Semi-parametric Models for Longitudinal Count, Binary and Multinomial Data

Abstract

In a longitudinal setup, the semi-parametric regression model contains a specified regression function in some suitable time dependent primary covariates and a non-parametric function in some other time dependent say secondary covariates. However, the functional form for such a semi-parametric regression model depends on the nature of the repeated responses collected from a large number of independent individuals. In cross sectional setup, these functional forms represent the marginal expectations of the responses, whereas in a longitudinal setup, in general, they represent the expectation of a response at a given time conditional on past responses. More specifically, these conditional expectations are modelled through certain dynamic relationships among the repeated responses which also specify the longitudinal correlation structure among these repeated responses. In this paper, we consider a lag 1 dynamic relationship among repeated responses whether they are linear, count, binary or multinomial, and exploit the underlying correlation structure for consistent and efficient estimation of the regression parameters involved in the specified regression function in primary covariates. Because the non-parametric function in secondary covariates is not of direct interest, for simplicity, we estimate this function consistently in all cases by using ‘working’ independence assumption for the repeated responses.