Testing structural equation models: The effect of kurtosis

Various chi-square statistics are used for testing structural equation models. A commonly used chi-square is the Satorra-Bentler (SB) statistic. This is a normal-theory chi-square corrected for non-normality in the observed variables. The relationship between the SB statistic and kurtosis is developed and it is shown that the SB statistic tends to decrease with increasing kurtosis if the model does not hold. It is shown that the population value of the correction factor for the SB statistic decreases toward zero with increasing kurtosis. Various chi-square statistics are used for testing structural equation models. If the model is fitted by the maximum likelihood (ML) method, one such chi-square statistic is obtained as n times the minimum of the ML fit function, where n is the sample size. An asymptotically equivalent chi-square statistic is n times the minimum value of a quadratic form using a weight matrix evaluated under multivariate normality of the observed variables. Following the notation in Jöreskog, Sörbom, Du Toit, & Du Toit (2003), these chi-square statistics are denoted c1 and c2, respectively. They are valid under multivariate normality of the observed variables and if the model holds. If the observed variables are non-normal, Satorra & Bentler (1988) proposed another chi-square statistic c3 (often called the SB statistic) which is c1 or c2 multiplied by a scale factor which is estimated from the sample and involves an estimate of the asymptotic covariance matrix (ACM) of the sample variances and covariances. Satorra & Bentler (1988) show that under multivariate non-normality, c1 and c2 have asymptotically a distribution which is a linear combination of chisquares with one degree of freedom. The scale factor is estimated such that c3 has an asymtotically correct mean even though it does not have an asymptotic chi-square distribution. In practice, c3 is conceived of as a way of correcting c1 or c2 for the effects of non-normality and c3 is often used as it performs better than the ADF test proposed by Browne (1984), particularly if n is not very large, see e.g., Hu, Bentler, & Kano (1992). In this paper we develop the relationship between c3 and kurtosis and we show that the c3 statistic tends to decrease with increasing kurtosis. The practical consequence of this is that models that do not hold tend to be accepted by the chi-square test if kurtosis is large. Thus, this test have low power for detecting misspecified models. Although the results developed here can be demonstrated by simulating and analyzing random samples, we will use a different approach. Curran, West and Finch (1996) presented a simulation study of these chi-square statitics and concluded “The most surprising findings are related to the behavior of the SB and ADF test statistics under simultanous conditions of misspecification and multivariate nonnormality (Models 3 and 4). The expected values of these test statistics markedly decreased with increasing nonnormality. That is, all else being equal, the SB and ADF test statistics were less likely to dectect a specification error given increasing departures from a multivariate normal distribution. The more severe the nonormality, the greater the corresponding loss of power. This result was unexpected, and we are not aware of any previous discussions of this finding” (Curran, West and Finch, 1996, p.25). This paper provides an explanation for their results. The Asymptotic Covariance Matrix under Non-Normality Browne & Shapiro (1988) considered the following general structure for an observable k×1 random vector z: z = μ + g ∑