Generalized linear mixed models with applications - Kalligeris, Emmanouil - Nektarios (2019)
This thesis is conducted at the Department of Mathematics, Division of Stat- istics and Actuarial-Financial Mathematics of the University of the Aegean, in the context of the MSc program in Statistics and Data Analysis. Its purpose is to analyze the class of Generalized Linear Mixed Models (GLMMs) and their implementation in real life problems, through a thorough study on influenza-like illness (ILI) rate data. More specifically, we focus on a special class of GLMMs, the class of periodic regression mixed models for modeling the ILI time series data. For the trend, linear, quadratic, cubic and quartic trends are considered while for the seasonal component, the most widely used periodicities are implemented, i.e. 12, 6, and 3 months. The class extends further to include first and second order AR and MA parts while environmental covariates potentially affecting the output are also included. The structure of the thesis consists of four Chapters. In Chapter 1, Generalized Linear (GLMs) and Generalized Linear Mixed Models are presented along with their properties. Some of the topics that will be discussed are, logistic regression model, maximum likelihood estimation and, test of hypotheses. Chapter 2, constitutes an introduction to univariate time series analysis. Important terms such as autocorrelation and white noise are defined, as well as the back- ward shift operator. At the end, various time series models (i.e., AR, (S)ARIMA, and, Periodic) are presented. Chapter 3, introduces four basic model selection criteria. Among them are, the Modified Divergence Information Criterion (MDIC) and R2, which are GLMM(m) being used for the selection of the ”best overall” model of the application that follows. Finally, in Chapter 4, an experimental study is applied on influenza-like illness (ILI) rate data, collected weekly through the sentinel surveillance system, provided by the Department of Epidemiological Surveillance and Intervention of the Hellenic Center for Disease Control and Prevention (H.C.D.C.P.). An exhaustive search process takes place, in order to provide guidelines for the selection of the optimal periodic regression mixed model for early and accurate outbreak detection in an epidemiological surveillance system, as well as for its proper use and implementation.