Contents

library(dplyr)
library(IHW)
library(fdrtool)
library(ggplot2)
library(cowplot)
theme_set(theme_cowplot())
library(tidyr)
library(scales)
library(latex2exp)

file_loc <- system.file("extdata","real_data", "hqtl_chrom1_chrom2", package = "IHWpaper")

First the two tables with the p-values corresponding to the two chromosomes. Note that only p-values <= 1e-4 are stored in these.

chr1_df <- readRDS(file.path(file_loc, "chr1_subset.Rds"))
chr2_df <- readRDS(file.path(file_loc, "chr2_subset.Rds"))
pval_threshold <- 10^(-4)

Also recall each hypothesis corresponds to a peak (which we call gene below) and a SNP. Hence let us load files about each of the SNPs and peaks:

snp_chr1 <- readRDS(file.path(file_loc, "snppos_chr1.Rds"))

peaks_chr1 <- dplyr::filter(all_peaks, chr=="chr1")
peaks_chr2 <- dplyr::filter(all_peaks, chr=="chr2")

We can use these both to infer how many hypotheses were conducted in total (or at a given distance), but also to calculate our covariates which are a function of SNP and peak (their distance).

Now let us attach the new column with the covariate (distance) to the data frames.

chr1_df <- left_join(chr1_df, select(snp_chr1, snp, pos), by=(c("SNP"="snp"))) %>%
left_join(peaks_chr1, by=(c("gene"="id"))) %>%
mutate( dist = pmin( abs(pos-start), abs(pos-end)))

chr2_df <- left_join(chr2_df, select(snp_chr2, snp, pos), by=(c("SNP"="snp"))) %>%
left_join(peaks_chr2, by=(c("gene"="id"))) %>%
mutate( dist = pmin( abs(pos-start), abs(pos-end)))

Now let us convert the distance to a categorical covariate by binning:

my_breaks <- c(-1,
seq(from=10000,to=290000, by=10000) ,
seq(from=300000, to=0.9*10^6, by=100000),
seq(from=10^6, to=25.1*10^7, by=10^7))
myf1 <- cut(chr1_df$dist, my_breaks) myf2 <- cut(chr2_df$dist, my_breaks)

To apply our method despite the fact that only small p-values are available, we will count how many hypotheses there are in each of the bins. The above code is not very efficient, so we have precomputed the results and do not run the below chunk.

cnt = 0
ms <- rep(0, length(levels(myf1)))
pb = txtProgressBar(min = 0, max = nrow(peaks_chr1), initial = 0)

for (i in 1:nrow(peaks_chr1)){
setTxtProgressBar(pb,i)
start_pos <- peaks_chr1$start[i] end_pos <- peaks_chr1$end[i]
dist_vec <- pmin( abs(snp_chr1$pos - start_pos), abs(snp_chr1$pos - end_pos) )
ms <- ms + table(cut(dist_vec, my_breaks))
}

saveRDS( ms, file = "m_groups_chr1.Rds" )

cnt = 0
ms_chr2 <- table(myf2)*0
pb = txtProgressBar(min = 0, max = nrow(peaks_chr2), initial = 0)

for (i in 1:nrow(peaks_chr2)){
setTxtProgressBar(pb,i)
start_pos <- peaks_chr1$start[i] end_pos <- peaks_chr1$end[i]
dist_vec <- pmin( abs(snp_chr2$pos - start_pos), abs(snp_chr2$pos - end_pos) )
ms_chr2 <- ms_chr2 + table(cut(dist_vec, my_breaks))
}

saveRDS( ms_chr2, file = "m_groups_chr2.Rds" )

Let us load the result from the above execution:

ms_chr1 <- readRDS(file.path(file_loc, "m_groups_chr1.Rds"))
ms_chr2 <- readRDS(file.path(file_loc, "m_groups_chr2.Rds"))

Let us put the data for the two chromosomes together:

chr1_chr2_df <- rbind(chr1_df, chr2_df)
chr1_chr2_groups <- as.factor(c(myf1,myf2))
folds_vec <- as.factor(c(rep(1, nrow(chr1_df)), rep(2, nrow(chr2_df))))
m_groups <- cbind(ms_chr1, ms_chr2)
m <- sum(m_groups) #total number of hypotheses
m
## [1] 15725016812

1 Histogram plots

Get our colors:

beyonce_colors <- c("#b72da0", "#7c5bd2", "#0097ed","#00c6c3",
"#9cd78a", "#f7f7a7", "#ebab5f", "#e24344",
"#04738d")#,"#d8cdc9")
beyonce_colors[6] <- c("#dbcb09") # thicker yellow
pretty_colors <- beyonce_colors[c(2,1,3:5)]
qs <- c(0.025, 0.05)
cutoffs <- c(0, quantile(chr1_chr2_df$dist,qs), Inf) cov_scatter_gg <- ggplot(chr1_chr2_df, aes(x=rank(dist)/nrow(chr1_chr2_df), y=-log10(pvalue))) + geom_bin2d(bins=150, drop=TRUE) + # geom_point(alpha=0.2, col=pretty_colors[1]) + geom_vline(xintercept=qs, linetype="dashed") + ylab(expression(paste(-log[10],"(p-value)"))) + xlab(expression(paste("Quantile of distance"))) + scale_fill_gradientn(trans="log10", colors=alpha(pretty_colors[1], c(0.2,1))) cov_scatter_gg ggsave(cov_scatter_gg, filename="cov_scatter_gg.pdf", width=4,height=3) chr1_chr2_df$cutoff_groups <- cut(chr1_chr2_df$dist, cutoffs) table(chr1_chr2_df$cutoff_groups)
##
##        (0,1.13e+05] (1.13e+05,2.34e+06]      (2.34e+06,Inf]
##               60404               60404             2295337

First let us plot the marginal histogram:

gg_marginal_hist <- ggplot(chr1_chr2_df, aes(x=pvalue*10^4)) +
geom_histogram(aes(y=..density..), alpha=0.5, binwidth=0.05, boundary = 0, colour="black",fill=pretty_colors[1]) +
scale_x_continuous(expand = c(0.02, 0), breaks=c(0,0.5,1)) +
scale_y_continuous(expand = c(0.02, 0), limits=c(0,2.5)) +
ylab(expression(paste("Density")))+
xlab(TeX("p-value ($\\times 10^{-4}$)"))
gg_marginal_hist

ggsave(gg_marginal_hist, filename="gg_marginal_hist.pdf", width=4,height=3)
gg_stratified_hist <- ggplot(chr1_chr2_df, aes(x=pvalue*10^4)) +
geom_histogram(aes(y=..density..), alpha=0.5, binwidth=0.05, boundary = 0, colour="black",fill=pretty_colors[1]) +
scale_x_continuous(expand = c(0.02, 0), breaks=c(0,0.5,1)) +
scale_y_continuous(expand = c(0.02, 0), limits=c(0,11)) +
ylab("Density")+
xlab(TeX("p-value ($\\times 10^{-4}$)")) +
facet_grid(~cutoff_groups) +
theme(strip.background = element_blank(), strip.text.y = element_blank()) +
theme(panel.spacing = unit(2, "lines"))
gg_stratified_hist      

ggsave(gg_stratified_hist, filename="gg_stratified_hist.pdf", width=7,height=3)

2 Apply IHW-BY and BY

We want to apply the Benjamini-Yekutieli at alpha=0.1, thus we will apply Benjamini-Hochberg at the corrected level:

alpha <- .01/(log(m)+1)

Now let us run the IHW procedure:

ihw_chr1_chr2 <- ihw(chr1_chr2_df$pvalue, chr1_chr2_groups, alpha, folds=folds_vec, m_groups=m_groups, lambdas=2000) Rejections of BY: sum(p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") <= alpha)
## [1] 9110

Rejections of IHW-BY:

rejections(ihw_chr1_chr2)
## [1] 21903

So we see that discoveries have more than doubled.

What if we had applied BH and IHW-BH instead of BY and IHW-BY?

alpha_bh <- 0.01
ihw_chr1_chr2_bh <- ihw(chr1_chr2_df$pvalue, chr1_chr2_groups, alpha_bh, folds=folds_vec, m_groups=m_groups, lambdas=2000) sum(p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") <= alpha_bh)
## [1] 19055
rejections(ihw_chr1_chr2_bh)
## [1] 52488

3 Hypotheses shown in table

For our table we show one hypothesis on Chromosome 1 that gets rejected both times (by BH and IHW):

idx <- which(rejected_hypotheses(ihw_chr1_chr2) & (p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") > alpha) & (covariates(ihw_chr1_chr2)==3) & (ihw_chr1_chr2@df$fold == 1))
idx_max <- which.max(pvalues(ihw_chr1_chr2)[idx])

ihw_chr1_chr2@df[idx[idx_max],]
## [1] pvalue          adj_pvalue      weight          weighted_pvalue
## [5] group           covariate       fold
## <0 rows> (or 0-length row.names)
chr1_df[idx[idx_max],]
## # A tibble: 0 Ã— 11
## # â„¹ 11 variables: SNP <chr>, gene <int>, beta <dbl>, tstat <dbl>, pvalue <dbl>,
## #   FDR <dbl>, pos <int>, chr <chr>, start <int>, end <int>, dist <int>

We show one hypothesis on Chromosome 1 that gets weight 0:

idx <- which( !rejected_hypotheses(ihw_chr1_chr2) & (p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") > alpha) & (covariates(ihw_chr1_chr2)==15) & (ihw_chr1_chr2@df$fold == 1))
idx_max <- which.max(pvalues(ihw_chr1_chr2)[idx])

ihw_chr1_chr2@df[idx[idx_max],]
## [1] pvalue          adj_pvalue      weight          weighted_pvalue
## [5] group           covariate       fold
## <0 rows> (or 0-length row.names)
chr1_df[idx[idx_max],]
## # A tibble: 0 Ã— 11
## # â„¹ 11 variables: SNP <chr>, gene <int>, beta <dbl>, tstat <dbl>, pvalue <dbl>,
## #   FDR <dbl>, pos <int>, chr <chr>, start <int>, end <int>, dist <int>

Next we find a hypothesis which gets rejected in both cases from Chr2 :

idx <- which( rejected_hypotheses(ihw_chr1_chr2) & (p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") <= alpha) & (covariates(ihw_chr1_chr2)==3) & (ihw_chr1_chr2@df$fold == 2))
ihw_chr1_chr2@df[idx[9],]
##    pvalue adj_pvalue weight weighted_pvalue group covariate fold
## NA     NA         NA     NA              NA  <NA>      <NA> <NA>
chr2_df[idx[9]-nrow(chr1_df),]
## # A tibble: 1 Ã— 11
##   SNP    gene  beta tstat pvalue   FDR   pos chr   start   end  dist
##   <chr> <int> <dbl> <dbl>  <dbl> <dbl> <int> <chr> <int> <int> <int>
## 1 <NA>     NA    NA    NA     NA    NA    NA <NA>     NA    NA    NA

And another one that only gets rejected in one case

idx <- which( rejected_hypotheses(ihw_chr1_chr2) & (p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") > alpha) & (covariates(ihw_chr1_chr2)==1) & (ihw_chr1_chr2@df$fold == 2))
idx_max <- which.max(pvalues(ihw_chr1_chr2)[idx])

ihw_chr1_chr2@df[idx[idx_max],]
## [1] pvalue          adj_pvalue      weight          weighted_pvalue
## [5] group           covariate       fold
## <0 rows> (or 0-length row.names)
chr2_df[idx[idx_max]-nrow(chr1_df),]
## # A tibble: 0 Ã— 11
## # â„¹ 11 variables: SNP <chr>, gene <int>, beta <dbl>, tstat <dbl>, pvalue <dbl>,
## #   FDR <dbl>, pos <int>, chr <chr>, start <int>, end <int>, dist <int>

4 Set out to do the weight/threshold/lfdr plots

First get the threshold below which BY rejects:

t_bh <- get_bh_threshold(chr1_chr2_df$pvalue, alpha, mtests = m) t_bh ## [1] 2.363825e-10 Next write a function to estimate the local fdr at a given threshold: get_local_fdr <- function(fold, group){ idx <- (chr1_chr2_groups == group) & (folds_vec == fold) pvals <- sort(chr1_chr2_df$pvalue[idx])
m_true <-  m_groups[group,fold]
gren <- IHW:::presorted_grenander(pvals, m_true)
myt <- thresholds(ihw_chr1_chr2, levels_only=TRUE)[group,fold]
id_ihw_myt <- which(myt < gren$x.knots)[1] local_fdr_ihw <- ifelse(myt == 0, 0, 1/gren$slope.knots[id_ihw_myt-1])

id_bh_thresh <- which(t_bh < gren$x.knots)[1] local_fdr_bh <- 1/gren$slope.knots[id_bh_thresh-1]

pi0 <- (m_true - length(pvals))/(1-10^(-4))/m_true
data.frame(fold=fold, group=group, pi0=pi0, t_ihw=myt,
local_fdr_ihw =  local_fdr_ihw,
local_fdr_bh = local_fdr_bh)
}
fold_groups <- expand.grid(1:62, 1:2)

Precompute the below too because it takes a while:

lfdrs <- bind_rows(mapply(get_local_fdr, fold_groups[[2]], fold_groups[[1]], SIMPLIFY = FALSE))
saveRDS(lfdrs,file="hqtl_estimated_lfdrs.Rds")
lfdrs <-  readRDS(file.path(file_loc, "hqtl_estimated_lfdrs.Rds"))
lfdrs <- mutate(lfdrs, Chromosome=paste0("chr", fold), stratum=group, t_bh=t_bh)

5 Start the plotting

breaks <-   my_breaks/10^3
breaks <- breaks[-1]
break_min <- 3000/10^3
breaks_left <- c(break_min,breaks[-length(breaks)])
stratum <- 1:62
step_df_weight <- data.frame(stratum=stratum, chr2=weights(ihw_chr1_chr2,levels_only=TRUE)[,1],
chr1=weights(ihw_chr1_chr2, levels_only=TRUE)[,2] ) %>%
gather(Chromosome, weight , -stratum)

step_df_threshold <- data.frame(stratum=stratum,
chr2=thresholds(ihw_chr1_chr2,levels_only=TRUE)[,2],
chr1=thresholds(ihw_chr1_chr2, levels_only=TRUE)[,1] ) %>%
gather(Chromosome, threshold , -stratum)

step_df <- left_join(step_df_weight, step_df_threshold) %>% left_join(lfdrs)
## Joining with by = join_by(stratum, Chromosome)
## Joining with by = join_by(stratum, Chromosome)
step_df <- step_df %>% mutate(break_left = breaks_left[stratum],
break_right = breaks[stratum],
break_ratio = break_right/break_left,
break_left_init = break_left,
break_right_init = break_right,
break_left =break_left * break_ratio^.2,
break_right = break_right *break_ratio^(-.2))

stratum_fun <- function(df, colname="weight"){
stratum <- df$stratum weight <- df[[colname]] stratum_left <- stratum[stratum != length(stratum)] weight_left <- weight[stratum_left] break_left <- df$break_right[stratum_left]
stratum_right <- stratum[stratum != 1]
weight_right <- weight[stratum_right]
break_right <- df\$break_left[stratum_right]
data.frame(stratum_left= stratum_left, weight_left= weight_left,
stratum_right = stratum_right, weight_right = weight_right,
break_left = break_left, break_right = break_right)
}

connecting_df_weights <- step_df %>% group_by(Chromosome) %>%
do(stratum_fun(.)) %>%
mutate(dashed = factor(ifelse(abs(weight_left - weight_right) > 0.5 , TRUE, FALSE),
levels=c(FALSE,TRUE)))

weights_panel <- ggplot(step_df, aes(x=break_left, xend=break_right,y=weight, yend=weight, col=Chromosome)) +
geom_segment(size=0.8)+
geom_segment(data= connecting_df_weights, aes(x=break_left, xend=break_right,
y=weight_left, yend=weight_right,
linetype=dashed),
size=0.8)+
scale_x_log10(breaks=c(10^4, 10^5,10^6,10^7,10^8),
labels = trans_format("log10", math_format(10^.x))) +
xlab("Genomic distance (bp)")+
ylab("Weight")+
theme(legend.position=c(0.8,0.6)) +
theme(plot.margin = unit(c(2, 1.5, 1, 2.5), "lines"))+
theme(axis.title = element_text(face="bold" ))+
scale_color_manual(values=pretty_colors)+
guides(linetype=FALSE)
## Warning: A numeric legend.position argument in theme() was deprecated in ggplot2
## 3.5.0.
## â„¹ Please use the legend.position.inside argument of theme() instead.
## This warning is displayed once every 8 hours.
## Call lifecycle::last_lifecycle_warnings() to see where this warning was
## generated.
## Warning: The <scale> argument of guides() cannot be FALSE. Use "none" instead as
## of ggplot2 3.3.4.
## This warning is displayed once every 8 hours.
## Call lifecycle::last_lifecycle_warnings() to see where this warning was
## generated.
weights_panel

# Weights chromosome 1

weights_panel_1 <- ggplot(filter(step_df, Chromosome == "chr1"), aes(x=break_left, xend=break_right,y=weight, yend=weight, col=Chromosome)) +
geom_segment(size=0.8,lineend="round")+
geom_segment(data= filter(connecting_df_weights, Chromosome=="chr1"), aes(x=break_left, xend=break_right,
y=weight_left, yend=weight_right,
linetype=dashed),
size=0.8,lineend="round")+
scale_x_log10(breaks=c(10, 10^2,10^3,10^4),
labels = trans_format("log10", math_format(10^.x))) +
scale_y_continuous(breaks=c(0,1000,2000))+
xlab(expression(paste("Distance (kbp)")))+
ylab(expression(paste("Weight")))+
theme(legend.position="none") +
theme(plot.margin = unit(c(2, 1.5, 1, 2.5), "lines"))+
theme(axis.title = element_text(face="bold" ))+
scale_color_manual(values=pretty_colors)+
guides(linetype=FALSE)

weights_panel_1

ggsave(weights_panel_1, filename="chr1_weights.pdf", width=3.5,height=2.5)
weights_panel_2 <- ggplot(filter(step_df, Chromosome == "chr2"), aes(x=break_left, xend=break_right,y=weight, yend=weight, col=Chromosome)) +
geom_segment(size=0.8, lineend="round")+
geom_segment(data= filter(connecting_df_weights, Chromosome=="chr2"), aes(x=break_left, xend=break_right,
y=weight_left, yend=weight_right,
linetype=dashed),
size=0.8, lineend="round")+
scale_x_log10(breaks=c(10, 10^2,10^3,10^4),
labels = trans_format("log10", math_format(10^.x))) +
scale_y_continuous(breaks=c(0,1000,2000))+
xlab(expression(paste("Distance (kbp)")))+
ylab(expression(paste("Weight")))+
theme(legend.position="none") +
theme(plot.margin = unit(c(2, 1.5, 1, 2.5), "lines"))+
theme(axis.title = element_text(face="bold" ))+
scale_color_manual(values=pretty_colors)+
guides(linetype=FALSE)

weights_panel_2

ggsave(weights_panel_2, filename="chr2_weights.pdf", width=3.5,height=2.5)
connecting_df_thresholds_ihw <- step_df %>% group_by(Chromosome) %>%
do(stratum_fun(., colname="t_ihw")) %>%
mutate(dashed = FALSE)#factor(ifelse(abs(weight_left - weight_right) > 10^{-7} , TRUE, FALSE),
# levels=c(FALSE,TRUE)))

thresholds_ihw_panel <- ggplot(step_df, aes(x=break_left, xend=break_right,y=t_ihw*10^6, yend=t_ihw*10^6, col=Chromosome)) +
geom_segment(size=0.8, lineend="round")+
geom_segment(data= connecting_df_thresholds_ihw, aes(x=break_left, xend=break_right,
y=weight_left*10^6, yend=weight_right*10^6,
linetype=dashed),
size=0.8, lineend="round")+
scale_x_log10(breaks=c(10, 10^2,10^3,10^4),
labels = trans_format("log10", math_format(10^.x))) +
scale_y_continuous(limits=c(0,1.8), breaks=c(0,1))+
xlab(expression(paste("Distance (kbp)")))+
ylab(expression(paste("IHW s(x) (",10^-6,")")))+
theme(legend.position=c(0.6,0.7), legend.title = element_blank()) +
theme(plot.margin = unit(c(2, 1.5, 1, 2.5), "lines"))+
theme(axis.title = element_text(face="bold" ))+
scale_color_manual(values=pretty_colors)+
guides(linetype=FALSE)

thresholds_ihw_panel

ggsave(thresholds_ihw_panel, filename="ihw_by_threshold.pdf", width=3.5,height=2.5)
connecting_df_thresholds_bh <- step_df %>% group_by(Chromosome) %>%
do(stratum_fun(., colname="t_bh")) %>%
mutate(dashed = FALSE)#factor(ifelse(abs(weight_left - weight_right) > 10^{-11} , TRUE, TRUE),
#levels=c(FALSE,TRUE)))

scientific_10 = function(x) {ifelse(x==0, "0", parse(text=gsub("[+]", "", gsub("e", " %*% 10^", scientific_format()(x)))))}
thresholds_bh_panel <- ggplot(step_df, aes(x=break_left, xend=break_right,y=10^10*t_bh, yend=10^10*t_bh, col=Chromosome)) +
geom_segment(size=0.8)+
geom_segment(data= connecting_df_thresholds_bh, aes(x=break_left, xend=break_right,
y=weight_left*10^10, yend=weight_right*10^10,
linetype=dashed),
size=0.8)+
scale_x_log10(breaks=c(10, 10^2,10^3,10^4),
labels = trans_format("log10", math_format(10^.x))) +
scale_y_continuous(limits=c(0,5), breaks=c(0,2,4))+
xlab(expression(paste("Distance (kbp)")))+
ylab(expression(paste("BY s(x) (",10^-10,")")))+
theme(legend.position=c(0.6,0.7), legend.title = element_blank()) +
theme(plot.margin = unit(c(2, 1.5, 1, 2.5), "lines"))+
theme(axis.title = element_text(face="bold" ))+
scale_color_manual(values=pretty_colors)+
guides(linetype=FALSE)

thresholds_bh_panel

ggsave(thresholds_bh_panel, filename="by_threshold.pdf", width=3.5,height=2.5)
connecting_df_lfdr_ihw <- step_df %>% group_by(Chromosome) %>%
do(stratum_fun(., colname="local_fdr_ihw")) %>%
mutate(dashed = FALSE)

lfdr_ihw_panel <- ggplot(step_df, aes(x=break_left, xend=break_right,y=10^1*local_fdr_ihw, yend=10^1*local_fdr_ihw, col=Chromosome)) +
geom_segment(size=0.8, lineend="round")+
geom_segment(data= connecting_df_lfdr_ihw, aes(x=break_left, xend=break_right,
y=10^1*weight_left, yend=10^1*weight_right,
linetype=dashed),
size=0.8,lineend="round")+
scale_x_log10(breaks=c(10, 10^2,10^3,10^4),
labels = trans_format("log10", math_format(10^.x))) +
xlab(expression(paste("Distance (kbp)")))+
ylab(expression(paste("IHW fdr(s(x) | x)")))+
theme(legend.position=c(0.6,0.7), legend.title = element_blank()) +
theme(plot.margin = unit(c(2, 1.5, 1, 2.5), "lines"))+
theme(axis.title = element_text(face="bold" ))+
scale_color_manual(values=pretty_colors)+
guides(linetype=FALSE)

lfdr_ihw_panel