ABSTRACT
CONTENTS 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 12.2 Univariate Exponential Dispersion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
12.2.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 12.3 Extensions to Longitudinal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 12.4 Generalized Estimating Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
12.4.1 Shortcomings of the GEE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 12.5 Gaussian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 12.6 Quasi-Least Squares Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 12.7 Asymptotic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 12.8 Analysis of Longitudinal Binary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 12.9 Analysis of Longitudinal Poisson Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
12.9.1 Correlation Bounds for Poisson Variables . . . . . . . . . . . . . . . . . . . 382 12.9.2 Multivariate Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
12.10 Epileptic Seizure Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 12.10.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
12.11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Statistical methods for analyzing longitudinal data are important tools in biomedical research. The phrase “repeated or longitudinal data” is used for data consisting of responses taken on subjects or experimental units at different timepoints or undermultiple treatments. Suchdata occur commonly inmany
C5777: “c5777_c012” — 2007/10/27 — 13:03 — page 372 — #2
in
studies. Although data can be viewed as multivariate data, there are some key differences between the two.Multivariate data is usually a snapshot of different variables taken at a single time point, whereas longitudinal data consists of snapshots of the same variable taken at different time points. Thus, even though there are similarities between the two types, the data analysis goals are usually not the same and each pose different challenges and require different approaches for statistical analysis. Much research has already been done on multivariate continuous response variables using linear models and the multivariate normal distribution. Modelling multivariate discrete data is difficult because there is not a singlemultivariate discrete distribution as prominent as themultivariate normal distribution that has nice properties. For discrete univariate outcomes, exponential dispersion families are among the most popular and widely studied probability models. These statistical models that relate the random outcomes or responses to covariates are useful to understand variation in responses as a function of the covariates. We begin this chapter with a brief discussion of these models and maximum likelihood (ML) estimation of model parameters.
