The MIXED procedure fits a variety of mixed linear models to data and enables you to use these fitted models to make statistical inferences about the data. A mixed linear model is a generalization of the standard linear model used in the GLM procedure, the generalization being that the data are permitted to exhibit correlation and nonconstant variability. The mixed linear model, therefore, provides you with the flexibility of modeling not only the means of your data (as in the standard linear model) but their variances and covariances as well.
The primary assumptions underlying the analyses performed by PROC MIXED are as follows:
The data are normally distributed (Gaussian).
The means (expected values) of the data are linear in terms of a certain set of parameters.
The variances and covariances of the data are in terms of a different set of parameters, and they exhibit a structure matching one of those available in PROC MIXED.
Since Gaussian data can be modeled entirely in terms of their means and variances/covariances, the two sets of parameters in a mixed linear model actually specify the complete probability distribution of the data. The parameters of the mean model are referred to as fixed-effects parameters, and the parameters of the variance-covariance model are referred to as covariance parameters.
The fixed-effects parameters are associated with known explanatory variables, as in the standard linear model. These variables can be either qualitative (as in the traditional analysis of variance) or quantitative (as in standard linear regression). However, the covariance parameters are what distinguishes the mixed linear model from the standard linear model.
The need for covariance parameters arises quite frequently in applications, the following being the two most typical scenarios:
The experimental units on which the data are measured can be grouped into clusters, and the data from a common cluster are correlated.
Repeated measurements are taken on the same experimental unit, and these repeated measurements are correlated or exhibit variability that changes.
The first scenario can be generalized to include one set of clusters nested within another. For example, if students are the experimental unit, they can be clustered into classes, which in turn can be clustered into schools. Each level of this hierarchy can introduce an additional source of variability and correlation. The second scenario occurs in longitudinal studies, where repeated measurements are taken over time. Alternatively, the repeated measures could be spatial or multivariate in nature.
PROC MIXED provides a variety of covariance structures to handle the previous two scenarios. The most common of these structures arises from the use of random-effects parameters, which are additional unknown random variables assumed to affect the variability of the data. The variances of the random-effects parameters, commonly known as variance components, become the covariance parameters for this particular structure. Traditional mixed linear models contain both fixed- and random-effects parameters, and, in fact, it is the combination of these two types of effects that led to the name mixed model. PROC MIXED fits not only these traditional variance component models but numerous other covariance structures as well.
PROC MIXED fits the structure you select to the data by using the method of restricted maximum likelihood (REML), also known as residual maximum likelihood. It is here that the Gaussian assumption for the data is exploited. Other estimation methods are also available, including maximum likelihood and MIVQUE0. The details behind these estimation methods are discussed in subsequent sections.
After a model has been fit to your data, you can use it to draw statistical inferences via both the fixed-effects and covariance parameters. PROC MIXED computes several different statistics suitable for generating hypothesis tests and confidence intervals. The validity of these statistics depends upon the mean and variance-covariance model you select, so it is important to choose the model carefully. Some of the output from PROC MIXED helps you assess your model and compare it with others.