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

Description

In this package, we implement Bayesian penalized quantile regression for the sparse linear model, binary LASSO, group LASSO and quantile varying coefficient (VC) models. The point-mass spike-and-slab priors and horseshoe family of (horseshoe, horseshoe+ and regularized horseshoe) priors have been incorporated in the Bayesian hierarchical models to facilitate Bayesian shrinkage estimation, variable selection and statistical inference. For the spike-and-slab priors, the four default methods are Bayesian regularized quantile regression with spike-and-slab priors under the sparse linear (i.e. LASSO), binary LASSO, group LASSO and VC model, correspondingly. In addition to the default methods, users can also choose models without robustness and/or spike–and–slab priors. Furthermore, under sparse linear models, we have implemented the horseshoe family of (horseshoe, horseshoe+ and regularized horseshoe) priors and the three non-robust alternatives. Currently, the horseshoe family of priors is only implemented under the sparse linear model.

Details

The user friendly, integrated interface pqrBayes() allows users to flexibly choose fitting models by specifying the following parameters:

The function pqrBayes() returns a pqrBayes object that stores the posterior estimates of regression 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 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, 107808 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 doi:10.1111/biom.13670

Fan, K. and Wu, C. (2025). A New Robust Binary Bayesian LASSO. Stat, 14 (3), e70078 doi:10.1002/sta4.70078

Fan, K., Srijana, S., Dissanayake, V. and Wu, C. (2025). Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors arXiv preprint doi:10.48550/arXiv.2507.10975

Wu, C., and Ma, S. (2015). A selective review of robust variable selection with applications in bioinformatics. Briefings in Bioinformatics, 16(5), 873–883 doi:10.1093/bib/bbu046

Zhou, F., Ren, J., Lu, X., Ma, S. and Wu, C. (2021). Gene–Environment Interaction: a Variable Selection Perspective. Epistasis. Methods in Molecular Biology. 2212:191–223 https://link.springer.com/protocol/10.1007/978-1-0716-0947-7_13

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 doi:10.1002/sim.8434

Ren, J., Zhou, F., Li, X., Wu, C. and Jiang, Y. (2019) spinBayes: Semi-Parametric Gene-Environment Interaction via Bayesian Variable Selection. R package version 0.1.0. https://CRAN.R-project.org/package=spinBayes

Wu, C., Jiang, Y., Ren, J., Cui, Y. and Ma, S. (2018). Dissecting gene-environment interactions: A penalized robust approach accounting for hierarchical structures. Statistics in Medicine, 37:437–456 doi:10.1002/sim.7518

Wu, C., Shi, X., Cui, Y. and Ma, S. (2015). A penalized robust semiparametric approach for gene-environment interactions. Statistics in Medicine, 34 (30): 4016–4030 doi:10.1002/sim.6609

Wu, C., Cui, Y., and Ma, S. (2014). Integrative analysis of gene–environment interactions under a multi–response partially linear varying coefficient model. Statistics in Medicine, 33(28), 4988–4998 doi:10.1002/sim.6287

Wu, C., Zhong, P.S. and Cui, Y. (2018). Additive varying–coefficient model for nonlinear gene–environment interactions. Statistical Applications in Genetics and Molecular Biology, 17(2) doi:10.1515/sagmb-2017-0008

Wu, C., Zhong, P.S. and Cui, Y. (2013). High dimensional variable selection for gene-environment interactions. Technical Report. Michigan State University.

See Also

pqrBayes

95% empirical coverage probability for a pqrBayes object

Description

Calculate 95% empirical coverage probabilities for regression coefficients under the sparse linear model, binary LASSO, group LASSO and VC models, respectively.

Usage

coverage(object,coefficient,u.grid=NULL,model="linear")

Arguments

Value

c

See Also

pqrBayes

Examples

## The quantile regression model
data(data)
data = data$data_linear
g=data$g
y=data$y
e=data$e
coeff = data$coeff
fit1=pqrBayes(g,y,e,d = NULL,quant=0.5,model="linear")
coverage=coverage(fit1,coeff,model="linear")

Simulated data under high-dimensional linear, binary, group LASSO and quantile varying coefficient models

Description

Simulated data under high-dimensional linear, binary, group LASSO and quantile varying coefficient models

Format

The data_linear object consists of 4 components: g, y, e and coeff. coeff contains the true values of parameters used for generating the response variable y. The data_binary object consists of 4 components: g, y, e and coeff. coeff contains the true values of parameters used for generating the response variable y. The data_group object consists of 4 components: g, y, e and coeff. coeff contains the true values of parameters used for generating the response variable y. The data_varying object consists of five components: g, y, u, e and coeff. coeff contains the true values of parameters used for generating the response variable y.

Details

Generating Y using a sparse linear (quantile) regression model

The true data generating model under sparse linear regression:

Y_i=\beta_0+\beta_{1}X_{i1}+\beta_{2}X_{i2}+\beta_{3}X_{i3}+\epsilon_i,

where \epsilon_i\sim N(0,1), \beta_{0}=0, \beta_{1}=1 , \beta_{2}=1.5 and \beta_3=2.

Generating Y using a sparse binary (quantile) regression model

The true data generating model under sparse linear regression:

\tilde{Y}_i=\beta_0+\beta_{1}X_{i1}+\beta_{2}X_{i2}+\beta_{3}X_{i3}+\epsilon_i,

where \epsilon_i\sim N(0,1), \beta_{0}=0, \beta_{1}=0.22 , \beta_{2}=0.18 and \beta_3=0.14.

Y_i=1 if \tilde{Y}_i>0 and Y_i=0 otherwise.

Generating Y using a high-dimensional group LASSO model

The true data generating model under a group LASSO model:

Y_i=\beta_0+\beta_{1}X_{i1}+\beta_{2}X_{i2}+\beta_{3}X_{i3}+\beta_{7}X_{i7}+\beta_{8}X_{i8}+\beta_{9}X_{i9}+\epsilon_i,

where \epsilon_i\sim N(0,1), \beta_{0}=0, \beta_{1}=0.6, \beta_{2}=0.7,\beta_{3}=0.8,\beta_{7}=0.65, \beta_{8}=0.75 and \beta_{9}=0.85.

Generating Y using a (quantile) varying coefficient model

Data generation under sparse (quantile) VC model:

Y_i=\gamma_0(v_i)+\gamma_1(v_i)X_{i1}+\gamma_2(v_i)X_{i2}+\gamma_3(v_i)X_{i3}+\epsilon_i,

where \epsilon_i\sim N(0,1), \gamma_{0}(v_i)=1.5\sin(0.2\pi*v_i), \gamma_{1}(v_i)=2\exp(0.2v_i-1)-1.5 , \gamma_{2}(v_i)=2-2v_i and \gamma_3(v_i)=-4+(v_i-2)^3/6.

See Also

pqrBayes

Examples

data(data)
data = data$data_linear
g=data$g
dim(g)
y=data$y
coeff=data$coeff
print(coeff)

data = data$data_binary
g=data$g
dim(g)
y=data$y
coeff=data$coeff
print(coeff)

data = data$data_group
g=data$g
dim(g)
y=data$y
coeff=data$coeff
print(coeff)

data = data$data_varying
g=data$g
dim(g)
coeff=data$coeff
print(coeff)

Estimation and estimation accuracy for a pqrBayes object

Description

Calculate estimated regression coefficients with estimation accuracy from the sparse linear model, binary LASSO, group LASSO and quantile VC models, respectively.

Usage

estimation.pqrBayes(object,coefficient,u.grid=NULL,model="linear")

Arguments

Value

an object of class ‘pqrBayes.est’ is returned, which is a list with components:

See Also

pqrBayes

Examples

## The quantile regression model
data(data)
data = data$data_linear
g=data$g
y=data$y
e=data$e
coeff = data$coeff
fit1=pqrBayes(g,y,e,d = NULL,quant=0.5,model="linear")
estimation=estimation.pqrBayes(fit1,coeff,model="linear")

fit Bayesian penalized quantile regression for the 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 the 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

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:

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 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, 107808 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 doi:10.1111/biom.13670

Fan, K. and Wu, C. (2025). A New Robust Binary Bayesian LASSO. Stat, 14 (3), e70078 doi:10.1002/sta4.70078

Fan, K., Srijana, S., Dissanayake, V. and Wu, C. (2025). Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors. arXiv preprint doi:10.48550/arXiv.2507.10975

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 doi:10.1002/sim.8434

Examples

## The sparse quantile (linear and binary) regression 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 example (LASSO)
fit2 <- pqrBayes(
  g = g, y = y, e = e, d = NULL,
  quant = 0.5,
  prior = "Laplace",
  model = "linear"
)

## Non-robust example (LASSO)
fit3 <- pqrBayes(
  g = g, y = y, e = e, d = NULL,
  quant = 0.5,
  robust = FALSE,
  model = "linear"
)


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

## Non-robust version
fit3 <- pqrBayes(
  g = g, y = y, e = e, d = 3,
  quant = 0.5,
  robust = FALSE,
  model = "group"
)

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

## Non-robust example
fit3 <- pqrBayes(
  g = g, y = y, e = e,
  quant = 0.5,
  robust = FALSE,
  model = "VC"
)

Variable selection for a pqrBayes object

Description

Variable selection for a pqrBayes object

Usage

pqrBayes.select(object,prior="SS",model="linear")

Arguments

Details

For class ‘Sparse’, the median probability model (MPM) (Barbieri and Berger, 2004) is used to identify predictors that are significantly associated with the response variable. For class ‘NonSparse’, variable selection is based on 95% credible interval. Please check the references for more details about the variable selection.

Value

an object of class ‘select’ is returned, which includes the indices of the selected predictors (e.g. genetic factors).

References

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 doi:10.1111/biom.13670

Barbieri, M.M. and Berger, J.O. (2004). Optimal predictive model selection. Ann. Statist, 32(3):870–897

See Also

pqrBayes

Examples

## The sparse quantile regression model
data(data)
data = data$data_linear
g=data$g
y=data$y
e=data$e

fit1=pqrBayes(g,y,e,d = NULL,quant=0.5,model="linear")
select=pqrBayes.select(obj = fit1,prior = "SS",model="linear")

## The quantile varying coefficient model
data(data)
data = data$data_varying
g=data$g
y=data$y
e=data$e
fit1=pqrBayes(g,y,e,d = NULL,quant=0.5,model="VC")
select=pqrBayes.select(obj = fit1,prior = "SS",model="VC")
select


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

select <- pqrBayes.select(obj = fit2, prior = "SS", model = "VC")
select

Make predictions from a pqrBayes object

Description

Make predictions from a pqrBayes object

Usage

predict_pqrBayes(object, g.new, u.new, e.new, y.new, quant, model, ...)

Arguments

Details

g.new (u.new) must have the same number of columns as g (u) used for fitting the model. By default, the clinical covariates are NULL unless provided. The predictions are made based on the posterior estimates of coefficients in the pqrBayes object.

Value

an object of class ‘pqrBayes.pred’ is returned, which is a list with components:

See Also

pqrBayes

Examples

## The quantile regression model
data(data)
data = data$data_linear
g=data$g
y=data$y
e=data$e
fit1=pqrBayes(g,y,e,d = NULL,quant=0.5,model="linear")
prediction=predict_pqrBayes(fit1,g,u.new=e,e.new = NULL, y.new = y,model="linear")

print a pqrBayes result

Description

Print a pqrBayes result

Usage

## S3 method for class 'pqrBayes'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Value

No return value, called for side effects.

See Also

pqrBayes

print a pqrBayes.pred object

Description

Print a summary of a pqrBayes.pred object

Usage

## S3 method for class 'pqrBayes.pred'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Value

No return value, called for side effects.

See Also

predict_pqrBayes

print a select.pqrBayes object

Description

Print a summary of a select.pqrBayes object

Usage

## S3 method for class 'pqrBayes.select'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Value

No return value, called for side effects.

See Also

pqrBayes.select