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Generalized Linear Model Diagnostics Using the Deviance and Single Case Deletions

1987; Oxford University Press; Volume: 36; Issue: 2 Linguagem: Inglês

10.2307/2347550

1467-9876

David A. Williams,

Advanced Statistical Methods and Models

SUMMARY This paper exploits the one step approximation, derived by Pregibon (1981), for the changes in the deviance of a generalized linear model when a single case is deleted from the data. This approximation suggests a particular set of residuals which can be used, not only to identify outliers and examine distributional assumptions, but also to calculate measures of the influence of single cases on various inferences that can be drawn from the fitted model using likelihood ratio statistics. Regression diagnostics for the Normal linear model are now well established in the literature. They are comprehensively surveyed by Cook and Weisberg (1982). Many of these diagnostics use statistics which measure the effects of deleting single cases from the data. These statistics exploit the exact algebraic relationship between the least squares fit of the linear model to a complete set of n cases, and the fit to the n - 1 cases remaining after the deletion of a single case. The maximum likelihood (ML) estimation of most generalized linear models (GLMs) requires iterative methods. The ML estimates from n - 1 cases cannot then be obtained as explicit functions of the results of the fit to all n cases. In an important paper Pregibon (1981) derives useful one step approximations for the changes in the ML estimate and the deviance of the model when a single case is deleted, and he discusses some diagnostic methods which use these approximations. Cook and Weisberg (1982) discuss GLM diagnostics briefly in Section 5.4 and they make use of Pregibon's results. McCullagh and Nelder (1983) discuss diagnostics in their chapter on model checking (chapter 11) but Pregibon's approximations are not mentioned. In this paper I describe some GLM diagnostics which all make use of Pregibon's one step approximations for the change in the components of the deviance when a single case is deleted. Section 2 establishes the notation of the paper, Section 3 states the one step approximations, and Section 4 shows how these approximations can be used to define residuals and to measure the influence of individual cases on different aspects of the fitted model. Throughout these three sections the computations will be exemplified using quantal assay data taken from Table V of Irwin (1937). These data were chosen for two reasons. Firstly they have been reanalysed by Copenhaver and Mielke (1977) and Morgan (1985) and they are known to contain some features of interest. Secondly, as they comprise only five cases, a variety of single case diagnostics can be completely tabulated and compared without taking up much space. These data are