The Cost of Generalizing Logistic Regression

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Abstract
Binary-response regression models in which the link function is a family defined by one or more unknown shape parameters are considered. Detailed attention is given to the two single-parameter families proposed by Aranda-Ordaz (1981) that incorporate the logistic, linear, and complementary log-log link functions as special cases. One model is a symmetric family of alternatives to the logistic shape and the second model is an asymmetric family. The increase in variance of quantities of interest due to the addition of an extra parameter is calculated using asymptotic methods. This inflation in variance is interpreted as the cost of adding an additional parameter to the model. In considering the seriousness of this cost the reduction in bias due to the estimation of the additional parameter should also be taken into account. Three quantities of interest—predicted probabilities for a fixed value of the explanatory variable, predicted value of the explanatory variable to obtain a specific probability,...

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