pqrBayes: fit Bayesian penalized quantile regression for a sparse...

In pqrBayes: Bayesian Penalized Quantile Regression

fit Bayesian penalized quantile regression for a sparse linear model, binary LASSO, group LASSO, or varying coefficient model based on spike-and-slab priors and/or the horseshoe family of priors

Description

fit Bayesian penalized quantile regression for a sparse linear model, binary LASSO, group LASSO, or varying coefficient model based on spike-and-slab priors and/or the horseshoe family of priors

Usage

pqrBayes(
  g,
  y,
  e,
  d = NULL,
  quant = 0.5,
  iterations = 10000,
  burn.in = NULL,
  robust = TRUE,
  prior = "SS",
  model = "linear",
  hyper = NULL,
  debugging = FALSE
)

Arguments

g	the matrix of predictors (subject to selection). Users do not need to specify an intercept which will be automatically included.
y	the response variable.
e	a matrix of clinical covariates not subject to selection.
d	a positive integer denotes the group size. When fitting a sparse linear or varying coefficient model, d = NULL.
quant	the quantile level specified by users. Required when robust = TRUE. Ignored (set to NULL) when robust = FALSE.The default value is 0.5.
iterations	the number of MCMC iterations. The default value is 10,000.
burn.in	the number of burn-in iterations. If NULL, the first half of MCMC iterations will be discarded as burn-ins.
robust	logical flag. If TRUE, robust methods are used. Otherwise, non-robust methods are used. The default value is TRUE.
prior	the type of prior used for variable selection. Users can specify "SS" for the spike-and-slab prior, "HS" for the horseshoe prior, "HS+" for the horseshoe plus prior, "RHS" for the regularized horseshoe prior and "Laplace" for the Laplace prior. The default value is "SS".
model	the model to be fitted. Users can specify "linear" for a sparse linear model (with a continuous response), "binary" for binary LASSO, "group" for group LASSO and "VC" for a varying coefficient model.
hyper	a named list of hyper-parameters. The default value is NULL.
debugging	logical flag. If TRUE, progress will be output to the console and extra information will be returned. The default value is FALSE.

Details

The sparse linear quantile regression model described in "data" is:

Y_{i}=\sum_{k=1}^{q} E_{ik} \beta_k +\sum_{j=0}^{p}X_{ij}\gamma_j +\epsilon_{i},

where \beta_k's are the regression coefficients for clinical covariates and \gamma_j's are the regression coefficients of \boldsymbol X.

The binary quantile regression model described in "data" is:

\tilde{Y}_{i}=\sum_{k=1}^{q} E_{ik} \beta_k +\sum_{j=0}^{p}X_{ij}\gamma_j +\epsilon_{i},

where \beta_k's are the regression coefficients for clinical covariates and \gamma_j's are the regression coefficients of \boldsymbol X.

Y_{i}=1 if \tilde{Y}_{i}>0 and Y_{i}=0 otherwise.

The group LASSO model described in "data" is:

Y_{i}=\sum_{k=1}^{q} E_{ik} \beta_k + X_{i0}\gamma_0+ \sum_{j=1}^{m}\boldsymbol{X_{ij}^\top}\boldsymbol{\gamma_j} +\epsilon_{i},

where \beta_k's are the regression coefficients for clinical covariates and \boldsymbol{\gamma_j} = (\gamma_{j1},\dots,\gamma_{jd})^\top is the vector of regression coefficients of the n \times d matrix \boldsymbol X_{j}.

The quantile varying coefficient model described in "data" is:

Y_{i}=\sum_{k=1}^{q} E_{ik} \beta_k +\sum_{j=0}^{p}\gamma_j(V_i)X_{ij} +\epsilon_{i},

where \beta_k's are the regression coefficients for the clinical covariates, and \gamma_j's are the varying intercept and varying coefficients for predictors (e.g. genetic factors), respectively.

Users can modify the hyper-parameters by providing a named list of hyper-parameters via the argument ‘hyper’. The list can have the following named components

a0, b0:

shape parameters of the Beta priors (\pi^{a_{0}-1}(1-\pi)^{b_{0}-1}) on \pi_{0}.

c1, c2:

the shape parameter and the rate parameter of the Gamma prior on \nu.

Please check the references for more details about the prior distributions.

Value

an object of class "pqrBayes" is returned, which is a list with components:

obj	a list of posterior samples from the MCMC and other parameters
coefficients	a list of posterior estimates of coefficients

References

Fan, K., Subedi, S., Yang, G., Lu, X., Ren, J. and Wu, C. (2024). Is Seeing Believing? A Practitioner's Perspective on High-dimensional Statistical Inference in Cancer Genomics Studies. Entropy, 26(9).794 \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3390/e26090794")}

Zhou, F., Ren, J., Ma, S. and Wu, C. (2023). The Bayesian regularized quantile varying coefficient model. Computational Statistics & Data Analysis, 187, 107808 \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.csda.2023.107808")}

Ren, J., Zhou, F., Li, X., Ma, S., Jiang, Y. and Wu, C. (2023). Robust Bayesian variable selection for gene-environment interactions. Biometrics, 79(2), 684-694 \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/biom.13670")}

Fan, K. and Wu, C. (2025). A New Robust Binary Bayesian LASSO. Stat, 14 (3), e70078 \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/sta4.70078")}

Fan, K., Srijana, S., Dissanayake, V. and Wu, C. (2026). Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors. Computational Statistics & Data Analysis, 219, 108358 \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.csda.2026.108358")}

Ren, J., Zhou, F., Li, X., Chen, Q., Zhang, H., Ma, S., Jiang, Y. and Wu, C. (2020) Semi-parametric Bayesian variable selection for gene-environment interactions. Statistics in Medicine, 39: 617– 638 \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/sim.8434")}

Examples

## The Bayesian regularized quantile regression under linear and binary model
data(data)
data = data$data_linear
g=data$g
y=data$y
e=data$e
data(data)
data_1=data$data_binary
g_1=data_1$g
y_1=data_1$y
e_1=data_1$e

fit1=pqrBayes(g,y,e,d=NULL,quant=0.5,model="linear")
fit1_1=pqrBayes(g_1,y_1,e_1,d=NULL,quant=0.5,model="binary")


## Non-sparse robust example (Bayesian Quantile LASSO)
fit2 <- pqrBayes(
  g = g, y = y, e = e, d = NULL,
  quant = 0.5,
  prior = "Laplace",
  model = "linear"
)

## Non-robust example (Bayesian LASSO)
fit3 <- pqrBayes(
  g = g, y = y, e = e, d = NULL,
  quant = NULL,
  robust = FALSE,
  model = "linear"
)


## The Bayesian quantile group LASSO model
data(data)
data = data$data_group
g=data$g
y=data$y
e=data$e

fit1=pqrBayes(g,y,e,d=3,quant=0.5,model="group")


## Non-sparse robust Bayesian version
fit2 <- pqrBayes(
  g = g, y = y, e = e, d = 3,
  quant = 0.5,
  prior = "Laplace",
  model = "group"
)

## Non-robust Bayesian version
fit3 <- pqrBayes(
  g = g, y = y, e = e, d = 3,
  quant = NULL,
  robust = FALSE,
  model = "group"
)

## The regularized Bayesian quantile varying coefficient model
data(data)
data = data$data_varying
g=data$g
y=data$y
e=data$e
fit1=pqrBayes(g,y,e,quant=0.5,model="VC")



## Non-sparse robust Bayesian example
fit2 <- pqrBayes(
  g = g, y = y, e = e,
  quant = 0.5,
  prior = "Laplace",
  model = "VC"
)

## Non-robust Bayesian example
fit3 <- pqrBayes(
  g = g, y = y, e = e,
  quant = NULL,
  robust = FALSE,
  model = "VC"
)

pqrBayes documentation built on March 15, 2026, 1:07 a.m.