This chapter further extends the discussion of models that are a marked departure from the independent errors models of Chapters 5 to 8. In the models that will be discussed in this chapter, there is a hierarchy of variation that corresponds to groupings within the data. The groups are nested. For example, students might be sampled from different classes, that in turn are sampled from different schools. Or, crop yields might be measured on multiple parcels of land at each of a number of different sites.

After fitting such models, predictions can be made at any of the given levels. For example, crop yield could be predicted at new sites, or at new parcels. Prediction for a new parcel at one of the existing sites is likely to be more accurate than a prediction for a totally new site. Multi-level models, i.e., models which have multiple error (or noise) terms, are able to account for such differences in predictive accuracy.

Repeated measures models are multi-level models where measurements consist of multiple profiles in time or space; each profile can be viewed as a time series. Such data may arise in a clinical trial, and animal or plant growth curves are common examples; each “individual” is measured at several different times. Typically, the data exhibit some form of time dependence that the model should accommodate.