For the Analysis of Multilevel Data

Methods for estimating and testing the fit of linear models, such as regression models with fixed regressor (explanatory) variables, or general models for linear structural relations, are well understood for the analysis of a simple random sample, or for the simultaneous analysis of random samples from distinct populations (see, for example, JSreskog & S6rbom, 1979; McArdle & McDonald, 1984). The extension of such methods to multilevel data, as given, for example, by samples of randomly drawn students from randomly drawn classes from randomly drawn schools, has presented both theoretical and practical difficulties. Aitkin and Longford, (1986), describing the application of a 2-level model to educational data, with students nested within schools, show that it is misleading either to aggregate the student variables to form school means for a "means-on-means" analysis, or to ignore the hierarchical structure of the data. It can be expected that the application of a factor model or a model for linear structural relations to multilevel data similarly will need to take into account the structure of the data. For example, existing models and computer software do not cater for the important case in which the factor structure of a set of tests given to random samples of students within a random sample of schools, results from distinct factor structures for the between-school covariance matrix and the between-student-within-school covariance matrix. Similarly, existing models for linear structural relations do not allow causal modeling of relationships between school, class and student characteristics. The general theory to be described covers such cases. In addition to the contribution from Aitkin and Longford (1986), in recent years a number of articles have described particular models for the analysis of certain kinds of multilevel data structures. Thus, Strenio, Weisberg, and Bryk (1983) show how maximum likelihood estimates can be obtained for growth curve data viewed as part of a two-level structure. Goldstein (1986a) shows how a general p-level multivariate mixed effects model