Generalized Linear Mixed Models: Part I

Abstract

For the most part, linear mixed models have been used in situations where the observations are continuous. However, there are cases in practice where the observations are discrete, or categorical. For example, the number of heart attacks of a potential patient during the past year takes the values 0, 1, 2, ..., and therefore is a discrete random variable. McCullagh and Nelder (1989) proposed an extension of linear models, called generalized linear models, or GLM. They noted that the key elements of a classical linear model, that is, a linear regression model, are (i) the observations are independent, (ii) the mean of the observation is a linear function of some covariates, and (iii) the variance of the observation is a constant. The extension to GLM consists of modification of (ii) and (iii) above; by (ii)′ the mean of the observation is associated with a linear function of some covariates through a link function; and (iii)′ the variance of the observation is a function of the mean. Note that (iii)′ is a result of (ii)′. See McCullagh and Nelder (1989) for details. Unlike linear models, GLMs include a variety of models that includes normal, binomial, Poisson, and multinomial as special cases. Therefore, these models are applicable to cases where the observations may not be continuous. The following is an example.