Gee-Smoothing Spline for Semiparametric Estimation of Longitudinal Categorical Data

In this thesis we propose estimation methods of semiparametric marginal models for longitudinal (correlated) categorical data, where the systematic component of the model consists of parametric and nonparametric forms. We develop GEE-Smoothing spline as a method to analyze semiparametric model for longitudinal data. The proposed methods are an extension of parametric generalized estimating equation (GEE) to semiparametric GEE by introducing smoothing spline into parametric GEE. We derive estimation method of GEE-Smoothing spline in the case of longitudinal binary, ordinal, and nominal data. Derivation of the estimating equation of GEE-Smoothing spline for these three types ofcategorical data is the same. However their estimating equations have different forms of the covariance and correlation matrices. In the estimation of the association (correlation) parameter for binary data, we use moment method of Liang & Zeger’s and method of Prentice’s. For ordinal and nominal data, we use different models of the covariance matrices than of binary data. These models need smaller number of the association parameter to be estimated which is different from the existing models of parametric GEE for ordinal data. We also derive and propose the methods to estimate the association parameter for these two types of data. The properties of the estimate for both parametric and nonparametric components of GEE-Smoothing spline are evaluated using simulation studies. We obtained that the estimates of parametric component for binary and ordinal data are unbiased. Whilst for nominal data, the estimates of parametric componentare almost unbiased. Meanwhile the estimates of the nonparametric component for all types of data are biased, with the bias decreases when the samplesize increases. The estimators of both parametric and nonparametric components are also consistent, and the consistency is not affected by the correct or incorrect working correlation used in model. This consistency property holds for correlated and independent data. The efficiency of the estimates of using independent or correlated working correlation in the estimation depends on the type of covariate, such as time varying, subject specific, or mean-balanced covariates. The estimates of both parametric and nonparametric components also follow the central limit theorem (CLT), for both independent and correlated data, and using correct or incorrect working correlation. Both components estimate have normal distribution.

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