109</p></td><td><p>.064</p></td><td><p><bold>.049</bold></p></td><td><p>.061</p></td><td><p>.081</p></td><td><p>.041</p></td><td><p><bold>.039</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.080</p></td><td><p>.107</p></td><td><p>.064</p></td><td><p><bold>.049</bold></p></td><td><p>.060</p></td><td><p>.080</p></td><td><p>.040</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.078</p></td><td><p>.102</p></td><td><p>.064</p></td><td><p><bold>.049</bold></p></td><td><p>.058</p></td><td><p>.078</p></td><td><p>.040</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.075</p></td><td><p>.098</p></td><td><p>.063</p></td><td><p><bold>.049</bold></p></td><td><p>.057</p></td><td><p>.076</p></td><td><p>.039</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.080</p></td><td><p>.106</p></td><td><p>.062</p></td><td><p><bold>.048</bold></p></td><td><p>.059</p></td><td><p>.078</p></td><td><p>.040</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.078</p></td><td><p>.103</p></td><td><p>.062</p></td><td><p><bold>.048</bold></p></td><td><p>.058</p></td><td><p>.078</p></td><td><p>.039</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.076</p></td><td><p>.100</p></td><td><p>.061</p></td><td><p><bold>.047</bold></p></td><td><p>.057</p></td><td><p>.076</p></td><td><p>.039</p></td><td><p><bold>.038</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.073</p></td><td><p>.095</p></td><td><p>.061</p></td><td><p><bold>.048</bold></p></td><td><p>.056</p></td><td><p>.075</p></td><td><p>.039</p></td><td><p><bold>.037</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.074</p></td><td><p>.098</p></td><td><p>.058</p></td><td><p><bold>.046</bold></p></td><td><p>.056</p></td><td><p>.074</p></td><td><p><bold>.037</bold></p></td><td><p><bold>.037</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.073</p></td><td><p>.097</p></td><td><p>.058</p></td><td><p><bold>.045</bold></p></td><td><p>.055</p></td><td><p>.074</p></td><td><p><bold>.037</bold></p></td><td><p><bold>.037</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.072</p></td><td><p>.095</p></td><td><p>.058</p></td><td><p><bold>.046</bold></p></td><td><p>.055</p></td><td><p>.074</p></td><td><p><bold>.037</bold></p></td><td><p><bold>.037</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.070</p></td><td><p>.091</p></td><td><p>.057</p></td><td><p><bold>.045</bold></p></td><td><p>.054</p></td><td><p>.072</p></td><td><p><bold>.037</bold></p></td><td><p><bold>.037</bold></p></td></tr></tbody></table> </ephtml> </p> <p>7 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>B3 Table RMSE Values for EAP Ability Estimates for Five Selected Examinees with Item Parameters Fixed to Values Obtained with Prior Settings of Varying IDM Prediction Accuracy under N=50$N=50$ and N=200$N=200$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">$N=50$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>200</mn></mrow><annotation encoding="application/x-tex">$N=200$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th></tr><tr><th /><th /><th 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</ephtml> </p> <p>8 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <hd id="AN0192629996-60">3 Appendix Study III: Bias of Item Parameter Estimates, Item Characteristic Curves, and Abili...</hd> <p>C1 Table Bias of Item Parameters a$a$, b$b$, and c$c$ Obtained with Varying IDM Prediction Accuracy under N=50$N=50$ and N=200$N=200$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">$N=50$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>200</mn></mrow><annotation encoding="application/x-tex">$N=200$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (.07)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (.06)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> (.03)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (.07)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (.06)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> 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ld></p></td><td><p><bold>−.003</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p><bold>.009</bold></p></td><td><p><bold>.003</bold></p></td><td><p><bold>−.023</bold></p></td><td><p><bold>.015</bold></p></td><td><p><bold>.019</bold></p></td><td><p><bold>−.011</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p><bold>.007</bold></p></td><td><p><bold>−.001</bold></p></td><td><p><bold>−.021</bold></p></td><td><p><bold>.011</bold></p></td><td><p><bold>.006</bold></p></td><td><p><bold>−.014</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p><bold>.017</bold></p></td><td><p><bold>.046</bold></p></td><td><p><bold>.003</bold></p></td><td><p><bold>.024</bold></p></td><td><p>.068</p></td><td><p><bold>.015</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p><bold>.010</bold></p></td><td><p><bold>.020</bold></p></td><td><p><bold>−.014</bold></p></td><td><p><bold>.017</bold></p></td><td><p><bold>.036</bold></p></td><td><p><bold>−.002</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p><bold>.007</bold></p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.018</bold></p></td><td><p><bold>.013</bold></p></td><td><p><bold>.018</bold></p></td><td><p><bold>−.009</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p><bold>.004</bold></p></td><td><p><bold>.001</bold></p></td><td><p><bold>−.018</bold></p></td><td><p><bold>.009</bold></p></td><td><p><bold>.008</bold></p></td><td><p><bold>−.013</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p><bold>.012</bold></p></td><td><p><bold>.029</bold></p></td><td><p><bold>.008</bold></p></td><td><p><bold>.018</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.013</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p><bold>.007</bold></p></td><td><p><bold>.014</bold></p></td><td><p><bold>−.007</bold></p></td><td><p><bold>.014</bold></p></td><td><p><bold>.025</bold></p></td><td><p><bold>−.000</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p><bold>.002</bold></p></td><td><p><bold>.006</bold></p></td><td><p><bold>−.014</bold></p></td><td><p><bold>.008</bold></p></td><td><p><bold>.014</bold></p></td><td><p><bold>−.008</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p><bold>−.001</bold></p></td><td><p><bold>.002</bold></p></td><td><p><bold>−.014</bold></p></td><td><p><bold>.005</bold></p></td><td><p><bold>.006</bold></p></td><td><p><bold>−.012</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p><bold>.013</bold></p></td><td><p><bold>.045</bold></p></td><td><p><bold>−.013</bold></p></td><td><p><bold>.019</bold></p></td><td><p>.075</p></td><td><p><bold>.008</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p><bold>.010</bold></p></td><td><p><bold>.013</bold></p></td><td><p><bold>−.022</bold></p></td><td><p><bold>.014</bold></p></td><td><p><bold>.040</bold></p></td><td><p><bold>−.005</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p><bold>.007</bold></p></td><td><p><bold>−.003</bold></p></td><td><p><bold>−.024</bold></p></td><td><p><bold>.011</bold></p></td><td><p><bold>.018</bold></p></td><td><p><bold>−.011</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p><bold>.006</bold></p></td><td><p><bold>−.007</bold></p></td><td><p><bold>−.021</bold></p></td><td><p><bold>.009</bold></p></td><td><p><bold>.004</bold></p></td><td><p><bold>−.014</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p><bold>.013</bold></p></td><td><p><bold>.048</bold></p></td><td><p><bold>−.005</bold></p></td><td><p><bold>.019</bold></p></td><td><p>.076</p></td><td><p><bold>.013</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p><bold>.010</bold></p></td><td><p><bold>.017</bold></p></td><td><p><bold>−.017</bold></p></td><td><p><bold>.014</bold></p></td><td><p><bold>.040</bold></p></td><td><p><bold>−.002</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p><bold>.006</bold></p></td><td><p><bold>.003</bold></p></td><td><p><bold>−.021</bold></p></td><td><p><bold>.010</bold></p></td><td><p><bold>.019</bold></p></td><td><p><bold>−.010</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.006</bold></p></td><td><p><bold>−.021</bold></p></td><td><p><bold>.008</bold></p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.014</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p><bold>.012</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.004</bold></p></td><td><p><bold>.017</bold></p></td><td><p><bold>.065</bold></p></td><td><p><bold>.015</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p><bold>.009</bold></p></td><td><p><bold>.021</bold></p></td><td><p><bold>−.011</bold></p></td><td><p><bold>.013</bold></p></td><td><p><bold>.038</bold></p></td><td><p><bold>.001</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p><bold>.005</bold></p></td><td><p><bold>.004</bold></p></td><td><p><bold>−.019</bold></p></td><td><p><bold>.009</bold></p></td><td><p><bold>.018</bold></p></td><td><p><bold>−.008</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p><bold>.003</bold></p></td><td><p><bold>−.001</bold></p></td><td><p><bold>−.017</bold></p></td><td><p><bold>.007</bold></p></td><td><p><bold>.007</bold></p></td><td><p><bold>−.012</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p><bold>.009</bold></p></td><td><p><bold>.025</bold></p></td><td><p><bold>.007</bold></p></td><td><p><bold>.014</bold></p></td><td><p><bold>.040</bold></p></td><td><p><bold>.014</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p><bold>.005</bold></p></td><td><p><bold>.014</bold></p></td><td><p><bold>−.006</bold></p></td><td><p><bold>.010</bold></p></td><td><p><bold>.025</bold></p></td><td><p><bold>.001</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p><bold>.003</bold></p></td><td><p><bold>.006</bold></p></td><td><p><bold>−.012</bold></p></td><td><p><bold>.007</bold></p></td><td><p><bold>.013</bold></p></td><td><p><bold>−.007</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p><bold>.001</bold></p></td><td><p><bold>.001</bold></p></td><td><p><bold>−.014</bold></p></td><td><p><bold>.004</bold></p></td><td><p><bold>.006</bold></p></td><td><p><bold>−.010</bold></p></td></tr></tbody></table> </ephtml> </p> <p>9 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>C2 Table Bias of Item Parameters a$a$, b$b$, and c$c$ Obtained with Varying IDM Prediction Accuracy under N=100$N=100$ and N=500$N=500$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (.07)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (.06)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> (.03)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math></p> (.07)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math></p> (.06)</th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math></p> 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></td><td><p>.60</p></td><td><p>.20</p></td><td><p><bold>.011</bold></p></td><td><p><bold>.032</bold></p></td><td><p><bold>−.002</bold></p></td><td><p><bold>.015</bold></p></td><td><p><bold>.046</bold></p></td><td><p><bold>.008</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p><bold>.008</bold></p></td><td><p><bold>.010</bold></p></td><td><p><bold>−.012</bold></p></td><td><p><bold>.011</bold></p></td><td><p><bold>.024</bold></p></td><td><p><bold>−.001</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p><bold>.006</bold></p></td><td><p><bold>.002</bold></p></td><td><p><bold>−.013</bold></p></td><td><p><bold>.008</bold></p></td><td><p><bold>.008</bold></p></td><td><p><bold>−.007</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p><bold>.013</bold></p></td><td><p><bold>.034</bold></p></td><td><p><bold>.015</bold></p></td><td><p><bold>.016</bold></p></td><td><p><bold>.048</bold></p></td><td><p><bold>.018</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p><bold>.008</bold></p></td><td><p><bold>.021</bold></p></td><td><p><bold>.000</bold></p></td><td><p><bold>.012</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.007</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p><bold>.005</bold></p></td><td><p><bold>.009</bold></p></td><td><p><bold>−.008</bold></p></td><td><p><bold>.009</bold></p></td><td><p><bold>.017</bold></p></td><td><p><bold>−.001</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p><bold>.003</bold></p></td><td><p><bold>.004</bold></p></td><td><p><bold>−.011</bold></p></td><td><p><bold>.006</bold></p></td><td><p><bold>.007</bold></p></td><td><p><bold>−.006</bold></p></td></tr></tbody></table> </ephtml> </p> <p>10 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>C3 Table Bias of ICCs over Different Ability Segments Obtained with Varying IDM Prediction Accuracy under N=50$N=50$ and N=200$N=200$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">$N=50$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>200</mn></mrow><annotation encoding="application/x-tex">$N=200$</annotation></semantics></math></p></th></tr><tr><th /><th /><th /><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th>all</th><th>lower</th><th>middle</th><th>upper</th><th>all</th><th>lower</th><th>middle</th><th>upper</th></tr><tr><th /><th /><th /><th>(.03)</th><th>(.04)</th><th>(.02)</th><th>(.03)</th><th>(.03)</th><th>(.04)</th><th>(.02)</th><th>(.03)</th></tr></thead><tbody><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.061</p></td><td><p>.090</p></td><td><p>.053</p></td><td><p>.036</p></td><td><p>.045</p></td><td><p>.066</p></td><td><p>.033</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.061</p></td><td><p>.088</p></td><td><p>.053</p></td><td><p>.036</p></td><td><p>.045</p></td><td><p>.065</p></td><td><p>.033</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.060</p></td><td><p>.086</p></td><td><p>.053</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.064</p></td><td><p>.033</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.059</p></td><td><p>.083</p></td><td><p>.053</p></td><td><p>.036</p></td><td><p>.043</p></td><td><p>.061</p></td><td><p>.033</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.060</p></td><td><p>.088</p></td><td><p>.052</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.064</p></td><td><p>.033</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.060</p></td><td><p>.087</p></td><td><p>.052</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.064</p></td><td><p>.033</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.059</p></td><td><p>.086</p></td><td><p>.052</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.063</p></td><td><p>.033</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.058</p></td><td><p>.083</p></td><td><p>.052</p></td><td><p>.036</p></td><td><p>.043</p></td><td><p>.061</p></td><td><p>.032</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.059</p></td><td><p>.085</p></td><td><p>.050</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.063</p></td><td><p>.032</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.059</p></td><td><p>.085</p></td><td><p>.050</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.063</p></td><td><p>.032</p></td><td><p><bold>.029</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.058</p></td><td><p>.084</p></td><td><p>.050</p></td><td><p>.036</p></td><td><p>.043</p></td><td><p>.063</p></td><td><p>.032</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.057</p></td><td><p>.081</p></td><td><p>.050</p></td><td><p>.036</p></td><td><p>.042</p></td><td><p>.061</p></td><td><p>.032</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.056</p></td><td><p>.080</p></td><td><p>.047</p></td><td><p>.035</p></td><td><p>.042</p></td><td><p>.059</p></td><td><p>.031</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.056</p></td><td><p>.080</p></td><td><p>.047</p></td><td><p>.035</p></td><td><p>.042</p></td><td><p>.059</p></td><td><p>.031</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.055</p></td><td><p>.079</p></td><td><p>.047</p></td><td><p>.035</p></td><td><p>.042</p></td><td><p>.059</p></td><td><p>.031</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.055</p></td><td><p>.078</p></td><td><p>.047</p></td><td><p>.035</p></td><td><p>.041</p></td><td><p>.058</p></td><td><p>.031</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.060</p></td><td><p>.088</p></td><td><p>.052</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.065</p></td><td><p>.033</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.060</p></td><td><p>.086</p></td><td><p>.052</p></td><td><p>.036</p></td><td><p>.044</p></td><td><p>.064</p></td><td><p>.033</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.058</p></td><td><p>.083</p></td><td><p>.052</p></td><td><p>.035</p></td><td><p>.043</p></td><td><p>.062</p></td><td><p>.033</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.057</p></td><td><p>.080</p></td><td><p>.052</p></td><td><p>.036</p></td><td><p>.042</p></td><td><p>.060</p></td><td><p>.032</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.059</p></td><td><p>.086</p></td><td><p>.051</p></td><td><p>.035</p></td><td><p>.043</p></td><td><p>.063</p></td><td><p>.033</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.059</p></td><td><p>.085</p></td><td><p>.051</p></td><td><p>.035</p></td><td><p>.043</p></td><td><p>.062</p></td><td><p>.032</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.058</p></td><td><p>.083</p></td><td><p>.051</p></td><td><p>.035</p></td><td><p>.043</p></td><td><p>.062</p></td><td><p>.032</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.056</p></td><td><p>.080</p></td><td><p>.051</p></td><td><p>.035</p></td><td><p>.042</p></td><td><p>.060</p></td><td><p>.032</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.058</p></td><td><p>.084</p></td><td><p>.050</p></td><td><p>.035</p></td><td><p>.043</p></td><td><p>.062</p></td><td><p>.032</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.057</p></td><td><p>.083</p></td><td><p>.050</p></td><td><p>.034</p></td><td><p>.043</p></td><td><p>.062</p></td><td><p>.032</p></td><td><p><bold>.028</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.057</p></td><td><p>.082</p></td><td><p>.049</p></td><td><p>.034</p></td><td><p>.042</p></td><td><p>.061</p></td><td><p>.032</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.056</p></td><td><p>.079</p></td><td><p>.049</p></td><td><p>.035</p></td><td><p>.041</p></td><td><p>.060</p></td><td><p>.031</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.054</p></td><td><p>.078</p></td><td><p>.046</p></td><td><p>.034</p></td><td><p>.040</p></td><td><p>.058</p></td><td><p>.030</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.055</p></td><td><p>.079</p></td><td><p>.046</p></td><td><p>.034</p></td><td><p>.041</p></td><td><p>.058</p></td><td><p>.030</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.054</p></td><td><p>.077</p></td><td><p>.046</p></td><td><p>.034</p></td><td><p>.040</p></td><td><p>.058</p></td><td><p>.030</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.053</p></td><td><p>.075</p></td><td><p>.046</p></td><td><p>.033</p></td><td><p>.040</p></td><td><p>.057</p></td><td><p>.030</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.059</p></td><td><p>.086</p></td><td><p>.052</p></td><td><p>.034</p></td><td><p>.043</p></td><td><p>.062</p></td><td><p>.033</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.058</p></td><td><p>.083</p></td><td><p>.052</p></td><td><p>.034</p></td><td><p>.042</p></td><td><p>.062</p></td><td><p>.032</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.056</p></td><td><p>.080</p></td><td><p>.051</p></td><td><p>.034</p></td><td><p>.041</p></td><td><p>.060</p></td><td><p>.032</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.055</p></td><td><p>.077</p></td><td><p>.051</p></td><td><p>.034</p></td><td><p>.041</p></td><td><p>.058</p></td><td><p>.031</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.058</p></td><td><p>.085</p></td><td><p>.051</p></td><td><p>.034</p></td><td><p>.042</p></td><td><p>.062</p></td><td><p>.032</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.057</p></td><td><p>.082</p></td><td><p>.051</p></td><td><p>.034</p></td><td><p>.042</p></td><td><p>.061</p></td><td><p>.032</p></td><td><p><bold>.027</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.056</p></td><td><p>.080</p></td><td><p>.051</p></td><td><p>.034</p></td><td><p>.041</p></td><td><p>.060</p></td><td><p>.031</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.054</p></td><td><p>.076</p></td><td><p>.050</p></td><td><p>.034</p></td><td><p>.040</p></td><td><p>.057</p></td><td><p>.031</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.056</p></td><td><p>.082</p></td><td><p>.049</p></td><td><p>.033</p></td><td><p>.041</p></td><td><p>.060</p></td><td><p>.031</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.056</p></td><td><p>.081</p></td><td><p>.049</p></td><td><p>.033</p></td><td><p>.041</p></td><td><p>.060</p></td><td><p>.031</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.055</p></td><td><p>.078</p></td><td><p>.049</p></td><td><p>.033</p></td><td><p>.040</p></td><td><p>.058</p></td><td><p>.031</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.053</p></td><td><p>.075</p></td><td><p>.048</p></td><td><p>.033</p></td><td><p>.040</p></td><td><p>.057</p></td><td><p>.031</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.053</p></td><td><p>.077</p></td><td><p>.046</p></td><td><p>.032</p></td><td><p>.039</p></td><td><p>.057</p></td><td><p>.030</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.053</p></td><td><p>.076</p></td><td><p>.046</p></td><td><p>.032</p></td><td><p>.039</p></td><td><p>.056</p></td><td><p>.030</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.053</p></td><td><p>.076</p></td><td><p>.046</p></td><td><p>.032</p></td><td><p>.039</p></td><td><p>.057</p></td><td><p>.030</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.051</p></td><td><p>.072</p></td><td><p>.046</p></td><td><p>.032</p></td><td><p>.038</p></td><td><p>.055</p></td><td><p>.029</p></td><td><p><bold>.026</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.057</p></td><td><p>.084</p></td><td><p>.051</p></td><td><p>.033</p></td><td><p>.041</p></td><td><p>.060</p></td><td><p>.032</p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.056</p></td><td><p>.081</p></td><td><p>.051</p></td><td><p>.033</p></td><td><p>.040</p></td><td><p>.059</p></td><td><p>.031</p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.054</p></td><td><p>.078</p></td><td><p>.050</p></td><td><p>.032</p></td><td><p>.039</p></td><td><p>.057</p></td><td><p>.031</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.052</p></td><td><p>.073</p></td><td><p>.050</p></td><td><p>.032</p></td><td><p>.038</p></td><td><p>.055</p></td><td><p>.030</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.056</p></td><td><p>.082</p></td><td><p>.050</p></td><td><p>.032</p></td><td><p>.040</p></td><td><p>.059</p></td><td><p>.031</p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.055</p></td><td><p>.080</p></td><td><p>.050</p></td><td><p>.032</p></td><td><p>.039</p></td><td><p>.058</p></td><td><p>.031</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.054</p></td><td><p>.077</p></td><td><p>.050</p></td><td><p>.032</p></td><td><p>.039</p></td><td><p>.057</p></td><td><p>.030</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.052</p></td><td><p>.073</p></td><td><p>.049</p></td><td><p>.032</p></td><td><p>.038</p></td><td><p>.055</p></td><td><p>.030</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.054</p></td><td><p>.080</p></td><td><p>.048</p></td><td><p>.031</p></td><td><p>.039</p></td><td><p>.058</p></td><td><p>.030</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.053</p></td><td><p>.077</p></td><td><p>.048</p></td><td><p>.031</p></td><td><p>.038</p></td><td><p>.056</p></td><td><p>.030</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.052</p></td><td><p>.075</p></td><td><p>.048</p></td><td><p>.031</p></td><td><p>.038</p></td><td><p>.056</p></td><td><p>.030</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.050</p></td><td><p>.071</p></td><td><p>.047</p></td><td><p>.031</p></td><td><p>.037</p></td><td><p>.054</p></td><td><p>.029</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.051</p></td><td><p>.074</p></td><td><p>.045</p></td><td><p><bold>.030</bold></p></td><td><p>.037</p></td><td><p>.054</p></td><td><p>.028</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.050</p></td><td><p>.073</p></td><td><p>.045</p></td><td><p><bold>.029</bold></p></td><td><p>.037</p></td><td><p>.054</p></td><td><p>.028</p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.050</p></td><td><p>.072</p></td><td><p>.045</p></td><td><p><bold>.030</bold></p></td><td><p>.037</p></td><td><p>.054</p></td><td><p>.029</p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.048</p></td><td><p>.069</p></td><td><p>.044</p></td><td><p><bold>.030</bold></p></td><td><p>.036</p></td><td><p>.052</p></td><td><p>.028</p></td><td><p><bold>.023</bold></p></td></tr></tbody></table> </ephtml> </p> <p>11 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>C4 Table Bias of ICCs over Different Ability Segments Obtained with Varying IDM Prediction Accuracy under N=10$N=10$ and N=500$N=500$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math></p></th></tr><tr><th /><th /><th /><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th><th>ICC</th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th>all</th><th>lower</th><th>middle</th><th>upper</th><th>all</th><th>lower</th><th>middle</th><th>upper</th></tr><tr><th /><th /><th /><th>(.03)</th><th>(.04)</th><th>(.02)</th><th>(.03)</th><th>(.03)</th><th>(.04)</th><th>(.02)</th><th>(.03)</th></tr></thead><tbody><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.053</p></td><td><p>.078</p></td><td><p>.042</p></td><td><p>.032</p></td><td><p><bold>.034</bold></p></td><td><p>.048</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.053</p></td><td><p>.077</p></td><td><p>.042</p></td><td><p>.032</p></td><td><p><bold>.034</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.052</p></td><td><p>.075</p></td><td><p>.042</p></td><td><p>.032</p></td><td><p><bold>.034</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.051</p></td><td><p>.074</p></td><td><p>.042</p></td><td><p>.032</p></td><td><p><bold>.033</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.052</p></td><td><p>.077</p></td><td><p>.041</p></td><td><p>.032</p></td><td><p><bold>.034</bold></p></td><td><p>.048</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.052</p></td><td><p>.076</p></td><td><p>.042</p></td><td><p>.032</p></td><td><p><bold>.034</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.052</p></td><td><p>.076</p></td><td><p>.042</p></td><td><p>.032</p></td><td><p><bold>.033</bold></p></td><td><p>.046</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.051</p></td><td><p>.073</p></td><td><p>.041</p></td><td><p>.032</p></td><td><p><bold>.033</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.051</p></td><td><p>.075</p></td><td><p>.040</p></td><td><p>.032</p></td><td><p><bold>.034</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.051</p></td><td><p>.075</p></td><td><p>.040</p></td><td><p>.032</p></td><td><p><bold>.033</bold></p></td><td><p>.046</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.025</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.051</p></td><td><p>.074</p></td><td><p>.041</p></td><td><p>.032</p></td><td><p><bold>.033</bold></p></td><td><p>.046</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.050</p></td><td><p>.072</p></td><td><p>.040</p></td><td><p>.032</p></td><td><p><bold>.033</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.049</p></td><td><p>.070</p></td><td><p>.038</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.049</p></td><td><p>.070</p></td><td><p>.038</p></td><td><p>.031</p></td><td><p><bold>.032</bold></p></td><td><p><bold>.045</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.049</p></td><td><p>.070</p></td><td><p>.038</p></td><td><p>.031</p></td><td><p><bold>.032</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.048</p></td><td><p>.069</p></td><td><p>.038</p></td><td><p>.031</p></td><td><p><bold>.032</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.052</p></td><td><p>.077</p></td><td><p>.042</p></td><td><p>.031</p></td><td><p><bold>.034</bold></p></td><td><p>.047</p></td><td><p><bold>.025</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.052</p></td><td><p>.076</p></td><td><p>.042</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.051</p></td><td><p>.074</p></td><td><p>.041</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.046</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.049</p></td><td><p>.071</p></td><td><p>.041</p></td><td><p>.031</p></td><td><p><bold>.032</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.051</p></td><td><p>.075</p></td><td><p>.041</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.051</p></td><td><p>.074</p></td><td><p>.041</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.050</p></td><td><p>.073</p></td><td><p>.041</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.046</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.049</p></td><td><p>.070</p></td><td><p>.041</p></td><td><p>.031</p></td><td><p><bold>.032</bold></p></td><td><p><bold>.045</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.050</p></td><td><p>.073</p></td><td><p>.040</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.024</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.050</p></td><td><p>.074</p></td><td><p>.040</p></td><td><p>.031</p></td><td><p><bold>.033</bold></p></td><td><p>.046</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.049</p></td><td><p>.072</p></td><td><p>.040</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.048</p></td><td><p>.070</p></td><td><p>.040</p></td><td><p><bold>.031</bold></p></td><td><p><bold>.032</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.047</p></td><td><p>.069</p></td><td><p>.037</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.032</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.048</p></td><td><p>.069</p></td><td><p>.038</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.032</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.048</p></td><td><p>.069</p></td><td><p>.038</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.046</p></td><td><p>.067</p></td><td><p>.038</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.050</p></td><td><p>.075</p></td><td><p>.041</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.033</bold></p></td><td><p>.047</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.050</p></td><td><p>.073</p></td><td><p>.041</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.046</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.049</p></td><td><p>.071</p></td><td><p>.040</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.047</p></td><td><p>.068</p></td><td><p>.040</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.050</p></td><td><p>.073</p></td><td><p>.040</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.033</bold></p></td><td><p>.046</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.049</p></td><td><p>.072</p></td><td><p>.040</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.046</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.023</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.049</p></td><td><p>.071</p></td><td><p>.040</p></td><td><p><bold>.030</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.047</p></td><td><p>.067</p></td><td><p>.040</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.049</p></td><td><p>.072</p></td><td><p>.039</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.046</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.048</p></td><td><p>.071</p></td><td><p>.039</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.047</p></td><td><p>.069</p></td><td><p>.039</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.046</p></td><td><p>.067</p></td><td><p>.039</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.046</p></td><td><p>.067</p></td><td><p>.037</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.046</p></td><td><p>.067</p></td><td><p>.037</p></td><td><p><bold>.028</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.046</p></td><td><p>.067</p></td><td><p>.037</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.043</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.021</bold></p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.045</p></td><td><p>.064</p></td><td><p>.037</p></td><td><p><bold>.028</bold></p></td><td><p><bold>.030</bold></p></td><td><p><bold>.042</bold></p></td><td><p><bold>.021</bold></p></td><td><p><bold>.022</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.049</p></td><td><p>.072</p></td><td><p>.040</p></td><td><p><bold>.029</bold></p></td><td><p><bold>.032</bold></p></td><td><p>.045</p></td><td><p><bold>.024</bold></p></td><td><p><bold>.021</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.048</p></td><td><p>.070</p></td><td><p>.040</p></td><td><p><bold>.028</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.023</bold></p></td><td><p><bold>.021</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.046</p></td><td><p>.068</p></td><td><p>.039</p></td><td><p><bold>.027</bold></p></td><td><p><bold>.030</bold></p></td><td><p><bold>.044</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.020</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.045</p></td><td><p>.065</p></td><td><p>.039</p></td><td><p><bold>.028</bold></p></td><td><p><bold>.029</bold></p></td><td><p><bold>.042</bold></p></td><td><p><bold>.022</bold></p></td><td><p><bold>.020</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.048</p></td><td><p>.071</p></td><td><p>.040</p></td><td><p><bold>.028</bold></p></td><td><p><bold>.031</bold></p></td><td><p>.045</p></td><td><p><bold>.023</bold></p></td><td><p><bold>.021</bold></p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.047</p></td><td><p>.069</p></td><td><p>.039</p></td><td><p><bold>.028</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bo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</ephtml> </p> <p>12 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>C5 Table Bias of EAP Ability Estimates for Five Selected Examinees with Item Parameters Fixed to Values Obtained with Prior Settings of Varying IDM Prediction Accuracy under N=50$N=50$ and N=200$N=200$ Compared to Default Prior Settings with N=1,000$N=1,000$.</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">$N=50$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>200</mn></mrow><annotation encoding="application/x-tex">$N=200$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th></tr><tr><th /><th /><th /><th>(.37)</th><th>(.03)</th><th>(.00)</th><th>(−.10)</th><th>(−.29)</th><th>(.37)</th><th>(.03)</th><th>(.00)</th><th>(−.10)</th><th>(−.29)</th></tr></thead><tbody><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.598</p></td><td><p>.157</p></td><td><p>.024</p></td><td><p>−.113</p></td><td><p>−.285</p></td><td><p>.495</p></td><td><p>.087</p></td><td><p>.018</p></td><td><p>−.093</p></td><td><p>−.265</p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.595</p></td><td><p>.160</p></td><td><p>.023</p></td><td><p><bold>−.119</bold></p></td><td><p>−.291</p></td><td><p>.490</p></td><td><p>.092</p></td><td><p>.015</p></td><td><p>−.094</p></td><td><p>−.265</p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.589</p></td><td><p>.155</p></td><td><p>.023</p></td><td><p><bold>−.121</bold></p></td><td><p>−.293</p></td><td><p>.487</p></td><td><p>.092</p></td><td><p>.013</p></td><td><p>−.096</p></td><td><p>−.264</p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.565</p></td><td><p>.150</p></td><td><p>.012</p></td><td><p><bold>−.128</bold></p></td><td><p>−.301</p></td><td><p>.474</p></td><td><p>.090</p></td><td><p>.008</p></td><td><p>−.099</p></td><td><p>−.264</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.569</p></td><td><p>.133</p></td><td><p>.016</p></td><td><p><bold>−.121</bold></p></td><td><p>−.289</p></td><td><p>.481</p></td><td><p>.082</p></td><td><p>.013</p></td><td><p>−.098</p></td><td><p>−.266</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.574</p></td><td><p>.150</p></td><td><p>.015</p></td><td><p><bold>−.125</bold></p></td><td><p>−.290</p></td><td><p>.481</p></td><td><p>.091</p></td><td><p>.011</p></td><td><p>−.100</p></td><td><p>−.264</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.567</p></td><td><p>.153</p></td><td><p>.012</p></td><td><p><bold>−.124</bold></p></td><td><p>−.288</p></td><td><p>.478</p></td><td><p>.092</p></td><td><p>.009</p></td><td><p>−.100</p></td><td><p>−.260</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.556</p></td><td><p>.141</p></td><td><p>.013</p></td><td><p><bold>−.128</bold></p></td><td><p>−.291</p></td><td><p>.478</p></td><td><p>.086</p></td><td><p>.009</p></td><td><p>−.100</p></td><td><p>−.258</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.537</p></td><td><p>.114</p></td><td><p><bold>−.001</bold></p></td><td><p><bold>−.130</bold></p></td><td><p>−.288</p></td><td><p>.467</p></td><td><p>.069</p></td><td><p><bold>.004</bold></p></td><td><p>−.105</p></td><td><p>−.268</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.550</p></td><td><p>.128</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.128</bold></p></td><td><p>−.280</p></td><td><p>.476</p></td><td><p>.082</p></td><td><p>.007</p></td><td><p>−.103</p></td><td><p>−.259</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.549</p></td><td><p>.144</p></td><td><p>.008</p></td><td><p><bold>−.127</bold></p></td><td><p>−.284</p></td><td><p>.473</p></td><td><p>.094</p></td><td><p>.007</p></td><td><p>−.100</p></td><td><p>−.259</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.536</p></td><td><p>.132</p></td><td><p><bold>.003</bold></p></td><td><p><bold>−.129</bold></p></td><td><p>−.284</p></td><td><p>.471</p></td><td><p>.086</p></td><td><p><bold>.005</bold></p></td><td><p>−.102</p></td><td><p>−.257</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.465</p></td><td><p>.077</p></td><td><p><bold>−.019</bold></p></td><td><p><bold>−.126</bold></p></td><td><p>−.272</p></td><td><p><bold>.434</bold></p></td><td><p>.055</p></td><td><p><bold>−.008</bold></p></td><td><p>−.105</p></td><td><p>−.258</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.493</p></td><td><p>.099</p></td><td><p><bold>−.007</bold></p></td><td><p><bold>−.119</bold></p></td><td><p>−.268</p></td><td><p>.449</p></td><td><p>.071</p></td><td><p><bold>−.002</bold></p></td><td><p>−.101</p></td><td><p>−.253</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.502</p></td><td><p>.107</p></td><td><p><bold>.001</bold></p></td><td><p><bold>−.119</bold></p></td><td><p>−.266</p></td><td><p>.456</p></td><td><p>.076</p></td><td><p><bold>.001</bold></p></td><td><p>−.101</p></td><td><p>−.248</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.484</p></td><td><p>.112</p></td><td><p><bold>−.000</bold></p></td><td><p><bold>−.117</bold></p></td><td><p>−.262</p></td><td><p>.447</p></td><td><p>.082</p></td><td><p>.006</p></td><td><p>−.094</p></td><td><p>−.243</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.587</p></td><td><p>.149</p></td><td><p>.020</p></td><td><p>−.113</p></td><td><p>−.280</p></td><td><p>.489</p></td><td><p>.083</p></td><td><p>.016</p></td><td><p>−.091</p></td><td><p>−.254</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.596</p></td><td><p>.150</p></td><td><p>.024</p></td><td><p><bold>−.119</bold></p></td><td><p>−.287</p></td><td><p>.491</p></td><td><p>.090</p></td><td><p>.016</p></td><td><p>−.092</p></td><td><p>−.258</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.567</p></td><td><p>.143</p></td><td><p>.018</p></td><td><p><bold>−.121</bold></p></td><td><p>−.285</p></td><td><p>.474</p></td><td><p>.087</p></td><td><p>.015</p></td><td><p>−.093</p></td><td><p>−.256</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.551</p></td><td><p>.143</p></td><td><p>.013</p></td><td><p><bold>−.127</bold></p></td><td><p>−.295</p></td><td><p>.472</p></td><td><p>.088</p></td><td><p>.010</p></td><td><p>−.097</p></td><td><p>−.259</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.558</p></td><td><p>.134</p></td><td><p>.007</p></td><td><p><bold>−.116</bold></p></td><td><p>−.277</p></td><td><p>.472</p></td><td><p>.073</p></td><td><p>.009</p></td><td><p>−.092</p></td><td><p>−.255</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.558</p></td><td><p>.145</p></td><td><p>.013</p></td><td><p><bold>−.124</bold></p></td><td><p>−.281</p></td><td><p>.469</p></td><td><p>.085</p></td><td><p>.009</p></td><td><p>−.097</p></td><td><p>−.256</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.552</p></td><td><p>.141</p></td><td><p>.014</p></td><td><p><bold>−.121</bold></p></td><td><p>−.283</p></td><td><p>.468</p></td><td><p>.085</p></td><td><p>.010</p></td><td><p>−.097</p></td><td><p>−.257</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.542</p></td><td><p>.138</p></td><td><p>.006</p></td><td><p><bold>−.121</bold></p></td><td><p>−.287</p></td><td><p>.468</p></td><td><p>.090</p></td><td><p>.007</p></td><td><p>−.096</p></td><td><p>−.257</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.520</p></td><td><p>.104</p></td><td><p><bold>−.002</bold></p></td><td><p><bold>−.125</bold></p></td><td><p>−.278</p></td><td><p>.456</p></td><td><p>.064</p></td><td><p><bold>.000</bold></p></td><td><p>−.101</p></td><td><p>−.260</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.546</p></td><td><p>.132</p></td><td><p>.007</p></td><td><p>−.114</p></td><td><p>−.274</p></td><td><p>.463</p></td><td><p>.084</p></td><td><p>.006</p></td><td><p>−.094</p></td><td><p>−.255</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.546</p></td><td><p>.127</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.120</bold></p></td><td><p>−.276</p></td><td><p>.466</p></td><td><p>.081</p></td><td><p><bold>.005</bold></p></td><td><p>−.097</p></td><td><p>−.253</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.525</p></td><td><p>.127</p></td><td><p><bold>.000</bold></p></td><td><p><bold>−.120</bold></p></td><td><p>−.274</p></td><td><p>.459</p></td><td><p>.084</p></td><td><p><bold>.001</bold></p></td><td><p>−.097</p></td><td><p>−.248</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.461</p></td><td><p>.078</p></td><td><p><bold>−.016</bold></p></td><td><p><bold>−.125</bold></p></td><td><p>−.272</p></td><td><p><bold>.420</bold></p></td><td><p>.054</p></td><td><p><bold>−.006</bold></p></td><td><p>−.105</p></td><td><p>−.257</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.482</p></td><td><p>.095</p></td><td><p><bold>−.005</bold></p></td><td><p><bold>−.118</bold></p></td><td><p>−.264</p></td><td><p><bold>.436</bold></p></td><td><p>.066</p></td><td><p><bold>−.000</bold></p></td><td><p>−.101</p></td><td><p>−.251</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.496</p></td><td><p>.103</p></td><td><p><bold>−.001</bold></p></td><td><p>−.111</p></td><td><p>−.256</p></td><td><p>.451</p></td><td><p>.075</p></td><td><p><bold>.001</bold></p></td><td><p>−.096</p></td><td><p>−.242</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.472</p></td><td><p>.099</p></td><td><p><bold>−.002</bold></p></td><td><p>−.112</p></td><td><p>−.260</p></td><td><p><bold>.437</bold></p></td><td><p>.073</p></td><td><p><bold>.002</bold></p></td><td><p>−.094</p></td><td><p>−.242</p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.583</p></td><td><p>.133</p></td><td><p>.016</p></td><td><p>−.106</p></td><td><p>−.270</p></td><td><p>.470</p></td><td><p>.071</p></td><td><p>.015</p></td><td><p>−.082</p></td><td><p>−.244</p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.566</p></td><td><p>.148</p></td><td><p>.017</p></td><td><p><bold>−.114</bold></p></td><td><p>−.271</p></td><td><p>.472</p></td><td><p>.087</p></td><td><p>.014</p></td><td><p>−.089</p></td><td><p>−.245</p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.552</p></td><td><p>.147</p></td><td><p>.014</p></td><td><p><bold>−.120</bold></p></td><td><p>−.275</p></td><td><p>.456</p></td><td><p>.090</p></td><td><p>.012</p></td><td><p>−.092</p></td><td><p>−.245</p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.540</p></td><td><p>.137</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.120</bold></p></td><td><p>−.279</p></td><td><p>.462</p></td><td><p>.089</p></td><td><p>.007</p></td><td><p>−.091</p></td><td><p>−.248</p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.551</p></td><td><p>.121</p></td><td><p>.006</p></td><td><p>−.111</p></td><td><p>−.271</p></td><td><p>.454</p></td><td><p>.062</p></td><td><p>.009</p></td><td><p>−.084</p></td><td><p>−.244</p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.558</p></td><td><p>.136</p></td><td><p>.008</p></td><td><p><bold>−.114</bold></p></td><td><p>−.269</p></td><td><p>.459</p></td><td><p>.084</p></td><td><p>.011</p></td><td><p>−.088</p></td><td><p>−.247</p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.530</p></td><td><p>.135</p></td><td><p><bold>.002</bold></p></td><td><p><bold>−.117</bold></p></td><td><p>−.279</p></td><td><p>.449</p></td><td><p>.084</p></td><td><p><bold>.004</bold></p></td><td><p>−.092</p></td><td><p>−.252</p></td></tr><tr><td><p>.40</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.534</p></td><td><p>.121</p></td><td><p>.006</p></td><td><p><bold>−.117</bold></p></td><td><p>−.279</p></td><td><p>.461</p></td><td><p>.077</p></td><td><p>.007</p></td><td><p>−.091</p></td><td><p>−.250</p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.509</p></td><td><p>.101</p></td><td><p><bold>−.003</bold></p></td><td><p><bold>−.117</bold></p></td><td><p>−.263</p></td><td><p><bold>.439</bold></p></td><td><p>.062</p></td><td><p><bold>.003</bold></p></td><td><p>−.092</p></td><td><p>−.244</p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.531</p></td><td><p>.114</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.115</bold></p></td><td><p>−.267</p></td><td><p>.445</p></td><td><p>.071</p></td><td><p><bold>.005</bold></p></td><td><p>−.094</p></td><td><p>−.247</p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.521</p></td><td><p>.124</p></td><td><p>.007</p></td><td><p>−.113</p></td><td><p>−.263</p></td><td><p>.446</p></td><td><p>.078</p></td><td><p>.006</p></td><td><p>−.092</p></td><td><p>−.243</p></td></tr><tr><td><p>.40</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.513</p></td><td><p>.118</p></td><td><p><bold>.003</bold></p></td><td><p><bold>−.114</bold></p></td><td><p>−.266</p></td><td><p>.447</p></td><td><p>.078</p></td><td><p>.007</p></td><td><p>−.091</p></td><td><p>−.243</p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.447</p></td><td><p>.069</p></td><td><p><bold>−.016</bold></p></td><td><p><bold>−.115</bold></p></td><td><p>−.265</p></td><td><p><bold>.409</bold></p></td><td><p>.047</p></td><td><p><bold>−.008</bold></p></td><td><p>−.098</p></td><td><p>−.250</p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.480</p></td><td><p>.092</p></td><td><p><bold>−.002</bold></p></td><td><p>−.106</p></td><td><p>−.255</p></td><td><p><bold>.425</bold></p></td><td><p>.061</p></td><td><p><bold>.001</bold></p></td><td><p>−.093</p></td><td><p>−.245</p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.488</p></td><td><p>.097</p></td><td><p><bold>.001</bold></p></td><td><p>−.104</p></td><td><p>−.247</p></td><td><p><bold>.438</bold></p></td><td><p>.068</p></td><td><p><bold>.002</bold></p></td><td><p>−.092</p></td><td><p>−.237</p></td></tr><tr><td><p>.40</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.481</p></td><td><p>.108</p></td><td><p><bold>.002</bold></p></td><td><p>−.108</p></td><td><p>−.247</p></td><td><p><bold>.442</bold></p></td><td><p>.081</p></td><td><p><bold>.003</bold></p></td><td><p>−.091</p></td><td><p>−.235</p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.562</p></td><td><p>.128</p></td><td><p>.012</p></td><td><p>−.106</p></td><td><p>−.259</p></td><td><p>.460</p></td><td><p>.071</p></td><td><p>.014</p></td><td><p>−.076</p></td><td><p>−.231</p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.557</p></td><td><p>.128</p></td><td><p><bold>.004</bold></p></td><td><p><bold>−.114</bold></p></td><td><p>−.273</p></td><td><p>.456</p></td><td><p>.078</p></td><td><p>.011</p></td><td><p>−.084</p></td><td><p>−.241</p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.541</p></td><td><p>.127</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.116</bold></p></td><td><p>−.267</p></td><td><p>.452</p></td><td><p>.079</p></td><td><p>.009</p></td><td><p>−.086</p></td><td><p>−.239</p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.517</p></td><td><p>.124</p></td><td><p><bold>.001</bold></p></td><td><p><bold>−.122</bold></p></td><td><p>−.274</p></td><td><p>.446</p></td><td><p>.083</p></td><td><p>.007</p></td><td><p>−.090</p></td><td><p>−.241</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.541</p></td><td><p>.107</p></td><td><p><bold>.003</bold></p></td><td><p>−.108</p></td><td><p>−.260</p></td><td><p><bold>.441</bold></p></td><td><p>.061</p></td><td><p>.007</p></td><td><p>−.082</p></td><td><p>−.236</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.537</p></td><td><p>.126</p></td><td><p><bold>.004</bold></p></td><td><p><bold>−.118</bold></p></td><td><p>−.263</p></td><td><p><bold>.440</bold></p></td><td><p>.074</p></td><td><p>.008</p></td><td><p>−.089</p></td><td><p>−.236</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.526</p></td><td><p>.130</p></td><td><p>.010</p></td><td><p><bold>−.114</bold></p></td><td><p>−.259</p></td><td><p><bold>.441</bold></p></td><td><p>.082</p></td><td><p>.008</p></td><td><p>−.087</p></td><td><p>−.233</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.518</p></td><td><p>.119</p></td><td><p><bold>.003</bold></p></td><td><p>−.111</p></td><td><p>−.265</p></td><td><p>.448</p></td><td><p>.078</p></td><td><p><bold>.005</bold></p></td><td><p>−.084</p></td><td><p>−.237</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.507</p></td><td><p>.086</p></td><td><p><bold>−.008</bold></p></td><td><p>−.111</p></td><td><p>−.254</p></td><td><p><bold>.423</bold></p></td><td><p>.054</p></td><td><p><bold>.004</bold></p></td><td><p>−.084</p></td><td><p>−.236</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.518</p></td><td><p>.109</p></td><td><p>.011</p></td><td><p>−.110</p></td><td><p>−.255</p></td><td><p><bold>.441</bold></p></td><td><p>.064</p></td><td><p>.010</p></td><td><p>−.089</p></td><td><p>−.233</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.517</p></td><td><p>.118</p></td><td><p>.009</p></td><td><p>−.109</p></td><td><p>−.253</p></td><td><p>.443</p></td><td><p>.072</p></td><td><p>.006</p></td><td><p>−.089</p></td><td><p>−.235</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.504</p></td><td><p>.112</p></td><td><p><bold>.002</bold></p></td><td><p>−.109</p></td><td><p>−.260</p></td><td><p>.447</p></td><td><p>.075</p></td><td><p><bold>.003</bold></p></td><td><p>−.086</p></td><td><p>−.237</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p>.449</p></td><td><p>.060</p></td><td><p><bold>−.012</bold></p></td><td><p>−.106</p></td><td><p>−.254</p></td><td><p><bold>.399</bold></p></td><td><p>.038</p></td><td><p><bold>−.007</bold></p></td><td><p>−.092</p></td><td><p>−.242</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.466</p></td><td><p>.084</p></td><td><p><bold>.000</bold></p></td><td><p>−.097</p></td><td><p>−.251</p></td><td><p><bold>.418</bold></p></td><td><p>.057</p></td><td><p><bold>−.001</bold></p></td><td><p>−.086</p></td><td><p>−.240</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.474</p></td><td><p>.089</p></td><td><p><bold>−.002</bold></p></td><td><p>−.098</p></td><td><p>−.246</p></td><td><p><bold>.426</bold></p></td><td><p>.064</p></td><td><p><bold>−.002</bold></p></td><td><p>−.085</p></td><td><p>−.235</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.473</p></td><td><p>.087</p></td><td><p>.006</p></td><td><p>−.106</p></td><td><p>−.243</p></td><td><p><bold>.430</bold></p></td><td><p>.066</p></td><td><p><bold>.004</bold></p></td><td><p>−.088</p></td><td><p>−.229</p></td></tr></tbody></table> </ephtml> </p> <p>13 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <p>C6 Table Bias of EAP Ability Estimates for Five Selected Examinees with Item Parameters Fixed to Values Obtained with Prior Settings of Varying IDM Prediction Accuracy under N=100$N=100$ and N=500$N=500$ Compared to Default Prior Settings with N=1,000$N=1,000$</p> <p> <ephtml> <table><thead><tr><th /><th /><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">$N=100$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>N</mi><mo>=</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">$N=500$</annotation></semantics></math></p></th></tr><tr><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = -2$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = -1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\theta = 0$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\theta = 1$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mi>θ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\theta = 2$</annotation></semantics></math></p></th></tr><tr><th /><th /><th /><th>(.37)</th><th>(.03)</th><th>(.00)</th><th>(−.10)</th><th>(−.29)</th><th>(.37)</th><th>(.03)</th><th>(.00)</th><th>(−.10)</th><th>(−.29)</th></tr></thead><tbody><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.542</p></td><td><p>.122</p></td><td><p>.018</p></td><td><p>−.105</p></td><td><p>−.273</p></td><td><p><bold>.407</bold></p></td><td><p>.054</p></td><td><p>.012</p></td><td><p>−.085</p></td><td><p>−.253</p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.538</p></td><td><p>.126</p></td><td><p>.014</p></td><td><p>−.111</p></td><td><p>−.279</p></td><td><p><bold>.407</bold></p></td><td><p>.056</p></td><td><p>.009</p></td><td><p>−.085</p></td><td><p>−.251</p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.529</p></td><td><p>.123</p></td><td><p>.015</p></td><td><p>−.111</p></td><td><p>−.279</p></td><td><p><bold>.406</bold></p></td><td><p>.055</p></td><td><p>.007</p></td><td><p>−.085</p></td><td><p>−.249</p></td></tr><tr><td><p>.00</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.510</p></td><td><p>.119</p></td><td><p>.007</p></td><td><p><bold>−.117</bold></p></td><td><p>−.283</p></td><td><p><bold>.403</bold></p></td><td><p>.056</p></td><td><p><bold>.003</bold></p></td><td><p>−.086</p></td><td><p>−.247</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.514</p></td><td><p>.105</p></td><td><p>.010</p></td><td><p>−.111</p></td><td><p>−.278</p></td><td><p><bold>.399</bold></p></td><td><p>.049</p></td><td><p>.010</p></td><td><p>−.088</p></td><td><p>−.254</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.530</p></td><td><p>.120</p></td><td><p>.011</p></td><td><p>−.114</p></td><td><p>−.277</p></td><td><p><bold>.404</bold></p></td><td><p>.057</p></td><td><p>.007</p></td><td><p>−.089</p></td><td><p>−.250</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.517</p></td><td><p>.120</p></td><td><p>.008</p></td><td><p><bold>−.115</bold></p></td><td><p>−.276</p></td><td><p><bold>.403</bold></p></td><td><p>.055</p></td><td><p><bold>.005</bold></p></td><td><p>−.087</p></td><td><p>−.245</p></td></tr><tr><td><p>.00</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.511</p></td><td><p>.114</p></td><td><p>.009</p></td><td><p><bold>−.118</bold></p></td><td><p>−.278</p></td><td><p><bold>.404</bold></p></td><td><p>.056</p></td><td><p><bold>.004</bold></p></td><td><p>−.086</p></td><td><p>−.244</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.496</p></td><td><p>.090</p></td><td><p><bold>−.001</bold></p></td><td><p><bold>−.119</bold></p></td><td><p>−.279</p></td><td><p><bold>.396</bold></p></td><td><p>.046</p></td><td><p><bold>.005</bold></p></td><td><p>−.091</p></td><td><p>−.254</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.512</p></td><td><p>.106</p></td><td><p><bold>.003</bold></p></td><td><p><bold>−.119</bold></p></td><td><p>−.273</p></td><td><p><bold>.407</bold></p></td><td><p>.053</p></td><td><p><bold>.005</bold></p></td><td><p>−.090</p></td><td><p>−.247</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.502</p></td><td><p>.119</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.117</bold></p></td><td><p>−.273</p></td><td><p><bold>.401</bold></p></td><td><p>.060</p></td><td><p><bold>.004</bold></p></td><td><p>−.087</p></td><td><p>−.245</p></td></tr><tr><td><p>.00</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.503</p></td><td><p>.111</p></td><td><p><bold>.004</bold></p></td><td><p><bold>−.119</bold></p></td><td><p>−.273</p></td><td><p><bold>.403</bold></p></td><td><p>.057</p></td><td><p><bold>.002</bold></p></td><td><p>−.088</p></td><td><p>−.241</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p><bold>.440</bold></p></td><td><p>.063</p></td><td><p><bold>−.016</bold></p></td><td><p><bold>−.119</bold></p></td><td><p>−.270</p></td><td><p><bold>.388</bold></p></td><td><p>.039</p></td><td><p><bold>−.002</bold></p></td><td><p>−.091</p></td><td><p>−.247</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.455</p></td><td><p>.082</p></td><td><p><bold>−.009</bold></p></td><td><p><bold>−.117</bold></p></td><td><p>−.266</p></td><td><p><bold>.393</bold></p></td><td><p>.050</p></td><td><p><bold>−.001</bold></p></td><td><p>−.090</p></td><td><p>−.242</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.471</p></td><td><p>.093</p></td><td><p><bold>−.002</bold></p></td><td><p><bold>−.115</bold></p></td><td><p>−.262</p></td><td><p><bold>.400</bold></p></td><td><p>.052</p></td><td><p><bold>.000</bold></p></td><td><p>−.089</p></td><td><p>−.239</p></td></tr><tr><td><p>.00</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.465</p></td><td><p>.098</p></td><td><p><bold>.001</bold></p></td><td><p>−.109</p></td><td><p>−.255</p></td><td><p><bold>.393</bold></p></td><td><p>.056</p></td><td><p><bold>.002</bold></p></td><td><p>−.083</p></td><td><p>−.232</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.00</p></td><td><p>.522</p></td><td><p>.114</p></td><td><p>.016</p></td><td><p>−.102</p></td><td><p>−.267</p></td><td><p><bold>.404</bold></p></td><td><p>.048</p></td><td><p>.012</p></td><td><p>−.083</p></td><td><p>−.246</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.526</p></td><td><p>.115</p></td><td><p>.014</p></td><td><p>−.109</p></td><td><p>−.273</p></td><td><p><bold>.411</bold></p></td><td><p>.050</p></td><td><p>.010</p></td><td><p>−.085</p></td><td><p>−.247</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.522</p></td><td><p>.118</p></td><td><p>.017</p></td><td><p>−.109</p></td><td><p>−.271</p></td><td><p><bold>.396</bold></p></td><td><p>.055</p></td><td><p>.008</p></td><td><p>−.083</p></td><td><p>−.242</p></td></tr><tr><td><p>.20</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.502</p></td><td><p>.112</p></td><td><p>.010</p></td><td><p>−.114</p></td><td><p>−.277</p></td><td><p><bold>.405</bold></p></td><td><p>.056</p></td><td><p><bold>.004</bold></p></td><td><p>−.086</p></td><td><p>−.243</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.509</p></td><td><p>.101</p></td><td><p><bold>.005</bold></p></td><td><p>−.104</p></td><td><p>−.267</p></td><td><p><bold>.393</bold></p></td><td><p>.044</p></td><td><p>.008</p></td><td><p>−.084</p></td><td><p>−.245</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.502</p></td><td><p>.115</p></td><td><p><bold>.005</bold></p></td><td><p><bold>−.115</bold></p></td><td><p>−.270</p></td><td><p><bold>.393</bold></p></td><td><p>.054</p></td><td><p>.006</p></td><td><p>−.088</p></td><td><p>−.244</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.510</p></td><td><p>.115</p></td><td><p>.008</p></td><td><p>−.112</p></td><td><p>−.273</p></td><td><p><bold>.397</bold></p></td><td><p>.052</p></td><td><p>.006</p></td><td><p>−.085</p></td><td><p>−.244</p></td></tr><tr><td><p>.20</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.497</p></td><td><p>.111</p></td><td><p>.007</p></td><td><p>−.113</p></td><td><p>−.274</p></td><td><p><bold>.398</bold></p></td><td><p>.059</p></td><td><p><bold>.003</bold></p></td><td><p>−.085</p></td><td><p>−.243</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.483</p></td><td><p>.084</p></td><td><p><bold>−.005</bold></p></td><td><p><bold>−.116</bold></p></td><td><p>−.275</p></td><td><p><bold>.392</bold></p></td><td><p>.038</p></td><td><p><bold>.002</bold></p></td><td><p>−.090</p></td><td><p>−.250</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.497</p></td><td><p>.107</p></td><td><p><bold>.002</bold></p></td><td><p>−.108</p></td><td><p>−.269</p></td><td><p><bold>.394</bold></p></td><td><p>.050</p></td><td><p><bold>.003</bold></p></td><td><p>−.085</p></td><td><p>−.243</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.500</p></td><td><p>.099</p></td><td><p><bold>.002</bold></p></td><td><p>−.111</p></td><td><p>−.267</p></td><td><p><bold>.399</bold></p></td><td><p>.051</p></td><td><p><bold>.002</bold></p></td><td><p>−.087</p></td><td><p>−.240</p></td></tr><tr><td><p>.20</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.484</p></td><td><p>.106</p></td><td><p><bold>−.002</bold></p></td><td><p>−.112</p></td><td><p>−.265</p></td><td><p><bold>.397</bold></p></td><td><p>.054</p></td><td><p><bold>.000</bold></p></td><td><p>−.086</p></td><td><p>−.236</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p><bold>.432</bold></p></td><td><p>.065</p></td><td><p><bold>−.016</bold></p></td><td><p><bold>−.120</bold></p></td><td><p>−.268</p></td><td><p><bold>.376</bold></p></td><td><p>.034</p></td><td><p><bold>−.002</bold></p></td><td><p>−.093</p></td><td><p>−.245</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p>.456</p></td><td><p>.079</p></td><td><p><bold>−.007</bold></p></td><td><p>−.112</p></td><td><p>−.261</p></td><td><p><bold>.383</bold></p></td><td><p>.042</p></td><td><p><bold>−.000</bold></p></td><td><p>−.089</p></td><td><p>−.241</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.463</p></td><td><p>.083</p></td><td><p><bold>−.004</bold></p></td><td><p>−.109</p></td><td><p>−.255</p></td><td><p><bold>.397</bold></p></td><td><p>.049</p></td><td><p><bold>−.001</bold></p></td><td><p>−.087</p></td><td><p>−.235</p></td></tr><tr><td><p>.20</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.450</p></td><td><p>.085</p></td><td><p><bold>−.002</bold></p></td><td><p>−.108</p></td><td><p>−.253</p></td><td><p><bold>.388</bold></p></td><td><p>.053</p></td><td><p><bold>.001</bold></p></td><td><p>−.086</p></td><td><p>−.233</p></td></tr><tr><td><p>.40</p></td><td><p>.20</p></td><td><p>.00</p></td>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></p></td><td><p>.039</p></td><td><p>.010</p></td><td><p>−.073</p></td><td><p>−.225</p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.20</p></td><td><p>.495</p></td><td><p>.099</p></td><td><p>.007</p></td><td><p>−.101</p></td><td><p>−.258</p></td><td><p><bold>.390</bold></p></td><td><p>.044</p></td><td><p>.007</p></td><td><p>−.079</p></td><td><p>−.232</p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.40</p></td><td><p>.484</p></td><td><p>.097</p></td><td><p>.006</p></td><td><p>−.102</p></td><td><p>−.255</p></td><td><p><bold>.389</bold></p></td><td><p>.048</p></td><td><p><bold>.005</bold></p></td><td><p>−.079</p></td><td><p>−.227</p></td></tr><tr><td><p>.60</p></td><td><p>.20</p></td><td><p>.60</p></td><td><p>.475</p></td><td><p>.102</p></td><td><p><bold>.002</bold></p></td><td><p>−.109</p></td><td><p>−.260</p></td><td><p><bold>.388</bold></p></td><td><p>.052</p></td><td><p><bold>.003</bold></p></td><td><p>−.083</p></td><td><p>−.230</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.00</p></td><td><p>.475</p></td><td><p>.083</p></td><td><p><bold>.003</bold></p></td><td><p>−.097</p></td><td><p>−.250</p></td><td><p><bold>.376</bold></p></td><td><p>.035</p></td><td><p><bold>.005</bold></p></td><td><p>−.077</p></td><td><p>−.228</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.20</p></td><td><p>.478</p></td><td><p>.098</p></td><td><p><bold>.005</bold></p></td><td><p>−.104</p></td><td><p>−.251</p></td><td><p><bold>.381</bold></p></td><td><p>.043</p></td><td><p><bold>.004</bold></p></td><td><p>−.082</p></td><td><p>−.228</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.40</p></td><td><p>.477</p></td><td><p>.105</p></td><td><p>.007</p></td><td><p>−.102</p></td><td><p>−.248</p></td><td><p><bold>.382</bold></p></td><td><p>.052</p></td><td><p><bold>.004</bold></p></td><td><p>−.081</p></td><td><p>−.225</p></td></tr><tr><td><p>.60</p></td><td><p>.40</p></td><td><p>.60</p></td><td><p>.479</p></td><td><p>.096</p></td><td><p><bold>.001</bold></p></td><td><p>−.098</p></td><td><p>−.254</p></td><td><p><bold>.393</bold></p></td><td><p>.050</p></td><td><p><bold>.001</bold></p></td><td><p>−.079</p></td><td><p>−.227</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.00</p></td><td><p>.450</p></td><td><p>.066</p></td><td><p><bold>−.001</bold></p></td><td><p>−.099</p></td><td><p>−.249</p></td><td><p><bold>.367</bold></p></td><td><p><bold>.031</bold></p></td><td><p><bold>.003</bold></p></td><td><p>−.079</p></td><td><p>−.228</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.20</p></td><td><p>.460</p></td><td><p>.076</p></td><td><p>.007</p></td><td><p>−.103</p></td><td><p>−.245</p></td><td><p><bold>.381</bold></p></td><td><p>.036</p></td><td><p>.006</p></td><td><p>−.083</p></td><td><p>−.227</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.40</p></td><td><p>.471</p></td><td><p>.090</p></td><td><p><bold>.004</bold></p></td><td><p>−.103</p></td><td><p>−.249</p></td><td><p><bold>.385</bold></p></td><td><p>.044</p></td><td><p><bold>.003</bold></p></td><td><p>−.082</p></td><td><p>−.226</p></td></tr><tr><td><p>.60</p></td><td><p>.60</p></td><td><p>.60</p></td><td><p>.469</p></td><td><p>.089</p></td><td><p><bold>−.000</bold></p></td><td><p>−.100</p></td><td><p>−.252</p></td><td><p><bold>.388</bold></p></td><td><p>.049</p></td><td><p><bold>−.001</bold></p></td><td><p>−.080</p></td><td><p>−.229</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.00</p></td><td><p><bold>.407</bold></p></td><td><p>.045</p></td><td><p><bold>−.014</bold></p></td><td><p>−.103</p></td><td><p>−.253</p></td><td><p><bold>.359</bold></p></td><td><p><bold>.024</bold></p></td><td><p><bold>−.004</bold></p></td><td><p>−.085</p></td><td><p>−.233</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.20</p></td><td><p><bold>.432</bold></p></td><td><p>.064</p></td><td><p><bold>−.006</bold></p></td><td><p>−.096</p></td><td><p>−.250</p></td><td><p><bold>.371</bold></p></td><td><p>.033</p></td><td><p><bold>−.003</bold></p></td><td><p>−.080</p></td><td><p>−.232</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.40</p></td><td><p>.446</p></td><td><p>.077</p></td><td><p><bold>−.006</bold></p></td><td><p>−.095</p></td><td><p>−.245</p></td><td><p><bold>.380</bold></p></td><td><p>.043</p></td><td><p><bold>−.003</bold></p></td><td><p>−.080</p></td><td><p>−.228</p></td></tr><tr><td><p>.60</p></td><td><p>.80</p></td><td><p>.60</p></td><td><p>.448</p></td><td><p>.076</p></td><td><p><bold>.003</bold></p></td><td><p>−.099</p></td><td><p>−.239</p></td><td><p><bold>.383</bold></p></td><td><p>.046</p></td><td><p><bold>.001</bold></p></td><td><p>−.082</p></td><td><p>−.222</p></td></tr></tbody></table> </ephtml> </p> <p>14 <emph>Note</emph>. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>a</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>b</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_b$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>R</mi><mi>c</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$R^2_c$</annotation></semantics></math> </ephtml> give simulated IDM prediction accuracies. Values printed in bold are comparable to the default baseline with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">$N=1,000$</annotation></semantics></math> </ephtml> given in brackets in the table header. Values with gray background indicate improved estimation accuracy.</p> <ref id="AN0192629996-61"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> This is markedly higher than standard textbook recommendations derived from synthesizing results from various simulation studies for the Rasch (100 to 200 examinees) and two‐parameter logistic model (500 examinees; DeMars, [16]).</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> A nonparametric approach for doing so has been suggested by Tsutakawa ([53]), who showed that the 3PL model can be reparameterized by specifying the probability <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>j</mi></msub><mrow><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>q</mi></msub><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">$p_j(\theta _q)$</annotation></semantics></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">$q=1,\dots,Q$</annotation></semantics></math> </ephtml> ) of a correct response for any <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Q</mi><annotation encoding="application/x-tex">$Q$</annotation></semantics></math> </ephtml> distinct points on the ability continuum, arguing that this parameterization is better identified than the standard 3PL parameterization. Tsutakawa ([53]) proposed using a constrained Dirichlet prior distribution on these probabilities and illustrated how these prior distributions can be informed by the item characteristic curves of the item pool.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref73" type="bt">3</bibl> <bibtext> We point out, however, that we (and other researchers; e.g., Štěpánek et al., [49]) found it challenging to evaluate the accuracy of these approaches as most studies compared the performance of different machine learning models against each other rather than some "gold‐standard" (e.g., MML with a large sample size; see McCarthy et al., [37], for an exception). Hence, IDM studies oftentimes only allow quantifying to which extent the best‐performing machine learning model outperformed its (possibly poorly performing) competitors, but do not allow to gauge and compare the proximity of the obtained IDM item parameter predictions to their large‐sample MML counterparts, and as such, their utility as substitutes.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref67" type="bt">4</bibl> <bibtext> Specifically, for each item, the following set of features was built using sentence embeddings: (1) The stimulus, question, answer, and distractors were mapped to a common vector space using a universal sentence encoder (Cer et al., [11]), resulting in four normalized vectors in a 512‐dimensional space— <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">s</mi><annotation encoding="application/x-tex">$\bm{s}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">q</mi><annotation encoding="application/x-tex">$\bm{q}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">d</mi><annotation encoding="application/x-tex">$\bm{d}$</annotation></semantics></math> </ephtml> . (2) The cosine similarity for all possible vector pairs was computed and the mean, standard deviation, minimum, and maximum were obtained as features. (3) The singular value decomposition of matrix <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">M</mi><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">s</mi><mo>,</mo><mi mathvariant="bold-italic">q</mi><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">d</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">$\bm{M} = [\bm{s}, \bm{q}, \bm{a}, \bm{d}]$</annotation></semantics></math> </ephtml> was computed, and four singular values were obtained as features. (4) Five dimensions were extracted from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">s</mi><annotation encoding="application/x-tex">$\bm{s}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">q</mi><annotation encoding="application/x-tex">$\bm{q}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">d</mi><annotation encoding="application/x-tex">$\bm{d}$</annotation></semantics></math> </ephtml> by means of principal component analysis.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref5" type="bt">5</bibl> <bibtext> Model 1 by Belov et al. ([8]) was a feedforward neural network with the following three layers: an input layer with 68 nodes to use the features described above, one hidden layer with 8 nodes, a ReLU activation function, and L1 regularization (Goodfellow et al., [23]), and an output layer with three nodes to provide point estimates for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>b</mi><annotation encoding="application/x-tex">$b$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="application/x-tex">$c$</annotation></semantics></math> </ephtml> . To train Model 1, Belov et al. 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Structural Equation Modeling: A Multidisciplinary Journal, 28(1), 40–50. https://doi.org/10.1080/10705511.2020.1752216</bibtext> </blist> </ref> <aug> <p>By Esther Ulitzsch; Dmitry Belov; Oliver Lüdtke and Alexander Robitzsch</p> <p>Reported by Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib27" firstref="ref3"></nolink> <nolink nlid="nl2" bibid="bib56" firstref="ref4"></nolink> <nolink nlid="nl3" bibid="bib55" firstref="ref6"></nolink> <nolink nlid="nl4" bibid="bib47" firstref="ref8"></nolink> <nolink nlid="nl5" bibid="bib49" firstref="ref9"></nolink> <nolink nlid="nl6" bibid="bib44" firstref="ref10"></nolink> <nolink nlid="nl7" bibid="bib46" firstref="ref13"></nolink> <nolink nlid="nl8" bibid="bib52" firstref="ref15"></nolink> <nolink nlid="nl9" bibid="bib50" firstref="ref16"></nolink> <nolink nlid="nl10" bibid="bib54" firstref="ref19"></nolink> <nolink nlid="nl11" bibid="bib34" firstref="ref20"></nolink> <nolink nlid="nl12" bibid="bib48" firstref="ref21"></nolink> <nolink nlid="nl13" bibid="bib13" firstref="ref22"></nolink> <nolink nlid="nl14" bibid="bib33" firstref="ref23"></nolink> <nolink nlid="nl15" bibid="bib32" firstref="ref24"></nolink> <nolink nlid="nl16" bibid="bib20" firstref="ref25"></nolink> <nolink nlid="nl17" bibid="bib14" firstref="ref26"></nolink> <nolink nlid="nl18" bibid="bib38" firstref="ref28"></nolink> <nolink nlid="nl19" bibid="bib29" firstref="ref29"></nolink> <nolink nlid="nl20" bibid="bib57" firstref="ref35"></nolink> <nolink nlid="nl21" bibid="bib42" firstref="ref37"></nolink> <nolink nlid="nl22" bibid="bib51" firstref="ref39"></nolink> <nolink nlid="nl23" bibid="bib28" firstref="ref40"></nolink> <nolink nlid="nl24" bibid="bib36" firstref="ref41"></nolink> <nolink nlid="nl25" bibid="bib43" firstref="ref50"></nolink> <nolink nlid="nl26" bibid="bib26" firstref="ref52"></nolink> <nolink nlid="nl27" bibid="bib18" firstref="ref53"></nolink> <nolink nlid="nl28" bibid="bib17" firstref="ref54"></nolink> <nolink nlid="nl29" bibid="bib37" firstref="ref72"></nolink> <nolink nlid="nl30" bibid="bib12" firstref="ref81"></nolink> <nolink nlid="nl31" bibid="bib45" firstref="ref88"></nolink> <nolink nlid="nl32" bibid="bib10" firstref="ref90"></nolink> <nolink nlid="nl33" bibid="bib24" firstref="ref91"></nolink> <nolink nlid="nl34" bibid="bib25" firstref="ref92"></nolink> <nolink nlid="nl35" bibid="bib40" firstref="ref93"></nolink> <nolink nlid="nl36" bibid="bib21" firstref="ref94"></nolink> <nolink nlid="nl37" bibid="bib22" firstref="ref95"></nolink> <nolink nlid="nl38" bibid="bib35" firstref="ref99"></nolink> <nolink nlid="nl39" bibid="bib39" firstref="ref100"></nolink> <nolink nlid="nl40" bibid="bib15" firstref="ref102"></nolink> <nolink nlid="nl41" bibid="bib41" firstref="ref105"></nolink> <nolink nlid="nl42" bibid="bib30" firstref="ref113"></nolink> <nolink nlid="nl43" bibid="bib31" firstref="ref114"></nolink> <nolink nlid="nl44" bibid="bib19" firstref="ref116"></nolink>
Header DbId: eric
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An: EJ1501461
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Using Item Parameter Predictions for Reducing Calibration Sample Requirements--A Case Study Based on a High-Stakes Admission Test
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Esther+Ulitzsch%22">Esther Ulitzsch</searchLink><br /><searchLink fieldCode="AR" term="%22Dmitry+Belov%22">Dmitry Belov</searchLink><br /><searchLink fieldCode="AR" term="%22Oliver+Lüdtke%22">Oliver Lüdtke</searchLink><br /><searchLink fieldCode="AR" term="%22Alexander+Robitzsch%22">Alexander Robitzsch</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Journal+of+Educational+Measurement%22"><i>Journal of Educational Measurement</i></searchLink>. 2026 63(1).
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 52
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2026
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22High+Stakes+Tests%22">High Stakes Tests</searchLink><br /><searchLink fieldCode="DE" term="%22Test+Items%22">Test Items</searchLink><br /><searchLink fieldCode="DE" term="%22Difficulty+Level%22">Difficulty Level</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Bayesian+Statistics%22">Bayesian Statistics</searchLink><br /><searchLink fieldCode="DE" term="%22Maximum+Likelihood+Statistics%22">Maximum Likelihood Statistics</searchLink><br /><searchLink fieldCode="DE" term="%22Sample+Size%22">Sample Size</searchLink><br /><searchLink fieldCode="DE" term="%22Accuracy%22">Accuracy</searchLink><br /><searchLink fieldCode="DE" term="%22Prediction%22">Prediction</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1111/jedm.12426
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0022-0655<br />1745-3984
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: In item difficulty modeling (IDM), item parameters are predicted from the items' linguistic features, aiming to ultimately render item calibration redundant. Current IDM applications, however, commonly do not yield the required prediction accuracy. To immediately exploit even somewhat inaccurate IDM predictions, we blend IDM with established Bayesian estimation techniques. We propose a two-step approach where (1) IDM predictions are obtained and (2) employed to construct informative prior distributions. We evaluate the approach in a case study on small-sample calibration of the 3PL in a high-stakes test. First, concerning implementation, we find computationally efficient penalized maximum likelihood estimation to be comparable to the best-performing MCMC-based approach. Second, we investigate sample size reductions achievable with state-of-the-art IDM predictions, finding negligible gains compared to merely considering the historical distribution of parameters. Third, we evaluate the prediction accuracy required for a targeted sample size reduction by gradually increasing simulated IDM prediction accuracies. We find that required accuracies can counterbalance each other, allowing calibration sample size to be reduced when either high-quality item difficulty predictions or good predictions of item discriminations and pseudo-guessing parameters are available. We argue that these evaluations provide new, portable IDM benchmarks quantifying performance in terms of achievable sample size reductions.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2026
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1501461
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1501461
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1111/jedm.12426
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 52
    Subjects:
      – SubjectFull: High Stakes Tests
        Type: general
      – SubjectFull: Test Items
        Type: general
      – SubjectFull: Difficulty Level
        Type: general
      – SubjectFull: Computation
        Type: general
      – SubjectFull: Bayesian Statistics
        Type: general
      – SubjectFull: Maximum Likelihood Statistics
        Type: general
      – SubjectFull: Sample Size
        Type: general
      – SubjectFull: Accuracy
        Type: general
      – SubjectFull: Prediction
        Type: general
    Titles:
      – TitleFull: Using Item Parameter Predictions for Reducing Calibration Sample Requirements--A Case Study Based on a High-Stakes Admission Test
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Esther Ulitzsch
      – PersonEntity:
          Name:
            NameFull: Dmitry Belov
      – PersonEntity:
          Name:
            NameFull: Oliver Lüdtke
      – PersonEntity:
          Name:
            NameFull: Alexander Robitzsch
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          Dates:
            – D: 01
              M: 03
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 0022-0655
            – Type: issn-electronic
              Value: 1745-3984
          Numbering:
            – Type: volume
              Value: 63
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: Journal of Educational Measurement
              Type: main
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