Constrained Maximum Likelihood Exemplified by Isotonic Convex Logistic Regression

Abstract

Abstract Maximum likelihood estimates in problems in which the likelihood is smooth and the parameter space is defined by linear or smooth nonlinear inequality constraints can be obtained using available nonlinear optimization packages, such as NPSOL. This class of models includes generalized linear models with order restrictions, convexity, or smoothness constraints on the parameters (smoothness constraints being in the form of bounds on finite differences of the regression function). Methods for such models are demonstrated by an analysis of data from four published studies of the incidence of Down's syndrome in single-year maternal age intervals. Constrained logistic regression of the incidence of Down's syndrome on maternal age shows that the data are fitted well by a nondecreasing convex function. P values for the likelihood ratio test of this model against alternatives are obtained by the parametric bootstrap and iterated parametric bootstrap. The iterated bootstrap is used as a diagnostic tool: It demonstrates that the bootstrap works in addition to providing a correction term. These methods provide a general approach applicable to most models specified by constraints on the parameter space.