Mid-quantile regression for discrete responses.

Area of Science:

• Statistics
• Econometrics
• Biostatistics

Background:

• Traditional quantile regression methods struggle with discrete response variables.
• Existing approaches often rely on jittering or latent variable constructs, which can introduce bias or complexity.
• There is a need for robust and flexible methods to analyze the distribution of discrete outcomes.

Purpose of the Study:

• To develop novel quantile regression methods specifically designed for discrete response variables.
• To extend Parzen's definition of marginal mid-quantiles to conditional mid-quantiles.
• To provide a computationally efficient and statistically sound estimator for discrete quantile regression.

Main Methods:

• Extending Parzen's definition of marginal mid-quantiles using interpolation.
• Defining the conditional mid-quantile function as the inverse of the conditional mid-distribution function.
• Proposing a two-step estimator: nonparametric estimation of conditional mid-probabilities followed by solving an implicit equation for regression coefficients.

Main Results:

• The proposed estimator is strongly consistent and asymptotically normal.
• A simulation study demonstrates satisfactory performance and an advantage over jittering-based methods.
• The methods are applicable to binary, ordinal, and count data, with demonstrated use in analyzing US prescription drug data.

Conclusions:

• The developed quantile regression methods offer a powerful tool for analyzing discrete outcomes.
• The application highlights potential gender-based disparities in medical treatment and the significant role of obesity in prescription drug usage.
• The methods are implemented in the R package Qtools for practical application.