On joint estimation of regression and overdispersion parameters in generalized linear models for longitudinal data
Longitudinal data comprise a set of multivariate responses \(y_i = (y_{i1}, \dots, y_{ij}, \dots, y_{it})^T\), for the \(i\) th \((i = 1, \dots, n)\) cluster, together with a \(t \times p\) matrix \(X_i = (x_{i1}, \dots, x_{ij}, \dots, x_{ik})^T\) of covariates associated with each response \(y_i\). Usually the \(j\) th outcome of the \(i\) th cluster, \(y_{ij}\), and the \(p \times 1\) vector of covariates, \(x_{ij}\), are observed at times \(j = 1, \dots, t\) for clusters \(i = 1, \dots, n\). For a given \(j\), when \(\{y_{ij}, i = 1, \dots, n\}\) represent count data or data in the form of proportions, routinely they are analyzed under a Poisson assumption in the former case and under a binomial assumption in the later case. The data of these types often exhibit overdispersion. The specific plan of the paper is as follows. In Section 2 we present the model. In Section 3, we first provide the joint score estimates for the regression and overdispersion parameters and derive an asymptotic score test for the regression, under the working independence assumption that the observations for a subject are independent. In the same section, we then exploit a general class of joint estimating equations for the regression and overdispersion parameters, under the same working independence assumption, and construct a chi-square test for testing the regression parameters. The test for overdispersion may be similarly constructed. Between the two tests, the likelihood-based score test is more powerful but this test appears to be computationally more cumbersome than the test based on the joint estimating equations. In Section 4, we consider the general case and provide the joint estimating equations for the regression and overdispersion parameters, under the condition that the observations in the cluster are correlated. Asymptotic tests are also discussed. An example is considered in Section 5 to illustrate the estimation and test procedures.
scientific article; zbMATH DE number 844624