An Overview on Regression Models for Discrete Longitudinal Responses

Abstract

In the longitudinal regression setup, interest may be focused primarily on the regression parameters for the marginal expectations of the longitudinal responses, the longitudinal correlation parameters being of secondary interest. Second, interest may be focused on both the regression and the longitudinal correlation parameters. Under the first setup, there exists a "working'' correlation matrix based generalized estimating equation (GEE) approach for the estimation of the regression parameters. Under the second setup, there exist two approaches for the joint estimation of the regression and the longitudinal correlations. In one approach, true longitudinal correlations are modeled and the regression and the true correlation parameters are jointly estimated based on a GEE approach. The second approach avoids the specification of the true longitudinal correlation structure and deals with the joint estimation of the regression and a vector of "working'' correlation parameters. In this second approach under the second setup, there again exist two joint estimation methods, one requiring moments up to order 4 and the other somehow using moments up to order 2 for the construction of the estimating equations for the "working'' correlation parameters. In this paper, we first provide an outline of the desirable features and drawbacks of each of these four existing approaches. By using a general autocorrelation structure to model the true longitudinal correlations, we then provide an outline of the advantages of three new approaches. In the first new approach, the true longitudinal correlations are estimated by the method of moments, whereas the regression estimates are obtained based on a generalized quasi-likelihood (GQL) estimation approach. The other two new approaches simultaneously estimate the regression and the true longitudinal correlation parameters. It is shown through a simulation study that, among these three new approaches, the first approach performs the best in estimating both the regression and the true correlation parameters, even though the longitudinal correlations are estimated separately by the method of moments.