178-2007: Repeated Measures Analysis with Clustered Subjects

Repeated measures can occur on subjects that are nested within clusters. The clusters may be considered a random effect. Then the covariance structure must accommodate between-cluster variance as well as within-cluster covariance. There also may be higher levels of clustering. In addition, there may be other fixed effect factors that are crossed with some or all of the random effects. The covariance structure must be built into a statistical for the data and the statistical model must be translated into the language of computer software that has capabilities to make computations for statistical data analysis. INTRODUCTION The term “repeated measures” refers to data that are obtained on multiple occasions on the same subject. The subject could be a person, animal, plant, machine, or other unit from which data may be obtained. The “unit” is usually called a “subject” in statistical terminology, due to the human subject heritage of repeated measures analysis methods. Most commonly, repeated measures are taken over time, but they also could be obtained over space. More generally, repeated measures could refer to a multivariate response of similar data from a subject. In these situations, repeated measures can be considered another “factor” with levels given by points in time, space, or measurement scale. One of the primary difficulties in analyzing repeated measures data is determining an appropriate covariance structure for the data. The covariance structure must address covariance within the subjects on which repeated are obtained. Also, the covariance structure must incorporate variance and covariance due to random factors, such as clusters in which the subjects might be nested. In addition, variance and covariance due to experimental or sampling design must be built into the statistical model. This process usually falls into the realm of “hierarchical” modeling, because random effects tend to follow a nested classification. In this paper we use an example from educational research to illustrate development of a statistical model. We also show how the statistical model can be implemented into the missed procedure of SAS. Along the way, we emphasize that model development utilizes statistical analytic principles combined with subjective professional skills of statistical practice. BUILDING A HIERARCHICAL LINEAR MODEL Nested models are special cases of hierarchical models. (See Raudenbush and Bryk, 2002.) In nested models, there is a hierarchy of random effects corresponding to increasingly larger units. In the more general hierarchical models, there is also a nesting of units. In addition, there may be fixed effects that are crossed with the random effects. The hierarchy comes from first developing a model for the smallest sized units. Then, the smallest units are declared to be samples from populations of larger sized units, and random effects are assigned accordingly. There are two steps in model development. First is the conceptualization step in which you think of the situation and develop the statistical model. Second is the programming step in which you translate the mathematical into a computer program that can provide data analysis according to the statistical model. CONCEPTUALIZATION STEP: THE STATISTICAL MODEL One of the most popular uses of hierarchical modeling is in educational research. Here is a common type of application. Students in a school district take a standardized exam and get scores on a quantitative and a qualitative scale. The students are in classes taught by teachers, so students may be considered “nested” within teachers. Likewise, teachers are nested within schools. In this example, we consider students and teachers to be nested random factors. Depending on the situation, the schools could be considered either fixed or random. In this application, we consider the schools fixed because we have data on all the school in the district, and we only wish to make inference about the schools in the district. Since the scales on the exam were the same for all students, teachers, and schools, the scale factor is crossed with students within teachers, with teachers within schools, and with schools. “Scale” is called an observation-level factor because the levels (qual and quan) change within the two observations from each student. In addition, the students are classified according to a social-economic (SES) criterion with two levels (0 and 1). This factor is crossed with teachers and schools. “SES” is called a student-level factor because the level is SES is the same for a given student, SAS Global Forum 2007 Statistics and Data Analysis