## Tests for goodness of fit in ordinal logistic regression models

We examine three approaches for testing goodness of fit in ordinal logistic regression models: an ordinal version of the Hosmer–Lemeshow test (${C}_{g}$), the Lipsitz test, and the Pulkstenis–Robinson (PR) tests. The properties of these tests have previously been investigated for the proportional odds model. Here, we extend the tests to two other commonly used models: the adjacent-category and the constrained continuation-ratio models. We use a simulation study to assess null distributions and power. All three tests work well and can detect several types of lack of fit under both the adjacent-category and constrained continuation-ratio models. The ${C}_{g}$ and Lipsitz tests are best suited to detect lack of fit associated with continuous covariates, whereas the PR tests excel at detecting lack of fit associated with categorical covariates. We illustrate the use of the tests with data from a study of aftercare placement of psychiatrically hospitalized adolescents. Based on the results here and previous research, we can make a joint recommendation for testing goodness of fit in proportional odds, adjacent-category, and constrained continuation-ratio logistic regression models: because the tests may detect different types of lack of fit, a thorough assessment of goodness of fit requires use of all three approaches.