hQTL analysis with IHW-Benjamini-Yekutieli
31 October 2024
Package
IHWpaper 1.34.0
Contents
library(dplyr)
library(IHW)
library(fdrtool)
library(ggplot2)
library(cowplot)
theme_set(theme_cowplot())
library(tidyr)
library(scales)
library(latex2exp)Let us start by loading in the data:
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"))
snp_chr2 <- readRDS(file.path(file_loc, "snppos_chr2.Rds"))
all_peaks <- readRDS(file.path(file_loc, "peak_locations.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] 157250168121 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 2295337First 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] 9110Rejections of IHW-BY:
rejections(ihw_chr1_chr2)## [1] 21903So 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] 19055rejections(ihw_chr1_chr2_bh)## [1] 524883 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 × 11
## # ℹ 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 × 11
## # ℹ 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 × 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 NAAnd 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 × 11
## # ℹ 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$pvalue, alpha, mtests = m)
t_bh## [1] 2.363825e-10Next write a function to estimate the local fdr at a given threshold:
get_local_fdr <- function(fold, group){
idx <- (chr1_chr2_groups == group) & (folds_vec == fold)
pvals <- sort(chr1_chr2_df$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.
## ℹ 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
ggsave(lfdr_ihw_panel, filename="ihw_by_fdr.pdf", width=3.5,height=2.5)connecting_df_lfdr_bh <- step_df %>% group_by(Chromosome) %>%
do(stratum_fun(., colname="local_fdr_bh")) %>%
mutate(dashed = FALSE)#factor(ifelse(abs(weight_left - weight_right) > 0.5*10^(-6) , TRUE, FALSE),
# levels=c(FALSE,TRUE)))
lfdr_bh_panel <- ggplot(step_df, aes(x=break_left, xend=break_right,y=local_fdr_bh, yend=local_fdr_bh, col=Chromosome)) +
geom_segment(size=0.8, lineend="round")+
geom_segment(data= connecting_df_lfdr_bh, 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_log10( labels = trans_format("log10", math_format(10^.x)))+
xlab(expression(paste("Distance (kbp)")))+
ylab(expression(paste("BY fdr(s(x) | x)")))+
theme(legend.position=c(0.6,0.4), 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_bh_panel
ggsave(lfdr_bh_panel, filename="by_fdr.pdf", width=3.5,height=2.5)