# Ordinal Regression

Ordinal Regression allows you to model the dependence of a polytomous ordinal response on a set of predictors, which can be factors or covariates. The design of Ordinal Regression is based on the methodology of McCullagh (1980, 1998), and the procedure is referred to as PLUM in the syntax.

Standard linear regression analysis involves minimizing the sum-of-squared differences between a response (dependent) variable and a weighted combination of predictor (independent) variables. The estimated coefficients reflect how changes in the predictors affect the response. The response is assumed to be numerical, in the sense that changes in the level of the response are equivalent throughout the range of the response. For example, the difference in height between a person who is 150 cm tall and a person who is 140 cm tall is 10 cm, which has the same meaning as the difference in height between a person who is 210 cm tall and a person who is 200 cm tall. These relationships do not necessarily hold for ordinal variables, in which the choice and number of response categories can be quite arbitrary.

Example. Ordinal Regression could be used to study patient reaction to drug dosage. The possible reactions may be classified as none, mild, moderate, or severe. The difference between a mild and moderate reaction is difficult or impossible to quantify and is based on perception. Moreover, the difference between a mild and moderate response may be greater or less than the difference between a moderate and severe response.

Statistics and plots. Observed and expected frequencies and cumulative frequencies, Pearson residuals for frequencies and cumulative frequencies, observed and expected probabilities, observed and expected cumulative probabilities of each response category by covariate pattern, asymptotic correlation and covariance matrices of parameter estimates, Pearson's chi-square and likelihood-ratio chi-square, goodness-of-fit statistics, iteration history, test of parallel lines assumption, parameter estimates, standard errors, confidence intervals, and Cox and Snell's, Nagelkerke's, and McFadden's R 2 statistics.

Ordinal Regression Data Considerations

Data. The dependent variable is assumed to be ordinal and can be numeric or string. The ordering is determined by sorting the values of the dependent variable in ascending order. The lowest value defines the first category. Factor variables are assumed to be categorical. Covariate variables must be numeric. Note that using more than one continuous covariate can easily result in the creation of a very large cell probabilities table.

Assumptions. Only one response variable is allowed, and it must be specified. Also, for each distinct pattern of values across the independent variables, the responses are assumed to be independent multinomial variables.

Related procedures. Nominal logistic regression uses similar models for nominal dependent variables.

Obtaining an Ordinal Regression

This feature requires the Statistics Base option.