UBC Theses and Dissertations
Markov generalized linear model and modeling of epileptic seizure data Luo, Lawrence Shou-Jun
How to capture the serial correlation has always been an important issue in the study of time series. Intuitively almost all time series are correlated. This topic has been extensively studied for linear time series models. In recent years many new models have been proposed to study nonlinear and nonnormal time series. For these new models some are restricted to specific types of data, e.g., longitudinal data, and some are computationally or conceptually complicated and may not be convenient for application use. The autoregressive Markov model, referred to as a Markov generalized linear model ( MGLM ) in this thesis, proposed by Zeger and Quaqish, however, requires little restriction on the type of data and are both conceptually and computationally attractive. It is a combination of the well studied generalized linear model and the classical linear autoregressive time series model and its computation is equivalent to that of the generalized linear model. The motivation of this thesis is to model epileptic seizure data. Our data were collected during 1990-1992 in B.C. Children's Hospital in a clinical trial of intravenous infusion of immunoglobulin G (IVIG). The controlled group used the best available treatments (BAT). The data of each subject are a time series with about 100 observations, but in each arm of study there are only 6-7 subjects. Moreover, the data of some subjects are serially correlated and there is a large variability in the data structures of different subjects. These characteristics prevent us from using the techniques for longitudinal data or techniques of random effects models. The MGLM is thus used to model this data set. To exploit the power of MGLM in capturing the correlation of time series we introduce a history information to "measure" the impact of the past behavior of a time series on its present outcome. Such a function treats all the past information as one factor and avoid using many past lags and thus helps reduce model complexity. A trend function is also used in the model to capture certain systematic pattern of data and hence also capture part of the dependence structure of data. The relationship between these two function in capturing the serial correlation of time series is clarified in the thesis. Models with both of these two functions, with trend only, with history information function only and with none of them are built to fit the data of individual subjects. It is found that for most subjects that either a trend function or a history information function is good enough to capture the dependence structure of data. However, models with none of these often have poor fitting. Some time series need both of them in their modeling. Only in the case where the data set has too little variability, that is, it contains only a few nonzero observations, the MGLM may not work satisfactorily. To explore the possibility of building a "super" MGLM for the whole data set we build such super models for different subsets. It turns out that the "quality" of such super models depends on the variability of data. If the trend functions of different subjects are similar, a super model works quite well. For the data analysis we model the data of each individual subject first, then analyze the whole data set in two ways:1) combine the results from the study of individual subjects to compare the treatment effects of BAT and IVIG; 2) build a "super" MGLM for the whole data set. The two methods arrive at the same conclusion that there is no difference between the treatment effects of BAT and IVIG. This conclusion is consistent with what we obtained through nonparametric methods in the initial data analysis. For the computation of MGLM since Splus did not work well for the super model, a program in C was written to carry out all the computation of MGLM . This program is more flexible in calculating any needed statistics and is much faster than Splus. The C codes are available at the end of this thesis.
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