Multiple simultaneous multilevel analyses

Question

Let's say I have a dataset containing students in classes in teachers. A good way to analyse the data then is to conduct 3-level-analyses. Let's further assume that most of the students in the dataset visit multiple courses and we have an ID for each student. What is a good way to implement this shared variance then, specifically if one wants to know how much of the variance of the student assessments are due to individual students, class, teacher, and situational factors?

Answer 1

What you describe is a multiple membership model with a 3-level hierarchy of students nested within classes within teachers, and where students are nested hierarchically within more than one class-within-teacher. Multiple membership models add a weighting to each lower-level unit's estimated fixed and random effects in a higher level unit. Such weightings might be based on, for example, proportion of time spent in one higher-level unit than another; in your case, possibly, each class attended by a student would receive equal weight, and each class not attended would receive zero weight.

References (The first provides a general treatment of these models, and the second is an example publication using them)

Browne, W. J., Goldstein, H., and Rasbash, J. (2001). Multiple membership multiple classification (MMMC) models. Statistical Modelling, 1:103–124.

Leyland, A. H. and Naess, Ø. (2009). The effect of area of residence over the life course on subsequent mortality. Journal of the Royal Statistical Society: Series A (Statistics in Society), 172(3):555–578.