The robustness of maximum likelihood estimation in structural equation models

Abstract

Introduction General methods for the analysis of covariance structures were introduced by Joreskog (1970, 1973). Within the general theoretical framework it is possible to estimate parameters and their corresponding standard errors and to test the goodness-of-fit of a linear structural equation system by means of maximum likelihood methods . Although other methods of estimating such models (least squares procedures, instrumental variable methods) are available, we do not discuss them here. For an introduction to the general model the reader is referred to Chapter 2 and for more detailed statistical discussions to Joreskog (1978, 1982a, b). The LISREL model considers a data matrix Z ( N × k ) of N observations on k random vaŕiables. It is assumed that the rows of Z are independently distributed, each having a multivariate normal distribution with the same mean vector μ and the same covariance matrix ∑; that is, each case in the data is independently sampled from the same population. In a specified model there are s independent model parameters co, to be estimated. For large samples the sampling distribution of the estimated parameters ŵ i and the sampling distribution of the likelihood ratio estimate for goodness-of-fit are approximately known, provided that the preceding assumptions hold. Under the same conditions the standard errors of the estimated parameters se ŵ i are also known asymptotically. In the following, standardized parameter estimates are defined by ŵ* i = (ŵ i − ω i )/ se ŵ i . For large samples the sampling distribution of ŵ* i is approximately standard normal, and the goodness-of-fit estimate has an approximate chi-square distribution with k ( k + 1)/2 – s degrees of freedom.