An Empirical Bayes Approach to Estimating Ordinal Treatment Effects

Abstract

Ordinal variables — categorical variables with a defined order to the categories, but without equal spacing between them — are frequently used in social science applications. Although a good deal of research exists on the proper modeling of ordinal response variables, there is not a clear directive as to how to model ordinal treatment variables. The usual approaches found in the literature for using ordinal treatment variables are either to use fully unconstrained, though additive, ordinal group indicators or to use a numeric predictor constrained to be continuous. Generalized additive models are a useful exception to these assumptions (Beck and Jackman 1998). In contrast to the generalized additive modeling approach, we propose the use of a Bayesian shrinkage estimator to model ordinal treatment variables. The estimator we discuss in this paper allows the model to contain both individual group level indicators and a continuous predictor. In contrast to traditionally used shrinkage models that pull the data toward a common mean, we use a linear model as the basis. Thus, each individual effect can be arbitrary, but the model “shrinks” the estimates toward a linear ordinal framework according to the data. We demonstrate the estimator on two political science examples: the impact of voter identification requirements on turnout (Alvarez, Bailey, and Katz 2007), and the impact of the frequency of religious service attendance on the liberality of abortion attitudes (e.g., Singh and Leahy 1978, Tedrow and Mahoney 1979, Combs and Welch 1982).