When fit indices and residuals are incompatible.

Standard chi-square-based fit indices for factor analysis and related models have a little known property: They are more sensitive to misfit when unique variances are small than when they are large.Consequently, very small correlation residuals indicating excellent fit can be accompanied by indications of bad fit by the fit indices when unique variances are small.An empirical example of this incompatibility between residuals and fit indices is provided.For illustrative purposes, an artificial example is provided that yields exactly the same correlation residuals as the empirical example but has larger unique variances.For this example, the fit indices indicate excellent fit.A theoretical explanation for this phenomenon is provided using relationships between unique variances and eigenvalues of the fitted correlation matrix.When assessing structural equation models, it is helpful to use a fit index as a single summary measure and to also inspect the matrix of residual covariances or correlations in case it suggests sources of lack of fit.In the hypothetical situation in which a model holds exactly, the chisquare goodness-of-fit statistic, on which standard fit indices are based, will have a value of zero and all residuals will also be zero (e.g., Bollen, 1989, p. 263).It is tempting to assume, therefore, that the standard fit indices and the usual residuals measure lack of fit of a structural equation model in a compatible manner.In this article, we show that this assumption is not true, particularly when the model fits closely, but not exactly, and when unique variances of manifest variables are very small.Small unique variances imply that both the error variances and specific variances of manifest variables are small so that the manifest variables yield accurate measures of the latent variables under consideration.It is disturbing that standard fit indices appear to indicate a bad fit by the model when very accurate measurements are used and even when all the residuals indicate a very good fit.This anomaly is examined in some detail.First, a numerical illustration of the problem is given.A matrix of correlations among eight variables from an empirical study in health psychology is subjected to an exploratory maximum-likelihood (ML) factor analysis extracting two factors.Values of several standard fit indices indicate an unsatisfactory fit.The usual residual correlation matrix is then presented.Elements are all small in magnitude, indicating a very good fit of the model and contradicting the fit indices.This illustrates that serious conflicts between the standard fit indices and the correlation residuals can occur in practice.