Robust Marginal Bayesian Variable Selection for Gene-Environment Interactions

Description

In this package, we provide a set of robust marginal Bayesian variable selection methods for gene-environment interaction analysis. A Bayesian formulation of the quantile regression has been adopted to accommodate data contamination and heavy-tailed distributions in the response. The proposed method conducts a robust marginal variable selection by accounting for structural sparsity. In particular, the spike-and-slab priors are imposed to identify important main and interaction effects. In addition to the default method, users can also choose different structures (robust or non-robust), methods without spike-and-slab priors.

Details

_PACKAGE

The user friendly, integrated interface marble() allows users to flexibly choose the fitting methods they prefer. There are two arguments in marble() that control the fitting method: robust: whether to use robust methods; sparse: whether to use the spike-and-slab priors to create sparsity. The function marble() returns a marble object that contains the posterior estimates of each coefficients. Moreover, it also provides a rank list of the genetic factors and gene-environment interactions. Functions GxESelection() and print.marble() are implemented for marble objects. GxESelection() takes a marble object and returns the variable selection results.

References

Lu, X., Fan, K., Ren, J., and Wu, C. (2021). Identifying Gene–Environment Interactions With Robust Marginal Bayesian Variable Selection. Frontiers in Genetics, 12:667074 doi:10.3389/fgene.2021.667074

Ren, J., Zhou, F., Li, X., Ma, S., Jiang, Y. and Wu, C. (2020). Robust Bayesian variable selection for gene-environment interactions. doi:10.1111/biom.13670

Zhou, F., Ren, J., Lu, X., Ma, S. and Wu, C. (2020). Gene–Environment Interaction: a Variable Selection Perspective. Epistasis. Methods in Molecular Biology. Humana Press (Accepted) https://arxiv.org/abs/2003.02930

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

Shi, X., Liu, J., Huang, J., Zhou, Y., Xie, Y. and Ma, S. (2014). A penalized robust method for identifying gene–environment interactions. Genetic epidemiology, 38(3), 220-230 doi:10.1002/gepi.21795

Chai, H., Zhang, Q., Jiang, Y., Wang, G., Zhang, S., Ahmed, S. E. and Ma, S. (2017). Identifying gene-environment interactions for prognosis using a robust approach. Econometrics and statistics, 4, 105-120 doi:10.1016/j.ecosta.2016.10.004

See Also

marble

Variable selection for a marble object

Description

Variable selection for a marble object

Usage

GxESelection(obj, sparse)

Arguments

Details

For class ‘Sparse’, the inclusion probability is used to indicate the importance of predictors. Here we use a binary indicator \phi to denote that the membership of the non-spike distribution. Take the main effect of the jth genetic factor, X_{j}, as an example. Suppose we have collected H posterior samples from MCMC after burn-ins. The jth G factor is included in the marginal G\timesE model at the jth MCMC iteration if the corresponding indicator is 1, i.e., \phi_j^{(h)} = 1. Subsequently, the posterior probability of retaining the jth genetic main effect in the final marginal model is defined as the average of all the indicators for the jth G factor among the H posterior samples. That is, p_j = \hat{\pi} (\phi_j = 1|y) = \frac{1}{H} \sum_{h=1}^{H} \phi_j^{(h)}, \; j = 1, \dots,p. A larger posterior inclusion probability of jth indicates a stronger empirical evidence that the jth genetic main effect has a non-zero coefficient, i.e., a stronger association with the phenotypic trait. Here, we use 0.5 as a cutting-off point. If p_j > 0.5, then the jth genetic main effect is included in the final model. Otherwise, the jth genetic main effect is excluded in the final model. 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 ‘GxESelection’ is returned, which is a list with components:

References

Lu, X., Fan, K., Ren, J., and Wu, C. (2021). Identifying Gene–Environment Interactions With Robust Marginal Bayesian Variable Selection. Frontiers in Genetics, 12:667074 doi:10.3389/fgene.2021.667074

See Also

marble

Examples

data(dat)
max.steps=5000
## sparse
fit=marble(X, Y, E, clin, max.steps=max.steps)
selected=GxESelection(fit,sparse=TRUE)
selected

## non-sparse
fit=marble(X, Y, E, clin, max.steps=max.steps, sparse=FALSE)
selected=GxESelection(fit,sparse=FALSE)
selected

simulated data for demonstrating the features of marble.

Description

Simulated gene expression data for demonstrating the features of marble.

Usage

data("dat")

Format

dat consists of four components: X, Y, E, clin.

Details

The data model for generating Y

Use subscript i to denote the ith subject. Let (Y_{i}, X_{i}, E_{i}, clin_{i}) (i=1,\ldots,n) be independent and identically distributed random vectors. Y_{i} is a continuous response variable representing the phenotype. X_{i} is the p–dimensional vector of genetic factors. The environmental factors and clinical factors are denoted as the q-dimensional vector E_{i} and the m-dimensional vector clin_{i}, respectively. The \epsilon follows some heavy-tailed distribution. For X_{ij} (j = 1,\ldots,p), the measurement of the jth genetic factor on the jth subject, considering the following model:

Y_{i} = \alpha_{0} + \sum_{k=1}^{q}\alpha_{k}E_{ik}+\sum_{t=1}^{m}\gamma_{t}clin_{it}+\beta_{j}X_{ij}+\sum_{k=1}^{q}\eta_{jk}X_{ij}E_{ik}+\epsilon_{i},

where \alpha_{0} is the intercept, \alpha_{k}'s and \gamma_{t}'s are the regression coefficients corresponding to effects of environmental and clinical factors, respectively. The \beta_{j}'s and \eta_{jk}'s are the regression coefficients of the genetic variants and G\timesE interactions effects, correspondingly. The G\timesE interactions effects are defined with W_{j} = (X_{j}E_{1},\ldots,X_{j}E_{q}). With a slight abuse of notation, denote \tilde{W} = W_{j}. Denote \alpha=(\alpha_{1}, \ldots, \alpha_{q})^{T}, \gamma=(\gamma_{1}, \ldots, \gamma_{m})^{T}, \beta=(\beta_{1}, \ldots, \beta_{p})^{T}, \eta=(\eta_{1}^{T}, \ldots, \eta_{p}^{T})^{T}, \tilde{W} = (\tilde{W_{1}}, \dots, \tilde{W_{p}}). Then model can be written as

Y_{i} = E_{i}\alpha + clin_{i}\gamma + X_{ij}\beta_{j} + \tilde{W}_{i}\eta_{j} + \epsilon_{i}.

See Also

marble

Examples

data(dat)
dim(X)

fit a robust Bayesian variable selection model for G×E interactions.

Description

fit a robust Bayesian variable selection model for G×E interactions.

Usage

marble(
  X,
  Y,
  E,
  clin,
  max.steps = 10000,
  robust = TRUE,
  sparse = TRUE,
  debugging = FALSE
)

Arguments

Details

Consider the data model described in "dat":

Y_{i} = \alpha_{0} + \sum_{k=1}^{q}\alpha_{k}E_{ik}+\sum_{t=1}^{m}\gamma_{t}clin_{it}+\beta_{j}X_{ij}+\sum_{k=1}^{q}\eta_{jk}X_{ij}E_{ik}+\epsilon_{i},

Where \alpha_{0} is the intercept, \alpha_{k}'s and \gamma_{t}'s are the regression coefficients corresponding to effects of environmental and clinical factors. And \beta_{j}'s and \eta_{jk}'s are the regression coefficients of the genetic variants and G\timesE interactions effects, correspondingly.

When sparse=TRUE (default), spike–and–slab priors are imposed to identify important main and interaction effects. If sparse=FALSE, Laplacian shrinkage will be used.

When robust=TRUE (default), the distribution of \epsilon_{i} is defined as a Laplace distribution with density f(\epsilon_{i}|\nu) = \frac{\nu}{2}\exp\left\{-\nu |\epsilon_{i}|\right\} , (i=1,\dots,n), which leads to a Bayesian formulation of LAD regression. If robust=FALSE, \epsilon_{i} follows a normal distribution.

Here, a rank list of the main and interaction effects is provided. For method incorporating spike-and-slab priors, the inclusion probability is used to indicate the importance of predictors. We use a binary indicator \phi to denote that the membership of the non-spike distribution. Take the main effect of the jth genetic factor, X_{j}, as an example. Suppose we have collected H posterior samples from MCMC after burn-ins. The jth G factor is included in the marginal G\timesE model at the jth MCMC iteration if the corresponding indicator is 1, i.e., \phi_j^{(h)} = 1. Subsequently, the posterior probability of retaining the jth genetic main effect in the final marginal model is defined as the average of all the indicators for the jth G factor among the H posterior samples. That is, p_j = \hat{\pi} (\phi_j = 1|y) = \frac{1}{H} \sum_{h=1}^{H} \phi_j^{(h)}, \; j = 1, \dots,p. A larger posterior inclusion probability jth indicates a stronger empirical evidence that the jth genetic main effect has a non-zero coefficient, i.e., a stronger association with the phenotypic trait. For method without spike-and-slab priors, variable selection is based on different level of credible intervals.

Both X, clin and E will be standardized before the generation of interaction terms to avoid the multicollinearity between main effects and interaction terms.

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

Value

an object of class ‘marble’ is returned, which is a list with component:

References

Lu, X., Fan, K., Ren, J., and Wu, C. (2021). Identifying Gene–Environment Interactions With Robust Marginal Bayesian Variable Selection. Frontiers in Genetics, 12:667074 doi:10.3389/fgene.2021.667074

See Also

GxESelection

Examples

data(dat)

## default method
max.steps=5000
fit=marble(X, Y, E, clin, max.steps=max.steps)

## coefficients of parameters
fit$coefficient

## Estimated values of main G effects 
fit$coefficient$G

## Estimated values of interactions effects 
fit$coefficient$GE

## Rank list of main G effects and interactions 
fit$ranklist


## alternative: robust selection
fit=marble(X, Y, E, clin, max.steps=max.steps, robust=TRUE, sparse=FALSE)
fit$coefficient
fit$ranklist

## alternative: non-robust sparse selection
fit=marble(X, Y, E, clin, max.steps=max.steps, robust=FALSE, sparse=FALSE)
fit$coefficient
fit$ranklist

print a GxESelection object

Description

Print a summary of a GxESelection object

Usage

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

Arguments

Value

No return value, called for side effects.

See Also

GxESelection

print a marble object

Description

Print a summary of a marble object

Usage

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

Arguments

Value

No return value, called for side effects.

See Also

marble