Pseudo-Maximum Likelihood Estimation of Mean and Covariance Structures with Missing Data

Abstract

Abstract A nonlinear mean- and covariance-structure model for one or more groups is constructed. The model subsumes the usual linear model considered in the literature. It is then shown how to estimate the parameters of the model and the asymptotic covariance matrix of the parameter estimates using pseudo-maximum likelihood (PML) estimation. The resulting estimates are strongly consistent under general regularity conditions, provided only that the model for the first two moments is correctly specified. Nevertheless, because the data are not necessarily drawn from a multivariate normal distribution, the usual likelihood ratio tests for model comparisons in mean- and covariance-structure models do not apply. Wald tests and Lagrange multiplier tests may be used to implement such comparisons. Next, the standard results on ML estimation with missing data are extended to the case of PML estimation with missing data, and the results are applied to the model. The approach to the missing-data problem adopted, which decomposes the pseudo-log-likelihood function from normal theory into a sum of individual components, cannot generally be implemented by using existing mean- and covariance-structure programs. In some important instances, however, the approach can be implemented by using one of the standard programs (e.g., LISREL). Finally, an example is used to illustrate the approach used. In particular, data from various sources are combined to circumvent an omitted-variables problem in a linear system of equations. The example is somewhat novel because there is no complete data sample from which the model could be estimated. Comments are made on other research situations where data can be combined from multiple sources in the absence of a complete data sample to estimate models that could not otherwise be considered.