R/ldsc.R
In MichelNivard/GenomicSEM: Structural equation modeling based on GWAS summary statistics
Defines functions
ldsc
Documented in
ldsc
ldsc <- function(traits, sample.prev, population.prev, ld, wld,
trait.names = NULL, sep_weights = FALSE, chr = 22,
n.blocks = 200, ldsc.log = NULL, stand = FALSE,select=FALSE,chisq.max = NA) {
time <- proc.time()
begin.time <- Sys.time()
if(is.null(ldsc.log)){
logtraits<-gsub(".*/","",traits)
log2<-paste(logtraits,collapse="_")
if(object.size(log2) > 200){
log2<-substr(log2,1,100)
}
log.file <- file(paste0(log2, "_ldsc.log"),open="wt")
}else{log.file<-file(paste0(ldsc.log, "_ldsc.log"),open="wt")}
.LOG("Multivariate ld-score regression of ", length(traits), " traits ", "(", paste(traits, collapse = " "), ")", " began at: ", begin.time, file=log.file)
if(select == "ODD" | select == "EVEN"){
odd<-seq(1,chr,2)
even<-seq(2,chr,2)
}
# Dimensions
n.traits <- length(traits)
n.V <- n.traits * (n.traits + 1) / 2
if(n.traits > 18){
n.blocks<-(((n.traits+1)*(n.traits+2))/2)+1
.LOG(" ", file=log.file, print = FALSE)
.LOG("Setting the number of blocks used to perform the block jacknife used to estimate the sampling covariance matrix (V) to ", n.blocks, file=log.file)
.LOG("This reflects the need to estimate V using at least one more block then their are nonredundant elements in the genetic covariance matrix that includes individual SNPs.", file=log.file)
.LOG("If the n.blocks is > 1000 you should carefully inspect output for any strange results, such as extremely significant Q_SNP estimates.", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
if(n.blocks > 1000){
warning("The number of blocks needed to estimate V is > 1000, which may result in sampling dependencies across the blocks used to estimate standard errors and can bias results.")
}
}
if(!(is.null(trait.names))){
check_names<-str_detect(trait.names, "-")
if(any(check_names))
warning("Your trait names specified include mathematical arguments (e.g., + or -) that will be misread by lavaan. Please rename the traits using the trait.names argument.")
}
if(length(traits)==1)
warning("Our version of ldsc requires 2 or more traits. Please include an additional trait.")
# Storage:
cov <- matrix(NA,nrow=n.traits,ncol=n.traits)
V.hold <- matrix(NA,nrow=n.blocks,ncol=n.V)
N.vec <- matrix(NA,nrow=1,ncol=n.V)
Liab.S <- rep(1, n.traits)
I <- matrix(NA,nrow=n.traits,ncol=n.traits)
######### READ LD SCORES:
.LOG("Reading in LD scores", file=log.file)
if(select == FALSE){
x <- do.call("rbind", lapply(1:chr, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "ODD"){
x <- do.call("rbind", lapply(odd, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "EVEN"){
x <- do.call("rbind", lapply(even, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(is.numeric(select)){
x <- do.call("rbind", lapply(select, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
x$CM <- NULL
x$MAF <- NULL
######### READ weights:
if(sep_weights){
if(select == FALSE){
w <- do.call("rbind", lapply(1:chr, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "EVEN"){
w <- do.call("rbind", lapply(even, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "ODD"){
w <- do.call("rbind", lapply(odd, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(is.numeric(select)){
w <- do.call("rbind", lapply(select, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
}else{w<-x}
w$CM <- NULL
w$MAF <- NULL
colnames(w)[ncol(w)] <- "wLD"
### READ M
if(select == FALSE){
m <- do.call("rbind", lapply(1:chr, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
if(select == "EVEN"){
m <- do.call("rbind", lapply(even, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
if(select == "ODD"){
m <- do.call("rbind", lapply(odd, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
if(is.numeric(select)){
m <- do.call("rbind", lapply(select, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
M.tot <- sum(m)
m <- M.tot
### READ ALL CHI2 + MERGE WITH LDSC FILES
s <- 0
all_y <- lapply(traits, function(chi1) {
## READ chi2
y1 <- suppressMessages(na.omit(read_delim(
chi1, delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE)))
.LOG("Read in summary statistics [", s <<- s + 1, "/", n.traits, "] from: ", chi1, file=log.file)
## Merge files
merged <- merge(y1[, c("SNP", "N", "Z", "A1")], w[, c("SNP", "wLD")], by = "SNP", sort = FALSE)
merged <- merge(merged, x, by = "SNP", sort = FALSE)
merged <- merged[with(merged, order(CHR, BP)), ]
.LOG("Out of ", nrow(y1), " SNPs, ", nrow(merged), " remain after merging with LD-score files", file=log.file)
## REMOVE SNPS with excess chi-square:
if(is.na(chisq.max)){
chisq.max <- max(0.001 * max(merged$N), 80)
}
rm <- (merged$Z^2 > chisq.max)
merged <- merged[!rm, ]
.LOG("Removing ", sum(rm), " SNPs with Chi^2 > ", chisq.max, "; ", nrow(merged), " remain", file=log.file)
merged
})
# count the total nummer of runs, both loops
s <- 1
for(j in 1:n.traits){
chi1 <- traits[j]
y1 <- all_y[[j]]
y1$chi1 <- y1$Z^2
for(k in j:length(traits)){
##### HERITABILITY code
if(j == k){
.LOG(" ", " ", file=log.file, print = FALSE)
.LOG("Estimating heritability [", s, "/", n.V, "] for: ", chi1, file=log.file)
samp.prev <- sample.prev[j]
pop.prev <- population.prev[j]
merged <- y1
n.snps <- nrow(merged)
## ADD INTERCEPT:
merged$intercept <- 1
merged$x.tot <- merged$L2
merged$x.tot.intercept <- 1
#### MAKE WEIGHTS:
tot.agg <- (M.tot*(mean(merged$chi1)-1))/mean(merged$L2*merged$N)
tot.agg <- max(tot.agg,0)
tot.agg <- min(tot.agg,1)
merged$ld <- pmax(merged$L2, 1)
merged$w.ld <- pmax(merged$wLD, 1)
merged$c <- tot.agg*merged$N/M.tot
merged$het.w <- 1/(2*(1+(merged$c*merged$ld))^2)
merged$oc.w <- 1/merged$w.ld
merged$w <- merged$het.w*merged$oc.w
merged$initial.w <- sqrt(merged$w)
merged$weights <- merged$initial.w/sum(merged$initial.w)
N.bar <- mean(merged$N)
## preweight LD and chi:
weighted.LD <- as.matrix(cbind(merged$L2,merged$intercept)*merged$weights)
weighted.chi <- as.matrix(merged$chi1*merged$weights)
## Perfrom analysis:
n.annot <- 1
select.from <- floor(seq(from=1,to=n.snps,length.out =(n.blocks+1)))
select.to <- c(select.from[2:n.blocks]-1,n.snps)
xty.block.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
xtx.block.values <- matrix(data=NA,nrow =((n.annot+1)* n.blocks),ncol =(n.annot+1))
colnames(xty.block.values)<- colnames(xtx.block.values)<- colnames(weighted.LD)
replace.from <- seq(from=1,to=nrow(xtx.block.values),by =(n.annot+1))
replace.to <- seq(from =(n.annot+1),to=nrow(xtx.block.values),by =(n.annot+1))
for(i in 1:n.blocks){
xty.block.values[i,] <- t(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.chi[select.from[i]:select.to[i],])
xtx.block.values[replace.from[i]:replace.to[i],] <- as.matrix(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.LD[select.from[i]:select.to[i],])
}
xty <- as.matrix(colSums(xty.block.values))
xtx <- matrix(data=NA,nrow =(n.annot+1),ncol =(n.annot+1))
colnames(xtx)<- colnames(weighted.LD)
for(i in 1:nrow(xtx)){xtx[i,] <- t(colSums(xtx.block.values[seq(from=i,to=nrow(xtx.block.values),by=ncol(weighted.LD)),]))}
reg <- solve(xtx, xty)
intercept <- reg[2]
coefs <- reg[1]/N.bar
reg.tot <- coefs*m
delete.from <- seq(from=1,to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.to <- seq(from=ncol(xtx.block.values),to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
colnames(delete.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){
xty.delete <- xty-xty.block.values[i,]
xtx.delete <- xtx-xtx.block.values[delete.from[i]:delete.to[i],]
delete.values[i,] <- solve(xtx.delete, xty.delete)
}
tot.delete.values <- delete.values[,1:n.annot]
pseudo.values <- matrix(data=NA,nrow=n.blocks,ncol=length(reg))
colnames(pseudo.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){pseudo.values[i,] <- (n.blocks*reg)-((n.blocks-1)* delete.values[i,])}
jackknife.cov <- cov(pseudo.values)/n.blocks
jackknife.se <- sqrt(diag(jackknife.cov))
intercept.se <- jackknife.se[length(jackknife.se)]
coef.cov <- jackknife.cov[1:n.annot,1:n.annot]/(N.bar^2)
cat.cov <- coef.cov*(m %*% t(m))
tot.cov <- sum(cat.cov)
tot.se <- sqrt(tot.cov)
V.hold[,s] <- pseudo.values[,1]
N.vec[1,s] <- N.bar
if(is.na(pop.prev)==F & is.na(samp.prev)==F){
conversion.factor <- (pop.prev^2*(1-pop.prev)^2)/(samp.prev*(1-samp.prev)* dnorm(qnorm(1-pop.prev))^2)
Liab.S[j] <- conversion.factor
.LOG(" ", file=log.file, print = FALSE)
.LOG("Please note that the results initially printed to the screen and log file reflect the NON-liability h2 and cov_g. However, a liability conversion is being used for trait ",
chi1, " when creating the genetic covariance matrix used as input for Genomic SEM and liability scale results are printed at the end of the log file.", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
}
cov[j,j] <- reg.tot
I[j,j] <- intercept
lambda.gc <- median(merged$chi1) / qchisq(0.5, df = 1)
mean.Chi <- mean(merged$chi1)
ratio <- (intercept - 1) / (mean.Chi - 1)
ratio.se <- intercept.se / (mean.Chi - 1)
.LOG("Heritability Results for trait: ", chi1, file=log.file)
.LOG("Mean Chi^2 across remaining SNPs: ", round(mean.Chi, 4), file=log.file)
.LOG("Lambda GC: ", round(lambda.gc, 4), file=log.file)
.LOG("Intercept: ", round(intercept, 4), " (", round(intercept.se, 4), ")", file=log.file)
.LOG("Ratio: ", round(ratio, 4), " (", round(ratio.se, 4), ")", file=log.file)
.LOG("Total Observed Scale h2: ", round(reg.tot, 4), " (", round(tot.se, 4), ")", file=log.file)
.LOG("h2 Z: ", format(reg.tot / tot.se, digits = 3), file=log.file)
}
##### GENETIC COVARIANCE code
if(j != k){
.LOG(" ", file=log.file, print = FALSE)
chi2 <- traits[k]
.LOG("Calculating genetic covariance [", s, "/", n.V, "] for traits: ", chi1, " and ", chi2, file=log.file)
# Reuse the data read in for heritability
y2 <- all_y[[k]]
y <- merge(y1, y2[, c("SNP", "N", "Z", "A1")], by = "SNP", sort = FALSE)
y$Z.x <- ifelse(y$A1.y == y$A1.x, y$Z.x, -y$Z.x)
y$ZZ <- y$Z.y * y$Z.x
y$chi2 <- y$Z.y^2
merged <- na.omit(y)
n.snps <- nrow(merged)
.LOG(n.snps, " SNPs remain after merging ", chi1, " and ", chi2, " summary statistics", file=log.file)
## ADD INTERCEPT:
merged$intercept <- 1
merged$x.tot <- merged$L2
merged$x.tot.intercept <- 1
#### MAKE WEIGHTS:
tot.agg <- (M.tot*(mean(merged$chi1)-1))/mean(merged$L2*merged$N.x)
tot.agg <- max(tot.agg,0)
tot.agg <- min(tot.agg,1)
merged$ld <- pmax(merged$L2, 1)
merged$w.ld <- pmax(merged$wLD, 1)
merged$c <- tot.agg*merged$N.x/M.tot
merged$het.w <- 1/(2*(1+(merged$c*merged$ld))^2)
merged$oc.w <- 1/merged$w.ld
merged$w <- merged$het.w*merged$oc.w
merged$initial.w <- sqrt(merged$w)
tot.agg2 <- (M.tot*(mean(merged$chi2)-1))/mean(merged$L2*merged$N.y)
tot.agg2 <- max(tot.agg2,0)
tot.agg2 <- min(tot.agg2,1)
merged$ld2 <- pmax(merged$L2, 1)
merged$w.ld2 <- pmax(merged$wLD, 1)
merged$c2 <- tot.agg2*merged$N.y/M.tot
merged$het.w2 <- 1/(2*(1+(merged$c2*merged$ld))^2)
merged$oc.w2 <- 1/merged$w.ld2
merged$w2 <- merged$het.w2*merged$oc.w2
merged$initial.w2 <- sqrt(merged$w2)
merged$weights_cov <- (merged$initial.w + merged$initial.w2)/sum(merged$initial.w + merged$initial.w2 )
N.bar <- sqrt(mean(merged$N.x)*mean(merged$N.y))
## preweight LD and chi:
weighted.LD <- as.matrix(cbind(merged$L2,merged$intercept)*merged$weights)
weighted.chi <- as.matrix(merged$ZZ *merged$weights_cov)
## Perfrom analysis:
n.annot <- 1
select.from <- floor(seq(from=1,to=n.snps,length.out =(n.blocks+1)))
select.to <- c(select.from[2:n.blocks]-1,n.snps)
xty.block.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
xtx.block.values <- matrix(data=NA,nrow =((n.annot+1)* n.blocks),ncol =(n.annot+1))
colnames(xty.block.values)<- colnames(xtx.block.values)<- colnames(weighted.LD)
replace.from <- seq(from=1,to=nrow(xtx.block.values),by =(n.annot+1))
replace.to <- seq(from =(n.annot+1),to=nrow(xtx.block.values),by =(n.annot+1))
for(i in 1:n.blocks){
xty.block.values[i,] <- t(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.chi[select.from[i]:select.to[i],])
xtx.block.values[replace.from[i]:replace.to[i],] <- as.matrix(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.LD[select.from[i]:select.to[i],])
}
xty <- as.matrix(colSums(xty.block.values))
xtx <- matrix(data=NA,nrow =(n.annot+1),ncol =(n.annot+1))
colnames(xtx)<- colnames(weighted.LD)
for(i in 1:nrow(xtx)){xtx[i,] <- t(colSums(xtx.block.values[seq(from=i,to=nrow(xtx.block.values),by=ncol(weighted.LD)),]))}
reg <- solve(xtx, xty)
intercept <- reg[2]
coefs <- reg[1]/N.bar
reg.tot <- coefs*m
delete.from <- seq(from=1,to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.to <- seq(from=ncol(xtx.block.values),to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
colnames(delete.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){
xty.delete <- xty-xty.block.values[i,]
xtx.delete <- xtx-xtx.block.values[delete.from[i]:delete.to[i],]
delete.values[i,] <- solve(xtx.delete, xty.delete)
}
tot.delete.values <- delete.values[,1:n.annot]
pseudo.values <- matrix(data=NA,nrow=n.blocks,ncol=length(reg))
colnames(pseudo.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){pseudo.values[i,] <- (n.blocks*reg)-((n.blocks-1)* delete.values[i,])}
jackknife.cov <- cov(pseudo.values)/n.blocks
jackknife.se <- sqrt(diag(jackknife.cov))
intercept.se <- jackknife.se[length(jackknife.se)]
coef.cov <- jackknife.cov[1:n.annot,1:n.annot]/(N.bar^2)
cat.cov <- coef.cov*(m %*% t(m))
tot.cov <- sum(cat.cov)
tot.se <- sqrt(tot.cov)
V.hold[, s] <- pseudo.values[, 1]
N.vec[1, s] <- N.bar
cov[k, j] <- cov[j, k] <- reg.tot
I[k, j] <- I[j, k] <- intercept
.LOG("Results for genetic covariance between: ", chi1, " and ", chi2, file=log.file)
.LOG("Mean Z*Z: ", round(mean(merged$ZZ), 4), file=log.file)
.LOG("Cross trait Intercept: ", round(intercept, 4), " (", round(intercept.se, 4), ")", file=log.file)
.LOG("Total Observed Scale Genetic Covariance (g_cov): ", round(reg.tot, 4), " (", round(tot.se, 4), ")", file=log.file)
.LOG("g_cov Z: ", format(reg.tot / tot.se, digits = 3), file=log.file)
.LOG("g_cov P-value: ", format(2 * pnorm(abs(reg.tot / tot.se), lower.tail = FALSE), digits = 5), file=log.file)
}
### Total count
s <- s + 1
}
}
## Scale V to N per study (assume m constant)
# /!\ crossprod instead of tcrossprod because N.vec is a one-row matrix
v.out <- cov(V.hold) / crossprod(N.vec * (sqrt(n.blocks) / m))
### Scale S and V to liability:
ratio <- tcrossprod(sqrt(Liab.S))
S <- cov * ratio
#calculate the ratio of the rescaled and original S matrices
scaleO <- gdata::lowerTriangle(ratio, diag = TRUE)
#rescale the sampling correlation matrix by the appropriate diagonals
V <- v.out * tcrossprod(scaleO)
#name traits according to trait.names argument
#use general format of V1-VX if no names provided
colnames(S) <- if (is.null(trait.names)) paste0("V", 1:ncol(S)) else trait.names
if(mean(Liab.S)!=1){
r<-nrow(S)
SE<-matrix(0, r, r)
SE[lower.tri(SE,diag=TRUE)] <-sqrt(diag(V))
.LOG(c(" ", " "), file=log.file, print = FALSE)
.LOG("Liability Scale Results", file=log.file)
for(j in 1:n.traits){
if(is.null(trait.names)){
chi1<-traits[j]
}else{chi1 <- trait.names[j]}
for(k in j:length(traits)){
if(j == k){
.LOG(" ", file=log.file, print = FALSE)
.LOG("Liability scale results for: ", chi1, file=log.file)
.LOG("Total Liability Scale h2: ", round(S[j, j], 4), " (", round(SE[j, j], 4), ")", file=log.file)
}
if(j != k){
if(is.null(trait.names)){
chi2<-traits[k]
}else{chi2 <- trait.names[k]}
.LOG("Total Liability Scale Genetic Covariance between ", chi1, " and ",
chi2, ": ", round(S[k, j], 4), " (", round(SE[k, j], 4), ")", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
}
}
}
}
if(all(diag(S) > 0)){
##calculate standardized results to print genetic correlations to log and screen
ratio <- tcrossprod(1 / sqrt(diag(S)))
S_Stand <- S * ratio
#calculate the ratio of the rescaled and original S matrices
scaleO <- gdata::lowerTriangle(ratio, diag = TRUE)
## MAke sure that if ratio in NaN (devision by zero) we put the zero back in
# -> not possible because of 'all(diag(S) > 0)'
# scaleO[is.nan(scaleO)] <- 0
#rescale the sampling correlation matrix by the appropriate diagonals
V_Stand <- V * tcrossprod(scaleO)
#enter SEs from diagonal of standardized V
r<-nrow(S)
SE_Stand<-matrix(0, r, r)
SE_Stand[lower.tri(SE_Stand,diag=TRUE)] <-sqrt(diag(V_Stand))
.LOG(c(" ", " "), file=log.file, print = FALSE)
.LOG("Genetic Correlation Results", file=log.file)
for(j in 1:n.traits){
if(is.null(trait.names)){
chi1<-traits[j]
}else{chi1 <- trait.names[j]}
for(k in j:length(traits)){
if(j != k){
if(is.null(trait.names)){
chi2<-traits[k]
}else{chi2 <- trait.names[k]}
.LOG("Genetic Correlation between ", chi1, " and ", chi2, ": ",
round(S_Stand[k, j], 4), " (", round(SE_Stand[k, j], 4), ")", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
}
}
}
}else{
warning("Your genetic covariance matrix includes traits estimated to have a negative heritability.")
.LOG("Your genetic covariance matrix includes traits estimated to have a negative heritability.", file=log.file, print = FALSE)
.LOG("Genetic correlation results could not be computed due to negative heritability estimates.", file=log.file)
}
end.time <- Sys.time()
total.time <- difftime(time1=end.time,time2=begin.time,units="sec")
mins <- floor(floor(total.time)/60)
secs <- floor(total.time-mins*60)
.LOG(" ", file=log.file, print = FALSE)
.LOG("LDSC finished running at ", end.time, file=log.file)
.LOG("Running LDSC for all files took ", mins, " minutes and ", secs, " seconds", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
flush(log.file)
close(log.file)
if(stand){
list(V=V,S=S,I=I,N=N.vec,m=m,V_Stand=V_Stand,S_Stand=S_Stand)
} else {
list(V=V,S=S,I=I,N=N.vec,m=m)
}
}
MichelNivard/GenomicSEM documentation built on Dec. 24, 2024, 3:23 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Defines functions ldsc
Documented in ldsc
ldsc <- function(traits, sample.prev, population.prev, ld, wld,
trait.names = NULL, sep_weights = FALSE, chr = 22,
n.blocks = 200, ldsc.log = NULL, stand = FALSE,select=FALSE,chisq.max = NA) {
time <- proc.time()
begin.time <- Sys.time()
if(is.null(ldsc.log)){
logtraits<-gsub(".*/","",traits)
log2<-paste(logtraits,collapse="_")
if(object.size(log2) > 200){
log2<-substr(log2,1,100)
}
log.file <- file(paste0(log2, "_ldsc.log"),open="wt")
}else{log.file<-file(paste0(ldsc.log, "_ldsc.log"),open="wt")}
.LOG("Multivariate ld-score regression of ", length(traits), " traits ", "(", paste(traits, collapse = " "), ")", " began at: ", begin.time, file=log.file)
if(select == "ODD" | select == "EVEN"){
odd<-seq(1,chr,2)
even<-seq(2,chr,2)
}
# Dimensions
n.traits <- length(traits)
n.V <- n.traits * (n.traits + 1) / 2
if(n.traits > 18){
n.blocks<-(((n.traits+1)*(n.traits+2))/2)+1
.LOG(" ", file=log.file, print = FALSE)
.LOG("Setting the number of blocks used to perform the block jacknife used to estimate the sampling covariance matrix (V) to ", n.blocks, file=log.file)
.LOG("This reflects the need to estimate V using at least one more block then their are nonredundant elements in the genetic covariance matrix that includes individual SNPs.", file=log.file)
.LOG("If the n.blocks is > 1000 you should carefully inspect output for any strange results, such as extremely significant Q_SNP estimates.", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
if(n.blocks > 1000){
warning("The number of blocks needed to estimate V is > 1000, which may result in sampling dependencies across the blocks used to estimate standard errors and can bias results.")
}
}
if(!(is.null(trait.names))){
check_names<-str_detect(trait.names, "-")
if(any(check_names))
warning("Your trait names specified include mathematical arguments (e.g., + or -) that will be misread by lavaan. Please rename the traits using the trait.names argument.")
}
if(length(traits)==1)
warning("Our version of ldsc requires 2 or more traits. Please include an additional trait.")
# Storage:
cov <- matrix(NA,nrow=n.traits,ncol=n.traits)
V.hold <- matrix(NA,nrow=n.blocks,ncol=n.V)
N.vec <- matrix(NA,nrow=1,ncol=n.V)
Liab.S <- rep(1, n.traits)
I <- matrix(NA,nrow=n.traits,ncol=n.traits)
######### READ LD SCORES:
.LOG("Reading in LD scores", file=log.file)
if(select == FALSE){
x <- do.call("rbind", lapply(1:chr, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "ODD"){
x <- do.call("rbind", lapply(odd, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "EVEN"){
x <- do.call("rbind", lapply(even, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(is.numeric(select)){
x <- do.call("rbind", lapply(select, function(i) {
suppressMessages(read_delim(
file.path(ld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
x$CM <- NULL
x$MAF <- NULL
######### READ weights:
if(sep_weights){
if(select == FALSE){
w <- do.call("rbind", lapply(1:chr, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "EVEN"){
w <- do.call("rbind", lapply(even, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(select == "ODD"){
w <- do.call("rbind", lapply(odd, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
if(is.numeric(select)){
w <- do.call("rbind", lapply(select, function(i) {
suppressMessages(read_delim(
file.path(wld, paste0(i, ".l2.ldscore.gz")),
delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE))
}))
}
}else{w<-x}
w$CM <- NULL
w$MAF <- NULL
colnames(w)[ncol(w)] <- "wLD"
### READ M
if(select == FALSE){
m <- do.call("rbind", lapply(1:chr, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
if(select == "EVEN"){
m <- do.call("rbind", lapply(even, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
if(select == "ODD"){
m <- do.call("rbind", lapply(odd, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
if(is.numeric(select)){
m <- do.call("rbind", lapply(select, function(i) {
suppressMessages(read_csv(file.path(ld, paste0(i, ".l2.M_5_50")), col_names = FALSE))
}))
}
M.tot <- sum(m)
m <- M.tot
### READ ALL CHI2 + MERGE WITH LDSC FILES
s <- 0
all_y <- lapply(traits, function(chi1) {
## READ chi2
y1 <- suppressMessages(na.omit(read_delim(
chi1, delim = "\t", escape_double = FALSE, trim_ws = TRUE, progress = FALSE)))
.LOG("Read in summary statistics [", s <<- s + 1, "/", n.traits, "] from: ", chi1, file=log.file)
## Merge files
merged <- merge(y1[, c("SNP", "N", "Z", "A1")], w[, c("SNP", "wLD")], by = "SNP", sort = FALSE)
merged <- merge(merged, x, by = "SNP", sort = FALSE)
merged <- merged[with(merged, order(CHR, BP)), ]
.LOG("Out of ", nrow(y1), " SNPs, ", nrow(merged), " remain after merging with LD-score files", file=log.file)
## REMOVE SNPS with excess chi-square:
if(is.na(chisq.max)){
chisq.max <- max(0.001 * max(merged$N), 80)
}
rm <- (merged$Z^2 > chisq.max)
merged <- merged[!rm, ]
.LOG("Removing ", sum(rm), " SNPs with Chi^2 > ", chisq.max, "; ", nrow(merged), " remain", file=log.file)
merged
})
# count the total nummer of runs, both loops
s <- 1
for(j in 1:n.traits){
chi1 <- traits[j]
y1 <- all_y[[j]]
y1$chi1 <- y1$Z^2
for(k in j:length(traits)){
##### HERITABILITY code
if(j == k){
.LOG(" ", " ", file=log.file, print = FALSE)
.LOG("Estimating heritability [", s, "/", n.V, "] for: ", chi1, file=log.file)
samp.prev <- sample.prev[j]
pop.prev <- population.prev[j]
merged <- y1
n.snps <- nrow(merged)
## ADD INTERCEPT:
merged$intercept <- 1
merged$x.tot <- merged$L2
merged$x.tot.intercept <- 1
#### MAKE WEIGHTS:
tot.agg <- (M.tot*(mean(merged$chi1)-1))/mean(merged$L2*merged$N)
tot.agg <- max(tot.agg,0)
tot.agg <- min(tot.agg,1)
merged$ld <- pmax(merged$L2, 1)
merged$w.ld <- pmax(merged$wLD, 1)
merged$c <- tot.agg*merged$N/M.tot
merged$het.w <- 1/(2*(1+(merged$c*merged$ld))^2)
merged$oc.w <- 1/merged$w.ld
merged$w <- merged$het.w*merged$oc.w
merged$initial.w <- sqrt(merged$w)
merged$weights <- merged$initial.w/sum(merged$initial.w)
N.bar <- mean(merged$N)
## preweight LD and chi:
weighted.LD <- as.matrix(cbind(merged$L2,merged$intercept)*merged$weights)
weighted.chi <- as.matrix(merged$chi1*merged$weights)
## Perfrom analysis:
n.annot <- 1
select.from <- floor(seq(from=1,to=n.snps,length.out =(n.blocks+1)))
select.to <- c(select.from[2:n.blocks]-1,n.snps)
xty.block.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
xtx.block.values <- matrix(data=NA,nrow =((n.annot+1)* n.blocks),ncol =(n.annot+1))
colnames(xty.block.values)<- colnames(xtx.block.values)<- colnames(weighted.LD)
replace.from <- seq(from=1,to=nrow(xtx.block.values),by =(n.annot+1))
replace.to <- seq(from =(n.annot+1),to=nrow(xtx.block.values),by =(n.annot+1))
for(i in 1:n.blocks){
xty.block.values[i,] <- t(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.chi[select.from[i]:select.to[i],])
xtx.block.values[replace.from[i]:replace.to[i],] <- as.matrix(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.LD[select.from[i]:select.to[i],])
}
xty <- as.matrix(colSums(xty.block.values))
xtx <- matrix(data=NA,nrow =(n.annot+1),ncol =(n.annot+1))
colnames(xtx)<- colnames(weighted.LD)
for(i in 1:nrow(xtx)){xtx[i,] <- t(colSums(xtx.block.values[seq(from=i,to=nrow(xtx.block.values),by=ncol(weighted.LD)),]))}
reg <- solve(xtx, xty)
intercept <- reg[2]
coefs <- reg[1]/N.bar
reg.tot <- coefs*m
delete.from <- seq(from=1,to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.to <- seq(from=ncol(xtx.block.values),to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
colnames(delete.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){
xty.delete <- xty-xty.block.values[i,]
xtx.delete <- xtx-xtx.block.values[delete.from[i]:delete.to[i],]
delete.values[i,] <- solve(xtx.delete, xty.delete)
}
tot.delete.values <- delete.values[,1:n.annot]
pseudo.values <- matrix(data=NA,nrow=n.blocks,ncol=length(reg))
colnames(pseudo.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){pseudo.values[i,] <- (n.blocks*reg)-((n.blocks-1)* delete.values[i,])}
jackknife.cov <- cov(pseudo.values)/n.blocks
jackknife.se <- sqrt(diag(jackknife.cov))
intercept.se <- jackknife.se[length(jackknife.se)]
coef.cov <- jackknife.cov[1:n.annot,1:n.annot]/(N.bar^2)
cat.cov <- coef.cov*(m %*% t(m))
tot.cov <- sum(cat.cov)
tot.se <- sqrt(tot.cov)
V.hold[,s] <- pseudo.values[,1]
N.vec[1,s] <- N.bar
if(is.na(pop.prev)==F & is.na(samp.prev)==F){
conversion.factor <- (pop.prev^2*(1-pop.prev)^2)/(samp.prev*(1-samp.prev)* dnorm(qnorm(1-pop.prev))^2)
Liab.S[j] <- conversion.factor
.LOG(" ", file=log.file, print = FALSE)
.LOG("Please note that the results initially printed to the screen and log file reflect the NON-liability h2 and cov_g. However, a liability conversion is being used for trait ",
chi1, " when creating the genetic covariance matrix used as input for Genomic SEM and liability scale results are printed at the end of the log file.", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
}
cov[j,j] <- reg.tot
I[j,j] <- intercept
lambda.gc <- median(merged$chi1) / qchisq(0.5, df = 1)
mean.Chi <- mean(merged$chi1)
ratio <- (intercept - 1) / (mean.Chi - 1)
ratio.se <- intercept.se / (mean.Chi - 1)
.LOG("Heritability Results for trait: ", chi1, file=log.file)
.LOG("Mean Chi^2 across remaining SNPs: ", round(mean.Chi, 4), file=log.file)
.LOG("Lambda GC: ", round(lambda.gc, 4), file=log.file)
.LOG("Intercept: ", round(intercept, 4), " (", round(intercept.se, 4), ")", file=log.file)
.LOG("Ratio: ", round(ratio, 4), " (", round(ratio.se, 4), ")", file=log.file)
.LOG("Total Observed Scale h2: ", round(reg.tot, 4), " (", round(tot.se, 4), ")", file=log.file)
.LOG("h2 Z: ", format(reg.tot / tot.se, digits = 3), file=log.file)
}
##### GENETIC COVARIANCE code
if(j != k){
.LOG(" ", file=log.file, print = FALSE)
chi2 <- traits[k]
.LOG("Calculating genetic covariance [", s, "/", n.V, "] for traits: ", chi1, " and ", chi2, file=log.file)
# Reuse the data read in for heritability
y2 <- all_y[[k]]
y <- merge(y1, y2[, c("SNP", "N", "Z", "A1")], by = "SNP", sort = FALSE)
y$Z.x <- ifelse(y$A1.y == y$A1.x, y$Z.x, -y$Z.x)
y$ZZ <- y$Z.y * y$Z.x
y$chi2 <- y$Z.y^2
merged <- na.omit(y)
n.snps <- nrow(merged)
.LOG(n.snps, " SNPs remain after merging ", chi1, " and ", chi2, " summary statistics", file=log.file)
## ADD INTERCEPT:
merged$intercept <- 1
merged$x.tot <- merged$L2
merged$x.tot.intercept <- 1
#### MAKE WEIGHTS:
tot.agg <- (M.tot*(mean(merged$chi1)-1))/mean(merged$L2*merged$N.x)
tot.agg <- max(tot.agg,0)
tot.agg <- min(tot.agg,1)
merged$ld <- pmax(merged$L2, 1)
merged$w.ld <- pmax(merged$wLD, 1)
merged$c <- tot.agg*merged$N.x/M.tot
merged$het.w <- 1/(2*(1+(merged$c*merged$ld))^2)
merged$oc.w <- 1/merged$w.ld
merged$w <- merged$het.w*merged$oc.w
merged$initial.w <- sqrt(merged$w)
tot.agg2 <- (M.tot*(mean(merged$chi2)-1))/mean(merged$L2*merged$N.y)
tot.agg2 <- max(tot.agg2,0)
tot.agg2 <- min(tot.agg2,1)
merged$ld2 <- pmax(merged$L2, 1)
merged$w.ld2 <- pmax(merged$wLD, 1)
merged$c2 <- tot.agg2*merged$N.y/M.tot
merged$het.w2 <- 1/(2*(1+(merged$c2*merged$ld))^2)
merged$oc.w2 <- 1/merged$w.ld2
merged$w2 <- merged$het.w2*merged$oc.w2
merged$initial.w2 <- sqrt(merged$w2)
merged$weights_cov <- (merged$initial.w + merged$initial.w2)/sum(merged$initial.w + merged$initial.w2 )
N.bar <- sqrt(mean(merged$N.x)*mean(merged$N.y))
## preweight LD and chi:
weighted.LD <- as.matrix(cbind(merged$L2,merged$intercept)*merged$weights)
weighted.chi <- as.matrix(merged$ZZ *merged$weights_cov)
## Perfrom analysis:
n.annot <- 1
select.from <- floor(seq(from=1,to=n.snps,length.out =(n.blocks+1)))
select.to <- c(select.from[2:n.blocks]-1,n.snps)
xty.block.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
xtx.block.values <- matrix(data=NA,nrow =((n.annot+1)* n.blocks),ncol =(n.annot+1))
colnames(xty.block.values)<- colnames(xtx.block.values)<- colnames(weighted.LD)
replace.from <- seq(from=1,to=nrow(xtx.block.values),by =(n.annot+1))
replace.to <- seq(from =(n.annot+1),to=nrow(xtx.block.values),by =(n.annot+1))
for(i in 1:n.blocks){
xty.block.values[i,] <- t(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.chi[select.from[i]:select.to[i],])
xtx.block.values[replace.from[i]:replace.to[i],] <- as.matrix(t(weighted.LD[select.from[i]:select.to[i],])%*% weighted.LD[select.from[i]:select.to[i],])
}
xty <- as.matrix(colSums(xty.block.values))
xtx <- matrix(data=NA,nrow =(n.annot+1),ncol =(n.annot+1))
colnames(xtx)<- colnames(weighted.LD)
for(i in 1:nrow(xtx)){xtx[i,] <- t(colSums(xtx.block.values[seq(from=i,to=nrow(xtx.block.values),by=ncol(weighted.LD)),]))}
reg <- solve(xtx, xty)
intercept <- reg[2]
coefs <- reg[1]/N.bar
reg.tot <- coefs*m
delete.from <- seq(from=1,to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.to <- seq(from=ncol(xtx.block.values),to=nrow(xtx.block.values),by=ncol(xtx.block.values))
delete.values <- matrix(data=NA,nrow=n.blocks,ncol =(n.annot+1))
colnames(delete.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){
xty.delete <- xty-xty.block.values[i,]
xtx.delete <- xtx-xtx.block.values[delete.from[i]:delete.to[i],]
delete.values[i,] <- solve(xtx.delete, xty.delete)
}
tot.delete.values <- delete.values[,1:n.annot]
pseudo.values <- matrix(data=NA,nrow=n.blocks,ncol=length(reg))
colnames(pseudo.values)<- colnames(weighted.LD)
for(i in 1:n.blocks){pseudo.values[i,] <- (n.blocks*reg)-((n.blocks-1)* delete.values[i,])}
jackknife.cov <- cov(pseudo.values)/n.blocks
jackknife.se <- sqrt(diag(jackknife.cov))
intercept.se <- jackknife.se[length(jackknife.se)]
coef.cov <- jackknife.cov[1:n.annot,1:n.annot]/(N.bar^2)
cat.cov <- coef.cov*(m %*% t(m))
tot.cov <- sum(cat.cov)
tot.se <- sqrt(tot.cov)
V.hold[, s] <- pseudo.values[, 1]
N.vec[1, s] <- N.bar
cov[k, j] <- cov[j, k] <- reg.tot
I[k, j] <- I[j, k] <- intercept
.LOG("Results for genetic covariance between: ", chi1, " and ", chi2, file=log.file)
.LOG("Mean Z*Z: ", round(mean(merged$ZZ), 4), file=log.file)
.LOG("Cross trait Intercept: ", round(intercept, 4), " (", round(intercept.se, 4), ")", file=log.file)
.LOG("Total Observed Scale Genetic Covariance (g_cov): ", round(reg.tot, 4), " (", round(tot.se, 4), ")", file=log.file)
.LOG("g_cov Z: ", format(reg.tot / tot.se, digits = 3), file=log.file)
.LOG("g_cov P-value: ", format(2 * pnorm(abs(reg.tot / tot.se), lower.tail = FALSE), digits = 5), file=log.file)
}
### Total count
s <- s + 1
}
}
## Scale V to N per study (assume m constant)
# /!\ crossprod instead of tcrossprod because N.vec is a one-row matrix
v.out <- cov(V.hold) / crossprod(N.vec * (sqrt(n.blocks) / m))
### Scale S and V to liability:
ratio <- tcrossprod(sqrt(Liab.S))
S <- cov * ratio
#calculate the ratio of the rescaled and original S matrices
scaleO <- gdata::lowerTriangle(ratio, diag = TRUE)
#rescale the sampling correlation matrix by the appropriate diagonals
V <- v.out * tcrossprod(scaleO)
#name traits according to trait.names argument
#use general format of V1-VX if no names provided
colnames(S) <- if (is.null(trait.names)) paste0("V", 1:ncol(S)) else trait.names
if(mean(Liab.S)!=1){
r<-nrow(S)
SE<-matrix(0, r, r)
SE[lower.tri(SE,diag=TRUE)] <-sqrt(diag(V))
.LOG(c(" ", " "), file=log.file, print = FALSE)
.LOG("Liability Scale Results", file=log.file)
for(j in 1:n.traits){
if(is.null(trait.names)){
chi1<-traits[j]
}else{chi1 <- trait.names[j]}
for(k in j:length(traits)){
if(j == k){
.LOG(" ", file=log.file, print = FALSE)
.LOG("Liability scale results for: ", chi1, file=log.file)
.LOG("Total Liability Scale h2: ", round(S[j, j], 4), " (", round(SE[j, j], 4), ")", file=log.file)
}
if(j != k){
if(is.null(trait.names)){
chi2<-traits[k]
}else{chi2 <- trait.names[k]}
.LOG("Total Liability Scale Genetic Covariance between ", chi1, " and ",
chi2, ": ", round(S[k, j], 4), " (", round(SE[k, j], 4), ")", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
}
}
}
}
if(all(diag(S) > 0)){
##calculate standardized results to print genetic correlations to log and screen
ratio <- tcrossprod(1 / sqrt(diag(S)))
S_Stand <- S * ratio
#calculate the ratio of the rescaled and original S matrices
scaleO <- gdata::lowerTriangle(ratio, diag = TRUE)
## MAke sure that if ratio in NaN (devision by zero) we put the zero back in
# -> not possible because of 'all(diag(S) > 0)'
# scaleO[is.nan(scaleO)] <- 0
#rescale the sampling correlation matrix by the appropriate diagonals
V_Stand <- V * tcrossprod(scaleO)
#enter SEs from diagonal of standardized V
r<-nrow(S)
SE_Stand<-matrix(0, r, r)
SE_Stand[lower.tri(SE_Stand,diag=TRUE)] <-sqrt(diag(V_Stand))
.LOG(c(" ", " "), file=log.file, print = FALSE)
.LOG("Genetic Correlation Results", file=log.file)
for(j in 1:n.traits){
if(is.null(trait.names)){
chi1<-traits[j]
}else{chi1 <- trait.names[j]}
for(k in j:length(traits)){
if(j != k){
if(is.null(trait.names)){
chi2<-traits[k]
}else{chi2 <- trait.names[k]}
.LOG("Genetic Correlation between ", chi1, " and ", chi2, ": ",
round(S_Stand[k, j], 4), " (", round(SE_Stand[k, j], 4), ")", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
}
}
}
}else{
warning("Your genetic covariance matrix includes traits estimated to have a negative heritability.")
.LOG("Your genetic covariance matrix includes traits estimated to have a negative heritability.", file=log.file, print = FALSE)
.LOG("Genetic correlation results could not be computed due to negative heritability estimates.", file=log.file)
}
end.time <- Sys.time()
total.time <- difftime(time1=end.time,time2=begin.time,units="sec")
mins <- floor(floor(total.time)/60)
secs <- floor(total.time-mins*60)
.LOG(" ", file=log.file, print = FALSE)
.LOG("LDSC finished running at ", end.time, file=log.file)
.LOG("Running LDSC for all files took ", mins, " minutes and ", secs, " seconds", file=log.file)
.LOG(" ", file=log.file, print = FALSE)
flush(log.file)
close(log.file)
if(stand){
list(V=V,S=S,I=I,N=N.vec,m=m,V_Stand=V_Stand,S_Stand=S_Stand)
} else {
list(V=V,S=S,I=I,N=N.vec,m=m)
}
}
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.