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Ordinal Logistic Regression

The binary logistic regression methods we have covered in this chapter apply when we have a categorical response of the simplest possible form - dichotomous. It is natural to consider methods for more categorical responses having more than two possible values. A variety of methods have been developed for covering the various possibilities. The best known and most highly developed are methods for ordinal response variables.

Recall that a categorical variable is considered ordinal if there is a natural ordering of the possible values, for example Low, Medium, and High. A number of proposed models for this type of data are extensions of the logistic regression model. The most well known of these ordinal logistic regression methods methods is called the proportional odds model.

The basic idea underlying the proportional odds model is re-expressing the categorical variable in terms of a number of binary variables based on internal cut-points in the ordinal scale. For example, if $y$ is a variable on a 4-point scale, we can define the corresponding binary variables, $y^*_c , c= 1, ..., 3$ by $ y^*_c = 1 $ if $y > c$ and $y^*_c$ if $ y \le c$.

If one has a set of explanatory variables, $x_j , j=1 , ... , k$ , then we can consider the 3 binary logistic models corresponding to regressing each of the $y^*_c$'s separately against the $x$'s. The proportional odds model assumes that the true $\beta$-values are the same in all three models, so that the only difference in models is the intercept terms, $\alpha_c , c = 1, 2, 3$. This means that the estimates from the three binary models can be pooled to provide just one set of $\beta$ estimates. By exponentiating the pooled estimate relative to a given predictor, i.e. taking $e^{\beta_j}$, we obtain an estimate of the common odds ratio that describes the relative odds for $y > c$ for values of $x_j$ differing by 1 unit.

Thus interpreting the proportional odds model is not much more difficult than a binary logistic regression. However, valid intrepretations will depend on the assumption that proportional odds assumption, which must be checked. See the hardcopy STATA documentation (Reference: ologit) for references.


next up previous
Next: Regression Models for Survival Up: Logistic Regression Previous: Sampling Design and the
Rollin Brant 2004-03-24