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In political science, data with heterogeneous units are used in many studies, such as those involving legislative proposals in different policy areas, electoral choices by different types of voters, and government formation in varying party systems. To disentangle decision-making mechanisms by units, traditional discrete choice models focus exclusively on the conditional mean and ignore the heterogeneous effects within a population. This paper proposes a conditional binary quantile model that goes beyond this limitation to analyze discrete response data with varying alternative-specific features. This model offers an in-depth understanding of the relationship between the explanatory and response variables. Compared to conditional mean-based models, the conditional binary quantile model relies on weak distributional assumptions and is more robust to distributional misspecification. The model also relaxes the assumption of the independence of irrelevant alternatives, which is often violated in practice. The method is applied to a range of political studies to show the heterogeneous effects of explanatory variables across the conditional distribution. Substantive interpretations from counterfactual scenarios are used to illustrate how the conditional binary quantile model captures unobserved heterogeneity, which extant models fail to do. The results point to the risk of averaging out the heterogeneous effects across units by conditional mean-based models.

Abstract
The interest in considering the relation among random variables in quantiles instead of the mean has emerged in various fields, and data collected from complex survey designs are of fundamental importance to different areas. Despite the extensive literature on survey data analysis and quantile regression models, research papers exploring quantile regression estimation accounting for an informative design have primarily been restricted to a frequentist framework. In this paper, we introduce different Bayesian methods relying on the survey-weighted estimator and the estimating equations. A model-based simulation study evaluates the proposed methods compared to alternative approaches and a naïve model fitting ignoring the informative sampling design under different scenarios. In addition, we illustrate and conduct a prior sensitivity analysis in a design-based simulation study that uses data from Prova Brasil 2011.

Abstract. In the context of a two–sample problem, a confidence interval for the difference of appropriate quantiles of the two survival distributions is described. This method is especially useful when the data include some right–censored observations. A relevant mathematical result is proved.

"This paper develops practical methods for relaxing the missing at random assumption when estimating models of conditional quantiles with missing outcome data and discrete covariates. We restrict the degree of non-ignorable selection governing the missingness process by imposing bounds on the Kolmogorov-Smirnov (KS) distance between the distribution of outcomes among missing observations and the overall (unselected) distribution. Two methods are developed for conducting inference in this environment. The first allows us to perform finite sample inference on the identified set and is well suited to tests of model specification. The second enables us to conduct inference on the parameters of potentially misspecified models. To illustrate our techniques, we revisit the results of Angrist, Chernozhukov, and Fernandez-Val (2006) regarding changes across Decennial Censuses in the quantile specific returns to schooling"--National Bureau of Economic Research web site