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Cited by 137Psychometric Methodologies and TestingStatistical Methods and Bayesian InferenceStatistical Distribution Estimation and Applications

The Noncentral Chi-square Distribution in Misspecified Structural Equation Models: Finite Sample Results from a Monte Carlo Simulation

Curran, Patrick J., Bollen, Kenneth A., Paxton, Pamela, Kirby, James, Chen, Feinian·2002·Multivariate Behavioral Research
DOI 10.1207/s15327906mbr3701_01
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Abstract

The noncentral chi-square distribution plays a key role in structural equation modeling (SEM). The likelihood ratio test statistic that accompanies virtually all SEMs asymptotically follows a noncentral chi-square under certain assumptions relating to misspecification and multivariate distribution. Many scholars use the noncentral chi-square distribution in the construction of fit indices, such as Steiger and Lind's (1980) Root Mean Square Error of Approximation (RMSEA) or the family of baseline fit indices (e.g., RNI, CFI), and for the computation of statistical power for model hypothesis testing. Despite this wide use, surprisingly little is known about the extent to which the test statistic follows a noncentral chi-square in applied research. Our study examines several hypotheses about the suitability of the noncentral chi-square distribution for the usual SEM test statistic under conditions commonly encountered in practice. We designed Monte Carlo computer simulation experiments to empirically test these research hypotheses. Our experimental conditions included seven sample sizes ranging from 50 to 1000, and three distinct model types, each with five specifications ranging from a correct model to the severely misspecified uncorrelated baseline model. In general, we found that for models with small to moderate misspecification, the noncentral chi-square distribution is well approximated when the sample size is large (e.g., greater than 200), but there was evidence of bias in both mean and variance in smaller samples. A key finding was that the test statistics for the uncorrelated variable baseline model did not follow the noncentral chi-square distribution for any model type across any sample size. We discuss the implications of our findings for the SEM fit indices and power estimation procedures that are based on the noncentral chi-square distribution as well as potential directions for future research.

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Authors
Curran, P. J., Bollen, K. A., Paxton, P., Kirby, J., & Chen, F.
ORCID
Venue
Multivariate Behavioral Research
Publisher
Informa UK Limited
Published
2002-01
Volume
37
Issue
1
Pages
1-36
ISSN
  • 0027-3171 (ISSN-L)
  • 1532-7906
DOI
10.1207/s15327906mbr3701_01
OpenAlex
W2150721786
Cited by
137
References
34
Citations by year
2026: 5 · 2025: 7 · 2024: 8 · 2023: 6 · 2022: 5 · 2021: 6 · 2020: 2 · 2019: 6 · 2018: 8 · 2017: 3 · 2016: 8 · 2015: 5 · 2014: 8 · 2013: 6 · 2012: 5
FWCI
7.68
Type
journal article
Language
en
Open access
closed
Locations
  • Multivariate Behavioral Research (publishedVersion)
  • PubMed (publishedVersion)
Topics
  • Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
Keywords
noncentral chi-square distribution, likelihood ratio test, finite sample properties, RMSEA, baseline model, model fit indices, statistical power, sample size, multivariate normality

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