Evaluation of Second- and Third-Level Variance Proportions in Multilevel Designs With Completely Observed Populations: A Note on a Latent Variable Modeling Procedure

Two- and three-level designs in educational and psychological research can involve entire populations of Level-3 and possibly Level-2 units, such as schools and educational districts nested within a given state, or neighborhoods and counties in a state. Such a design is of increasing relevance in empirical research owing to the growing popularity of large-scale studies in these and cognate disciplines. The present note discusses a readily applicable procedure for point-and-interval estimation of the proportions of second- and third-level variances in such multilevel settings, which may also be employed in model choice considerations regarding ensuing analyses for response variables of interest. The method is developed within the framework of the latent variable modeling methodology, is readily utilized with widely used software, and is illustrated with an example.

Design effects for sample size computation in three-level designs

Experiments with multiple nested levels where randomization can take place at any level bring challenges to the computation of sample sizes. Formulas derived under simple single-level experiments must be adjusted using multiplicative factors or design effects. In this work, we take a unified approach to finding the design effects in terms of intracluster correlations and present formulas to compute sample sizes of different levels. Equal cluster sample sizes and homogeneous within cluster variances are assumed.