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# Ordinal Logistic Regression

Up till now we have only considered binary logistic regression where the response variable is binary - for example infected or uninfected. It is possible to do a logistic regression where the response variable has more than two categories (= levels). Polytomous logistic regression (also termed multinomial logistic regression) is used when the response variable has multiple but unordered categories. Ordinal logistic regression is used when there is a natural ordering of the levels of the response variable - for example low, medium and high.

There are several models for this type of data, but the most commonly used is the proportional odds model (also termed the cumulative logit model). Let us assume that we have graded disease severity on a four point scale from 1 (lowest) to 4 (highest) with a single explanatory variable. Under the proportional odds model, the odds ratio for category 1 versus categories 2-4 combined should be the same as for categories 1-2 combined versus 3-4 combined, and the same as for categories 1-3 combined versus category 4. So if we did three separate binary logistic regression for these comparisons we would be estimating the same odds ratio.

Before using this model it is important to verify whether the proportional odds assumption is met. Testing the homogeneity of the odds ratio generally uses the score test, referred to by Hosmer & Lemeshow as the parallel regression test.

If the proportional odds assumption is not met, a simple and valid approach to analyze such data is to dichotomize the ordinal response variable by means of several cutoff points. Separate binary logistic regression models are then used for each dichotomized response (see Bender & Grouven (1998)). Alternatively the (more complicated) partial proportional odds model can be used.