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#
# Copyright 2007-2019 by the individuals mentioned in the source code history
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#This script (by Rob K.) demonstrates the use of the GREML feature in a simple but realistic example.
#It first simulates a genomic-relatedness matrix (GRM), a phenotype, and a null covariate. Then, it
#fits a simple GREML model to estimate additive-genetic variance, residual variance, and heritability.
require(OpenMx)
options(mxCondenseMatrixSlots=TRUE) #<--Saves memory
require(mvtnorm)
#Generate data:
set.seed(476)
A <- matrix(0,1000,1000) #<--Empty GRM
A[lower.tri(A)] <- runif(499500, -0.025, 0.025)
A <- A + t(A)
diag(A) <- runif(1000,0.95,1.05) #<--GRM now complete
y <- t(rmvnorm(1,sigma=A*0.5)) #<--Phenotype 'y' has a "population" variance of 1 and h2 of 0.5
y <- y + rnorm(1000,sd=sqrt(0.5))
x <- rnorm(1000) #<--Covariate 'x' is actually independent of the phenotype.
#Merge variables into data matrix:
dat <- cbind(y,x)
colnames(dat) <- c("y","x") #<--Column names
#The GREML expectation tells OpenMx that the model-expected covariance matrix is named 'V', that the one
#phenotype is has column label 'y' in the dataset, that the one covariate has column label 'x' in the dataset,
#and that a lead column of ones needs to be appended to the 'X' matrix (for the intercept):
ge <- mxExpectationGREML(V="V",yvars="y", Xvars="x", addOnes=T)
#The GREML fitfunction tells OpenMx that the derivative of 'V' with respect to free parameter
#'va'(the additive-genetic variance) is a matrix named 'A', and that the derivative of 'V' w/r/t free parameter
#'#'ve' is a matrix named 'I'. At runtime, the GREML fitfunction will use these derivatives to help with
#'#optimization and compute standard errors:
gff <- mxFitFunctionGREML(dV=c(va="A",ve="I"))
#This is a custom compute plan. It is necessary because we want to use the Newton-Raphson optimizer, which
#can use analytic first and second derivatives of the GREML fitfunction to speed up convergence. It looks
#especially messy here because we want a profile-likelihood confidence interval for the heritability:
plan <- mxComputeSequence(steps=list(
mxComputeNewtonRaphson(fitfunction="fitfunction"),
mxComputeOnce('fitfunction', c('fit','gradient','hessian','ihessian')),
mxComputeConfidenceInterval(
plan=mxComputeGradientDescent(
fitfunction="GREML_1GRM_1trait.fitfunction", nudgeZeroStarts=FALSE, maxMajorIter=150),
fitfunction="GREML_1GRM_1trait.fitfunction"),
mxComputeStandardError(),
mxComputeReportDeriv(),
mxComputeReportExpectation()
))
#The MxData object. N.B. use of 'sort=FALSE' is CRITICALLY IMPORTANT, because the rows and columns of dataset
#'dat' and the rows and columns of GRM 'A' are already properly ordered:
mxdat <- mxData(observed = dat, type="raw", sort=FALSE)
#We will create some of the necessary objects inside the mxModel() statement. We mainly want to avoid creating
#more copies of the GRM than we need to:
testmod <- mxModel(
"GREML_1GRM_1trait", #<--Model name
#1x1 matrix containing residual variance component:
mxMatrix(type = "Full", nrow = 1, ncol=1, free=T, values = var(y)/2, labels = "ve", lbound = 0.0001,
name = "Ve"),
#1x1 matrix containing additive-genetic variance component:
mxMatrix(type = "Full", nrow = 1, ncol=1, free=T, values = var(y)/2, labels = "va", name = "Va"),
#1000x1000 identity matrix--the "relatedness matrix" for the residuals:
mxMatrix("Iden",nrow=1000,name="I"),
#The GRM:
mxMatrix("Symm",nrow=1000,free=F,values=A,name="A"),
#The model-expected covariance matrix:
mxAlgebra((A%x%Va) + (I%x%Ve), name="V"),
#An MxAlgebra for the heritability:
mxAlgebra(Va/(Va+Ve), name="h2"),
mxCI("h2"), #<--Request confidence interval for heritability
mxdat, #<--MxData object
ge, #<--GREML expectation
gff, #<--GREML fitfunction
plan #<--Custom compute plan
)
testrun <- mxRun(testmod,intervals = T) #<--Run model (Status Red is OK in this case)
summary(testrun) #<--Model summary
#Obtain SE of h2 from delta-method approximation (e.g., Lynch & Walsh, 1998, Appendix 1):
scm <- chol2inv(chol(testrun$output$hessian/2)) #<--Sampling covariance matrix for ve and va
pointest <- testrun$output$estimate #<--Point estimates of ve and va
h2se <- sqrt(
(pointest[2]/(pointest[1]+pointest[2]))^2 * (
(scm[2,2]/pointest[2]^2) - (2*scm[1,2]/pointest[1]/(pointest[1]+pointest[2])) +
(sum(scm)*(pointest[1]+pointest[2])^-2)
))
#Compare:
mxEval(h2,testrun,T)[1,1] + 2*c(-h2se,h2se)
testrun$output$confidenceIntervals
#Test for regressions in how GREML handles its analytic derivatives (for OpenMx developer use):
omxCheckTrue(testrun$output$hessian[1,2]>0)
omxCheckCloseEnough(testrun$output$gradient,c(0,0),epsilon=0.15)
#Diagonalize the problem: ###
eigenA <- eigen(A) #<--Eigen decomposition of the GRM
#We "rotate out" the dependence among participants by premultiplying the 'y' and 'X' matrices by the
#eigenvectors of the GRM:
yrot <- t(eigenA$vectors) %*% y
xrot <- t(eigenA$vectors) %*% cbind(1,x)
datrot <- cbind(yrot,xrot)
colnames(datrot) <- c("y","x0","x1")
#Make a new MxModel:
testmod2 <- mxModel(
"GREMLtest_1GRM_1trait_diagonalized",
mxData(observed = datrot, type="raw", sort=FALSE),
mxExpectationGREML(V="V",yvars="y", Xvars=c("x0","x1"), addOnes=F),
mxMatrix(type = "Full", nrow = 1, ncol=1, free=T, values = var(y)/2, labels = "ve", lbound = 0.0001,
name = "Ve"),
mxMatrix(type = "Full", nrow = 1, ncol=1, free=T, values = var(y)/2, labels = "va", name = "Va"),
mxMatrix("Iden",nrow=1000,name="I"),
mxMatrix("Diag",nrow=1000,free=F,values=eigenA$values,name="A"),
mxAlgebra((A%x%Va) + (I%x%Ve), name="V"),
mxAlgebra(Va/(Va+Ve), name="h2"),
gff,
#We'll do without the CI this time:
mxComputeSequence(steps=list(
mxComputeNewtonRaphson(fitfunction="fitfunction"),
mxComputeOnce('fitfunction', c('fit','gradient','hessian','ihessian')),
mxComputeStandardError(),
mxComputeReportDeriv(),
mxComputeReportExpectation()
))
)
testrun2 <- mxRun(testmod2)
#Results are substantially equivalent to those from the previous MxModel:
summary(testrun2)
mxEval(h2,testrun2,T)
#OpenMx developer tests:
omxCheckCloseEnough(testrun2$output$gradient,c(0,0),1e-3)
omxCheckCloseEnough(testrun$output$hessian,testrun2$output$hessian,epsilon=1)
omxCheckCloseEnough(testrun$output$estimate,testrun2$output$estimate,0.001)
omxCheckCloseEnough(testrun$output$standardErrors,testrun2$output$standardErrors,0.0001)
omxCheckCloseEnough(testrun$expectation$b,testrun2$expectation$b,1e-5)
omxCheckCloseEnough(testrun$expectation$bcov,testrun2$expectation$bcov,1e-6)
# Reparametrize the problem in terms of total variance and heritability: ###
gff3 <- mxFitFunctionGREML(dV=c(h2="dVdH2",vp="dVdVp")) #<--Need new fitfunction object
#Need new compute plan:
plan3 <- mxComputeSequence(steps=list(
mxComputeNewtonRaphson(fitfunction="fitfunction"),
mxComputeOnce('fitfunction', c('fit','gradient','hessian','ihessian')),
mxComputeConfidenceInterval(
plan=mxComputeGradientDescent(
fitfunction="GREML_1GRM_1trait_altparam.fitfunction", nudgeZeroStarts=FALSE, maxMajorIter=150),
fitfunction="GREML_1GRM_1trait_altparam.fitfunction"),
mxComputeStandardError(),
mxComputeReportDeriv(),
mxComputeReportExpectation()
))
testmod3 <- mxModel(
"GREML_1GRM_1trait_altparam", #<--Model name
#1x1 matrix containing heritability:
mxMatrix(type = "Full", nrow = 1, ncol=1, free=T, values = 0.5, labels = "h2", lbound = 0.0001, ubound=0.9999,
name = "H2"),
#1x1 matrix containing total phenotypic variance:
mxMatrix(type = "Full", nrow = 1, ncol=1, free=T, values = var(y), labels = "vp", name = "Vp"),
#1000x1000 identity matrix--the "relatedness matrix" for the residuals:
mxMatrix("Iden",nrow=1000,name="I"),
#The GRM:
mxMatrix("Symm",nrow=1000,free=F,values=A,name="A"),
#MxAlgebra for additive-genetic variance:
mxAlgebra(H2*Vp, name="Va"),
#MxAlgebra for residual variance:
mxAlgebra((1-H2)*Vp, name="Ve"),
#The model-expected covariance matrix:
mxAlgebra( (A%x%Va) + (I%x%Ve), name="V"),
#MxAlgebras for derivatives of V w/r/t free parameters:
mxAlgebra((A-I)%x%Vp, name="dVdH2"),
mxAlgebra(I + (A-I)%x%H2, name="dVdVp"),
mxCI("h2"), #<--Request confidence interval for heritability
mxdat, #<--MxData object
ge, #<--GREML expectation
gff3, #<--GREML fitfunction
plan3 #<--Custom compute plan
)
testrun3 <- mxRun(testmod3, intervals = T)
summary(testrun3)
#Compare:
mxEval(h2,testrun3,T)[1,1] + 2*c(-0.07824315,0.07824315) #<--0.07824315 is the SE of h2
testrun3$output$confidenceIntervals
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