---
title: "hQTL analysis with IHW-Benjamini-Yekutieli"
author: "Nikos Ignatiadis"
date: "`r doc_date()`"
package: "`r pkg_ver('IHWpaper')`"
output: BiocStyle::html_document
bibliography: bibliography.bib
vignette: >
  %\VignetteIndexEntry{"Stats paper: hQTL analysis with IHW-Benjamini-Yekutieli"}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r warning=FALSE, message=FALSE}
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:

```{r}
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.
```{r}
chr1_df <- readRDS(file.path(file_loc, "chr1_subset.Rds"))
chr2_df <- readRDS(file.path(file_loc, "chr2_subset.Rds"))
```

```{r}
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:

```{r}
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.

```{r}
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:

```{r}
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.

```{r eval=FALSE}
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:

```{r}
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:

```{r}
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)
```
```{r}
m <- sum(m_groups) #total number of hypotheses
m
```

# Histogram plots

Get our colors:
```{r}
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)]
```

  
```{r}
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
```



```{r eval=FALSE}
ggsave(cov_scatter_gg, filename="cov_scatter_gg.pdf", width=4,height=3)
```


```{r}
chr1_chr2_df$cutoff_groups <- cut(chr1_chr2_df$dist, cutoffs)
table(chr1_chr2_df$cutoff_groups)
```





First let us plot the marginal histogram:

```{r}
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
```
```{r eval=FALSE}
ggsave(gg_marginal_hist, filename="gg_marginal_hist.pdf", width=4,height=3)
```


```{r}
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      
```

```{r eval=FALSE}
ggsave(gg_stratified_hist, filename="gg_stratified_hist.pdf", width=7,height=3)
```


# 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:

```{r}
alpha <- .01/(log(m)+1)
```

Now let us run the IHW procedure:
```{r}
ihw_chr1_chr2 <- ihw(chr1_chr2_df$pvalue, chr1_chr2_groups, alpha, folds=folds_vec,
                     m_groups=m_groups, lambdas=2000)
```


Rejections of BY:

```{r}
sum(p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") <= alpha)
```

Rejections of IHW-BY:

```{r}
rejections(ihw_chr1_chr2)
```

So we see that discoveries have more than doubled.


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

```{r}
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)
```

```{r}
sum(p.adjust(chr1_chr2_df$pvalue, n = m, method="BH") <= alpha_bh)
```
```{r}
rejections(ihw_chr1_chr2_bh)
```

# Hypotheses shown in table

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

```{r}
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],]
```
```{r}
chr1_df[idx[idx_max],]
```

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

```{r}
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],]
```
```{r}
chr1_df[idx[idx_max],]
```

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

```{r}
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))
```

```{r}
ihw_chr1_chr2@df[idx[9],]
```

```{r}
chr2_df[idx[9]-nrow(chr1_df),]
```


And another one that only gets rejected in one case
```{r}
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],]
```

```{r}
chr2_df[idx[idx_max]-nrow(chr1_df),]
```

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


First get the threshold below which BY rejects:

```{r}
t_bh <- get_bh_threshold(chr1_chr2_df$pvalue, alpha, mtests = m)
t_bh
```

Next write a function to estimate the local fdr at a given threshold:
```{r}
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)
}
```

```{r}
fold_groups <- expand.grid(1:62, 1:2)
```

Precompute the below too because it takes a while:
```{r eval=FALSE}
lfdrs <- bind_rows(mapply(get_local_fdr, fold_groups[[2]], fold_groups[[1]], SIMPLIFY = FALSE))
saveRDS(lfdrs,file="hqtl_estimated_lfdrs.Rds")
```

```{r}
lfdrs <-  readRDS(file.path(file_loc, "hqtl_estimated_lfdrs.Rds"))
```

```{r}
lfdrs <- mutate(lfdrs, Chromosome=paste0("chr", fold), stratum=group, t_bh=t_bh)
```

# Start the plotting

```{r}
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)

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)

weights_panel

```
# Weights chromosome 1

```{r}
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
```
```{r eval=FALSE}
ggsave(weights_panel_1, filename="chr1_weights.pdf", width=3.5,height=2.5)
```

```{r}
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
```
```{r eval=FALSE}
ggsave(weights_panel_2, filename="chr2_weights.pdf", width=3.5,height=2.5)
```

```{r}
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
```

```{r eval=FALSE}
ggsave(thresholds_ihw_panel, filename="ihw_by_threshold.pdf", width=3.5,height=2.5)
```

```{r}
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
```
```{r}
ggsave(thresholds_bh_panel, filename="by_threshold.pdf", width=3.5,height=2.5)
```

```{r}
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
```
```{r eval=FALSE}
ggsave(lfdr_ihw_panel, filename="ihw_by_fdr.pdf", width=3.5,height=2.5)
```


```{r}
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
```

```{r eval=FALSE}
ggsave(lfdr_bh_panel, filename="by_fdr.pdf", width=3.5,height=2.5)
```




