Sparse Group Variable Selection for Gene-Environment Interactions in the Longitudinal Study

Description

In this package, we provide a set of regularized variable selection methods tailored for longitudinal studies of gene- environment interactions. The proposed method conducts sparse group variable selection by accounting for bi-level sparsity. Specifically, the individual and group level penalties have been simultaneously imposed to identify important main and interaction effects under three working correlation structures (exchangeable , AR-1 and independence), based on either the quadratic inference function (QIF) or generalized estimating equation (GEE). In addition, only the individual or group level selection in the longitudinal setting can also be conducted using springer. In total, springer provides 18 (=3\times3\times2) methods. Among them, sparse group variable selection for longitudinal studies have been developed for the first time. Please read the Details below for how to configure the method used.

Details

Users can flexibly choose the methods to fit the model by specifying the three arguments in the user interface springer():

The function springer() returns a springer object that contains the estimated coefficients.

References

Zhou, F., Liu, Y., Ren, J., Wang, W., and Wu, C. (2023). Springer: An R package for bi-level variable selection of high-dimensional longitudinal data. Frontiers in Genetics, 14, 1088223 doi:10.3389/fgene.2023.1088223

Zhou, F., Lu, X., Ren, J., Fan, K., Ma, S. and Wu, C. (2022). Sparse Group Variable Selection for Gene-Environment Interactions in the Longitudinal Study. Genetic Epidemiology, 46(5-6), 317-340 doi:10.1002/gepi.22461

Zhou, F., Ren, J., Lu, X., Ma, S. and Wu, C. (2021). Gene-Environment Interaction: a Variable Selection Perspective. Epistasis: Methods and Protocols, Springer US doi:10.1007/978-1-0716-0947-7_13

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

Zhou, F., Ren, J., Li, X., Wu, C. and Jiang, Y. (2019) interep: Interaction Analysis of Repeated Measure Data. R package version 0.3.1. https://CRAN.R-project.org/package=interep

Ren, J., Du, Y., Li, S., Ma, S., Jiang, Y. and Wu, C. (2019). Robust network-based regularization and variable selection for high-dimensional genomic data in cancer prognosis. Genetic epidemiology, 43(3), 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. Journal of multivariate analysis, 168, 119-130 doi:10.1016/j.jmva.2018.06.009

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., 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

Ren, J., He, T., Li, Y., Liu, S., Du, Y., Jiang, Y. and Wu, C. (2017). Network-based regularization for high dimensional SNP data in the case-control study of Type 2 diabetes. BMC genetics, 18(1), 44 doi:10.1186/s12863-017-0495-5

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. and Cui, Y. (2014). Boosting signals in gene-based association studies via efficient SNP selection. Briefings in bioinformatics, 15(2), 279-291 doi:10.1093/bib/bbs087

Wu, C., Zhong, P.S. and Cui, Y. (2013). High dimensional variable selection for gene-environment interactions. Technical Report. Michigan State University.

See Also

springer

k-folds cross-validation for springer

Description

This function conducts k-fold cross-validation for springer and returns the optimal values of the tuning parameters.

Usage

cv.springer(
  clin = NULL,
  e,
  g,
  y,
  beta0,
  lambda1,
  lambda2,
  nfolds,
  func,
  corr,
  structure,
  maxits = 30,
  tol = 0.001
)

Arguments

Details

For bi-level sparse group selection, cv.springer returns two optimal tuning parameters, \lambda_{1} and \lambda_{2}; for group-level selection, this function returns the optimal \lambda_{2} with \lambda_{1}=0; for individual-level selection, this function returns the optimal \lambda_{1} with \lambda_{2}=0.

Value

an object of class "cv.springer" is returned, with is a list with components below:

simulated data for demonstrating the usage of springer

Description

Simulated gene expression data for demonstrating the usage of springer.

Usage

data("dat")

Format

The dat file consists of five components: e, g, y, clin and coeff. The coefficients are the true values of parameters used for generating Y.

Details

The data model for generating Y

Consider a longitudinal case study with n subjects and k_i measurements over time for the ith subject (i=1,\ldots,n). Let Y_{ij} be the response of the jth observation for the ith subject (i=1,\ldots,n, j=1,\ldots,k_i), X_{ij}=(X_{ij1},...,X_{ijp})^\top be a p-dimensional vector of covariates denoting p genetic factors, E_{ij}=(E_{ij1},...,E_{ijq})^\top be a q-dimensional environmental factor and Clin_{ij}=(Clin_{ij1},...,Clin_{ijt})^\top be a t-dimensional clinical factor. There is time dependence among measurements on the same subject, but we assume that the measurements between different subjects are independent. The model we used for hierarchical variable selection for gene–environment interactions is given as:

Y_{ij}= \alpha_0 + \sum_{m=1}^{t}\theta_m Clin_{ijm} + \sum_{u=1}^{q}\alpha_u E_{iju} + \sum_{v=1}^{p}(\gamma_v X_{ijv} + \sum_{u=1}^{q}h_{uv} E_{iju} X_{ijv})+\epsilon_{ij},

where \alpha_{0} is the intercept and the marginal density of Y_{ij} belongs to a canonical exponential family defined in Liang and Zeger (1986). Define \eta_v=(\gamma_v, h_{1v}, ..., h_{qv})^\top, which is a vector of length q+1 and Z_{ijv}=(X_{ijv}, E_{ij1}X_{ijv}, ..., E_{ijq}X_{ijv})^\top, which contains the main genetic effect of the vth SNP from the jth measurement on the ith subject and its interactions with all the q environmental factors. The model can be written as:

Y_{ij}= \alpha_0 + \sum_{m=1}^{t}\theta_m Clin_{ijm} + \sum_{u=1}^{q}\alpha_u E_{iju} + \sum_{v=1}^{p}\eta_v^\top Z_{ijv}+\epsilon_{ij},

where Z_{ijv} is the vth genetic factor and its interactions with the q environment factors for the jth measurement on the ith subject, and \eta_{v} is the corresponding coefficient vector of length 1+q. The random error \epsilon_{i}=(\epsilon_{i1},...,\epsilon_{ik_i})^{T}, which is assumed to follow a multivariate normal distribution with \Sigma_i as the covariance matrix for the repeated measurements of the ith subject among the k_i time points.

See Also

springer

The first order derivative function of MCP (Minimax Concave Penalty)

Description

The first order derivative function of MCP (Minimax Concave Penalty)

Usage

dmcp(theta, lambda, gamma = 3)

Arguments

Details

The regularization parameter \gamma for MCP should be obtained via a data-driven approach in a rigorous way. Among the published studies, it is suggested to check several choices, such as 1.4, 3, 4.2, 5.8, 6.9, and 10, then fix the value. We examined this sequence in our study and found that the results are not sensitive to the choice of value for \gamma. Therefore, we set the value to 3. To be prudent, other values should also be examined in practice.

Value

the first order derivative of the MCP function.

References

Ren, J., Du, Y., Li, S., Ma, S., Jiang, Y. and Wu, C. (2019). Robust network-based regularization and variable selection for high-dimensional genomic data in cancer prognosis. Genetic epidemiology, 43(3), 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. Journal of multivariate analysis, 168, 119-130 doi:10.1016/j.jmva.2018.06.009

Ren, J., He, T., Li, Y., Liu, S., Du, Y., Jiang, Y. and Wu, C. (2017). Network-based regularization for high dimensional SNP data in the case-control study of Type 2 diabetes. BMC genetics, 18(1), 44 doi:10.1186/s12863-017-0495-5

Examples

theta=runif(30,-4,4)
lambda=1
dmcp(theta,lambda,gamma=3)

This function provides the penalty functions. Users can choose one of the three penalties: sparse group MCP, group MCP and MCP.

Description

This function provides the penalty functions. Users can choose one of the three penalties: sparse group MCP, group MCP and MCP.

Usage

penalty(x, n, t, p, q, beta, lam1, structure, p1, lam2)

Arguments

Details

When structure="bilevel", sparse group MCP is adopted and variable selection for longitudinal data including both genetic main effects and gene-environment interactions will be conducted on both individual and group levels (bi-level selection):

• Group-level selection: If the vth genetic factor has any effect at all (associated with the response or not) can be determined by whether ||\eta_{v}||_{2}=0.

• Individual-level selection: whether the vth genetic variant has main effect, G\timesE interaction or both can be determined by the nonzero componet.

If structure="group", group MCP will be used and only group-level selection will be conducted on ||\eta_{v}||_{2}; if structure="individual", MCP will be adopted and only individual-level selection will be conducted on each \eta_{vu}, (u=1,\ldots,q).

The minimax concave penalty (MCP) is adopted as the baseline penalty function in the springer package. Methods based on other popular choices, such as SCAD and LASSO, will be examined in the future.

Value

print a springer result

Description

Print a springer result

Usage

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

Arguments

See Also

springer

This function changes the format of the longitudinal data from wide format to long format

Description

This function changes the format of the longitudinal data from wide format to long format

Usage

reformat(k, y, x)

Arguments

fit the model with given tuning parameters

Description

This function performs penalized variable selection for longitudinal data based on generalized estimating equation (GEE) or quadratic inference functions (QIF) with a given value of lambda. Typical usage is to first obtain the optimal lambda using cross validation, then provide it to the springer function.

Usage

springer(
  clin = NULL,
  e,
  g,
  y,
  beta0,
  func,
  corr,
  structure,
  lam1,
  lam2,
  maxits = 30,
  tol = 0.001
)

Arguments

Details

Look back to the data model described in "dat":

Y_{ij}= \alpha_0 + \sum_{m=1}^{t}\theta_m Clin_{ijm} + \sum_{u=1}^{q}\alpha_u E_{iju} + \sum_{v=1}^{p}\eta_v^\top Z_{ijv}+\epsilon_{ij},

where Z_{ijv} contains the vth genetic main factor and its interactions with the q environment factors for the jth measurement on the ith subject and \eta_{v} is the corresponding coefficient vector of length 1+q.

When structure="bilevel", variable selection for genetic main effects and gene-environment interactions under the longitudinal response will be conducted on both individual and group levels (bi-level selection):

• Group-level selection: by determining whether ||\eta_{v}||_{2}=0, we can know if the vth genetic variant has any effect at all.

• Individual-level selection: investigate whether the vth genetic variant has main effect, G\timesE interaction or both, by determining which components in \eta_{v} has non-zero values.

If structure="group", only group-level selection will be conducted on ||\eta_{v}||_{2}; if structure="individual", only individual-level selection will be conducted on each \eta_{vu}, (u=1,\ldots,q).

This function also provides choices for the framework that is used. If func="QIF", variable selection will be conducted within the quadratic inference functions framework; if func="GEE", variable selection will be conducted within the generalized estimating equation framework.

There are three options for the choice of the working correlation. If corr="exchangeable", the exchangeable working correlation will be applied; if corr="AR-1", the AR-1 working correlation will be adopted; if corr="independence", the independence working correlation will be used. Please check the references for more details.

Value

Examples

data("dat")
##load the clinical covariates, environment factors, genetic factors and response from the
##"dat" file
clin=dat$clin
if(is.null(clin)){t=0} else{t=dim(clin)[2]}
e=dat$e
u=dim(e)[2]
g=dat$g
y=dat$y
##initial coefficient
beta0=dat$coef
##true nonzero coefficients
index=dat$index
beta = springer(clin=clin, e, g, y,beta0,func="GEE",corr="independence",structure="bilevel",
lam1=dat$lam1, lam2=dat$lam2,maxits=30,tol=0.01)
##only focus on the genetic main effects and gene-environment interactions
beta[1:(1+t+u)]=0
##effects that have nonzero coefficients
pos = which(beta != 0)
##true positive and false positive
tp = length(intersect(index, pos))
fp = length(pos) - tp
list(tp=tp, fp=fp)