Approximate Methods For Analyzing Semi-Parametric Longitudinal Models With Non-Ignorable Missing Responses

In this thesis, we suggest and explore semi-parametric generalized partially linear mixed models for longitudinal data with non-ignorable and non-monotone missing responses. The key subject of our attention is the estimation of mean response parameters and variance components using a semi-parametric Monte Carlo EM method, where the conditional mean response is semi-parametric. We first discuss the penalized regression spline method, which is often referred to as P-splines, for linear mixed model. We investigate the connection between P-splines and linear mixed model through incorporating the non-parametric mean functions into longitudinal linear mixed model. An extensive simulation study using different semi-parametric mean response functions are presented. Our simulation study reports that when the true underlying model is partially linear, the penalized spline method provides unbiased and efficient estimators. On the other hand, when the mean response is a correctly specified linear model, the P-spline still provides reliable estimates of the model parameters. Next, we present semi-parametric generalized partially linear mixed models for longitudinal data with non-ignorable missing responses. In this situation, we introduce a parametric model for non-ignorable missing data and incorporate it into the likelihood function. We obtain the asymptotic variances of the proposed estimators by the method of Louis (cf. [2], [7]). In addition, we propose and explore a semi-parametric Monte Carlo EM (MCEM) algorithm for simultaneous estimation of the regression parameters and variance components in partially linear mixed models with non-ignorable and non-monotone missing responses. In simulations, the empirical properties of the proposed method are evaluated. The simulation study shows that our proposed semi-parametric method performs well even under a large proportion of non-ignorable missing responses. Finally, the proposed semi-parametric MCEM method are applied to some actual longitudinal data obtained from a health survey, referred to as the Health and Retirement Study (HRS). The data showed strong evidence of a non-linear trend in the mean response function. It is evident from this application that our proposed methods can be used to improve the efficiency of the estimates in a partially linear mixed model for longitudinal data with non-ignorable missing responses.