Package 'mixedBayes'

Bayesian Longitudinal Regularized Quantile Mixed Model

Description

In this package, we provide a set of Bayesian regularized variable selection methods under the mixed effect models (random intercept and slope model, random intercept model) to dissect important gene - environment interactions for longitudinal studies. A Bayesian quantile regression has been adopted to accommodate data contamination and heavy-tailed distributions in the response/ phenotype. The default method (the proposed method) conducts variable selection by accounting the group level selection on the interaction effects under random intercept and slope model. In particular, the spike–and–slab priors are imposed on both individual and group levels to identify important main and interaction effects. In addition to the default method, users can also choose different selection structures for the interaction effects (group-level or individual-level), random intercept model, methods without spike–and–slab priors and non-robust methods. In total, mixedBayes provides 16 different methods (8 robust and 8 non-robust) under both mixed effects models. Among them, robust methods with spike–and–slab priors and the robust method for both individual level selection and group level selection under both mixed effects models have been developed for the first time. Please read the Details below for how to configure the method used.

Details

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

slope:	whether to use random intercept and slope model.
robust:	whether to use robust methods for modelling.
quant:	to specify different quantiles when using robust methods.
structure:	structure for interaction effects.
sparse:	whether to use the spike-and-slab priors to impose sparsity.

The function mixedBayes() returns a mixedBayes object that contains the posterior estimates of each coefficients. S3 generic functions selection()and print() are implemented for mixedBayes objects. selection() takes a mixedBayes object and returns the variable selection results.

References

Fan, K., Wang, W. and Wu, C. (2022). Bayesian Regularized Quantile Mixed Model for Lipid - Environment Interactions in a Longitudinal Study. (to be submitted)

Zhou, F., Ren, J., Li, G., Jiang, Y., Li, X., Wang, W. and Wu, C. (2019). Penalized Variable Selection for Lipid-Environment Interactions in a Longitudinal Lipidomics Study. Genes, 10(12), 1002 doi:10.3390/genes10121002

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

Ren, J., Zhou, F., Li, X., Ma, S., Jiang, Y. and Wu, C. (2020). roben: Robust Bayesian Variable Selection for Gene-Environment Interactions. R package version 0.1.1. https://CRAN.R-project.org/package=roben

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., 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. (2013). High dimensional variable selection for gene-environment interactions. Technical Report. Michigan State University.

See Also

mixedBayes

---

simulated data for demonstrating the features of mixedBayes

Description

Simulated gene expression data for demonstrating the features of mixedBayes.

Format

The data object consists of seven components: y, e, X, g, w ,k and coeff. coeff contains the true values of parameters used for generating Y.

Details

The data and model setting

Consider a longitudinal study on $n$ subjects with $k$ repeated measurement for each subject. Let $Y_{ij}$ be the measurement for the $i$th subject at each time point $j$($1\leq i \leq n, 1\leq j \leq k$) .We use a $m$-dimensional vector $G_{ij}$ to denote the genetics factors, where $G_{ij} = (G_{ij1},...,G_{ijm})^\top$. Also, we use $p$-dimensional vector $E_{ij}$ to denote the environment factors, where $E_{ij} = (E_{ij1},...,E_{ijp})^\top$. $X_{ij} = (1, T_{ij})^\top$, where $T_{ij}^\top$ is a vector of time effects . $Z_{ij}$ is a $h \times 1$ covariate associated with random effects and $\alpha_{i}$ is a $h\times 1$ vector of random effects. At the beginning, the interaction effects is modeled as the product of genomics features and environment factors with 4 different levels. After representing the environment factors as three dummy variables, the identification of the gene by environment interaction needs to be performed as group level. Combing the genetics factors, environment factors and their interactions that associated with the longitudinal phenotype, we have the following mixed-effects model:

$Y_{ij} = X_{ij}^\top\gamma_{0}+E_{ij}^\top\gamma_{1}+G_{ij}^\top\gamma_{2}+(G_{ij}\bigotimes E_{ij})^\top\gamma_{3}+Z_{ij}^\top\alpha_{i}+\epsilon_{ij}.$

where $\gamma_{1}$,$\gamma_{2}$,$\gamma_{3}$ are $p$,$m$ and $mp$ dimensional vectors that represent the coefficients of the environment effects, the genetics effects and interactions effects, respectively. Accommodating the Kronecker product of the $m$ - dimensional vector $G_{ij}$ and the $p$-dimensional vector $E_{ij}$, the interactions between genetics and environment factors can be expressed as a $mp$-dimensional vector, denoted as the following form:

$G_{ij}\bigotimes E_{ij} = [E_{ij1}E_{ij1},E_{ij2}E_{ij2},...,E_{ij1}E_{ijp},E_{ij2}E_{ij1},...,E_{ijm}E_{ijp}]^\top.$

For random intercept and slope model, $Z_{ij}^\top = (1,j)$ and $\alpha_{i} = (\alpha_{i1},\alpha_{i2})^\top$. For random intercept model, $Z_{ij}^\top = 1$ and $\alpha_{i} = \alpha_{i1}$.

See Also

mixedBayes

Examples

data(data)
length(y)
dim(g)
dim(e)
dim(w)
print(k)
print(X)
print(coeff)

---

fit a Bayesian longitudinal regularized quantile mixed model

Description

fit a Bayesian longitudinal regularized quantile mixed model

Usage

mixedBayes(
  y,
  e,
  X,
  g,
  w,
  k,
  iterations = 10000,
  burn.in = NULL,
  slope = TRUE,
  robust = TRUE,
  quant = 0.5,
  sparse = TRUE,
  structure = c("group", "individual")
)

Arguments

y	the vector of response variable. The current version of mixedBayes only supports continuous response.
e	the matrix of a group of dummy environmental factors variables.
X	the matrix of the intercept and time effects (time effects are optional).
g	the matrix of predictors (genetic factors) without intercept. Each row should be an observation vector.
w	the matrix of interactions between genetic factors and environmental factors.
k	the total number of time points.
iterations	the number of MCMC iterations.
burn.in	the number of iterations for burn-in.
slope	logical flag. If TRUE, random intercept and slope model will be used.
robust	logical flag. If TRUE, robust methods will be used.
quant	specify different quantiles when applying robust methods.
sparse	logical flag. If TRUE, spike-and-slab priors will be used to shrink coefficients of irrelevant covariates to zero exactly.
structure	structure for interaction effects, two choices are available. "group" for selection on group-level only. "individual" for selection on individual-level only.

Details

Consider the data model described in "data":

$Y_{ij} = X_{ij}^\top\gamma_{0}+E_{ij}^\top\gamma_{1}+\sum_{l=1}^{p}G_{ijl}\gamma_{2l}+\sum_{l=1}^{p}W_{ijl}^\top\gamma_{3l}+Z_{ij}^\top\alpha_{i}+\epsilon_{ij}.$

where $\gamma_{2l}$ is the main effect of the $l$th genetic variant. The interaction effects is corresponding to the coefficient vector $\gamma_{3l}=(\gamma_{3l1}, \gamma_{3l2},\ldots,\gamma_{3lm})^\top$.

When 'structure="group"', group-level selection will be conducted on $||\gamma_{3l}||_{2}$. If 'structure="individual"', individual-level selection will be conducted on each $\gamma_{3lq}$, ($q=1,\ldots,m$).

When 'slope=TRUE' (default), random intercept and slope model will be used as the mixed effects model.

When 'sparse=TRUE' (default), spike-and-slab priors are imposed on individual and/or group levels to identify important main and interaction effects. Otherwise, Laplacian shrinkage will be used.

When 'robust=TRUE' (default), the distribution of $\epsilon_{ij}$ is defined as a Laplace distribution with density.

$f(\epsilon_{ij}|\theta,\tau) = \theta(1-\theta)\exp\left\{-\tau\rho_{\theta}(\epsilon_{ij})\right\}$, ($i=1,\dots,n,j=1,\dots,k$), which leads to a Bayesian formulation of quantile regression. If 'robust=FALSE', $\epsilon_{ij}$ follows a normal distribution.

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

Value

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

posterior	the posteriors of coefficients.
coefficient	the estimated coefficients.
burn.in	the total number of burn-ins.
iterations	the total number of iterations.

See Also

data

Examples

data(data)

## default method
fit = mixedBayes(y,e,X,g,w,k,structure=c("group"))
fit$coefficient

## Compute TP and FP
b = selection(fit,sparse=TRUE)
index = which(coeff!=0)
pos = which(b != 0)
tp = length(intersect(index, pos))
fp = length(pos) - tp
list(tp=tp, fp=fp)

## alternative: robust individual selection
fit = mixedBayes(y,e,X,g,w,k,structure=c("individual"))
fit$coefficient

## alternative: non-robust group selection
fit = mixedBayes(y,e,X,g,w,k,robust=FALSE, structure=c("group"))
fit$coefficient

## alternative: robust group selection under random intercept model
fit = mixedBayes(y,e,X,g,w,k,slope=FALSE, structure=c("group"))
fit$coefficient

---

Variable selection for a mixedBayes object

Description

Variable selection for a mixedBayes object

Usage

selection(obj, sparse)

Arguments

obj	mixedBayes object.
sparse	logical flag. If TRUE, spike-and-slab priors will be used to shrink coefficients of irrelevant covariates to zero exactly..

Details

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

Value

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

inde	a vector of indicators of selected effects.

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

mixedBayes

Examples

data(data)
## sparse
fit = mixedBayes(y,e,X,g,w,k,structure=c("group"))
selected=selection(fit,sparse=TRUE)
selected


## non-sparse
fit = mixedBayes(y,e,X,g,w,k,sparse=FALSE,structure=c("group"))
selected=selection(fit,sparse=FALSE)
selected

---