Standard Errors and Sample Sizes for Two-Level Research

The hierarchical linear model approach to a two-level design is considered, some variables at the lower level having fixed and others having random regression coefficients.An approximation is derived to the covariance matrix of the estimators of the fixed regression coefficients (for variables at the lower and the higher level) under the assumption that the sample sizes at either level are large enough.This covariance matrix is expressed as a function of parameters occurring in the model.If a research planner can make a reasonable guess as to these parameters, this approximation can be used as a guide to the choice of sample sizes at either level.Multilevel and, in particular, two-level designs are used frequently in educational and social research.Hierarchical linear models incorporating both random and fixed effects provide a useful statistical paradigm for situations where nesting is an obvious and direct consequence of multistage sampling as well as situations with nested sources of random variability.(See Bryk see Raudenbush, 1988, for a review.)Some applications of such models follow.1. Nesting of microunits within macrounits-for instance, students within schools (e.g., Aitkin & Longford, 1986).The sampling design can be, but does not need to be, a multistage sample.2. Multivariate analysis, with randomly sampled measurements on a latent construct where the measurements are nested within randomly sampled units (e.g., Goldstein, 1987, p. 61). 3. Assessment of change in a repeated measurements design, with the repeated measures assumed to be random samples within units, the samples being ordered in time (e.g., Bryk & Raudenbush, 1987).