# Mid-quantile regression for discrete responses

## TL;DR

This paper introduces a new mid-quantile regression method for discrete responses that improves estimation accuracy over existing jittering approaches, applicable to binary, ordinal, and count data.

## Contribution

We develop a novel interpolation-based mid-quantile regression approach for discrete responses, with a two-step estimator that is consistent and asymptotically normal.

## Key findings

- Estimator performs well in simulations
- Reveals gender inequality in prescription data
- Identifies obesity as a key driver of medication use

## Abstract

We develop quantile regression methods for discrete responses by extending Parzen's definition of marginal mid-quantiles. As opposed to existing approaches, which are based on either jittering or latent constructs, we use interpolation and define the conditional mid-quantile function as the inverse of the conditional mid-distribution function. We propose a two-step estimator whereby, in the first step, conditional mid-probabilities are obtained nonparametrically and, in the second step, regression coefficients are estimated by solving an implicit equation. When constraining the quantile index to a data-driven admissible range, the second-step estimating equation has a least-squares type, closed-form solution. The proposed estimator is shown to be strongly consistent and asymptotically normal. A simulation study shows that our estimator performs satisfactorily and has an advantage over a competing alternative based on jittering. Our methods can be applied to a large variety of discrete responses, including binary, ordinal, and count variables. We show an application using data on prescription drugs in the United States and discuss two key findings. First, our analysis suggests a possible differential medical treatment that worsens the gender inequality among the most fragile segment of the population. Second, obesity is a strong driver of the number of prescription drugs and is stronger for more frequent medications users. The proposed methods are implemented in the R package Qtools.