Mixed models are often applied in situations where data are clustered, grouped, or otherwise hierarchically organized. For example, observations might be collected by randomly selecting schools in a school district, then randomly selecting classrooms within schools, followed by selecting students within the classroom. A longitudinal study might randomly select individuals and take repeatedly measurements on them. In the first example, a school is a cluster of observations, which consists of smaller clusters (classrooms) and so on. In the longitudinal example the observations for a particular individual form a cluster. Mixed models are popular analysis tools for hierarchically organized data for the following reasons:

• The selection of groups is often performed randomly, so that the associated effects are random effects.

• The data from different clusters are independent by virtue of the random selection or by assumption.

• The observations from the same cluster are often correlated, such as the repeated observations in a repeated measures or longitudinal study.

• It is often believed that there is heterogeneity in model parameters across subjects; for example, slopes and intercepts might differ across individuals in a longitudinal growth study. This heterogeneity, if due to stochastic sources, can be modeled with random effects.

A linear mixed models with clustered, hierarchical structure can be written as a special case of the general linear mixed model by introducing appropriate subscripts. For example, a mixed model with one type of clustering and clusters can be written as

In SAS/STAT software, the clusters are referred to as subjects, and the effects that define clusters in your data can be specified with the SUBJECT= option in the GLIMMIX, HPMIXED, MIXED, and NLMIXED procedures. The vector collects the observations for the th subject. In certain disciplines, the organization of a hierarchical model is viewed in a bottom-up form, where the measured observations represent the first level, these are collected into units at the second level, and so forth. In the school data example, the bottom-up approach considers a student’s score as the level-1 observation, the classroom as the level-2 unit, and the school district as the level-3 unit (if these were also selected from a population of districts).

The following points are noteworthy about mixed models with SUBJECT= specification:

• A SUBJECT= option is available in the RANDOM statements of the GLIMMIX, HPMIXED, MIXED, and NLMIXED procedures and in the REPEATED statement of the MIXED procedure.

• A SUBJECT= specification is required with the NLMIXED procedure. It is not required with any other mixed modeling procedure in SAS/STAT software.

• Specifying models with subjects is usually more computationally efficient in the MIXED and GLIMMIX procedures, especially if the SUBJECT= effects are identical or contained within each other. The computational efficiency of the HPMIXED procedure is not dependent on SUBJECT= effects in the manner in which the MIXED and GLIMMIX procedures are affected.

• There is no limit to the number of SUBJECT= effects with the MIXED, HPMIXED, and GLIMMIX procedures—that is, you can achieve an arbitrary depth of the nesting.