Increasing the power: A practical approach to goodness-of-fit test for logistic regression models with continuous predictors

When continuous predictors are present, classical Pearson and deviance goodness-of-fit tests to assess logistic model fit break down. The Hosmer-Lemeshow test can be used in these situations. While simple to perform and widely used, it does not have desirable power in many cases and provides no further information on the source of any detectable lack of fit. Tsiatis proposed a score statistic to test for covariate regional effects. While conceptually elegant, its lack of a general rule for how to partition the covariate space has, to a certain degree, limited its popularity. We propose a new method for goodness-of-fit testing that uses a very general partitioning strategy (clustering) in the covariate space and either a Pearson statistic or a score statistic. Properties of the proposed statistics are discussed, and a simulation study demonstrates increased power to detect model misspecification in a variety of settings. An application of these different methods on data from a clinical trial illustrates their use. Discussions on further improvement of the proposed tests and extending this new method to other data situations, such as ordinal response regression models are also included.