Diagnostics for Assessing Regression Models

• Published in: Journal of the American Statistical Association Vol. 86; no. 415; pp. 684 - 692
• Main Authors: Staniswalis, Joan G., Severini, Thomas A.
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis Group 01-09-1991; American Statistical Association
• Subjects: Approximation; Estimators; Exact sciences and technology; Kernel estimator; Lack of fit; Linear inference, regression; Linear regression; Local likelihood; Mathematical functions; Mathematical independent variables; Mathematical lattices; Mathematics; Maximum likelihood estimation; Nonparametric deviance; Parametric models; Probability and statistics; Random variables; Regression analysis; Sciences and techniques of general use; Statistics; Theory and Methods; Weighted likelihood; Limit distribution; Test statistic; Regression model; Goodness of fit test; Kernel method
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.1991.10475095

We concern ourselves with diagnostics for checking the overall and local goodness of fit of a model s(x) used in the regression of Y on x ∈ U = [0, 1]
d
. The model for s(x) is a functional form that depends on a finite number of unknown parameters. Two statistics, Λ and Λ
w
, are proposed that measure the level of agreement between the model fit to the data and the nonparametric kernel estimator on m preselected points in U. Conditions are given under which Λ and λ
w
, are asymptotically equivalent. Both of these statistics measure overall lack of fit and are related to the deviance. Their asymptotic distribution under the null model and under local alternatives is derived. This work is motivated by the local mean deviance plot of Landwehr, Pregibon, and Shoemaker for assessing overall lack of fit in logistic regression. Their plot is summarized by our test statistics and is extended to other likelihood based regressions of Y on x.