UBC Theses and Dissertations
Some statistical models for the multivariate analysis of longitudinal data Song, Peter Xue-Kun
This thesis develops some statistical models for the multivariate analysis of longitudinal data on the basis of the dispersion models of J0rgensen (1987a, 1996), consisting of three topics: multivariate dispersion models and their application to regression analysis, stationary time series models with non-normal margins and state space models with Markov latent processes. The goal of the thesis is to develop statistical models which can accommodate features of both trend and dependence for longitudinal data. This thesis focusses mainly on the following three types of longitudinal data, namely (1) many short time series, (2) a few long stationary time series and (3) a few long non-stationary time series with time-varying covariates. A class of multivariate dispersion models is proposed to deal with data of type (1), in a spirit similar to multivariate analysis based on the multivariate normal distribution. Under these multivariate parametric models, population-averaged models (Diggle et al., 1994) are revisited, where approximate inferences for regression parameters are presented, including the generalized estimating equation (GEE) of Liang and Zeger (1986) as a special case. The thesis also presents a class of stationary autoregressive moving-average (ARMA) models with exponential dispersion model margins for data of type (2). The class of ARMA models is defined as a special case of a class of stationary infinite order moving average processes constructed by means of the thinning operation of Joe (1996a). For analysis of type (3) data, two classes of state space models, including one with stationary latent processes and another with non-stationary latent processes, are proposed. To estimate regression parameters in both classes of models, we develop an algorithm for solving the so-called Kalman estimating equation (KEE), corresponding to a modified EM-algorithm where the E-step is approximated by the Kalman smoother that estimates the latent process via the best linear unbiased predictor (BLUP). Two simulation studies are conducted in the thesis based on Poisson-gamma models. One is for the comparison of the efficiency of the K E E approach and the Monte Carlo E M (MCEM) algorithm. The other simulation study is for the examination of the utility of the model diagnosis for detecting the misspecification of stationarity and non-stationarity for latent process. The thesis contains two data analyses. One data set consists of daily counts of emergency room visits for respiratory diseases to the hospital of Prince George, British Columbia, along with covariates of air pollution variables and meteorological variables. These data are analyzed through state space models to investigate the relationship between air pollution and respiratory morbidity. The other data set, consisting of the monthly number of poliomyelitis cases in the USA from 1970 to 1983, is analyzed based on the Poisson stationary-gamma model to study whether or not there is an evidence of a decreasing trend in the rate of polio infections in the USA.
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