Hosmer–Lemeshow test

The Hosmer–Lemeshow test is a statistical test for goodness of fit for logistic regression models. It is used frequently in risk prediction models. The test assesses whether or not the observed event rates match expected event rates in subgroups of the model population. The Hosmer–Lemeshow test specifically identifies subgroups as the deciles of fitted risk values. Models for which expected and observed event rates in subgroups are similar are called well calibrated. Logistic regression models provide an estimate of the probability of an outcome, usually designated as a "success". It is desirable that the estimated probability of success be close to the true probability. Consider the following example. A researcher wishes to know if caffeine improves performance on a memory test. Volunteers consume different amounts of caffeine from 0 to 500 mg, and their score on the memory test is recorded. The results are shown in the table below. The table has the following columns. group: identifier for the 11 treatment groups, each receiving a different dose caffeine: mg of caffeine for volunteers in a treatment group n.volunteers: number of volunteers in a treatment group A.grade: the number of volunteers who achieved an A grade in the memory test (success) proportion.A: the proportion of volunteers who achieved an A grade The researcher performs a logistic regression, where "success" is a grade of A in the memory test, and the explanatory (x) variable is dose of caffeine. The logistic regression indicates that caffeine dose is significantly associated with the probability of an A grade (p < 0.001). However, the plot of the probability of an A grade versus mg caffeine shows that the logistic model (red line) does not accurately predict the probability seen in the data (black circles). The logistic model suggests that the highest proportion of A scores will occur in volunteers who consume zero mg caffeine, when in fact the highest proportion of A scores occurs in volunteer consuming in the range of 100 to 150 mg.