Package 'emBayes'

Robust Bayesian Variable Selection via Expectation-Maximization

Description

This package provides the implementation of the spike-and-slab quantile LASSO (ssQLASSO) and spike-and-slab quantile group LASSO varying coefficient mixed model (ssQVCM) which combines the strength of Bayesian robust variable selection and the Expectation-Maximization (EM) coordinate descent approach. The alternative methods spike-and-slab LASSO (ssLASSO) and spike-and-slab group LASSO varying coefficient mixed model (ssVCM) are also included in the package.

Details

Two user friendly, integrated interface cv.emBayes() and emBayes() allows users to flexibly choose the variable selection method by specifying the following parameter:

quant:	to specify different quantiles when using robust methods.
func:	the model to perform variable selection. Four choices are available:
	"ssLASSO", "ssQLASSO", "ssVCM" and "ssQVCM".
error:	to specify the difference between expectations of likelihood of two
	consecutive iterations. It can be used to determine convergence.
maxiter:	to specify the maximum number of iterations.

Function cv.emBayes() returns cross-validation errors based on the check loss, least squares loss and Schwarz Information Criterion along with the corresponding optimal tuning parameters. Function emBayes() returns the estimated intercept, clinical coefficients, beta coefficients, scale parameter, probability parameter, number of iterations and expectation of likelihood at each iteration.

References

Liu, Y., Ren, J., Ma, S., and Wu, C. (2024). The Spike-and-Slab Quantile LASSO for Robust Variable Selection in Cancer Genomics Studies. Statistics in Medicine.

Ren, J., Zhou, F., Li, X., Ma, S., Jiang, Y., and Wu, C. (2022). Robust Bayesian variable selection for gene–environment interactions. Biometrics. doi:10.1111/biom.13670

Ren, J., Du, Y., Li, S., Ma, S., Jiang,Y. and Wu, C. (2019). Robust network-based regularization and variable selection for high dimensional genomics data in cancer prognosis. Genet. Epidemiol., 43:276-291 doi:10.1002/gepi.22194

Wu, C., Zhang, Q., Jiang,Y. and Ma, S. (2018). Robust network-based analysis of the associations between (epi)genetic measurements. J Multivar Anal., 168:119-130 doi:10.1016/j.jmva.2018.06.009

Tang, Z., Shen, Y., Zhang, X., and Yi, N. (2017). The spike-and-slab lasso generalized linear models for prediction and associated genes detection. Genetics, 205(1), 77-88 doi:10.1534/genetics.116.192195

Tang, Z., Shen, Y., Zhang, X., and Yi, N. (2017). The spike-and-slab lasso Cox model for survival prediction and associated genes detection. Bioinformatics, 33(18), 2799-2807 doi:10.1093/bioinformatics/btx300

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, Y. H., Ni, Z. X., and Li, Y. (2014). Quantile regression via the EM algorithm. Communications in Statistics-Simulation and Computation, 43(10), 2162-2172 doi:10.1080/03610918.2012.746980

Ročková, V., and George, E. I. (2014). EMVS: The EM approach to Bayesian variable selection. Journal of the American Statistical Association, 109(506), 828-846 doi:10.1080/01621459.2013.869223

Li, Q., Lin, N., and Xi, R. (2010). Bayesian regularized quantile regression. Bayesian Analysis, 5(3), 533-556 doi:10.1214/10-BA521

George, E. I., and McCulloch, R. E. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88(423), 881-889 doi:10.1080/01621459.1993.10476353

See Also

cv.emBayes emBayes

---

k-folds cross-validation for 'emBayes'

Description

This function performs cross-validation and returns the optimal values of the tuning parameters.

Usage

cv.emBayes(
  y,
  clin = NULL,
  X,
  W = NULL,
  nt = NULL,
  group = NULL,
  quant,
  t0,
  t1,
  k,
  func,
  error = 0.01,
  maxiter = 100
)

Arguments

y	a vector of response variable.
clin	a matrix of clinical factors. It has default value NULL.
X	a matrix of genetic factors.
W	a matrix of random factors.
nt	a vector of number of repeated measurements for each subject. They can be same or different.
group	a vector of group sizes. They can be same or different.
quant	value of quantile.
t0	a user-supplied sequence of the spike scale $s_{0}$.
t1	a user-supplied sequence of the slab scale $s_{1}$.
k	number of folds for cross-validation.
func	methods to perform variable selection. Four choices are available. For non longitudinal analysis: "ssLASSO" and "ssQLASSO". For longitudinal varying-coefficient analysis: "ssVCM" and "ssQVCM".
error	cutoff value for determining convergence. The algorithm reaches convergence if the difference in the expected log-likelihood of two iterations is less than the value of error. The default value is 0.01.
maxiter	the maximum number of iterations that is used in the estimation algorithm. The default value is 200.

Details

When performing cross-validation for emBayes, function cv.emBayes returns two sets of optimal tuning parameters and their corresponding cross-validation error matrices. The spike scale parameter $CL.s0$ and the slab scale parameter $CL.s1$ are obtained based on the quantile check loss. The spike scale parameter $SL.s0$ and the slab scale parameter $SL.s1$ are obtained based on the least squares loss. The spike scale parameter $SIC.s0$ and the slab scale parameter $SIC.s1$ are obtained based on the Schwarz Information Criterion (SIC). Corresponding error matrices $CL.CV$, $SL.CV$ and $SIC.CV$ can also be obtained from the output.

Schwarz Information Criterion has the following form:

$SIC=\log\sum_{i=1}^nL(y_i-\hat{y_i})+\frac{\log n}{2n}edf$

where $L(\cdot)$ is the check loss and $edf$ is the number of close to zero residuals $(\leq 0.001)$. For non-robust method “ssLASSO”, one should use least squares loss for tuning selection. For robust method “ssQLASSO”, one can either use quantile check loss or SIC for tuning selection. We suggest using SIC, since it has been extensively utilized for tuning selection in high-dimensional quantile regression, as documented in numerous literature sources.

Value

A list with components:

CL.s0	the optimal spike scale under check loss.
CL.s1	the optimal slab scale under check loss.
SL.s0	the optimal slab scale under least squares loss.
SL.s1	the optimal slab scale under least squares loss.
SIC.s0	the optimal slab scale under SIC.
SIC.s1	the optimal slab scale under SIC.
CL.CV	cross-validation error matrix under check loss.
SL.CV	cross-validation error matrix under least squares loss.
SIC.CV	cross-validation error matrix under SIC.

---

simulated gene expression example data

Description

Simulated gene expression data for demonstrating the usage of emBayes.

Usage

data(data)

Format

The data file consists of five components: y, clin, X, quant, coef and clin.coe. The coefficients and clinical coefficients are the true values of parameters used for generating response y. They can be used for performance evaluation.

Details

The data model for generating response

Let $y_{i}$ be the response of the $i$-th subject (1$\leq$ i$\leq$ n). We have $z_{i}=(1,z_{i1},\dots,z_{iq})^{\top}$ being a $(q+1)$-dimensional vector of which the last $q$ components indicate clinical factors and $x_{i}=(x_{i1},\dots,x_{ip})^{\top}$ denoting a $p$-dimensional vector of genetic factors. The linear quantile regression model for the $\tau$-th quantile $(0<\tau<1)$ is:

$y_i=z_i^\top\alpha+x_i^\top\beta+\epsilon_i$

where $\alpha=(\alpha_0,\cdots,\alpha_q)^\top$ contains the intercept and the regression coefficients for the clinical covariates. $\beta=(\beta_1,\cdots,\beta_p)^\top$ are the regression coefficients and random error $\epsilon_{i}=(\epsilon_{1},...,\epsilon_{n})^\top$ is set to follow a T2 distribution and has value $0$ at its $\tau$-th quantile.

See Also

emBayes

---

fit a model with given tuning parameters

Description

This function performs penalized variable selection based on spike-and-slab quantile LASSO (ssQLASSO), spike-and-slab LASSO (ssLASSO), spike-and-slab quantile group LASSO varying coefficient mixed model (ssQVCM) and spike-and-slab group LASSO varying coefficient mixed model (ssVCM). Typical usage is to first obtain the optimal spike scale and slab scale using cross-validation, then specify them in the 'emBayes' function.

Usage

emBayes(
  y,
  clin = NULL,
  X,
  W = NULL,
  nt = NULL,
  group = NULL,
  quant,
  s0,
  s1,
  func,
  error = 0.01,
  maxiter = 100
)

Arguments

y	a vector of response variable.
clin	a matrix of clinical factors. It has default value NULL.
X	a matrix of genetic factors.
W	a matrix of random factors.
nt	a vector of number of repeated measurements for each subject. They can be same or different.
group	a vector of group sizes. They can be same or different.
quant	value of quantile.
s0	value of the spike scale $s_{0}$.
s1	value of the slab scale $s_{1}$.
func	methods to perform variable selection. Four choices are available. For non longitudinal analysis: "ssLASSO" and "ssQLASSO". For longitudinal varying-coefficient analysis: "ssVCM" and "ssQVCM".
error	cutoff value for determining convergence. The algorithm reaches convergence if the difference in the expected log-likelihood of two iterations is less than the value of error. The default value is 0.01.
maxiter	the maximum number of iterations that is used in the estimation algorithm. The default value is 200.

Details

The current version of emBayes supports four types of methods: "ssLASSO", "ssQLASSO", "ssVCM" and "ssQVCM".

• ssLASSO: spike-and-slab LASSO fits a Bayesian linear regression through the EM algorithm.

• ssQLASSO: spike-and-slab quantile LASSO fits a Bayesian quantile regression (based on asymmetric Laplace distribution) through the EM algorithm.

• ssVCM: spike-and-slab group LASSO varying coefficient mixed model fits a Bayesian linear mixed model through the EM algorithm.

• ssQVCM: spike-and-slab quantile group LASSO varying coefficient mixed model fits a Bayesian quantile mixed model through the EM algorithm.

Users can choose the desired method by setting func="ssLASSO", "ssQLASSO", "ssVCM" or "ssQVCM".

Value

A list with components:

alpha	a vector containing the estimated intercept and clinical coefficients.
intercept	value of the estimated intercept.
clin.coe	a vector of estimated clinical coefficients.
r	a vector of estimated ranodm effect coefficients.
beta	a vector of estimated beta coefficients.
sigma	value of estimated asymmetric Laplace distribution scale parameter $\sigma$.
theta	value of estimated probability parameter $\theta$.
iter	value of number of iterations.
ll	a vector of expectation of likelihood at each iteration.

Examples

data(data)
##load the clinical factors, genetic factors, response and quantile data
clin=data$clin
X=data$X
y=data$y
quant=data$quant

##generate tuning vectors of desired range 
t0 <- seq(0.01,0.015,length.out=2)
t1 <- seq(0.1,0.5,length.out=2)

##perform cross-validation and obtain tuning parameters based on check loss
CV <- cv.emBayes(y,clin,X,W=NULL,nt=NULL,group=NULL,quant,t0,t1,k=5,
func="ssQLASSO",error=0.01,maxiter=200)
s0 <- CV$CL.s0
s1 <- CV$CL.s1

##perform BQLSS under optimal tuning and calculate value of TP and FP for selecting beta 
EM <- emBayes(y,clin,X,W=NULL,nt=NULL,group=NULL,quant,s0,s1,func="ssQLASSO",
error=0.01,maxiter=200)
fit <- EM$beta
coef <- data$coef
tp <- sum(fit[coef!=0]!=0)
fp <- sum(fit[coef==0]!=0)
list(tp=tp,fp=fp)

---

print an cv.emBayes result

Description

Print a summary of an 'cv.emBayes' result

Usage

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

Arguments

x	cv.emBayes result
digits	significant digits in printout.
...	other print arguments

Value

Print a list of output from a cv.emBayes object.

See Also

cv.emBayes

---

print an emBayes result

Description

Print a summary of an 'emBayes' result

Usage

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

Arguments

x	emBayes result
digits	significant digits in printout.
...	other print arguments

Value

Print a list of output from a emBayes object.

See Also

emBayes

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