Additive Gene by Environment (GxE) Interaction Tests Under the Trend Effect of genotypes
_{Nilotpal Sanyal, Matthieu Rochemonteix, Summer Han}

• Introduction
• Installation of CGEN
• Import Data
• Additive GxE tests under the trend of effect of genotypes
  • 1. Without G-E independence assumption (Prospective likelihood) (indep=FALSE)
  • 2. Under G-E independence assumption (Retrospective likelihood) (indep=TRUE)
• References

Introduction

This page serves as a tutorial for the additive gene by environment (GxE) interaction tests under the trend effect of genotypes that have been proposed by Rochemonteix et al. (2020) and Sanyal et al. (2021) and implemented in the additive.test function of the R package CGEN. Currently, these tests are available only for binary environmental variables.

Specifically, under the trend effect of genotypes, we illustrate the following tests:

• Likelihood ratio tests (LRTs) without/with gene-environment independence assumption (Rochemonteix et al. (2020)).

• Wald tests without/with (UML/CML) gene-environment independence assumption, and a robust Wald test (EB) based on an empirical Bayes-type shrinkage estimator that combines estimates from the former Wald tests (UML and CML) (Sanyal et al. (2021)).

The tests that use the gene-environment independence assumption are based on the ‘retrospective likelihood’ function and constrained maximum likelihood (CML) estimates where the constraint is imposed by the independence assumption. When that assumption is not made, the use of the retrospective likelihood function along with the corresponding unconstrained maximum likelihood estimates (UML) is equivalent to the use of traditionally used ‘propsective likelihood’ function and the estimates obtained therefrom.

We start by loading the package.

Installation of CGEN

The R package CGEN can be installed from its Bioconductor repository using:

if (!require("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("CGEN")

library(CGEN)

Import Data

For the illustration of the tests, we use the Xdata2 dataset which comes inside the CGEN package. This dataset contains sample covariate and outcome data that have been taken from a lung cancer study. Let us load the data and look at its contents.

data(Xdata2, package="CGEN") 

str(Xdata2)
#'data.frame':    11449 obs. of  8 variables:
# $ case.control: int  0 0 0 0 0 0 0 1 0 0 ...
# $ SNP         : int  0 0 0 1 1 0 2 1 1 0 ...
# $ smoking     : int  1 1 1 1 1 0 0 0 1 1 ...
# $ cov1        : num  25 25 15 35 5 25 5 15 15 5 ...
# $ cov2        : int  1 1 1 1 0 0 0 0 1 1 ...
# $ cov3        : int  4 8 3 4 4 4 8 0 8 4 ...
# $ cov4        : num  3.22 3.22 2.71 3.56 1.61 ...
# $ study       : int  4 4 3 5 2 4 2 3 3 2 ...

So, the data contain, for 11449 subjects, case-control type 0-1 outcomes, genotype values of a SNP (0,1,2), values of a binary (0-1) environmental variable smoking, values of four covariates and a study variable indicating which study subjects are taken from. We test for additive SNP x smoking interaction in this data.

Additive GxE tests under the trend of effect of genotypes

We use the additive.test function to conduct additive GxE interaction tests under the assumption of an additive genetic model, i.e., the presence of the trend effect of genotypes. The additive genetic model is specified by the option genetic.model = 0. Further, the gene-environment independence assumption is specified by setting indep=TRUE. By default, indep=FALSE which corresponds to the prospective likelihood-based analysis—which we illustrate first.

1. Without G-E independence assumption (Prospective likelihood) (indep=FALSE)

The following code performs tests for additive SNP x smoking interaction under the trend effect of genotypes without any independence assumption. In the output, we get

• LRT p value and test statistic under prospective likelihood.
• Wald test p value and RERI estimates under prospective likelihood.

test_noindep <- additive.test(data = Xdata2, response.var = "case.control",
                snp.var = "SNP", exposure.var = "smoking", main.vars = c("cov1", "cov2",
                "cov3", "cov4", "study"), strata.var = "study", op = list(genetic.model = 0))

names(test_noindep)
#[1] "additive"   "model.info"

names(test_noindep$additive)
#[1] "pval.add.LRT" "pval.add.UML" "LRT.add"      "RERI.UML"    

# => p value of the LRT
test_noindep$additive$pval.add.LRT
#[1] 0.03197455

# => test statistic of the LRT
test_noindep$additive$LRT.add
#[1] 4.599861

# => p value of the Wald (UML) test
test_noindep$additive$pval.add.UML
#[1] 0.03924117

# => RERI, standard error and gradient estimates (UML)
test_noindep$additive$RERI.UML
#$value
#[1] -0.1530518
#
#$variance
#[1] 0.005511217
#
#$gradient
#     x11.UML     x2_1.UML x11:x2_1.UML 
#   0.5597640   -0.2059929    1.5068229

Additionally, the estimates of the additive risk model parameters, their standard errors and covariance matrix, estimates of the odds ratio table, the fitted maximum likelihood value and some other information related to the model specification and fit are included in the model.info component of the output as pointed below.

names(test_noindep$model.info)
# [1] "response.var"     "snp.var"          "exposure.var"     "main.vars"        "strata.var"      
# [6] "op"               "GxEtable"         "method"           "parms.lm.UML"     "cov.lm.UML"      
#[11] "full.loglike.UML" "DF"               "OR.table.UML"    

# => Estimates of the additive risk model parameters
test_noindep$model.info$parms.lm.UML
#$parms.lm.UML
#             Estimate  Std. Error    z value     Pr(>|z|)
#Intercept -1.63045022 0.118039297 -13.812775 2.134487e-43
#cov1       0.02247682 0.003151272   7.132618 9.847750e-13
#cov2       0.12581973 0.039654108   3.172931 1.509086e-03
#cov3      -0.05452759 0.008103140  -6.729193 1.706071e-11
#cov4       0.21096033 0.050196679   4.202675 2.637791e-05
#study      0.20136331 0.027637750   7.285807 3.197506e-13
#x2_1       0.53813867 0.060343158   8.917973 4.748992e-19
#x11       -0.05439399 0.037462923  -1.451942 1.465178e-01
#x11:x2_1  -0.07374131 0.051971696  -1.418874 1.559356e-01
#x12       -0.11191773 0.079362236  -1.410214 1.584765e-01
#x12:x2_1  -0.16321845 0.112151620  -1.455337 1.455759e-01

# => Estimates of the covariance matrix of the additive risk model parameters
test_noindep$model.info$cov.lm.UML
#              Intercept          cov1          cov2          cov3          cov4         study
#Intercept  0.0152983986  2.378421e-04 -9.331093e-04 -1.761301e-04 -5.349101e-03 -6.958840e-04
#cov1       0.0002378421  9.557136e-06  1.989700e-06 -1.005618e-07 -1.142051e-04 -3.248776e-05
#cov2      -0.0009331093  1.989700e-06  1.572155e-03  6.407017e-06  1.153846e-05 -6.320078e-06
#cov3      -0.0001761301 -1.005618e-07  6.407017e-06  6.467444e-05 -6.432421e-06 -6.995111e-06
#cov4      -0.0053491011 -1.142051e-04  1.153846e-05 -6.432421e-06  2.929254e-03 -2.564966e-04
#study     -0.0006958840 -3.248776e-05 -6.320078e-06 -6.995111e-06 -2.564966e-04  7.311637e-04
#x2_1      -0.0013610431  2.190878e-06  1.196209e-04 -6.171406e-05  5.203345e-05 -1.785638e-04
#x11       -0.0012307572 -6.155937e-07  2.037488e-05  3.100335e-06 -1.440166e-06  6.894449e-06
#x11:x2_1   0.0013213374  6.863978e-07 -4.557594e-05 -2.898525e-06 -2.603386e-05 -8.483745e-06
#                   x2_1           x11      x11:x2_1
#Intercept -1.361043e-03 -1.230757e-03  1.321337e-03
#cov1       2.190878e-06 -6.155937e-07  6.863978e-07
#cov2       1.196209e-04  2.037488e-05 -4.557594e-05
#cov3      -6.171406e-05  3.100335e-06 -2.898525e-06
#cov4       5.203345e-05 -1.440166e-06 -2.603386e-05
#study     -1.785638e-04  6.894449e-06 -8.483745e-06
#x2_1       3.668713e-03  1.202591e-03 -2.307680e-03
#x11        1.202591e-03  1.464546e-03 -1.464767e-03
#x11:x2_1  -2.307680e-03 -1.464767e-03  2.720610e-03

# => Fitted maximum likelihood value
test_noindep$model.info$full.loglike.UML
#[1] -7305.029

# => Estimates of the odds ratio table
test_noindep$model.info$OR.table.UML
#          0        1
#0 1.0000000 1.712816
#1 0.9470589 1.506823
#2 0.8941178 1.300830

2. Under G-E independence assumption (Retrospective likelihood) (indep=TRUE)

Under the assumption that SNP and smoking are independent (specified by indep=TRUE), the following code performs tests for additive SNP x smoking interaction under the trend effect of genotypes. From the output, we get

• LRT p value and test statistic under retrospective likelihood.
• Wald test p values and RERI estimates under retrospective likelihood.

test_indep <- additive.test(data = Xdata2, response.var = "case.control",
              snp.var = "SNP", exposure.var = "smoking", main.vars = c("cov1", "cov2",
              "cov3", "cov4", "study"), strata.var = "study", op = list(genetic.model = 0,
              indep = TRUE))

names(test_indep)
#[1] "additive"   "model.info"

names(test_indep$additive)
#[1] "pval.add.LRT" "pval.add.UML" "pval.add.CML" "pval.add.EB"  "LRT.add"      "RERI.UML"    
#[7] "RERI.CML"     "RERI.EB"     

# => p value of the LRT
test_indep$additive$pval.add.LRT
#[1] 0.4595864

# => test statistic of the LRT
test_indep$additive$LRT.add
#[1] 0.5469018

# => p value of the Wald UML test
test_indep$additive$pval.add.UML
#[1] 0.03924117

# => p value of the Wald CML test
test_indep$additive$pval.add.CML
#[1] 0.4661798

# => p value of the Wald EB test
test_indep$additive$pval.add.EB
#[1] 0.131338

# => RERI, standard error and gradient estimates for UML
test_indep$additive$RERI.UML
#$value
#[1] -0.1530518
#
#$variance
#[1] 0.005511217
#
#$gradient
#     x11.UML     x2_1.UML x11:x2_1.UML 
#   0.5597640   -0.2059929    1.5068229 
   
# => RERI, standard error and gradient estimates for CML
test_indep$additive$RERI.CML
#$value
#[1] -0.036034
#
#$variance
#[1] 0.002445213
#
#$gradient
#     x11.CML     x2_1.CML x11:x2_1.CML 
#    0.557127    -0.119424     1.473737
    
# => RERI, standard error and gradient estimates for EB
test_indep$additive$RERI.EB
#$value
#[1] -0.1194704
#
#$gradient
#     x11.UML     x2_1.UML x11:x2_1.UML      x11.CML     x2_1.CML x11:x2_1.CML 
#  0.62820373  -0.23117872   1.69105517  -0.06811736   0.01460142  -0.18018706 
#
#$variance
#[1] 0.006269466

Further, the estimates of the additive risk model parameters, their standard errors and covariance matrix under UML and CML, estimates of the odds ratio table, the fitted maximum likelihood value and some other information related to the model specification and fit are included in the model.info component of the output as pointed below.

names(test_indep$model.info)
# [1] "response.var"     "snp.var"          "exposure.var"     "main.vars"        "strata.var"      
# [6] "op"               "GxEtable"         "method"           "parms.lm.UML"     "parms.lm.CML"    
#[11] "cov.lm.UML"       "cov.lm.CML"       "full.loglike.UML" "full.loglike.CML" "DF"              
#[16] "OR.table.CML"

# => UML estimates of the additive risk model parameters
test_indep$model.info$parms.lm.UML
#             Estimate  Std. Error    z value     Pr(>|z|)
#Intercept -1.63045022 0.118039297 -13.812775 2.134487e-43
#cov1       0.02247682 0.003151272   7.132618 9.847750e-13
#cov2       0.12581973 0.039654108   3.172931 1.509086e-03
#cov3      -0.05452759 0.008103140  -6.729193 1.706071e-11
#cov4       0.21096033 0.050196679   4.202675 2.637791e-05
#study      0.20136331 0.027637750   7.285807 3.197506e-13
#x2_1       0.53813867 0.060343158   8.917973 4.748992e-19
#x11       -0.05439399 0.037462923  -1.451942 1.465178e-01
#x11:x2_1  -0.07374131 0.051971696  -1.418874 1.559356e-01
#x12       -0.11191773 0.079362236  -1.410214 1.584765e-01
#x12:x2_1  -0.16321845 0.112151620  -1.455337 1.455759e-01

# => UML estimates of the covariance matrix of the additive risk model parameters
test_indep$model.info$cov.lm.UML
#              Intercept          cov1          cov2          cov3          cov4         study
#Intercept  0.0152983986  2.378421e-04 -9.331093e-04 -1.761301e-04 -5.349101e-03 -6.958840e-04
#cov1       0.0002378421  9.557136e-06  1.989700e-06 -1.005618e-07 -1.142051e-04 -3.248776e-05
#cov2      -0.0009331093  1.989700e-06  1.572155e-03  6.407017e-06  1.153846e-05 -6.320078e-06
#cov3      -0.0001761301 -1.005618e-07  6.407017e-06  6.467444e-05 -6.432421e-06 -6.995111e-06
#cov4      -0.0053491011 -1.142051e-04  1.153846e-05 -6.432421e-06  2.929254e-03 -2.564966e-04
#study     -0.0006958840 -3.248776e-05 -6.320078e-06 -6.995111e-06 -2.564966e-04  7.311637e-04
#x2_1      -0.0013610431  2.190878e-06  1.196209e-04 -6.171406e-05  5.203345e-05 -1.785638e-04
#x11       -0.0012307572 -6.155937e-07  2.037488e-05  3.100335e-06 -1.440166e-06  6.894449e-06
#x11:x2_1   0.0013213374  6.863978e-07 -4.557594e-05 -2.898525e-06 -2.603386e-05 -8.483745e-06
#                   x2_1           x11      x11:x2_1
#Intercept -1.361043e-03 -1.230757e-03  1.321337e-03
#cov1       2.190878e-06 -6.155937e-07  6.863978e-07
#cov2       1.196209e-04  2.037488e-05 -4.557594e-05
#cov3      -6.171406e-05  3.100335e-06 -2.898525e-06
#cov4       5.203345e-05 -1.440166e-06 -2.603386e-05
#study     -1.785638e-04  6.894449e-06 -8.483745e-06
#x2_1       3.668713e-03  1.202591e-03 -2.307680e-03
#x11        1.202591e-03  1.464546e-03 -1.464767e-03
#x11:x2_1  -2.307680e-03 -1.464767e-03  2.720610e-03

# => CML estimates of the additive risk model parameters
test_indep$model.info$parms.lm.CML
#                  Estimate  Std. Error     z value     Pr(>|z|)
#Intercept     -1.600452023 0.116190346 -13.7743976 3.634018e-43
#cov1           0.022386415 0.003142912   7.1228256 1.057366e-12
#cov2           0.124318932 0.039614500   3.1382179 1.699785e-03
#cov3          -0.054459013 0.008099091  -6.7240895 1.766938e-11
#cov4           0.210126279 0.050155451   4.1895003 2.795694e-05
#study          0.202221660 0.027600193   7.3268206 2.356770e-13
#x2_1           0.465720112 0.051206243   9.0949869 9.459789e-20
#x11           -0.087073182 0.032456121  -2.6827970 7.300930e-03
#x11:x2_1       0.009154455 0.036774022   0.2489381 8.034087e-01
#Allele_freq.1 -0.295012198 0.035331048  -8.3499418 6.829661e-17
#Allele_freq.2 -0.396743240 0.033161918 -11.9638209 5.497296e-33
#Allele_freq.3 -0.333456711 0.025535852 -13.0583744 5.693046e-39
#Allele_freq.4 -0.322738880 0.035995826  -8.9660085 3.074542e-19
#Allele_freq.5 -0.278420402 0.051568950  -5.3989930 6.701599e-08
#x12           -0.182457534 0.071408762  -2.5551141 1.061530e-02
#x12:x2_1       0.020031786 0.076765420   0.2609480 7.941326e-01

# => CML estimates of the covariance matrix of the additive risk model parameters
test_indep$model.info$cov.lm.CML
#                  Intercept          cov1          cov2          cov3          cov4         study
#Intercept      0.0149183836  2.363420e-04 -9.125414e-04 -1.731255e-04 -5.334602e-03 -6.826112e-04
#cov1           0.0002363420  9.541184e-06  1.974250e-06 -8.948280e-08 -1.141744e-04 -3.236240e-05
#cov2          -0.0009125414  1.974250e-06  1.569414e-03  6.289523e-06  1.105356e-05 -6.401576e-06
#cov3          -0.0001731255 -8.948280e-08  6.289523e-06  6.457902e-05 -6.436362e-06 -7.070684e-06
#cov4          -0.0053346020 -1.141744e-04  1.105356e-05 -6.436362e-06  2.928927e-03 -2.570816e-04
#study         -0.0006826112 -3.236240e-05 -6.401576e-06 -7.070684e-06 -2.570816e-04  7.303818e-04
#x2_1          -0.0007571200  3.053459e-06  8.075863e-05 -6.400035e-05  2.419504e-05 -1.825082e-04
#x11           -0.0008418266  9.247539e-07  8.709930e-09 -3.636213e-08 -1.515403e-05 -6.750915e-06
#x11:x2_1       0.0005868225 -3.481024e-07 -6.797736e-08  1.968512e-08  4.930166e-06 -3.399926e-06
#Allele_freq.1  0.0002239115 -9.120136e-07  7.422754e-08  1.249674e-07  1.562366e-05 -7.342697e-06
#Allele_freq.2  0.0002918164  8.344021e-07  2.731239e-08  3.441251e-08 -1.378131e-05 -3.833865e-06
#Allele_freq.3  0.0002535507 -6.584542e-07 -1.184019e-07 -1.608495e-07  1.067051e-05  6.117895e-06
#Allele_freq.4  0.0002856738 -6.334723e-07  2.534818e-07  2.813931e-07  1.127990e-05  1.337965e-05
#Allele_freq.5  0.0004026789 -2.774734e-07 -4.649136e-08 -6.133013e-08  4.875884e-06  1.892906e-05
#                       x2_1           x11      x11:x2_1 Allele_freq.1 Allele_freq.2 Allele_freq.3
#Intercept     -7.571200e-04 -8.418266e-04  5.868225e-04  2.239115e-04  2.918164e-04  2.535507e-04
#cov1           3.053459e-06  9.247539e-07 -3.481024e-07 -9.120136e-07  8.344021e-07 -6.584542e-07
#cov2           8.075863e-05  8.709930e-09 -6.797736e-08  7.422754e-08  2.731239e-08 -1.184019e-07
#cov3          -6.400035e-05 -3.636213e-08  1.968512e-08  1.249674e-07  3.441251e-08 -1.608495e-07
#cov4           2.419504e-05 -1.515403e-05  4.930166e-06  1.562366e-05 -1.378131e-05  1.067051e-05
#study         -1.825082e-04 -6.750915e-06 -3.399926e-06 -7.342697e-06 -3.833865e-06  6.117895e-06
#x2_1           2.611665e-03  5.905199e-04 -1.099144e-03 -1.306548e-04  3.069330e-07  7.026481e-05
#x11            5.905199e-04  1.056882e-03 -7.109619e-04 -2.621026e-04 -2.957803e-04 -3.300160e-04
#x11:x2_1      -1.099144e-03 -7.109619e-04  1.335450e-03  1.491739e-04 -1.038123e-06 -8.665475e-05
#Allele_freq.1 -1.306548e-04 -2.621026e-04  1.491739e-04  1.240445e-03  7.988810e-05  9.174099e-05
#Allele_freq.2  3.069330e-07 -2.957803e-04 -1.038123e-06  7.988810e-05  1.114959e-03  1.644835e-04
#Allele_freq.3  7.026481e-05 -3.300160e-04 -8.665475e-05  9.174099e-05  1.644835e-04  6.825037e-04
#Allele_freq.4  9.547743e-05 -3.953258e-04 -1.222033e-04  1.102479e-04  2.011962e-04  2.633754e-04
#Allele_freq.5 -2.032411e-05 -5.361886e-04  9.522093e-06  1.438327e-04  2.319524e-04  2.940179e-04
#              Allele_freq.4 Allele_freq.5
#Intercept      2.856738e-04  4.026789e-04
#cov1          -6.334723e-07 -2.774734e-07
#cov2           2.534818e-07 -4.649136e-08
#cov3           2.813931e-07 -6.133013e-08
#cov4           1.127990e-05  4.875884e-06
#study          1.337965e-05  1.892906e-05
#x2_1           9.547743e-05 -2.032411e-05
#x11           -3.953258e-04 -5.361886e-04
#x11:x2_1      -1.222033e-04  9.522093e-06
#Allele_freq.1  1.102479e-04  1.438327e-04
#Allele_freq.2  2.011962e-04  2.319524e-04
#Allele_freq.3  2.633754e-04  2.940179e-04
#Allele_freq.4  1.251823e-03  3.599500e-04
#Allele_freq.5  3.599500e-04  2.673354e-03

# => The fitted maximum likelihood value under UML
test_indep$model.info$full.loglike.UML
#[1] -7305.029

# => The fitted maximum likelihood value under CML
test_indep$model.info$full.loglike.CML
#[1] -18923.57

# => Estimates of the odds ratio table under CML
test_indep$model.info
#$OR.table.CML
#        0        1
#0 1.00000 1.593161
#1 0.91661 1.473737
#2 0.83322 1.354313

References

De Rochemonteix, M., Napolioni, V., Sanyal, N., Belloy, M.E., Caporaso, N.E., Landi, M.T., Greicius, M.D., Chatterjee, N. and Han, S.S., 2021. A likelihood ratio test for gene-environment interaction based on the trend effect of genotype under an additive risk model using the gene-environment independence assumption. American journal of epidemiology, 190(1), pp.129-141. https://doi.org/10.1093/aje/kwaa132.

Sanyal, N., Napolioni, V., de Rochemonteix, M., Belloy, M.E., Caporaso, N.E., Landi, M.T., Greicius, M.D., Chatterjee, N. and Han, S.S., 2021. A Robust Test for Additive Gene-Environment Interaction Under the Trend Effect of Genotype Using an Empirical Bayes-Type Shrinkage Estimator. American journal of epidemiology, 190(9), pp.1948-1960. https://doi.org/10.1093/aje/kwab124.