18: Nominal and Ordinal Dependent Variables

This chapter concludes our exploration of dependent variables in regression analysis. Having previously examined continuous, count, and binary outcomes, we now turn our attention to the final major category: categorical dependent variables.

Categorical variables describe qualities or characteristics that place observations into distinct groups. This chapter specifically addresses two fundamental types:

• Nominal variables: Categories without any inherent order (e.g., species of pet, brand of car, type of cuisine).
• Ordinal variables: Categories with a meaningful sequence or ranking (e.g., Likert-scale satisfaction ratings, education levels, disease severity stages).

The critical distinction between these types is not merely semantic; it has direct and significant implications for statistical modeling. Applying a technique designed for one type to the other can lead to inefficient models, loss of statistical power, or fundamentally incorrect interpretations. For instance, treating an ordered rating scale as a simple set of unordered labels discards valuable information about progression, while forcing arbitrary numeric values onto unordered categories can produce misleading results.

Therefore, the core objective of this chapter is to equip you with the appropriate regression frameworks for each scenario. We begin by extending the principles of logistic regression to handle nominal outcomes with more than two categories, focusing on how to model the probability of each category relative to a baseline. We then advance to specialized models for ordinal outcomes, which uniquely leverage the inherent ordering to provide more parsimonious and insightful results (Oooh! A GRE word!!).

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One of the most pervasive research questions in Political Science is to predict a person's vote based on demographic information. In other words, if you know a person's age, gender, income, education, and religion, how well can you predict how that individual will vote in the upcoming election?

At first glance, this question appears to be a binary dependent variable problem. After all, there are only two parties, right? Well, even if you ignore third parties, there is a third option: abstention. In each Ruritanian parliamentary election, a sizable number of registered voters decide not to vote. For instance, in the 2016 election, while Kuzněcov (of the royalist Král a Země party) received 48% of the vote cast and Ivanović (of the republican Republikánská Strana) received 46%, a full 45.3% of the eligible voters did not vote. Thus, the distribution of votes in this election is 26.3% Kuzněcov, 25.2% Ivanović, 3.2% other, and 45.3% none of the above. As such, conclusions based on those models that assume a binary outcome have definite issues with generalization to the voting public at large. They are ignoring important information.

A better alternative is to specifically add in "abstention" and model the three possible outcomes at once (or "abstention" and "other" and model the four). Such a regression model is called a nominal regression model or a multinomial regression model, because there is no inherent ordering among the levels of the dependent variable.

There is a second type of dependent variable that is closely related to the nominal case – the ordinal dependent variable. The difference between the nominal and the ordinal is that the ordinal has more information contained in it. There is no ordering in the nominal case, whereas there is an implicit ordering in the ordinal case. Examples of ordinal variables include ratings and indices.

If we just use our logistic regression methods (Chapter 15: Binary Dependent Variables), we come up with some odd results. If we force a nominal variable into just two categories, we lose information in the data. If we treat ordinal dependent variables simply as nominal, information is also lost. If we treat them as continuous, our conclusions may not match reality.

Thus, both nominal and ordinal dependent variables need their own modeling methods. This chapter examines how to model both the nominal dependent variable and the ordinal dependent variable more properly.