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Are the Signs of Factor Loadings Arbitrary in Confirmatory Factor Analysis? Problems and Solutions

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Title: Are the Signs of Factor Loadings Arbitrary in Confirmatory Factor Analysis? Problems and Solutions
Language: English
Authors: Dandan Tang (ORCID 0009-0007-3453-9660), Steven M. Boker, Xin Tong
Source: Structural Equation Modeling: A Multidisciplinary Journal. 2025 32(1):142-154.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 13
Publication Date: 2025
Sponsoring Agency: National Institutes of Health (NIH) (DHHS)
Contract Number: 4R37HD05830527
Document Type: Journal Articles
Reports - Research
Descriptors: Test Validity, Factor Analysis, Replication (Evaluation), Social Science Research, Behavioral Science Research, Test Reliability, Error of Measurement, Statistical Significance, Robustness (Statistics), Structural Equation Models
DOI: 10.1080/10705511.2024.2351102
ISSN: 1070-5511
1532-8007
Abstract: The replication crisis in social and behavioral sciences has raised concerns about the reliability and validity of empirical studies. While research in the literature has explored contributing factors to this crisis, the issues related to analytical tools have received less attention. This study focuses on a widely used analytical tool - confirmatory factor analysis (CFA) - and investigates one issue that is typically overlooked in practice: accurately estimating factor-loading signs. Incorrect loading signs can distort the relationship between observed variables and latent factors, leading to unreliable or invalid results in subsequent analyses. Our study aims to investigate and address the estimation problem of factor-loading signs in CFA models. Based on an empirical demonstration and Monte Carlo simulation studies, we found current methods have drawbacks in estimating loading signs. To address this problem, three solutions are proposed and proven to work effectively. The applications of these solutions are discussed and elaborated.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1457215
Database: ERIC
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  Value: <anid>AN0182192618;7mz01jan.25;2025Jan15.04:19;v2.2.500</anid> <title id="AN0182192618-1">Are the Signs of Factor Loadings Arbitrary in Confirmatory Factor Analysis? Problems and Solutions </title> <p>The replication crisis in social and behavioral sciences has raised concerns about the reliability and validity of empirical studies. While research in the literature has explored contributing factors to this crisis, the issues related to analytical tools have received less attention. This study focuses on a widely used analytical tool - confirmatory factor analysis (CFA) - and investigates one issue that is typically overlooked in practice: accurately estimating factor-loading signs. Incorrect loading signs can distort the relationship between observed variables and latent factors, leading to unreliable or invalid results in subsequent analyses. Our study aims to investigate and address the estimation problem of factor-loading signs in CFA models. Based on an empirical demonstration and Monte Carlo simulation studies, we found current methods have drawbacks in estimating loading signs. To address this problem, three solutions are proposed and proven to work effectively. The applications of these solutions are discussed and elaborated.</p> <p>Keywords: Confirmatory factor analysis; factor loading signs; identification methods; replication crisis</p> <hd id="AN0182192618-2">1. Introduction</hd> <p>The replication crisis in psychology has aroused great attention to the lack of reproducibility in empirical studies across social and behavioral sciences, as well as in other scientific disciplines. This raises concerns about the reliability and validity of empirical research within these fields (Cockburn et al., [<reflink idref="bib8" id="ref1">8</reflink>]; Oberauer & Lewandowsky, [<reflink idref="bib27" id="ref2">27</reflink>]). Traditionally, replication, the practice of repeating a study to confirm its finding, is the bedrock of scientific validation. Recently, however, more and more previously accepted findings have proven difficult or impossible to replicate (Camerer et al., [<reflink idref="bib7" id="ref3">7</reflink>]; Collaboration, [<reflink idref="bib9" id="ref4">9</reflink>]; Youyou et al., [<reflink idref="bib43" id="ref5">43</reflink>]). Open Science Collaboration ([<reflink idref="bib9" id="ref6">9</reflink>]) found a replication rate of 36% among 97 experiments from papers published in 2008 in three high-ranking psychology journals. Nosek et al. ([<reflink idref="bib26" id="ref7">26</reflink>]) found only 64% of 307 experiments replicated. Even some famous studies, like the social psychological study" elderly-walking" conducted by social psychologist John Bargh and colleagues, and its relevant studies failed to replicate, casting doubts on the theory of the goal priming effect (Bargh et al., [<reflink idref="bib5" id="ref8">5</reflink>]; Harris et al., [<reflink idref="bib14" id="ref9">14</reflink>]; Muthukrishna & Henrich, [<reflink idref="bib25" id="ref10">25</reflink>]; Pashler et al., [<reflink idref="bib28" id="ref11">28</reflink>]).</p> <p>Many studies have investigated the problem of replication, identifying factors that may contribute to it, including p-hacking or Cherry-Picking results, publication bias, low statistical power, questionable research practices, inadequate research training, failure to share data and methods, pressure to publish, complexity of scientific research, etc. For instance, <emph>p</emph>-hacking, manipulating data of statistical analyses to achieve a significant result (<emph>p</emph> <.05; Crane, [<reflink idref="bib10" id="ref12">10</reflink>]), may lead to non-replicable research outcomes. A researcher may analyze data in various ways but only report a significant result or stop collecting data as soon as a significant result is found. Klein et al. ([<reflink idref="bib17" id="ref13">17</reflink>]) discovered that among 28 classic and contemporary published findings, 54% of the replications had statistically significant results with a significance level of <emph>p</emph> <.05. Publication bias means that journals tend to publish papers with significant/positive results rather than non-significant/negative results (Wagner, [<reflink idref="bib41" id="ref14">41</reflink>]). This leads researchers to pursue significant outcomes, sometimes regardless of rigorous methodology. Franco et al. ([<reflink idref="bib12" id="ref15">12</reflink>]) found that only 20% of the studies published in social sciences journals reported non-significant results, and 60% of the studies discovered non-significant results but never reported them. In practice, although a study may yield significant findings, the associated statistical power could be low (Anderson & Maxwell, [<reflink idref="bib2" id="ref16">2</reflink>]), meaning there is a high probability of failing to reject a false null hypothesis in the future. A meta-analysis of 44 reviews of statistical power observed a mean statistical power of 0.24 to detect a small effect size (d =.20) with a type I error rate of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mn>.05</mn></mrow></math> </ephtml> (Smaldino & McElreath, [<reflink idref="bib33" id="ref17">33</reflink>]).</p> <p>Although many factors leading to the replication crisis have been investigated, few studies have considered the issues with analytical tools (Van Lissa et al., [<reflink idref="bib39" id="ref18">39</reflink>]). Analytical tools, comprising statistical methods and software, are the backbone of empirical research, allowing scientists to discover patterns, make inferences, and establish relationships among variables based on the collected data (Ali & Bhaskar, [<reflink idref="bib1" id="ref19">1</reflink>]; Zhong-Lin et al., [<reflink idref="bib45" id="ref20">45</reflink>]). Reliable and consistent use of analytical tools ensures robust research outcomes. If a study uses flawed or inappropriate statistical methods, its findings could be artifacts of the analysis rather than true reflections of underlying phenomena (Tang & Tong, [<reflink idref="bib36" id="ref21">36</reflink>]; Wen et al., [<reflink idref="bib42" id="ref22">42</reflink>]). Marcoulides and Yuan ([<reflink idref="bib23" id="ref23">23</reflink>]) highlighted this issue by demonstrating that many structural equation models, previously considered having 'good fit' based on model fit criteria in published psychological studies, failed to reproduce when re-evaluated through equivalence testing.</p> <p>In this study, we focus on one of the most popular analytical tools in social and behavioral sciences, confirmatory factor analysis (CFA), which has been implemented in every structural equation modeling program, such as Mplus, lavaan, and OpenMx (Asparouhov & Muthen, [<reflink idref="bib4" id="ref24">4</reflink>]; Boker et al., [<reflink idref="bib6" id="ref25">6</reflink>]; Rosseel, [<reflink idref="bib32" id="ref26">32</reflink>]). This technique is particularly important in the development and validation of measurement instruments, such as psychological tests and surveys, because it can test theoretical expectations about the relationships between variables and validate the structure of a test or survey (Lin et al., [<reflink idref="bib21" id="ref27">21</reflink>]). Based on CFA models, we can test measurement invariance to check whether the measurement of the psychological construct is invariant across groups or times (Vandenberg & Lance, [<reflink idref="bib40" id="ref28">40</reflink>]). CFA is broadly applied in practice, as reflected in the growing number of studies in databases. A search for" confirmatory factor analysis" in the PsychINFO database yields 456 articles from 1961-1990, 8820 articles from 1991-2010, and 23,652 articles from 2011 to now.</p> <p>Despite the CFA's popularity and well-established methodological foundations, there are still potential problems with this analytical tool. When testing the measurement invariance (MI) using CFA, one important step is to check factor loading invariance, which requires that factor loadings are the same across groups or time. For instance, it is discussed in the literature that the Children's Depression Inventory (CDI) is valid to have a one-factor structure with CFA (Stumper et al., [<reflink idref="bib35" id="ref29">35</reflink>]) among adolescents, and the factor loadings of the one-factor structure are invariant between 13-year-old and 16-year-old adolescents. However, when we replicated the study to test whether the structure of CDI is invariant in the same data but without missingness, the factor loadings in the 13-year-old group were all negative, but in the 16-year-old group, they were all positive. The absolute values of the corresponding loadings between the two groups were very close. However, according to the definition, we were not able to conclude that these loadings are invariant between the two groups. The natural question is, do factor loading signs matter or not? This question serves as the motivation for our study.</p> <p>In CFA, factor loadings represent the strength and direction of the relationship between observed variables and latent factors. A positive loading indicates that the observed score increases as the latent factor score increases. A negative loading indicates that the observed variable score decreases as the latent factor score increases. Thus, an incorrect sign can misinterpret the relationship between an observed variable and the latent factor. If the loading sign in a CFA model is inconsistent or" incorrect" (e.g., not in line with theoretical expectations or prior findings), subsequent analyses based on this model can produce unreliable or invalid results. Furthermore, if other researchers attempt to replicate the study and find different signs of factor loadings or can't reproduce the results, they may not trust the original findings.</p> <p>Several previous studies have noticed the change of factor loading signs from the perspective of mathematical reasoning. Jöreskog ([<reflink idref="bib15" id="ref30">15</reflink>], [<reflink idref="bib16" id="ref31">16</reflink>]) proposed some constraints to achieve "local" identification except sign changes in the CFA model. Peeters ([<reflink idref="bib29" id="ref32">29</reflink>]) proposed an additional sign constraint that fixes factor loadings, either strict positivity or strict negativity, to achieve global identification. This constraint has since been adopted in innovative latent factor models, including Bayesian inequality-constrained CFA (Peeters et al., [<reflink idref="bib30" id="ref33">30</reflink>]), factor analysis neural drift diffusion models (Turner et al., [<reflink idref="bib37" id="ref34">37</reflink>]), and meta-analytic structural equation models (Uanhoro, [<reflink idref="bib38" id="ref35">38</reflink>]). However, CFA has two primary fixed methods of model identification: fixed factor variance and fixed factor loading (Steiger, [<reflink idref="bib34" id="ref36">34</reflink>]). The sign constraint primarily addresses the method involving fixed factor variance, neglecting the alternative approach (Graves & Merkle, [<reflink idref="bib13" id="ref37">13</reflink>]). Moreover, most software packages capable of conducting CFA analyses do not account for the potential issue of sign reversal, potentially resulting in inconsistent or unexpected signs of factor loadings.</p> <p>This article aims to investigate the reasons behind the factor loading sign issue in CFA, considering both model identification methods, and propose solutions to improve the reliability and validity of empirical research findings by ensuring consistency of factor loading signs. Furthermore, using both empirical demonstration and Monte Carlo simulation studies, this article will not only highlight the limitations of current analytical tools in handling sign reversals but also advance the understanding of the underlying causes of sign inconsistencies. The proposed solutions will be tested and validated in simulation and empirical studies and implemented in most CFA software packages, offering researchers practical strategies to minimize the impact of sign reversal on the interpretation of CFA results.</p> <p>The outline of this article is as follows. We begin with a real-data example to demonstrate the factor loading sign problem in CFA models. Next, we will briefly introduce the estimation method for CFA models, investigate the reasons for the sign reversal problem, and propose solutions. Thereafter, two simulations will be conducted to evaluate the impact of the problem and the effectiveness of these solutions. Then, the empirical example will be re-analyzed to illustrate these solutions in most CFA software packages. Last, the article discusses the prevalence of the factor loading sign problem and provides practical recommendations for researchers.</p> <hd id="AN0182192618-3">2. An Empirical Example</hd> <p>In this section, we provide a detailed illustration of the estimation problem in a CFA model using a real data example from the study conducted by Stumper et al. ([<reflink idref="bib35" id="ref38">35</reflink>]). The factor structure of the CDI was investigated using a sample of 227 adolescents aged approximately 13 at baseline (T1) and 16 at follow-up (T2). The CDI has 27 items, scaling 0-2. Stumper et al. ([<reflink idref="bib35" id="ref39">35</reflink>]) used a weighted least square mean and variance (WLSMV) estimation method in <emph>Mplus</emph> (Muthén & Muthén, [<reflink idref="bib24" id="ref40">24</reflink>]) to analyze the data because the observed variables were binary. To replicate the analysis, we first analyzed the full data, including missing values, using diagonally weighted least squares (DWLS; (Rosseel, [<reflink idref="bib32" id="ref41">32</reflink>])) in the <emph>lavaan</emph> package in R (see the codes in our GitHub), which in <emph>Mplus</emph> is given by the WLSMV estimators (Asparouhov & Muthen, [<reflink idref="bib4" id="ref42">4</reflink>]). As shown in Table 1, the loadings from the one-factor model at both T1 and T2 were positive. However, after removing 12 subjects who did not finish the CDI at age 16, the loadings at T1 became negative, but at T2, they were still positive. Note that although this example refers to missing data, it aims to provide two datasets, illustrating the opposite signs of the factor loadings between groups.</p> <p>Table 1. Factor loadings for a one-factor model of the CDI at time 1 and time 2.</p> <p> <ephtml> <table><thead><tr><td /><td>With missingness</td><td>Without missingness</td></tr><tr><td>Item</td><td>T1</td><td>T2</td><td>T1</td><td>T2</td></tr></thead><tbody valign="top"><tr><td>CDI1</td><td char=".">0.772</td><td char=".">0.746</td><td char=".">−0.773</td><td char=".">0.742</td></tr><tr><td>CDI2</td><td char=".">0.649</td><td char=".">0.649</td><td char=".">−0.639</td><td char=".">0.660</td></tr><tr><td>CDI3</td><td char=".">0.618</td><td char=".">0.659</td><td char=".">−0.614</td><td char=".">0.655</td></tr><tr><td>CDI4</td><td char=".">0.534</td><td char=".">0.716</td><td char=".">−0.525</td><td char=".">0.741</td></tr><tr><td>CDI5</td><td char=".">0.552</td><td char=".">0.692</td><td char=".">−0.567</td><td char=".">0.688</td></tr><tr><td>CDI6</td><td char=".">0.477</td><td char=".">0.544</td><td char=".">−0.464</td><td char=".">0.538</td></tr><tr><td>CDI7</td><td char=".">0.747</td><td char=".">0.829</td><td char=".">−0.747</td><td char=".">0.826</td></tr><tr><td>CDI8</td><td char=".">0.604</td><td char=".">0.740</td><td char=".">−0.637</td><td char=".">0.761</td></tr><tr><td>CDI9</td><td char=".">0.675</td><td char=".">0.741</td><td char=".">−0.690</td><td char=".">0.738</td></tr><tr><td>CDI10</td><td char=".">0.822</td><td char=".">0.770</td><td char=".">−0.823</td><td char=".">0.766</td></tr><tr><td>CDI11</td><td char=".">0.769</td><td char=".">0.708</td><td char=".">−0.765</td><td char=".">0.703</td></tr><tr><td>CDI12</td><td char=".">0.571</td><td char=".">0.551</td><td char=".">−0.577</td><td char=".">0.546</td></tr><tr><td>CDI13</td><td char=".">0.503</td><td char=".">0.626</td><td char=".">−0.532</td><td char=".">0.618</td></tr><tr><td>CDI14</td><td char=".">0.536</td><td char=".">0.700</td><td char=".">−0.541</td><td char=".">0.696</td></tr><tr><td>CDI15</td><td char=".">0.451</td><td char=".">0.576</td><td char=".">−0.457</td><td char=".">0.580</td></tr><tr><td>CDI16</td><td char=".">0.650</td><td char=".">0.638</td><td char=".">−0.644</td><td char=".">0.631</td></tr><tr><td>CDI17</td><td char=".">0.692</td><td char=".">0.578</td><td char=".">−0.694</td><td char=".">0.569</td></tr><tr><td>CDI18</td><td char=".">0.416</td><td char=".">0.554</td><td char=".">−0.404</td><td char=".">0.559</td></tr><tr><td>CDI19</td><td char=".">0.400</td><td char=".">0.462</td><td char=".">−0.399</td><td char=".">0.455</td></tr><tr><td>CDI20</td><td char=".">0.819</td><td char=".">0.827</td><td char=".">−0.814</td><td char=".">0.824</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr><tr><td>CDI27</td><td char=".">0.673</td><td char=".">0.714</td><td char=".">−0.691</td><td char=".">0.711</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note:</emph> The table only lists part of the results because of limited space. For complete results, please see our GitHub.</p> <p>When testing measurement invariance, researchers could not conclude that these loadings are invariant between T1 and T2 because, for the data containing missing values, the loadings are in the same direction, whereas for the data without missingness, they have different signs. Such an observation could lead researchers to presume inaccuracies in the estimation of loadings at T1, particularly when the absolute values of these loadings appear similar across time points, potentially refusing further metric invariance testing. Moreover, while this example presents similar absolute values of these loadings despite the sign inconsistency, it is important to acknowledge the possibility of encountering practical scenarios where both the absolute values differ significantly, and the signs are inconsistent. In such instances, researchers might be even more inclined to give up metric invariance testing.</p> <p>One of the most popular methods to evaluate measurement invariance is multiple-group CFA. With this method, metric invariance is often assessed by a chi-square difference test or by comparing fit indices (such as the Comparative Fit Index and Root Mean Square Error of Approximation). However, the chi-square difference test may not be a valid and reliable approach for assessing MI. Yuan and Chan ([<reflink idref="bib44" id="ref43">44</reflink>]) found that the chi-square difference test performs poorly and fails to control either type I or type II errors in MI testing. Furthermore, for fit indices, some researchers proposed different criteria, and some researchers questioned fit indices entirely because of their lack of precision (Putnick & Bornstein, [<reflink idref="bib31" id="ref44">31</reflink>]). There is no consensus about the best fit indices or cutoff values for alternative fit indices under all conditions. Thus, this method may not work well in testing MI.</p> <p>Additionally, the data with missingness and without missingness at T1 are from the same population, but the signs of the loadings are opposite. This indicates that results from a later analysis may not confirm the previous analysis. Researchers may even suspect that removing the 12 subjects with missing values changed the sample distribution.</p> <hd id="AN0182192618-4">3. Estimation Method</hd> <p>We will first introduce the estimation method for CFA models to understand the problem of the factor loading signs. If the observed variables <bold>x</bold> are continuous, a CFA model can be expressed as:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">x</mi><mo>=</mo><mi>τ</mi><mo>+</mo><mi>Λ</mi><mi>η</mi><mo>+</mo><mi>ϵ</mi><mo>,</mo></mrow></math> </ephtml> </p> <p>where Λ is a vector of factor loadings,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>η</mi></mrow></math> </ephtml> is a vector of latent factors,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>τ</mi></mrow></math> </ephtml> is a vector of latent intercepts, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow></math> </ephtml> is a vector of measurement errors. It is assumed that the measurement error is normally distributed with mean</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn mathvariant="bold">0</mn></mrow></math> </ephtml> and variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Θ</mi></mrow><mo>,</mo></math> </ephtml> namely,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo>,</mo><mi mathvariant="bold">Θ</mi><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> </p> <p>If the observed variables <bold>x</bold> are categorical and have <emph>C</emph> categories, the CFA model can be expressed as:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold">x</mi></mrow><mo>∗</mo></msup><mo>=</mo><mi>τ</mi><mo>+</mo><mi>Λ</mi><mi>η</mi><mo>+</mo><mi>ϵ</mi><mo>,</mo></mrow></math> </ephtml> </p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold">x</mi></mrow><mo>∗</mo></msup></mrow></math> </ephtml> is an underlying continuous variable that is related to <bold>x</bold> though a set of <emph>C</emph> + 1 thresholds,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">v</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>C</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>v</mi></mrow><mn>0</mn></msub><mo>=</mo><mo>−</mo><mi>∞</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>C</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>∞</mi></mrow><mo>.</mo></math> </ephtml> The probability of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">x</mi><mo>=</mo><mi>c</mi></mrow></math> </ephtml> is given as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo>=</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mi>c</mi></msub><mo>≤</mo><msup><mrow><mi>x</mi></mrow><mo>∗</mo></msup><mo>≤</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>C</mi></mrow><mo>.</mo></math> </ephtml> The covariance structure of the CFA model is</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Σ</mi><mo>=</mo><mi mathvariant="bold">Λ</mi><mi mathvariant="bold">Φ</mi><mi mathvariant="bold">Λ</mi><mo>′</mo><mo>+</mo><mi mathvariant="bold">Θ</mi><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib1" id="ref45">1</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Σ</mi></mrow></math> </ephtml> is the covariance matrix implied by the CFA model, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Φ</mi></mrow></math> </ephtml> is the covariance of the latent factors (Liu et al., [<reflink idref="bib22" id="ref46">22</reflink>]).</p> <p>Maximum Likelihood Estimation (MLE) is often applied to estimate model parameters when the observed variables are continuous and normally distributed. MLE aims to find the parameter values that maximize the likelihood function, meaning the parameter values maximize the probability of observing the current sample data (Li, [<reflink idref="bib19" id="ref47">19</reflink>]; Tang & Tong, [<reflink idref="bib36" id="ref48">36</reflink>]). In the CFA model, the likelihood function is given as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">F</mi><mo>=</mo><mtext mathvariant="italic">ln</mtext><mo>|</mo><mi mathvariant="bold">Σ</mi><mo>|</mo><mo>−</mo><mtext mathvariant="italic">ln</mtext><mo>|</mo><mi mathvariant="bold">S</mi><mo>|</mo><mo>+</mo><mtext mathvariant="italic">tr</mtext><mo stretchy="false">(</mo><mi mathvariant="bold">S</mi><msup><mrow><mi mathvariant="bold">Σ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>−</mo><mi>p</mi><mo>,</mo></mrow></math> </ephtml> </p> <p>where <emph>p</emph> is the number of the observed variables, and <bold>S</bold> is the covariance matrix of the observed variables if the observed variables <bold>x</bold> are continuous, or is the covariance matrix of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold">x</mi></mrow><mo>∗</mo></msup></mrow></math> </ephtml> if the observed variables <bold>x</bold> are categorical.</p> <p>When the observed variables are categorical, weighted least square (WLS) estimation is often applied to estimate model parameters. WLS aims to find the parameter values that minimize the fit function, meaning the parameter values minimize the difference between the observed data and the theoretical model based on a weight matrix (Asparouhov & Muthen, [<reflink idref="bib4" id="ref49">4</reflink>]; Li, [<reflink idref="bib19" id="ref50">19</reflink>]). In the CFA model, the fit function can be expressed by</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">F</mi></mrow><mrow><mtext mathvariant="bold">wls</mtext></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>′</mo><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold">W</mi></mrow><mrow><mo>−</mo><mn mathvariant="bold">1</mn></mrow></msup><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>where <emph>θ</emph> is the vector of model parameters, <bold>W</bold> is the weight matrix,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is the model-implied vector containing the nonredundant, vectorized elements of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>,</mo></math> </ephtml> and <emph>s</emph> is the vector containing the unique elements of sample statistics (i.e., threshold and polychoric correlation estimates; (Li, [<reflink idref="bib19" id="ref51">19</reflink>])). If <bold>W</bold> is a diagonal matrix, where off-diagonal entries are 0, and diagonal entries remain the same, WLS will become WLSMV (Asparouhov & Muthen, [<reflink idref="bib4" id="ref52">4</reflink>]).</p> <hd id="AN0182192618-5">3.1. Problems and Solutions</hd> <p>Because the latent factor itself does not have a natural scale, Equation (<reflink idref="bib1" id="ref53">1</reflink>) is not identified. To identify it, two approaches are often used in popular software for CFA, such as <emph>lavaan</emph>, <emph>Mplus</emph>, and <emph>OpenMx</emph> (Asparouhov & Muthen, [<reflink idref="bib4" id="ref54">4</reflink>]; Boker et al., [<reflink idref="bib6" id="ref55">6</reflink>]; Rosseel, [<reflink idref="bib32" id="ref56">32</reflink>]). One approach is to fix the factor variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>η</mi><mn>2</mn></msubsup></mrow><mo>,</mo></math> </ephtml> a diagonal element of the covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Φ</mi></mrow><mo>,</mo></math> </ephtml> to 1, and the factor means</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>μ</mi></mrow><mi>η</mi></msub></mrow></math> </ephtml> to 0, that is, fixed factor variance. Another approach is to fix one of the factor loadings on each factor to 1, that is, fixed factor loading.</p> <p>From Equation (<reflink idref="bib1" id="ref57">1</reflink>), we obtain</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub><mo stretchy="false">)</mo><mtext> or</mtext><mo /><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mi>i</mi><mo>∗</mo></msubsup><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mi>j</mi><mo>∗</mo></msubsup><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mi>i</mi></msub><msub><mrow><mi>λ</mi></mrow><mi>j</mi></msub><msubsup><mrow><mi>σ</mi></mrow><mi>η</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mtext mathvariant="italic">ij</mtext></mrow><mn>2</mn></msubsup><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib2" id="ref58">2</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>p</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>p</mi></mrow><mo>.</mo></math> </ephtml> Without loss of generality, we assume that when <emph>i</emph> =<emph> j</emph>,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mtext mathvariant="italic">ij</mtext></mrow><mn>2</mn></msubsup><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mi>i</mi><mn>2</mn></msubsup></mrow><mo>;</mo></math> </ephtml> otherwise,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mtext mathvariant="italic">ij</mtext></mrow><mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math> </ephtml> When using the first fixing method to identify the model, Equation (<reflink idref="bib2" id="ref59">2</reflink>) can be written as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub><mo stretchy="false">)</mo><mtext> or</mtext><mo /><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mi>i</mi><mo>∗</mo></msubsup><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mi>j</mi><mo>∗</mo></msubsup><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mi>i</mi></msub><msub><mrow><mi>λ</mi></mrow><mi>j</mi></msub><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mtext mathvariant="italic">ij</mtext></mrow><mn>2</mn></msubsup><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib3" id="ref60">3</reflink>)</p> <p>If the covariance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub><mo stretchy="false">)</mo><mtext> or</mtext><mo /><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mi>i</mi><mo>∗</mo></msubsup><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mi>j</mi><mo>∗</mo></msubsup><mo stretchy="false">)</mo></mrow></math> </ephtml> is positive, Equation (<reflink idref="bib3" id="ref61">3</reflink>) indicates that <emph>λ<subs>i</subs></emph> and <emph>λ<subs>j</subs></emph> should be both positive or negative. Whether positive or negative, the loadings are correctly estimated, but the direction of the estimated loadings could be opposite. This explains why the loadings at T1 from the data without missing values are all negative in the empirical example. However, when factor loading signs are opposite, the interpretation of the factor also has opposite meanings.</p> <p>When using the second fixing method to identify the model, Equation (<reflink idref="bib2" id="ref62">2</reflink>) can be written as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub><mo stretchy="false">)</mo><mtext> or</mtext><mo /><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mi>j</mi><mo>∗</mo></msubsup><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mi>j</mi></msub><msubsup><mrow><mi>σ</mi></mrow><mi>η</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mn>1</mn><mi>j</mi></mrow><mn>2</mn></msubsup><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib4" id="ref63">4</reflink>)</p> <p>Equation (<reflink idref="bib4" id="ref64">4</reflink>) indicates that <emph>λ<subs>j</subs></emph> should keep the same sign as the covariance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub><mo stretchy="false">)</mo><mtext> or</mtext><mo /><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mi>j</mi><mo>∗</mo></msubsup><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> However, if the true value of the first loading is negative but fixed to 1, this will also lead to a direction problem in the estimation. To solve this estimation problem, we propose to fix a positive loading to 1, which can help other loadings to be of the expected sign.</p> <p>To fit a CFA model using MLE or WLS, an optimization algorithm iteratively adjusts the parameter estimates to find the values that maximize the likelihood or minimize the fit function (Kochenderfer & Wheeler, [<reflink idref="bib18" id="ref65">18</reflink>]). As shown by the symmetric curve in Figure 1, the optimal solution can fall in the negative or positive range of the x-axis. To solve this estimation problem, we propose to set the lower or upper bounds of the loadings to be larger (smaller) than 0 when the true values of the loadings are expected to be larger (smaller) than 0.</p> <p>Graph: Figure 1. The sample graph of the likelihood and fit function 1.</p> <p>Furthermore, the optimization algorithm needs a starting value for each parameter. The starting value is an initial guess for the parameter, and then the estimation procedure iteratively refines this starting guess to converge toward the parameter's most optimal value (Kochenderfer & Wheeler, [<reflink idref="bib18" id="ref66">18</reflink>]). Sometimes, algorithms can get stuck in" local" solutions, which are specific to the region around the starting values, rather than finding the" global" solution, which is the best solution overall (Arora et al., [<reflink idref="bib3" id="ref67">3</reflink>]). Figure 2 shows parameter estimates can be negative when algorithms get stuck in" local" solutions at the negative x-axis. To solve this estimation problem, we propose adjusting starting values to obtain factor loading estimates with expected signs. In addition, a large positive or small negative starting value may help obtain positive or negative estimates.</p> <p>Graph: Figure 2. The sample graph of the likelihood and fit function 2.</p> <p>We conducted two Monte Carlo simulations to understand the estimation problem and proposed solutions comprehensively. The first simulation study will assess how the estimation problem affects the signs and consistency of factor loadings across the above CFA software under various situations. The second simulation study will further assess our solutions to the problem of factor loading signs in CFA.</p> <hd id="AN0182192618-6">4. Simulation 1: Investigating the Estimation Problem</hd> <p></p> <hd id="AN0182192618-7">4.1. Simulation Design</hd> <p>We conducted a Monte Carlo simulation study to systematically investigate the estimation problems with different fixed methods in CFA. Regardless of continuous or categorical observed variables, the CFA model has the same estimation problem. Thus, the simulation focuses on continuous observed variables using MLE.</p> <p>To illustrate the impact of the estimation problem on the direction of factor loadings in the simplest form, this simulation used the simplest and perfectly identified model, a one-factor CFA model with three continuous indicators, as the population model (see Figure 3) from which the data were generated. This is because if such a fundamental model has the problem of loadings' signs related to starting values, it logically follows that more complex and potentially unidentified models would encounter similar challenges. In the population model, the absolute values of all the loadings</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo>|</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> were 0.7, the variance of the indicators</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>k</mi><mn>2</mn></msubsup></mrow></math> </ephtml> was 1, the variance of the latent factor</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>η</mi><mn>2</mn></msubsup></mrow></math> </ephtml> was 1, and the intercepts were set at 0.</p> <p>DIAGRAM: Figure 3. Path diagram of a confirmatory factor analysis model.</p> <p>In the simulation, we varied the loadings in terms of their signs, creating four distinct conditions:</p> <p></p> <ulist> <item> Condition 1: All loadings are positive.</item> <p></p> <item> Condition 2: All loadings are negative.</item> <p></p> <item> Condition 3: One of the three loadings is negative.</item> <p></p> <item> Condition 4: Two of the three loadings are negative.</item> </ulist> <p>The sample size was set at 200 for the simulation study. For each condition, 500 datasets were generated. We fitted the population model to each dataset in three popular CFA software packages, <emph>lavaan 0.6-16</emph>, <emph>Mplus 8.10</emph>, and <emph>OpenMx 2.21.8</emph>. These analyses used various fixing techniques. In the fixing factor variance approach, we used both the default starting values for factor loadings provided by the software packages and also manually set these starting values. Therefore, to thoroughly assess the impact of starting values on the direction of factor loadings, we used the default starting value, a positive starting value of 1, and a negative starting value of −1. For the fixed factor loading approach, the common default in most software is to fix the first-factor loading at 1. As a result, the four methods used in this simulation are:</p> <p></p> <ulist> <item> Method 1: Fixing factor variance with default starting values of the factor loadings.</item> <p></p> <item> Method 2: Fixing factor variance with starting values of the factor loadings set to 1.</item> <p></p> <item> Method 3: Fixing factor variance with starting values of the factor loadings set to -1.</item> <p></p> <item> Method 4: Fixing the first loading at 1.</item> </ulist> <hd id="AN0182192618-8">4.2. Results</hd> <p>To evaluate the estimation problem with the fixed methods, we calculated the directional consistency rate (DCR) to measure the possibility that the direction of the loading estimates is the same as the direction of the true values among 500 replicates. DCR can be expressed by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">DCR</mtext><mo>=</mo><mfrac><mi>M</mi><mi>N</mi></mfrac><mo>×</mo><mn>100</mn><mi>%</mi></mrow><mo>,</mo></math> </ephtml> where <emph>M</emph> is the number of factor loading estimates with the same directional sign as the true values, and <emph>N</emph> is the total number of factor loading estimates across all replicates. Table 2 presents the directional consistency rates for four fixing methods under various conditions in <emph>lavaan</emph>. When using Method 4, where the first-factor loading is fixed at 1, DCR was not computed for this loading as it was fixed and cannot be freely estimated. Under various conditions, the DCR for each loading was either 100% or 0%. A DCR of 100% implies that across the replicates, the estimated factor loadings have the same directional sign (either all positive or all negative) as the true factor loadings, which is complete consistency. Conversely, a DCR of 0% indicates that each estimated loading is directionally opposite to its true value, which is a complete lack of consistency.</p> <p>Table 2. Directional consistency rates (%) for simulation 1 with four fixing methods (<emph>lavaan</emph>).</p> <p> <ephtml> <table><thead><tr><td>Fixed method</td><td>Condition 1</td><td>Condition 2</td><td>Condition 3</td><td>Condition 4</td></tr><tr><td>loading</td><td>0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td></tr></thead><tbody valign="top"><tr><td>Method 1</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td></tr><tr><td>Method 2</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td></tr><tr><td>Method 3</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">100</td></tr><tr><td>Method 4</td><td /><td char=".">100</td><td char=".">100</td><td /><td char=".">0</td><td char=".">0</td><td /><td char=".">0</td><td char=".">0</td><td /><td char=".">0</td><td char=".">0</td></tr></tbody></table> </ephtml> </p> <p>Method 1 could lead to loading estimates with inconsistent signs. For Methods 2 and 3, if the sign of the starting values was not consistent with true factor loadings, these loading estimates could have incorrect signs. For Method 4, the same directional inconsistency could arise when the first loading was fixed to 1, but its population value was negative. This implies that fixing a negative loading to 1 could mistakenly estimate positive loadings as negative or vice versa. Thus, the simulation findings underscore the potential drawbacks of these four fixing methods, particularly the risk of obtaining estimated loadings with unexpected signs.</p> <p>We further validated our findings by running the simulation using <emph>Mplus 8.10</emph> and <emph>OpenMx 2.21.8</emph>. Tables 3 and 4 display results largely consistent with those obtained using <emph>lavaan</emph>. Although some difference is observed between Method 1 and 2, the overall results from <emph>Mplus</emph> and <emph>OpenMx</emph> further validate our findings that these four fixing methods may lead to loading estimates with inconsistent signs. Additionally, despite the unknown default starting values in these three software packages, Method 1 still can not guarantee the loading estimates in the expected directions.</p> <p>Table 3. Directional consistency rates (%) for simulation 1 with four fixing methods (<emph>mplus</emph>).</p> <p> <ephtml> <table><thead><tr><td>Fixed methods</td><td>Condition 1</td><td>Condition 2</td><td>Condition 3</td><td>Condition 4</td></tr><tr><td>loading</td><td>0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td></tr></thead><tbody valign="top"><tr><td>Method 1</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td></tr><tr><td>Method 2</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td></tr><tr><td>Method 3</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">100</td></tr><tr><td>Method 4</td><td /><td char=".">100</td><td char=".">100</td><td /><td char=".">0</td><td char=".">0</td><td /><td char=".">0</td><td char=".">0</td><td /><td char=".">0</td><td char=".">100</td></tr></tbody></table> </ephtml> </p> <p>Table 4. Directional consistency rates (%) for simulation 1 with four fixing methods (<emph>OpenMx</emph>).</p> <p> <ephtml> <table><thead><tr><td>Fixed methods</td><td>Condition 1</td><td>Condition 2</td><td>Condition 3</td><td>Condition 4</td></tr><tr><td>loading</td><td>0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td></tr></thead><tbody valign="top"><tr><td>Method 1</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td></tr><tr><td>Method 2</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">0</td></tr><tr><td>Method 3</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">0</td><td char=".">0</td><td char=".">0</td><td char=".">100</td><td char=".">100</td><td char=".">100</td></tr><tr><td>Method 4</td><td /><td char=".">100</td><td char=".">100</td><td /><td char=".">0</td><td char=".">0</td><td /><td char=".">0</td><td char=".">0</td><td /><td char=".">0</td><td char=".">0</td></tr></tbody></table> </ephtml> </p> <hd id="AN0182192618-9">5. Simulation 2: Solutions to the Estimation Problem</hd> <p></p> <hd id="AN0182192618-10">5.1. Simulation Design</hd> <p>To address the issue of directional inconsistency in CFA factor loading estimates, we will propose two practical solutions and one potential solution that can yield estimates in a more interpretable and consistent direction. Furthermore, a Monte Carlo simulation was conducted to assess the effectiveness of these three solutions to the estimation problems for CFA. This simulation used the same population model as Simulation 1, with a sample size of 200 and 500 datasets generated for each condition.</p> <hd id="AN0182192618-11">5.1.1. Solution 1: Fixing a Positive Factor Loading at 1</hd> <p>Our previous simulation findings, particularly with Method 4, indicated that fixing a negative loading at 1 led to inconsistencies in the directions of other loadings. Conversely, fixing a positive loading at 1 aligned other loadings in the expected directions. Therefore, we used this solution in <emph>lavaan 0.6-16</emph>, <emph>Mplus 8.10</emph>, and <emph>OpenMx 2.21.8</emph> for the population model under Conditions 1, 3, and 4, where not all loadings are negative.</p> <hd id="AN0182192618-12">5.1.2. Solution 2: Setting Bounds for Factor Loadings</hd> <p>This solution is to fix the factor variance and set a lower bound for the inherently positive loadings and an upper bound for the inherently negative loadings. For instance, specifying 0 as the lower and upper bounds for positive and negative loadings, respectively. This solution is uniquely implementable in the OpenMx R package, which allows setting the same bound for all loading and a separate bound for each loading. Consequently, we used this solution in <emph>OpenMx 2.21.8</emph>, using 0 as the bound under Conditions 1 and 2, where each condition has the factor loadings with uniform signs. Specifically, we set the lower bound of 0 for all loadings under Condition 1 and an upper bound of 0 under Condition 2.</p> <hd id="AN0182192618-13">5.1.3. Solution 3: Adjusting Starting Values of Factor Loadings</hd> <p>The potential solution is to fix the factor variance and set the starting values of the factor loadings in the same direction as the true loadings. As suggested by our simulation results for Methods 2 and 3, setting a start value consistent with the direction of true loadings could steer the estimates toward the desired direction. Therefore, we used this solution in <emph>lavaan 0.6-16</emph>, <emph>Mplus 8.10</emph>, and <emph>OpenMx 2.21.8</emph> for each dataset under Conditions 1 to 4. Specifically, we set a large positive start value of 1 for a true positive loading or a small negative start value of −1 for a true negative loading.</p> <hd id="AN0182192618-14">5.2. Results</hd> <p>The DCR was again computed to evaluate the effectiveness of three Solutions. Tables 5–7 present the DCR results for three solutions under various conditions. Solution 1, which fixes a positive factor loading at 1, successfully aligned other loading estimates in their correct direction. In Solution 2, fixing factor variance with a lower bound of 0 or an upper bound of 0 for all loadings effectively ensured that all loadings were consistently positive or negative. Solution 3, fixing the factor variance and setting the starting values to 1 or −1, proved effective in shifting the loadings toward the desired direction. From these findings, we concluded that Solutions 1 and 2 can solve the issue of directional inconsistency in factor loading, and Solution 3 can obtain the preferred direction of factor loadings.</p> <p>Table 5. Directional consistency rates (%) for Solution 1.</p> <p> <ephtml> <table><thead><tr><td>Fixed method</td><td>Condition 1</td><td>Condition 3</td><td>Condition 4</td></tr><tr><td>loading</td><td>0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td></tr></thead><tbody valign="top"><tr><td>Solution 1 (<italic>lavaan</italic>)</td><td char=".">100</td><td char=".">100</td><td /><td char=".">100</td><td char=".">100</td><td /><td char=".">100</td><td char=".">100</td><td /></tr><tr><td>Solution 1 (<italic>Mplus</italic>)</td><td char=".">100</td><td char=".">100</td><td /><td char=".">100</td><td char=".">100</td><td /><td char=".">100</td><td char=".">100</td><td /></tr><tr><td>Solution 1 (<italic>OpenMx</italic>)</td><td char=".">100</td><td char=".">100</td><td /><td char=".">100</td><td char=".">100</td><td /><td char=".">100</td><td char=".">100</td><td /></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note:</emph> Under each condition, the last factor loading was fixed at 1.</p> <p>Table 6. Directional consistency rates (%) for Solution 2.</p> <p> <ephtml> <table><thead><tr><td>Fixed method</td><td>Condition 1</td><td>Condition 2</td></tr><tr><td>loading</td><td>0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td></tr></thead><tbody valign="top"><tr><td>Solution 2 (<italic>OpenMX</italic>)</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td></tr></tbody></table> </ephtml> </p> <p>Table 7. Directional consistency rates (%) for Solution 3.</p> <p> <ephtml> <table><thead><tr><td>Fixed method</td><td>Condition 1</td><td>Condition 2</td><td>Condition 3</td><td>Condition 4</td></tr><tr><td>loading</td><td>0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td><td>0.7</td><td>−0.7</td><td>−0.7</td><td>0.7</td></tr></thead><tbody valign="top"><tr><td>Solution 3 (<italic>lavaan</italic>)</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td></tr><tr><td>Solution 3 (<italic>Mplus</italic>)</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td></tr><tr><td>Solution 3 (<italic>OpenMX</italic>)</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td><td char=".">100</td></tr></tbody></table> </ephtml> </p> <hd id="AN0182192618-15">6. Reanalysis of the Empirical Example with the Solutions</hd> <p>In practice, we can use the three solutions to handle the issue of directional inconsistency in factor loading estimates. However, not all proposed solutions can be applied to popular CFA software packages. Specifically, <emph>lavaan</emph> and <emph>OpenMx</emph> support the first solution for both continuous and categorical variables, while <emph>Mplus</emph> only supports continuous variables. The second solution can only be applied in <emph>OpenMx</emph>. The third solution can be used in these three software. However, <emph>OpenMx</emph> can not handle more than 20 categorical observed variables. Detailed implementation guidelines and codes are available in the Appendix 1.</p> <p>Since there are more than 20 categorical observed variables in the empirical example, the second solution can not be applied in this example. Only Solutions 1 and 3 are used through <emph>lavaan</emph> and <emph>Mplus</emph> (see the codes in our GitHub). For Solution 1, the first-factor loading is fixed at 1 because the first item, Mood (Questions related to feelings of sadness or happiness) and the latent factor of Children's Depression have a positive relationship given substantive psychological theory. For Solution 3, a positive start value of 1 is applied because all the items have a positive relationship with the latent factor of Children's Depression. The re-analysis results are presented in Table 8. All loadings at T1 in the data without missingness are positive. This implies that Solutions 1 and 3 work for this example, and the previous non-replicated results were due to how CFA was implemented.</p> <p>Table 8. Factor loadings for a one-factor model of the CDI at time 1 in the data without missingness.</p> <p> <ephtml> <table><thead><tr><td /><td><italic>lavaan</italic></td><td><italic>Mplus</italic></td></tr><tr><td>Item</td><td>Solution 1</td><td>Solution 3</td><td>Solution 1</td><td>Solution 3</td></tr></thead><tbody valign="top"><tr><td>CDI1</td><td char=".">1.000</td><td char=".">0.773</td><td char=".">1.000</td><td char=".">0.775</td></tr><tr><td>CDI2</td><td char=".">0.827</td><td char=".">0.639</td><td char=".">0.820</td><td char=".">0.635</td></tr><tr><td>CDI3</td><td char=".">0.795</td><td char=".">0.614</td><td char=".">0.793</td><td char=".">0.615</td></tr><tr><td>CDI4</td><td char=".">0.679</td><td char=".">0.525</td><td char=".">0.675</td><td char=".">0.523</td></tr><tr><td>CDI5</td><td char=".">0.733</td><td char=".">0.567</td><td char=".">0.732</td><td char=".">0.567</td></tr><tr><td>CDI6</td><td char=".">0.600</td><td char=".">0.464</td><td char=".">0.614</td><td char=".">0.476</td></tr><tr><td>CDI7</td><td char=".">0.967</td><td char=".">0.747</td><td char=".">0.898</td><td char=".">0.696</td></tr><tr><td>CDI8</td><td char=".">0.824</td><td char=".">0.637</td><td char=".">0.822</td><td char=".">0.637</td></tr><tr><td>CDI9</td><td char=".">0.892</td><td char=".">0.690</td><td char=".">0.890</td><td char=".">0.689</td></tr><tr><td>CDI10</td><td char=".">1.064</td><td char=".">0.823</td><td char=".">1.062</td><td char=".">0.823</td></tr><tr><td>CDI11</td><td char=".">0.990</td><td char=".">0.765</td><td char=".">1.002</td><td char=".">0.777</td></tr><tr><td>CDI12</td><td char=".">0.747</td><td char=".">0.577</td><td char=".">0.717</td><td char=".">0.555</td></tr><tr><td>CDI13</td><td char=".">0.689</td><td char=".">0.532</td><td char=".">0.685</td><td char=".">0.531</td></tr><tr><td>CDI14</td><td char=".">0.701</td><td char=".">0.541</td><td char=".">0.694</td><td char=".">0.538</td></tr><tr><td>CDI15</td><td char=".">0.592</td><td char=".">0.457</td><td char=".">0.592</td><td char=".">0.459</td></tr><tr><td>CDI16</td><td char=".">0.833</td><td char=".">0.644</td><td char=".">0.832</td><td char=".">0.645</td></tr><tr><td>CDI17</td><td char=".">0.898</td><td char=".">0.694</td><td char=".">0.894</td><td char=".">0.693</td></tr><tr><td>CDI18</td><td char=".">0.522</td><td char=".">0.404</td><td char=".">0.531</td><td char=".">0.412</td></tr><tr><td>CDI19</td><td char=".">0.516</td><td char=".">0.399</td><td char=".">0.500</td><td char=".">0.388</td></tr><tr><td>CDI20</td><td char=".">1.053</td><td char=".">0.814</td><td char=".">1.050</td><td char=".">0.813</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr><tr><td>CDI27</td><td char=".">0.895</td><td char=".">0.691</td><td char=".">0.894</td><td char=".">0.693</td></tr></tbody></table> </ephtml> </p> <p>3 <emph>Note:</emph> The table only lists part of the results because of limited space. For complete results, please see our GitHub.</p> <hd id="AN0182192618-16">7. Prevalence of the Issue and Practical Suggestions</hd> <p>From a mathematical standpoint, estimated factor loading directions in CFA models appear to be no issue because the sign of the corresponding factor is also reversed. However, when we consider the practical implications of factor loadings, the two fixed methods may result in inconsistent signs. This inconsistency can pose challenges for applied researchers who rely on these loadings for theoretical explanations or aim to replicate previous findings in a meaningful way. However, The issue of estimating factor loading direction in CFA models has often been overlooked in practice. Yet, its significance cannot be overstated, especially in comparative and longitudinal studies. Comparative research requires close attention to factor loading signs, particularly in measurement invariance tests. As the practical example shows, ignoring the sign of factor loadings makes it difficult to determine whether the loadings are invariant across groups or over time. Furthermore, in longitudinal research, which tracks individual changes over time (Fitzmaurice et al., [<reflink idref="bib11" id="ref68">11</reflink>]), inconsistent loading signs may distort true within-individual changes. Additionally, the issue extends to cross-sectional studies where mismatched loading signs can make it difficult to validate the structure of a scale (Lin et al., [<reflink idref="bib20" id="ref69">20</reflink>]).</p> <p>Although the simulation studies used the simplest model to illustrate the impact of the estimation problem on the direction of factor loadings and the effectiveness of three solutions, we expanded our investigation to include two additional models: a two-factor CFA model and a three-factor CFA model, each with three indicators per factor. Our simulation results across these varied model complexities consistently demonstrate that the starting values' impact on the direction of factor loadings remains unchanged, and three solutions still work well. To make these supplementary results accessible, we have uploaded the additional simulations to our GitHub repository for reference. Therefore, any study using CFA should consider the problem of factor loading signs in case the results of this study can not be replicated due to loading signs. Furthermore, this estimation problem is not confined to CFA models but is also prevalent in other structural equation models, as they employ similar estimation techniques. It is imperative to address the issue in all SEM models to avoid a potential replication dilemma.</p> <p>To address the problem of loading signs, we proposed three solutions to reflect the true relationship between observed variables and latent factors. Even if this article only discussed it in CFA models, the solutions can be applied in all structural equation models. The first solution, fixing a positive loading at 1, can ensure that other loadings in the model are in the expected direction. Using substantive theory, researchers can identify an observed variable with a positive correlation with a latent factor and fix the loading on this variable to 1. However, researchers must identify and fix one positive loading for each latent factor. The second solution, fixing the factor variance and setting a bound for loadings, can ensure that all loadings are in the correct direction. However, to set the right bound for each loading, researchers must determine all the relationships between observed variables and latent factors. The third solution, fixing the factor variance and setting the starting values of loadings at larger values with the same direction as the loadings, leads the loadings to the desired direction but does not guarantee it. Like the second solution, it requires a comprehensive understanding of the relationships between observed variables and latent factors.</p> <hd id="AN0182192618-17">Acknowledgement</hd> <p>The authors gratefully thank Dr. Caspar van Lissa for the valuable guidance on the empirical example.</p> <hd id="AN0182192618-18">Author Contributions</hd> <p>Dandan Tang: Formal Analysis, Investigation, Methodology, Visualization, Writing-original draft; Steven M. Boker: Methodology, Supervision, Writing-review & editing; Xin Tong: Supervision, Writing-review & editing.</p> <hd id="AN0182192618-19">Disclosure Statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <hd id="AN0182192618-20">Appendices</hd> <p></p> <hd id="AN0182192618-21">Continuous variables</hd> <p>Suppose a one-factor model with three continuous variables. This is the R code for <bold>Solution 1</bold>, a positive loading at 1 using <emph>lavaan</emph> package. If the true value of the first loading is positive, the first loading can be fixed at 1 by "auto.fix.first = TRUE"</p> <p> <emph> # load package</emph> <bold>library</bold>(lavaan)# read data<bold>data</bold> <-<bold> read.csv</bold>('data.csv',header = T)# CFA model for T1CFA <- 'F=∼x1 + x2 + x3'# The true value of the first loading is# positive, so the first loading is# fixed at 1 by "auto.fix.first = TRUE"result <- cfa(CFA, <bold>data = data,</bold>auto.<bold>fix.</bold>first = TRUE, <bold>ordered</bold> = FALSE)# print result<bold>summary</bold>(result)</p> <p>If the first loading is not positive, we can move a positive loading to the first place. for example, if the loading on item x2 is positive, we can set the CFA model as follows:</p> <p>CFA_reloaction <- 'F=∼x2 + x1 + x3'</p> <p>This is the R code for <bold>Solution 3</bold> using <emph>lavaan</emph> package. The factor variance can be fixed by "std.lv = TRUE", and the starting values of the loadings can be set at 1 or −1 by "start(<reflink idref="bib1" id="ref70">1</reflink>)*" or "start(-1)*".</p> <p> <emph> # set start values of the</emph># loadings at 1 by "start(<reflink idref="bib1" id="ref71">1</reflink>)*"CFA_start <- 'F=∼start(<reflink idref="bib1" id="ref72">1</reflink>)*x1 + start(<reflink idref="bib1" id="ref73">1</reflink>)*x2+start(<reflink idref="bib1" id="ref74">1</reflink>)*x3'# fix the factor variance by "std.lv = TRUE"result <- cfa(CFA<bold>_start</bold>, <bold>data = data,</bold>std.lv = TRUE, <bold>ordered</bold> = FALSE)# print results<bold>summary</bold>(result)</p> <p>This is <emph>Mplus</emph> code for <bold>Solution 1</bold>, fixing a positive loading at 1. If the true value of the first loading is positive, the first loading is fixed at 1 by "@1".</p> <p> <bold>! read data</bold>DATA: FILE = data.dat;! name observed variablesVARIABLE: NAMES = x1-x3;Model:! fix the first loading at 1 by "@1"F by x1@1 x2 x3;</p> <p>If the first loading is not positive, we can move a positive loading to the first place. for example, if the loading on item x2 is positive, we can set its loadings at 1 by "@1".</p> <p>DATA: FILE = <bold>data.</bold>dat; VARIABLE: NAMES = x1-x3; Model:<bold>! fix</bold> the second loading at 1 <bold>by</bold> "@1" F <bold>by</bold> x1 x2@1 x3;</p> <p>This is <emph>Mplus</emph> code for <bold>Solution 3</bold>, fixing the factor variance by "@1" and setting the starting values of the loadings to 1 or −1 by "*1" or "*-1".</p> <p>DATA: FILE = <bold>data.</bold>dat; VARIABLE: NAMES = x1-x3; Model:<bold>! set</bold> the starting values of! the loadings to 1 <bold>by</bold> "<bold>*</bold>1"F <bold>by</bold> x1-x3*1;<bold>! fix</bold> the <bold>factor</bold> variance <bold>by</bold> "@1" F@1;</p> <p>This is the R code for <bold>Solution 1</bold>, a positive loading at 1 using <emph>OpenMx</emph> package. If the true value of the first loading is positive, the first loading is fixed at 1 by "mxPath(from = c("F"), to = c("x1"), arrows = 1, free = FALSE, values = 1)".</p> <p> <emph># load package</emph> <bold>library</bold>(OpenMx)# read data<bold>data <- read.csv</bold>("data.csv")# name variables and parametersindicators <- <bold>names(data)</bold>latents <- c("F")loadingLabels <-<bold> paste</bold>("b_",indicators, sep="")uniqueLabels <-<bold> paste</bold>("U_",indicators, sep="")meanLabels <-<bold> paste</bold>("M_", indicators, sep="")factorVarLabels <-<bold> paste</bold>("Var_", latents, sep="")# build modeloneFactorRaw1 <- mxModel("Single factor Model with Fixed Loading",type= "RAM",manifestVars = indicators,latentVars = latents,mxPath(from = latents, to = indicators,<bold>arrows </bold>=<bold></bold>1, connect= "unique.bivariate",free = TRUE, values=.2,<bold>labels</bold>=loadingLabels),mxPath(from = c("F"), to = c("x1"),<bold>arrows </bold>=<bold></bold>1, free = FALSE, values = 1),mxPath(from = indicators,<bold>arrows </bold>=<bold></bold>2,free = TRUE, values=.8,<bold>labels</bold>=uniqueLabels),mxPath(from = latents,<bold>arrows </bold>=<bold></bold>2,free = TRUE, values=.8,<bold>labels</bold>=factorVarLabels),mxPath(from="one", to = indicators,arrows = 1, free = TRUE, values=.1,labels = meanLabels),mxData(observed = data, type="raw"))oneFactorRaw1Out <- mxRun(oneFactorRaw1)# print results<bold>summary</bold>(oneFactorRaw1Out)</p> <p>If the first loading is not positive, we can move a positive loading to the first place. for example, if the loading on item x2 is positive, we can set its loadings at 1 by "mxPath(from = c("F"), to = c("x2"), arrows = 1, free = FALSE, values = 1)".</p> <p> <emph> # build model</emph>oneFactorRaw1 <- mxModel("Single factor Model with Fixed Loading",type= "RAM",manifestVars = indicators,latentVars = latents,mxPath(from = latents, to = indicators,<bold>arrows </bold>=<bold></bold>1, connect= "unique.bivariate",free = TRUE, values=.2,<bold>labels</bold>=loadingLabels),mxPath(from = c("F"), to = c("x2"),<bold>arrows </bold>=<bold></bold>1, free = FALSE, values = 1),mxPath(from = indicators,<bold>arrows </bold>=<bold></bold>2,free = TRUE, values=.8,<bold>labels</bold>=uniqueLabels),mxPath(from = latents,<bold>arrows </bold>=<bold></bold>2,free = TRUE, values=.8,<bold>labels</bold>=factorVarLabels),mxPath(from="one", to = indicators,<bold>arrows </bold>=<bold></bold>1, free = TRUE, values=.1,<bold>labels</bold>=meanLabels),mxData(observed=<bold>data</bold>, type="raw"))</p> <p>This is the R code for <bold>Solution 3</bold> using <emph>OpenMx</emph> package. The factor variance is fixed to 1 by setting "free = FALSE and values = 1" and the starting values of the loadings are set at 1 by "free = TRUE and values = 1".</p> <p> <emph> # build model</emph>oneFactorRaw1 <- mxModel("Single Factor model with Fixed Variance",type= "RAM",manifestVars = indicators,latentVars = latents,mxPath(from = latents, to = indicators,<bold>arrows </bold>=<bold></bold>1, connect= "unique.bivariate",# set the starting values of# the loadings at 1.free = TRUE, values = 1,<bold>labels</bold>=loadingLabels),mxPath(from = indicators,<bold>arrows </bold>=<bold></bold>2,free = TRUE, values=.8,<bold>labels</bold>=uniqueLabels),mxPath(from = latents,<bold>arrows </bold>=<bold></bold>2,# fixing factor variance to 1.free = FALSE, values = 1,<bold>labels</bold>=factorVarLabels),mxPath(from="one", to = indicators,<bold>arrows </bold>=<bold></bold>1, free = TRUE, values=.1,<bold>labels</bold>=meanLabels),mxData(observed = data, type="raw"))</p> <p>This is the R code for <bold>Solution 2</bold>, a positive loading at 1 using <emph>OpenMx</emph> package. We can set the lower bound of 0 by "lbound = 0", or the upper bound of 0 by "ubound = 0"</p> <p>oneFactorRaw1 <- mxModel("Single Factor model with Fixed Variance",type= "RAM",manifestVars = indicators,latentVars = latents,mxPath(from = latents, to = indicators,<bold>arrows </bold>=<bold></bold>1, connect= "unique.bivariate",<emph># setting the lower</emph><emph># bound of the loadings >0.</emph>free = TRUE, lbound = 0,<bold>labels</bold>=loadingLabels),mxPath(from = indicators,<bold>arrows </bold>=<bold></bold>2,free = TRUE, values=.8,<bold>labels</bold>=uniqueLabels),mxPath(from = latents,<bold>arrows </bold>=<bold></bold>2,<emph># fixing factor variance to 1.</emph>free = FALSE, values = 1,<bold>labels</bold>=factorVarLabels),mxPath(from="one", to = indicators,<bold>arrows </bold>=<bold></bold>1, free = TRUE, values=.1,<bold>labels</bold>=meanLabels),mxData(observed=<bold>data</bold>, type="raw"))</p> <hd id="AN0182192618-22">Categorical variables</hd> <p>Suppose a one-factor model with three categorical variables. If using <emph>lavaan</emph>, "ordered = FALSE" should be changed into "ordered = TRUE" and the rest part should be kept the same.</p> <p>This is <emph>Mplus</emph> code for <bold>Solution 1</bold>, fixing a positive loading at 1. If the true value of the first loading is positive, the first loading is fixed at 1 by "@1".</p> <p>DATA: FILE = data.dat; VARIABLE: NAMES = x1-x3; CATEGORICAL ARE x1-x3; ! The estimation method is WLSMV ANALYSIS: ESTIMATOR = WLSMV; Model: !! fix the first loading at 1 by "@1" F by x1@1 x2-x3*1; OUTPUT: TECH1 TECH8;</p> <p>This is <emph>Mplus</emph> code for <bold>Solution 3</bold>, fixing the factor variance by "@1" and setting the starting values of the loadings to 1 or −1 by "*1" or "*-1".</p> <p>DATA: FILE = data.dat; VARIABLE: NAMES = x1-x3; CATEGORICAL ARE x1-x3; ! 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An: EJ1457215
AccessLevel: 3
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Are the Signs of Factor Loadings Arbitrary in Confirmatory Factor Analysis? Problems and Solutions
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Dandan+Tang%22">Dandan Tang</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0007-3453-9660">0009-0007-3453-9660</externalLink>)<br /><searchLink fieldCode="AR" term="%22Steven+M%2E+Boker%22">Steven M. Boker</searchLink><br /><searchLink fieldCode="AR" term="%22Xin+Tong%22">Xin Tong</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Structural+Equation+Modeling%3A+A+Multidisciplinary+Journal%22"><i>Structural Equation Modeling: A Multidisciplinary Journal</i></searchLink>. 2025 32(1):142-154.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 13
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: SourceSuprt
  Label: Sponsoring Agency
  Group: SrcSuprt
  Data: National Institutes of Health (NIH) (DHHS)
– Name: NumberContract
  Label: Contract Number
  Group: NumCntrct
  Data: 4R37HD05830527
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Test+Validity%22">Test Validity</searchLink><br /><searchLink fieldCode="DE" term="%22Factor+Analysis%22">Factor Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Replication+%28Evaluation%29%22">Replication (Evaluation)</searchLink><br /><searchLink fieldCode="DE" term="%22Social+Science+Research%22">Social Science Research</searchLink><br /><searchLink fieldCode="DE" term="%22Behavioral+Science+Research%22">Behavioral Science Research</searchLink><br /><searchLink fieldCode="DE" term="%22Test+Reliability%22">Test Reliability</searchLink><br /><searchLink fieldCode="DE" term="%22Error+of+Measurement%22">Error of Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Significance%22">Statistical Significance</searchLink><br /><searchLink fieldCode="DE" term="%22Robustness+%28Statistics%29%22">Robustness (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Structural+Equation+Models%22">Structural Equation Models</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/10705511.2024.2351102
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1070-5511<br />1532-8007
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The replication crisis in social and behavioral sciences has raised concerns about the reliability and validity of empirical studies. While research in the literature has explored contributing factors to this crisis, the issues related to analytical tools have received less attention. This study focuses on a widely used analytical tool - confirmatory factor analysis (CFA) - and investigates one issue that is typically overlooked in practice: accurately estimating factor-loading signs. Incorrect loading signs can distort the relationship between observed variables and latent factors, leading to unreliable or invalid results in subsequent analyses. Our study aims to investigate and address the estimation problem of factor-loading signs in CFA models. Based on an empirical demonstration and Monte Carlo simulation studies, we found current methods have drawbacks in estimating loading signs. To address this problem, three solutions are proposed and proven to work effectively. The applications of these solutions are discussed and elaborated.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1457215
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1457215
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/10705511.2024.2351102
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 13
        StartPage: 142
    Subjects:
      – SubjectFull: Test Validity
        Type: general
      – SubjectFull: Factor Analysis
        Type: general
      – SubjectFull: Replication (Evaluation)
        Type: general
      – SubjectFull: Social Science Research
        Type: general
      – SubjectFull: Behavioral Science Research
        Type: general
      – SubjectFull: Test Reliability
        Type: general
      – SubjectFull: Error of Measurement
        Type: general
      – SubjectFull: Statistical Significance
        Type: general
      – SubjectFull: Robustness (Statistics)
        Type: general
      – SubjectFull: Structural Equation Models
        Type: general
    Titles:
      – TitleFull: Are the Signs of Factor Loadings Arbitrary in Confirmatory Factor Analysis? Problems and Solutions
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Dandan Tang
      – PersonEntity:
          Name:
            NameFull: Steven M. Boker
      – PersonEntity:
          Name:
            NameFull: Xin Tong
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 1070-5511
            – Type: issn-electronic
              Value: 1532-8007
          Numbering:
            – Type: volume
              Value: 32
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: Structural Equation Modeling: A Multidisciplinary Journal
              Type: main
ResultId 1