Semi-parametric χ2 Testing in Mean Structure and Covariance Structure Analysis Using Projections
Mean structures form a basis for mean, covariance, and other forms of moment structure analysis including structural equation modeling. It is shown how to analyze mean structures using projections. These are used to derive a simple general goodness of fit test statistic that is asymptotically chi-squared and robust to departures from normality. Projections are also used to derive two goodness of fit test statistics for mean structures that are substructures of a more general mean structure. One of these uses the difference of two goodness of fit test statistics, one for the general structure and one for the substructure. It is shown how to use the mean structure results for covariance structure analysis. Best generalized least squares, or ADF estimates are not required. Any asymptotically normal estimates may be use. The primary methods used for testing mean and covariance structures are orthogonal complement methods. A basic difficulty with using these is identified. Specific examples show how the general results may be applied to generalized nonlinear regression and to autoregression with measurement errors. Simulation studies investigate the type one errors and power of the test statistics involved. An appendix contains a review of the basic asymptotic and projection methods used. It also gives conditions that lead to the commonly made assumption that the asymptotic covariance matrix of a vectorized form of a sample covariance matrix is positive definite and that this is a very mild assumption.