The real relationship from which data are drawn may not adhere to a proposed model for this structure. Goodness-of-fit (GOF) tests help to identify when a model is a poor fit for a given dataset. A special GOF test for logistic regression models, the Hosmer-Lemeshow test, is used extensively in applications in many disciplines. It has been extensively studied, and some of its properties are somewhat surprising. There is evidence that the test functions poorly under certain circumstances. We will explore the origins of some of these properties and suggest ways to improve the test based on these findings. For example, the test statistic consists of a sum of squared Pearson residuals computed on grouped data. Preliminary evidence suggests that using adjusted Pearson residuals may allow the test to maintain its size better under certain circumstances. We will examine this, and other possible corrections to the test. Also, we will discuss the performance of a parallel test for Poisson regression models.