SOME ALGEBRA AND GEOMETRY FOR HIERARCHICAL-MODELS, APPLIED TO DIAGNOSTICS

Recent advances in computing make it practical to use complex hierarch ical models. However, the complexity makes it difficult to see how fea tures of the data determine the fitted model. This paper describes an approach to diagnostics for hierarchical models, specifically linear h ierarchical models with additive normal or t-errors. The key is to exp ress hierarchical models in the form of ordinary linear models by addi ng artificial 'cases' to the data set corresponding to the higher leve ls of the hierarchy. The error term of this linear model is not homosc edastic, but its covariance structure is much simpler than that usuall y used in variance component or random effects models. The re-expressi on has several advantages. First, it is extremely general, covering dy namic linear models, random effect and mixed effect models, and pairwi se difference models, among others. Second, it makes more explicit the geometry of hierarchical models, by analogy with the geometry of line ar models. Third, the analogy with linear models provides a rich sourc e of ideas for diagnostics for all the parts of hierarchical models. T his paper gives diagnostics to examine candidate added variables, tran sformations, collinearity, case influence and residuals.