View in EDS HTML Full Text PDF Full Text

Enhancing Model Fit Evaluation in SEM: Practical Tips for Optimizing Chi-Square Tests

Saved in:
Bibliographic Details
Title: Enhancing Model Fit Evaluation in SEM: Practical Tips for Optimizing Chi-Square Tests
Language: English
Authors: Bang Quan Zheng (ORCID 0000-0003-2614-2501), Peter M. Bentler (ORCID 0000-0002-9440-721X)
Source: Structural Equation Modeling: A Multidisciplinary Journal. 2025 32(1):136-141.
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: 6
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Descriptors: Monte Carlo Methods, Structural Equation Models, Goodness of Fit, Robustness (Statistics), Evaluation Methods, Social Science Research, Behavioral Science Research, Error of Measurement, Test Reliability
DOI: 10.1080/10705511.2024.2354802
ISSN: 1070-5511
1532-8007
Abstract: This paper aims to advocate for a balanced approach to model fit evaluation in structural equation modeling (SEM). The ongoing debate surrounding chi-square test statistics and fit indices has been characterized by ambiguity and controversy. Despite the acknowledged limitations of relying solely on the chi-square test, its careful application can enhance its effectiveness in evaluating model fit and specification. To illustrate this point, we present three common scenarios relevant to social and behavioral science research using Monte Carlo simulations, where fit indices may inadequately address concerns regarding goodness-of-fit, while the chi-square statistic can offer valuable insights. Our recommendation is to report both the chi-square test and fit indices, prioritizing precise model specification to ensure the reliability of model fit indicators.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1457158
Database: ERIC
Full text is not displayed to guests.
FullText Links:
  – Type: pdflink
    Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwFEprFA-76ZBCfmBOx6U3xEAAAA4zCB4AYJKoZIhvcNAQcGoIHSMIHPAgEAMIHJBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDKNPbZfXDswcwIdKqgIBEICBm5c09kc_wF0axAN0iALhNAak9Fl0iueYmPl7oxv3E93bOWE5YKz6hlkQdAUUS-nO66IK82cgW9jr11Uj0bRbHGcHVYwGn-mbpJzP4Bfp5Q-uQX42FTO68Qif618aWoc8xlBHy2tFgkkOELb4ORbi3QhkJrttI3iMeU2SePOIQbsoDSZfxdJLwq_bIECGcIKLrYf6Ago19QT9VbLh
Text:
  Availability: 1
  Value: <anid>AN0182192619;7mz01jan.25;2025Jan15.04:19;v2.2.500</anid> <title id="AN0182192619-1">Enhancing Model Fit Evaluation in SEM: Practical Tips for Optimizing Chi-Square Tests </title> <p>This paper aims to advocate for a balanced approach to model fit evaluation in structural equation modeling (SEM). The ongoing debate surrounding chi-square test statistics and fit indices has been characterized by ambiguity and controversy. Despite the acknowledged limitations of relying solely on the chi-square test, its careful application can enhance its effectiveness in evaluating model fit and specification. To illustrate this point, we present three common scenarios relevant to social and behavioral science research using Monte Carlo simulations, where fit indices may inadequately address concerns regarding goodness-of-fit, while the chi-square statistic can offer valuable insights. Our recommendation is to report both the chi-square test and fit indices, prioritizing precise model specification to ensure the reliability of model fit indicators.</p> <p>Keywords: Chi-square test; goodness-of-fit; Monte Carlo simulation; SEM</p> <hd id="AN0182192619-2">1. Introduction</hd> <p>Social and behavioral scientists often grapple with complicated and abstract concepts such as democracy, value, ideology, identity, trust, political tolerance, and political conceptualization, among others (Acock et al., [<reflink idref="bib1" id="ref1">1</reflink>]; Davidov, [<reflink idref="bib9" id="ref2">9</reflink>]; Feldman, [<reflink idref="bib10" id="ref3">10</reflink>]; Goren, [<reflink idref="bib11" id="ref4">11</reflink>]; Sullivan et al., [<reflink idref="bib19" id="ref5">19</reflink>]; Zheng, [<reflink idref="bib23" id="ref6">23</reflink>]). They often utilize Structural Equation Modeling (SEM) with latent variables, such as confirmatory factor analysis, to estimate statistical models that amalgamate indicators, aiming to gauge the underlying latent concepts. SEM's appeal lies in its dual capability to assess hypotheses regarding the influences of latent and observable variables on other variables, while also enabling simultaneous modeling of measurement error (Yuan & Liu, [<reflink idref="bib22" id="ref7">22</reflink>]).</p> <p>The crux of SEM lies in assessing model fit, which heavily relies on chi-square test statistics and fit indices, such as the normed fit index (NFI), comparative fit index (CFI), Tucker-Lewis fit index (TLI), root mean square error approximation (RMSEA), etc. Yet, there are currently no clear guidelines for interpreting model fit integrating chi-square test statistics and fit indices. Scholars have debated the applicability of fit indices due to the absence of a unanimous consensus on which fit indices to employ. In this discourse, there are those who contend that fit indices might possess limited practical utility (Barrett, [<reflink idref="bib3" id="ref8">3</reflink>]), stressing the singular interpretation of the chi-square statistic. They express concerns that fit indices could potentially lead researchers to assert adequacy for a model that is incorrectly specified (Stone, [<reflink idref="bib18" id="ref9">18</reflink>]). Some advocate against relying solely on preset cutoff values for fit indices, as these can be deceptive and misused. This perspective also highlights the issue of "cherry-picking," whereby researchers selectively choose a fit index that conforms to their preconceived viewpoint, thereby supporting a poorly fitting model (Jackson et al., [<reflink idref="bib14" id="ref10">14</reflink>]; Kline, [<reflink idref="bib17" id="ref11">17</reflink>]; Stone, [<reflink idref="bib18" id="ref12">18</reflink>]).</p> <p>While SEM has made significant advancements in recent decades, this paper does not delve into those methodological advancements. Rather, its aim is to raise awareness and underscore the importance of adopting a well-balanced approach when evaluating model fit, while also providing guidance on its implementation. The argument presented is that the difficulties linked to chi-square tests don't solely arise from their constraints, but frequently result from a lack of proper understanding about their appropriate application. To exemplify, we will delineate three noteworthy scenarios frequently encountered by social and behavioral scientists when applying SEM, where fit indices might fall short in adequately addressing goodness-of-fit concerns, while the chi-square statistic can be effectively utilized. Scenario 1 involves misspecification, where the model used in the analysis poorly fits the data. Scenario 2 pertains to small sample sizes. A limitation of asymptotics is its lack of consideration for a statistic's behavior in small samples. Scenario 3 addresses non-normal data, where the maximum likelihood (ML) approach may not be effective. By employing alternative methods like the Lagrange Multiplier (LM) test, reweighted least squares, and Satorra-Bentler scaled robust estimators, we can achieve more accurate assessments of model fit. Furthermore, a proper interpretation and understanding of chi-square results, alongside other fit indices, are crucial for drawing accurate conclusions from data analysis. Our contention is that fit indices should not be solely relied upon as a cutoff point for model assessment; instead, researchers are strongly encouraged to report both chi-square and fit indices, with a greater emphasis on correctly specifying the model to ensure the trustworthiness of fit indices as meaningful indicators of model fit.</p> <hd id="AN0182192619-3">2. Review of Chi-Square Test and Fit Indices</hd> <p>In SEM, the model's parameters are held in the observed variable's covariance matrix</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> which can be written as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Σ</mi><mo>=</mo><mi mathvariant="bold">Λ</mi><mi mathvariant="bold">Φ</mi><msup><mrow><mi mathvariant="bold">Λ</mi></mrow><mrow><mi mathvariant="bold">′</mi></mrow></msup><mo>+</mo><mi mathvariant="bold">Ψ</mi><mo>.</mo></math> </ephtml> Here,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Λ</mi></mrow></math> </ephtml> is a matrix of factor loadings,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Φ</mi></mrow></math> </ephtml> is a matrix of factor covariances, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Ψ</mi></mrow></math> </ephtml> represents unique scores' covariances. In SEM, the expected structure of the population covariance matrix</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 denoted as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Σ</mi><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">θ</mi></mrow></math> </ephtml> includes free parameters. Given that the sample covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">S</mi></mrow></math> </ephtml> is an unbiased estimator 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> an objective function</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>[</mo><mi mathvariant="bold">Σ</mi><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>]</mo></math> </ephtml> gauges the difference between</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Σ</mi><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">S</mi></mrow><mo>.</mo></math> </ephtml> Our aim is to find</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>̂</mo></mover></mrow></mrow><mo>,</mo></math> </ephtml> the estimated value of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">θ</mi></mrow></math> </ephtml> that minimizes</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>[</mo><mi mathvariant="bold">Σ</mi><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>]</mo><mo>.</mo></math> </ephtml> This involves iterative nonlinear programming, where we begin with an initial guess</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mo>{</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>}</mo></math> </ephtml> and iteratively generate a sequence until it converges to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>̂</mo></mover></mrow></mrow><mo>,</mo></math> </ephtml> assuming smoothness in the partial derivatives of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>[</mo><mi mathvariant="bold">Σ</mi><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo>]</mo></math> </ephtml> with respect to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>.</mo></math> </ephtml> </p> <p>In covariance structure analysis with multivariate normally distributed variables, the most common method for evaluating goodness-of-fit is maximum likelihood (ML) (Jöreskog, [<reflink idref="bib15" id="ref13">15</reflink>]). Equation (<reflink idref="bib1" id="ref14">1</reflink>) fits the model implied covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Σ</mi><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> to the sample covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">S</mi></mrow></math> </ephtml> using the Wishart likelihood function.</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub><mo stretchy="true">(</mo><mrow><mi>θ</mi></mrow><mo stretchy="true">)</mo><mo>=</mo><mrow><mrow><mtext mathvariant="normal">log</mtext></mrow><mo /><mrow><mo stretchy="true">|</mo><mrow><mi mathvariant="bold">Σ</mi><mo /><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo><mo>|</mo><mo>−</mo><mtext mathvariant="normal">log</mtext></mrow><mo stretchy="true">|</mo><mi mathvariant="bold-italic">S</mi><mo>|</mo><mo>+</mo><mtext mathvariant="italic">tr</mtext><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">S</mi><msup><mrow><mi mathvariant="bold">Σ</mi><mo stretchy="true">(</mo><mrow><mi mathvariant="bold">θ</mi></mrow><mo stretchy="true">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo stretchy="true">)</mo><mo>−</mo><mi>p</mi></mrow></mrow></math> </ephtml> (<reflink idref="bib1" id="ref15">1</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub><mo>=</mo><mtext mathvariant="italic">argmin</mtext><mi mathvariant="normal" /><msub><mrow><mi>F</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub><mo>(</mo><mi mathvariant="bold-italic">θ</mi><mo>)</mo></math> </ephtml> (<reflink idref="bib2" id="ref16">2</reflink>)</p> <p>As shown in Equation (<reflink idref="bib2" id="ref17">2</reflink>), at the minimum of the fit function</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo stretchy="true">)</mo><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> contains parameter estimates</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Λ</mi></mrow><mo>̂</mo></mover></mrow></mrow><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Φ</mi></mrow><mo>̂</mo></mover></mrow></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Ψ</mi></mrow><mo>̂</mo></mover></mrow></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Λ</mi></mrow><mo>̂</mo></mover></mrow></mrow></math> </ephtml> is a matrix of estimated factor loadings,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Φ</mi></mrow><mo>̂</mo></mover></mrow></mrow></math> </ephtml> gives estimated factor covariances, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Ψ</mi></mrow><mo>̂</mo></mover></mrow></mrow></math> </ephtml> is the covariance matrix of error variables.</p> <p>Moreover, the goodness-of-fit test statistic is defined by</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub><mo>=</mo><mo stretchy="true">(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="true">)</mo><msub><mrow><mi>F</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub><mo stretchy="true">(</mo><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>̂</mo></mover></mrow></mrow></mrow><mo stretchy="true">)</mo><mo>,</mo></math> </ephtml> (<reflink idref="bib3" id="ref18">3</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> represents a test statistic calculated using</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub><mo stretchy="true">(</mo><mrow><mi>θ</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> based on the final parameter estimates and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi></mrow></math> </ephtml> is the sample size. As</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>→</mo><mi>∞</mi><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> is referred to a chi-square distribution with degrees of freedom</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">df</mtext><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>−</mo><mi>q</mi><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi></mrow></math> </ephtml> is the number of free parameters.</p> <hd id="AN0182192619-4">2.1. Limitations of Chi-Square Test</hd> <p>Like any statistical test, the chi-square test has its limitations. For social science research, the major limitations are under-specification. Fitting a structural equation model often necessitates a larger number of items. However, unlike in psychology, social science research typically faces limitations in item availability, leading to model under-specification. Consequently, the chi-square test statistics tend to be substantially larger than the degrees of freedom (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">df</mtext></mrow></math> </ephtml> ), resulting in <emph>p</emph>-values approaching zero.</p> <p>Second, the issue of statistical power, which refers to the ability to reject the null hypothesis, becomes particularly relevant when dealing with large sample sizes in SEM. Social science studies often involve relatively larger sample sizes. In SEM, the null hypothesis is that the covariance matrix Σ is equal to the model-implied covariance matrix Σ(θ). To establish a plausible structural relationship, a <emph>p</emph>-value of 0.05 or greater is typically required, which is in contrast to regression analysis, where we expect a <emph>p</emph>-value of 0.05 or less to reject the null hypothesis. Yet, owing to the properties of the chi-square test as sample size increases, the likelihood of rejecting the null hypothesis strengthens. That is, in larger samples, the model's statistical power becomes large, potentially leading to null hypothesis rejection even when the model possesses minor inaccuracies. Fit indices have been developed to provide alternative measures of model fit.</p> <p>Third, with small samples,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> may not be reliable. Based on the assumption of multivariate normality, ML method provides the most widely used estimator in SEM (Bollen, [<reflink idref="bib7" id="ref19">7</reflink>]; Hu et al., [<reflink idref="bib12" id="ref20">12</reflink>]; Jöreskog, [<reflink idref="bib15" id="ref21">15</reflink>]). The behavior of this statistic is based on asymptotic properties, that is,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi></mrow></math> </ephtml> must be sufficiently large. Previous research has found that a small sample</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi></mrow></math> </ephtml> is the main contributor to failure of asymptotic theory, but a large number of variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi></mrow></math> </ephtml> and/or parameters</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi></mrow><mo>,</mo></math> </ephtml> a small number of indicator loadings per factor, and small ratio of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi></mrow></math> </ephtml> to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">df</mtext></mrow></math> </ephtml> also contribute to spurious goodness-of-fit model rejections (Arruda & Bentler, [<reflink idref="bib2" id="ref22">2</reflink>]; Yuan & Bentler, [<reflink idref="bib21" id="ref23">21</reflink>]; Zheng & Bentler, [<reflink idref="bib26" id="ref24">26</reflink>], [<reflink idref="bib25" id="ref25">25</reflink>]).</p> <p>Fourth,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> does not work for non-normal data. To follow a chi-square distribution ML is predicated on the assumption of multivariate normality, that is, normally distributed data. Nevertheless, in real-world data analysis, violations of these assumptions are common occurrences. Attempting to fit non-normal data using ML can result in the rejection of the null hypothesis, even when the model is correct. Owing to these limitations, the interpretability of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> can be compromised. To address this issue, researchers have formulated a set of fit indices to enhance the evaluation of models.</p> <hd id="AN0182192619-5">2.2. Fit Indices</hd> <p>Among the array of fit indices, the NFI, CFI, TLI, and RMSEA emerge as the most commonly used measures for assessing model fit. All fit indices but RMESA build upon a series of nested models,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo></math> </ephtml> spanning from the most constrained to the least constrained (the saturated model). Correspondingly, their associated chi-square test statistics are</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>.</mo></math> </ephtml> Each index contributes its own set of metrics and limitations to the evaluation process, but all quantify the fit of the proposed model compared to the fit of the most constrained model. Values close to 1.0 are ideal. Bentler and Bonett ([<reflink idref="bib6" id="ref26">6</reflink>]) introduced normed fit index (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="normal">NFI</mtext></mrow></math> </ephtml> ), which is defined as:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">NFI</mtext><mo>=</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>,</mo></math> </ephtml> (<reflink idref="bib4" id="ref27">4</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> </ephtml> represents the baseline chi-square value of an independence model, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></math> </ephtml> represents the chi-square value of the particular model. Both</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></math> </ephtml> are derived from a specific fitting function, such as ML. When</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="normal">NFI</mtext></mrow></math> </ephtml> =1, signifying a perfect fit. Conversely, when</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></math> </ephtml> deviates from the expected values,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="normal">NFI</mtext><mo><</mo><mn>1</mn><mo>.</mo></math> </ephtml> Bentler ([<reflink idref="bib4" id="ref28">4</reflink>]) also proposed CFI, which is an incremental fit index that also compares the fit of the hypothesized model with that of a baseline model,</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">CFI</mtext><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>,</mo></math> </ephtml> (<reflink idref="bib5" id="ref29">5</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> </ephtml> are population noncentral parameters of hypothesized model and baseline models in practice estimated by difference between the test statistic and its degrees of freedom. Their sizes can be considered as indictors of model misspecification (Bentler, [<reflink idref="bib4" id="ref30">4</reflink>]). Like CFI, TLI is defined as:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">TLI</mtext><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>/</mo><msub><mrow><mtext mathvariant="italic">df</mtext></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><msub><mrow><mtext mathvariant="italic">df</mtext></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>,</mo></math> </ephtml> (<reflink idref="bib6" id="ref31">6</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> </ephtml> are the same as in NFI,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mtext mathvariant="italic">df</mtext></mrow><mrow><mi>k</mi></mrow></msub></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mtext mathvariant="italic">df</mtext></mrow><mrow><mi>i</mi></mrow></msub></math> </ephtml> are their respective degrees of freedom. Finally, an absolute fit index is given by</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">RMSEA</mtext><mo>=</mo><msqrt><mi mathvariant="normal">m</mi><mtext mathvariant="normal">ax</mtext><mo>⁡</mo><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><msub><mrow><mtext mathvariant="italic">df</mtext></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi>n</mi><mo>∙</mo><msub><mrow><mtext mathvariant="italic">df</mtext></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>,</mo><mn>0</mn></mrow><mo stretchy="true">)</mo></msqrt><mo>.</mo></math> </ephtml> (<reflink idref="bib7" id="ref32">7</reflink>)</p> <p>Whose size quantifies the extent of lack of fit. The better the fit, the closer the RMSEA value is to zero. The NFI has been found to be sensitive to the influence of small sample sizes, whereas variations in sample sizes have minimal impact on the CFI and TLI. Similarly, RMSEA, like NFI, is influenced by both model complexity and sample size. When the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">df</mtext></mrow></math> </ephtml> increases while</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi></mrow></math> </ephtml> decreases, RMSEA values tend to rise (Bentler, [<reflink idref="bib5" id="ref33">5</reflink>]; Kenny et al., [<reflink idref="bib16" id="ref34">16</reflink>]). A simulation study by Hu and Bentler ([<reflink idref="bib13" id="ref35">13</reflink>]) delved into the impact of various cutoff values for RMSEA, CFI, and TLI on rejection rates within accurate and misspecified models. Their findings suggested that, in general, a model can be deemed to exhibit relatively good fit when the RMSEA falls below 0.06, and both the CFI and TLI surpass 0.95.</p> <p>A limitation of these fit indices lies in their strong reliance on metrics developed under the ML estimator (Xia & Yang, [<reflink idref="bib20" id="ref36">20</reflink>]). Furthermore, if the chi-square test statistic encounters issues, the fit indices relying on them may also face challenges (Bentler, [<reflink idref="bib5" id="ref37">5</reflink>]). In this context, the primary factor for obtaining dependable model fit evaluations for both chi-square tests and fit indices is ensuring the accurate specification of models.</p> <hd id="AN0182192619-6">3. Empirical Strategy and Simulation</hd> <p>This section delves into three key scenarios relevant to social science research where fit indices might lack in sufficiently tackling goodness-of-fit concerns. We present suggestions to optimize the utilization of the chi-square statistic, enabling its effective application in addressing these challenges. We conducted Monte Carlo simulations across varying sample sizes to visually illustrate their performances. To achieve this, we begin by establishing a population model from which we draw samples. Specifically, we opt for a confirmatory factor model represented as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">Λ</mi><msub><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>=</mo><mo stretchy="true">(</mo><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mtext mathvariant="italic">ip</mtext></mrow></msub></mrow><mo stretchy="true">)</mo><mo>′</mo></math> </ephtml> is a vector of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi></mrow></math> </ephtml> observations on person</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi></mrow></math> </ephtml> in a population, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>N</mi><mo>.</mo></math> </ephtml> Under standard assumptions, this formulation leads to the covariance structure</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Σ</mi><mo>=</mo><mi mathvariant="bold">Λ</mi><mi mathvariant="bold">Φ</mi><msup><mrow><mi mathvariant="bold">Λ</mi></mrow><mrow><mi mathvariant="bold">′</mi></mrow></msup><mo>+</mo><mi mathvariant="bold">Ψ</mi><mo>.</mo></math> </ephtml> We chose a 3-factor model, with each factor being measured by 5 indicators, resulting in a total of 15 indicators. This model entails 33 free parameters and 87</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">df</mtext></mrow><mo>.</mo></math> </ephtml> The factor loading</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">Λ</mi></mrow><mrow><mi mathvariant="bold">′</mi></mrow></msup></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Φ</mi></mrow></math> </ephtml> are defined as:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">Λ</mi></mrow><mrow><mi mathvariant="bold">′</mi></mrow></msup><mo>=</mo><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mo>.</mo><mn>7</mn></mtd><mtd><mo>.</mo><mn>7</mn></mtd><mtd><mo>.</mo><mn>75</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mo>.</mo><mn>8</mn></mtd><mtd><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>.</mo><mn>7</mn></mtd><mtd><mo>.</mo><mn>75</mn></mtd><mtd><mo>.</mo><mn>8</mn><mi mathvariant="normal" /></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>.</mo><mn>7</mn></mtd><mtd><mo>.</mo><mn>7</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>.</mo><mn>75</mn></mtd><mtd><mo>.</mo><mn>8</mn></mtd><mtd><mo>.</mo><mn>8</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Φ</mi><mo>=</mo><mo stretchy="true">[</mo><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>.</mo><mn>3</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>.</mo><mn>4</mn></mtd><mtd><mo>.</mo><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo stretchy="true">]</mo></math> </ephtml> </p> <p>The data generating process consists of two steps. For a given <emph>N</emph>, a sample</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mtext> is drawn </mtext></math> </ephtml> from a covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Φ</mi></mrow><mo>,</mo></math> </ephtml> while the unique factors</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">ϵ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> </ephtml> are drawn from a multivariate normal distribution with covariance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Ψ</mi></mrow><mo>.</mo></math> </ephtml> For multivariate normal data,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mi mathvariant="normal" /></math> </ephtml> =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">Φ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>ξ</mi></mrow></msub></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo /></math> </ephtml> =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">Ψ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">Φ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="bold">Φ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="bold">Φ</mi><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">Ψ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="bold">Ψ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="bold">Ψ</mi><mo>,</mo></math> </ephtml> and both</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>ξ</mi></mrow></msub></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> </ephtml> followed a standard normal distribution</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">N</mi></mrow></math> </ephtml> (0, 1). This process was replicated 500 times across a range of sample sizes from 50 to 10,000. The performance of both the chi-square test and fit indices was computed over the 500 repetitions for each sample size.</p> <hd id="AN0182192619-7">3.1. Misspecified Model</hd> <p>When the model is misspecified,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> may not work well. In such cases, we can use the LM test for specification search and make appropriate modifications to the model accordingly. To illustrate, we need to assess the performance of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> in cases where the models are incorrectly specified and compare it with a modified model based on the LM test.</p> <p>To generate a misspecified model we modified the population model by adding an extra parameter to factors one and two respectively and set the factor loadings at the values of.3 and.2 respectively. The analysis model remains no change. Thus, the new factor loading matrix is defined as:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Λ</mi></mrow><mo>˜</mo></mover></mrow></mrow></mrow><mrow><mi mathvariant="bold">′</mi></mrow></msup><mo>=</mo><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>0.7</mn></mtd><mtd><mn>0.7</mn></mtd><mtd><mn>0.75</mn></mtd></mtr><mtr><mtd><mn>0.2</mn></mtd><mtd><mn>0.2</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mn>0.8</mn></mtd><mtd><mn>0.8</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0.7</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0.3</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0.7</mn></mtd><mtd><mn>0.75</mn></mtd><mtd><mi mathvariant="normal" /><mn>0.8</mn><mi mathvariant="normal" /></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0.3</mn></mtd></mtr><mtr><mtd><mn>0.8</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0.7</mn></mtd><mtd><mn>0.7</mn></mtd></mtr></mtable><mi mathvariant="normal" /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0.75</mn></mtd><mtd><mn>0.8</mn></mtd><mtd><mn>0.8</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo><mo>.</mo></math> </ephtml> </p> <p>Besides model misspecification, large sample sizes can also be a challenge for social science research. As the sample size increases, the tendency is for any model structure null hypothesis to face rejection. Consequently, many researchers find the large chi-square values to lack meaningful interpretation, prompting them to rely on fit indices for statistical justification in such scenarios. However, we contend that even when dealing with a large sample size, it is crucial to examine both chi-square and the LM test, along with fit indices.</p> <p>Consider the following scenario: Initially, we have a model with a chi-square value of 900, based on a</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo></math> </ephtml> 3,000 and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">df</mtext><mo>=</mo><mn>5</mn><mo>.</mo></math> </ephtml> This result is deemed rejectable, indicating a poor fit for the model. However, the LM test suggests that adding an additional parameter might be beneficial. Acting on this advice, we introduce the extra parameter, resulting in a revised chi-square value of 600 with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">df</mtext><mo>=</mo><mn>6</mn><mo>.</mo></math> </ephtml> Although the fit is still not satisfactory, the model's chi-square has dropped by 300 points for just 1</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">df</mtext></mrow><mo>,</mo></math> </ephtml> representing a highly significant improvement. This improvement through the addition of the extra parameter might hold valuable insights or meaningful relationships that were previously unidentified. Without this step, we would never have known about this potential improvement.</p> <hd id="AN0182192619-8">3.2. Small Samples</hd> <p>Earlier research has noted that the ML estimator tends to exhibit an elevated rate of null hypothesis rejection in scenarios involving small sample sizes. Addressing this, the reweighted least squares (RLS) estimator has proven to be the most effective choice for such cases. The foundation of RLS is rooted in the normal-distribution GLS function initially introduced by Browne ([<reflink idref="bib8" id="ref38">8</reflink>]),</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mtext mathvariant="italic">GLS</mtext></mrow></msub><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="normal" /><mi>t</mi><mi mathvariant="normal">r</mi><mo stretchy="true">[</mo><mrow><msup><mrow><mo stretchy="true">{</mo><mrow><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">S</mi><mo>−</mo><mi mathvariant="bold">Σ</mi><mo>(</mo><mi mathvariant="bold-italic">θ</mi><mo>)</mo></mrow><mo stretchy="true">)</mo><mi mathvariant="bold-italic">V</mi></mrow><mo stretchy="true">}</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo stretchy="true">]</mo><mo>,</mo></math> </ephtml> (<reflink idref="bib8" id="ref39">8</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-italic">V</mi></mrow></math> </ephtml> is a consistent estimator of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">Σ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>.</mo></math> </ephtml> In practice,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">V</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>.</mo></math> </ephtml> To obtain the RLS function we first need compute the ML estimator</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> (see Equation (<reflink idref="bib2" id="ref40">2</reflink>)) and the associated</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>⌢</mo></mover></mrow></mrow></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow><mrow><mo /></mrow></msubsup><mo>.</mo></math> </ephtml> Then, also using (<reflink idref="bib8" id="ref41">8</reflink>),</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">RLS</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mi mathvariant="normal" /><mtext mathvariant="italic">tr</mtext><msup><mrow><mo stretchy="true">{</mo><mrow><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">S</mi><mo>−</mo><msub><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></mrow><mo stretchy="true">)</mo><msubsup><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><mo stretchy="true">}</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></math> </ephtml> (<reflink idref="bib9" id="ref42">9</reflink>)</p> <p>Hence, the estimator is ML, but the GLS function (<reflink idref="bib8" id="ref43">8</reflink>) is evaluated with weight matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">V</mi><mo>=</mo><msubsup><mrow><mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">Σ</mi></mrow><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>.</mo></math> </ephtml> Simulations show that RLS outperforms ML across varied sample sizes within the framework of a normal distribution (Zheng & Bentler, [<reflink idref="bib26" id="ref44">26</reflink>]). An open-source R package for fitting the proposed models, 'RLS', can be found on GitHub: install_github("bqzheng/RLS", dependencies = TRUE) (Zheng, [<reflink idref="bib24" id="ref45">24</reflink>]).</p> <hd id="AN0182192619-9">3.3. Non-Normal Data (Elliptical Distributions)</hd> <p>Non-normal data are common in real-world social science data analysis, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> is not appropriate for situations where data deviate from normal distribution. As a solution, we can employ robust estimators. As an example, we have chosen the Satorra-Bentler scaled test. This estimator is specifically designed to address issues related to non-normality. For illustration purposes, we can generate simulations of non-normal data following an elliptical distribution, which represents symmetric distributions with heavy tails, based on the original population model. In the elliptical distribution condition</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">ξ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mi mathvariant="normal" /></math> </ephtml> =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>r</mi><msup><mrow><mi mathvariant="bold">Φ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi mathvariant="bold-italic">ξ</mi></mrow></msub><mo>,</mo></math> </ephtml> and <bold>ε</bold> =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><msup><mrow><mi mathvariant="bold">Ψ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi mathvariant="bold-italic">ε</mi></mrow></msub></math> </ephtml> with r</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>∼</mo><mo stretchy="false">(</mo><mn>3</mn><mo>/</mo><msubsup><mrow><mi>χ</mi></mrow><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Φ</mi><mo>=</mo><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><mi mathvariant="normal">ξ</mi><mo stretchy="false">)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Ψ</mi><mo>=</mo><mtext mathvariant="italic">cov</mtext><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ε</mi><mo stretchy="false">)</mo><mo>.</mo></math> </ephtml> </p> <hd id="AN0182192619-10">4. Monte Carlo Simulation Results</hd> <p>Figure 1 displays chi-square test statistics and fit indices for both a misspecified model and a LM modified model across various sample sizes. The chi-square test statistics of the misspecified model rise as the sample size N increases, indicating poor fit, while the LM modified model consistently demonstrates good fit. However, the fit indices in the right panel inaccurately suggest a good fit.</p> <p>PHOTO (COLOR): Figure 1. Chi-square statistics and fit indices.</p> <p>Figure 2 displays non-normal data (elliptical distribution).</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> deviated from the anticipated value of 87, and the fit indices in the right panel incorrectly demonstrated a good fit. Nonetheless, this concern can be rectified by utilizing a suitable robust estimator. In this instance, we employed the Satorra-Bentler scaled test. Notably, as Figure 3 shows, the chi-square test statistics rooted in Satorra-Bentler scaled chi-square maintain remarkable stability across nearly all sample sizes.</p> <p>PHOTO (COLOR): Figure 2. Chi-square statistics of non-normal data and fit indices.</p> <p>PHOTO (COLOR): Figure 3. Chi-square statistics on small samples and fit indices.</p> <p>Figure 3 illustrates that when sample sizes are small (<emph>N</emph> < 200), the default ML estimator becomes ineffective. In such instances, an alternative estimator is necessary for achieving precise outcomes. Figure 3 also highlights that with small sample sizes,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mtext mathvariant="italic">ML</mtext></mrow></msub></math> </ephtml> exceeds the expected value of 87, and the fit indices on the right panel also inaccurately showed a good fit. However, employing the RLS method yields notably consistent chi-square test statistics, resulting in enhanced accuracy and reliability in such scenarios.</p> <hd id="AN0182192619-11">5. Discussion and Conclusion</hd> <p>SEM entails numerous subtleties in statistical interpretations, accentuating the intricate nature of this methodology. In our study employing a series of Monte Carlo simulations, we have demonstrated that chi-square test statistics provide insights beyond those from the measurement of goodness-of-fit, encompassing vital insights into the overall performance and reliability of the model. Fit indices do not provide the same level of comprehensive evaluation as an appropriate chi-square test. Each fit index carries its distinct theoretical and analytical emphasis, limiting its ability to offer a holistic assessment of model fit. As such, relying solely on fit indices may obscure important misspecifications that could be of significant interest.</p> <p>In sum, the chi-square test serves as a fundamental indicator of the model's overall fit quality and its compatibility with the observed data. Therefore, we advocate for the inclusion of chi-square test statistics in research reports, alongside fit indices, to present a more comprehensive and robust evaluation of the SEM model's fit. This approach allows for a balanced interpretation, taking into account both the nuanced insights provided by fit indices and the fundamental assessment offered by the chi-square test statistics.</p> <ref id="AN0182192619-12"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Acock, A., Clarke, H. D., & Stewart, M. C. (1985). A new model for old measures: A covariance structure analysis of political efficacy. Journal of Politics, 47, 1062 – 1084. https://doi.org/10.2307/2130807</bibtext> </blist> <blist> <bibl id="bib2" idref="ref16" type="bt">2</bibl> <bibtext> Arruda, E. H., & Bentler, P. (2017). A regularized GLS for structural equation modeling. Structural Equation Modeling, 24, 657 – 665.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref8" type="bt">3</bibl> <bibtext> Barrett, P. (2007). Structural equation modeling: Adjudging model fit. Personality and Individual Differences, 42 (, 815 – 824. https://doi.org/10.1016/j.paid.2006.09.018</bibtext> </blist> <blist> <bibl id="bib4" idref="ref27" type="bt">4</bibl> <bibtext> Bentler, P. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238 – 246. https://doi.org/10.1037/0033-2909.107.2.238</bibtext> </blist> <blist> <bibl id="bib5" idref="ref29" type="bt">5</bibl> <bibtext> Bentler, P. (2006). EQS 6 structural equations program manual. Multivariate Software, Inc.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref26" type="bt">6</bibl> <bibtext> Bentler, P., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588 – 606. https://doi.org/10.1037/0033-2909.88.3.588</bibtext> </blist> <blist> <bibl id="bib7" idref="ref19" type="bt">7</bibl> <bibtext> Bollen, K. A. (1989). Structural equations with latent variables. John Wiley & Sons.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref38" type="bt">8</bibl> <bibtext> Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariances. South African Statistical Journal, 8, 1 – 24.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref2" type="bt">9</bibl> <bibtext> Davidov, E. (2009). Measurement equivalence of nationalism and constructive patrioticsm in the ISSP: 34 Countries in a comparative perspectives. Political Analysis, 17, 64 – 82. https://doi.org/10.1093/pan/mpn014</bibtext> </blist> <blist> <bibtext> Feldman, S. (1988). Structure and consistency in public opinion: The role of core beliefs and values. American Journal of Political Science, 32, 416 – 440. https://doi.org/10.2307/2111130</bibtext> </blist> <blist> <bibtext> Goren, P. (2005). Party identification and core political values. American Journal of Political Science, 49, 881 – 896. https://doi.org/10.1111/j.1540-5907.2005.00161.x</bibtext> </blist> <blist> <bibtext> Hu, L.-T., Bentler, P., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted? Psychological Bulletin, 112, 351 – 362. https://doi.org/10.1037/0033-2909.112.2.351</bibtext> </blist> <blist> <bibtext> Hu, L-t., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1 – 55. https://doi.org/10.1080/10705519909540118</bibtext> </blist> <blist> <bibtext> Jackson, D. L., Gillaspy, A. J., & Purc-Stephenson, R. (2009). Reporting practices in confirmatory factor analysis: An overview and some recommendations. Psychological Methods, 14, 6 – 23. https://doi.org/10.1037/a0014694</bibtext> </blist> <blist> <bibtext> Jöreskog, K. G. (1969). Some contribution to maximum likelihood factor analysis. Psychometrika, 34, 183 – 202. https://doi.org/10.1007/BF02289343</bibtext> </blist> <blist> <bibtext> Kenny, D. A., Kaniskan, B., & McCoach, D. B. (2015). The performance of RMSEA in models with small degrees of freedom. Sociological Methods & Research, 44, 486 – 507. https://doi.org/10.1177/0049124114543236</bibtext> </blist> <blist> <bibtext> Kline, R. B. (2015). Principles and practice of structural equation modeling. Guilford publications.</bibtext> </blist> <blist> <bibtext> Stone, B. (2021). The ethnical use of fit indices in structural equation modeling: Recommendations for psychologists. Frontiers in Psychology, 12, 783226. https://doi.org/10.3389/fpsyg.2021.783226</bibtext> </blist> <blist> <bibtext> Sullivan, J., Marcus, G., Feldman, S., & Piereson, J. (1981). The source of political tolerance: A multivariate analysis. American Political Science Review, 75, 92 – 106. https://doi.org/10.2307/1962161</bibtext> </blist> <blist> <bibtext> Xia, Y., & Yang, Y. (2019). RMSEA, CFI, and TLI in structural equation modeling with ordered categorical data: The story they tell depends on the estimation methods. Behavior Research Methods, 51, 409 – 428. https://doi.org/10.3758/s13428-018-1055-2</bibtext> </blist> <blist> <bibtext> Yuan, K. H., & Bentler, P. (1999). On asymptotic distribution s of normal theory MLE in covariance structure analysis under some nonnormal distributions. Statistics and Probability Letters, 42, 107 – 113. https://doi.org/10.1016/S0167-7152(98)00171-0</bibtext> </blist> <blist> <bibtext> Yuan, K. H., & Liu, F. (2021). Which method is more reliable in performing model modification: Lasso regularization or Lagrange multiplier test? Structural Equation Modeling, 28, 69 – 81.</bibtext> </blist> <blist> <bibtext> Zheng, B. Q. (2023). Asian and Latino American political conceptualization: A dual-concept model. American Politics Research, 51, 182–196. https://doi.org/10.1177/1532673X221132479</bibtext> </blist> <blist> <bibtext> Zheng, B. Q. (2024). R package 'RLS': Reweighted least squares estimator. GitHub Repository. https://github.com/bqzheng/RLS</bibtext> </blist> <blist> <bibtext> Zheng, B. Q., & Bentler, P. M. (2021). Testing mean and covariance structures with reweighted least squares. Structural Equation Modeling: A Multidisciplinary Journal, 29, 259 – 266. https://doi.org/10.1080/10705511.2021.1977649</bibtext> </blist> <blist> <bibtext> Zheng, B. Q., & Bentler, P. M. (2023). RGLS and RLS in covariance structure analysis. Structural Equation Modeling, 30, 234 – 244. https://doi.org/10.1080/10705511.2022.2117182</bibtext> </blist> </ref> <aug> <p>By Bang Quan Zheng and Peter M. Bentler</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref3"></nolink> <nolink nlid="nl2" bibid="bib11" firstref="ref4"></nolink> <nolink nlid="nl3" bibid="bib19" firstref="ref5"></nolink> <nolink nlid="nl4" bibid="bib23" firstref="ref6"></nolink> <nolink nlid="nl5" bibid="bib22" firstref="ref7"></nolink> <nolink nlid="nl6" bibid="bib18" firstref="ref9"></nolink> <nolink nlid="nl7" bibid="bib14" firstref="ref10"></nolink> <nolink nlid="nl8" bibid="bib17" firstref="ref11"></nolink> <nolink nlid="nl9" bibid="bib15" firstref="ref13"></nolink> <nolink nlid="nl10" bibid="bib12" firstref="ref20"></nolink> <nolink nlid="nl11" bibid="bib21" firstref="ref23"></nolink> <nolink nlid="nl12" bibid="bib26" firstref="ref24"></nolink> <nolink nlid="nl13" bibid="bib25" firstref="ref25"></nolink> <nolink nlid="nl14" bibid="bib16" firstref="ref34"></nolink> <nolink nlid="nl15" bibid="bib13" firstref="ref35"></nolink> <nolink nlid="nl16" bibid="bib20" firstref="ref36"></nolink> <nolink nlid="nl17" bibid="bib24" firstref="ref45"></nolink>
Header DbId: eric
DbLabel: ERIC
An: EJ1457158
AccessLevel: 3
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Enhancing Model Fit Evaluation in SEM: Practical Tips for Optimizing Chi-Square Tests
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Bang+Quan+Zheng%22">Bang Quan Zheng</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-2614-2501">0000-0003-2614-2501</externalLink>)<br /><searchLink fieldCode="AR" term="%22Peter+M%2E+Bentler%22">Peter M. Bentler</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-9440-721X">0000-0002-9440-721X</externalLink>)
– 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):136-141.
– 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: 6
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Monte+Carlo+Methods%22">Monte Carlo Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Structural+Equation+Models%22">Structural Equation Models</searchLink><br /><searchLink fieldCode="DE" term="%22Goodness+of+Fit%22">Goodness of Fit</searchLink><br /><searchLink fieldCode="DE" term="%22Robustness+%28Statistics%29%22">Robustness (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Evaluation+Methods%22">Evaluation Methods</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="%22Error+of+Measurement%22">Error of Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Test+Reliability%22">Test Reliability</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/10705511.2024.2354802
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1070-5511<br />1532-8007
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: This paper aims to advocate for a balanced approach to model fit evaluation in structural equation modeling (SEM). The ongoing debate surrounding chi-square test statistics and fit indices has been characterized by ambiguity and controversy. Despite the acknowledged limitations of relying solely on the chi-square test, its careful application can enhance its effectiveness in evaluating model fit and specification. To illustrate this point, we present three common scenarios relevant to social and behavioral science research using Monte Carlo simulations, where fit indices may inadequately address concerns regarding goodness-of-fit, while the chi-square statistic can offer valuable insights. Our recommendation is to report both the chi-square test and fit indices, prioritizing precise model specification to ensure the reliability of model fit indicators.
– 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: EJ1457158
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1457158
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/10705511.2024.2354802
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 6
        StartPage: 136
    Subjects:
      – SubjectFull: Monte Carlo Methods
        Type: general
      – SubjectFull: Structural Equation Models
        Type: general
      – SubjectFull: Goodness of Fit
        Type: general
      – SubjectFull: Robustness (Statistics)
        Type: general
      – SubjectFull: Evaluation Methods
        Type: general
      – SubjectFull: Social Science Research
        Type: general
      – SubjectFull: Behavioral Science Research
        Type: general
      – SubjectFull: Error of Measurement
        Type: general
      – SubjectFull: Test Reliability
        Type: general
    Titles:
      – TitleFull: Enhancing Model Fit Evaluation in SEM: Practical Tips for Optimizing Chi-Square Tests
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Bang Quan Zheng
      – PersonEntity:
          Name:
            NameFull: Peter M. Bentler
    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