Selection In Quantile Regression Research Articles

Bayesian effect selection in structured additive quantile regression

Bayesian structured additive quantile regression is an established tool for regressing outcomes with unknown distributions on a set of explanatory variables and/or when interest lies with effects on the more extreme values of the outcome. Even though variable selection for quantile regression exists, its scope is limited. We propose the use of the Normal Beta Prime Spike and Slab (NBPSS) prior in Bayesian quantile regression to aid the researcher in not only variable but also effect selection. We compare the Bayesian NBPSS approach to statistical boosting for quantile regression, a current standard in automated variable selection in quantile regression, in a simulation study with varying degrees of model complexity and illustrate both methods on an example of childhood malnutrition in Nigeria. The NBPSS prior shows good performance in variable and effect selection as well as prediction compared to boosting and can thus be recommended as an additional tool for quantile regression model building.

Compositional variable selection in quantile regression for microbiome data with false discovery rate control

AbstractAdvancement in high‐throughput sequencing technologies has stimulated intensive research interests to identify specific microbial taxa that are associated with disease conditions. Such knowledge is invaluable both from the perspective of understanding biology and from the biomedical perspective of therapeutic development, as the microbiome is inherently modifiable. Despite availability of massive data, analysis of microbiome compositional data remains difficult. The nature that relative abundances of all components of a microbial community sum to one poses challenges for statistical analysis, especially in high‐dimensional settings, where a common research theme is to select a small fraction of signals from amid many noisy features. Motivated by studies examining the role of microbiome in host transcriptomics, we propose a novel approach to identify microbial taxa that are associated with host gene expressions. Besides accommodating compositional nature of microbiome data, our method both achieves FDR‐controlled variable selection, and captures heterogeneity due to either heteroscedastic variance or non‐location‐scale covariate effects displayed in the motivating dataset. We demonstrate the superior performance of our method using extensive numerical simulation studies and then apply it to real‐world microbiome data analysis to gain novel biological insights that are missed by traditional mean‐based linear regression analysis.

Bayesian variable selection in quantile regression with random effects: an application to Municipal Human Development Index

According to the Atlas of Human Development in Brazil, the income dimension of Municipal Human Development Index (MHDI-I) is an indicator that shows the population's ability in a municipality to ensure a minimum standard of living to provide their basic needs, such as water, food and shelter. In public policy, one of the research objectives is to identify social and economic variables that are associated with this index. Due to the income inequality, evaluate these associations in quantiles, instead of the mean, could be more interest. Thus, in this paper, we develop a Bayesian variable selection in quantile regression models with hierarchical random effects. In particular, we assume a likelihood function based on the Generalized Asymmetric Laplace distribution, and a spike-and-slab prior is used to perform variable selection. The Generalized Asymmetric Laplace distribution is a more general alternative than the Asymmetric Laplace one, which is a common approach used in quantile regression under the Bayesian paradigm. The performance of the proposed method is evaluated via a comprehensive simulation study, and it is applied to the MHDI-I from municipalities located in the state of Rio de Janeiro.

Efficient information-based criteria for model selection in quantile regression

Efficient information-based criteria for model selection in quantile regression

Atan Regularized in Quantile Regression for High Dimensional Data

In this article, we proposed using a new version of penalty functions to estimate of parameters and variable selection in quantile regression. This version of the penalty depends on arctangent function called the atan penalty. Ridge and elastic-net are the penalties that are used to compare with the proposed estimator in quantile regression. A simulation study and the real data application showed that the proposed estimator is the best compred to the other estimators.

Modified check loss for efficient estimation via model selection in quantile regression

The check loss function is used to define quantile regression. In cross-validation, it is also employed as a validation function when the true distribution is unknown. However, our empirical study indicates that validation with the check loss often leads to overfitting the data. In this work, we suggest a modified or L2-adjusted check loss which rounds the sharp corner in the middle of check loss. This has the effect of guarding against overfitting to some extent. The adjustment is devised to shrink to zero as sample size grows. Through various simulation settings of linear and nonlinear regressions, the improvement due to modification of the check loss by quadratic adjustment is examined empirically.

Bayesian Quantile Regression Method to Construct the Low Birth Weight Model

This study aims to implement Bayesian quantile regression method in constructing the model of Low Birth Weight. The data of Low Birth Weight is violated of nonnormal assumption for error terms. This study considers quantile regression approach and use Gibbs sampling algorithm from Bayesian method for fitting the quantile regression model. This study explores the performance of the asymmetric Laplace distribution for working likelihood in posterior estimation process. This study also compare the result of variable selection in quantile regression and Bayesian quantile regression for Low Birth Weight model. This study. proved that Bayesan quantile method produced better model than just quantile approach. Bayesian quantile method proved that it can handle the nonnormal problem although using moderate size of data.

Screening and selection for quantile regression using an alternative measure of variable importance

We propose a variable importance measure called partial quantile utility (PQU). We then introduce a quantile forward regression algorithm (QFR) that uses PQU-based ranking to screen important variables from a potential set whose dimension can be substantially larger than the sample size. We prove that QFR-based screening can identify all the important variables in a small number of steps. To remove noise variables from the screening step, we further implement variable selection by adopting a modified Bayesian information criterion. We show that the smaller selected set also contains all the important variables with overwhelming probability. Using simulation designs that are intentionally chosen to show its capability in identifying jointly but not marginally important variables and detecting heterogeneous associations, we extensively investigate its finite-sample performance with regard to screening, selection and out-of-sample prediction. To further illustrate the merit of our proposal, we provide an application to the problem of identifying risk factors that are associated with childhood malnutrition in India.

Group Identification and Variable Selection in Quantile Regression

Using the Pairwise Absolute Clustering and Sparsity (PACS) penalty, we proposed the regularized quantile regression QR method (QR-PACS). The PACS penalty achieves the elimination of insignificant predictors and the combination of predictors with indistinguishable coefficients (IC), which are the two issues raised in the searching for the true model. QR-PACS extends PACS from mean regression settings to QR settings. The paper shows that QR-PACS can yield promising predictive precision as well as identifying related groups in both simulation and real data.

Hierarchically penalized quantile regression with multiple responses

We study variable selection in quantile regression with multiple responses. Instead of applying conventional penalized quantile regression to each response separately, it is desired to solve them simultaneously when the sparsity patterns of the regression coefficients for different responses are similar, which is often the case in practice. In this paper, we propose employing a hierarchical penalty that enables us to detect a common sparsity pattern shared between different responses as well as additional sparsity patterns within the selected variables. We establish the oracle property of the proposed method and demonstrate it offers better performance than existing approaches.

Bayesian Quantile Regression Based on the Empirical Likelihood with Spike and Slab Priors

In this paper, we consider nonparametric Bayesian variable selection in quantile regression. The Bayesian model is based on the empirical likelihood, and the prior is chosen as the “spike-and-slab” prior–a mixture of a point mass at zero and a normal distribution. We show that the posterior distribution of the zero coefficients converges to a point mass at zero and that of the nonzero coefficients converges to a normal distribution. To further address the problem of low statistical efficiency in extreme quantile regression, we extend the Bayesian model such that it can integrate information at multiple quantiles to provide more accurate inference of extreme quantiles for homogenous error models. Simulation studies demonstrate that the proposed methods outperform or perform equally well compared with existing methods. We apply this Bayesian method to study the role of microRNAs on regulating gene expression and find that the regulation of microRNA may have a positive effect on the gene expression variation.

Bayesian model selection in ordinal quantile regression

A Bayesian stochastic search variable selection (BSSVS) method is presented for variable selection in quantile regression (QReg) for ordinal models. A Markov Chain Monte Carlo (MCMC) method is adopted to draw the unknown quantities from the full posteriors. Through simulations and analysis of an educational attainment dataset, the performance of the proposed approach is compared with some existing approaches, showing that the proposed approach performs quite good in comparison to some other methods.

Bayesian variable selection in quantile regression using the Savage–Dickey density ratio

In this paper we propose a Bayesian variable selection method in quantile regression based on the Savage–Dickey density ratio of Dickey (1976). The Bayes factor of a model containing a subset of variables against an encompassing model is given as the ratio of the marginal posterior and the marginal prior density of the corresponding subset of regression coefficients under the encompassing model. Posterior samples are generated from the encompassing model via a Gibbs sampling algorithm and the Bayes factors of all candidate models are computed simultaneously using one set of posterior samples from the encompassing model. The performance of the proposed method is investigated via simulation examples and real data sets.

Variable selection in quantile regression when the models have autoregressive errors

This paper considers a problem of variable selection in quantile regression with autoregressive errors. Recently, Wu and Liu (2009) investigated the oracle properties of the SCAD and adaptive-LASSO penalized quantile regressions under non identical but independent error assumption. We further relax the error assumptions so that the regression model can hold autoregressive errors, and then investigate theoretical properties for our proposed penalized quantile estimators under the relaxed assumption. Optimizing the objective function is often challenging because both quantile loss and penalty functions may be non-differentiable and/or non-concave. We adopt the concept of pseudo data by Oh et al. (2007) to implement a practical algorithm for the quantile estimate. In addition, we discuss the convergence property of the proposed algorithm. The performance of the proposed method is compared with those of the majorization-minimization algorithm (Hunter and Li, 2005) and the difference convex algorithm (Wu and Liu, 2009) through numerical and real examples.

Segmented Model Selection in Quantile Regression Using the Minimum Description Length Principle

This article proposes new model-fitting techniques for quantiles of an observed data sequence, including methods for data segmentation and variable selection. The main contribution, however, is in providing a means to perform these two tasks simultaneously. This is achieved by matching the data with the best-fitting piecewise quantile regression model, where the fit is determined by a penalization derived from the minimum description length principle. The resulting optimization problem is solved with the use of genetic algorithms. The proposed, fully automatic procedures are, unlike traditional break point procedures, not based on repeated hypothesis tests, and do not require, unlike most variable selection procedures, the specification of a tuning parameter. Theoretical large-sample properties are derived. Empirical comparisons with existing break point and variable selection methods for quantiles indicate that the new procedures work well in practice.

Focused model selection in quantile regression

We consider the problem of model selection for quantile regression analysis where a particular purpose of the modeling procedure has to be taken into account. Typical examples include estimation of the area under the curve in pharmacokinetics or estimation of the minimum eective dose in phase II clinical trials. A focused information criterion for quantile regression is developed, analyzed and investigated by means of a simulation study.

Focused model selection in quantile regression

Focused model selection in quantile regression

Interquantile shrinkage and variable selection in quantile regression

Examination of multiple conditional quantile functions provides a comprehensive view of the relationship between the response and covariates. In situations where quantile slope coefficients share some common features, estimation efficiency and model interpretability can be improved by utilizing such commonality across quantiles. Furthermore, elimination of irrelevant predictors will also aid in estimation and interpretation. These motivations lead to the development of two penalization methods, which can identify the interquantile commonality and nonzero quantile coefficients simultaneously. The developed methods are based on a fused penalty that encourages sparsity of both quantile coefficients and interquantile slope differences. The oracle properties of the proposed penalization methods are established. Through numerical investigations, it is demonstrated that the proposed methods lead to simpler model structure and higher estimation efficiency than the traditional quantile regression estimation.

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study.

Focused Model Selection in Quantile Regression

We consider the problem of model selection for quantile regression analysis where a particular purpose of the modeling procedure has to be taken into account. Typical examples include estimation of the area under the curve in pharmacokinetics or estimation of the minimum eff ective dose in phase II clinical trials. A focused information criterion for quantile regression is developed, analyzed and investigated by means of a simulation study and data analysis.