On the Analysis of Categorical Variables by Linear Models having Singular Covariance Matrices

Abstract

The approach of Bonett, Woodward, and Bentler (1985, Biometrics 41, 745-750) for the analysis of categorical variables is examined. An asymptotically best linear unbiased estimator (ABLUE) is obtained by appealing to the optimal theory for linear models having singular covariance matrices. It is shown that the only model of interest for which their estimator is ABLUE in general is the loglinear model. It is also ABLUE for the power law model, which includes the linear case, provided the sampling is multinomial. An example involving product-binomial sampling is given to demonstrate substantial suboptimality of their estimator even in the linear case. The ABLUE is also shown to be BAN for the special case of multinomial sampling. Tests of hypotheses, and extensions to exact or stochastic linear constraints on the parameters are also given.