Multilevel diagnostics for mixed and hierarchical linear models

Abstract

In this dissertation, I develop a multilevel approach to diagnosing and assessing t in
mixed linear models and hierarchical linear models, which may be extended to other
generalizations of mixed models. Since these models include multiple sources of error,
I de ne several di erent types of residuals. Most residuals are confounded in the sense
that they are subject to extraneous sources of error. The confounding of residuals
reduces the analyst's power to detect violations of modeling assumptions. I present
various approaches for reducing the confounding in residuals. I give a construction
for uncorrelated standardized residuals which are also least confounded. I argue that
a multilevel approach to diagnostics can overcome some confounding. Furthermore,
the multilevel approach simpli es the diagnostician's task by justifying the use of well
known procedures on within-unit models. I discuss the problem of identifying and
discriminating between in
uential cases and units. The case-deletion approach moti-vates the generalization of several common diagnostic measures. Unfortunately, the
discrete approach to case-deletion, which I clarify, is imprecise relative to approaches
bases on analytic approximations. Also I give some attention to the graphical presentation
and interpretation of diagnostics while discussing various examples.