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.
翻译:我们通过扩展Parzen对边缘中度反应的定义,为离散反应制定了四分位回归法。与以偏差或潜伏结构为基础的现有方法相反,我们使用内插和定义有条件中度功能作为条件中度函数的反差,我们提出一个两步偏差估计法,在第一步,以非对称方式获得有条件中度概率,在第二步,通过解决隐性方程式估算回归系数。在将量化指数限制在数据驱动可接受范围时,第二阶段估算方程式有一个最差方形的封闭式解决方案。拟议的估计方程式显示,与有条件中度函数的中度函数相反,与有条件的中度函数相反,与有条件的中度函数相对立。模拟研究表明,在第一步,有条件的中位概率的中位概率是非对等变量。我们提出的方法可以用大量离差的回答方法,包括二进制、或分数和数变量来估算。我们用处方药物的数据显示,在最常态的方位类型中,最常态的中位数分析显示,最易发生性别比例分析。