Mis-specification and goodness-of-fit in logistic regression

Abstract

The logistic regression model has become a standard model for binary outcomes in many areas of application and is widely used in medical statistics. Much work has been carried out to examine the asymptotic behaviour of the distribution of Maximum Likelihood Estimates (MLE) for the logistic regression model, although the most widely known properties apply only if the assumed model is correct. There has been much work on goodness-of-fit tests to address the last point. The first part of this thesis investigates the behaviour of the asymptotic distribution of the (MLE) under a form of model mis-specification, namely when covariates from the true model are omitted from the fitted model. When the incorrect model is fitted the maximum likelihood estimates converge to the least false values. In this work, key integrals cannot be evaluated explicitly but we use properties of the skew-Normal distribution and the approximation of the Logit by a suitable Probit function to obtain a good approximation for the least false values. The second part of the thesis investigates the assessment of a particular goodness-of-fit test namely the information matrix test (IM) test as applied to binary data models. Kuss (2002), claimed that the IM test has reasonable power compared with other statistics. In this part of the thesis we investigate this claim, consider the distribution of the moments of the IM statistic and the asymptotic distribution of the IM test (IMT) statistic. We had dificulty in reproducing the results claimed by Kuss (2002) and considered that this was probably due to the near singularity of the variance of IMT. We define a new form of the IMT statistic, IMTR, which addresses this issue.