psychometrika—vol. 82, no. 3, 533–558
ASSESSING THE SIZE OF MODEL MISFIT IN STRUCTURAL EQUATION MODELS
UNIVERSITY OF BARCELONA
UNIVERSITY OF SOUTH CAROLINA
When a statistically signiﬁcant mean difference is found, the magnitude of the difference is judged
qualitatively using an effect size such as Cohen’s d. In contrast, in a structural equation model (SEM), the
result of the statistical test of model ﬁt is often disregarded if signiﬁcant, and inferences are drawn using
“close” models retained based on point estimates of sample statistics (goodness-of-ﬁt indices). However,
when a SEM cannot be retained using a test of exact ﬁt, all substantive inferences drawn from it are suspect.
It is therefore important to determine the size of the model misﬁt. Standardized residual covariances and
residual correlations providestandardized effectsizes of the misﬁt of SEM models. Theycan be summarized
using the Standardized Root Mean Squared Residual (SRMSR) and the Correlation Root Mean Squared
Residual (CRMSR) which can be used as overall effect sizes of the misﬁt. Statistical theory is provided
that allows the construction of conﬁdence intervals and tests of close ﬁt based on the SRMSR and CRMSR.
It is hoped that the use of standardized effect sizes of misﬁt will help reconcile current practices in SEM
and elsewhere in statistics.
Key words: goodness-of-ﬁt, RMSEA, effect size.
Goodness-of-ﬁt assessment refers to how well our model reproduces the data-generating
process. When ﬁtting a regression model, researchers assess the goodness-of-ﬁt of their model
by inspecting a plot of the standardized residuals versus standardized ﬁtted values in order to
check the linearity and homoscedasticity assumption of the model. Failure to meet the model
assumptions will likely result in incorrect parameter estimates. Simply put, when the ﬁtted model
is incorrectly speciﬁed, all inferences based on it are suspect.
In structural equation modeling (SEM), we ﬁt a system of regression-like equations to our
data (possibly involving latent variables), and procedures similar to those used in regression can
be used to assess each of the equations in the model (Bollen & Arminger, 1991; Coffman &
Millsap, 2006; Hildreth, 2013; Yuan & Hayashi, 2010). However, they hardly seem to be used
in applications. Rather, for historical reasons, the assessment of model ﬁt in structural equation
models has relied on the use of grouped data (residual means, covariances, and correlations) as
opposed to individual residual observations. There is evidence that residual covariances (i.e., the
differences between observed and ﬁtted covariances) are sensitive to linear mis-speciﬁcations
(Raykov, 2000) but I am not aware of any research that has examined whether they are sensitive
to the presence of heteroscedasticity. Residual summary statistics such as residual covariances
can be summarized into a single test statistic to assess the overall goodness-of-ﬁt of a struc-
tural equations model. Although overall test statistics are invariably reported in applications, they
are most often ignored as they usually suggest that the model is incorrectly speciﬁed. Rather,
Presidential Address to the Psychometric Society, delivered at the annual meeting in Madison (WI), July 2014.
This research was supported by an ICREA-Academia Award and Grant SGR 2014 1500 from the Catalan Government
and Grant PSI2012-33601 from the Spanish Ministry of Education. I am indebted to Peter Bentler, Ke-Hai Yuan, Albert
Satorra, Jim Steiger, Haruhiko Ogasawara, and Yves Rosseel for their helpful comments. I am also most thankful to Yves
Rosseel for implementing these methods in the Lavaan package in R.
Correspondence should be made to Alberto Maydeu-Olivares, Department of Psychology, University of South
Carolina, Barnwell College, 1512 Pendleton St., Columbia, SC 29208, USA. Email: firstname.lastname@example.org
© 2017 The Psychometric Society