powerAnalysis.Rmd

---
title: "Conducting power analysis for eQTL mapping"
author: "Anthony Hung"
date: "2021-01-19"
output: workflowr::wflow_html
editor_options:
  chunk_output_type: console
---

# Introduction

Here, we plot power curves under the assumptions of different sample sizes for an eQTL study. We also look at eQTL effect sizes from previously published response-eQTL studies to contextualize the power curves.

Power Curve derivations based on Abhishek Sarkar's calculations: https://users.rcc.uchicago.edu/~aksarkar/hypoxia/reqtl.html

# Plot a power curve

First, we need to specify the heritability for our model.

```{r power curve}
heritability <- 0.16 # median cis-heritability based on Gusev et al. 2016, Wheeler et al. 2016
lambda <- sqrt(heritability / (1-heritability))
```

Next, we make the plot for sample sizes ranging from 3 to 100.

```{r curve}
colors <- cbPalette <- c("#999999", "#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7")
samp_size <- c(3, 10, 20, 40, 58, 100) # sample sizes for power curve
alpha <- 5e-6 #for FWER of 0.05 and a list of 10000 genes, we have an adjusted alpha (based on bonferroni correction) of 5e-6 

power_function <- function(x, n){ 
     pnorm(qnorm(alpha/2) + x * sqrt(n))
}

#The False positive rate for calling an eQTL discovered in one condition (out of two total condition) as a condition specific eQTL (a.k.a. a "response eQTL") is computed by this function.
FPR_function <- function(x, n){ 
     1 - pnorm(qnorm(alpha/2) + x * sqrt(n))
}

p <- ggplot(data.frame(x = c(0, 2)), aes(x = x)) +
     stat_function(fun = power_function, args = list(n = 3),
                      aes(colour = "3", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 3),
                      aes(colour = "3", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 10),
                      aes(colour = "10", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 10),
                      aes(colour = "10", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 30),
                      aes(colour = "30", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 30),
                      aes(colour = "30", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 58),
                      aes(colour = "58", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 58),
                      aes(colour = "58", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 100),
                      aes(colour = "100", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 100),
                      aes(colour = "100", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     
     # annotate("rect", xmin = 0.25340626, xmax = 0.2652244, ymin = -Inf, ymax = Inf, alpha = .2, fill = "blue") +
     annotate("segment", x = 0.2593153, xend = 0.2593153, y = -Inf, yend = Inf, alpha = 0.3, color = "blue", size = 1) + 
     # annotate("rect", xmin = 0.7232331, xmax = 0.8596644, ymin = -Inf, ymax = Inf, alpha = .2, fill = "purple") +
     annotate("segment", x = 0.793114125, xend = 0.793114125, y = -Inf, yend = Inf, alpha = 0.3, color = "purple", size = 1) + 
     #annotate("rect", xmin = 0.6035850, xmax = 0.6097121, ymin = -Inf, ymax = Inf, alpha = .2, fill = "brown") +
     annotate("segment", x = 0.605953025, xend = 0.605953025, y = -Inf, yend = Inf, alpha = 0.3, color = "brown", size = 1) + 
     # stat_function(fun = function(x) x^2/(1+x^2),
     #                  aes(colour = "heritability"), size = 1, alpha = 0.5) +
     scale_x_continuous(name = "Standardized Effect Size",
                              limits=c(0, 2)) +
     scale_y_continuous(name = "Power",
                           limits = c(0,1)) +
     ggtitle("eQTL Power Curves") +
     scale_colour_manual("Sample Size", breaks = c("3", "10", "30", "58", "100"), values = colors) +
     scale_linetype_manual("Curve Type", breaks = c("Power", "dynamic QTL FPR"), values = c("dynamic QTL FPR" = "dotted", "Power" = "solid")) +
     theme_bw() +
     geom_hline(yintercept = .8, linetype = "dashed", color = "red")
p
```

# Compute the SES thresholds for 80% power at each sample size

```{r finding SES}
alpha <- 5e-6 #for FWER of 0.05 (assuming 10,000 genes)
#for Power = 0.8, what is the Standardized effect size?
#n = 100
n100_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(100)
#n = 58
n58_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(58)
#n = 30
n30_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(30)
#n = 10
n10_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(10)
```

# Import outside data on eQTLs to see distributions of standardized effect sizes

After plotting our power curve, we would like to get a sense of the range of standardized effect sizes in several previously published response-eQTL papers. 

```{r ecdf and cdf functions}
plot_cdf <- function(ash_obj){
     grid_of_effects <- seq(-1, 1, length.out = 1e5)
     vector_of_percentiles <- cdf.ash(ash_obj, grid_of_effects)
     percentiles_df <- tibble(latent = vector_of_percentiles$x, percentile = as.vector(vector_of_percentiles$y))
     plot <- ggplot(percentiles_df, aes(x = latent, y = percentile)) +
               geom_point()
     print(plot)
}

get_ecdf <- function(vector_of_SES){
     SES_ecdf <- ecdf(vector_of_SES)
     print(quantile(SES_ecdf, c(.75, .95, .99)))
     print(plot(SES_ecdf, ylim = c(-2,2)))
}
```

## Michelle Ward's Hypoxia study 

```{r hypoxia}
#michelle hypoxia reQTL study
conditionA <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/A-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))
conditionB <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/B-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))
conditionC <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/C-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))
conditionD <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/D-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))


zz <- gzfile('data/poweranalysis/ward_et_al_2020/reqtls.txt.gz','rt')  
hypoxia <- as_tibble(read.table(zz, fill = TRUE, stringsAsFactors = FALSE))
library(tidyverse)

names(hypoxia) <- hypoxia %>% 
     dplyr::slice(1) %>% 
     unlist()
hypoxia <- hypoxia %>% 
     dplyr::slice(-1)



## find SNP-gene pairs that were tested in all conditions and compute allelic effect sizes
common <- sort(Reduce(intersect, list(paste0(conditionA$TEST.SNP.CHROM, conditionA$TEST.SNP.POS, " ", conditionA$PHENO),
                           paste0(conditionB$TEST.SNP.CHROM, conditionB$TEST.SNP.POS, " ", conditionB$PHENO),
                           paste0(conditionC$TEST.SNP.CHROM, conditionC$TEST.SNP.POS, " ", conditionC$PHENO),
                           paste0(conditionD$TEST.SNP.CHROM, conditionD$TEST.SNP.POS, " ", conditionD$PHENO)
                           )))
length(common)

conditionA <- conditionA %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
conditionB <- conditionB %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
conditionC <- conditionC %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
conditionD <- conditionD %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
dim(conditionA)
dim(conditionB)
dim(conditionC)
dim(conditionD)

# obtain SE for effect sizes using bootstrapping


# Plot standardized effect sizes (adjust for the sqrt(n) term in the SE equation)
hist(conditionA$zscore/sqrt(15))
hist(conditionB$zscore/sqrt(15))
hist(conditionC$zscore/sqrt(15))
hist(conditionD$zscore/sqrt(15))

#conditionA
get_ecdf(conditionA$zscore/sqrt(15))
#conditionB
get_ecdf(conditionB$zscore/sqrt(15))
#conditionC
get_ecdf(conditionC$zscore/sqrt(15))
#conditionD
get_ecdf(conditionD$zscore/sqrt(15))

# See how many genes have at least one test that meets each of the 3 SES thresholds determined above in each condition
n_egenes_ward <- matrix(NA, nrow = 4, ncol = 4)
colnames(n_egenes_ward) <- c("n=100", "n=58", "n=30", "n=10")
rownames(n_egenes_ward) <- c("conditionA", "conditionB", "conditionC", "conditionD")
i <- 1
for(condition in list(conditionA, conditionB, conditionC, conditionD)){
     j <- 1
     for(threshold in c(n100_ses_threshold, n58_ses_threshold, n30_ses_threshold, n10_ses_threshold)){
         n_egenes_ward[i,j] <- length(unique(condition %>% dplyr::filter(zscore/sqrt(15) > threshold | zscore/sqrt(15) < -threshold) %>% pull(PHENO)))
         j <- j + 1
     }
     i <- i + 1
}
n_egenes_ward
```

## Alasoo et al 2019 IFN gamma and salmonella reQTL study

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6548559/#FN5
data on Zenodo: https://zenodo.org/record/1158560

```{r ifn_salmonella}
#load in data from eQTL analysis downloaded using wget from zenodo (wget https://zenodo.org/record/1158560/files/RNA_FastQTL_SL1344_500kb_pvalues.sorted.txt.gz https://zenodo.org/record/1158560/files/RNA_FastQTL_SL1344_500kb_permuted.txt.gz https://zenodo.org/record/1158560/files/RNA_FastQTL_naive_500kb_pvalues.sorted.txt.gz https://zenodo.org/record/1158560/files/RNA_FastQTL_naive_500kb_permuted.txt.gz https://zenodo.org/record/1158560/files/RNA_FastQTL_IFNg_SL1344_500kb_pvalues.sorted.txt.gz https://zenodo.org/record/1158560/files/RNA_FastQTL_IFNg_SL1344_500kb_permuted.txt.gz https://zenodo.org/record/1158560/files/RNA_FastQTL_IFNg_500kb_pvalues.sorted.txt.gz https://zenodo.org/record/1158560/files/RNA_FastQTL_IFNg_500kb_permuted.txt.gz)

n_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_naive_500kb_pvalues.sorted.txt.gz"))

sal_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_SL1344_500kb_pvalues.sorted.txt.gz"))

IFN_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_IFNg_500kb_pvalues.sorted.txt.gz"))

salIFN_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_IFNg_SL1344_500kb_pvalues.sorted.txt.gz"))

#for sorted data, 
#V1 = gene 
#V2 = chromosome
#V3 = location of SNP
#V4 = SNP
#V5 = distance to gene from SNP?
#V6 = p-value
#v7 = slope of regression
head(n_sorted)

#We can use the p-value to obtain the zscore (pvalue -> Zscore)
#then use slopes and zscores to obtain SE (slope/zscore = SE)
n_sorted <- n_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)
sal_sorted <- sal_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)
IFN_sorted <- IFN_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)
salIFN_sorted <- salIFN_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)


# use ashr to estimate true effect sizes from the data in order to compute SES
# n_sorted_ash_out <- ash(betahat = n_sorted$V7, sebetahat = n_sorted$SE, df = 84)
# saveRDS(n_sorted_ash_out, "output/ash_power_analysis/alasoo_naive_ash_out.rds")
# sal_sorted_ash_out <- ash(betahat = sal_sorted$V7, sebetahat = sal_sorted$SE, df = 84)
# saveRDS(sal_sorted_ash_out, "output/ash_power_analysis/alasoo_sal_ash_out.rds")
# IFN_sorted_ash_out <- ash(betahat = IFN_sorted$V7, sebetahat = IFN_sorted$SE, df = 84)
# saveRDS(IFN_sorted_ash_out, "output/ash_power_analysis/alasoo_IFN_ash_out.rds")
# salIFN_sorted_ash_out <- ash(betahat = salIFN_sorted$V7, sebetahat = salIFN_sorted$SE, df = 84)
# saveRDS(salIFN_sorted_ash_out, "output/ash_power_analysis/alasoo_salIFN_ash_out.rds")
n_sorted_ash_out <- readRDS("output/ash_power_analysis/alasoo_naive_ash_out.rds")
sal_sorted_ash_out <- readRDS("output/ash_power_analysis/alasoo_sal_ash_out.rds")
IFN_sorted_ash_out <- readRDS("output/ash_power_analysis/alasoo_IFN_ash_out.rds")
salIFN_sorted_ash_out <- readRDS("output/ash_power_analysis/alasoo_salIFN_ash_out.rds")

#add columns to datasets corresponding to the SES computed from ash betas and sds
n_sorted$SES <- n_sorted_ash_out$result$PosteriorMean/n_sorted_ash_out$result$PosteriorSD
sal_sorted$SES <- sal_sorted_ash_out$result$PosteriorMean/sal_sorted_ash_out$result$PosteriorSD
IFN_sorted$SES <- IFN_sorted_ash_out$result$PosteriorMean/IFN_sorted_ash_out$result$PosteriorSD
salIFN_sorted$SES <- salIFN_sorted_ash_out$result$PosteriorMean/salIFN_sorted_ash_out$result$PosteriorSD

#naive
get_ecdf(n_sorted$SES)
get_ecdf(n_sorted_ash_out$result$PosteriorMean)

#sal
get_ecdf(sal_sorted$SES)
#IFN
get_ecdf(IFN_sorted$SES)
#sal_IFN
get_ecdf(salIFN_sorted$SES)

alpha <- 0.05/15786 #for FWER of 0.05 (assuming 15786 genes)
#for Power = 0.8, what is the SES?
#n = 100
n100_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(100)
#n = 58
n58_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(58)
#n = 30
n30_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(30)
#n = 10
n10_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(10)

# See how many genes have at least one test that meets each of the 3 SES thresholds determined above in each condition
n_egenes_alasoo <- matrix(NA, nrow = 4, ncol = 4)
colnames(n_egenes_alasoo) <- c("n=100", "n=58", "n=30", "n=10")
rownames(n_egenes_alasoo) <- c("naive", "IFN", "salmonella", "IFN + salmonella")
i <- 1
for(condition in list(n_sorted, IFN_sorted, sal_sorted, salIFN_sorted)){
     j <- 1
     for(threshold in c(n100_ses_threshold, n58_ses_threshold, n30_ses_threshold, n10_ses_threshold)){
         n_egenes_alasoo[i,j] <- length(unique(condition %>% dplyr::filter(SES > threshold | SES < -threshold) %>% pull(V1)))
         j <- j + 1
     }
     i <- i + 1
}
n_egenes_alasoo
```

## Caliskan et al 2015 Rhinovirus reQTL study

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4395341/

Data output from MatrixeQTL kindly provided by Dr. Minal Caliskan

```{r rhinovirus}
rhinovirus_data <- as.tibble(read.table("data/poweranalysis/caliskan_et_al/eQTL_results_cis_response.txt.gz", 
                              sep = "\t",
                              header = TRUE))

#obtain standard errors by using beta and t.stat
rhinovirus_data <- rhinovirus_data %>% 
     dplyr::mutate(se = beta/t.stat)

#add column to denote which condition the association comes from; separate into two datasets
rhinovirus_data <- rhinovirus_data %>% 
     mutate(condition = ifelse(grepl("_f", SNP), "f", "r"))

rhinovirus_data_f <- rhinovirus_data %>% 
     dplyr::filter(condition == "f")

rhinovirus_data_r <- rhinovirus_data %>% 
     dplyr::filter(condition == "r")

# use ashr to estimate true effect sizes from the data in order to compute SES (separately for each half of the data)
rhinovirus_f_ashr_out <- ash(betahat = rhinovirus_data_f$beta, sebetahat = rhinovirus_data_f$se, df = 96)
rhinovirus_r_ashr_out <- ash(betahat = rhinovirus_data_r$beta, sebetahat = rhinovirus_data_r$se, df = 96)

#look for the effect sizes that correspond to percentiles of interest

#add column for SES (based on ash output)
rhinovirus_data_f$SES <- rhinovirus_f_ashr_out$result$PosteriorMean/rhinovirus_f_ashr_out$result$PosteriorSD
rhinovirus_data_r$SES <- rhinovirus_r_ashr_out$result$PosteriorMean/rhinovirus_r_ashr_out$result$PosteriorSD

#uninfected
get_ecdf(rhinovirus_data_f$SES)
get_ecdf(rhinovirus_f_ashr_out$result$PosteriorMean)

#rhinovirus
get_ecdf(rhinovirus_data_r$SES)
get_ecdf(rhinovirus_r_ashr_out$result$PosteriorMean)


alpha <- 0.05/length(unique(rhinovirus_data_f$gene)) #for FWER of 0.05 (assuming 10893 genes)
#for Power = 0.8, what is the SES?
#n = 100
n100_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(100)
#n = 58
n58_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(58)
#n = 30
n30_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(30)
#n = 10
n10_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(10)

# See how many genes have at least one test that meets each of the 3 SES thresholds determined above in each condition
n_egenes_caliskan <- matrix(NA, nrow = 2, ncol = 4)
colnames(n_egenes_caliskan) <- c("n=100", "n=58", "n=30", "n=10")
rownames(n_egenes_caliskan) <- c("f", "r")
i <- 1
for(condition in list(rhinovirus_data_f, rhinovirus_data_r)){
     j <- 1
     for(threshold in c(n100_ses_threshold, n58_ses_threshold, n30_ses_threshold, n10_ses_threshold)){
         n_egenes_caliskan[i,j] <- length(unique(condition %>% dplyr::filter(SES > threshold | SES < -threshold) %>% pull(gene)))
         j <- j + 1
     }
     i <- i + 1
}
n_egenes_caliskan
```

# Number of genes meeting the SES threshold

For the supplementary table, we are interested in the number of eQTLs that meet the SES threshold in each study for each sample size.

```{r}
n_egenes_ward
n_egenes_alasoo
n_egenes_caliskan
```