Generalized logistic models

Abstract A class of models indexed by two shape parameters is introduced, both to extend the scope of the standard logistic model to asymmetric probability curves and improve the fit in the noncentral probability regions. One-parameter subclasses can be used to examine symmetric or asymmetric deviations from the logistic model. The delta algorithm is adapted to obtain maximum likelihood estimates of the parameters. A review is made of other proposed generalizations. The standard linear logistic model is widely used for modeling the dependence of binary data on explanatory variables. Its success is due to its broad applicability, simplicity of form, and ease of interpretation. This model works well for many common applications; however, it assumes that the expected probability curve μ(η) is skew-symmetric about μ = 1 2 and that the shape of μ(η) is the cumulative distribution function of the logistic distribution. Symmetric data with a shallower or steeper slope of ascent may not be fitted well by this model, nor is there any provision for treating the two tails of the estimated curve μ(η) asymmetrically or fitting different distributions for μ(η). This article introduces a class of models, indexed by one or two shape parameters, that encompasses a wider range of situations than the standard logistic model (although the standard model is included). The shape parameters have been specifically designed to modify the behavior of the curve in the extreme-probability regions where problems of lack of fit may occur, while allowing for asymmetric treatment of the two tails. Members of this family approximate the Gaussian, Laplace, and extreme minimum and maximum distributions up to the first four moments. The model can be collapsed to several simpler one-parameter symmetric and asymmetric formulations.