Regression Models For Ordinal Data

Ordinal-level variables (such as LIKERT SCALES from survey questions, some measures of social class, etc.) are commonly used as DEPENDENT VARIABLES in the social sciences. Such data can be modeled in several ways within a regression framework: (a) The ordering of the categories can be ignored, allowing the use of binary or MULTINOMIAL LOGIT models; (b) the categories can be treated as though they were measured on an interval scale, enabling the use of ordinary LEAST SQUARES (ols); and (c) models developed specifically for ordinal dependent variables can be used. Each of these methods has strengths and limitations.

The first strategy—ignoring the ordered nature of the dependent variable—is most feasible when there are few categories to the dependent variable. For example, if the categories of the dependent variable can be legitimately collapsed into two categories without losing too much information, simple binary logit models (LOGISTIC REGRESSION) can be used. For these models, ODDS RATIOS can be used to determine differences between categories. The limitation of this approach, however, is that by collapsing categories, we lose possibly important information about the dependent variable. Although multinomial logit models can be used if we choose not to collapse the categories, these models are inefficient and become increasingly difficult to interpret as the number of categories in the dependent variable rises because of the large number of parameters that are estimated.

The strength of the second strategy—using OLS—is that the parameter estimates are easily interpreted (see slope). OLS can be legitimately used if the distribution of the dependent variable appears to be roughly normal and the ordered categories accurately represent an underlying continuous distribution. If the variable has many categories, a HISTOGRAM can give a rough assessment of the distribution. On the other hand, if the variable is measured using only a few categories, the variable will be characterized by such a high degree of measurement error that this assumption cannot be reasonably assessed. These models can also be problematic if prediction is a main goal of the research, because predicted values outside the range of the variable are possible. Nonetheless, because the variable is knowingly measured with error, having predicted values outside the range of the variable is not necessarily problematic, especially if the research is primarily concerned with determining relationships rather than making predictions.

Because we often cannot assume that the distances between categories of ordered variables are interval—meaning that they do not satisfy the criteria of OLS (in particular, the assumption of constant error variance)—regression models specifically for ordinal data have been developed. These models include the ordered logit model, which is a generalization of logistic regression, and the ordered PROBIT ANALYSIS model, which is a generalization of the binomial probit model. Here, we will deal with the most commonly used of the ordered logit models: the proportional-odds model.

Logits can directly incorporate the ordering of categories in a dependent variable, resulting in a model with a simpler interpretation and potentially greater power than multinomial logit models, which are typically used for unordered categorical dependent variables. Consider a latent continuous variable ξ, which is a linear function of the explanatory variables, the Xs, plus a random

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