Hosmer-Lemeshow Goodness of Fit Test
Hypothesis Testing
Let’s refer to the model obtained through logistic regression as $M$.
- $H_{0}$: $M$ is appropriate.
- $H_{1}$: $M$ is not appropriate.
Description
The Hosmer-Lemeshow goodness of fit test is a representative hypothesis test used to determine the adequacy of logistic regression models.
Although it’s a very simple test, the null hypothesis and the alternative hypothesis can be confusing. While it’s true that there is no good or bad in hypothesis testing, honestly speaking, if you are performing regression analysis, it’s usually to understand some correlation, and thus, one hopes to reject the null hypothesis of the F-test. This aspect is similar in t-tests, where learners familiar enough with regression analysis to study logistic regression end up adopting the non-intuitive intuition that ‘a smaller p-value equals success’.
Therefore, there can be moments of surprise when the results of the Hosmer-Lemeshow goodness of fit test deviate from ’that intuition’, despite the analysis being properly conducted. This is why it’s necessary to precisely check what the null hypothesis and the alternative hypothesis are.
Caution
Recently, weaknesses of the Hosmer-Lemeshow goodness of fit test have been pointed out , and it is said to be not highly recommended1.