Appendix B: Simulation

Last updated: 2025-11-09

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Introduction

The simulated datasets look like the following:

library(fashr)
library(ggplot2)
set.seed(12345)
sigma_vec = c(0.1, 0.3, 0.5)

data_sim_list_A <- lapply(1:3, function(i) simulate_process(sd_poly = 1, type = "nondynamic", sd = sigma_vec, normalize = F))

data_sim_list_B <- lapply(1:3, function(i) simulate_process(sd_poly = 1, type = "linear", sd = sigma_vec, normalize = F))

data_sim_list_C <- lapply(1:3, function(i) simulate_process(sd_poly = 0, type = "nonlinear", sd = sigma_vec, sd_fun = 5, p = 2, normalize = F))

datasets <- c(data_sim_list_A, data_sim_list_B, data_sim_list_C)
labels <- c(rep("A", 3), rep("B", 3), rep("C", 3))
indices_A <- 1:3
indices_B <- 4:6
indices_C <- 7:9

dataset_labels <- rep(as.character(NA),9)
dataset_labels[indices_A] <- paste0("A",seq(1,length(indices_A)))
dataset_labels[indices_B] <- paste0("B",seq(1,length(indices_B)))
dataset_labels[indices_C] <- paste0("C",seq(1,length(indices_C)))
names(datasets) <- dataset_labels

par(mfrow = c(3, 3))
for(i in indices_A[1:3]){
  plot(datasets[[i]]$x, datasets[[i]]$y, 
       type = "p", col = "black", lwd = 1, pch = 20,
       xlab = "Time", ylab = "Effect Size", 
       ylim = c(min(sapply(datasets[indices_A], function(d) min(d$y - 2*d$sd))) - 0.5,
                max(sapply(datasets[indices_A], function(d) max(d$y + 2*d$sd)))) + 0.5,
       main = paste("Category A: ", i))
  lines(datasets[[i]]$x, datasets[[i]]$truef, col = "red", lwd = 2)
   arrows(
    datasets[[i]]$x,
    datasets[[i]]$y - 2 * datasets[[i]]$sd,
    datasets[[i]]$x,
    datasets[[i]]$y + 2 * datasets[[i]]$sd,
    length = 0.05,
    angle = 90,
    code = 3,
    col = "black"
  )
}

for(i in indices_B[1:3]){
  plot(datasets[[i]]$x, datasets[[i]]$y, type = "p", col = "black", 
       lwd = 1, pch = 20,
       xlab = "Time", ylab = "Effect Size", 
       ylim = c(min(sapply(datasets[indices_B], function(d) min(d$y - 2*d$sd))) - 0.5,
                max(sapply(datasets[indices_B], function(d) max(d$y + 2*d$sd)))) + 0.5,
       main = paste("Category B: ", i))
  lines(datasets[[i]]$x,
        datasets[[i]]$truef,
        col = "red",
        lwd = 2)
  arrows(
    datasets[[i]]$x,
    datasets[[i]]$y - 2 * datasets[[i]]$sd,
    datasets[[i]]$x,
    datasets[[i]]$y + 2 * datasets[[i]]$sd,
    length = 0.05,
    angle = 90,
    code = 3,
    col = "black"
  )
}

for(i in indices_C[1:3]){
  plot(datasets[[i]]$x, datasets[[i]]$y, type = "p", 
       col = "black", lwd = 1, pch = 20,
       xlab = "Time", ylab = "Effect Size", 
       ylim = c(min(sapply(datasets[indices_C], function(d) min(d$y - 2*d$sd))) - 0.5,
                max(sapply(datasets[indices_C], function(d) max(d$y + 2*d$sd)))) + 0.5,
       main = paste("Category C: ", i))
  lines(datasets[[i]]$x,
        datasets[[i]]$truef,
        col = "red",
        lwd = 2)
  arrows(
    datasets[[i]]$x,
    datasets[[i]]$y - 2 * datasets[[i]]$sd,
    datasets[[i]]$x,
    datasets[[i]]$y + 2 * datasets[[i]]$sd,
    length = 0.05,
    angle = 90,
    code = 3,
    col = "black"
  )
}

par(mfrow = c(1, 1))

Define the functions to be used for simulation:

get_one_set_of_datasets <- function(J, pho0 = 0.1, pho1 = 0.05, sigma_vec = c(0.05, 0.1, 0.2)){
  # check if pho0 > pho1
  if(pho0 <= pho1){
    stop("pho0 must be greater than pho1")
  }
  propA <- 1 - pho0
  propB <- pho0 - pho1
  propC <- pho1
  
  sizeA <- J * propA
  data_sim_list_A <- lapply(1:sizeA, function(i) simulate_process(sd_poly = 1, type = "nondynamic", sd = sigma_vec, normalize = F))
  
  sizeB <- J * propB
  if(sizeB > 0){
    data_sim_list_B <- lapply(1:sizeB, function(i) simulate_process(sd_poly = 1, type = "linear", sd = sigma_vec, normalize = F))
  }else{
    data_sim_list_B <- list()
  }
  
  sizeC <- J * propC
  data_sim_list_C <- lapply(1:sizeC, function(i) simulate_process(sd_poly = 0, type = "nonlinear", sd = sigma_vec, sd_fun = 5, p = 2, normalize = F))
  
  datasets <- c(data_sim_list_A, data_sim_list_B, data_sim_list_C)
  labels <- c(rep("A", sizeA), rep("B", sizeB), rep("C", sizeC))
  indices_A <- 1:sizeA
  indices_B <- (sizeA + 1):(sizeA + sizeB)
  indices_C <- (sizeA + sizeB + 1):(sizeA + sizeB + sizeC)
  
  dataset_labels <- rep(as.character(NA),100)
  dataset_labels[indices_A] <- paste0("A",seq(1,length(indices_A)))
  dataset_labels[indices_B] <- paste0("B",seq(1,length(indices_B)))
  dataset_labels[indices_C] <- paste0("C",seq(1,length(indices_C)))
  names(datasets) <- dataset_labels
  
  return(datasets)
}

get_result_once <- function(J, pho0 = 0.1, pho1 = 0.05, sigma_vec = c(0.05, 0.1, 0.2), 
                            grid = sort(c(0, exp(-0.5*seq(0,10, by = 0.2)))),
                            penalty = 10, num_basis = 20, num_cores = 1){
  
  pi00 <- 1 - pho0
  pi01 <- 1 - pho1
  
  datasets <- get_one_set_of_datasets(J, pho0, pho1, sigma_vec)
  
  fash_fit1 <- fash(Y = "y", smooth_var = "x", S = "sd", data_list = datasets, order = 1,
                    verbose = FALSE, num_cores = num_cores,
                    grid = grid, num_basis = num_basis, penalty = penalty)
  fash_fit2 <- fash(Y = "y", smooth_var = "x", S = "sd", data_list = datasets, order = 2,
                    verbose = FALSE, num_cores = num_cores,
                    grid = grid, num_basis = num_basis, penalty = penalty)
  
  hat_pi_00 <- fash_fit1$prior_weights$prior_weight[1]
  hat_pi_01 <- fash_fit2$prior_weights$prior_weight[1]
  
  fash_fit1_update <- BF_update(fash_fit1, plot = FALSE)
  fash_fit2_update <- BF_update(fash_fit2, plot = FALSE)
  
  tilde_pi_00 <- fash_fit1_update$prior_weights$prior_weight[1]
  tilde_pi_01 <- fash_fit2_update$prior_weights$prior_weight[1]
  
  data.frame(pi_00 = pi00, pi_01 = pi01,
             hat_pi_00 = hat_pi_00, hat_pi_01 = hat_pi_01,
             tilde_pi_00 = tilde_pi_00, tilde_pi_01 = tilde_pi_01)
  
}

Simulation

Dense grid

In the first setting, consider a relatively dense grid:

## Setting A:
set.seed(12345)
pho_vec <- seq(0.05, 0.5, by = 0.01)

result_all <- lapply(pho_vec, function(pho0){
  pho1 <- pho0 / 2
  get_result_once(J = 300, pho0 = pho0, pho1 = pho1, sigma_vec = c(0.1, 0.3, 0.5), 
                  grid = sort(c(0, exp(-0.5*seq(0,10, by = 0.1)))),
                  penalty = 1, num_cores = 5,
                  num_basis = 20)
})
result_df <- do.call(rbind, result_all)
save(result_df, file = "data/simulation_result_denser_grid.RData")

load("data/appendixB/simulation_result_denser_grid.RData")

par(mfrow = c(1, 2))
plot(result_df$pi_00, result_df$hat_pi_00, 
     xlab = expression("True " * pi[0]), 
     ylab = expression("Estimated " * pi[0]),
     main = expression("Estimated vs True " * pi[0] * " (Order 1)"),
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.5,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_00, result_df$tilde_pi_00, pch = 17, col = rgb(0,0,1,0.5))
legend("topleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
plot(result_df$pi_01, result_df$hat_pi_01, 
     xlab = expression("True " * pi[0]), 
     ylab = expression("Estimated " * pi[0]),
     main = expression("Estimated vs True " * pi[0] * " (Order 2)"),
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.75,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_01, result_df$tilde_pi_01, pch = 17, col = rgb(0,0,1,0.5))

# legend("bottomleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
par(mfrow = c(1, 1))

pdf("output/appendixB/simulation_result_denser_grid.pdf", width = 5, height = 5)
plot(result_df$pi_00, result_df$hat_pi_00, 
     xlab = expression("True " * pi[0]), 
     ylab = expression("Estimated " * pi[0]),
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.5,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_00, result_df$tilde_pi_00, pch = 17, col = rgb(0,0,1,0.5))
legend("topleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
dev.off()

pdf("output/appendixB/simulation_result_denser_grid_order2.pdf", width = 5, height = 5)
plot(result_df$pi_01, result_df$hat_pi_01, 
     xlab = expression("True " * pi[0]), 
     ylab = expression("Estimated " * pi[0]),
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.75,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_01, result_df$tilde_pi_01, pch = 17, col = rgb(0,0,1,0.5))
# legend("bottomleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
dev.off()

Loose grid

In the second setting, consider a relatively loose grid:

## Setting B:
set.seed(12345)
pho_vec <- seq(0.05, 0.5, by = 0.01)

result_all <- lapply(pho_vec, function(pho0){
  pho1 <- pho0 / 2
  get_result_once(J = 300, pho0 = pho0, pho1 = pho1, sigma_vec = c(0.1, 0.3, 0.5), 
                  grid = sort(c(0, exp(-0.5*seq(0,10, by = 0.2)))),
                  penalty = 1, num_cores = 5,
                  num_basis = 20)
})
result_df <- do.call(rbind, result_all)
save(result_df, file = "data/simulation_result_dense_grid.RData")

load("data/appendixB/simulation_result_dense_grid.RData")

par(mfrow = c(1, 2))
plot(result_df$pi_00, result_df$hat_pi_00, 
     xlab = expression("True " * pi[0]), 
     ylab = expression("Estimated " * pi[0]),
     main = expression("Estimated vs True " * pi[0] * " (Order 1)"),
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.5,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_00, result_df$tilde_pi_00, pch = 17, col = rgb(0,0,1,0.5))
legend("topleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
plot(result_df$pi_01, result_df$hat_pi_01, 
     xlab = expression("True " * pi[0]), 
     ylab = expression("Estimated " * pi[0]),
     main = expression("Estimated vs True " * pi[0] * " (Order 2)"),
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.75,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_01, result_df$tilde_pi_01, pch = 17, col = rgb(0,0,1,0.5))

# legend("bottomleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
par(mfrow = c(1, 1))

pdf("output/appendixB/simulation_result_dense_grid.pdf", width = 5, height = 5)
plot(result_df$pi_00, result_df$hat_pi_00, 
     xlab = "True pi0", 
     ylab = "Estimated pi0", 
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.5,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_00, result_df$tilde_pi_00, pch = 17, col = rgb(0,0,1,0.5))
legend("topleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
dev.off()

pdf("output/appendixB/simulation_result_dense_grid_order2.pdf", width = 5, height = 5)
plot(result_df$pi_01, result_df$hat_pi_01, 
     xlab = "True pi0", 
     ylab = "Estimated pi0", 
     pch = 16, col = rgb(0,0,0,0.5), ylim = c(0,1), xlim = c(0.75,1))
abline(0,1,col='red',lty=2, lwd = 2)
points(result_df$pi_01, result_df$tilde_pi_01, pch = 17, col = rgb(0,0,1,0.5))
# legend("bottomleft", legend = c("Before BF update", "After BF update"), pch = c(16,17), col = c(rgb(0,0,0,0.5), rgb(0,0,1,0.5)), bty = "n")
dev.off()

Focusing on one particular replication

We will fix \(\pi_0 = 0.2\) and \(\pi_1 = 0.1\), and focus on one particular replication to illustrate the inference using FASH.

set.seed(12345)
J = 1200; pho0 = 0.2; pho1 = 0.1;
datasets <- get_one_set_of_datasets(J = J, pho0 = pho0, pho1 = pho1, sigma_vec = c(0.1, 0.3, 0.5))

log_prec <- seq(0,10, by = 0.2)
fine_grid <- sort(c(0, exp(-0.5*log_prec)))
num_cores <- 4

fash_fit1 <- fash(Y = "y", smooth_var = "x", S = "sd", data_list = datasets,
                  num_basis = 20, order = 1, betaprec = 0,
                  pred_step = 1, penalty = 10, grid = fine_grid,
                  num_cores = num_cores, verbose = TRUE)
fash_fit1_update <- BF_update(fash_fit1)


fash_fit2 <- fash(Y = "y", smooth_var = "x", S = "sd", data_list = datasets,
                  num_basis = 20, order = 2, betaprec = 0,
                  pred_step = 1, penalty = 10, grid = fine_grid,
                  num_cores = num_cores, verbose = TRUE)
fash_fit2_update <- BF_update(fash_fit2)

save(fash_fit1, fash_fit1_update, fash_fit2, fash_fit2_update, file = "data/appendixB/fash_fit_example.RData")

load("data/appendixB/fash_fit_example.RData")

Testing dynamic eQTLs:

We will first focus on testing dynamic eQTLs:

alpha <- 0.05
test1 <- fdr_control(fash_fit1, alpha = alpha, plot = F)

210 datasets are significant at alpha level 0.05. Total datasets tested: 1200.

test1_corrected <- fdr_control(fash_fit1_update, alpha = alpha, plot = F)

205 datasets are significant at alpha level 0.05. Total datasets tested: 1200.

What datasets are called significant?

alpha_vec = seq(0.01, 0.2, by = 0.01)
FDR0 <- c(); FDR0_corrected <- c()
Power0 <- c(); Power0_corrected <- c()

for (alpha in alpha_vec) {
  index1 <- test1$fdr_results$index[test1$fdr_results$FDR <= alpha]
  index1_corrected <- test1_corrected$fdr_results$index[test1_corrected$fdr_results$FDR <= alpha]
  
  # True FDR
  FDR0 <- c(FDR0, mean(index1 <= (J * (1 - pho0))))
  FDR0_corrected <- c(FDR0_corrected, mean(index1_corrected <= (J * (1 - pho0))))
  
  # Power
  Power0 <- c(Power0, sum(index1 > (J * (1 - pho0))) / (J * pho0))
  Power0_corrected <- c(Power0_corrected, sum(index1_corrected > (J * (1 - pho0))) / (J * pho0))
}

pdf("output/appendixB/power_fdr_order1.pdf", width = 5, height = 5)
# Power plot
plot(alpha_vec, Power0, type = "o", pch = 16, col = "blue",
     lty = 1, lwd = 1.5, 
     xlab = expression(alpha), ylab = "Power", ylim = c(0.75,0.9),
     # main = "Power vs alpha (Order 1)")
     )
points(alpha_vec, Power0_corrected, type = "o", pch = 17, col = "red",
       lty = 2, lwd = 1.5)
legend("bottomright", 
       legend = c("Before BF update", "After BF update"),
       pch = c(16,17), col = c("blue", "red"), lty = c(1,2), bty = "n")
dev.off()

# FDR plot
pdf("output/appendixB/fdr_plot_order1.pdf", width = 5, height = 5)
plot(alpha_vec, FDR0, type = "o", pch = 16, col = "blue",
     lty = 1, lwd = 1.5,
     xlab = expression(alpha), ylab = "true FDR", 
     ylim = c(0,0.3),
     # main = "FDR vs alpha (Order 1)"
     )
points(alpha_vec, FDR0_corrected, type = "o", pch = 17, col = "red",
       lty = 2, lwd = 1.5)
abline(0,1,col='black',lty=2, lwd = 1)
legend("topleft", 
       legend = c("Before BF update", "After BF update"),
       pch = c(16,17), col = c("blue", "red"), lty = c(1,2), bty = "n")
dev.off()

Showing the cumulative FDR plot:

lfdr <- fash_fit1$posterior_weights[,1]
sizeA <- J * (1 - pho0); sizeB <- J * (pho0 - pho1); sizeC <- J * pho1
fdr_df <- data.frame(eQTL = 1:length(lfdr), fdr = lfdr, type = rep(c("A", "B", "C"), times = c(sizeA, sizeB, sizeC)))
fdr_df <- fdr_df[order(fdr_df$fdr), ] # ordering it
fdr_df$cumulative_fdr <- cumsum(fdr_df$fdr)/seq_along(fdr_df$fdr)
fdr_df$rank <- 1:length(lfdr)

ggplot(fdr_df, aes(x = 1:length(lfdr), y = cumulative_fdr, col = type)) +
  geom_point() +
  geom_hline(yintercept = 0.05, linetype = "dashed", color = "purple") +
  labs(x = "Ordered eQTLs", y = "Cumulative FDR", col = "Type") +
  theme_minimal() +
  # ggtitle("Cumulative FDR Plot") +
  scale_color_manual(values = c("red", "blue", "green")) +
  theme(
    axis.title = element_text(size = 16),   # xlab size
    axis.text  = element_text(size = 14),   # x lim size
    legend.key.size = unit(1.2, 'lines'), # key size
    legend.title = element_text(size = 14), # legend title size
    legend.text = element_text(size = 12),  # legend text size
    legend.position = c(0.8, 0.4),            # move inside
    legend.background = element_rect(fill = alpha("white", 0.6)) # background
  )

ggsave("output/appendixB/cumulative_fdr_order1.pdf", width = 5, height = 5)

lfdr <- fash_fit1_update$posterior_weights[,1]
fdr_df <- data.frame(eQTL = 1:length(lfdr), fdr = lfdr, type = rep(c("A", "B", "C"), times = c(sizeA, sizeB, sizeC)))
fdr_df <- fdr_df[order(fdr_df$fdr), ] # ordering it
fdr_df$cumulative_fdr <- cumsum(fdr_df$fdr)/seq_along(fdr_df$fdr)
fdr_df$rank <- 1:length(lfdr)

ggplot(fdr_df, aes(x = 1:length(lfdr), y = cumulative_fdr, col = type)) +
  geom_point() +
  geom_hline(yintercept = 0.05, linetype = "dashed", color = "purple") +
  labs(x = "Ordered eQTLs", y = "Cumulative FDR", col = "Type") +
  theme_minimal() +
  # ggtitle("Cumulative FDR Plot") +
  scale_color_manual(values = c("red", "blue", "green")) +
  theme(
    axis.title = element_text(size = 16),   # xlab size
    axis.text  = element_text(size = 14),   # x lim size
    legend.key.size = unit(1.2, 'lines'), # key size
    legend.title = element_text(size = 14), # legend title size
    legend.text = element_text(size = 12),  # legend text size
    legend.position = c(0.8, 0.4),            # move inside
    legend.background = element_rect(fill = alpha("white", 0.6)) # background
  )

ggsave("output/appendixB/cumulative_fdr_order1_corrected.pdf", width = 5, height = 5)

A few examples of most significant datasets:

set.seed(1234)
most_significant_indices <- sample(test1_corrected$fdr_results$index[test1_corrected$fdr_results$FDR <= 0.05], 4)
pdf("output/appendixB/fitted_curves_order1.pdf", width = 10, height = 8)
par(mfrow = c(2, 2))
for (i in most_significant_indices) {
  fitted_result <- predict(fash_fit1_update,
                           index = i,
                           smooth_var = seq(0, 16, by = 0.1))
  
  plot(datasets[[i]]$x, datasets[[i]]$y, type = "p", col = "black", 
       lwd = 1, pch = 20,
       # increase font size
       cex.axis = 1.5, cex.lab = 1.5, cex.main = 1.5,
       xlab = "Time", ylab = "Effect Size", 
       ylim = c(min(datasets[[i]]$y) - 0.5,
                max(datasets[[i]]$y) + 0.5),
       main = paste("Dataset:", names(datasets)[i]))
  lines(datasets[[i]]$x,
        datasets[[i]]$truef,
        col = "blue",
        lwd = 1, lty = 2)
  arrows(
    datasets[[i]]$x,
    datasets[[i]]$y - 2 * datasets[[i]]$sd,
    datasets[[i]]$x,
    datasets[[i]]$y + 2 * datasets[[i]]$sd,
    length = 0.05,
    angle = 90,
    code = 3,
    col = "black"
  )
  lines(fitted_result$x,
        fitted_result$mean,
        col = "red",
        lwd = 1.2)
  polygon(
    c(fitted_result$x, rev(fitted_result$x)),
    c(fitted_result$lower, rev(fitted_result$upper)),
    col = rgb(1, 0, 0, 0.1),
    border = NA
  )
}
par(mfrow = c(1, 1))
dev.off()

quartz_off_screen 
                2

Testing non-linear eQTLs:

We now focus on testing non-linear dynamic eQTLs:

alpha <- 0.05
test2 <- fdr_control(fash_fit2, alpha = alpha, plot = F)

106 datasets are significant at alpha level 0.05. Total datasets tested: 1200.

test2_corrected <- fdr_control(fash_fit2_update, alpha = alpha, plot = F)

98 datasets are significant at alpha level 0.05. Total datasets tested: 1200.