Dual Regression

We propose dual regression as an alternative to the quantile regression process for the global estimation of conditional distribution functions under minimal assumptions. Dual regression provides all the interpretational power of the quantile regression process while avoiding the need for repairing the intersecting conditional quantile surfaces that quantile regression often produces in practice. Our approach introduces a mathematical programming characterization of conditional distribution functions which, in its simplest form, is the dual program of a simultaneous estimator for linear location-scale models. We apply our general characterization to the specification and estimation of a flexible class of conditional distribution functions, and present asymptotic theory for the corresponding empirical dual regression process.Comment: Version accepted for publication, 39 pages, 4 figure

Variable selection for BART: An application to gene regulation

We consider the task of discovering gene regulatory networks, which are defined as sets of genes and the corresponding transcription factors which regulate their expression levels. This can be viewed as a variable selection problem, potentially with high dimensionality. Variable selection is especially challenging in high-dimensional settings, where it is difficult to detect subtle individual effects and interactions between predictors. Bayesian Additive Regression Trees [BART, Ann. Appl. Stat. 4 (2010) 266-298] provides a novel nonparametric alternative to parametric regression approaches, such as the lasso or stepwise regression, especially when the number of relevant predictors is sparse relative to the total number of available predictors and the fundamental relationships are nonlinear. We develop a principled permutation-based inferential approach for determining when the effect of a selected predictor is likely to be real. Going further, we adapt the BART procedure to incorporate informed prior information about variable importance. We present simulations demonstrating that our method compares favorably to existing parametric and nonparametric procedures in a variety of data settings. To demonstrate the potential of our approach in a biological context, we apply it to the task of inferring the gene regulatory network in yeast (Saccharomyces cerevisiae). We find that our BART-based procedure is best able to recover the subset of covariates with the largest signal compared to other variable selection methods. The methods developed in this work are readily available in the R package bartMachine.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS755 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org