The goal of TScML is to perform the Two-stage Constrained Maximum Likelihood (2ScML) method. In the following example, we apply 2ScML in TWAS simulation to illustrate how to use our software.
You can install the development version of TScML from GitHub with:
# install.packages("devtools")
devtools::install_github("xue-hr/TScML")We need three R packages:
- MASS: This package is available on CRAN. We use
MASS::mvrnorm()function to generate random variables. - devtools: This package is available on CRAN. We use
devtools::install_githubto install package from GitHub. - lasso2: This package is not available on CRAN for R 4.2.2, it is
availale on GitHub at
https://github.com/cran/lasso2. We uselasso2::l1ceto solve constrained lasso problem.
if(!require("MASS"))
{
install.packages("MASS")
library(MASS)
}
#> Loading required package: MASS
if(!require("devtools"))
{
install.packages("devtools")
library(devtools)
}
#> Loading required package: devtools
#> Loading required package: usethis
if(!require("lasso2"))
{
devtools::install_github("cran/lasso2")
library(lasso2)
}
#> Loading required package: lasso2
#> R Package to solve regression problems while imposing
#> an L1 constraint on the parameters. Based on S-plus Release 2.1
#> Copyright (C) 1998, 1999
#> Justin Lokhorst <jlokhors@stats.adelaide.edu.au>
#> Berwin A. Turlach <bturlach@stats.adelaide.edu.au>
#> Bill Venables <wvenable@stats.adelaide.edu.au>
#>
#> Copyright (C) 2002
#> Martin Maechler <maechler@stat.math.ethz.ch>
library(TScML)Parameters in our simulation are set as below:
load("MAFB_SNP_May16.Rdata") # data used in simulation
p = 56 # number of SNPs
n1 = 500 # first sample size
n2 = 50000 # second sample size
n.ref = 10000 # reference panel size
random.error.size = 2 # random error size
beta.true = 0 # true causal effect
gamma.vec = c(0,rep(1,7),rep(0,p-8))*1 # effects from SNPs to exposure
alpha.vec = c(1,rep(0,5),rep(1,3),rep(0,p-9))*1 # direct effects from SNPs to outcome
sigma.12 = 0.5 # correlation between error terms
Sigma.Err =
matrix(c(1,sigma.12,sigma.12,1),
nrow = 2)*random.error.size # covariance matrix of errorsWe first generate individual level
set.seed(123)
### generate all Z
Z.Stage1 = SNP_BED[sample(1:408339,n1,replace = T),]
Z.Stage2 = SNP_BED[sample(1:408339,n2,replace = T),]Generate the first sample:
### generate stage 1 sample
Z = Z.Stage1
Random_Error = mvrnorm(n1, mu = c(0,0), Sigma = Sigma.Err)
D = Z%*%gamma.vec + Random_Error[,1]
Y = beta.true*D + Z%*%alpha.vec + Random_Error[,2]
Z1 = scale(Z,scale = F)
D1 = scale(D,scale = F)
Y1 = scale(Y,scale = F)Generate the second sample:
### generate stage 2 sample
Z = Z.Stage2
Random_Error = mvrnorm(n2, mu = c(0,0), Sigma = Sigma.Err)
D = Z%*%gamma.vec + Random_Error[,1]
Y = beta.true*D + Z%*%alpha.vec + Random_Error[,2]
Z2 = scale(Z,scale = F)
D2 = scale(D,scale = F)
Y2 = scale(Y,scale = F)Generate reference panel:
### generate reference panel
Z.ref = sample(1:408339,n.ref,replace = T)
Z.ref.original = SNP_BED[Z.ref,]
cor.Z.ref.original = cor(Z.ref.original) + diag(0.00001,p)Calculate summary data:
### Input
cor.D1Z1.original = as.numeric(cor(D1,Z1))
cor.Y2Z2.original = as.numeric(cor(Y2,Z2))We apply 2ScML and Oracle methods to simulated data. We fit the model in first stage:
### Stage1 with individual-level data
Stage1FittedModel =
TScMLStage1(cor.D1Z1 = cor.D1Z1.original,
Cap.Sigma.stage1 = cor(Z1),
n1 = n1,
p = p,
ind.stage1 = 2:8)Apply the Oracle model in second stage:
### Oracle Stage 2 with summary data and reference panel
Est.Sigma1Square =
as.numeric(1 - cor.D1Z1.original%*%solve(cor.Z.ref.original,tol=0)%*%cor.D1Z1.original)
Est.Sigma2Square =
as.numeric(1 - cor.Y2Z2.original%*%solve(cor.Z.ref.original,tol=0)%*%cor.Y2Z2.original)
if(Est.Sigma1Square<=0)
{
Est.Sigma1Square = 1
}
if(Est.Sigma2Square<=0)
{
Est.Sigma2Square = 1
}
OracleStage2.ref =
OracleStage2(gamma.hat.stage1 = Stage1FittedModel,
cor.Y2Z2 = cor.Y2Z2.original,
Estimated.Sigma = cor.Z.ref.original,
n1 = n1,
n2 = n2,
p = p,
ind.stage2 = c(1,2,8,9,10),
Est.Sigma1Square = Est.Sigma1Square,
Est.Sigma2Square = Est.Sigma2Square)
Oracle.Summary.Var =
TScMLVar(Z.ref.original = Z.ref.original,
Stage1FittedModel = Stage1FittedModel,
betaalpha.hat.stage2 = OracleStage2.ref$betaalpha.hat.stage2,
Est.Sigma1Square = Est.Sigma1Square,
Est.Sigma2Square = Est.Sigma2Square,
n1 = n1,n2 = n2,n.ref = n.ref)Apply 2ScML in the second stage:
### TScML Stage2 with summary data and reference panel
Est.Sigma1Square =
as.numeric(1 - cor.D1Z1.original%*%solve(cor.Z.ref.original,tol=0)%*%cor.D1Z1.original)
Est.Sigma2Square =
as.numeric(1 - cor.Y2Z2.original%*%solve(cor.Z.ref.original,tol=0)%*%cor.Y2Z2.original)
if(Est.Sigma1Square<=0)
{
Est.Sigma1Square = 1
}
if(Est.Sigma2Square<=0)
{
Est.Sigma2Square = 1
}
start.time = Sys.time()
TScMLStage2.Ref =
TScMLStage2(gamma.hat.stage1 = Stage1FittedModel,
cor.Y2Z2 = cor.Y2Z2.original,
Estimated.Sigma = cor.Z.ref.original,
n1 = n1,
n2 = n2,
p = p,
K.vec.stage2 = 0:10,
Est.Sigma1Square = Est.Sigma1Square,
Est.Sigma2Square = Est.Sigma2Square)
end.time = Sys.time()
TScML.Summary.Var =
TScMLVar(Z.ref.original = Z.ref.original,
Stage1FittedModel = Stage1FittedModel,
betaalpha.hat.stage2 = TScMLStage2.Ref$betaalpha.hat.stage2,
Est.Sigma1Square = Est.Sigma1Square,
Est.Sigma2Square = Est.Sigma2Square,
n1 = n1,n2 = n2,n.ref = n.ref)We record the run time of our main function TScMLStage2. Different
from methods that use individual-level data, sample size does not
influence run time of our main function TScMLStage2 as we only use
summary data. The run time of TScMLStage2 mainly depends on two
values, the first one is the number of instruments, i.e. $p$; the second
one is the set of candidate
run.time = end.time - start.time
run.time
#> Time difference of 0.2383361 secsNow we can show the results from Oracle and 2ScML.
# Oracle Estimate
OracleStage2.ref$betaalpha.hat.stage2[1]
#> [1] -0.01266214
# Uncorrected Variance of Oracle Estimate
OracleStage2.ref$Asympt.Var.BetaHat
#> [1] 0.0001017406
# Corrected Variance of Oracle Estimate
Oracle.Summary.Var
#> [1] 0.000291072
# 2ScML Estimate
TScMLStage2.Ref$betaalpha.hat.stage2[1]
#> [1] -0.01266214
# Uncorrected Variance of 2ScML Estimate
TScMLStage2.Ref$Asympt.Var.BetaHat
#> [1] 0.0001017406
# Corrected Variance of 2ScML Estimate
TScML.Summary.Var
#> [1] 0.000291072We can see, in this simulation, the proposed 2ScML method gives same result as oracle method.
Here is the R session information. We performed the example in the latest R release 4.2.2.
sessionInfo()
#> R version 4.2.2 (2022-10-31)
#> Platform: aarch64-apple-darwin20 (64-bit)
#> Running under: macOS Monterey 12.5
#>
#> Matrix products: default
#> BLAS: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRblas.0.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRlapack.dylib
#>
#> locale:
#> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] TScML_0.0.0.9000 lasso2_1.2-22 devtools_2.4.5 usethis_2.1.6
#> [5] MASS_7.3-58.2
#>
#> loaded via a namespace (and not attached):
#> [1] Rcpp_1.0.9 urlchecker_1.0.1 compiler_4.2.2 later_1.3.0
#> [5] remotes_2.4.2 prettyunits_1.1.1 profvis_0.3.7 tools_4.2.2
#> [9] digest_0.6.30 pkgbuild_1.4.0 pkgload_1.3.2 evaluate_0.18
#> [13] memoise_2.0.1 lifecycle_1.0.3 rlang_1.0.6 shiny_1.7.4
#> [17] cli_3.4.1 rstudioapi_0.14 yaml_2.3.6 xfun_0.35
#> [21] fastmap_1.1.0 stringr_1.4.1 knitr_1.41 htmlwidgets_1.5.4
#> [25] fs_1.5.2 glue_1.6.2 R6_2.5.1 processx_3.8.0
#> [29] rmarkdown_2.18 sessioninfo_1.2.2 purrr_0.3.5 callr_3.7.3
#> [33] magrittr_2.0.3 promises_1.2.0.1 ps_1.7.2 ellipsis_0.3.2
#> [37] htmltools_0.5.4 mime_0.12 xtable_1.8-4 httpuv_1.6.6
#> [41] stringi_1.7.8 miniUI_0.1.1.1 cachem_1.0.6 crayon_1.5.2Haoran Xue, Xiaotong Shen & Wei Pan (2023) Causal Inference in Transcriptome-Wide Association Studies with Invalid Instruments and GWAS Summary Data, Journal of the American Statistical Association, DOI: 10.1080/01621459.2023.2183127
Feel free to contact the author at xuexx268@umn.edu for any comments!