Logistic Regression, a review

A review is given of the development of logistic regression as a multi-purpose statistical tool.
A historical introduction shows several lines culminating in the unifying paper of Cox (1966), in which theory as developed in the field of bio-assay is shown to be applicable to designs as discriminant-analysis and case-control study. A review is given of several designs all leading to the same analysis. The link is made with epidemiological literature.
Several optimization criteria are discussed that can be used in the case of more observations per cell, namely maximum likelihood, minimum chi-square and weighted regression on the observed logits. Recent literature on the goodness of fit problem is reviewed and finally, comments are made about the non-parametric approach to logistic regression which is still in rapid development.     

Predicting survival using disease history: a model combining relative survival and frailty

Information on disease history and comorbidity of patients can often be of great value to predict survival, for example in cancer research. In this paper a model is presented that accommodates such information by combining relative survival and frailty. Relative survival is used to model the excess risk of dying from recent concurrent diseases. Individual frailty allows estimation of a 'selection effect', which occurs if patients who have survived much hazard in the past are tougher and therefore tend to live longer than those who have survived less. Results are shown to be independent of the chosen family of frailty distributions if heterogeneity is small and to lead to a simple proportional excess hazards model. The model is applied to data from the Leiden University Medical Center on patients with head/neck tumors using information on previous tumors.     

Building logistic models by means of a non parametric goodness of fit test: a case study

A sequence of logistic models is fitted to data from a Dutch follow-up study on preterm infants (POPS). To examine the adequacy of the model, a recently developed non parametric method to check goodness of fit is applied (le Cessie and Van Houwelingen (1991)). This method uses a test statistic based upon kernel regression methods.
In this paper the problem of choosing a "best" bandwidth, corresponding to the greatest power of the test statistic, is avoided by computing the test statistic for a range of different bandwidths. Testing is then based upon the asymptotic distribution of the maximum of the test statistics.
The testing method is used as a goodness of fit criterion, and the contribution of each individual observation to the test statistic is used as a diagnostic tool to localize deviations of the model, and to determine directions in which the model can be improved.