Ordinal logistic regression

In statistics, the ordered logit model (also ordered logistic regression or proportional odds model), is an ordinal regression model—that is, a regression model for ordinal dependent variables—first considered by Peter McCullagh. For example, if one question on a survey is to be answered by a choice among "poor", "fair", "good", "very good", and "excellent", and the purpose of the analysis is to see how well that response can be predicted by the responses to other questions, some of which may be quantitative, then ordered logistic regression may be used. It can be thought of as an extension of the logistic regression model that applies to dichotomous dependent variables, allowing for more than two (ordered) response categories.

The model only applies to data that meet the proportional odds assumption, the meaning of which can be exemplified as follows. Suppose the proportions of members of the statistical population who would answer "poor", "fair", "good", "very good", and "excellent" are respectively p₁, p₂, p₃, p₄, p₅. Then the logarithms of the odds (not the logarithms of the probabilities) of answering in certain ways are:

The proportional odds assumption is that the number added to each of these logarithms to get the next is the same in every case. In other words, these logarithms form an arithmetic sequence. The model states that the number in the last column of the table—the number of times that that logarithm must be added—is some linear combination of the other observed variables.

The coefficients in the linear combination cannot be consistently estimated using ordinary least squares. They are usually estimated using maximum likelihood. The maximum-likelihood estimates are computed by using iteratively reweighted least squares.

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