Catálogo de publicaciones - libros

The best way to understand a linear mixed model, or mixed linear model in some earlier literature, is to first recall a linear regression model. The latter can be expressed as = + , where is a vector of observations, is a matrix of known covariates, is a vector of unknown regression coefficients, and is a vector of (unobservable random) errors. In this model, the regression coefficients are considered fixed. However, there are cases in which it makes sense to assume that some of these coefficients are random. These cases typically occur when the observations are correlated. For example, in medical studies observations are often collected from the same individuals over time. It may be reasonable to assume that correlations exist among the observations from the same individual, especially if the times at which the observations are collected are relatively close. In animal breeding, lactation yields of dairy cows associated with the same sire may be correlated. In educational research, test scores of the same student may be related.

The previous section dealt with point estimation and related problems in linear mixed models. In this section, we consider a different type of inference, namely, tests in linear mixed models. Section 2.1.1 discusses statistical tests in Gaussian mixed models. As shown, exact -tests can often be derived under Gaussian ANOVA models. Furthermore, in some special cases, optimal tests such as uniformly most powerful unbiased (UMPU) tests exist and coincide with the exact -tests. Section 2.1.2 considers tests in non-Gaussian linear mixed models. In such cases, exact/optimal tests typically do not exist. Therefore, statistical tests are usually developed based on asymptotic theory.

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.

As mentioned in Section 3.4, the likelihood function under a GLMM typically involves integrals with no analytic expressions, and therefore is difficult to evaluate. For relatively simple models, the likelihood function may be evaluated by numerical integration techniques. See, for example, Hinde (1982), and Crouch and Spiegelman (1990). Such a technique is tractable if the integrals involved are low-dimensional. The following is an example.