Package 'pqrBayes'

Bayesian penalized quantile regression for linear and varying coefficient models

Description

In this package, we implement Bayesian penalized quantile regression for the linear regression model and quantile varying coefficient (VC) model. Point-mass spike-and-slab priors have been incorporated in the Bayesian hierarchical models to facilitate Bayesian shrinkage estimation with exact sparsity in both models. The two default methods are Bayesian regularized quantile regression with spike-and-slab priors under the linear model and VC model, correspondingly. In addition to default methods, users can also choose methods without robustness and/or spike–and–slab priors.

Details

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

robust:	whether to fit a robust sparse quantile regression model or a quantile varying coefficient
	model or their non-robust counterparts.
sparse:	whether to use the spike-and-slab priors to impose exact sparsity.
model:	whether to fit a linear model or a varying coefficient model.

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

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 linear and VC models, respectively.

Usage

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

Arguments

object	the pqrBayes object.
coefficient	the vector of true regression coefficients under a linear model or the matrix of true varying coefficients evaluated on the grid points under a varying coefficient model.
u.grid	the vector of grid points under a varying coefficient model. When assessing empirical coverage probabilities under a linear model, u.grid = NULL.
model	the model to be fitted. Users can also choose "linear" for a linear model or "VC" for a varying coefficient model.

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,u=NULL,e,quant=0.5,spline=NULL,model="linear")
coverage=coverage(fit1,coeff,model="linear")

---

simulated data under sparse linear and quantile varying coefficient models

Description

Simulated data under sparse linear 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_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 (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$data_varying
g=data$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 linear and quantile VC models, respectively.

Usage

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

Arguments

object	an object of class ‘pqrBayes’.
coefficient	the vector of quantile regression coefficients under a linear model or the matrix of true varying coefficients evaluated on the grid points under a varying coefficient model.
u.grid	the vector of grid points under a varying coefficient model. When fitting a linear (quantile) regression model, u.grid = NULL.
model	the model to be fitted. Users can choose "linear" for a linear model or "VC" for a varying coefficient model.

Value

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

error	mean square error or integrated mean square errors and total integrated mean square error.
coeff.est	estimated values of the regression coefficients or the varying coefficients.

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,u=NULL,e,quant=0.5,spline=NULL,model="linear")
estimation=estimation.pqrBayes(fit1,coeff,model="linear")

---

fit Bayesian penalized quantile regression for linear and varying coefficient models

Description

fit Bayesian penalized quantile regression for linear and varying coefficient models

Usage

pqrBayes(
  g,
  y,
  u = NULL,
  e,
  quant = 0.5,
  iterations = 10000,
  burn.in = NULL,
  spline,
  robust = TRUE,
  sparse = TRUE,
  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. The current version only supports the continuous response.
u	a vector of effect modifying variable of the quantile varying coefficient model. When fitting a linear model, u = NULL.
e	a matrix of clinical covariates not subject to selection.
quant	the quantile level specified by users. 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 used as burn-ins.
spline	a list of the number of interior knots (kn) for B-spline and the degree of B-spline basis (degree). When fitting a linear model, spline = NULL.
robust	logical flag. If TRUE, robust methods will be used. Otherwise, non-robust methods will be used. The default value is TRUE.
sparse	logical flag. If TRUE, spike-and-slab priors will be adopted to impose exact sparsity on regression coefficients. Otherwise, Laplacian shrinkage will be adopted. The default value is TRUE.
model	the model to be fitted. Users can specify "linear" for a linear model 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 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 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 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

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

fit1=pqrBayes(g,y,u=NULL,e,quant=0.5,spline=NULL,model="linear")


## non-sparse
sparse=FALSE
fit2=pqrBayes(g,y,u=NULL,e,quant=0.5,spline=NULL,sparse = sparse,model="linear")

## non-robust
robust = FALSE
fit3=pqrBayes(g,y,u=NULL,e,quant=0.5,spline=NULL,robust = robust,model="linear")



## The quantile varying coefficient model
data(data)
data = data$data_varying
g=data$g
y=data$y
u=data$u
e=data$e

spline = list(kn=2,degree=2)
fit1=pqrBayes(g,y,u,e,quant=0.5,spline = spline, model="VC")



## non-sparse
sparse=FALSE
fit2=pqrBayes(g,y,u,e,quant=0.5,spline = spline,sparse = sparse,model="VC")

## non-robust
robust = FALSE
fit3=pqrBayes(g,y,u,e,quant=0.5,spline = spline,robust = robust,model="VC")

---

Variable selection for a pqrBayes object

Description

Variable selection for a pqrBayes object

Usage

pqrBayes.select(object,sparse=T,model="linear")

Arguments

object	a pqrBayes object.
sparse	logical flag. If TRUE, the sparse model is used for variable selection. The default value is TRUE.
model	the model to be fitted. Users can also choose "linear" for a linear model or "VC" for a varying coefficient model.

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

fit1=pqrBayes(g,y,u=NULL,e,quant=0.5,spline=NULL,model="linear")
sparse=TRUE
select=pqrBayes.select(obj = fit1,sparse = sparse,model="linear")

## The quantile varying coefficient model
data(data)
data = data$data_varying
g=data$g
y=data$y
u=data$u
e=data$e
spline = list(kn=2,degree=2)
fit1=pqrBayes(g,y,u,e,quant=0.5,spline = spline, model="VC")
sparse=TRUE
select=pqrBayes.select(obj = fit1,sparse = sparse,model="VC")
select


## non-sparse
sparse=FALSE
spline = list(kn=2,degree=2)
fit2=pqrBayes(g,y,u,e,quant=0.5,spline = spline,sparse = sparse,model="VC")
select=pqrBayes.select(obj=fit2,sparse=FALSE,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

object	a pqrBayes object.
g.new	a matrix of new predictors (e.g. genetic factors) at which predictions are to be made. When being applied to the linear model, g.new = g.
u.new	a vector of new environmental factor at which predictions are to be made. When being applied to the linear model, u.new = NULL.
e.new	a vector or matrix of new clinical covariates at which predictions are to be made. When being applied to the linear model, e.new = e.
y.new	a vector of the response of new observations. When being applied to the linear model, y.new = y.
quant	the quantile level. The default is 0.5.
model	the model to be fitted. The default is "VC" for a quantile varying coefficient model. Users can also specify "linear" for a linear model.
...	other predict 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:

error	prediction error.
y.pred	predicted values of the new observations.

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,u=NULL,e,quant=0.5,spline=NULL,model="linear")
prediction=predict_pqrBayes(fit1,g,u.new=NULL,e.new = e, 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

x	pqrBayes result.
digits	significant digits in printout.
...	other print 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

x	pqrBayes.pred object.
digits	significant digits in printout.
...	other print 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

x	pqrBayes.select object.
digits	significant digits in printout.
...	other print arguments

Value

No return value, called for side effects.

See Also

pqrBayes.select

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