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Modeling Missing Response Data in Item Response Theory: Addressing Missing Not at Random Mechanism with Monotone Missing Characteristics

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Title: Modeling Missing Response Data in Item Response Theory: Addressing Missing Not at Random Mechanism with Monotone Missing Characteristics
Language: English
Authors: Jiwei Zhang, Jing Lu, Zhaoyuan Zhang
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: 34
Publication Date: 2026
Document Type: Journal Articles
Reports - Research
Education Level: Secondary Education
Descriptors: Item Response Theory, Research Problems, Models, Bayesian Statistics, Evaluation Criteria, Sampling, Algorithms, Achievement Tests, Foreign Countries, International Assessment, Secondary School Students
Assessment and Survey Identifiers: Program for International Student Assessment
DOI: 10.1111/jedm.12428
ISSN: 0022-0655
1745-3984
Abstract: Item nonresponses frequently occurs in educational and psychological assessments, and if not appropriately handled, it can undermine the reliability of the results. This study introduces a missing data model based on the missing not at random (MNAR) mechanism, incorporating the monotonic missingness assumption to capture individual-level missingness patterns and behavioral dynamics. In specific, the cumulative number of missing indicators allows to consider the tendency of current item's missingness based on the previous missingnesses, which reduces the number of nuisance parameters for modeling missing data mechanisms. Two Bayesian model evaluation criteria were developed to distinguish between missing at random (MAR) and MNAR mechanisms by imposing specific parameter constraints. Additionally, the study introduces a highly efficient Bayesian slice sampling algorithm to estimate the model parameters. Four simulation studies were conducted to show the performance of the proposed model. The PISA 2015 science data was carried out to further illustrate the application of the proposed approach.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1501423
Database: ERIC
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  Value: <anid>AN0192629997;mea01mar.26;2026Apr01.06:22;v2.2.500</anid> <title id="AN0192629997-1">Modeling Missing Response Data in Item Response Theory: Addressing Missing Not at Random Mechanism with Monotone Missing Characteristics </title> <sbt id="AN0192629997-2">Introduction</sbt> <p>Item nonresponses frequently occurs in educational and psychological assessments, and if not appropriately handled, it can undermine the reliability of the results. This study introduces a missing data model based on the missing not at random (MNAR) mechanism, incorporating the monotonic missingness assumption to capture individual‐level missingness patterns and behavioral dynamics. In specific, the cumulative number of missing indicators allows to consider the tendency of current item's missingness based on the previous missingnesses, which reduces the number of nuisance parameters for modeling missing data mechanisms. Two Bayesian model evaluation criteria were developed to distinguish between missing at random (MAR) and MNAR mechanisms by imposing specific parameter constraints. Additionally, the study introduces a highly efficient Bayesian slice sampling algorithm to estimate the model parameters. Four simulation studies were conducted to show the performance of the proposed model. The PISA 2015 science data was carried out to further illustrate the application of the proposed approach.</p> <p>Item nonresponse data is a common and inevitable issue in educational and psychological assessments. For instance, the Programme for International Student Assessment (PISA) 2009 study had an average missing data rate of 10% (Rose et al., [<reflink idref="bib44" id="ref1">44</reflink>]). According to Blossfeld et al. ([<reflink idref="bib5" id="ref2">5</reflink>]), the National Educational Panel Study (NEPS) reported an overall item nonresponse rate of 18.8% on a reading test, consisting of 5.4% omitted items and 13.5% not‐reached items. However, it is important to note that the rate of missing data can vary significantly depending on the item, subtest, and subgroup. For example, the PISA 2006 study found that the percentage of missing response items for each OECD country ranged from 4% to 15% (Rose et al., [<reflink idref="bib45" id="ref3">45</reflink>], [<reflink idref="bib44" id="ref4">44</reflink>]). Improper handling of nonresponse data or treating it as an incorrect response can lead to inaccurate model parameter estimates. Therefore, identifying missing data mechanisms and developing appropriate missing models are crucial to obtain precise parameter estimates (Huisman, [<reflink idref="bib26" id="ref5">26</reflink>]; Lord, [<reflink idref="bib32" id="ref6">32</reflink>]).</p> <p>Rubin ([<reflink idref="bib46" id="ref7">46</reflink>]) classifies missing data mechanisms into three types: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). MCAR occurs when missingness is unrelated to both observed and unobserved data, posing no bias to statistical results. MAR refers to cases where missingness depends on observed data but not unobserved variables. To maintain valid statistical inferences under MAR, relevant covariates associated with missingness should be correctly identified and included. In contrast, MNAR occurs when missingness depends on unobserved variables, making it nonignorable. To address this, additional methods must be employed to directly model the relationship between the missing data and its latent variables.</p> <p>When item nonresponse may be attributed to low ability or high item difficulty, missingness is classified as MNAR. Imputation methods are widely employed to handle missing data, ranging from simple imputation (e.g., mean, median, or mode) to more advanced techniques like multiple imputation and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>k</mi><annotation encoding="application/x-tex">$k$</annotation></semantics></math> </ephtml> ‐nearest neighbors (KNN). While these techniques can improve inference, they may introduce uncertainty and bias (Finch, [<reflink idref="bib11" id="ref8">11</reflink>]; Pohl et al., [<reflink idref="bib39" id="ref9">39</reflink>]; Sinharay, [<reflink idref="bib47" id="ref10">47</reflink>]). To better handle nonignorable missing data, various models have been developed, including selection models (SLMs; Heckman, [<reflink idref="bib23" id="ref11">23</reflink>], [<reflink idref="bib24" id="ref12">24</reflink>]), pattern mixture models (PMMs; Little, [<reflink idref="bib30" id="ref13">30</reflink>]), and multidimensional item response theory (MIRT) models (O'Muircheartaigh & Moustaki, [<reflink idref="bib38" id="ref14">38</reflink>]). SLMs are extensively used to model the dependency of the missing indicator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> on item response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> (Du et al., [<reflink idref="bib10" id="ref15">10</reflink>]; Finch, [<reflink idref="bib12" id="ref16">12</reflink>]; Galimard et al., [<reflink idref="bib13" id="ref17">13</reflink>]; Robitzsch, [<reflink idref="bib41" id="ref18">41</reflink>]). It decomposes the joint distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> into two parts: the distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Y</mi><annotation encoding="application/x-tex">$Y$</annotation></semantics></math> </ephtml> (the item response model) and the conditional distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> (the missingness model), that is, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">R</mi><mfenced separators="" open="|" close=""><mrow><mi mathvariant="bold">Λ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold">Δ</mi></mrow></mfenced></mfenced><mo linebreak="badbreak">=</mo><mi>P</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">Y</mi><mfenced separators="" open="|" close=""><mrow><mi mathvariant="bold">Λ</mi></mrow></mfenced></mfenced><mi>P</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">R</mi><mfenced separators="" open="|" close=""><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mrow><mi mathvariant="bold">Δ</mi></mrow></mfenced></mfenced><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} P{\left(\bm{Y},\bm{R}{\left|\bm{\Lambda },\bm{\Delta }\right.} \right)} =P{\left(\bm{Y}{\left|\bm{\Lambda }\right.} \right)} P{\left(\bm{R}{\left|\bm{Y},\bm{\Delta }\right.} \right)}, \end{equation*}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Λ</mi></mrow><annotation encoding="application/x-tex">$\bm{\Lambda }$</annotation></semantics></math> </ephtml> is the parameter vector of the item response model and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Δ</mi></mrow><annotation encoding="application/x-tex">$\bm{\Delta }$</annotation></semantics></math> </ephtml> is the parameter vector of the missing data model. In the missing data model, due to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> being dependent on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> , imputing the missing response data <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> becomes crucial. In contrast to SLMs, some educational psychometricians employ PMMs to account for the dependency of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> (Glas & Pimentel, [<reflink idref="bib19" id="ref19">19</reflink>]; Holman & Glas, [<reflink idref="bib25" id="ref20">25</reflink>]; Zhang et al., [<reflink idref="bib57" id="ref21">57</reflink>]). PMMs decompose the joint distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> into the distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> and the distribution of <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> , that is, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">R</mi><mfenced separators="" open="|" close=""><mrow><mi mathvariant="bold">Λ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold">Δ</mi></mrow></mfenced></mfenced><mo linebreak="badbreak">=</mo><mi>P</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">Y</mi><mfenced separators="" open="|" close=""><mi mathvariant="bold-italic">R</mi><mo>,</mo><mrow><mi mathvariant="bold">Λ</mi></mrow></mfenced></mfenced><mi>P</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">R</mi><mfenced separators="" open="|" close=""><mrow><mi mathvariant="bold">Δ</mi></mrow></mfenced></mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} P{\left(\bm{Y},\bm{R}{\left|\bm{\Lambda },\bm{\Delta }\right.} \right)} =P{\left(\bm{Y}{\left|\bm{R},\bm{\Lambda }\right.} \right)} P{\left(\bm{R}{\left|\bm{\Delta }\right.} \right)}. \end{equation*}$$</annotation></semantics></math> </ephtml> Since <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> depends on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> , when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$R=1$</annotation></semantics></math> </ephtml> indicates that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Y</mi><annotation encoding="application/x-tex">$Y$</annotation></semantics></math> </ephtml> is observed, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><mi>Y</mi><mo>|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo>,</mo><mrow><mi mathvariant="bold">Λ</mi></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(Y|R=1,\bm{\Lambda })$</annotation></semantics></math> </ephtml> is the traditional item response model, and when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$R=0$</annotation></semantics></math> </ephtml> indicates that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Y</mi><annotation encoding="application/x-tex">$Y$</annotation></semantics></math> </ephtml> is missing, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><mi>Y</mi><mo>|</mo><mi>R</mi><mo>=</mo><mn>0</mn><mo>,</mo><mrow><mi mathvariant="bold">Λ</mi></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(Y|R=0,\bm{\Lambda })$</annotation></semantics></math> </ephtml> is a degenerate distribution of a constant. MIRT models has been widely used to analyze the MNAR mechanism (e.g., Holman & Glas, [<reflink idref="bib25" id="ref22">25</reflink>]; Glas & Pimentel, [<reflink idref="bib19" id="ref23">19</reflink>]; Glas et al., [<reflink idref="bib20" id="ref24">20</reflink>]; Rose, [<reflink idref="bib43" id="ref25">43</reflink>]; Rose et al., [<reflink idref="bib45" id="ref26">45</reflink>]). However, both O'Muircheartaigh and Moustaki ([<reflink idref="bib38" id="ref27">38</reflink>]) and Rose ([<reflink idref="bib43" id="ref28">43</reflink>]) have demonstrated that the MIRT models can be viewed as a special case of PMM or derived from SLM in different ways. In addition, alternative modeling techniques, such as latent regression models (LRM; Rose et al., [<reflink idref="bib44" id="ref29">44</reflink>]) and multiple‐group IRT models, have been extensively investigated as effective manifest approaches to address nonignorable missing data (Pohl et al., [<reflink idref="bib39" id="ref30">39</reflink>]; Rose, [<reflink idref="bib43" id="ref31">43</reflink>]; Rose et al., [<reflink idref="bib45" id="ref32">45</reflink>], [<reflink idref="bib44" id="ref33">44</reflink>]).</p> <p>The widespread use of computer‐based assessments has facilitated the collection of response time (RT) data, which includes the time to answer items and the time spent on omitted items (i.e., omission time; Ulitzsch et al., [<reflink idref="bib51" id="ref34">51</reflink>]). Several RT‐based methods have been proposed to model omissions based on RTs. Ulitzsch et al. ([<reflink idref="bib51" id="ref35">51</reflink>]) developed a joint model that combines response and omission behaviors, incorporating latent factors like omission propensity and speed. In contrast, Lu and Wang ([<reflink idref="bib33" id="ref36">33</reflink>]) and Lu et al. ([<reflink idref="bib34" id="ref37">34</reflink>]) focused on the relationship between RTs and omission time, without making distributional assumptions about omission time. Their methods assume independence between RTs and omissions, making them particularly suitable for ignorable omitted items when RTs are considered. Additionally, some methods have addressed missingness from not‐reached items. Pohl et al. ([<reflink idref="bib40" id="ref38">40</reflink>]) and Lu and Wang ([<reflink idref="bib33" id="ref39">33</reflink>]) used van der Linden ([<reflink idref="bib52" id="ref40">52</reflink>]) hierarchical framework to jointly model response accuracy and RTs for observed items to address missingness resulting from not‐reached items due to time limits. Guo et al. ([<reflink idref="bib21" id="ref41">21</reflink>]) argued that the initial not‐reached items provide valuable information since the responses are censored when the RTs exceed the remaining test time.</p> <p>The RT‐based models for handling missing data are specific to computer‐based testing and do not apply to traditional paper‐and‐pencil assessments. This study aims to address the common issue of missing response data in large‐scale testing. We propose an innovative approach, that is, the one‐parameter logistic missing data process (1PLMDP) model, which adopts the SLM modeling strategy to account for the dependence of missing indicators on item responses. This method is specifically designed to effectively handle MNAR items, including those that contain only omitted items, only not‐reached items, or both types of missing items simultaneously. Our modeling approach differs from Rose et al.'s (2017) study, which use the MIRT model and latent regression model, respectively, to fit omitted and not‐reached items. However, their approach can be excessively complicated, making it less feasible and appealing for use in applied research. The proposed method does not distinguish between the two types of missing data but instead adopts a unified modeling framework to treat both types of missing responses as a single category. The advantage of this approach is that it avoids the complexity that may arise in certain practical applications due to excessive assumptions or overly detailed distinctions between missing data types. This innovation is particularly important in practice, where distinguishing between omitted and not‐reached items is often challenging. The unified modeling approach improves the model's flexibility and applicability.</p> <p>In addition, the proposed model is the first study to introduce the assumption of monotonic missingness into the modeling framework to characterize missingness patterns at the individual level. This assumption suggests that if an examinee has missing responses on earlier items, they are likely to have missing responses on later items as well. Conversely, if they have no missing responses on earlier items, the likelihood of missing responses on later items decreases. This approach leverages the cumulative effect of an examinee's missing indicators from earlier items to describe their potential effect on the probability of missing indicator on current item. This idea is inspired by autoregressive models in time series analysis, where past observations are used to predict future outcomes. Similarly, the monotonicity assumption applies a recursive structure to capture the relationships between missingness patterns across items, providing theoretical supports for understanding such behaviors.</p> <p>Although Rose et al. ([<reflink idref="bib44" id="ref42">44</reflink>]) considered the cumulative effects of missing indicators from not‐reached items in their latent regression model, their use of these indicators was not designed to capture monotonic missingness explicitly. The key difference between their approach and this study lies in the handling of missing indicator data. In Rose et al.'s (2017, Equation 24) model, ability and missing propensity parameters are explained through a function of the missing indicators, meaning that the function of missing indicators directly influences ability and missing propensity parameters. In contrast, the proposed missing data model not only utilizes cumulative missing indicators to capture the characteristics of monotonic missingness but also incorporates additional critical factors: individual‐level missing propensity parameters, item‐specific missing item parameters, and the current item response. Note that we assume that the current item response influences the current missing indicator. This idea is also addressed in Robitzsch's (2023) approach, which considers the dependency of the missing indicator on the corresponding item response. These factors included in our model interact to create a more systematic framework for predicting the probability of missing indicators. Additionally, in Robitzsch ([<reflink idref="bib41" id="ref43">41</reflink>]), missing responses are treated as ignorable latent variables by using the sum score of response indicators across all items, assuming that the total score is a sufficient statistic for response propensity. Our model, however, explicitly considers the cumulative effect of previous missing responses on the probability of missing responses for the current item. Compared to Rose et al. ([<reflink idref="bib44" id="ref44">44</reflink>]) and Robitzsch ([<reflink idref="bib42" id="ref45">42</reflink>]), our method offers a more detailed and dynamic understanding of missingness, capturing the complex interactions between various factors, and provides a more robust theoretical framework for analyzing missing data.</p> <p>In contrast to Glas and Pimentel's (2008) study, they modeled not‐reached missingness by adjusting item difficulty parameters and assumed that the probability of a missing response increases with item position, particularly in speeded tests where not‐reached responses are more likely toward the end of the test due to time limits. However, our proposed model emphasizes the internal dynamics of an examinee's behavior, which captures the cumulative trend of missing behaviors, reflecting the dynamic characteristics of the examinee's behavioral patterns and state changes, rather than relying solely on external factors, such as item position or time limits. This difference makes our model more flexible, as it can be applied to various test settings without being limited by specific item positions or time limits.</p> <p>When addressing missing data, it's crucial to understand the underlying missing data mechanism, as this influences model selection and impacts the reliability of the results. The missing data mechanism, that is, whether MAR or MNAR, determines how missing data should be handled. Accurately identifying the mechanism is essential for ensuring valid inferences. In our study, we propose a flexible framework for modeling missing data that can accommodate both MAR and MNAR mechanisms. This framework imposes specific constraints on model parameters, allowing the model to adjust according to the characteristics of the real data to align with the missing mechanism. To determine whether the real data fits the MAR or MNAR mechanism, we develop two Bayesian model evaluation criteria. These criteria evaluate the constraints on the model parameters to infer the most likely missing data mechanism.</p> <p>The structure of this article is as follows: "Models and Model Identifications" section 2 introduces models for item responses and missing data mechanisms, followed by model identification. "Bayesian Estimation Strategy" section offers an overview of the slice sampling algorithm. In "Bayesian Model Assessment" section, two Bayesian model assessment criteria are introduced based on Markov chain Monte Carlo (MCMC) posterior samples. "Simulation Studies" section presents four simulation studies to evaluate the algorithm's performance. The proposed methodology is further demonstrated through an empirical example in "Empirical Example" section. Finally, "Conclusion" section presents concluding remarks.</p> <hd id="AN0192629997-3">Models and Model Identifications</hd> <p></p> <hd id="AN0192629997-4">Two Parameter Logistic IRT Model</hd> <p>Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> be a two‐dimensional item response data matrix with the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">$(i,j)$</annotation></semantics></math> </ephtml> element <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">$Y_{ij}$</annotation></semantics></math> </ephtml> , where examinees are indexed as <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 are indexed as <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> . <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$Y_{ij} = 1$</annotation></semantics></math> </ephtml> indicates that the <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> th examinee answered the <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> th item correctly, while <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$Y_{ij} = 0$</annotation></semantics></math> </ephtml> indicates that the <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> th examinee answered the <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> th item incorrectly. Additionally, a missing indicator data matrix <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> is defined, which has the same dimensions as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> , with the element <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$R_{ij}=1$</annotation></semantics></math> </ephtml> if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">$Y_{ij}$</annotation></semantics></math> </ephtml> is a missing observation, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$R_{ij}=0$</annotation></semantics></math> </ephtml> otherwise. Based on the observed response data <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> and the missing indicator data <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> , the following IRT model and the missing indicator model are constructed. The correct response probability is expressed as: 1 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo linebreak="badbreak">=</mo><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn><mfenced separators="" open="|" close=""><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>a</mi><mi>j</mi></msub><mo>,</mo><msub><mi>b</mi><mi>j</mi></msub><mo>,</mo></mfenced></mfenced><mo linebreak="goodbreak">=</mo><mfrac><mrow><mi>exp</mi><mfenced separators="" open="[" close="]"><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>−</mo><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mfenced separators="" open="[" close="]"><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>−</mo><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow></mfenced></mrow></mfrac><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} P_{ij}=P{\left(Y_{ij}=1{\left|\theta _{i}, a_{j}, b_{j}, \right.} \right)} =\frac{\exp {\left[ a_{j}(\theta _{i}-b_{j})\right]} }{1+\exp {\left[ a_{j}(\theta _{i}-b_{j})\right]} }, \end{equation}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>θ</mi><mi>i</mi></msub><annotation encoding="application/x-tex">$\theta _{i}$</annotation></semantics></math> </ephtml> is the latent ability of examinee <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><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>n</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">$(i=1,\ldots,n)$</annotation></semantics></math> </ephtml> , and <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> and <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> are the discrimination and difficulty parameters, respectively.</p> <hd id="AN0192629997-5">Missing Data Mechanism</hd> <p>The missing data process is modeled using a one parameter logistic model, which is appropriate for situations where there is a small amount of missing response data and few parameters are needed to estimate the model (Lord, [<reflink idref="bib32" id="ref46">32</reflink>]). The one parameter logistic missing data process (1PLMDP) model is expressed as follows: 2 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>π</mi><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></msub><mo linebreak="badbreak">=</mo><mi>P</mi><mrow><mo>(</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo linebreak="goodbreak">=</mo><mn>1</mn><mfenced separators="" open="|" close=""><msub><mi>τ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ς</mi><mi>j</mi></msub><mo>,</mo><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>,</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mfenced><mo>)</mo></mrow><mo linebreak="goodbreak">=</mo><mfrac><mrow><mi>exp</mi><mrow><mo>[</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>−</mo><msub><mi>τ</mi><mi>i</mi></msub><mo>+</mo><msub><mi>ς</mi><mi>j</mi></msub><mo>+</mo><mi>g</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>+</mo><mi>l</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>]</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mrow><mo>[</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>−</mo><msub><mi>τ</mi><mi>i</mi></msub><mo>+</mo><msub><mi>ς</mi><mi>j</mi></msub><mo>+</mo><mi>g</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>+</mo><mi>l</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>]</mo></mrow></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \pi _{R_{ij}}=P{(R_{ij}=1{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R}_{ij-1},\bm{Y}_{ij}\right.})} =\frac{\exp {[ \gamma _{0}-\tau _{i}+\varsigma _{j}+g{(\bm{R}_{ij-1},\gamma _{1})} +l{(\bm{Y} _{ij},\bm{\gamma }_{2})} ]} }{1+\exp {[ \gamma _{0} -\tau _{i}+\varsigma _{j}+g{(\bm{R}_{ij-1},\gamma _{1})} +l{(\bm{Y}_{ij},\bm{\gamma }_{2})} ]} }. \end{equation}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo>(</mo><msub><mi>R</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>,</mo><mtext>...</mtext><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\bm{R}_{ij-1}=(R_{i1},R_{i2},\ldots,R_{ij-1})$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><msub><mi>Y</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>Y</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>,</mo><mtext>...</mtext><mo>,</mo><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\bm{Y}_{ij}=(Y_{i1},Y_{i2},\ldots,Y_{ij})$</annotation></semantics></math> </ephtml> . <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>0</mn></msub><annotation encoding="application/x-tex">$\gamma _0$</annotation></semantics></math> </ephtml> is a fixed unknown intercept parameter, where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>0</mn></msub><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\gamma _0<0$</annotation></semantics></math> </ephtml> controls the baseline proportion of missing data. This constraint reflects observations from real‐world datasets, where missing responses are typically less than 50% and occur much less frequently than valid item responses. The missing propensity parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>τ</mi><mi>i</mi></msub><annotation encoding="application/x-tex">$\tau _i$</annotation></semantics></math> </ephtml> and the missing item parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>ς</mi><mi>j</mi></msub><annotation encoding="application/x-tex">$\varsigma _j$</annotation></semantics></math> </ephtml> are the factors influencing the missing indicator. The fixed unknown slope parameter for the missing indicator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><annotation encoding="application/x-tex">$\bm{R}_{ij-1}$</annotation></semantics></math> </ephtml> is denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _1$</annotation></semantics></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\gamma _1>0$</annotation></semantics></math> </ephtml> . <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">$g(\cdot)$</annotation></semantics></math> </ephtml> is a function of the missing indicators that represents the effect of previously missing response items on the current item. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mo>(</mo><mrow><mo>·</mo></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$l(\bm{\cdot })$</annotation></semantics></math> </ephtml> is a function of the item response variables, which characterizes the effect of previous item response variables on the current item. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold-italic">γ</mi><mn>2</mn></msub><annotation encoding="application/x-tex">$\bm{\gamma }_{2}$</annotation></semantics></math> </ephtml> is a fixed unknown slope parameter vector of the response variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">$\bm{Y}_{ij}$</annotation></semantics></math> </ephtml> causing missing. We set <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>(</mo><mo>·</mo><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$g(\cdot)=0$</annotation></semantics></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$j=1$</annotation></semantics></math> </ephtml> because there are no previous missing indicators <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><annotation encoding="application/x-tex">$\bm{R}_{ij-1}$</annotation></semantics></math> </ephtml> . The joint distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> is constructed using a sequence of one dimensional conditional distribution modeling method proposed by Ibrahim et al. ([<reflink idref="bib27" id="ref47">27</reflink>], [<reflink idref="bib28" id="ref48">28</reflink>]). That is, 3 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>R</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo linebreak="goodbreak">=</mo><msub><mi>r</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo linebreak="goodbreak">=</mo><msub><mi>r</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>,</mo><mtext>...</mtext><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>J</mi></mrow></msub><mo>=</mo><msub><mi>r</mi><mrow><mi>i</mi><mi>J</mi></mrow></msub><mfenced separators="" open="|" close=""><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">,</mi><mspace width="0.33em" /><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mo>,</mo><mspace width="0.33em" /><mrow><mi mathvariant="bold-italic">ς</mi></mrow><mo>,</mo><mspace width="0.33em" /><msub><mi mathvariant="bold-italic">y</mi><mrow><mi>i</mi><mi>J</mi></mrow></msub></mfenced></mfenced><mo linebreak="badbreak">=</mo><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>J</mi></munderover><msubsup><mi>π</mi><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mrow><mn mathvariant="bold">1</mn></mrow><mfenced separators="" open="(" close=")"><msub><mi>r</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mfenced></mrow></msubsup><msup><mfenced separators="" open="(" close=")"><mn>1</mn><mo>−</mo><msub><mi>π</mi><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></msub></mfenced><mrow><mrow><mn mathvariant="bold">1</mn></mrow><mfenced separators="" open="(" close=")"><msub><mi>r</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mfenced></mrow></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} P{\left(R_{i1}=r_{i1},R_{i2}=r_{i2},\ldots,R_{iJ}=r_{iJ}{\left|\bm{\gamma }\text{, }\bm{\tau },\text{ }\bm{\varsigma },\text{ }\bm{y}_{iJ}\right.} \right)} =\prod \limits _{j=1}^{J} \pi _{R_{ij}}^{\bm{1}{\left(r_{ij}=1\right)} }{\left(1-\pi _{R_{ij} }\right)} ^{\bm{1}{\left(r_{ij}=0\right)} }. \end{equation}$$</annotation></semantics></math> </ephtml><bold>Remark 1</bold>: To model the missingness process, we use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>γ</mi><mn>1</mn></msub><munderover><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>h</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$g(\bm{R} _{ij-1},\bm{\gamma }_{1}) =\gamma _{1}\sum \limits _{h=1} ^{j-1}\bm{R}_{ih}$</annotation></semantics></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$j\ge 1$</annotation></semantics></math> </ephtml> , which depends on all the previous missing indicators. The new covariate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>h</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$\sum \limits _{h=1}^{j-1}\bm{R}_{ih}$</annotation></semantics></math> </ephtml> captures the cumulative number of missing indicators and reduces the number of nuisance parameters for modeling the missing data mechanism. In Equation 2, we define <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>γ</mi><mn>2</mn></msub><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$l(\bm{Y}_{ij},\bm{\gamma } _{2}) =\gamma _{2}Y_{ij}$</annotation></semantics></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\gamma _{2}<0$</annotation></semantics></math> </ephtml> . This approach builds on Wu et al.'s (2018) work on intermittent and dropout missingness, applying their methods from Remarks 2 and 3 (please see Wu et al.'s, 2018, paper for details) to construct the functions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>(</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>1</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$g(\bm{R}_{ij-1}, \bm{\gamma }_{1})$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mo>(</mo><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>2</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$l(\bm{Y}_{ij}, \bm{\gamma }_{2})$</annotation></semantics></math> </ephtml> , which provide strong theoretical support for our model. If the latent 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> and the missing propensity parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> are independent, and the missing indicator does not depend on the response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Y</mi><annotation encoding="application/x-tex">$Y$</annotation></semantics></math> </ephtml> , the missing mechanism is MAR. However, if the missing data depends on the latent 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> , missing propensity parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> , and missing response variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Y</mi><annotation encoding="application/x-tex">$Y$</annotation></semantics></math> </ephtml> , the missing mechanism is MNAR.</p> <p>Two models are utilized to characterize two distinct types of missing data mechanisms: MAR and MNAR (Little & Rubin, [<reflink idref="bib31" id="ref49">31</reflink>]).</p> <hd1 id="AN0192629997-6">MAR model</hd1> <p>The joint distribution of the MAR model can be expressed as: 4 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mn>1</mn></msub><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><mfenced separators="" open="(" close=")"><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="" open="|" close=""><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>a</mi><mi>j</mi></msub><mo>,</mo><msub><mi>b</mi><mi>j</mi></msub></mfenced></mfenced><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="" open="|" close=""><msub><mi>τ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ς</mi><mi>j</mi></msub><mo>,</mo><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>,</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub></mfenced></mfenced><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>θ</mi><mi>i</mi></msub></mfenced><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>τ</mi><mi>i</mi></msub></mfenced><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} Q_{1}=\prod \limits _{i=1}^{N}\prod \limits _{j=1}^{J}P{\left(Y_{ij}{\left|\theta _{i},a_{j},b_{j}\right.} \right)} P{\left(R_{ij}{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R}_{ij-1}\right.} \right)} P{\left(\theta _{i}\right)} P{\left(\tau _{i}\right)}, \end{equation}$$</annotation></semantics></math> </ephtml> where the distributions of <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> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> are denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$p(\theta _{i})$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>(</mo><msub><mi>τ</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$p(\tau _{i})$</annotation></semantics></math> </ephtml> , respectively. Under the MAR, the missingness of data is random, but conditional on the observed data. This means that although the missing data mechanism might be related to the observed data, it does not depend on the values of the missing data itself. Therefore, under the MAR assumption, if the observed data can be appropriately used in the analysis, direct modeling of the missing data is not necessary. This means that we can estimate the item response model without taking into account the missing data process <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mi>τ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ς</mi><mi>j</mi></msub><mo>,</mo><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>,</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(R_{ij}|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R} _{ij-1})$</annotation></semantics></math> </ephtml> . It should be noted that the latent variables for the observed data and the missing data process are independent in the MAR model.</p> <hd1 id="AN0192629997-7">MNAR model</hd1> <p>In MNAR model, the latent ability parameter ( <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> ) is correlated with the missing propensity parameter ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> ) associated with the missing indicators through the covariance matrix <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Σ</mi></mrow><annotation encoding="application/x-tex">$\bm{\Sigma }$</annotation></semantics></math> </ephtml> . This implies that the missing data mechanism indirectly influenced by <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> , because the missing data relies on the missing propensity parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> , which is correlated with <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> . Furthermore, the missing indicator is influenced by the item response itself. In the 1PLMPD model, the item response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> consists of two components: the observed response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{obs}$</annotation></semantics></math> </ephtml> and the missing response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> . Therefore, under the MNAR mechanism, the model depends both on the latent variables and the missing item response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> . The joint distribution of the MNAR model can be expressed as: 5 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mn>2</mn></msub><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><mfenced separators="" open="(" close=")"><msub><mi>y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="" open="|" close=""><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>a</mi><mi>j</mi></msub><mo>,</mo><msub><mi>b</mi><mi>j</mi></msub></mfenced></mfenced><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="" open="|" close=""><msub><mi>τ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ς</mi><mi>j</mi></msub><mo>,</mo><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>,</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mfenced></mfenced><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi><mi>i</mi></msub><mfenced separators="" open="|" close=""><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold">Σ</mi></mrow></mfenced></mfenced><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} Q_{2}=\prod \limits _{i=1}^{N}\prod \limits _{j=1}^{J}P{\left(y_{ij}{\left|\theta _{i},a_{j},b_{j}\right.} \right)} P{\left(R_{ij}{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R}_{ij-1},\bm{Y}_{ij}\right.} \right)} P{\left(\theta _{i},\tau _{i}{\left|\bm{\mu },\bm{\Sigma }\right.} \right)}, \end{equation}$$</annotation></semantics></math> </ephtml> where the latent variables for both the person parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">U</mi><mi>i</mi></msub><mo>=</mo><msup><mrow><mo>(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>′</mo></msup></mrow><annotation encoding="application/x-tex">$\bm{U}_{i}=(\theta _{i},\tau _{i}) ^{\prime }$</annotation></semantics></math> </ephtml> are assumed to be correlated and follow a bivariate normal distribution, denoted by 6 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">U</mi><mi>i</mi></msub><mo>∼</mo><mi>p</mi><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold-italic">U</mi><mi>i</mi></msub><mfenced separators="" open="|" close=""><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold">Σ</mi></mrow></mfenced></mfenced><mo linebreak="goodbreak">=</mo><mfrac><msup><mfenced separators="" open="|" close="|"><msup><mi mathvariant="normal">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mfenced><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mi>exp</mi><mfenced separators="" open="[" close="]"><mo linebreak="badbreak">−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold-italic">U</mi><mi>i</mi></msub><mo>−</mo><mrow><mi mathvariant="bold-italic">μ</mi></mrow></mfenced><mo>'</mo></msup><msup><mi mathvariant="bold">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold-italic">U</mi><mi>i</mi></msub><mo>−</mo><mrow><mi mathvariant="bold-italic">μ</mi></mrow></mfenced></mfenced></mrow><annotation encoding="application/x-tex">$$\begin{equation} \bm{U}_{i}\sim p{\left(\bm{U}_{i}{\left|\bm{\mu },\bm{\Sigma }\right.} \right)} =\frac{{\left|\Sigma ^{-1} \right|}^{1/2}}{2\pi }\exp {\left[ -\frac{1}{2}{\left(\bm{U} _{i}-\bm{\mu }\right)} ^{'}\bm{\Sigma }^{-1}{\left(\bm{U}_{i}-\bm{\mu }\right)} \right]} \end{equation}$$</annotation></semantics></math> </ephtml> with mean vector <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mo linebreak="badbreak">=</mo><msup><mfenced separators="" open="(" close=")"><msub><mi>μ</mi><mi>θ</mi></msub><mo>,</mo><msub><mi>μ</mi><mi>τ</mi></msub></mfenced><mo>′</mo></msup><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} \bm{\mu }={\left(\mu _{\theta },\mu _{\tau }\right)} ^{\prime }, \end{equation*}$$</annotation></semantics></math> </ephtml> and covariance matrix <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold">Σ</mi></mrow><mo linebreak="badbreak">=</mo><mfenced separators="" open="(" close=")">σθ2σθτστθστ2</mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} \bm{\Sigma }={\left(\def\eqcellsep{&}\begin{array}{ll}\sigma _{\theta }^{2} & \sigma _{\theta \tau }\\[3pt] \sigma _{\tau \theta } & \sigma _{\tau }^{2} \end{array} \right)}. \end{equation*}$$</annotation></semantics></math> </ephtml></p> <hd id="AN0192629997-8">Model Identifications</hd> <p>The linear components of the 2PL model and 1PLMDP model in Equations 1 and 2 can be represented as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mfenced separators="" open="(" close=")"><msub><mi>θ</mi><mi>i</mi></msub><mo>−</mo><msub><mi>b</mi><mi>j</mi></msub></mfenced><mspace width="0.33em" /><mtext>and</mtext><mspace width="0.33em" /><msub><mi>γ</mi><mn>0</mn></msub><mo linebreak="badbreak">−</mo><msub><mi>τ</mi><mi>i</mi></msub><mo linebreak="goodbreak">+</mo><msub><mi>ς</mi><mi>j</mi></msub><mo linebreak="goodbreak">+</mo><mi>g</mi><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>γ</mi><mn>1</mn></msub></mfenced><mo linebreak="goodbreak">+</mo><mi>l</mi><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>2</mn></msub></mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} a_{j}{\left(\theta _{i}-b_{j}\right)} \text{ and }\gamma _{0}-\tau _{i}+\varsigma _{j}+g{\left(\bm{R}_{ij-1},\gamma _{1}\right)} +l{\left(\bm{Y}_{ij},\bm{\gamma }_{2}\right)}. \end{equation*}$$</annotation></semantics></math> </ephtml> To remove the trade‐off between <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>θ</mi><annotation encoding="application/x-tex">$\theta$</annotation></semantics></math> </ephtml> in location for the 2PL model, the mean of the population level of ability is set to zero ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>θ</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\mu _{\theta }=0$</annotation></semantics></math> </ephtml> ). Similarly, to remove the trade‐off between <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 <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> in scale for the 2PL model, the population variance of ability is set to one ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mi>θ</mi><mn>2</mn></msubsup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$\sigma _{\theta }^{2}=1$</annotation></semantics></math> </ephtml> ). For the 1PLMDP model, to remove the trade‐offs among <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ς</mi><annotation encoding="application/x-tex">$\varsigma$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>0</mn></msub><annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math> </ephtml> in location, we can fix the population mean levels of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ς</mi><annotation encoding="application/x-tex">$\varsigma$</annotation></semantics></math> </ephtml> to zero ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>τ</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\mu _{\tau }=0$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>ς</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\mu _{\varsigma }=0$</annotation></semantics></math> </ephtml> ). Specifically, we define <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>γ</mi><mn>1</mn></msub><munderover><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi mathvariant="bold-italic">R</mi><mrow><mi>i</mi><mi>h</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$g(\bm{R}_{ij-1},\gamma _{1}) =\gamma _{1}\sum \limits _{h=1}^{j-1}\bm{R}_{ih}$</annotation></semantics></math> </ephtml> , for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">$j\ge 1$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="bold-italic">γ</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>γ</mi><mn>2</mn></msub><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$l(\bm{Y}_{ij},\bm{\gamma }_{2}) =\gamma _{2}Y_{ij}$</annotation></semantics></math> </ephtml> in the 1PLMDP model. Note that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">$Y_{ij}$</annotation></semantics></math> </ephtml> is either the observed response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msubsup><annotation encoding="application/x-tex">$Y_{ij}^{obs}$</annotation></semantics></math> </ephtml> or the missing response <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msubsup><annotation encoding="application/x-tex">$Y_{ij}^{mis}$</annotation></semantics></math> </ephtml> imputed through Bayesian algorithms. Therefore, we treat the imputed <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> as known values, which, together with the observed <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{obs}$</annotation></semantics></math> </ephtml> , form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> . This allows us to perform statistical inference on the parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> , ensuring its estimability. In fact, the parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>2</mn></msub><annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math> </ephtml> do not need to be constrained to identify the 1PLMDP model.</p> <hd id="AN0192629997-9">Bayesian Estimation Strategy</hd> <p>In this study, we use a highly efficient slice sampling algorithm (SSA; Bishop, [<reflink idref="bib4" id="ref50">4</reflink>]; Damien et al., [<reflink idref="bib9" id="ref51">9</reflink>]; Neal, [<reflink idref="bib37" id="ref52">37</reflink>]; Lu et al., [<reflink idref="bib35" id="ref53">35</reflink>]) within a fully Bayesian framework to estimate the proposed model. We chose this algorithm for the following reasons: First, it is highly flexible and capable of handling various discrete response types due to the known distribution and convenient mathematical properties of the augmented data likelihood. Second, unlike the Metropolis Hastings sampling algorithm (MH; Chen, et al., 2000; Chib & Greenberg, [<reflink idref="bib8" id="ref54">8</reflink>]; Hastings, [<reflink idref="bib22" id="ref55">22</reflink>]; Metropolis et al., [<reflink idref="bib36" id="ref56">36</reflink>]), SSA does not require retrospective tuning, making it more efficient and less sensitive to variations in step size, ensuring that all samples are accepted and thus improving sampling efficiency. Third, while Gibbs sampling algorithm (Albert, [<reflink idref="bib1" id="ref57">1</reflink>]; Béguin & Glas, [<reflink idref="bib3" id="ref58">3</reflink>]; Gelfand & Smith, [<reflink idref="bib16" id="ref59">16</reflink>]; Tanner & Wong, [<reflink idref="bib50" id="ref60">50</reflink>]) is effective for drawing samples from the posterior distribution, it is limited to conjugate priors and is not suitable for nonconjugate models. However, SSA can provide accurate estimation results for any nonconjugate prior distributions. Fourth, SSA is robust to prior misspecification, maintaining reliable performance even when prior distributions are not perfectly specified.</p> <p>Several comparative studies have highlighted the efficiency and accuracy of the slice sampling algorithm (SSA) over the traditional Metropolis‐Hastings (MH) algorithm across different IRT models. For example, Lu et al. ([<reflink idref="bib35" id="ref61">35</reflink>]) demonstrated the SSA's superiority in estimating multilevel IRT models through both theoretical analysis and simulation studies. Similarly, Zhang et al. ([<reflink idref="bib55" id="ref62">55</reflink>]) showed the SSA's efficiency and accuracy in estimating the graded response model (GRM) in simulation studies. In the context of cognitive diagnostic assessment, Zhang et al. ([<reflink idref="bib56" id="ref63">56</reflink>]) applied the SSA to estimate the reparameterized DINA model and confirmed its better performance compared to the MH algorithm in simulation studies.</p> <p>Before using the SSA, we first introduce the auxiliary variables in both the 2PL model and 1PLMDP model. For the 2PL model, the auxiliary variables are denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">δ</mi></mrow><mo>=</mo><msub><mrow><mo>(</mo><msub><mi>δ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>N</mi><mo>×</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$\bm{\delta }=(\delta _{ij}) _{N\times J}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">λ</mi><mo>=</mo></mrow><msub><mrow><mo>(</mo><msub><mi>λ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>N</mi><mo>×</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$\bm{\lambda =}(\lambda _{ij}) _{N\times J}$</annotation></semantics></math> </ephtml> , while for the 1PLMDP model, they are <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">φ</mi></mrow><mo>=</mo><msub><mrow><mo>(</mo><msub><mi>φ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>N</mi><mo>×</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$\bm{\varphi }=(\varphi _{ij}) _{N\times J}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">ψ</mi><mo>=</mo></mrow><msub><mrow><mo>(</mo><msub><mi>ψ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>N</mi><mo>×</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$\bm{\psi =}(\psi _{ij}) _{N\times J}$</annotation></semantics></math> </ephtml> . These auxiliary variables are introduced to facilitate the implementation of the SSA. By integrating out these auxiliary variables, the model reverts to its traditional likelihood function form. Next, we provide the specific expression of the joint posterior distribution of all parameters according to the augmented latent variables. The joint posterior distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">δ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">φ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">b</mi><mo>,</mo><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">ζ</mi></mrow><mo>,</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup></mrow><annotation encoding="application/x-tex">$\bm{\delta },\bm{\lambda },\bm{\varphi },\bm{\psi },\bm{\theta },\bm{a},\bm{b},\bm{\tau },\bm{\gamma },\bm{\zeta },\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> can be written as 7 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>Pδ,λ,φ,ψ,θ,a,b,τ,γ,ζ,YmisYobs,R∝Pδ,λθ,a,b,Ymis,YobsPφ,ψτ,γ,ζ,Yobs,Ymis,R×PYmisθ,a,b,τ,γ,ζ,Yobs,RPapbPζpγPθ,τμ,ΣPμ,Σ∝∏i=1N∏j=1JIYij=1I0<ζij≤Pij+IYij=0I0<λij≤qij×∏i=1N∏j=1JIRij=1I0<φij≤dij+IRij=0I0<ψij≤tij×∏i=1N∏j=1JPYijmisθi,aj,bj,τi,ζj,γ0,γ1,γ2,Yobs,R×∏j=1JPajPbjPζj∏i=1NPθi,τiμ,Σ∏k=02PγkPμ,Σ.<annotation encoding="application/x-tex">$$\begin{align} & P{\left(\bm{\delta },\bm{\lambda },\bm{\varphi },\bm{\psi },\bm{\theta },\bm{a},\bm{b},\bm{\tau },\bm{\gamma },\bm{\zeta },\bm{Y} ^{mis}{\left|\bm{Y}^{obs},\bm{R}\right.} \right)} \nonumber \\[6pt] & \propto P{\left(\bm{\delta },\bm{\lambda }{\left|\bm{\theta },\bm{a},\bm{b},\bm{Y} ^{mis},\bm{Y}^{obs}\right.} \right)} P{\left(\bm{\varphi },\bm{\psi }{\left|\bm{\tau },\bm{\gamma },\bm{\zeta },\bm{Y}^{obs},\bm{Y}^{mis},\bm{R} \right.} \right)} \nonumber \\[6pt] & \times P{\left(\bm{Y}^{mis}{\left|\bm{\theta },\bm{a},\bm{b},\bm{\tau },\bm{\gamma },\bm{\zeta },\bm{Y}^{obs},\bm{R}\right.} \right)} P{\left(\bm{a}\right)} p{\left(\bm{b}\right)} P{\left(\bm{\zeta }\right)} p{\left(\bm{\gamma }\right)} P{\left(\bm{\theta },\bm{\tau }{\left|\bm{\mu },\bm{\Sigma }\right.} \right)} P{\left(\bm{\mu },\bm{\Sigma }\right)} \nonumber \\[6pt] & \propto {\displaystyle \prod \limits _{i=1}^{N}} {\displaystyle \prod \limits _{j=1}^{J}} {\left[ \mathrm{I}{\left(Y_{ij}=1\right)} \mathrm{I}{\left(0<\zeta _{ij}\le P_{ij}\right)} +\mathrm{I}{\left(Y_{ij}=0\right)} \mathrm{I}{\left(0<\lambda _{ij}\le q_{ij}\right)} \right]} \nonumber \\[6pt] & \times {\displaystyle \prod \limits _{i=1}^{N}} {\displaystyle \prod \limits _{j=1}^{J}} {\left[ \mathrm{I}{\left(R_{ij}=1\right)} \mathrm{I}{\left(0<\varphi _{ij}\le d_{ij}\right)} +\mathrm{I}{\left(R_{ij}=0\right)} \mathrm{I}{\left(0<\psi _{ij}\le t_{ij}\right)} \right]} \nonumber \\[6pt] & \times {\left[ {\displaystyle \prod \limits _{i=1}^{N}} {\displaystyle \prod \limits _{j=1}^{J}} P{\left(Y_{ij}^{mis}{\left|\theta _{i},a_{j},b_{j},\tau _{i},\zeta _{j},\gamma _{0},\gamma _{1},\gamma _{2},\bm{Y}^{obs},\bm{R} \right.} \right)} \right]} \nonumber \\[6pt] & \times {\left[ {\displaystyle \prod \limits _{j=1}^{J}} P{\left(a_{j}\right)} P{\left(b_{j}\right)} P{\left(\zeta _{j}\right)} \right]} {\left[ {\displaystyle \prod \limits _{i=1}^{N}} P{\left(\theta _{i},\tau _{i}{\left|\bm{\mu },\bm{\Sigma }\right.} \right)} \right]} {\left[ {\displaystyle \prod \limits _{k=0}^{2}} P{\left(\gamma _{k}\right)} \right]} P{\left(\bm{\mu },\bm{\Sigma }\right)}. \end{align}$$</annotation></semantics></math> </ephtml> The response data is represented as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">Y</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">$\bm{Y=}$</annotation></semantics></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mrow><mo>(</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><mo>,</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><mo>)</mo></mrow><mo>′</mo></msup><annotation encoding="application/x-tex">$(\bm{Y}^{mis},\bm{Y}^{obs}) ^{\prime }$</annotation></semantics></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{obs}$</annotation></semantics></math> </ephtml> represent the missing and observed response data, respectively. Note that we treated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> as latent variables and estimated them using Bayesian sampling algorithm, thereby "imputing" the missing responses <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> . This imputation allowed us to obtain complete data to estimate the model parameters more accurately. Please see the sampling step 5 in the Online Supplement. The bivariate normal distribution with mean vector <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">μ</mi></mrow><annotation encoding="application/x-tex">$\bm{\mu }$</annotation></semantics></math> </ephtml> and covariance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Σ</mi></mrow><annotation encoding="application/x-tex">$\bm{\Sigma }$</annotation></semantics></math> </ephtml> is represented as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mo>|</mo><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(\bm{\theta },\bm{\tau }|\bm{\mu },\bm{\Sigma })$</annotation></semantics></math> </ephtml> . In the IRT model, the prior distributions for the discrimination and difficulty parameters are denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>a</mi><mi>j</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(a_{j})$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(b_{j})$</annotation></semantics></math> </ephtml> , respectively, while the prior distribution of difficulty parameters in the 1PLMDP model is represented as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>ς</mi><mi>j</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(\varsigma _{j})$</annotation></semantics></math> </ephtml> . The prior distribution of intercept parameter in the 1PLMDP model is represented as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(\gamma _{0})$</annotation></semantics></math> </ephtml> , and the prior distributions of slope parameters in the 1PLMDP model are represented as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(\gamma _{1})$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>γ</mi><mn>2</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(\gamma _{2})$</annotation></semantics></math> </ephtml> . The detailed slice sampling algorithm is provided in the Online Supplementary Material.</p> <hd id="AN0192629997-10">Bayesian Model Assessment</hd> <p></p> <hd id="AN0192629997-11">Bayesian Model Assessment Criteria</hd> <p>Two Bayesian model evaluation criteria were developed to distinguish between MAR and MNAR mechanisms by imposing parameter constraints. Many studies have employed similar approaches. For instance, Holman and Glas ([<reflink idref="bib25" id="ref64">25</reflink>]) differentiated ignorable and nonignorable models by controlling whether the latent variables, that is, ability and missing propensity, are independent or jointly normally distributed. This was achieved by constraining the covariance parameter of a bivariate normal distribution, where a zero covariance implies statistical independence. Similarly, Wu et al. ([<reflink idref="bib53" id="ref65">53</reflink>]) applied this principle in medical statistics, determining whether a missing data mechanism is ignorable or nonignorable based on specific parameter constraints. Note that in MCAR and MAR mechanisms, missingness does not depend on unobserved data, making them ignorable. However, all nonignorable missing mechanisms can be classified as MNAR, but not all MNAR cases depend solely on unobserved data. Indeed, MNAR scenarios might involve missing data that depends on other latent variables, extending beyond merely nonignorable missing mechanisms. Next, we will demonstrate how to obtain the joint distributions under both MAR and MNAR by imposing specific constraints on the model parameters. In the case of MAR, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\gamma _{2}=0$</annotation></semantics></math> </ephtml> (Wu et al., [<reflink idref="bib53" id="ref66">53</reflink>]), and <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> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> are independent (Holman & Glas, [<reflink idref="bib25" id="ref67">25</reflink>]; Equation 3). Therefore, the joint distribution under MAR can be expressed as follows: 8 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>Q1=PY,RΩ1PθPτ=PYobs,Ymis,RΩ1PθPτ=∏i=1N∏j=1JPYijθi,aj,bjPRijτi,ςj,γ0,γ1,Rij−1PθiPτi,<annotation encoding="application/x-tex">$$\begin{align} Q_{1}=P{\left(\bm{Y},\bm{R}{\left|\bm{\Omega }_{1}\right.}\right)}P{\left(\bm{\theta }\right)}P{\left(\bm{\tau }\right)} & =P{\left(\bm{Y}^{obs},\bm{Y} ^{mis},\bm{R}{\left|\bm{\Omega }_{1}\right.} \right)} P{\left(\bm{\theta }\right)} P{\left(\bm{\tau }\right)} \nonumber \\ & =\prod \limits _{i=1}^{N}\prod \limits _{j=1}^{J}P{\left(Y_{ij}{\left|\theta _{i}, a_{j},b_{j}\right.} \right)} P{\left(R_{ij}{\left|\tau _{i},\varsigma _{j},\gamma _{0},\gamma _{1},\bm{R}_{ij-1}\right.} \right)} P{\left(\theta _{i}\right)} P{\left(\tau _{i}\right)}, \end{align}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Ω</mi><mn>1</mn></msub><mrow><mo>=</mo></mrow><mrow><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mrow><mn>1</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mspace width="0.33em" /><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>N</mi><mo>.</mo><mspace width="0.33em" /><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>J</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\bm{\Omega }_{1}\bm{=}(\bm{\Omega }_{1,ij},\text{ }i=1,\ldots,N.\text{ }j=1,\ldots,J)$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Ω</mi><mrow><mn>1</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msup><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>ς</mi><mi>j</mi></msub><mo>,</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>μ</mi><mi>θ</mi></msub><mo>,</mo><msubsup><mi>σ</mi><mi>θ</mi><mn>2</mn></msubsup><mo>,</mo><msub><mi>μ</mi><mi>τ</mi></msub><mo>,</mo><msubsup><mi>σ</mi><mi>τ</mi><mn>2</mn></msubsup><mo>)</mo></mrow><mo>′</mo></msup></mrow><annotation encoding="application/x-tex">$\bm{\Omega }_{1,ij} = (a_{j}, b_{j}, \varsigma _{j}, \gamma _{0}, \gamma _{1}, \mu _{\theta }, \sigma _{\theta }^{2}, \mu _{\tau }, \sigma _{\tau }^{2})^{\prime }$</annotation></semantics></math> </ephtml> . Here, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> includes both the observed responses <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{obs}$</annotation></semantics></math> </ephtml> and the missing responses <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{Y}^{mis}$</annotation></semantics></math> </ephtml> , and each element <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">$Y_{ij}$</annotation></semantics></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">Y</mi><annotation encoding="application/x-tex">$\bm{Y}$</annotation></semantics></math> </ephtml> is either the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msubsup><annotation encoding="application/x-tex">$Y_{ij}^{obs}$</annotation></semantics></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msubsup><annotation encoding="application/x-tex">$Y_{ij}^{mis}$</annotation></semantics></math> </ephtml> , which is imputed through the SSA.</p> <p>The MNAR occurs when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\gamma _{2}\ne 0$</annotation></semantics></math> </ephtml> (Wu et al., [<reflink idref="bib53" id="ref68">53</reflink>]) and there is a dependence between <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> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> (Holman & Glas, [<reflink idref="bib25" id="ref69">25</reflink>]; Equation 4). As a result, the joint distribution under the MNAR can be expressed as follows: 9 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>Q2=PY,RΩ2Pθ,τ=PYobs,Ymis,RΩ2Pθ,τ=∏i=1N∏j=1JPYijθi,aj,bjPRijτi,ςj,γ0,γ1,γ2,Rij−1,YijPθi,τiμ,Σ,<annotation encoding="application/x-tex">$$\begin{align} Q_{2} & =P{\left(\bm{Y},\bm{R}{\left|\bm{\Omega }_{2}\right.} \right)} P{\left(\bm{\theta },\bm{\tau }\right)} =P{\left(\bm{Y}^{obs},\bm{Y}^{mis},\bm{R}{\left|\bm{\Omega }_{2}\right.} \right)} P{\left(\bm{\theta },\bm{\tau }\right)} \nonumber \\ & =\prod \limits _{i=1}^{N}\prod \limits _{j=1}^{J}P{\left(Y_{ij}{\left|\theta _{i},a_{j},b_{j}\right.} \right)} P{\left(R_{ij}{\left|\tau _{i},\varsigma _{j},\gamma _{0},\gamma _{1},\gamma _{2},\bm{R} _{ij-1},Y_{ij}\right.} \right)} P{\left(\theta _{i},\tau _{i}{\left|\bm{\mu },\bm{\Sigma }\right.} \right)}, \end{align}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Ω</mi><mn>2</mn></msub><mrow><mo>=</mo></mrow><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Ω</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mspace width="0.33em" /><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>N</mi><mo>.</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>J</mi></mfenced></mrow><annotation encoding="application/x-tex">$\bm{\Omega }_{2}\bm{=}\left(\bm{\Omega }_{2,ij},\text{ }i=1,\ldots,N. j=1,\ldots,J\right)$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Ω</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msup><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>ς</mi><mi>j</mi></msub><mo>,</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>γ</mi><mn>2</mn></msub><mo>,</mo><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>)</mo></mrow><mo>′</mo></msup></mrow><annotation encoding="application/x-tex">$\bm{\Omega }_{2,ij}=(a_{j}, b_{j}, \varsigma _{j},\gamma _{0},\gamma _{1},\gamma _{2}, \bm{\mu },\bm{\Sigma })^{\prime }$</annotation></semantics></math> </ephtml> . <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> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> have a bivariate normal distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi><mi>i</mi></msub><mo>|</mo><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$P(\theta _{i},\tau _{i}|\bm{\mu },\bm{\Sigma })$</annotation></semantics></math> </ephtml> .</p> <p>Spiegelhalter et al. ([<reflink idref="bib48" id="ref70">48</reflink>]) proposed the deviance information criterion (DIC) to evaluate the goodness of fit of a model using Bayesian posterior estimates, while balancing the complexity of the model and the adequacy of its fit. This criterion considers the trade‐off between the model's goodness of fit and the number of its parameters. Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mrow><mo>=</mo></mrow><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Ω</mi><mrow><mn>3</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mspace width="0.33em" /><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>N</mi><mo>.</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>J</mi><mo>.</mo></mfenced></mrow><annotation encoding="application/x-tex">$\bm{\Omega }_{3}\bm{=}\left(\bm{\Omega }_{3,ij}, \text{ }i=1,\ldots,N. j=1,\ldots,J.\right)$</annotation></semantics></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">Ω</mi><mrow><mn>3</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>=</mo><msup><mfenced separators="" open="(" close=")"><msubsup><mi>θ</mi><mi>i</mi><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msubsup><mi>a</mi><mi>j</mi><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msubsup><mi>b</mi><mi>j</mi><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msubsup><mi>ς</mi><mi>j</mi><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msubsup><mi>γ</mi><mn>0</mn><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msubsup><mi>γ</mi><mn>1</mn><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msubsup><mi>γ</mi><mn>2</mn><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msubsup><mi>τ</mi><mi>i</mi><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup><mo>,</mo><msup><mi mathvariant="bold-italic">μ</mi><mfenced open="(" close=")"><mi>m</mi></mfenced></msup><mo>,</mo><msup><mi mathvariant="bold">Σ</mi><mfenced open="(" close=")"><mi>m</mi></mfenced></msup></mfenced><mo>′</mo></msup></mrow><annotation encoding="application/x-tex">$\bm{\Omega }_{3,ij}^{\left(m\right) }=\left(\theta _{i}^{\left(m\right) },a_{j}^{\left(m\right) }, b_{j}^{\left(m\right) },\varsigma _{j}^{\left(m\right) },\gamma _{0}^{\left(m\right) },\gamma _{1}^{\left(m\right) },\gamma _{2}^{\left(m\right) },\tau _{i}^{\left(m\right) },\bm{\mu }^{\left(m\right) },\bm{\Sigma }^{\left(m\right) }\right) ^{\prime }$</annotation></semantics></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">$m=1,\ldots,M$</annotation></semantics></math> </ephtml> , denote an MCMC sample from the posterior distribution. The logarithm for the the joint likelihood function of item response and missing indicator data evaluated at <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi mathvariant="bold">Ω</mi><mn>3</mn><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup><annotation encoding="application/x-tex">$\bm{\Omega }_{3}^{(m) }$</annotation></semantics></math> </ephtml> is denoted by: 10 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mi>P</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">R</mi><mfenced separators="" open="|" close=""><msubsup><mi mathvariant="bold">Ω</mi><mn>3</mn><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup></mfenced></mfenced><mo linebreak="badbreak">=</mo><mi>log</mi><mi>P</mi><mfenced separators="" open="(" close=")"><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><mo>,</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">R</mi><mfenced separators="" open="|" close=""><msubsup><mi mathvariant="bold">Ω</mi><mn>3</mn><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup></mfenced></mfenced><mo linebreak="goodbreak">=</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>log</mi><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="" open="|" close=""><msubsup><mi mathvariant="bold">Ω</mi><mrow><mn>3</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup></mfenced></mfenced><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \log P{\left(\bm{Y},\bm{R}{\left|\bm{\Omega } _{3}^{{\left(m\right)} }\right.} \right)} =\log P{\left(\bm{Y} ^{obs},\bm{Y}^{mis},\bm{R}{\left|\bm{\Omega } _{3}^{{\left(m\right)} }\right.} \right)} =\sum \limits _{i=1}^{N} \sum \limits _{j=1}^{J}\log P{\left(Y_{ij},R_{ij}{\left|\bm{\Omega } _{3,ij}^{{\left(m\right)} }\right.} \right)}, \end{equation}$$</annotation></semantics></math> </ephtml> where 11 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>PYij,RijΩ3,ij=P(Yij=1θi,aj,bj)PRij=1τi,ςj,γ,Rij−1,Yij=1Rij.Yij×P(Yij=0θi,aj,bj)PRij=1τi,ςj,γ,Rij−1,Yij=01−Rij.Yij×P(Yij=1θi,aj,bj)PRij=0τi,ςj,γ,Rij−1,Yij=11−Rij.Yij×P(Yij=0θi,aj,bj)PRij=0τi,ςj,γ,Rij−1,Yij=01−Rij.1−Yij.<annotation encoding="application/x-tex">$$\begin{align} P{\left(Y_{ij},R_{ij}{\left|\bm{\Omega }_{3,ij}\right.} \right)} & ={\left[ P(Y_{ij}=1{\left|\theta _{i},a_{j}, b_{j}\right.})P{\left(R_{ij} =1{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R} _{ij-1},Y_{ij}=1\right.} \right)} \right]} ^{R_{ij}.Y_{ij}}\nonumber \\ & \times {\left[ P(Y_{ij}=0{\left|\theta _{i}, a_{j}, b_{j}\right.})P{\left(R_{ij}=1{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R} _{ij-1},Y_{ij}=0\right.} \right)} \right]} ^{{\left(1-R_{ij}\right)}.Y_{ij} }\nonumber \\ & \times {\left[ P(Y_{ij}=1{\left|\theta _{i},a_{j}, b_{j}\right.})P{\left(R_{ij}=0{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R} _{ij-1},Y_{ij}=1\right.} \right)} \right]} ^{{\left(1-R_{ij}\right)}.Y_{ij} }\nonumber \\ & \times {\left[ P(Y_{ij}=0{\left|\theta _{i},a_{j}, b_{j}\right.})P{\left(R_{ij}=0{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R} _{ij-1},Y_{ij}=0\right.} \right)} \right]} ^{{\left(1-R_{ij}\right)}.{\left(1-Y_{ij}\right)} }. \end{align}$$</annotation></semantics></math> </ephtml> Since the log‐likelihood function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mi>P</mi><mo>(</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><mo>,</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">R</mi><mo>|</mo><msubsup><mi mathvariant="bold">Ω</mi><mn>3</mn><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup><mo>)</mo></mrow><annotation encoding="application/x-tex">$\log P(\bm{Y} ^{obs},\bm{Y}^{mis},\bm{R}|\bm{\Omega } _{3}^{(m)})$</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> , <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> , are readily available from the R outputs, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mi>P</mi><mo>(</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><mo>,</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">R</mi><mo>|</mo><msubsup><mi mathvariant="bold">Ω</mi><mn>3</mn><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup><mo>)</mo></mrow><annotation encoding="application/x-tex">$\log P (\bm{Y} ^{obs},\bm{Y}^{mis},\bm{R}|\bm{\Omega } _{3}^{(m) })$</annotation></semantics></math> </ephtml> in (<reflink idref="bib10" id="ref71">10</reflink>) is easy to compute. The DIC can be calculated as follows: 12 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>DIC</mi><mo linebreak="badbreak">=</mo><mover accent="true"><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>̂</mo></mover><mo linebreak="goodbreak">+</mo><mn>2</mn><msub><mi>P</mi><mi>D</mi></msub><mo linebreak="goodbreak">=</mo><mover accent="true"><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>̂</mo></mover><mo linebreak="goodbreak">+</mo><mn>2</mn><mfenced separators="" open="[" close="]"><mover><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>¯</mo></mover><mo>−</mo><mover accent="true"><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>̂</mo></mover></mfenced><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \mathrm{DIC}=\widehat{\mathrm{D} (\bm{\Omega }_{3})}+2P_{D}=\widehat{\mathrm{D}(\bm{\Omega }_{3})}+2{\left[ \overline{\mathrm{D}(\bm{\Omega }_{3})} -\widehat{\mathrm{D}(\bm{\Omega }_{3})}\right]}, \end{equation}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>D(Ω3)̂=−2max1≤m≤MlogPYobs,Ymis,RΩ3m,D(Ω3)¯=−2M∑m=1MlogPYobs,Ymis,RΩ3m.<annotation encoding="application/x-tex">$$\begin{align*} \widehat{\mathrm{D}(\bm{\Omega }_{3})} & =-2\max _{1\le m\le M}\log P{\left(\bm{Y}^{obs},\bm{Y}^{mis},\bm{R}{\left|\bm{\Omega }_{3}^{{\left(m\right)}}\right.} \right)},\\ \overline{\mathrm{D}(\bm{\Omega }_{3})} & =-\frac{2}{M} \sum \limits _{m=1}^{M}\log P{\left(\bm{Y}^{obs},\bm{Y} ^{mis},\bm{R}{\left|\bm{\Omega }_{3}^{{\left(m\right)} }\right.}\right)}. \end{align*}$$</annotation></semantics></math> </ephtml> Here, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover accent="true"><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>̂</mo></mover><annotation encoding="application/x-tex">$\widehat{\mathrm{D}(\bm{\Omega }_{3})}$</annotation></semantics></math> </ephtml> is an approximation of the deviance function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">D</mi><mrow><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn><mi>log</mi><mi>P</mi><mrow><mo>(</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup><mo>,</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">R</mi><mo>|</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\mathrm{D} (\bm{\Omega }_{3})=-2\log P(\bm{Y}^{obs},\bm{Y}^{mis},\bm{R} |\bm{\Omega }_{3})$</annotation></semantics></math> </ephtml> at the posterior mode <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mrow><mover accent="true"><mi mathvariant="bold">Ω</mi><mo>̂</mo></mover></mrow><mn>3</mn></msub><annotation encoding="application/x-tex">$\bm{\hat{\Omega }}_{3}$</annotation></semantics></math> </ephtml> . <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>¯</mo></mover><annotation encoding="application/x-tex">$\overline{\mathrm{D}(\bm{\Omega }_{3})}$</annotation></semantics></math> </ephtml> is a Monte‐Carlo estimate of the posterior expectation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">$\mathrm{D} (\bm{\Omega }_{3})$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>D</mi></msub><mo>=</mo><mover><mrow><mi>Dev</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>¯</mo></mover><mo>−</mo><mover accent="true"><mrow><mi mathvariant="normal">D</mi><mo>(</mo><msub><mi mathvariant="bold">Ω</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>̂</mo></mover></mrow><annotation encoding="application/x-tex">$P_{D}=\overline{\mathrm{Dev}(\bm{\Omega }_{3})}-\widehat{\mathrm{D}(\bm{\Omega }_{3})}$</annotation></semantics></math> </ephtml> represents the effective number of parameters. The model with a lower DIC value is considered to be a better fit for the data.</p> <p>Another approach to compare the fit of two models is the logarithm of the pseudomarginal likelihood (LPML; Geisser & Eddy, [<reflink idref="bib14" id="ref72">14</reflink>]; Ibrahim et al., [<reflink idref="bib27" id="ref73">27</reflink>]), which involves calculating the conditional predictive ordinates (CPO; Chen et al., [<reflink idref="bib7" id="ref74">7</reflink>]; Gelfand et al., [<reflink idref="bib15" id="ref75">15</reflink>]) index. To compute the CPO, we use a Monte‐Carlo estimate given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mrow><mi>i</mi><mi>j</mi><mo>,</mo><mi>max</mi></mrow></msub><mo>=</mo><munder><mi>max</mi><mrow><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>M</mi></mrow></munder><mrow><mo>{</mo><mo>−</mo><mi>log</mi><mi>P</mi><mrow><mo>(</mo><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msubsup><mi mathvariant="normal">Ω</mi><mrow><mn>3</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup><mo>)</mo></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">$H_{ij,\max }=\underset{1\le m\le M}{\max }\lbrace -\log P(Y_{ij},R_{ij}|\Omega _{3,ij}^{(m) }) \rbrace $</annotation></semantics></math> </ephtml> . Then, we can estimate the CPO using the formula: 13 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mover accent="true"><mrow><mo>(</mo><msub><mi>CPO</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>̂</mo></mover><mo linebreak="badbreak">=</mo><mo>−</mo><msub><mi>H</mi><mrow><mi>i</mi><mi>j</mi><mo>,</mo><mi>max</mi></mrow></msub><mo linebreak="goodbreak">−</mo><mi>log</mi><mfenced separators="" open="[" close="]"><mfrac><mn>1</mn><mi>M</mi></mfrac><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mi>exp</mi><mfenced separators="" open="{" close="}"><mo linebreak="badbreak">−</mo><mi>log</mi><mi>P</mi><mfenced separators="" open="(" close=")"><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="" open="|" close=""><msubsup><mi mathvariant="normal">Ω</mi><mrow><mn>3</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow><mfenced open="(" close=")"><mi>m</mi></mfenced></msubsup></mfenced></mfenced><mo>−</mo><msub><mi>H</mi><mrow><mi>i</mi><mi>j</mi><mo>,</mo><mi>max</mi></mrow></msub></mfenced></mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \log \widehat{(\mathrm{CPO}_{ij})}=-H_{ij,\max }-\log {\left[\frac{1}{M} \sum \limits _{m=1}^{M}\exp {\left\lbrace -\log P{\left(Y_{ij},R_{ij}{\left|\Omega _{3,ij}^{{\left(m\right)}}\right.}\right)}-H_{ij,\max }\right\rbrace} \right]}. \end{equation}$$</annotation></semantics></math> </ephtml> In Equation 13, it is worth noting that adjusting the maximum value in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mover accent="true"><mrow><mo>(</mo><mi>CPO</mi><msub><mrow /><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>̂</mo></mover></mrow><annotation encoding="application/x-tex">$\log \widehat{(\mathrm{CPO}{_{ij}})}$</annotation></semantics></math> </ephtml> is crucial to ensure numerical stability while calculating <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>exp</mi><mo>{</mo><mo>−</mo><mi>log</mi><mi>P</mi><mrow><mo>(</mo><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>R</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msubsup><mi mathvariant="normal">Ω</mi><mrow><mn>3</mn><mo>,</mo><mi>i</mi><mi>j</mi></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup><mo>)</mo></mrow><mo>−</mo><msub><mi>H</mi><mrow><mi>i</mi><mi>j</mi><mo>,</mo><mi>max</mi></mrow></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">$\exp \lbrace -\log P(Y_{ij},R_{ij}|\Omega _{3,ij}^{(m)}) -H_{ij,\max }\rbrace $</annotation></semantics></math> </ephtml> . The sum of the logarithms of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover accent="true"><msub><mi>CPO</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>̂</mo></mover><annotation encoding="application/x-tex">$\widehat{\mathrm{CPO}_{ij}}$</annotation></semantics></math> </ephtml> , known as the LPML, is a summary statistic. It can be expressed as 14 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>LPML</mi><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>log</mi><mover accent="true"><mrow><mo>(</mo><msub><mi>CPO</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>̂</mo></mover><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation} \mathrm{LPML}=\sum \limits _{i=1}^{N} \sum \limits _{j=1}^{J}\log \widehat{(\mathrm{CPO}_{ij})}. \end{equation}$$</annotation></semantics></math> </ephtml> The goodness of fit of the model can be determined by evaluating the value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>LPML</mi><annotation encoding="application/x-tex">$\mathrm{LPML}$</annotation></semantics></math> </ephtml> , where a higher value indicates a better fit to the data.</p> <hd id="AN0192629997-12">Posterior Predictive Probability</hd> <p>Posterior predictive model checking (PPMC; Gelman et al., [<reflink idref="bib17" id="ref76">17</reflink>]) was employed to evaluate the goodness of fit between model and data. Posterior predictive probability (PPP) values approximately equal to 0.5 imply that there are no substantial systematic differences between the real and predicted values, thereby suggesting that the model fits the data adequately. In this study, we evaluated the model fit to the data separately for both the item response model and the missing model. First, we evaluate the fit of the item response model. The sum of the squared Pearson residuals for the IRT model with respect to the examinee <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> and 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> is <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>DY∼,Ξ=∑i=1N∑j=1JDY∼ij;θi,aj,bj=∑i=1N∑j=1JY∼ij−PY∼ij=1θi,aj,bjPY∼ij=1θi,aj,bj1−PY∼ij=1θi,aj,bj2.<annotation encoding="application/x-tex">$$\begin{align*} D{\left(\widetilde{\bm{Y}},\bm{\Xi }\right)} & = {\displaystyle \sum \limits _{i=1}^{N}} {\displaystyle \sum \limits _{j=1}^{J}} D{\left(\widetilde{Y}_{ij};\theta _{i},a_{j},b_{j}\right)} \\ & = {\displaystyle \sum \limits _{i=1}^{N}} {\displaystyle \sum \limits _{j=1}^{J}} {\left(\frac{\widetilde{Y}_{ij}-P{\left(\widetilde{Y}_{ij}=1{\left|\theta _{i},a_{j},b_{j}\right.} \right)} }{\sqrt {P{\left(\widetilde{Y} _{ij}=1{\left|\theta _{i},a_{j},b_{j}\right.} \right)} {\left[ 1-P{\left(\widetilde{Y}_{ij}=1{\left|\theta _{i},a_{j},b_{j}\right.} \right)} \right]} }}\right)} ^{2}. \end{align*}$$</annotation></semantics></math> </ephtml> This discrepancy measure, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>(</mo><mover accent="true"><mi mathvariant="bold-italic">Y</mi><mo>∼</mo></mover><mo>,</mo><mrow><mi mathvariant="bold">Ξ</mi></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$D(\widetilde{\bm{Y}},\bm{\Xi })$</annotation></semantics></math> </ephtml> , is used to assess the overall fit of the item response model (Almond et al., 2015; Yan et al., 2003). Here, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover accent="true"><mi mathvariant="bold-italic">Y</mi><mo>∼</mo></mover><annotation encoding="application/x-tex">$\widetilde{\bm{Y}}$</annotation></semantics></math> </ephtml> is the observed response data and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold">Ξ</mi><mo>=</mo></mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">b</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">$\bm{\Xi =}(\bm{\theta },\bm{a},\bm{b})$</annotation></semantics></math> </ephtml> represents the unknown parameters in terms of the 2PL model. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mover accent="true"><mi mathvariant="bold-italic">Y</mi><mo>∼</mo></mover><mrow><mi>r</mi><mi>e</mi><mi>p</mi></mrow></msup><annotation encoding="application/x-tex">$\widetilde{\bm{Y}}^{rep}$</annotation></semantics></math> </ephtml> is the predictive value of the item response data, derived from the predictive distribution for each simulated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Ξ</mi></mrow><annotation encoding="application/x-tex">$\bm{\Xi }$</annotation></semantics></math> </ephtml> . Thus, if we have <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> simulations from the posterior density of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Ξ</mi></mrow><annotation encoding="application/x-tex">$\bm{\Xi }$</annotation></semantics></math> </ephtml> , we draw one <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mover accent="true"><mi mathvariant="bold-italic">Y</mi><mo>∼</mo></mover><mrow><mi>r</mi><mi>e</mi><mi>p</mi><mo>.</mo><mi>m</mi></mrow></msup><annotation encoding="application/x-tex">$\widetilde{\bm{Y}}^{rep.m}$</annotation></semantics></math> </ephtml> from the predictive distribution for the <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> th simulated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold">Ξ</mi><mi>m</mi></msup><annotation encoding="application/x-tex">$\bm{\Xi }^{m}$</annotation></semantics></math> </ephtml> , thereby obtaining PPP value based on the observed item response data is as follows: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>P</mi><msub><mi>P</mi><mi>Y</mi></msub><mo linebreak="badbreak">=</mo><mi>P</mi><mfenced separators="" open="(" close=")"><mi>D</mi><mfenced separators="" open="(" close=")"><mover accent="true"><mi mathvariant="bold-italic">Y</mi><mo>∼</mo></mover><mo>,</mo><msup><mi mathvariant="bold">Ξ</mi><mi>m</mi></msup></mfenced><mo>≤</mo><mi>D</mi><mfenced separators="" open="(" close=")"><msup><mover accent="true"><mi mathvariant="bold-italic">Y</mi><mo>∼</mo></mover><mrow><mi>r</mi><mi>e</mi><mi>p</mi><mo>.</mo><mi>m</mi></mrow></msup><mo>,</mo><msup><mi mathvariant="bold">Ξ</mi><mi>m</mi></msup></mfenced></mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} PPP_{Y}=P{\left(D{\left(\widetilde{\bm{Y}},\bm{\Xi } ^{m}\right)} \le D{\left(\widetilde{\bm{Y}}^{rep.m},\bm{\Xi }^{m}\right)} \right)}. \end{equation*}$$</annotation></semantics></math> </ephtml> Next, we evaluate the fit of the missing data model. The sum of the squared Pearson residuals for the 1PLMDP model with respect to the examinee <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> and 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> is <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>DR,Π=∑i=1N∑j=1JDRij;τi,ςj,γ,Rij−1,Yij=∑i=1N∑j=1JRij−PRij=1τi,ςj,γ,Rij−1,YijPRij=1τi,ςj,γ,Rij−1,Yij1−PRij=1τi,ςj,γ,Rij−1,Yij2,<annotation encoding="application/x-tex">$$\begin{align*} D{\left(\bm{R},\bm{\Pi }\right)} & = {\displaystyle \sum \limits _{i=1}^{N}} {\displaystyle \sum \limits _{j=1}^{J}} D{\left(R_{ij};\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R} _{ij-1},\bm{Y}_{ij}\right)} \\ & = {\displaystyle \sum \limits _{i=1}^{N}} {\displaystyle \sum \limits _{j=1}^{J}} {\left(\frac{R_{ij}-P{\left(R_{ij}=1{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R}_{ij-1},\bm{Y}_{ij}\right.} \right)} }{\sqrt {P{\left(R_{ij}=1{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R}_{ij-1},\bm{Y}_{ij}\right.} \right)} {\left[ 1-P{\left(R_{ij}=1{\left|\tau _{i},\varsigma _{j},\bm{\gamma },\bm{R}_{ij-1},\bm{Y}_{ij}\right.} \right)} \right]} }}\right)} ^{2}, \end{align*}$$</annotation></semantics></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>(</mo><mi mathvariant="bold-italic">R</mi><mo>,</mo><mrow><mi mathvariant="bold">Π</mi></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$D(\bm{R},\bm{\Pi })$</annotation></semantics></math> </ephtml> is utilized as a discrepancy measure to assess overall fit of the missing model. Here, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">R</mi><annotation encoding="application/x-tex">$\bm{R}$</annotation></semantics></math> </ephtml> is the observed missing indicator and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold">Π</mi><mo>=</mo></mrow><mo>(</mo><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">ς</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>,</mo><msup><mi mathvariant="bold-italic">Y</mi><mrow><mi>m</mi><mi>i</mi><mi>s</mi></mrow></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">$\bm{\Pi =}(\bm{\tau },\bm{\varsigma },\bm{\gamma },\bm{Y}^{mis})$</annotation></semantics></math> </ephtml> represents the unknown parameters in terms of the 1PLMDP model. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">R</mi><mrow><mi>r</mi><mi>e</mi><mi>p</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{R}^{rep}$</annotation></semantics></math> </ephtml> is the predictive value of the missing indicator, derived from the predictive distribution for each simulated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Π</mi></mrow><annotation encoding="application/x-tex">$\bm{\Pi }$</annotation></semantics></math> </ephtml> . Thus, if we have <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> simulations from the posterior density of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Π</mi></mrow><annotation encoding="application/x-tex">$\bm{\Pi }$</annotation></semantics></math> </ephtml> , we draw one <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold-italic">R</mi><mrow><mi>r</mi><mi>e</mi><mi>p</mi><mo>,</mo><mi>m</mi></mrow></msup><annotation encoding="application/x-tex">$\bm{R}^{rep,m}$</annotation></semantics></math> </ephtml> from the predictive distribution for the <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> th simulated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi mathvariant="bold">Π</mi><mi>m</mi></msup><annotation encoding="application/x-tex">$\bm{\Pi }^{m}$</annotation></semantics></math> </ephtml> , thereby obtaining PPP value based on the missing indicator is as follows: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>P</mi><msub><mi>P</mi><mi>R</mi></msub><mo linebreak="badbreak">=</mo><mi>P</mi><mfenced separators="" open="(" close=")"><mi>D</mi><mfenced separators="" open="(" close=")"><mi mathvariant="bold-italic">R</mi><mo>,</mo><msup><mi mathvariant="bold">Π</mi><mi>m</mi></msup></mfenced><mo>≤</mo><mi>D</mi><mfenced separators="" open="(" close=")"><msup><mi mathvariant="bold-italic">R</mi><mrow><mi>r</mi><mi>e</mi><mi>p</mi><mo>,</mo><mi>m</mi></mrow></msup><mo>,</mo><msup><mi mathvariant="bold">Π</mi><mi>m</mi></msup></mfenced></mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} PPP_{R}=P{\left(D{\left(\bm{R},\bm{\Pi }^{m}\right)} \le D{\left(\bm{R}^{rep,m},\bm{\Pi }^{m}\right)} \right)}. \end{equation*}$$</annotation></semantics></math> </ephtml></p> <hd id="AN0192629997-13">Simulation Studies</hd> <p></p> <hd id="AN0192629997-14">Simulation Study 1</hd> <p>This simulation study aims to assess the performance of the Bayesian sampling algorithm to recover parameters under different simulation conditions.</p> <hd1 id="AN0192629997-15">Simulation Designs</hd1> <p>We consider the following factors: (a) test length, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi><mo>=</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">$J=40$</annotation></semantics></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi><mo>=</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">$J=60$</annotation></semantics></math> </ephtml> ; and (b) sample size, <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> or <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> . By fully crossing these factors, we obtain four simulation conditions. In the 2PLIRT model, we generate true values of <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> from a normal distribution truncated at zero, i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mi mathvariant="normal">I</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mi>∞</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$a_{j}\sim N(0,1) \mathrm{I}(0,+\infty)$</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><mn>2</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">$j=1,2,\ldots,J$</annotation></semantics></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">I</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">$\mathrm{I}(A)$</annotation></semantics></math> </ephtml> is an indicator function. <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>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$b_{j}\sim N(0,1)$</annotation></semantics></math> </ephtml> . In the 1PLMDP model, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ς</mi><mi>j</mi></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\varsigma _{j}\sim N(0,1)$</annotation></semantics></math> </ephtml> . The specific true values of the item parameters are presented in the Online Supplementary Material. The true values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>0</mn></msub><annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>2</mn></msub><annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math> </ephtml> are set to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$-2$</annotation></semantics></math> </ephtml> , 0.02, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>0.2</mn></mrow><annotation encoding="application/x-tex">$-0.2$</annotation></semantics></math> </ephtml> , respectively. In order to represent the correlation between persons under the MNAR missing mechanism, the true values of person parameters are generated from a bivariate normal distribution, that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo>(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>′</mo></msup><mo>∼</mo><mi>M</mi><mi>V</mi><mi>N</mi><mfenced separators="" open="(" close=")"><mrow><mn mathvariant="bold">0</mn></mrow><mo>,</mo><mspace width="0.33em" /><mfenced separators="" open="(" close=")">10.50.51</mfenced></mfenced></mrow><annotation encoding="application/x-tex">$(\theta _{i},\tau _{i}) ^{\prime }\sim MVN\left(\bm{0},\text{ }\left(\def\eqcellsep{&}\begin{array}{cc} 1 & 0.5\\ 0.5 & 1 \end{array} \right) \right)$</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><mn>2</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">$i=1,2,\ldots,N$</annotation></semantics></math> </ephtml> . The proportions of missing responses for each simulation condition is presented in Table 1. We use noninformative prior distributions for all model parameters, specifically, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mn>10</mn><mn>5</mn></msup><mo>)</mo></mrow><mi mathvariant="normal">I</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mi>∞</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$a_j \sim N(0, 10^5)\mathrm{I}(0,+\infty)$</annotation></semantics></math> </ephtml> , <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>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mn>10</mn><mn>5</mn></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$b_j \sim N(0, 10^5)$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ς</mi><mi>j</mi></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mn>10</mn><mn>5</mn></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\varsigma _j \sim N(0, 10^5)$</annotation></semantics></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">$j=1,2,\ldots,J$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>0</mn></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mn>10</mn><mn>5</mn></msup><mo>)</mo></mrow><mi mathvariant="normal">I</mi><mrow><mo>(</mo><msub><mi>γ</mi><mn>0</mn></msub><mo><</mo><mn>0</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _0 \sim N(0, 10^5)\mathrm{I}(\gamma _0<0)$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>1</mn></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mn>10</mn><mn>5</mn></msup><mo>)</mo></mrow><mi mathvariant="normal">I</mi><mrow><mo>(</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>></mo><mn>0</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _1 \sim N(0, 10^5)\mathrm{I}(\gamma _1>0)$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mn>10</mn><mn>5</mn></msup><mo>)</mo></mrow><mi mathvariant="normal">I</mi><mrow><mo>(</mo><msub><mi>γ</mi><mn>2</mn></msub><mo><</mo><mn>0</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _2 \sim N(0, 10^5)\mathrm{I}(\gamma _2<0)$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>σ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mn>10</mn><mn>5</mn></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\sigma _{\theta \tau }\sim N(0,10^5)$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mi>τ</mi><mn>2</mn></msubsup><mo>∼</mo><mi>Inv</mi></mrow><annotation encoding="application/x-tex">$\sigma _{\tau }^{2}\sim \mathrm{Inv}$</annotation></semantics></math> </ephtml> ‐ <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>a</mi><mi>m</mi><mi>m</mi><mi>a</mi><mo>(</mo><mn>0.0001</mn><mo>,</mo><mn>0.0001</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">$Gamma(0.0001,0.0001)$</annotation></semantics></math> </ephtml> . We perform 50 replications for each simulation condition.</p> <p>1 Table The Proportions of Missing Responses in Simulation Study 1</p> <p> <ephtml> <table><thead><tr><th align="center">The Proportions of Missing Responses</th></tr><tr><th align="center">Person 500</th><th align="center">Person 1,000</th></tr></thead><tbody><tr><td>Test length 40</td><td>Test length 60</td><td>Test length 40</td><td>Test length 60</td></tr><tr><td>18.2%</td><td>16.7%</td><td>15.4%</td><td>17.3%</td></tr></tbody></table> </ephtml> </p> <hd1 id="AN0192629997-16">Bayesian Convergence Assessments</hd1> <p>Due to space limitations, we present the convergence analysis for the condition with a test length of 40 and a sample size of 500. Two methods were used for convergence evaluation: (<reflink idref="bib1" id="ref77">1</reflink>) visual inspection of the historical plots of the generated sequences and (<reflink idref="bib2" id="ref78">2</reflink>) the Gelman‐Rubin method (Brooks & Gelman, [<reflink idref="bib6" id="ref79">6</reflink>]; Gelman & Rubin, [<reflink idref="bib18" id="ref80">18</reflink>]) to check the convergence of the estimated parameters. We rigorously monitored the trace plots of parameters across consecutive sequences of 20,000 iterations, with the initial 10,000 iterations serving as the burn‐in period. For each replication, we executed four chains, using overdispersed starting values. The trace plots of three randomly selected items, including fixed effects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">γ</mi></mrow><annotation encoding="application/x-tex">$\bm{\gamma }$</annotation></semantics></math> </ephtml> and variance‐covariance parameters ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mi>τ</mi><mn>2</mn></msubsup><mo>,</mo><msub><mi>σ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">$\sigma _{\tau }^{2}, \sigma _{\theta \tau }$</annotation></semantics></math> </ephtml> ), are depicted in Figure S1. Additionally, the posterior histograms of item parameters, randomly chosen based on 10,000 simulated values following 10,000 burn‐in iterations, are illustrated in Figure S2. Moreover, all parameters demonstrate potential scale reduction factor (PSRF; Brooks & Gelman, [<reflink idref="bib6" id="ref81">6</reflink>]) values of less than 1.1, indicating that all chains have converged as expected. The PSRF values for all item parameters are presented in Figure S3.</p> <hd1 id="AN0192629997-17">Item Parameters Recovery</hd1> <p>To evaluate the accuracy of parameter estimates, five criteria were employed, namely Bias, Mean Squared Error (MSE), Standard Error (SE), Standard Deviation (SD), and the coverage probability (CP) of the 95% Highest Posterior Density (HPD) intervals. The interested parameter was denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>η</mi><annotation encoding="application/x-tex">$\eta$</annotation></semantics></math> </ephtml> , and a total of 50 datasets were generated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mi>R</mi><mo>=</mo><mn>50</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">$(R=50)$</annotation></semantics></math> </ephtml> . For each simulated dataset <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext>...</mtext><mo>,</mo><mi>R</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">$(r=1,\ldots,R)$</annotation></semantics></math> </ephtml> , the posterior mean and posterior standard deviation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>η</mi><annotation encoding="application/x-tex">$\eta$</annotation></semantics></math> </ephtml> were estimated as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mover accent="true"><mi>η</mi><mo>̂</mo></mover><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msup><annotation encoding="application/x-tex">$\widehat{\eta }^{(r)}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>SD</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\mathrm{SD}^{(r)}(\eta)$</annotation></semantics></math> </ephtml> , respectively.</p> <p>Bias of parameter is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Bias</mi><mfenced open="(" close=")"><mi>η</mi></mfenced><mo linebreak="badbreak">=</mo><mfrac><mn>1</mn><mi>R</mi></mfrac><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></munderover><mfenced separators="" open="(" close=")"><msup><mover accent="true"><mi>η</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>r</mi></mfenced></msup><mo>−</mo><mi>η</mi></mfenced><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} \mathrm{Bias}{\left(\eta \right)} =\frac{1}{R}\sum \limits _{r=1}^{R}{\left(\widehat{\eta }^{{\left(r\right)} }-\eta \right)}, \end{equation*}$$</annotation></semantics></math> </ephtml> and MSE of parameter is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>MSE</mi><mfenced open="(" close=")"><mi>η</mi></mfenced><mo linebreak="badbreak">=</mo><mfrac><mn>1</mn><mi>R</mi></mfrac><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></munderover><msup><mfenced separators="" open="(" close=")"><msup><mover accent="true"><mi>η</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>r</mi></mfenced></msup><mo>−</mo><mi>η</mi></mfenced><mn>2</mn></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} \mathrm{MSE}{\left(\eta \right)} =\frac{1}{R}\sum \limits _{r=1}^{R}{\left(\widehat{\eta }^{{\left(r\right)} }-\eta \right)} ^{2}. \end{equation*}$$</annotation></semantics></math> </ephtml> SE can be defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Simulation</mi><mspace width="0.33em" /><mi>SE</mi><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow><mo linebreak="badbreak">=</mo><msqrt><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><msubsup><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></msubsup><msup><mfenced separators="" open="(" close=")"><msup><mover accent="true"><mi>η</mi><mo>̂</mo></mover><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msup><mo>−</mo><mfrac><mn>1</mn><mi>R</mi></mfrac><msubsup><mo>∑</mo><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></msubsup><msup><mover accent="true"><mi>η</mi><mo>̂</mo></mover><mrow><mo>(</mo><mi>ℓ</mi><mo>)</mo></mrow></msup></mfenced><mn>2</mn></msup></mrow></msqrt><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} \mathrm{Simulation\ SE}(\eta)=\sqrt {\frac{1}{R}\sum _{r=1}^{R}{\left(\widehat{\eta }^{(r)}-\frac{1}{R}\sum _{\ell =1}^{R}\widehat{\eta }^{(\ell)}\right)} ^{2}}, \end{equation*}$$</annotation></semantics></math> </ephtml> and SD can be defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>SD</mi><mfenced open="(" close=")"><mi>η</mi></mfenced><mo linebreak="badbreak">=</mo><mfrac><mn>1</mn><mi>R</mi></mfrac><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></munderover><msup><mi>SD</mi><mfenced open="(" close=")"><mi>r</mi></mfenced></msup><mfenced open="(" close=")"><mi>η</mi></mfenced><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} \mathrm{SD}{\left(\eta \right)} =\frac{1}{R}\sum \limits _{r=1}^{R} \mathrm{SD}^{{\left(r\right)} }{\left(\eta \right)}. \end{equation*}$$</annotation></semantics></math> </ephtml> CP can be defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>CP</mi><mfenced open="(" close=")"><mi>η</mi></mfenced><mo linebreak="badbreak">=</mo><mfrac><mrow><mo>#</mo><mspace width="0.33em" /><mtext>of</mtext><mspace width="0.33em" /><mn>95</mn><mo>%</mo><mspace width="0.33em" /><mtext>HPD intervals containing</mtext><mspace width="0.33em" /><mi>η</mi><mspace width="0.33em" /><mtext>in</mtext><mspace width="0.33em" /><mi>R</mi><mspace width="0.33em" /><mtext>simulated data sets</mtext></mrow><mi>R</mi></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation*} \mathrm{CP}{\left(\eta \right)} =\frac{\# \text{ of }95\% \text{ HPD intervals containing }\eta \mbox{ in }R\mbox{ simulated data sets}}{R}. \end{equation*}$$</annotation></semantics></math> </ephtml></p> <p>Table 2 presents the average Bias, MSE, SD, SE, and CP for the item parameters, fixed effect parameters, and variance‐covariance parameters under four simulation conditions. Several conclusions can be drawn from these results. First, by maintaining a constant total test length and increasing the number of examinees from 500 to 1,000, the average values of MSE, SD, and SE for the discrimination, difficulty, and missing difficulty parameters decrease. Second, when the number of items is fixed, the accuracy of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>σ</mi><mi>τ</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math> </ephtml> increases as the number of examinees increases. Third, the average CP of the discrimination, difficulty, missing difficulty, fixed effect, and variance‐covariance parameters is approximately 0.9500 across all simulation conditions. Fourth, the stability of the Bayesian algorithm is demonstrated by the recovery results of discrimination, difficulty, and missing difficulty parameters being similar when the number of examinees is fixed at 500 (<reflink idref="bib1" id="ref82">1</reflink>,000) and the number of items is fixed at 40, compared to the scenario where the total test length is 60. This implies that the Bayesian algorithm can provide accurate estimates of item parameters, fixed effect parameters, and variance‐covariance parameters, regardless of the number of examinees and items. In summary, the Bayesian sampling algorithm is reliable and consistent across different simulation conditions.</p> <p>2 Table Assessment of the Accuracy of Estimating Item Parameters across Four Simulation Conditions in Simulation Study 1</p> <p> <ephtml> <table><thead><tr><th /><th align="center">Test Length 40</th></tr><tr><th /><th align="center">Person 500</th><th align="center">Person 1,000</th></tr><tr><th>Parameter</th><th align="center">Bias</th><th align="center">MSE</th><th align="center">SD</th><th align="center">SE</th><th align="center">CP</th><th align="center">Bias</th><th align="center">MSE</th><th align="center">SD</th><th align="center">SE</th><th align="center">CP</th></tr></thead><tbody><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math></p></td><td>0.0055</td><td>0.0310</td><td>0.1475</td><td>0.1638</td><td>0.9896</td><td>−0.0497</td><td>0.0166</td><td>0.0945</td><td>0.1101</td><td>0.9815</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math></p></td><td>−0.0557</td><td>0.0326</td><td>0.1403</td><td>0.1551</td><td>0.9866</td><td>−0.0444</td><td>0.0282</td><td>0.1108</td><td>0.1384</td><td>0.9846</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">ς</mi><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math></p></td><td>0.0014</td><td>0.0330</td><td>0.1390</td><td>0.1386</td><td>0.9220</td><td>0.1962</td><td>0.0232</td><td>0.1085</td><td>0.1171</td><td>0.9572</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>0<annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math></p></td><td>−0.0369</td><td>0.0019</td><td>0.0132</td><td>0.0235</td><td>0.9325</td><td>−0.2005</td><td>0.0405</td><td>0.0174</td><td>0.0192</td><td>0.9420</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>1<annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math></p></td><td>−0.0078</td><td>0.0001</td><td>0.0019</td><td>0.0003</td><td>0.9545</td><td>−0.0083</td><td>0.0001</td><td>0.0015</td><td>0.0003</td><td>0.9738</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>2<annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math></p></td><td>0.0633</td><td>0.0040</td><td>0.0291</td><td>0.0020</td><td>0.9486</td><td>0.0893</td><td>0.0079</td><td>0.0080</td><td>0.0002</td><td>0.9682</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>θ</mi><mi>τ</mi><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math></p></td><td>−0.0964</td><td>0.0099</td><td>0.0497</td><td>0.0258</td><td>0.9542</td><td>−0.1057</td><td>0.0114</td><td>0.0354</td><td>0.0157</td><td>0.9745</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>τ</mi>2<annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math></p></td><td>0.0146</td><td>0.0002</td><td>0.0652</td><td>0.0023</td><td>0.9664</td><td>0.0076</td><td>0.0001</td><td>0.0459</td><td>0.0017</td><td>0.9462</td></tr></tbody></table> </ephtml> </p> <p></p> <p> <ephtml> <table><thead><tr><th /><th align="center">Test Length 60</th></tr><tr><th /><th align="center">Person 500</th><th align="center">Person 1,000</th></tr><tr><th>Parameter</th><th align="center">Bias</th><th align="center">MSE</th><th align="center">SD</th><th align="center">SE</th><th align="center">CP</th><th align="center">Bias</th><th align="center">MSE</th><th align="center">SD</th><th align="center">SE</th><th align="center">CP</th></tr></thead><tbody><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math></p></td><td>−0.0078</td><td>0.0262</td><td>0.1447</td><td>0.1520</td><td>0.9453</td><td>−0.0483</td><td>0.0172</td><td>0.0927</td><td>0.1098</td><td>0.9625</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math></p></td><td>−0.0315</td><td>0.0287</td><td>0.1385</td><td>0.1521</td><td>0.9616</td><td>−0.0163</td><td>0.0230</td><td>0.1046</td><td>0.1320</td><td>0.9843</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">ς</mi><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math></p></td><td>0.1221</td><td>0.0436</td><td>0.1436</td><td>0.1487</td><td>0.9811</td><td>0.0212</td><td>0.0281</td><td>0.1040</td><td>0.1092</td><td>0.9743</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>0<annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math></p></td><td>−0.0789</td><td>0.0076</td><td>0.0098</td><td>0.0374</td><td>0.9042</td><td>−0.0530</td><td>0.0029</td><td>0.0078</td><td>0.0115</td><td>0.9458</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>1<annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math></p></td><td>−0.0046</td><td>0.0000</td><td>0.0039</td><td>0.0033</td><td>0.9462</td><td>−0.0092</td><td>0.0001</td><td>0.0006</td><td>0.0002</td><td>0.9574</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>2<annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math></p></td><td>−0.0827</td><td>0.0068</td><td>0.1755</td><td>0.0735</td><td>0.9446</td><td>0.0802</td><td>0.0064</td><td>0.0156</td><td>0.0008</td><td>0.9680</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>θ</mi><mi>τ</mi><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math></p></td><td>−0.0828</td><td>0.0072</td><td>0.0486</td><td>0.0205</td><td>0.9740</td><td>−0.0689</td><td>0.0048</td><td>0.0330</td><td>0.0119</td><td>0.9348</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>τ</mi>2<annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math></p></td><td>0.0141</td><td>0.0002</td><td>0.0651</td><td>0.0022</td><td>0.9584</td><td>0.0071</td><td>0.0001</td><td>0.0454</td><td>0.0022</td><td>0.9552</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. The Bias, MSE, SD, SE and CP indicate the mean values of Bias, MSE, SD, SE and CP, respectively, for the parameters. The vector <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> refers to all discrimination parameters in the item response model, the vector <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math> </ephtml> refers to all difficulty parameters in the item response model, and the vector <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ς</mi></mrow><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math> </ephtml> denotes all difficulty parameters in the missing‐process model.</p> <hd1 id="AN0192629997-18">Person Parameters Recovery</hd1> <p>We assess the accuracy of estimating the person parameters using five evaluation criteria, as presented in Table 3. The results demonstrate that increasing the number of items from 40 to 60 led to a decrease in the average MSE, SD, and SE for both ability and missing propensity parameters, while the average CP of the ability parameters remains around 0.9500 across all simulation conditions. These findings provide further evidence of the effectiveness and accuracy for the Bayesian sampling algorithm in estimating the parameters under different simulation conditions.</p> <p>3 Table Assessment of the Accuracy of Estimating Person Parameters across Four Simulation Conditions in Simulation Study 1</p> <p> <ephtml> <table><thead><tr><th>Test Length</th><th align="center">Parameter</th><th align="center">Person</th><th align="center">Bias</th><th align="center">MSE</th><th align="center">SD</th><th align="center">SE</th><th align="center">CP</th></tr></thead><tbody><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">θ</mi><annotation encoding="application/x-tex">$\bm{\theta }$</annotation></semantics></math></p></td><td>500</td><td>−0.0311</td><td>0.1313</td><td>0.3507</td><td>0.3259</td><td>0.9411</td></tr><tr><td>40</td><td /><td>1,000</td><td>0.0072</td><td>0.1471</td><td>0.3939</td><td>0.3550</td><td>0.9542</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">τ</mi><annotation encoding="application/x-tex">$\bm{\tau }$</annotation></semantics></math></p></td><td>500</td><td>−0.0287</td><td>0.2024</td><td>0.4508</td><td>0.3659</td><td>0.9494</td></tr><tr><td /><td /><td>1,000</td><td>−0.0352</td><td>0.2290</td><td>0.4520</td><td>0.3503</td><td>0.9437</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">θ</mi><annotation encoding="application/x-tex">$\bm{\theta }$</annotation></semantics></math></p></td><td>500</td><td>−0.0311</td><td>0.0906</td><td>0.2962</td><td>0.2763</td><td>0.9515</td></tr><tr><td>60</td><td /><td>1000</td><td>0.0072</td><td>0.1000</td><td>0.3205</td><td>0.2999</td><td>0.9520</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">τ</mi><annotation encoding="application/x-tex">$\bm{\tau }$</annotation></semantics></math></p></td><td>500</td><td>−0.0287</td><td>0.1554</td><td>0.3915</td><td>0.3319</td><td>0.9428</td></tr><tr><td /><td /><td>1,000</td><td>−0.0352</td><td>0.1532</td><td>0.3664</td><td>0.3130</td><td>0.9388</td></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note</emph>. The Bias, MSE, SD, SE, and CP values pertaining to the person parameters indicate the respective average values of Bias, MSE, SD, SE, and CP.</p> <p>We also conducted an additional simulation study A1, which explores the sensitivity of the proposed slice algorithm to various priors for fixed effects and item parameters, please see simulation study A1 in the Online Supplementary Material.</p> <hd id="AN0192629997-19">Simulation Study 2</hd> <p>This simulation study 2 evaluates the performance of the proposed missing mechanism models based on three factors using Bayesian sampling algorithm.</p> <hd1 id="AN0192629997-20">Simulation Designs</hd1> <p>For this simulation study, we fixed the number of examinees and items at 500 and 40, respectively, and used the same true values for item parameters as in simulation study 1 for both the 2PL model and the 1PLMDP model. Three factors were considered: First, correction coefficient, (<reflink idref="bib1" id="ref83">1</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0$</annotation></semantics></math> </ephtml> (implying that <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> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> are independent), (<reflink idref="bib2" id="ref84">2</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math> </ephtml> (weak correlation), (<reflink idref="bib3" id="ref85">3</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math> </ephtml> (moderate correlation), and (<reflink idref="bib4" id="ref86">4</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math> </ephtml> (strong correlation). Second, proportion of missing data, by adjusting the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">γ</mi></mrow><annotation encoding="application/x-tex">$\bm{\gamma }$</annotation></semantics></math> </ephtml> parameters in the 1PLMDP model, different proportions of item nonresponses can be obtained: (<reflink idref="bib1" id="ref87">1</reflink>) Low missing proportion (LMP) where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">$\gamma _{0}=-3$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>1</mn></msub><mo>=</mo><mn>0.01</mn></mrow><annotation encoding="application/x-tex">$\gamma _{1}=0.01$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mn>0.1</mn></mrow><annotation encoding="application/x-tex">$\gamma _{2}=-0.1$</annotation></semantics></math> </ephtml> ; the LMP under various simulation conditions is approximately 9.1%. (<reflink idref="bib2" id="ref88">2</reflink>) High missing proportion (HMP) where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$\gamma _{0}=-2$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>1</mn></msub><mo>=</mo><mn>0.02</mn></mrow><annotation encoding="application/x-tex">$\gamma _{1}=0.02$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mn>0.2</mn></mrow><annotation encoding="application/x-tex">$\gamma _{2}=-0.2$</annotation></semantics></math> </ephtml> . The HMP under various simulation conditions is approximately 18.1%. Third, the missing data mechanism: (<reflink idref="bib1" id="ref89">1</reflink>) MAR; (<reflink idref="bib2" id="ref90">2</reflink>) MNAR. Additionally, the true values of <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> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">$\tau$</annotation></semantics></math> </ephtml> were generated from a bivariate normal distribution with mean vector <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">$\bm{0}$</annotation></semantics></math> </ephtml> , variances of 1, and a correlation coefficient of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><annotation encoding="application/x-tex">$\rho _{\theta \tau }$</annotation></semantics></math> </ephtml> . The Bayesian sampling algorithm is run for 20,000 iterations, with the first 10,000 iterations discarded as burn‐in. Each simulation condition is replicated 50 times. The PSRF values for all parameters in each replication condition are less than 1.1.</p> <p>Table 4 displays the performance of model parameters <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> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ζ</mi></mrow><annotation encoding="application/x-tex">$\bm{\zeta }$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">γ</mi></mrow><annotation encoding="application/x-tex">$\bm{\gamma }$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>σ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>σ</mi><mi>τ</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math> </ephtml> under eight simulation conditions. From Table 4, the average MSE, SD, and SE values for discrimination and difficulty parameters tend to increase as the proportion of missing data rises from LMP to HMP. This pattern is consistent under both MAR and MNAR conditions, regardless of the chosen correction coefficient ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau } =$</annotation></semantics></math> </ephtml> 0, 0.3, 0.5, or 0.8). On the other hand, the average MSE, SD, and SE values for missing difficulty parameters and fixed effect parameters tend to decrease under both MAR and MNAR conditions. This may be due to the reduced number of item responses, which decreases the accuracy of discrimination and difficulty parameter estimates in the 2PL model. However, in the 1PLMDM model, the accuracy of difficulty and fixed effect parameter estimates improves as the number of missing responses increases. The results show that under both missing mechanisms, regardless of weak and strong correlations ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0$</annotation></semantics></math> </ephtml> or 0.3 and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math> </ephtml> , respectively), the average MSE, SD and SE of the item, fixed effect, and variance‐covariance parameters ( <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> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ζ</mi></mrow><annotation encoding="application/x-tex">$\bm{\zeta }$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">γ</mi></mrow><annotation encoding="application/x-tex">$\bm{\gamma }$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>σ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>σ</mi><mi>τ</mi><mn>2</mn></msubsup><annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math> </ephtml> ) are similar. This suggests that the correlation strength has little impact on estimating these parameters using different missing mechanisms. Overall, the Bayesian algorithm was able to accurately estimate these parameters in various simulation conditions. Due to space limitation, the evaluation of the performance of the proposed missing mechanism models under the condition of negative correlation coefficients is provided in simulation study A2 of the online supplement.</p> <p>4 Table Evaluating the Performance of the Proposed Missing Mechanism Models Using Slice Sampling Algorithm in Simulation Study 2</p> <p> <ephtml> <table><thead><tr><th /><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><msub><mi>γ</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=\gamma _{2}=0$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math></p></th><th align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math></p></th></tr><tr><th /><th align="center">LMP</th><th align="center">HMP</th><th align="center">LMP</th><th align="center">HMP</th><th align="center">LMP</th><th align="center">HMP</th><th align="center">LMP</th><th align="center">HMP</th></tr><tr><th /><th align="center">MAR</th><th align="center">MNAR</th><th align="center">MNAR</th><th align="center">MNAR</th></tr><tr><th>Bias</th><th align="center" /><th align="center" /><th align="center" /><th align="center" /><th align="center" /><th align="center" /><th align="center" /><th align="center" /></tr></thead><tbody><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math></p></td><td>0.0050</td><td>0.0028</td><td>−0.0460</td><td>−0.0470</td><td>−0.0478</td><td>−0.0497</td><td>−0.0416</td><td>−0.0510</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math></p></td><td>0.0446</td><td>0.0474</td><td>−0.0187</td><td>−0.0314</td><td>−0.0271</td><td>−0.0444</td><td>−0.0366</td><td>−0.0636</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">ς</mi><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math></p></td><td>0.1962</td><td>0.1962</td><td>0.1962</td><td>0.1962</td><td>0.1962</td><td>0.1962</td><td>0.1962</td><td>0.1962</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>0<annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math></p></td><td>−0.1795</td><td>−0.1598</td><td>−0.2149</td><td>−0.2022</td><td>−0.2124</td><td>−0.2005</td><td>−0.2165</td><td>−0.1967</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>1<annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math></p></td><td>−0.0053</td><td>−0.0084</td><td>−0.0051</td><td>−0.0083</td><td>−0.0051</td><td>−0.0083</td><td>−0.0050</td><td>−0.0083</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>2<annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.0899</td><td>0.0891</td><td>0.0901</td><td>0.0893</td><td>0.0902</td><td>0.0895</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>θ</mi><mi>τ</mi><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>−0.0694</td><td>−0.0695</td><td>−0.1111</td><td>−0.1057</td><td>−0.1733</td><td>−0.1577</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>τ</mi>2<annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math></p></td><td>0.0048</td><td>0.0051</td><td>0.0074</td><td>0.0080</td><td>0.0078</td><td>0.0076</td><td>0.0060</td><td>0.0059</td></tr><tr><td>MSE</td><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math></p></td><td>0.0128</td><td>0.0145</td><td>0.0150</td><td>0.0171</td><td>0.0150</td><td>0.0166</td><td>0.0148</td><td>0.0176</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math></p></td><td>0.0191</td><td>0.0218</td><td>0.0222</td><td>0.0264</td><td>0.0239</td><td>0.0282</td><td>0.0245</td><td>0.0333</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">ς</mi><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math></p></td><td>0.1719</td><td>0.0676</td><td>0.1940</td><td>0.0724</td><td>0.1938</td><td>0.0730</td><td>0.1967</td><td>0.0721</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>0<annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math></p></td><td>0.0329</td><td>0.0258</td><td>0.0467</td><td>0.0413</td><td>0.0460</td><td>0.0405</td><td>0.0478</td><td>0.0391</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>1<annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math></p></td><td>0.0001</td><td>0.0000</td><td>0.0001</td><td>0.0000</td><td>0.0001</td><td>0.0000</td><td>0.0001</td><td>0.0000</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>2<annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.0084</td><td>0.0063</td><td>0.0079</td><td>0.0068</td><td>0.0082</td><td>0.0076</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>θ</mi><mi>τ</mi><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.0051</td><td>0.0051</td><td>0.0126</td><td>0.0114</td><td>0.0302</td><td>0.0250</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>τ</mi>2<annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math></p></td><td>0.0000</td><td>0.0001</td><td>0.0000</td><td>0.0001</td><td>0.0000</td><td>0.0001</td><td>0.0000</td><td>0.0001</td></tr><tr><td>SD</td><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math></p></td><td>0.0917</td><td>0.0968</td><td>0.0897</td><td>0.0943</td><td>0.0902</td><td>0.0945</td><td>0.0902</td><td>0.0947</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math></p></td><td>0.0965</td><td>0.1028</td><td>0.1047</td><td>0.1106</td><td>0.1049</td><td>0.1108</td><td>0.1053</td><td>0.1124</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">ς</mi><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math></p></td><td>0.1483</td><td>0.1065</td><td>0.1508</td><td>0.1086</td><td>0.1507</td><td>0.1085</td><td>0.1510</td><td>0.1083</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>0<annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math></p></td><td>0.0146</td><td>0.0144</td><td>0.0158</td><td>0.0152</td><td>0.0174</td><td>0.0152</td><td>0.0166</td><td>0.0153</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>1<annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math></p></td><td>0.0044</td><td>0.0014</td><td>0.0046</td><td>0.0015</td><td>0.0046</td><td>0.0015</td><td>0.0047</td><td>0.0015</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>2<annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.0081</td><td>0.0075</td><td>0.0080</td><td>0.0074</td><td>0.0078</td><td>0.0073</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>θ</mi><mi>τ</mi><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.0410</td><td>0.0375</td><td>0.0396</td><td>0.0354</td><td>0.0349</td><td>0.0304</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>τ</mi>2<annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math></p></td><td>0.0451</td><td>0.0447</td><td>0.0454</td><td>0.0453</td><td>0.0465</td><td>0.0459</td><td>0.0440</td><td>0.0440</td></tr><tr><td>SE</td><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math></p></td><td>0.1071</td><td>0.1130</td><td>0.1056</td><td>0.1128</td><td>0.1048</td><td>0.1101</td><td>0.1060</td><td>0.1133</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math></p></td><td>0.1196</td><td>0.1281</td><td>0.1268</td><td>0.1369</td><td>0.1288</td><td>0.1384</td><td>0.1293</td><td>0.1447</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">ς</mi><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math></p></td><td>0.1556</td><td>0.1137</td><td>0.1546</td><td>0.1156</td><td>0.1527</td><td>0.1171</td><td>0.1531</td><td>0.1156</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>0<annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math></p></td><td>0.0267</td><td>0.0182</td><td>0.0229</td><td>0.0203</td><td>0.0301</td><td>0.0192</td><td>0.0303</td><td>0.0211</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>1<annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math></p></td><td>0.0007</td><td>0.0006</td><td>0.0006</td><td>0.0003</td><td>0.0006</td><td>0.0003</td><td>0.0006</td><td>0.0002</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>2<annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.0002</td><td>0.0001</td><td>0.0002</td><td>0.0001</td><td>0.0002</td><td>0.0001</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>θ</mi><mi>τ</mi><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.0193</td><td>0.0168</td><td>0.0181</td><td>0.0157</td><td>0.0149</td><td>0.0109</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>τ</mi>2<annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math></p></td><td>0.0012</td><td>0.0011</td><td>0.0017</td><td>0.0019</td><td>0.0016</td><td>0.0017</td><td>0.0037</td><td>0.0029</td></tr><tr><td>CP</td><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">a</mi><annotation encoding="application/x-tex">$\bm{a}$</annotation></semantics></math></p></td><td>0.9869</td><td>0.9869</td><td>0.9822</td><td>0.9804</td><td>0.9818</td><td>0.9815</td><td>0.9826</td><td>0.9800</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">b</mi><annotation encoding="application/x-tex">$\bm{b}$</annotation></semantics></math></p></td><td>0.9806</td><td>0.9812</td><td>0.9861</td><td>0.9853</td><td>0.9854</td><td>0.9846</td><td>0.9846</td><td>0.9816</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi mathvariant="bold-italic">ς</mi><annotation encoding="application/x-tex">$\bm{\varsigma }$</annotation></semantics></math></p></td><td>0.9731</td><td>0.9559</td><td>0.9761</td><td>0.9576</td><td>0.9758</td><td>0.9572</td><td>0.9762</td><td>0.9574</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>0<annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math></p></td><td>0.9472</td><td>0.9338</td><td>0.9546</td><td>0.9363</td><td>0.9451</td><td>0.9420</td><td>0.9458</td><td>0.9545</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>1<annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math></p></td><td>0.9731</td><td>0.9786</td><td>0.9693</td><td>0.9798</td><td>0.9748</td><td>0.9738</td><td>0.9881</td><td>0.9826</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>γ</mi>2<annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.9657</td><td>0.9685</td><td>0.9661</td><td>0.9682</td><td>0.9701</td><td>0.9733</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>θ</mi><mi>τ</mi><annotation encoding="application/x-tex">$\sigma _{\theta \tau }$</annotation></semantics></math></p></td><td align="center">—</td><td align="center">—</td><td>0.9681</td><td>0.9754</td><td>0.9804</td><td>0.9745</td><td>0.9766</td><td>0.9741</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>σ</mi><mi>τ</mi>2<annotation encoding="application/x-tex">$\sigma _{\tau }^{2}$</annotation></semantics></math></p></td><td>0.9446</td><td>0.9498</td><td>0.9490</td><td>0.9473</td><td>0.9468</td><td>0.9462</td><td>0.9447</td><td>0.9463</td></tr></tbody></table> </ephtml> </p> <hd id="AN0192629997-21">Simulation 3</hd> <p>The aim of this simulation study is to examine the impact of fitting data under distinct missing mechanisms on Bayesian model assessment. Specifically, we employ two Bayesian model assessment techniques, DIC and LPML, to evaluate and distinguish between various missing mechanism models.</p> <hd1 id="AN0192629997-22">Simulation Designs</hd1> <p>As an illustration, we use a sample of 500 examinees and 40 test items in both the 2PL and 1PLMDP models. The true values of item parameters are the same as those used in simulation study 1. The primary factor under consideration is the correlation coefficient, with three different values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><annotation encoding="application/x-tex">$\rho _{\theta \tau }$</annotation></semantics></math> </ephtml> : (<reflink idref="bib1" id="ref91">1</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math> </ephtml> , (<reflink idref="bib2" id="ref92">2</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math> </ephtml> , and (<reflink idref="bib3" id="ref93">3</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math> </ephtml> . The second factor includes different missing proportions, that is, low missing proportions (LMP) with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>0</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>3</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _{0}(-3)$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>0.01</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _{1}(0.01)$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>0.1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _{2}(-0.1)$</annotation></semantics></math> </ephtml> , and high missing proportions (HMP) with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>0</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _{0}(-2)$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>0.02</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _{1}(0.02)$</annotation></semantics></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>0.2</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">$\gamma _{2}(-0.2)$</annotation></semantics></math> </ephtml> . Finally, we consider different missing mechanisms, including MAR and MNAR. Consequently, the true models and fitted models are provided as follows:</p> <p></p> <ulist> <item> <bold> (i) True model: _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> _ht_ 1PLMDP with LMP in MNAR ( _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math> _ht_ ). _B_v.s</bold>. Fitted model: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with LMP in MAR, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with LMP in MNAR ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau } =0.3$</annotation></semantics></math> </ephtml> );</item> <p></p> <item> <bold> (ii) True model: _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> _ht_ 1PLMDP with HMP in MNAR ( _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math> _ht_ ). _B_v.s</bold>. Fitted model: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with HMP in MAR, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with HMP in MNAR ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau } =0.3$</annotation></semantics></math> </ephtml> );</item> <p></p> <item> <bold> (iii) True model: _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> _ht_ 1PLMDP with LMP in MNAR ( _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math> _ht_ ). _B_v.s</bold>. Fitted model: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with LMP in MAR, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with LMP in MNAR ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau } =0.5$</annotation></semantics></math> </ephtml> );</item> <p></p> <item> <bold> (iv) True model: _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> _ht_ 1PLMDP with HMP in MNAR ( _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math> _ht_ ). _B_v.s</bold>. Fitted model: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with HMP in MAR, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with HMP in MNAR ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau } =0.5$</annotation></semantics></math> </ephtml> );</item> <p></p> <item> <bold> (v) True model: _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> _ht_ 1PLMDP with LMP in MNAR ( _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math> _ht_ ). _B_v.s</bold>. Fitted model: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with LMP in MAR, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with LMP in MNAR ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau } =0.8$</annotation></semantics></math> </ephtml> );</item> <p></p> <item> <bold> (vi) True model: _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> _ht_ 1PLMDP with HMP in MNAR ( _HT_ <math display="inline" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math> _ht_ ). _B_v.s</bold>. Fitted model: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with HMP in MAR, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with HMP in MNAR ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math> </ephtml> ).</item> </ulist> <p>The same noninformative priors from simulation study 1 are employed. The MCMC sampling algorithm is implemented with chains of length 20,000 and an initial burn‐in period of 10,000. Each simulation condition is replicated 50 times. All item and person parameters have PSRF values less than 1.1 for each simulation condition.</p> <p>From Table 5, both Bayesian model assessment criteria are effective in correctly identifying the true models, regardless of the proportion of missing responses or the strength of correlation ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><annotation encoding="application/x-tex">$\rho _{\theta \tau }$</annotation></semantics></math> </ephtml> ). Specifically, even under the condition of weak correlation and low missing proportion, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with LMP in MNAR fits the data best when it is the true model. This is consistent with our expectation. The median of DIC for the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with LMP in MNAR(144007.78) is smaller than that for the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with LMP in MAR(144066.86), and the median of LPML for the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with LMP in MNAR(−79768.85) is larger than that for the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with LMP in MAR(−80152.69). In the case of a high missing proportion and strong correlation, when the true model is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with HMP in MNAR, the same model provides the best fit to the data. The differences in medians of DIC and LPML are −113.67 and 80.05, respectively, between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with HMP in MNAR and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM with HMP in MAR. In conclusion, the Bayesian assessment criteria are effective in identifying the correct models and can be used in empirical example analysis.</p> <p>5 Table The Results of Bayesian Model Assessment in Simulation Study 3.</p> <p> <ephtml> <table><thead><tr valign="bottom"><th /><th>Fitted Model</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with LMP in MAR</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with LMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math></p>)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with HMP in MAR</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with HMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math></p>)</th></tr></thead><tbody><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>143972.90</td><td>143893.91</td><td>–</td><td>–</td></tr><tr><td /><td /><td>DIC</td><td>Median</td><td>144066.86</td><td>144007.78</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>144178.48</td><td>144111.87</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td>IQR</td><td>205.58</td><td>217.96</td><td>–</td><td>–</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with‐</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>−80254.43</td><td>−79936.48</td><td>–</td><td>–</td></tr><tr><td /><td>LMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ρ</mi><mi>θ</mi><mi>τ</mi><mo>=</mo>0.3<annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math></p>)</td><td>LPML</td><td>Median</td><td>−80152.69</td><td>−79768.85</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>−80093.56</td><td>−79669.21</td><td>–</td><td>–</td></tr><tr><td>True</td><td /><td /><td>IQR</td><td>160.87</td><td>267.27</td><td>–</td><td>–</td></tr><tr><td>Model</td><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>148975.03</td><td>148935.46</td></tr><tr><td /><td /><td>DIC</td><td>Median</td><td>–</td><td>–</td><td>149109.10</td><td>149078.64</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>149236.12</td><td>149212.61</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with</td><td /><td>IQR</td><td>–</td><td>–</td><td>261.08</td><td>277.15</td></tr><tr><td /><td>HMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ρ</mi><mi>θ</mi><mi>τ</mi><mo>=</mo>0.3<annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math></p>)</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>−81377.06</td><td>−81198.33</td></tr><tr><td /><td /><td>LPML</td><td>Median</td><td>–</td><td>–</td><td>−81279.18</td><td>−81107.51</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>−81145.02</td><td>−81030.40</td></tr><tr><td /><td /><td /><td>IQR</td><td>–</td><td>–</td><td>232.04</td><td>167.93</td></tr></tbody></table> </ephtml> </p> <p></p> <p> <ephtml> <table><thead><tr valign="bottom"><th /><th>Fitted Model</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with LMP in MAR</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with LMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math></p>)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>1</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$1{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with HMP in MAR</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with HMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math></p>)</th></tr></thead><tbody><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>143966.91</td><td>143891.12</td><td>–</td><td>–</td></tr><tr><td /><td /><td>DIC</td><td>Median</td><td>144114.04</td><td>144021.78</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>144238.87</td><td>144131.325</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td>IQR</td><td>271.95</td><td>240.19</td><td>–</td><td>–</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>−80309.21</td><td>−79898.05</td><td>–</td><td>–</td></tr><tr><td /><td>LMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ρ</mi><mi>θ</mi><mi>τ</mi><mo>=</mo>0.5<annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math></p>)</td><td>LPML</td><td>Median</td><td>−80145.31</td><td>−79698.99</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>−80047.70</td><td>−79625.56</td><td>–</td><td>–</td></tr><tr><td>True</td><td /><td /><td>IQR</td><td>261.50</td><td>272.48</td><td>–</td><td>–</td></tr><tr><td>Model</td><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>149021.57</td><td>148902.24</td></tr><tr><td /><td /><td>DIC</td><td>Median</td><td>–</td><td>–</td><td>149149.07</td><td>149095.85</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>149263.53</td><td>149211.65</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with</td><td /><td>IQR</td><td>–</td><td>–</td><td>241.95</td><td>252.32</td></tr><tr><td /><td>HMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ρ</mi><mi>θ</mi><mi>τ</mi><mo>=</mo>0.5<annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.5$</annotation></semantics></math></p>)</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>−81433.62</td><td>−81232.95</td></tr><tr><td /><td /><td>LPML</td><td>Median</td><td>–</td><td>–</td><td>−81287.72</td><td>−81099.48</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>−81190.84</td><td>−80978.35</td></tr><tr><td /><td /><td /><td>IQR</td><td>–</td><td>–</td><td>242.78</td><td>254.99</td></tr></tbody></table> </ephtml> </p> <p></p> <p> <ephtml> <table><thead><tr valign="bottom"><th /><th>Fitted Model</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with LMP in MAR</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with LMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math></p>)</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with HMP in MAR</th><th><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP with HMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math></p>)</th></tr></thead><tbody><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>144079.76</td><td>143943.66</td><td>–</td><td>–</td></tr><tr><td /><td /><td>DIC</td><td>Median</td><td>144165.29</td><td>144053.92</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>144276.78</td><td>144190.54</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td>IQR</td><td>197.02</td><td>264.87</td><td>–</td><td>–</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>2PLMDP</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>−80326.74</td><td>−79849.40</td><td>–</td><td>–</td></tr><tr><td /><td>with LMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ρ</mi><mi>θ</mi><mi>τ</mi><mo>=</mo>0.8<annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math></p>)</td><td>LPML</td><td>Median</td><td>−80189.11</td><td>−79601.75</td><td>–</td><td>–</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>−80034.77</td><td>−79492.40</td><td>–</td><td>–</td></tr><tr><td>True</td><td /><td /><td>IQR</td><td>291.96</td><td>357.00</td><td>–</td><td>–</td></tr><tr><td>Model</td><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>149091.18</td><td>149036.86</td></tr><tr><td /><td /><td>DIC</td><td>Median</td><td>–</td><td>–</td><td>149228.81</td><td>149115.14</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>149374.57</td><td>149326.20</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDP</td><td /><td>IQR</td><td>–</td><td>–</td><td>283.39</td><td>289.34</td></tr><tr><td /><td>with HMP in MNAR(<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ρ</mi><mi>θ</mi><mi>τ</mi><mo>=</mo>0.8<annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.8$</annotation></semantics></math></p>)</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>1<annotation encoding="application/x-tex">$Q_{1}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>−81490.17</td><td>−81433.73</td></tr><tr><td /><td /><td>LPML</td><td>Median</td><td>–</td><td>–</td><td>−81355.07</td><td>−81275.02</td></tr><tr><td /><td /><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>Q</mi>3<annotation encoding="application/x-tex">$Q_{3}$</annotation></semantics></math></p></td><td>–</td><td>–</td><td>−81255.14</td><td>−81108.57</td></tr><tr><td /><td /><td /><td>IQR</td><td>–</td><td>–</td><td>235.03</td><td>325.15</td></tr></tbody></table> </ephtml> </p> <p>3 <emph>Note</emph>. The <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP in MAR denotes IRT model is the 2PL model, missing indicator model is the1PLMDP model, and the missing mechanism is MAR missing. The <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDP with LMP (HMP) in MNAR ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mrow><annotation encoding="application/x-tex">$\rho _{\theta \tau }=0.3$</annotation></semantics></math> </ephtml> , 0.5, and 0.8) denotes under the MNAR missing and low (high) missing proportion condition, the IRT model and missing process model are specified as 2PL and 1PLMDP models, respectively, and there is a correlation of 0.3 (0.5 and 0.8) between the ability and the missing propensity parameters.</p> <hd id="AN0192629997-23">Empirical Example</hd> <p>The real data set is from the 2015 PISA science assessment for the Dominican Republic (DOM). The data was collected through a computer‐based testing platform, where students responded to a set of items from a test booklet. Initially, 545 students participated, but 18 were excluded due to invalid or inapplicable responses (original code 95 or 5 and 97 or 7, respectively). Missing data includes "Not Response" (99/9) and "Not Reached" (96/6). After these exclusions, the final sample consisted of 527 students who completed a test with 17 items. Table 6 provides the descriptive statistics of this data set.</p> <p>6 Table The Descriptive Statistics for PISA 2015 Released Computer‐Based Sciences Items</p> <p> <ephtml> <table><thead><tr><th>Item</th><th>Question</th><th>Response Category</th><th>Missing Proportion</th><th>Correct Rate</th></tr></thead><tbody><tr><td>DS465Q01C</td><td>Different Climates—Q01</td><td>Polytomous</td><td>29.6%</td><td>14.8%</td></tr><tr><td>CS465Q02S</td><td>Different Climates—Q02</td><td>Binary</td><td>14.2%</td><td>35.7%</td></tr><tr><td>CS465Q04S</td><td>Different Climates—Q04</td><td>Binary</td><td>12.3%</td><td>23.9%</td></tr><tr><td>DS131Q02C</td><td>Good Vibrations—Q02</td><td>Binary</td><td>20.3%</td><td>4.9%</td></tr><tr><td>DS131Q04C</td><td>Good Vibrations—Q04</td><td>Binary</td><td>23.7%</td><td>8.2%</td></tr><tr><td>CS428Q01S</td><td>Bacteria in Milk—Q01</td><td>Binary</td><td>9.7%</td><td>19.9%</td></tr><tr><td>CS428Q03S</td><td>Bacteria in Milk—Q03</td><td>Binary</td><td>10.2%</td><td>22.0%</td></tr><tr><td>DS428Q05C</td><td>Bacteria in Milk—Q05</td><td>Binary</td><td>27.5%</td><td>3.2%</td></tr><tr><td>DS514Q02C</td><td>Development and Disaster—Q02</td><td>Binary</td><td>24.7%</td><td>39.8%</td></tr><tr><td>DS514Q03C</td><td>Development and Disaster—Q03</td><td>Binary</td><td>16.9%</td><td>7.8%</td></tr><tr><td>DS514Q04C</td><td>Development and Disaster—Q04</td><td>Binary</td><td>19.8%</td><td>6.1%</td></tr><tr><td>CS438Q01S</td><td>Green Parks—Q01</td><td>Binary</td><td>19.7%</td><td>30.7%</td></tr><tr><td>CS438Q02S</td><td>Green Parks—Q02</td><td>Binary</td><td>23.9%</td><td>26.8%</td></tr><tr><td>DS438Q03C</td><td>Green Parks—Q03</td><td>Binary</td><td>45.9%</td><td>3.0%</td></tr><tr><td>CS415Q02S</td><td>Solar Power Generation (Solar Panels)—Q02</td><td>Binary</td><td>27.9%</td><td>33.4%</td></tr><tr><td>CS415Q07S</td><td>Solar Power Generation (Solar Panels)—Q07</td><td>Binary</td><td>24.7%</td><td>37.6%</td></tr><tr><td>CS415Q08S</td><td>Solar Power Generation (Solar Panels)—Q08</td><td>Binary</td><td>27.5%</td><td>19.7%</td></tr></tbody></table> </ephtml> </p> <p>4 <emph>Note</emph>. For the calculation of the correct rate, missing responses are considered as incorrect answers. For polytomous items, a response is considered partially or completely correct if it is recorded as 1, otherwise it is recorded as 0.</p> <p>The lowest correct rates were observed for DS438Q03C (at 3.0%), DS131Q02C (at 4.9%), and DS514Q04C (at 6.1%). For DS438Q03C and DS514Q04C, missing response proportions are especially high at 45.9% and 29.8%, respectively. It implies that the overall correct rate may have been decreased because the missing response were likely coded as incorrect. On the other hand, the highest correct rates belong to DS514Q02C (39.8%), CS415Q07S (37.6%), and CS465Q02C (35.7%), respectively. Figure 1 displays the frequency histogram of the correct rate for the 527 examinees, along with the correct rate and missing proportion for each item.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12428-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12428-fig-0001.jpg" title="1 Left: the frequency histogram of the correct rate for 527 examinees. Right: the correct rate and missing proportion for each item." /> </p> <p></p> <hd id="AN0192629997-25">Bayesian Model Assessment</hd> <p>We applied two models, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM in MAR and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM in MNAR, to fit the empirical example data. All posterior estimates are calculated using 20,000 MCMC samples after a burn‐in of 10,000 iterations for each model. The convergence of the chains was checked by PSRF, and all item and person parameters had PSRF values of less than 1.1.</p> <p>Table 7 displays the results of model assessment for the empirical example data. Based on the Bayesian model evaluation criteria, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM model is identified as the best fit within the MNAR framework. Subsequently, we calculate the absolute fit indices for this optimal model using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>P</mi><msub><mi>P</mi><mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">$PPP_{Y}$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>P</mi><msub><mi>P</mi><mi>R</mi></msub></mrow><annotation encoding="application/x-tex">$PPP_{R}$</annotation></semantics></math> </ephtml> for the item response and missing indicator data. The results are as follows: the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>P</mi><msub><mi>P</mi><mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">$PPP_{Y}$</annotation></semantics></math> </ephtml> value for the item response model is 0.5860, and the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>P</mi><msub><mi>P</mi><mi>R</mi></msub></mrow><annotation encoding="application/x-tex">$PPP_{R}$</annotation></semantics></math> </ephtml> value for the 1PLMDP model is 0.4251. A PPP value near 0.5 suggests that the model‐generated data is statistically similar to the observed data, implying that the model effectively captures the underlying patterns in the data without significant bias or deviation. In addition, the PPP values of Rose et al. ([<reflink idref="bib44" id="ref94">44</reflink>]) and Holman and Glas (2017) models deviate more substantially from 0.5 compared to the PPP values of our model. As a result, these findings demonstrate that both the 2PL and 1PLMDP models effectively fit the PISA dataset. Consequently, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>PL</mi><mi>⊕</mi></mrow><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math> </ephtml> 1PLMDM in MNAR is selected for further analysis of the empirical dataset.</p> <p>7 Table The Results of Bayesian Model Assessment for the PISA Data</p> <p> <ephtml> <table><thead><tr><th>Model</th><th align="center">DIC</th><th align="center">LPML</th></tr></thead><tbody><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDM in MAR</td><td>54970.22</td><td>−30582.63</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2<mi>PL</mi><mi>⊕</mi><annotation encoding="application/x-tex">$2{\rm PL}\oplus$</annotation></semantics></math></p>1PLMDM in MNAR</td><td>54687.41</td><td>−30174.71</td></tr></tbody></table> </ephtml> </p> <hd id="AN0192629997-26">Analysis of Item Parameters</hd> <p>According to Table 8, the estimated item parameters indicate that 14 discrimination parameters have expected a posteriori (EAP) estimates above 0.5, suggesting that these items effectively differentiate among ability levels. The three items with the lowest discrimination are item 3 (CS465Q04S), item 2 (CS465Q02S), and item 16 (CS415Q07S). Moreover, five items exhibit higher difficulty levels, with EAP estimates exceeding 2. These items, in descending order of difficulty, are item 3 (CS465Q04S), item 4 (DS131Q02C), item 14 (DS438Q03C), item 8 (DS428Q05C), and item 10 (DS514Q03C), with respective difficulty estimates of 3.2671, 2.6952, 2.5046, 2.4165, and 2.2368. From Table 7, their corresponding correct rates are 23.9%, 4.9%, 3.0%, 3.2%, and 7.8%, respectively. As expected, except for item 3 (CS465Q04S), the most difficult items exhibit low correct rates. However, item 3 not only has the highest difficulty but also the lowest discrimination, meaning it does not effectively distinguish between ability levels. Its relatively high correct rate may be due to guessing. The missing indicator model shows that item 14 (DS438Q03C) and item 1 (CS428Q03S) have the highest missing difficulty parameters, with EAP estimates of 2.2731 and 0.9844, respectively. As the estimated missing difficulty increases, the proportion of missing responses also rises. The missing response rates for items 14 and 1 are 45.9% and 29.8% respectively. The SD for the discrimination parameters range from 0.0012 to 0.1129, while those for the difficulty parameters range from 0.0107 to 0.1929. For the missing difficulty parameters, the SD values range from 0.0102 to 0.0560.</p> <p>8 Table The Results of Estimating Item Parameters for the PISA Data</p> <p> <ephtml> <table><thead><tr><th>PARM</th><th align="center">EAP</th><th align="center">SD</th><th align="center">HPDI</th><th align="center">PARM</th><th align="center">EAP</th><th align="center">SD</th><th align="center">HPDI</th><th align="center">PARM</th><th align="center">EAP</th><th align="center">SD</th><th align="center">HPDI</th></tr></thead><tbody><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>1<annotation encoding="application/x-tex">$a_{1}$</annotation></semantics></math></p></td><td>0.9393</td><td>0.0274</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.6629<mo>,</mo>1.2568<annotation encoding="application/x-tex">$\left[ 0.6629,1.2568\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>1<annotation encoding="application/x-tex">$b_{1}$</annotation></semantics></math></p></td><td>1.6582</td><td>0.0583</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.2955<mo>,</mo>2.1253<annotation encoding="application/x-tex">$\left[ 1.2955,2.1253\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>1<annotation encoding="application/x-tex">$\varsigma _{1}$</annotation></semantics></math></p></td><td>0.9844</td><td>0.0102</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.7887<mo>,</mo>1.1850<annotation encoding="application/x-tex">$\left[ 0.7887,1.1850\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>2<annotation encoding="application/x-tex">$a_{2}$</annotation></semantics></math></p></td><td>0.3572</td><td>0.0188</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.0931<mo>,</mo>0.6128<annotation encoding="application/x-tex">$\left[ 0.0931,0.6128\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>2<annotation encoding="application/x-tex">$b_{2}$</annotation></semantics></math></p></td><td>0.4942</td><td>0.0935</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.1194<mo>,</mo>1.0690<annotation encoding="application/x-tex">$\left[ -0.1194,1.0690\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>2<annotation encoding="application/x-tex">$\varsigma _{2}$</annotation></semantics></math></p></td><td>−0.7553</td><td>0.0280</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>1.1015<mo>,</mo><mo>−</mo>0.4535<annotation encoding="application/x-tex">$\left[ -1.1015,-0.4535\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>3<annotation encoding="application/x-tex">$a_{3}$</annotation></semantics></math></p></td><td>0.2231</td><td>0.0012</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.1587<mo>,</mo>0.2986<annotation encoding="application/x-tex">$\left[ 0.1587,0.2986\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>3<annotation encoding="application/x-tex">$b_{3}$</annotation></semantics></math></p></td><td>3.2671</td><td>0.1929</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2.4034<mo>,</mo>4.1082<annotation encoding="application/x-tex">$\left[ 2.4034,4.1082\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>3<annotation encoding="application/x-tex">$\varsigma _{3}$</annotation></semantics></math></p></td><td>−1.1784</td><td>0.0448</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>1.5743<mo>,</mo><mo>−</mo>0.7504<annotation encoding="application/x-tex">$\left[ -1.5743,-0.7504\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>4<annotation encoding="application/x-tex">$a_{4}$</annotation></semantics></math></p></td><td>1.1762</td><td>0.0153</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.9258<mo>,</mo>1.3862<annotation encoding="application/x-tex">$\left[ 0.9258,1.3862\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>4<annotation encoding="application/x-tex">$b_{4}$</annotation></semantics></math></p></td><td>2.6952</td><td>0.0398</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2.3490<mo>,</mo>3.1125<annotation encoding="application/x-tex">$\left[ 2.3490,3.1125\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>4<annotation encoding="application/x-tex">$\varsigma _{4}$</annotation></semantics></math></p></td><td>−0.1400</td><td>0.0199</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.3928<mo>,</mo>0.1809<annotation encoding="application/x-tex">$\left[ -0.3928,0.1809\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>5<annotation encoding="application/x-tex">$a_{5}$</annotation></semantics></math></p></td><td>1.6902</td><td>0.0550</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.2083<mo>,</mo>2.1165<annotation encoding="application/x-tex">$\left[ 1.2083,2.1165\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>5<annotation encoding="application/x-tex">$b_{5}$</annotation></semantics></math></p></td><td>1.7944</td><td>0.0183</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.5558<mo>,</mo>2.0627<annotation encoding="application/x-tex">$\left[ 1.5558,2.0627\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>5<annotation encoding="application/x-tex">$\varsigma _{5}$</annotation></semantics></math></p></td><td>0.2718</td><td>0.0114</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.0752<mo>,</mo>0.5101<annotation encoding="application/x-tex">$\left[ 0.0752,0.5101\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>6<annotation encoding="application/x-tex">$a_{6}$</annotation></semantics></math></p></td><td>1.2618</td><td>0.0227</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.9874<mo>,</mo>1.5847<annotation encoding="application/x-tex">$\left[ 0.9874,1.5847\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>6<annotation encoding="application/x-tex">$b_{6}$</annotation></semantics></math></p></td><td>1.1919</td><td>0.0125</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.0070<mo>,</mo>1.4373<annotation encoding="application/x-tex">$\left[ 1.0070,1.4373\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>6<annotation encoding="application/x-tex">$\varsigma _{6}$</annotation></semantics></math></p></td><td>−1.6886</td><td>0.0560</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>2.1438<mo>,</mo><mo>−</mo>1.2536<annotation encoding="application/x-tex">$\left[ -2.1438,-1.2536\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>7<annotation encoding="application/x-tex">$a_{7}$</annotation></semantics></math></p></td><td>1.2049</td><td>0.0239</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.9509<mo>,</mo>1.5186<annotation encoding="application/x-tex">$\left[ 0.9509,1.5186\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>7<annotation encoding="application/x-tex">$b_{7}$</annotation></semantics></math></p></td><td>1.0632</td><td>0.0137</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.8483<mo>,</mo>1.2941<annotation encoding="application/x-tex">$\left[ 0.8483,1.2941\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>7<annotation encoding="application/x-tex">$\varsigma _{7}$</annotation></semantics></math></p></td><td>−1.6311</td><td>0.0576</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>2.1109<mo>,</mo><mo>−</mo>1.2079<annotation encoding="application/x-tex">$\left[ -2.1109,-1.2079\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>8<annotation encoding="application/x-tex">$a_{8}$</annotation></semantics></math></p></td><td>1.8867</td><td>0.1129</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.1450<mo>,</mo>2.4528<annotation encoding="application/x-tex">$\left[ 1.1450,2.4528\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>8<annotation encoding="application/x-tex">$b_{8}$</annotation></semantics></math></p></td><td>2.4165</td><td>0.0608</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2.0521<mo>,</mo>2.9896<annotation encoding="application/x-tex">$\left[ 2.0521,2.9896\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>8<annotation encoding="application/x-tex">$\varsigma _{8}$</annotation></semantics></math></p></td><td>0.5949</td><td>0.0288</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.1956<mo>,</mo>0.9097<annotation encoding="application/x-tex">$\left[ 0.1956,0.9097\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>9<annotation encoding="application/x-tex">$a_{9}$</annotation></semantics></math></p></td><td>0.9899</td><td>0.0328</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.6744<mo>,</mo>1.3413<annotation encoding="application/x-tex">$\left[ 0.6744,1.3413\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>9<annotation encoding="application/x-tex">$b_{9}$</annotation></semantics></math></p></td><td>−0.2163</td><td>0.0107</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.4105<mo>,</mo><mo>−</mo>0.0084<annotation encoding="application/x-tex">$\left[ -0.4105,-0.0084\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>9<annotation encoding="application/x-tex">$\varsigma _{9}$</annotation></semantics></math></p></td><td>0.2424</td><td>0.0282</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.1100<mo>,</mo>0.5732<annotation encoding="application/x-tex">$\left[ -0.1100,0.5732\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>10<annotation encoding="application/x-tex">$a_{10}$</annotation></semantics></math></p></td><td>1.2603</td><td>0.0429</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.8457<mo>,</mo>1.6762<annotation encoding="application/x-tex">$\left[ 0.8457,1.6762\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>10<annotation encoding="application/x-tex">$b_{10}$</annotation></semantics></math></p></td><td>2.2368</td><td>0.0699</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.7124<mo>,</mo>2.7141<annotation encoding="application/x-tex">$\left[ 1.7124,2.7141\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>10<annotation encoding="application/x-tex">$\varsigma _{10}$</annotation></semantics></math></p></td><td>−0.6747</td><td>0.0248</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.9720<mo>,</mo><mo>−</mo>0.3700<annotation encoding="application/x-tex">$\left[ -0.9720,-0.3700\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>11<annotation encoding="application/x-tex">$a_{11}$</annotation></semantics></math></p></td><td>2.2139</td><td>0.1080</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.6459<mo>,</mo>2.8849<annotation encoding="application/x-tex">$\left[ 1.6459,2.8849\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>11<annotation encoding="application/x-tex">$b_{11}$</annotation></semantics></math></p></td><td>1.8136</td><td>0.0165</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.5453<mo>,</mo>2.0696<annotation encoding="application/x-tex">$\left[ 1.5453,2.0696\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>11<annotation encoding="application/x-tex">$\varsigma _{11}$</annotation></semantics></math></p></td><td>0.6805</td><td>0.0180</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.4148<mo>,</mo>0.9297<annotation encoding="application/x-tex">$\left[ 0.4148,0.9297\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>12<annotation encoding="application/x-tex">$a_{12}$</annotation></semantics></math></p></td><td>0.8676</td><td>0.0388</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.5080<mo>,</mo>1.2259<annotation encoding="application/x-tex">$\left[ 0.5080,1.2259\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>12<annotation encoding="application/x-tex">$b_{12}$</annotation></semantics></math></p></td><td>0.4736</td><td>0.0225</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.2194<mo>,</mo>0.7614<annotation encoding="application/x-tex">$\left[ 0.2194,0.7614\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>12<annotation encoding="application/x-tex">$\varsigma _{12}$</annotation></semantics></math></p></td><td>−0.2164</td><td>0.0230</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.4923<mo>,</mo>0.1008<annotation encoding="application/x-tex">$\left[ -0.4923,0.1008\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>13<annotation encoding="application/x-tex">$a_{13}$</annotation></semantics></math></p></td><td>0.6908</td><td>0.0145</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.4402<mo>,</mo>0.8933<annotation encoding="application/x-tex">$\left[ 0.4402,0.8933\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>13<annotation encoding="application/x-tex">$b_{13}$</annotation></semantics></math></p></td><td>0.8995</td><td>0.0333</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.5573<mo>,</mo>1.2393<annotation encoding="application/x-tex">$\left[ 0.5573,1.2393\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>13<annotation encoding="application/x-tex">$\varsigma _{13}$</annotation></semantics></math></p></td><td>0.1279</td><td>0.0188</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.1648<mo>,</mo>0.3873<annotation encoding="application/x-tex">$\left[ -0.1648,0.3873\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>14<annotation encoding="application/x-tex">$a_{14}$</annotation></semantics></math></p></td><td>1.6459</td><td>0.0649</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.2042<mo>,</mo>2.1380<annotation encoding="application/x-tex">$\left[ 1.2042,2.1380\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>14<annotation encoding="application/x-tex">$b_{14}$</annotation></semantics></math></p></td><td>2.5046</td><td>0.0386</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">2.1328<mo>,</mo>2.8754<annotation encoding="application/x-tex">$\left[ 2.1328,2.8754\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>14<annotation encoding="application/x-tex">$\varsigma _{14}$</annotation></semantics></math></p></td><td>2.2731</td><td>0.0265</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">1.9858<mo>,</mo>2.6225<annotation encoding="application/x-tex">$\left[ 1.9858,2.6225\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>15<annotation encoding="application/x-tex">$a_{15}$</annotation></semantics></math></p></td><td>0.8030</td><td>0.0192</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.5318<mo>,</mo>1.0586<annotation encoding="application/x-tex">$\left[ 0.5318,1.0586\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>15<annotation encoding="application/x-tex">$b_{15}$</annotation></semantics></math></p></td><td>0.0439</td><td>0.0317</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.3124<mo>,</mo>0.3933<annotation encoding="application/x-tex">$\left[ -0.3124,0.3933\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>15<annotation encoding="application/x-tex">$\varsigma _{15}$</annotation></semantics></math></p></td><td>0.4496</td><td>0.0157</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.2112<mo>,</mo>0.6947<annotation encoding="application/x-tex">$\left[ 0.2112,0.6947\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>16<annotation encoding="application/x-tex">$a_{16}$</annotation></semantics></math></p></td><td>0.4496</td><td>0.0269</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.0657<mo>,</mo>0.7162<annotation encoding="application/x-tex">$\left[ 0.0657,0.7162\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>16<annotation encoding="application/x-tex">$b_{16}$</annotation></semantics></math></p></td><td>−0.2923</td><td>0.1328</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.9888<mo>,</mo>0.4438<annotation encoding="application/x-tex">$\left[ -0.9888,0.4438\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>16<annotation encoding="application/x-tex">$\varsigma _{16}$</annotation></semantics></math></p></td><td>0.2131</td><td>0.0263</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>−</mo>0.0453<mo>,</mo>0.6826<annotation encoding="application/x-tex">$\left[ -0.0453,0.6826\right]$</annotation></semantics></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>a</mi>17<annotation encoding="application/x-tex">$a_{17}$</annotation></semantics></math></p></td><td>0.7003</td><td>0.0174</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.4375<mo>,</mo>0.9355<annotation encoding="application/x-tex">$\left[ 0.4375,0.9355\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>b</mi>17<annotation encoding="application/x-tex">$b_{17}$</annotation></semantics></math></p></td><td>1.4430</td><td>0.1062</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.9601<mo>,</mo>2.1270<annotation encoding="application/x-tex">$\left[ 0.9601,2.1270\right]$</annotation></semantics></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>ς</mi>17<annotation encoding="application/x-tex">$\varsigma _{17}$</annotation></semantics></math></p></td><td>0.4468</td><td>0.0183</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns="">0.1834<mo>,</mo>0.6773<annotation encoding="application/x-tex">$\left[ 0.1834,0.6773\right]$</annotation></semantics></math></p></td></tr></tbody></table> </ephtml> </p> <p>5 <emph>Note</emph>. PARM refers to the parameter, EAP stands for expected a posteriori estimation, and HPDI stands for the highest probability density interval.</p> <hd id="AN0192629997-27">Evaluation of Fixed Effect Parameters</hd> <p>For the analysis of fixed effect parameters, the EAP estimate of parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>0</mn></msub><annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math> </ephtml> is −2.2036, with a 95% HPDI of [–2.2360, −2.1660]. The negative <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>0</mn></msub><annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math> </ephtml> indicates a low proportion of missing data, with significantly fewer missing responses than valid ones. This result suggests that parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>0</mn></msub><annotation encoding="application/x-tex">$\gamma _{0}$</annotation></semantics></math> </ephtml> effectively reflects the characteristic of a relatively small amount of missing responses in the overall dataset. For <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> , the estimate is 0.0819, with a 95% HPDI of [0.0137, 0.1819]. Since its HPDI does not include zero, indicating that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> is significantly nonzero. The positive value of parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> suggests that if examinees have missing responses on earlier items, they are more likely to have missing responses on subsequent items. This finding validates the assumption of monotonicity in missing data, implying that missing responses tend to accumulate over the sequence of items. Furthermore, as a fixed effect parameter, parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> quantifies the strength and direction of the impact of the cumulative missingness indicator on the probability of missing responses for the current item. The positive value of parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> further indicates that as the cumulative missingness indicator increases (i.e., the number of missing responses on previous items increases), the probability of missing responses for the current item also increases. This relationship strongly supports the presence of monotonic missingness, confirming the validity of this assumption in the model. Finally, for parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>2</mn></msub><annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math> </ephtml> , its EAP estimate is −0.1362, with a 95% HPDI of [−0.3708, −0.0028]. The negative value of parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>2</mn></msub><annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math> </ephtml> indicates that if an examinee correctly answers the current item, the probability of missing responses on this item significantly decreases. In other words, the missingness probability indicated by the current items missingness indicator decreases as the examinee answers correctly. This suggests that parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>2</mn></msub><annotation encoding="application/x-tex">$\gamma _{2}$</annotation></semantics></math> </ephtml> effectively characterizes the negative correlation between response correctness and the probability of missing responses. Therefore, these parameters support the theoretical foundation and empirical value of the model, especially by using parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>γ</mi><mn>1</mn></msub><annotation encoding="application/x-tex">$\gamma _{1}$</annotation></semantics></math> </ephtml> to confirm the existence of monotonic missingness.</p> <hd id="AN0192629997-28">Evaluation of Person Parameters</hd> <p>Figure 2 displays the histograms and scatter plot of the EAP estimates of person parameters. The distribution of ability estimates shows a slight positive skew. This is supported by a calculated skewness of 0.0529. While a few examinees have estimated abilities between 0.5 and 2.5, most fall within the range of −1.5 to 0.5. The frequency distribution of correct rates, as demonstrated in Figure 1, aligns with the histogram of ability estimates. Specifically, there are more examinees with low correct rates than those with high correct rates, further validating the accuracy of our estimation. Similarly, the majority of missing propensity estimates are around 0.5. However, there are more examinees with high missing propensity than those with low missing propensity, with a few values below one. Additionally, the estimated correlation coefficient ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>ρ</mi><mrow><mi>θ</mi><mi>τ</mi></mrow></msub><annotation encoding="application/x-tex">$\rho _{\theta \tau }$</annotation></semantics></math> </ephtml> ) between the ability and missing propensity parameter is −0.2681. The scatter plot illustrates that as ability estimates decrease, the estimates of missing propensity parameters increase.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/MEA/01mar26/jedm12428-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jedm12428-fig-0002.jpg" title="2 Frequency histograms and a scatter plot of the EAP estimates of person parameters for the empirical example data." /> </p> <p></p> <hd id="AN0192629997-30">Conclusion</hd> <p>Handling missing data is a crucial challenge in educational and psychological assessments. If not handled properly, missing data can negatively impact the validity and reliability of the test results. Therefore, researchers need to develop statistical models that can account for different types of missing data mechanisms to ensure the accuracy of their findings.</p> <p>Although various models have been proposed for modeling missing data, recent research has increasingly focused on distinguishing different types of item nonresponse, such as not‐reached and omitted items, and examining them under different missing data mechanisms. Unlike these studies, we propose a novel and general‐purpose model, that is, the 1PLMDP model, that provides a unified approach to handling missing data. This model reduces the risk of over‐assumptions or misclassification of missing data types. Moreover, when a particular type of missing data is rare, methods that explicitly differentiate between missing mechanisms can introduce unnecessary complexity. The 1PLMDP model effectively accommodates item nonresponse under various MNAR mechanisms while incorporating the monotonicity assumption, offering a more flexible and comprehensive approach to modeling missing data.</p> <p>Our monotonicity assumption differs from that of Glas and Pimentel ([<reflink idref="bib19" id="ref95">19</reflink>]), which relies solely on item position or time constraints. Instead, our approach captures the cumulative nature of missing behaviors, reflecting the dynamic characteristics of examinees' response patterns and state changes. Specifically, if an examinee fails to respond to a given item, the likelihood of missing subsequent items increases. This phenomenon is distinct from a simple decline in response rates due to item difficulty, as item difficulty alone does not directly influence subsequent missing behaviors. In contrast, the monotonicity assumption reflects the continuity of missing responses during the test‐taking process; once an examinee chooses to not respond, the probability of missing subsequent items further increases. Compared to traditional MIRT models, the 1PLMDP model not only retains the advantages of traditional models, such as considering both person and item factors in missing responses, but also overcomes their limitations like the assumption of local independence among item responses. Additionally, our research introduces a flexible framework for modeling missing data that can accommodate both MAR and MNAR mechanisms by applying constraints to model parameters. These constraints enable the model to adapt to real‐world data, ensuring alignment with the appropriate missing data mechanism and enhancing its applicability. Two Bayesian model evaluation criteria have been developed to identify the most likely missing data mechanism.</p> <p>The 1PLMDP model has broad applications beyond educational testing, extending to personality assessments. In such assessments, certain items may be skipped due to their difficulty or sensitive nature. The model helps identify these missing response patterns, offering insights into examinee behavior. For instance, questions involving personal or sensitive topics may lead some respondents to withhold answers due to privacy concerns. By analyzing these missing data patterns, the 1PLMDP model can reveal underlying response tendencies. A key feature of the model is the monotonicity assumption, which suggests that if an examinee has missing responses on earlier items, they are more likely to miss subsequent items as well. This assumption is particularly relevant in personality assessments, where individuals may consistently avoid certain types of content. For example, if a respondent tends to skip highly difficult or sensitive items, they are more likely to omit similar items later in the test. This characteristic allows researchers to examine not only the occurrence of missing data but also the psychological and behavioral factors that contribute to item nonresponses. Future research could further extend the model by incorporating covariates to explore how different factors influence missing responses, enhancing the understanding of missing data mechanisms.</p> <p>While the 1PLMDP model offers a flexible approach to handling missing data, it is essential to acknowledge its limitations and areas for improvement. The monotonic missingness assumption may not always hold in practical testing environments, particularly in computer‐adaptive testing. In such testing, the selection of subsequent items is dynamically determined based on the examinee's ability and previous responses, resulting in individualized test sequences. This differs from the assumption of a fixed pattern of item nonresponses, potentially limiting the model's applicability to adaptive assessments. Moreover, implementing the model within this assumption requires specialized statistical techniques, which may present challenges for researchers without extensive expertise in Bayesian methods or missing data analysis. Lu et al. ([<reflink idref="bib35" id="ref96">35</reflink>]) have demonstrated the efficiency of the SSA in estimating model parameters through theoretical analysis and extensive simulation studies. Compared to the traditional MH algorithm, the SSA exhibits significant advantages, particularly in eliminating the need for tuning acceptance probabilities and avoiding reliance on conjugate prior distributions. The SSA has been successfully applied in various IRT‐based models (Zhang et al., [<reflink idref="bib55" id="ref97">55</reflink>], [<reflink idref="bib56" id="ref98">56</reflink>]), proving its practical value. However, despite its strengths, SSA can become computationally intensive when applied to large‐scale datasets with numerous examinees, items, or missing responses. To address this issue, future research should focus on optimizing computational efficiency. One potential solution is to develop a dedicated R package with Fortran or C++ integration to support large‐scale applications, reducing computational load. However, such optimizations alone may not fully resolve the complexity of SSA for large datasets. Further research is needed to refine the algorithm and explore alternative Bayesian estimation techniques that balance computational demands with efficiency, ultimately enhancing the practicality of SSA in large‐scale assessments.</p> <hd id="AN0192629997-31">Acknowledgments</hd> <p>This research was supported by the general projects of National Social Science Fund of China on Statistics (Grant No. 23BTJ067).</p> <hd id="AN0192629997-32">Conflict of Interest Statement</hd> <p>The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.</p> <hd id="AN0192629997-33">Ethics Statement</hd> <p>Our research does not involve animal and human studies</p> <p>GRAPH: Supporting Information</p> <ref id="AN0192629997-34"> <title> References </title> <blist> <bibl id="bib1" idref="ref57" type="bt">1</bibl> <bibtext> Albert, J. H. (1992). 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Journal of the Korean Statistical Society, 50, 955–996.</bibtext> </blist> </ref> <aug> <p>By Jiwei Zhang; Jing Lu and Zhaoyuan Zhang</p> <p>Reported by Author; Author; Author</p> <p></p> <p>JIWEI ZHANG is an Associate Professor at Faculty of Education, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun, Jilin, China; email: zhangjw713@nenu.edu.cn.</p> <p>JING LU is an Associate Professor at School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Key Laboratory of Big Data Analysis of Jilin Province, Northeast Normal University, Changchun, Jilin, China; email: luj282@nenu.edu.cn.</p> <p>ZHAOYUAN ZHANG is an Associate Professor at School of Mathematics and Statistics, Institute of Applied Mathematics, Yili Normal University, Yining, China; email: zhangzy328@nenu.edu.cn.</p> </aug> <nolink nlid="nl1" bibid="bib44" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib45" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib26" firstref="ref5"></nolink> <nolink nlid="nl4" bibid="bib32" firstref="ref6"></nolink> <nolink nlid="nl5" bibid="bib46" firstref="ref7"></nolink> <nolink nlid="nl6" bibid="bib11" firstref="ref8"></nolink> <nolink nlid="nl7" bibid="bib39" firstref="ref9"></nolink> <nolink nlid="nl8" bibid="bib47" firstref="ref10"></nolink> <nolink nlid="nl9" bibid="bib23" firstref="ref11"></nolink> <nolink nlid="nl10" bibid="bib24" firstref="ref12"></nolink> <nolink nlid="nl11" bibid="bib30" firstref="ref13"></nolink> <nolink nlid="nl12" bibid="bib38" firstref="ref14"></nolink> <nolink nlid="nl13" bibid="bib10" firstref="ref15"></nolink> <nolink nlid="nl14" bibid="bib12" firstref="ref16"></nolink> <nolink nlid="nl15" bibid="bib13" firstref="ref17"></nolink> <nolink nlid="nl16" bibid="bib41" firstref="ref18"></nolink> <nolink nlid="nl17" bibid="bib19" firstref="ref19"></nolink> <nolink nlid="nl18" bibid="bib25" firstref="ref20"></nolink> <nolink nlid="nl19" bibid="bib57" firstref="ref21"></nolink> <nolink nlid="nl20" bibid="bib20" firstref="ref24"></nolink> <nolink nlid="nl21" bibid="bib43" firstref="ref25"></nolink> <nolink nlid="nl22" bibid="bib51" firstref="ref34"></nolink> <nolink nlid="nl23" bibid="bib33" firstref="ref36"></nolink> <nolink nlid="nl24" bibid="bib34" firstref="ref37"></nolink> <nolink nlid="nl25" bibid="bib40" firstref="ref38"></nolink> <nolink nlid="nl26" bibid="bib52" firstref="ref40"></nolink> <nolink nlid="nl27" bibid="bib21" firstref="ref41"></nolink> <nolink nlid="nl28" bibid="bib42" firstref="ref45"></nolink> <nolink nlid="nl29" bibid="bib27" firstref="ref47"></nolink> <nolink nlid="nl30" bibid="bib28" firstref="ref48"></nolink> <nolink nlid="nl31" bibid="bib31" firstref="ref49"></nolink> <nolink nlid="nl32" bibid="bib37" firstref="ref52"></nolink> <nolink nlid="nl33" bibid="bib35" firstref="ref53"></nolink> <nolink nlid="nl34" bibid="bib22" firstref="ref55"></nolink> <nolink nlid="nl35" bibid="bib36" firstref="ref56"></nolink> <nolink nlid="nl36" bibid="bib16" firstref="ref59"></nolink> <nolink nlid="nl37" bibid="bib50" firstref="ref60"></nolink> <nolink nlid="nl38" bibid="bib55" firstref="ref62"></nolink> <nolink nlid="nl39" bibid="bib56" firstref="ref63"></nolink> <nolink nlid="nl40" bibid="bib53" firstref="ref65"></nolink> <nolink nlid="nl41" bibid="bib48" firstref="ref70"></nolink> <nolink nlid="nl42" bibid="bib14" firstref="ref72"></nolink> <nolink nlid="nl43" bibid="bib15" firstref="ref75"></nolink> <nolink nlid="nl44" bibid="bib17" firstref="ref76"></nolink> <nolink nlid="nl45" bibid="bib18" firstref="ref80"></nolink>
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  Label: Abstract
  Group: Ab
  Data: Item nonresponses frequently occurs in educational and psychological assessments, and if not appropriately handled, it can undermine the reliability of the results. This study introduces a missing data model based on the missing not at random (MNAR) mechanism, incorporating the monotonic missingness assumption to capture individual-level missingness patterns and behavioral dynamics. In specific, the cumulative number of missing indicators allows to consider the tendency of current item's missingness based on the previous missingnesses, which reduces the number of nuisance parameters for modeling missing data mechanisms. Two Bayesian model evaluation criteria were developed to distinguish between missing at random (MAR) and MNAR mechanisms by imposing specific parameter constraints. Additionally, the study introduces a highly efficient Bayesian slice sampling algorithm to estimate the model parameters. Four simulation studies were conducted to show the performance of the proposed model. The PISA 2015 science data was carried out to further illustrate the application of the proposed approach.
– 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: EJ1501423
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1501423
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1111/jedm.12428
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 34
    Subjects:
      – SubjectFull: Item Response Theory
        Type: general
      – SubjectFull: Research Problems
        Type: general
      – SubjectFull: Models
        Type: general
      – SubjectFull: Bayesian Statistics
        Type: general
      – SubjectFull: Evaluation Criteria
        Type: general
      – SubjectFull: Sampling
        Type: general
      – SubjectFull: Algorithms
        Type: general
      – SubjectFull: Achievement Tests
        Type: general
      – SubjectFull: Foreign Countries
        Type: general
      – SubjectFull: International Assessment
        Type: general
      – SubjectFull: Secondary School Students
        Type: general
      – SubjectFull: Program for International Student Assessment
        Type: general
    Titles:
      – TitleFull: Modeling Missing Response Data in Item Response Theory: Addressing Missing Not at Random Mechanism with Monotone Missing Characteristics
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Jiwei Zhang
      – PersonEntity:
          Name:
            NameFull: Jing Lu
      – PersonEntity:
          Name:
            NameFull: Zhaoyuan Zhang
    IsPartOfRelationships:
      – BibEntity:
          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
ResultId 1