Multilevel Diagnostics in Mixed and Hierarchical Linear Models
Advisor: Jan de Leeuw
In this dissertation, I develop a multilevel approach to diagnosing and assessing fit 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 define several different 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 simplifies 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 influential cases and units. The case-deletion approach motivates 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.