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abcd_baseline_np_analyses_final.R
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abcd_baseline_np_analyses_final.R
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#############################################################################################################
#############################################################################################################
## Code for: ##
## Thompson, W. K., Barch, D., Bjork, J., Gonzalez, R., Nagel, B., Nixon, S. J., & Luciana, M. (2018). ##
## The Structure of Cognition in 9 and 10 year-old Children and Associations with Problem Behaviors: ##
## Findings from the ABCD Study’s Baseline Neurocognitive Battery. Developmental Cognitive Neuroscience. ##
## ##
## Wes Thompson (wes.stat@gmail.com) ##
## Dec 22, 2018 ##
#############################################################################################################
#############################################################################################################
####################
####################
## Load libraries ##
####################
####################
library(mvtnorm)
library(tableone)
library(parallel)
library(rstan)
library(loo)
library(gamm4)
library(Hmisc)
library(FactoMineR)
library(nFactors)
library(reshape2)
library(psych)
library(data.table)
library(mice)
library(abind)
library(cvTools)
library(modEvA)
####################################
####################################
## Load and manipulate nda17 data ##
####################################
####################################
## Read Rds file from DEAP (ABCD NDA version 1.1 release)
## scripts for creating the file "nda17_release1.1.Rds" can be found at: https://github.com/ABCD-STUDY/analysis-nda17
nda17 = readRDS("data/nda17_release1.1.Rds")
#########################################
#########################################
## Select & process data for analyses ##
#########################################
#########################################
## Select demographics
ind_demog = c(which(names(nda17)=="age"),which(names(nda17)=="female"),which(names(nda17)=="race.ethnicity"),
which(names(nda17)=="high.educ"),which(names(nda17)=="married"),which(names(nda17)=="household.income"))
names(nda17)[ind_demog]
summary(nda17[,ind_demog])
## Select nesting variables
ind_nest = c(which(names(nda17)=="abcd_site"),which(names(nda17)=="rel_family_id"));summary(nda17[,ind_nest])
nda17$abcd_site = as.character(nda17$abcd_site)
nda17$abcd_site[nda17$abcd_site=="site22"] = "site07"
nda17$abcd_site = factor(nda17$abcd_site)
## Select neuropsychological measures
ind_pea_ravlt = c(which(names(nda17)=="pea_ravlt_sd_trial_i_tc"),which(names(nda17)=="pea_ravlt_sd_trial_ii_tc"),
which(names(nda17)=="pea_ravlt_sd_trial_iii_tc"),which(names(nda17)=="pea_ravlt_sd_trial_iv_tc"),
which(names(nda17)=="pea_ravlt_sd_trial_v_tc")); names(nda17)[ind_pea_ravlt]; summary(nda17[,ind_pea_ravlt])
nda17$pea_ravlt_ld = apply(nda17[,ind_pea_ravlt],1,sum)
par(mfrow=c(1,2))
hist(nda17$pea_ravlt_ld)
hist(nda17$lmt_scr_perc_correct)
ind_np = c(which(names(nda17)=="nihtbx_picvocab_uncorrected"),which(names(nda17)=="nihtbx_flanker_uncorrected"),
which(names(nda17)=="nihtbx_list_uncorrected"),which(names(nda17)=="nihtbx_cardsort_uncorrected"),
which(names(nda17)=="nihtbx_pattern_uncorrected"),which(names(nda17)=="nihtbx_picture_uncorrected"),
which(names(nda17)=="nihtbx_reading_uncorrected"),which(names(nda17)=="pea_ravlt_ld"),
which(names(nda17)=="lmt_scr_perc_correct")); names(nda17)[ind_np]
## Select dependent measures
ind_dv = c(which(names(nda17)=="cbcl_scr_syn_external_r"),
which(names(nda17)=="cbcl_scr_syn_internal_r"),which(names(nda17)=="cbcl_scr_07_stress_r"))
names(nda17)[ind_dv]
######################
######################
## Rename variables ##
######################
######################
names(nda17)[which(names(nda17)=="age")] = "Age"; nda17$Age = nda17$Age/12
names(nda17)[which(names(nda17)=="female")] = "Female"
names(nda17)[which(names(nda17)=="race.ethnicity")] = "RaceEthnicity"
names(nda17)[which(names(nda17)=="high.educ")] = "HighestParentalEducation"
names(nda17)[which(names(nda17)=="married")] = "HouseholdMaritalStatus"
names(nda17)[which(names(nda17)=="household.income")] = "HouseholdIncome"
names(nda17)[which(names(nda17)=="abcd_site")] = "Site"
names(nda17)[which(names(nda17)=="rel_relationship")] = "Relationship"
names(nda17)[which(names(nda17)=="nihtbx_picvocab_uncorrected")] = "PicVocab"
names(nda17)[which(names(nda17)=="nihtbx_flanker_uncorrected")] = "Flanker"
names(nda17)[which(names(nda17)=="nihtbx_list_uncorrected")] = "List"
names(nda17)[which(names(nda17)=="nihtbx_cardsort_uncorrected")] = "CardSort"
names(nda17)[which(names(nda17)=="nihtbx_pattern_uncorrected")] = "Pattern"
names(nda17)[which(names(nda17)=="nihtbx_picture_uncorrected")] = "Picture"
names(nda17)[which(names(nda17)=="nihtbx_reading_uncorrected")] = "Reading"
names(nda17)[which(names(nda17)=="pea_ravlt_ld")] = "RAVLT"
names(nda17)[which(names(nda17)=="lmt_scr_perc_correct")] = "LMT"
names(nda17)[which(names(nda17)=="pea_wiscv_tss")] = "WISC-V"
names(nda17)[which(names(nda17)=="cbcl_scr_syn_external_r")] = "Externalizing"
names(nda17)[which(names(nda17)=="cbcl_scr_syn_internal_r")] = "Internalizing"
names(nda17)[which(names(nda17)=="cbcl_scr_07_stress_r")] = "Stress"
nda17$compl = "Incomplete"
nda17$compl[complete.cases(nda17[,c(ind_nest,ind_np)])] = "Complete"
table(nda17$compl)
## Create Table 1
vars1 <- c("Age", "Female", "RaceEthnicity", "HighestParentalEducation", "HouseholdMaritalStatus",
"HouseholdIncome","Relationship","Site")
tab1 <- CreateTableOne(vars = vars1, data = nda17, strata = "compl")
tabAsStringMatrix <- print(tab1, printToggle = FALSE, noSpaces = TRUE)
tab1 = knitr::kable(tabAsStringMatrix)
## Create Table 2
vars2 <- c("PicVocab", "Flanker", "List", "CardSort", "Pattern",
"Picture","Reading","RAVLT","WISC-V",
"LMT","Externalizing","Internalizing","Stress")
tab2 <- CreateTableOne(vars = vars2, data = nda17,strata = "compl")
tabAsStringMatrix <- print(tab2, printToggle = FALSE, noSpaces = TRUE)
tab2 = knitr::kable(tabAsStringMatrix)
## Create SM Figures 1 & 2
pdf("figures_and_tables/fig_sm1.pdf")
par(mfrow=c(3,3))
for(p in 1:9){
hist(scale(nda17[,ind_np[p]]), xlab="", freq=FALSE,main=names(nda17)[ind_np][p])
}
dev.off()
pdf("figures_and_tables/fig_sm2.pdf")
par(mfrow=c(2,2))
for(p in 1:3){
hist(scale(nda17[,ind_dv[p]]), xlab="", freq=FALSE,main=names(nda17)[ind_dv][p])
}
dev.off()
###################################
###################################
## Subset variables for analyses ##
###################################
###################################
data = nda17[,c(1,ind_nest,ind_demog,which(names(nda17)=="rel_relationship"),which(names(nda17)=="rel_group_id"),ind_np,ind_dv)]
names(data);dim(data)
data$src_subject_id = as.character(data$src_subject_id)
names(data)[names(data)=="src_subject_id"] = "pid"
data$site_num = as.numeric(substr(data$Site,5,6))
data$fam_num = 0
ind=0
for(i in sort(unique(data$rel_family_id))){
ind = ind+1
data$fam_num[data$rel_family_id==i & !is.na(data$rel_family_id)] = ind
}
data = data[order(data$site_num,data$fam_num,data$rel_group_id),]
####################
####################
## Fit usual PCA ##
####################
####################
## PCA on unimputed data
ind_Y = c(11:19); names(data)[ind_Y]
Y = as.matrix(scale(data[complete.cases(data[,c(ind_Y)]),ind_Y]))
ev = eigen(cor(Y))
ap = parallel(subject=nrow(Y),var=ncol(Y),rep=100,cent=.05)
nS = nScree(x=ev$values,aparallel=ap$eigen$qevpea)
plotnScree(nS)
ncomp = 3
#y.pca = psych::principal(Y, rotate="promax", nfactors=ncomp, scores=TRUE)
#y.pca$loadings
y.pca = psych::principal(Y, rotate="varimax", nfactors=ncomp, scores=TRUE)
y.pca$loadings
#########################################
#########################################
## Fit 9-variable PCA Model using Stan ##
#########################################
#########################################
###############
## Prep data ##
###############
ind_Y = 11:19; names(data)[ind_Y]
ind_demog = c(4:9); names(data)[ind_demog]
ind_nest = c(2,3); names(data)[ind_nest]
data1 = data[complete.cases(data[,c(ind_Y)]),]; names(data1); dim(data1)
site_num = rep(NA,length(data1$site_num))
for(s in 1:length(unique(data1$site_num))){
site_num[data1$site_num == unique(data1$site_num)[s]] = s
}
data1$site_num = site_num; rm(site_num)
data1$id_fam = 0
data1$fam_size = 0
ind=0
for(s in 1:length(unique(data1$site_num))){
data_s = data1[data1$site_num == s, ]
for(f in 1:length(unique(data_s$rel_family_id))){
data_s$fam_size[data_s$rel_family_id == unique(data_s$rel_family_id)[f]] =
sum(data_s$rel_family_id == unique(data_s$rel_family_id)[f])
if(sum(data_s$fam_size[data_s$rel_family_id == unique(data_s$rel_family_id)[f]])>1){
ind=ind+1
data_s$id_fam[data_s$rel_family_id == unique(data_s$rel_family_id)[f]] = ind
}
}
data1[data1$site_num == s, ] = data_s
}
data1 = data1[order(data1$site_num,data1$id_fam),]
Site = data1$site_num
Fam = data1$id_fam
ns = length(unique(Site))
ns_s = rep(0,ns)
for(s in 1:ns){
ns_s[s] = sum(Site==s)
}
nf_s = rep(0,ns)
for(s in 1:ns){
nf_s[s] = length(unique(Fam[Site==s]))-1
}
nf = sum(nf_s)
Y = (as.matrix(scale(data1[,ind_Y]))); summary(Y)
N = nrow(Y)
P = ncol(Y)
####################
## Run stan model ##
####################
Nsamples = 1000
Nchains = 3
model_file = "stan_code/bppca.stan"
smod = stan_model(model_file)
D_max = 5
sa.list = list()
log_lik.list = list()
looic.list = list()
for(d in 1:D_max){
print(d)
pca_data <- list(Y = Y, N = N, P = P, D = d, Fam = Fam, Site = Site, ns = ns, nf = nf)
set.seed(314)
sa.list[[d]] = sampling(smod, data= pca_data, iter=Nsamples, chains=Nchains,init="random")
#log_lik.list[[d]] <- extract_log_lik(sa.list[[d]])
log_lik.list[[d]] <- extract(sa.list[[d]],"log_lik_marg")[[1]]
looic.list[[d]] = loo(log_lik.list[[d]])
save(sa.list,log_lik.list,looic.list,file="results/bppca_results.RData")
print("###############################")
}
#################################
## Model selection using LOOIC ##
#################################
load("results/bppca_results.RData")
looic.obj = compare(looic.list[[1]],looic.list[[2]],looic.list[[3]],looic.list[[4]],looic.list[[5]])
print(looic.obj)
d=3
print(sa.list[[d]], pars=c("Q"), probs=c(.025,.5,.975))
print(sa.list[[d]], pars=c("sigma_eps","sigma2_a","sigma2_b","sigma_c","sigma_d"), probs=c(.025,.5,.975))
## Create SM Figure 3
jpeg("figures_and_tables/fig_sm3.jpeg")
traceplot(sa.list[[d]], pars=c("sigma_eps","sigma2_a","sigma2_b","sigma_c","sigma_d"))
dev.off()
#######################################
## Extract parameters from the model ##
#######################################
sa = sa.list[[d]]
Q = extract(sa,"Q",permuted=FALSE)
Theta = extract(sa,"Theta",permuted=FALSE)
Lambda = extract(sa,"Lambda",permuted=FALSE)
sigma2_a = extract(sa,"sigma2_a",permuted=FALSE)
sigma2_b = extract(sa,"sigma2_b",permuted=FALSE)
sigma_c = extract(sa,"sigma_c",permuted=FALSE)
sigma_d = extract(sa,"sigma_d",permuted=FALSE)
sigma_eps = extract(sa,"sigma_eps",permuted=FALSE)
par(mfrow=c(2,2))
hist(sigma2_a,main="Var(a)",xlim=c(0,.12),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma2_a,.5),3)))
hist(sigma2_b,main="Var(b)",xlim=c(.35,.65),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma2_b,.5),3)))
hist(sigma_c^2,main="Var(c)",xlim=c(0,.012),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma_c^2,.5),3)))
hist(sigma_d^2,main="Var(d)",xlim=c(.06,.13),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma_d^2,.5),3)))
vc_tab = array(0, dim=c(4,3))
vc_tab[1,] = quantile(sigma2_a,c(.025,.5,.975))
vc_tab[2,] = quantile(sigma_c^2,c(.025,.5,.975))
vc_tab[3,] = quantile(sigma2_b,c(.025,.5,.975))
vc_tab[4,] = quantile(sigma_d^2,c(.025,.5,.975))
## Reshape parameters and reorient loadings with PCA rotation
Q_new = array(0, dim=c(P,P,Nchains*Nsamples/2))
R_new = array(0, dim=c(P,P,Nchains*Nsamples/2))
S_new = array(0, dim=c(d,d,Nchains*Nsamples/2))
W_new = array(0, dim=c(d,d,Nchains*Nsamples/2))
Theta_old = array(0, dim=c(d,N,Nchains*Nsamples/2))
Theta_new = array(0, dim=c(d,N,Nchains*Nsamples/2))
Lambda_old = array(0, dim=c(P,d,Nchains*Nsamples/2))
Lambda_new = array(0, dim=c(P,d,Nchains*Nsamples/2))
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind+ 1
Theta_old[,,ind] = t(array(Theta[i,j,],dim=c(N,d)))
Lambda_old[,,ind] = array(Lambda[i,j,],dim=c(P,d))
Q_new[,,ind] = array(Q[i,j,],dim=c(P,P))
W_new[,,ind] = diag(eigen(Q_new[,,ind] - diag(rep(sigma_eps[i,j,]^2,P)))$values[1:d]^.5)
R_new[,,ind] = diag(diag(Q_new[,,ind])^-.5)%*%Q_new[,,ind]%*%diag(diag(Q_new[,,ind])^-.5)
S_new[,,ind] = diag(eigen(Q_new[,,ind])$values[1:d]^.5)
Lambda_new[,,ind] = eigen(Q_new[,,ind])$vectors[,1:d]%*%S_new[,,ind]
}
}
## Fix signs of factors and rotate factor scores
ll = rep(0,d)
ll_mx = rep(0,d)
for(k in 1:d){
for(j in 1:(Nsamples*Nchains/2)){
if(max(abs(Lambda_new[,k,j]))>ll_mx[k])
ll[k] = which(abs(Lambda_new[,k,j])==max(abs(Lambda_new[,k,j])))
ll_mx[k] = max(abs(Lambda_new[,k,j]))
}
}
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
for(k in 1:d){
Lambda_new[,k,ind] = sign(Lambda_new[ll[k],k,ind]) * Lambda_new[,k,ind]
}
Theta_new[,,ind] = solve(S_new[,,ind])%*%W_new[,,ind]%*%t(Lambda_new[,,ind]%*%solve(S_new[,,ind])%*%W_new[,,ind])%*%
Lambda_old[,,ind]%*%solve(t(Lambda_old[,,ind])%*%Lambda_old[,,ind])%*%Theta_old[,,ind]
}
}
## Varimax & Promax Rotations
Lambda_vmx = Lambda_new
Theta_vmx = Theta_new
Rot_vmx = array(0, dim = c(d,d,Nchains*Nsamples/2))
Lambda_pmx = Lambda_new
Theta_pmx = Theta_new
Rot_pmx = array(0, dim = c(d,d,Nchains*Nsamples/2))
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
tmp = varimax(Lambda_new[,,ind])
Rot_vmx[,,ind] = tmp$rot
Lambda_vmx[,,ind] = Lambda_new[,,ind]%*%Rot_vmx[,,ind]
Theta_vmx[,,ind] = t(Rot_vmx[,,ind])%*%t(scale(t(Theta_new[,,ind])))
tmp = psych::Promax(Lambda_new[,,ind])
Rot_pmx[,,ind] = tmp$rot
Lambda_pmx[,,ind] = Lambda_new[,,ind]%*%Rot_pmx[,,ind]
Theta_pmx[,,ind] = t(Rot_pmx[,,ind])%*%t(scale(t(Theta_new[,,ind])))
}
}
## Permute Varimax factors
tr=principal(Y,nfactors=d,rotate="varimax")$loadings[,1:d]
for(d1 in 1:d){
tr[,d1] = sign(tr[abs(tr[,d1])==max(abs(tr[,d1])),d1])*tr[,d1]
}
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
ll=rep(0,d)
for(d1 in 1:d){
tmp1=apply((Lambda_vmx[,d1,ind] - tr[,])^2,2,sum)
tmp2=apply((Lambda_vmx[,d1,ind] + tr[,])^2,2,sum)
if(min(tmp1) < min(tmp2)) ll[d1]=which(tmp1==min(tmp1))
if(min(tmp1) > min(tmp2)) ll[d1]=which(tmp2==min(tmp2))
}
Lambda_vmx[,,ind] = Lambda_vmx[,ll,ind]
Theta_vmx[,,ind] = Theta_vmx[ll,,ind]
Rot_vmx[,,ind] = Rot_vmx[ll,ll,ind]
for(d1 in 1:d){
Theta_vmx[d1,,ind] = Theta_vmx[d1,,ind]*sign(Lambda_vmx[abs(Lambda_vmx[,d1,ind])==max(abs(Lambda_vmx[,d1,ind])),d1,ind])
Rot_vmx[,d1,ind] = Rot_vmx[,d1,ind]*sign(Lambda_vmx[abs(Lambda_vmx[,d1,ind])==max(abs(Lambda_vmx[,d1,ind])),d1,ind])
Lambda_vmx[,d1,ind] = Lambda_vmx[,d1,ind]*sign(Lambda_vmx[abs(Lambda_vmx[,d1,ind])==max(abs(Lambda_vmx[,d1,ind])),d1,ind])
}
}
}
## Permute Promax factors
tr=principal(Y,nfactors=d,rotate="promax")$loadings[,1:d]
for(d1 in 1:d){
tr[,d1] = sign(tr[abs(tr[,d1])==max(abs(tr[,d1])),d1])*tr[,d1]
}
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
ll=rep(0,d)
for(d1 in 1:d){
tmp1=apply((Lambda_pmx[,d1,ind] - tr[,])^2,2,sum)
tmp2=apply((Lambda_pmx[,d1,ind] + tr[,])^2,2,sum)
if(min(tmp1) < min(tmp2)) ll[d1]=which(tmp1==min(tmp1))
if(min(tmp1) > min(tmp2)) ll[d1]=which(tmp2==min(tmp2))
}
Lambda_pmx[,,ind] = Lambda_pmx[,ll,ind]
Theta_pmx[,,ind] = Theta_pmx[ll,,ind]
Rot_pmx[,,ind] = Rot_pmx[ll,ll,ind]
for(d1 in 1:d){
Theta_pmx[d1,,ind] = Theta_pmx[d1,,ind]*sign(Lambda_pmx[abs(Lambda_pmx[,d1,ind])==max(abs(Lambda_pmx[,d1,ind])),d1,ind])
Rot_pmx[,d1,ind] = Rot_pmx[,d1,ind]*sign(Lambda_pmx[abs(Lambda_pmx[,d1,ind])==max(abs(Lambda_pmx[,d1,ind])),d1,ind])
Lambda_pmx[,d1,ind] = Lambda_pmx[,d1,ind]*sign(Lambda_pmx[abs(Lambda_pmx[,d1,ind])==max(abs(Lambda_pmx[,d1,ind])),d1,ind])
}
}
}
## Variance explained by retained factors
var_new = array(0, dim=c(d,Nchains*Nsamples/2))
var_vmx = array(0, dim=c(d,Nchains*Nsamples/2))
var_pmx = array(0, dim=c(d,Nchains*Nsamples/2))
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
var_new[,ind] = diag(t(Lambda_new[,,ind])%*%(Lambda_new[,,ind]))
var_vmx[,ind] = diag(t(Lambda_vmx[,,ind])%*%(Lambda_vmx[,,ind]))
var_pmx[,ind] = diag(t(Lambda_pmx[,,ind])%*%(Lambda_pmx[,,ind]))
}
}
## Communalities & Uniquenesses
comm_uniq_new = array(0, dim=c(P,2,Nchains*Nsamples/2))
comm_uniq_vmx = array(0, dim=c(P,2,Nchains*Nsamples/2))
comm_uniq_pmx = array(0, dim=c(P,2,Nchains*Nsamples/2))
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
comm_uniq_new[,,ind] = cbind(diag((Lambda_new[,,ind])%*%t(Lambda_new[,,ind])),1-diag((Lambda_new[,,ind])%*%t(Lambda_new[,,ind])))
comm_uniq_vmx[,,ind] = cbind(diag((Lambda_vmx[,,ind])%*%t(Lambda_vmx[,,ind])),1-diag((Lambda_vmx[,,ind])%*%t(Lambda_vmx[,,ind])))
comm_uniq_pmx[,,ind] = cbind(diag((Lambda_pmx[,,ind])%*%t(Lambda_pmx[,,ind])),1-diag((Lambda_pmx[,,ind])%*%t(Lambda_pmx[,,ind])))
}
}
## Compute posterior median estimates and 95% posterior credible intervals
Q_m = array(0,dim=c(P,P,3))
for(p1 in 1:P){
for(p2 in 1:P){
Q_m[p1,p2,] = quantile(Q_new[p1,p2,],probs=c(.025,.5,.975))
}
}
comm_uniq_new_m = array(0,dim=c(P,2,3))
comm_uniq_vmx_m = array(0,dim=c(P,2,3))
comm_uniq_pmx_m = array(0,dim=c(P,2,3))
Lambda_new_m = array(0,dim=c(P,d,3))
Lambda_vmx_m = array(0,dim=c(P,d,3))
Lambda_pmx_m = array(0,dim=c(P,d,3))
for(p in 1:P){
comm_uniq_new_m[p,1,] = quantile(comm_uniq_new[p,1,],probs=c(.025,.5,.975))
comm_uniq_vmx_m[p,1,] = quantile(comm_uniq_vmx[p,1,],probs=c(.025,.5,.975))
comm_uniq_pmx_m[p,1,] = quantile(comm_uniq_pmx[p,1,],probs=c(.025,.5,.975))
comm_uniq_new_m[p,2,] = quantile(comm_uniq_new[p,2,],probs=c(.025,.5,.975))
comm_uniq_vmx_m[p,2,] = quantile(comm_uniq_vmx[p,2,],probs=c(.025,.5,.975))
comm_uniq_pmx_m[p,2,] = quantile(comm_uniq_pmx[p,2,],probs=c(.025,.5,.975))
for(k in 1:d){
Lambda_new_m[p,k,] = quantile(Lambda_new[p,k,],probs=c(.025,.5,.975))
Lambda_vmx_m[p,k,] = quantile(Lambda_vmx[p,k,],probs=c(.025,.5,.975))
Lambda_pmx_m[p,k,] = quantile(Lambda_pmx[p,k,],probs=c(.025,.5,.975))
}
}
Theta_new_m = array(0,dim=c(d,N,3))
Theta_vmx_m = array(0,dim=c(d,N,3))
Theta_pmx_m = array(0,dim=c(d,N,3))
for(k in 1:d){
for(i in 1:N){
Theta_new_m[k,i,] = quantile(Theta_new[k,i,],probs=c(.025,.5,.975))
Theta_vmx_m[k,i,] = quantile(Theta_vmx[k,i,],probs=c(.025,.5,.975))
Theta_pmx_m[k,i,] = quantile(Theta_pmx[k,i,],probs=c(.025,.5,.975))
}
}
var_new_m = array(0,dim=c(d,3))
var_vmx_m = array(0,dim=c(d,3))
var_pmx_m = array(0,dim=c(d,3))
for(k in 1:d){
var_new_m[k,] = quantile(var_new[k,],c(.025,.5,.975))
var_vmx_m[k,] = quantile(var_vmx[k,],c(.025,.5,.975))
var_pmx_m[k,] = quantile(var_pmx[k,],c(.025,.5,.975))
}
var_expl_new = apply(var_new_m,2,sum)/P
var_expl_vmx = apply(var_vmx_m,2,sum)/P
var_expl_pmx = apply(var_pmx_m,2,sum)/P
## Create SM Table 1A
lambda.tab = as.data.frame(array(0,dim=c(P+1,d*3)))
names(lambda.tab) = c("","PC1","","","PC2","","","PC3","")
rownames(lambda.tab) = c("",names(data)[ind_Y])
lambda.tab[1,] = rep(c(".025","0.50",".975"),d)
for(d1 in 1:d){
lambda.tab[2:(P+1),3*(d1-1)+1]=round(Lambda_new_m[,d1,1],3)
lambda.tab[2:(P+1),3*(d1-1)+2]=round(Lambda_new_m[,d1,2],3)
lambda.tab[2:(P+1),3*(d1-1)+3]=round(Lambda_new_m[,d1,3],3)
}
tabAsStringMatrix = print(lambda.tab, printToggle = FALSE, noSpaces = TRUE)
tab_sm1a = knitr::kable(tabAsStringMatrix)
## Create SM Table 1B
lambda.tab = as.data.frame(array(0,dim=c(P+1,d*3)))
names(lambda.tab) = c("","PC1","","","PC2","","","PC3","")
rownames(lambda.tab) = c("",names(data)[ind_Y])
lambda.tab[1,] = rep(c(".025","0.50",".975"),d)
for(d1 in 1:d){
lambda.tab[2:(P+1),3*(d1-1)+1]=round(Lambda_pmx_m[,d1,1],3)
lambda.tab[2:(P+1),3*(d1-1)+2]=round(Lambda_pmx_m[,d1,2],3)
lambda.tab[2:(P+1),3*(d1-1)+3]=round(Lambda_pmx_m[,d1,3],3)
}
tabAsStringMatrix = print(lambda.tab, printToggle = FALSE, noSpaces = TRUE)
tab_sm1b = knitr::kable(tabAsStringMatrix)
## Create SM Table 3
comm_uniq.tab = as.data.frame(array(0,dim=c(P+1,6)))
names(comm_uniq.tab) = c("","Communality","","","Uniqueness","")
rownames(comm_uniq.tab) = c("Quantiles:",names(data)[ind_Y])
comm_uniq.tab[1,] = rep(c(".025","0.50",".975"),2)
comm_uniq.tab[2:(P+1),c(1,4)]=sprintf("%.3f", round(comm_uniq_vmx_m[,,1],3))
comm_uniq.tab[2:(P+1),c(2,5)]=sprintf("%.3f", round(comm_uniq_vmx_m[,,2],3))
comm_uniq.tab[2:(P+1),c(3,6)]=sprintf("%.3f", round(comm_uniq_vmx_m[,,3],3))
tabAsStringMatrix = print(comm_uniq.tab, printToggle = FALSE, noSpaces = TRUE)
tab_sm3 = knitr::kable(tabAsStringMatrix)
## Create Table 3
lambda.tab = as.data.frame(array(0,dim=c(P+1,d*3)))
names(lambda.tab) = c("","PC1","","","PC2","","","PC3","")
rownames(lambda.tab) = c("",names(data)[ind_Y])
lambda.tab[1,] = rep(c(".025","0.50",".975"),d)
for(d1 in 1:d){
lambda.tab[2:(P+1),3*(d1-1)+1]=round(Lambda_vmx_m[,d1,1],3)
lambda.tab[2:(P+1),3*(d1-1)+2]=round(Lambda_vmx_m[,d1,2],3)
lambda.tab[2:(P+1),3*(d1-1)+3]=round(Lambda_vmx_m[,d1,3],3)
}
tabAsStringMatrix = print(lambda.tab, printToggle = FALSE, noSpaces = TRUE)
tab3 = knitr::kable(tabAsStringMatrix)
## Add PC scores into dataset
data1$pc1 = Theta_vmx_m[1,,2]
data1$pc2 = Theta_vmx_m[2,,2]
data1$pc3 = Theta_vmx_m[3,,2]
##########################
##########################
## Impute Missing Data ##
##########################
##########################
data = merge(data,data1[,c(1,which(names(data1)=="pc1"),which(names(data1)=="pc2"),which(names(data1)=="pc3"))],by="pid",all.x=TRUE)
# Number of multiple imputed datasets & maximum number of iterations
n.imp = 5
n.iter = 5
var.ls <- c("RaceEthnicity","HighestParentalEducation","HouseholdMaritalStatus","HouseholdIncome","PicVocab","Flanker",
"List","CardSort","Pattern","Picture","Reading","RAVLT","LMT","pc1","pc2","pc3")
dat0 = data.table(data)
dat0 <- dat0[, var.ls, with = FALSE ]
Hmisc::describe(dat0)
# Initialize model
ini = mice( dat0, m = 1, maxit = 0 )
meth = ini$meth
meth["RaceEthnicity"] = "polyreg"
meth["HighestParentalEducation"] = "polyreg"
meth["HouseholdMaritalStatus"] = "polyreg"
meth["HouseholdIncome"] = "polyreg"
meth["PicVocab"] = "norm.predict"
meth["Flanker"] = "norm.predict"
meth["List"] = "norm.predict"
meth["CardSort"] = "norm.predict"
meth["Pattern"] = "norm.predict"
meth["Picture"] = "norm.predict"
meth["Reading"] = "norm.predict"
meth["RAVLT"] = "norm.predict"
meth["LMT"] = "norm.predict"
meth["pc1"] = "norm.predict"
meth["pc2"] = "norm.predict"
meth["pc3"] = "norm.predict"
pred = ini$pred
set.seed(314)
post = mice( dat0, meth = meth, pred = pred, seed = 111,
m = 1, maxit = 0)$post
dat.imp = mice( dat0, meth = meth, pred = pred, post = post,
seed = 1111, m = n.imp, maxit = n.iter)
rm(dat0)
# Convert imputed data to long format
dat.mi <- complete(dat.imp, action = "long", include = TRUE)
rm(dat.imp)
names(dat.mi)[1:2] <- c("imp", "id")
dat.mi$pid = rep(data$pid,n.imp+1)
# Adding variables
dat = data[,c(1,2,3,4,5,20:22)]
dat.mi <- merge( dat.mi, dat, by = "pid", all.x = TRUE )
dat.mi$id = NULL
names(dat.mi)
######################################################
######################################################
## Association of factor scores with CBCL variables ##
######################################################
######################################################
d_rsq = array(NA, dim = c(n.imp, 3))
gamm.results = list()
for(j in 1:n.imp){
dat.j = dat.mi[dat.mi$imp==j,]
ind_y = c(which(names(dat.j)=="Externalizing"),which(names(dat.j)=="Internalizing"),which(names(dat.j)=="Stress")); names(dat.j)[ind_y]
dat.j[,ind_y] = dat.j[,ind_y]+.1
## Run GAMMs
p.values = array(NA, dim=c(1,length(ind_y)))
t.values = array(NA, dim=c(1,length(ind_y)))
results = list()
for(i in 1:length(ind_y)){
show(c(j,i))
results[[i]] = list()
form0 = paste(names(dat.j)[ind_y[i]],"~ ")
for(k in 1:length(ind_demog)){
form0 = paste(form0,"+",names(data1)[ind_demog[k]])
}
form0 = formula(form0)
form = paste(names(dat.j)[ind_y[i]],"~ pc1+pc2+pc3")
for(k in 1:length(ind_demog)){
form = paste(form,"+",names(data1)[ind_demog[k]])
}
form = formula(form)
ran = paste("~(1|",names(data1)[ind_nest[1]])
for(k in 2:length(ind_nest)){
ran = paste(ran,"/",names(data1)[ind_nest[k]])
}
ran = paste(ran,")")
ran = formula(ran)
gamm0 = gamm4(formula = form0, random = ran, data = dat.j, family = Gamma(link = "log"))
gamm1 = gamm4(formula = form, random = ran, data = dat.j, family = Gamma(link = "log"))
results[[i]][[1]] = gamm0
results[[i]][[2]] = gamm1
p.values[1,i] = summary(results[[i]][[2]]$gam)$p.pv[2]
t.values[1,i] = summary(results[[i]][[2]]$gam)$p.t[2]
print(summary(results[[i]][[1]]$gam))
print(summary(results[[i]][[2]]$gam))
print(anova(results[[i]][[2]]$gam))
print("######################################")
}
d_rsq[j,] = rep(0,length(ind_y))
for(p in 1:length(ind_y)){
d_rsq[j,p] = summary(results[[p]][[2]]$gam)$r.sq - summary(results[[p]][[1]]$gam)$r.sq
}
gamm.results[[j]] = results
}
d_rsq = apply(d_rsq,2,mean)
## Create Table 4
cbcl.gamm.tab = as.data.frame(array(0,dim=c(15+d,length(ind_y)*3)))
rownames(cbcl.gamm.tab) = c("Variable",names(summary(results[[1]][[2]]$gam)$p.coeff))
names(cbcl.gamm.tab) = c(names(data1)[ind_y][1],names(data1)[ind_y][1],names(data1)[ind_y][1],
names(data1)[ind_y][2],names(data1)[ind_y][2],names(data1)[ind_y][2],
names(data1)[ind_y][3],names(data1)[ind_y][3],names(data1)[ind_y][3])
cbcl.gamm.tab[1,] = rep(c("coef ","se ","p_value"),length(ind_y))
ind = 0
for(k in 1:length(ind_y)){
ind = ind + 1
coeff = summary(gamm.results[[1]][[k]][[2]]$gam)$p.table[,1]
for(b in 2:n.imp){
coeff = coeff + summary(gamm.results[[b]][[k]][[2]]$gam)$p.table[,1]
}
coeff = coeff/n.imp
cbcl.gamm.tab[2:(15+d),ind] = coeff
ind = ind + 1
se = summary(gamm.results[[1]][[k]][[2]]$gam)$p.table[,2]
for(b in 2:n.imp){
se = se + summary(gamm.results[[b]][[k]][[2]]$gam)$p.table[,2]
}
se = se/n.imp
imp.err = 0*se
for(b in 1:n.imp){
imp.err = imp.err + (summary(gamm.results[[b]][[k]][[2]]$gam)$p.table[,1]-coeff)^2
}
imp.err = imp.err/n.imp
se = se + imp.err
cbcl.gamm.tab[2:(15+d),ind] = round(se,2)
ind = ind + 1
cbcl.gamm.tab[2:(15+d),ind] = 2*(1-pnorm(abs(coeff),0,se))
}
cbcl.gamm.tab
tabAsStringMatrix = print(cbcl.gamm.tab, printToggle = FALSE, noSpaces = TRUE)
tab4 = knitr::kable(tabAsStringMatrix)
#############################
## K-fold Cross Validation ##
#############################
K = 10
set.seed(314)
cv = cvFolds(n=dim(data)[1], K = K, R = 1, type = c("random"))
gamm.cv.results = list()
for(k in 1:K){
dat.k = data[!cv$which==k,]
dat.test = data[cv$which==k,]
ind_y = c(which(names(dat.k)=="Externalizing"),which(names(dat.k)=="Internalizing"),which(names(dat.k)=="Stress")); names(dat.k)[ind_y]
dat.k[,ind_y] = dat.k[,ind_y]+.1
## Run GAMMs
results = list()
for(i in 1:length(ind_y)){
show(c(k,i))
results[[i]] = list()
form0 = paste(names(dat.k)[ind_y[i]],"~ ")
for(kk in 1:length(ind_demog)){
form0 = paste(form0,"+",names(data1)[ind_demog[kk]])
}
form0 = formula(form0)
form = paste(names(dat.k)[ind_y[i]],"~ pc1+pc2+pc3")
for(kk in 1:length(ind_demog)){
form = paste(form,"+",names(data1)[ind_demog[kk]])
}
form = formula(form)
ran = paste("~(1|",names(data1)[ind_nest[1]])
for(kk in 2:length(ind_nest)){
ran = paste(ran,"/",names(data1)[ind_nest[kk]])
}
ran = paste(ran,")")
ran = formula(ran)
gamm0 = gamm4(formula = form0, random = ran, data = dat.k, family = Gamma(link = "log"))
gamm1 = gamm4(formula = form, random = ran, data = dat.k, family = Gamma(link = "log"))
results[[i]][[1]] = gamm0
results[[i]][[2]] = gamm1
print(summary(results[[i]][[1]]$gam))
print(summary(results[[i]][[2]]$gam))
print("######################################")
}
gamm.cv.results[[k]] = results
}
d_rsq.cv = array(0, dim = c(K, 3))
for(k in 1:K){
results = gamm.cv.results[[k]]
dat.k = data[!cv$which==k,]
dat.test = data[cv$which==k,]
for(p in 1:length(ind_y)){
d_rsq.cv[k,p] = cor.test(predict(results[[p]][[2]]$gam, newdata = dat.test),dat.test[,ind_y[p]],method = "spearman")$estimate^2 -
cor.test(predict(results[[p]][[1]]$gam, newdata = dat.test),dat.test[,ind_y[p]],method = "spearman")$estimate^2
}
}
d_rsq.cv = apply(d_rsq.cv,2,mean)
##########################################################################################################
##########################################################################################################
##########################################################################################################
##########################################################################################################
##########################################
##########################################
## Fit NIH Toolbox PCA Model using Stan ##
##########################################
##########################################
###############
## Prep data ##
###############
ind_Y = c(11:17); names(data)[ind_Y]
ind_demog = c(4:9); names(data)[ind_demog]
ind_nest = c(2,3); names(data)[ind_nest]
data2 = data[complete.cases(data[,c(ind_Y,ind_nest)]),]; names(data2); dim(data2)
site_num = rep(NA,length(data2$site_num))
for(s in 1:length(unique(data2$site_num))){
site_num[data2$site_num == unique(data2$site_num)[s]] = s
}
data2$site_num = site_num; rm(site_num)
data2$id_fam = 0
data2$fam_size = 0
ind=0
for(s in 1:length(unique(data2$site_num))){
data_s = data2[data2$site_num == s, ]
for(f in 1:length(unique(data_s$rel_family_id))){
data_s$fam_size[data_s$rel_family_id == unique(data_s$rel_family_id)[f]] =
sum(data_s$rel_family_id == unique(data_s$rel_family_id)[f])
if(sum(data_s$fam_size[data_s$rel_family_id == unique(data_s$rel_family_id)[f]])>1){
ind=ind+1
data_s$id_fam[data_s$rel_family_id == unique(data_s$rel_family_id)[f]] = ind
}
}
data2[data2$site_num == s, ] = data_s
}
data2 = data2[order(data2$site_num,data2$id_fam),]
Site = data2$site_num
Fam = data2$id_fam
ns = length(unique(Site))
ns_s = rep(0,ns)
for(s in 1:ns){
ns_s[s] = sum(Site==s)
}
nf_s = rep(0,ns)
for(s in 1:ns){
nf_s[s] = length(unique(Fam[Site==s]))-1
}
nf = sum(nf_s)
Y = (as.matrix(scale(data2[,ind_Y]))); summary(Y)
N = nrow(Y)
P = ncol(Y)
####################
## Run stan model ##
####################
Nsamples = 1000
Nchains = 3
model_file = "stan_code/bppca.stan"
smod = stan_model(model_file)
D_max = 4
sa.nihtb.list = list()
log_lik.nihtb.list = list()
looic.nihtb.list = list()
set.seed(314)
for(d in 1:D_max){
print(d)
pca_data <- list(Y = Y, N = N, P = P, D = d, Fam = Fam, Site = Site, ns = ns, nf = nf)
set.seed(314)
sa.nihtb.list[[d]] = sampling(smod, data= pca_data, iter=Nsamples, chains=Nchains,init="random")
log_lik.nihtb.list[[d]] <- extract(sa.nihtb.list[[d]],"log_lik_marg")[[1]]
looic.nihtb.list[[d]] = loo(log_lik.nihtb.list[[d]])
save(sa.nihtb.list,log_lik.nihtb.list,looic.nihtb.list,file="results/bppca_nihtb_results.RData")
print("###############################")
}
#################################
## Model selection using LOOIC ##
#################################
load("results/bppca_nihtb_results.Rdata")
looic.nihtb.obj = compare(looic.nihtb.list[[1]],looic.nihtb.list[[2]],looic.nihtb.list[[3]],looic.nihtb.list[[4]])
print(looic.nihtb.obj)
d=3
## Create SM Figure 4
pdf("figures_and_tables/fig_sm4.pdf")
traceplot(sa.nihtb.list[[d]], pars=c("sigma_eps","sigma2_a","sigma2_b","sigma_c","sigma_d"))
dev.off()
#######################################
## Extract parameters from the model ##
#######################################
sa = sa.nihtb.list[[d]]
Q = extract(sa,"Q",permuted=FALSE)
Theta = extract(sa,"Theta",permuted=FALSE)
Lambda = extract(sa,"Lambda",permuted=FALSE)
sigma2_a = extract(sa,"sigma2_a",permuted=FALSE)
sigma2_b = extract(sa,"sigma2_b",permuted=FALSE)
sigma_c = extract(sa,"sigma_c",permuted=FALSE)
sigma_d = extract(sa,"sigma_d",permuted=FALSE)
sigma_eps = extract(sa,"sigma_eps",permuted=FALSE)
par(mfrow=c(2,2))
hist(sigma2_a,main="Var(a)",xlim=c(0,.12),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma2_a,.5),3)))
hist(sigma2_b,main="Var(b)",xlim=c(.35,.65),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma2_b,.5),3)))
hist(sigma_c^2,main="Var(c)",xlim=c(0,.012),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma_c^2,.5),3)))
hist(sigma_d^2,main="Var(d)",xlim=c(.06,.13),freq=FALSE,ylab = "Proportion",xlab=paste("median =",round(quantile(sigma_d^2,.5),3)))
vc_nihtb_tab = array(0, dim=c(4,3))
vc_nihtb_tab[1,] = quantile(sigma2_a,c(.025,.5,.975))
vc_nihtb_tab[2,] = quantile(sigma_c^2,c(.025,.5,.975))
vc_nihtb_tab[3,] = quantile(sigma2_b,c(.025,.5,.975))
vc_nihtb_tab[4,] = quantile(sigma_d^2,c(.025,.5,.975))
## Reshape parameters and reorient loadings with PCA rotation
Q_new = array(0, dim=c(P,P,Nchains*Nsamples/2))
R_new = array(0, dim=c(P,P,Nchains*Nsamples/2))
S_new = array(0, dim=c(d,d,Nchains*Nsamples/2))
W_new = array(0, dim=c(d,d,Nchains*Nsamples/2))
Theta_old = array(0, dim=c(d,N,Nchains*Nsamples/2))
Theta_new = array(0, dim=c(d,N,Nchains*Nsamples/2))
Lambda_old = array(0, dim=c(P,d,Nchains*Nsamples/2))
Lambda_new = array(0, dim=c(P,d,Nchains*Nsamples/2))
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind+ 1
Theta_old[,,ind] = t(array(Theta[i,j,],dim=c(N,d)))
Lambda_old[,,ind] = array(Lambda[i,j,],dim=c(P,d))
Q_new[,,ind] = array(Q[i,j,],dim=c(P,P))
W_new[,,ind] = diag(eigen(Q_new[,,ind] - diag(rep(sigma_eps[i,j,]^2,P)))$values[1:d]^.5)
R_new[,,ind] = diag(diag(Q_new[,,ind])^-.5)%*%Q_new[,,ind]%*%diag(diag(Q_new[,,ind])^-.5)
S_new[,,ind] = diag(eigen(Q_new[,,ind])$values[1:d]^.5)
Lambda_new[,,ind] = eigen(Q_new[,,ind])$vectors[,1:d]%*%S_new[,,ind]
}
}
## Fix signs of factors and rotate factor scores
ll = rep(0,d)
ll_mx = rep(0,d)
for(k in 1:d){
for(j in 1:(Nsamples*Nchains/2)){
if(max(abs(Lambda_new[,k,j]))>ll_mx[k])
ll[k] = which(abs(Lambda_new[,k,j])==max(abs(Lambda_new[,k,j])))
ll_mx[k] = max(abs(Lambda_new[,k,j]))
}
}
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
for(k in 1:d){
Lambda_new[,k,ind] = sign(Lambda_new[ll[k],k,ind]) * Lambda_new[,k,ind]
}
Theta_new[,,ind] = solve(S_new[,,ind])%*%W_new[,,ind]%*%t(Lambda_new[,,ind]%*%solve(S_new[,,ind])%*%W_new[,,ind])%*%
Lambda_old[,,ind]%*%solve(t(Lambda_old[,,ind])%*%Lambda_old[,,ind])%*%Theta_old[,,ind]
}
}
## Varimax & Promax Rotations
Lambda_vmx = Lambda_new
Theta_vmx = Theta_new
Rot_vmx = array(0, dim = c(d,d,Nchains*Nsamples/2))
Lambda_pmx = Lambda_new
Theta_pmx = Theta_new
Rot_pmx = array(0, dim = c(d,d,Nchains*Nsamples/2))
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
tmp = varimax(Lambda_new[,,ind])
Rot_vmx[,,ind] = tmp$rot
Lambda_vmx[,,ind] = Lambda_new[,,ind]%*%Rot_vmx[,,ind]
Theta_vmx[,,ind] = t(Rot_vmx[,,ind])%*%t(scale(t(Theta_new[,,ind])))
tmp = psych::Promax(Lambda_new[,,ind])
Rot_pmx[,,ind] = tmp$rot
Lambda_pmx[,,ind] = Lambda_new[,,ind]%*%Rot_pmx[,,ind]
Theta_pmx[,,ind] = t(Rot_pmx[,,ind])%*%t(scale(t(Theta_new[,,ind])))
}
}
## Permute Varimax factors
tr=principal(Y,nfactors=d,rotate="varimax")$loadings[,1:d]
tr = tr[,c(2,1,3)]
for(d1 in 1:d){
tr[,d1] = sign(tr[abs(tr[,d1])==max(abs(tr[,d1])),d1])*tr[,d1]
}
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1
ll=rep(0,d)
for(d1 in 1:d){
tmp1=apply((Lambda_vmx[,d1,ind] - tr[,])^2,2,sum)
tmp2=apply((Lambda_vmx[,d1,ind] + tr[,])^2,2,sum)
if(min(tmp1) < min(tmp2)) ll[d1]=which(tmp1==min(tmp1))
if(min(tmp1) > min(tmp2)) ll[d1]=which(tmp2==min(tmp2))
}
Lambda_vmx[,,ind] = Lambda_vmx[,ll,ind]
Theta_vmx[,,ind] = Theta_vmx[ll,,ind]
Rot_vmx[,,ind] = Rot_vmx[ll,ll,ind]
for(d1 in 1:d){
Theta_vmx[d1,,ind] = Theta_vmx[d1,,ind]*sign(Lambda_vmx[abs(Lambda_vmx[,d1,ind])==max(abs(Lambda_vmx[,d1,ind])),d1,ind])
Rot_vmx[,d1,ind] = Rot_vmx[,d1,ind]*sign(Lambda_vmx[abs(Lambda_vmx[,d1,ind])==max(abs(Lambda_vmx[,d1,ind])),d1,ind])
Lambda_vmx[,d1,ind] = Lambda_vmx[,d1,ind]*sign(Lambda_vmx[abs(Lambda_vmx[,d1,ind])==max(abs(Lambda_vmx[,d1,ind])),d1,ind])
}
}
}
## Permute Promax factors
tr=principal(Y,nfactors=d,rotate="promax")$loadings[,1:d]
for(d1 in 1:d){
tr[,d1] = sign(tr[abs(tr[,d1])==max(abs(tr[,d1])),d1])*tr[,d1]
}
ind = 0
for(i in 1:dim(Q)[1]){
for(j in 1:dim(Q)[2]){
ind = ind + 1