ABSTRACT
Often the expected value of the y-variable of a regression has a restricted range. For example, an expected count is positive and expected probability is within the range (0,1). In generalized linear models, this restriction is implemented with a link function. In addition, a theoretically justified distribution is assumed for the response. This chapter discusses especially analysis of binary, grouped binary and count data using the Bernoulli, binomial and Poisson distributions. With these models, the mean and variance are related, which enables goodness-of-fit tests of the models. Often, the assumed mean-variance relationship does not hold in empirical data. This phenomena, called overdispersion or underdispersion, can be modeled in various ways. The estimation of generalized linear models is based on maximum likelihood. However, because only the variance-mean ratio is used in estimation, the model can be seen as a special case of the nonlinear model of Chapter 7. Generalized linear mixed-effects model is developed through inclusion of random effects to the model. Several alternative estimation methods for such models are demonstrated. The effect of random effects on the prediction is discussed and a related bias-correction is demonstrated. A total of 14 examples illustrate the concepts and their use with real-life data.
