A Lack-of-Fit Test for the Mean Function in a Generalized Linear Model

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Abstract
A supremum-type statistic, based on partial sums of residuals, is proposed to test the validity of the mean function of the response variable in a generalized linear model. The proposed test does not need a partition of the space of covariates to handle the case of nonreplication. The new test is consistent against the alternative, under which the deterministic part of the model is not the one specified in the null hypothesis. The asymptotic null distribution of the test statistic can be approximated through simulations. This approximation is valid even if the variance of the response variable is misspecified in the model. For practical sample sizes, the adequacy of this large-sample approximation to the null distribution of the test statistic is carefully examined. Power comparisons with other lack-of-fit tests are also performed to show the advantage of the test. In particular, we find that the new procedure is rather sensitive to detect a misspecified mean function, if the assumed mean function and the true one do not intersect too frequently in the space of the independent variable. The new test is illustrated with examples for the logistic and linear regression models.

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