An Empirical Bayes Approach to Estimating Ordinal Treatment Effects

Working Paper No.: 
Date Published: 
Jonathan N. Katz
Delia Bailey

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).

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