  Unlike standard linear regression, quantile regression captures the
relationship between covariates and the conditional response distribution as a
whole, rather than only the relationship between covariates and the expected
value of the conditional response. However, while there are well-established
quantile regression methods for continuous variables and some forms of discrete
data, there is no widely accepted method for ordinal variables, despite their
importance in many medical contexts. In this work, we describe two existing
ordinal quantile regression methods and demonstrate their weaknesses. We then
propose a new method, Bayesian ordinal quantile regression with a partially
collapsed Gibbs sampler (BORPS). We show superior results using BORPS versus
existing methods on an extensive set of simulations. We further illustrate the
benefits of our method by applying BORPS to the Fragile Families and Child
Wellbeing Study data to tease apart associations with early puberty among both
genders. Software is available at: GitHub.com/igrabski/borps.
