Czado, Claudia; Munk, Axel
(1996):
Noncanonical Links in Generalized Linear Models  When is the Effort Justified?
Collaborative Research Center 386, Discussion Paper 22

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Abstract
Generalized linear models (GLM) allow for a wide range of statistical models for regression data. In particular, the logistic model is usually applied for binomial observations. Canonical links for GLM's such as the logit link in the binomial case, are often used because in this case sufficient statistics for the regression parameter exist which allow for simple interpretation of the results. However, in some applications, the overall fit as measured by the pvalues of goodness of fit statistics (as the residual deviance) can be improved significantly by the use of a noncanonical link. In this case, the interpretation of the influence of the covariables is more complicated compared to GLM's with canonical link functions. It will be illustrated through simulation that the pvalue associated with the common goodness of link tests is not appropriate to quantify the changes to mean response estimates and other quantities of interest when switching to a noncanonical link. In particular, the rate of misspecifications becomes considerably large, when the inverse information value associated with the underlying parametric link model increases. This shows that the classical tests are often too sensitive, in particular, when the number of observations is large. The consideration of a generalized pvalue function is proposed instead, which allows the exact quantification of a suitable distance to the canonical model at a controlled error rate. Corresponding tests for validating or discriminating the canonical model can easily performed by means of this function.