Multilevel and Related Models for Longitudinal Data

Longitudinal data, often called repeated measurements in medicine and panel data in the social sciences, arise when units provide responses on multiple occasions. Such data can be thought of as clustered or two-level data with occasions i at level 1 and units j at level 2. One feature distinguishing longitudinal data from other types of clustered data is the chronological ordering of the responses, implying that level-1 units cannot be viewed as exchangeable. Another feature of longitudinal data is that they often consist of a large number of small clusters. A typical aim in longitudinal analysis is to investigate the effects of co-variates both on the overall level of the responses and on changes of the responses over time. An important merit of longitudinal designs is that they allow the separation of cross-sectional and longitudinal effects. They also allow the investigation of heterogeneity across units both in the overall level of the response and in the development over time. Heterogeneity not captured by observed covariates produces dependence among responses even after controlling for those covariates. This violates the typical assumptions of ordinary regression models and must be accommodated to avoid invalid inference. It is useful to distinguish between longitudinal data with balanced and unbalanced occasions. The occasions are balanced if all units are measured at the same time points t i , i = 1,. .. , n, and unbalanced if units are measured at different time points, t ij , i = 1,. .. , n j. In the case of balanced occasions, the data can also be viewed as single-level multivariate data where responses at different occasions are treated as different variables. One advantage of the univariate multilevel approach taken here is that unbalanced occasions and