Package 'pqrBayes'

Regularized Bayesian Quantile Varying Coefficient Model

Description

In this package, we implement a Bayesian quantile varying coefficient model for non-linear gene-environment interaction analysis. The varying coefficient functions capture the possible non-linear gene-environment interactions and they are approximated using linear combinations of B-spline basis. Quantile regression is adopted as it's robust to long-tailed distributions in the response/phenotype and provides the capability of describing the relationship between the response variable and predictors at different quantiles of the response variable. The default method (the proposed method) conducts variable selection by accounting for sparsity. In particular, the spike–and–slab priors are adopted to shrink the coefficients of unimportant effects to exactly zero. In addition to the default method, users can also choose the method without spike–and–slab priors.

Details

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

sparse:

whether to use the spike-and-slab priors to impose sparsity.

The function pqrBayes() returns a pqrBayes object that contains the posterior estimates of each coefficients.

References

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

---

simulated data for demonstrating the features of pqrBayes

Description

Simulated gene expression data for demonstrating the features of pqrBayes.

Format

The data 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

The model for generating Y

Use subscript $i$ to denote the $i$th subject. Let $(\boldsymbol X_{i}, Y_{i}, V_{i}, \boldsymbol E_{i})$, ($i=1,\ldots,n$) be independent and identically distributed random vectors. $Y_{i}$ is a continuous response variable representing the disease phenotype. $\boldsymbol X_{i}=(X_{i0},...,X_{ip})^\top$ denotes a $(1+p)$–dimensional vector of predictors (e.g. genetic factors) with the first element $X_{i0}=1$. The environmental factor $V_i \in \rm I\!R^1$ is a univariate index variable. $\boldsymbol E_{i}=(E_{i1},...,E_{iq})^\top$ is the $q$-dimensional vector of clinical covariates. At a given quantile level $\tau$, considering the following quantile varying coefficient model:

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

where $\beta_{k,\tau}$'s are the regression coefficients for the clinical covariates and $\gamma_{j,\tau}(\cdot)$'s are unknown smooth varying-coefficient functions. The regression coefficients of $\boldsymbol X$ vary with the univariate index variable $\boldsymbol v=(v_1,...,v_n)^\top$. The $\epsilon_{i,\tau}$ is the random error. For simplicity of notation, the quantile level $\tau$ has been suppressed hereafter.

The true model that we used to generate Y:

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

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

See Also

pqrBayes

Examples

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

---

fit a regularized Bayesian quantile varying coefficient model

Description

fit a regularized Bayesian quantile varying coefficient model

Usage

pqrBayes(
  g,
  y,
  u,
  e = NULL,
  quant = 0.5,
  iterations = 10000,
  kn = 2,
  degree = 2,
  sparse = TRUE,
  hyper = NULL,
  debugging = FALSE
)

Arguments

g	the matrix of predictors (subject to selection) without intercept.
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.
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.
kn	the number of interior knots for B-spline.
degree	the degree of B-spline basis.
sparse	logical flag. If TRUE, spike-and-slab priors will be used to shrink coefficients of irrelevant covariates to zero exactly.
hyper	a named list of hyperparameters.
debugging	logical flag. If TRUE, progress will be output to the console and extra information will be returned.

Details

The 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 coefficients for the intercept and predictors (e.g. genetic factors).

When sparse=TRUE (default), spike–and–slab priors are adopted. Otherwise, Laplacian shrinkage will be used. 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:

posterior	posterior samples from the MCMC
coefficients	a list of posterior estimates of coefficients

References

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

data(data)
g=data$g
y=data$y
u=data$u
e=data$e

## default method
fit1=pqrBayes(g,y,u,e,quant=0.5)
fit1



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

---

make predictions from a pqrBayes object

Description

make predictions from a pqrBayes object

Usage

## S3 method for class 'pqrBayes'
predict(
  object,
  g.new,
  u.new,
  e.new = NULL,
  y.new = NULL,
  quant = 0.5,
  kn = 2,
  degree = 2,
  ...
)

Arguments

object	pqrBayes object.
g.new	a matrix of new predictors (e.g. genetic factors) at which predictions are to be made.
u.new	a vector of new environmental factor at which predictions are to be made.
e.new	a vector or matrix of new clinic covariates at which predictions are to be made.
y.new	a vector of the response of new observations. If provided, the prediction error will be computed based on Y.new.
quant	the quantile for the response variable. The default is 0.5.
kn	the number of interior knots for B-spline.
degree	the degree of B-spline basis.
...	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 clinic covariates are NULL unless provided. The predictions are made based on the posterior estimates of coefficients in the pqrBayes object.

If y.new is provided, the prediction error will be computed based on the check loss.

Value

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

error	prediction error. error is NULL if y.new=NULL.
y.pred	predicted values of the new observations.

See Also

pqrBayes

---

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.VC object

Description

Print a summary of a select.VC object

Usage

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

Arguments

x	VCselect object.
digits	significant digits in printout.
...	other print arguments

Value

No return value, called for side effects.

See Also

VCselect

---

Variable selection for a pqrBayes object

Description

Variable selection for a pqrBayes object

Usage

VCselect(obj, sparse, iterations = 10000, kn = 2, degree = 2)

Arguments

obj	pqrBayes object.
sparse	logical flag.
iterations	the number of MCMC iterations.
kn	the number of interior knots for B-spline.
degree	the degree of B-spline basis.

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 ‘VCselect’ 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. (2022). Robust Bayesian variable selection for gene-environment interactions. Biometrics, (in press) 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

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