The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis.

Abstract

Monte Carlo computer simulations were used to investigate the performance of three X 2 test statistics in confirmatory factor analysis (CFA). Normal theory maximum likelihood )~2 (ML), Browne's asymptotic distribution free X 2 (ADF), and the Satorra-Bentler rescaled X 2 (SB) were examined under varying conditions of sample size, model specification, and multivariate distribution. For properly specified models, ML and SB showed no evidence of bias under normal distributions across all sample sizes, whereas ADF was biased at all but the largest sample sizes. ML was increasingly overestimated with increasing nonnormality, but both SB (at all sample sizes) and ADF (only at large sample sizes) showed no evidence of bias. For misspecified models, ML was again inflated with increasing nonnormality, but both SB and ADF were underestimated with increasing nonnormality. It appears that the power of the SB and ADF test statistics to detect a model misspecification is attenuated given nonnormally distributed data.Confirmatory factor analysis (CFA) has become an increasingly popular method of investigating the structure of data sets in psychology. In contrast to traditional exploratory factor analysis that does not place strong a priori restrictions on the structure of the model being tested, CFA requires the investigator to specify both the number of factors and the specific pattern of loadings of each of the measured variables on the underlying set of factors. In typical simple CFA models, each measured variable is hypothesized to load on only one factor, and positive, negative, or zero (orthogonal) correlations are specified between the factors. Such models can provide strong evidence about the convergent and discriminant validity of a set of measured variables and allow tests among a set of theories of measurement structure. More complicated CFA models may specify more complex patterns of factor loadings, correlations among errors or specific factors, or both. In all cases, CFA models set restrictions on the factor loadings, the correlations between factors, and the correlations between errors of measurement that permit tests of the fit of the hypothesized model to the data.There are two general classes of assumptions that underlie the statistical methods used to estimate CFA models: distributional and structural (Satorra, 1990). Normal theory maximum likelihood (ML) estimation has been used to analyze the majority of CFA models. ML makes the distributional assumption that the measured variables have a multivariate normal distribution in the pop-16

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“…The risk is to wrongly reject a proper model (type 1 error) [45]. We used the SB estimation due to questionable normality of section B in particular.…”

“…We used the SB estimation due to questionable normality of section B in particular. The asymptotic distribution free method is applicable for non-continuous data, but was not applied as it may reject properly specified models if sample sizes are small (N < 500) or deviation from normality is minimal [45]. It has been suggested that non-normality is not problematic for the maximum likelihood method until univariate skewness and kurtosis approach 2.0 and 10.0, respectively [45]; our data is below these limits (Table 2).…”

“…In field test II (step 18) we conducted a CFA for section A and B separately, using maximum likelihood with the Satorra-Bentler (SB) estimation, which is robust to non-normality [45,46]. This was relevant since neither section showed multivariate normality.…”