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In this paper, goodness-of-fit test statistics for ordinal regression models are proposed, which have approximate X2-distributions when the model has been correctly specified. The statistics proposed can be viewed as extensions of the Hosmer-Lemeshow statistic to ordinal categorical data and can be easily calculated by using existing statistical software for analysing ordinal response data The methods are illustrated by using data from an arthritis clinical trial comparing the drug auranofin and placebo therapy for the treatment of rheumatoid arthritis, in which the response is a self-assessment of arthritis, classified as poor, fair and good. The covariates of interest are age, gender, treatment and base-line response. A proportional odds model is fitted to the data, and the proposed goodness-of-fit statistics are applied to the fitted model. Also, the small sample properties of the proposed goodness-of-fit statistics are compared in a simulation study.
Lipsitz et al. (Mon,) studied this question.