1 Summary

In this set of simulations, we consider settings with both null and non-null tests with varying distribution of effect sizes under the non-null (alternative) setting. Both informative and uninformative covariates are included in the setting as described in simulations-informative-cubic.Rmd. The effect sizes for non-null tests are sampled from unimodal distributions composed of a mixture of normal distributions, as described in the original adaptive shrinkage (ASH) manuscript (Stephens, 2016).

This set of simulations is similar to those presented in simulations-uasettings.Rmd. However, t-distributed test statistics with 11 degrees of freedom are considered rather than normally distributed test statistics.

2 Workspace Setup

library(dplyr)
library(ggplot2)
library(SummarizedBenchmark)
library(parallel)

## load helper functions
for (f in list.files("../R", "\\.(r|R)$", full.names = TRUE)) {
    source(f)
}

## project data/results folders
resdir <- "results"
dir.create(resdir, showWarnings = FALSE, recursive = TRUE)

## intermediary files we create below
spiky_file <- file.path(resdir, "uasettings-t-benchmark-spiky.rds")
flattop_file <- file.path(resdir, "uasettings-t-benchmark-flattop.rds")
skew_file <- file.path(resdir, "uasettings-t-benchmark-skew.rds")
bimodal_file <- file.path(resdir, "uasettings-t-benchmark-bimodal.rds")

## number of cores for parallelization
cores <- 20
B <- 100

## define bechmarking design
bd <- initializeBenchDesign()

As described in simulations-null.Rmd, we include Scott’s FDR Regression in the analysis for simulations with Gaussian or t-distributed test statistics. Again, we include both nulltype = "empirical" and nulltype = "theoretical". Since all settings in this series of simulations use t-distributed test statistics, we include Scott’s FDR Regression in all of the comparisons.

bdplus <- bd
bdplus <- addBMethod(bdplus, "fdrreg-t",
                     FDRreg::FDRreg,
                     function(x) { x$FDR },
                     z = test_statistic,
                     features = model.matrix( ~  splines::bs(ind_covariate, df = 3) - 1),
                     nulltype = 'theoretical',
                     control = list(lambda = 0.01))
bdplus <- addBMethod(bdplus, "fdrreg-e",
                     FDRreg::FDRreg,
                     function(x) { x$FDR },
                     z = test_statistic,
                     features = model.matrix( ~  splines::bs(ind_covariate, df = 3) - 1),
                     nulltype = 'empirical',
                     control = list(lambda = 0.01))

Since all simulation settings in this case study use t-distributed test statistics, we must specify the number of degrees of freedom for ASH. We add an additional parameter to the ashq method with the corresponding degrees of freedom of the test statistics.

bdplus <- modifyBMethod(bdplus, "ashq", df = 11)

All simulation settings will share the following parameters.

m <- 20000                          # integer: number of hypothesis tests
pi0 <- pi0_cubic(0.90)              # numeric: proportion of null hypotheses
ts_dist <- rt_perturber(11)         # functional: sampling dist/noise for test stats
null_dist <- rt_2pvaluer(11)        # functional: dist to calc p-values
icovariate <- runif                 # functional: independent covariate

Simulation results will be presented excluding a subset of methods, and for certain plots (upset plots), a single alpha cutoff will be used.

excludeSet <- c("unadjusted", "bl-df02", "bl-df04", "bl-df05")
ualpha <- 0.05

3 Spiky Setting

First, we consider the setting where the effect sizes under the alternative are distributed according to a “spiky” unimodal distribution centered around zero, as defined in the ASH simulations.

3.1 Data Simulation

es_dist <- function(n) { 2*sampler_spiky(n) }       # functional: dist of alternative test stats
seed <- 778

We next run the simulations.

if (file.exists(spiky_file)) {
    res <- readRDS(spiky_file)
} else {
    res <- mclapply(X = 1:B, FUN = simIteration, bench = bdplus, m = m,
                    pi0 = pi0, es_dist = es_dist, icovariate = icovariate,
                    ts_dist = ts_dist, null_dist = null_dist,
                    seed = seed, mc.cores = cores)
    saveRDS(res, file = spiky_file)
}
res_i <- lapply(res, `[[`, "informative")
res_u <- lapply(res, `[[`, "uninformative")

3.2 Covariate Diagnostics

Here, we show the relationship between the independent covariate and p-values for a single replication of the experiment.

onerun <- simIteration(bdplus, m = m, pi0 = pi0, es_dist = es_dist, ts_dist = ts_dist,
                       icovariate = icovariate, null_dist = null_dist, execute = FALSE)
rank_scatter(onerun, pvalue = "pval", covariate = "ind_covariate")

strat_hist(onerun, pvalue = "pval", covariate = "ind_covariate", maxy = 10, numQ = 3)
## 
## Attaching package: 'cowplot'
## The following object is masked from 'package:ggplot2':
## 
##     ggsave

3.3 Benchmark Metrics

We plot the averaged results across 100 replications.

resdf <- plotsim_standardize(res_i, alpha = seq(0.01, 0.10, 0.01))

plotsim_average(resdf, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE)

We also take a look at the distribution of rejects for each method as a function of the effect size and independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "effect_size")

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate")

Finally, (if enough methods produce rejections at 0.05) we take a look at the overlap of rejections between methods.

if (numberMethodsReject(resdf, alphacutoff = ualpha, filterSet = excludeSet) >= 3) {
    aggupset(res_i, alpha = ualpha, supplementary = FALSE, return_list = FALSE)
} else {
    message("Not enough methods found rejections at alpha ", ualpha, 
            "; skipping upset plot")
}

We also compare the simulation results with and without an informative covariate.

resdfu <- plotsim_standardize(res_u, alpha = seq(0.01, 0.10, 0.01))

resdfiu <- dplyr::full_join(select(resdf, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            select(resdfu, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            by = c("rep", "blabel", "param.alpha", "key",
                                   "performanceMetric", "alpha"),
                            suffix = c(".info", ".uninfo"))
resdfiu <- dplyr::mutate(resdfiu, value = value.info - value.uninfo)

plotsim_average(resdfiu, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

4 Flat-Top Setting

Next, we consider the setting where the effect sizes under the alternative are distributed according to a “flat top” unimodal distribution centered around zero, as defined in the ASH simulations.

4.1 Data Simulation

es_dist <- function(n) { 2*sampler_flat_top(n) }       # functional: dist of alternative test stats
seed <- 980

We next run the simulations.

if (file.exists(flattop_file)) {
    res <- readRDS(flattop_file)
} else {
    res <- mclapply(X = 1:B, FUN = simIteration, bench = bdplus, m = m,
                    pi0 = pi0, es_dist = es_dist, icovariate = icovariate,
                    ts_dist = ts_dist, null_dist = null_dist,
                    seed = seed, mc.cores = cores)
    saveRDS(res, file = flattop_file)
}
res_i <- lapply(res, `[[`, "informative")
res_u <- lapply(res, `[[`, "uninformative")

4.2 Covariate Diagnostics

Here, we show the relationship between the independent covariate and p-values for a single replication of the experiment.

onerun <- simIteration(bdplus, m = m, pi0 = pi0, es_dist = es_dist, ts_dist = ts_dist,
                       icovariate = icovariate, null_dist = null_dist, execute = FALSE)
rank_scatter(onerun, pvalue = "pval", covariate = "ind_covariate")

strat_hist(onerun, pvalue = "pval", covariate = "ind_covariate", maxy = 10, numQ = 3)

4.3 Benchmark Metrics

We plot the averaged results across 100 replications.

resdf <- plotsim_standardize(res_i, alpha = seq(0.01, 0.10, 0.01))

plotsim_average(resdf, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE)

We also take a look at the distribution of rejects for each method as a function of the effect size and independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "effect_size")

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate")

Finally, (if enough methods produce rejections at 0.05) we take a look at the overlap of rejections between methods.

if (numberMethodsReject(resdf, alphacutoff = ualpha, filterSet = excludeSet) >= 3) {
    aggupset(res_i, alpha = ualpha, supplementary = FALSE, return_list = FALSE)
} else {
    message("Not enough methods found rejections at alpha ", ualpha, 
            "; skipping upset plot")
}

We also compare the simulation results with and without an informative covariate.

resdfu <- plotsim_standardize(res_u, alpha = seq(0.01, 0.10, 0.01))

resdfiu <- dplyr::full_join(select(resdf, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            select(resdfu, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            by = c("rep", "blabel", "param.alpha", "key",
                                   "performanceMetric", "alpha"),
                            suffix = c(".info", ".uninfo"))
resdfiu <- dplyr::mutate(resdfiu, value = value.info - value.uninfo)

plotsim_average(resdfiu, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

5 Skewed Setting

Next, we consider the setting where the effect sizes under the alternative are distributed according to a skewed unimodal distribution not centered at zero, as defined in the ASH simulations.

5.1 Data Simulation

es_dist <- function(n) { 2*sampler_skew(n) }       # functional: dist of alternative test stats
seed <- 206

We next run the simulations.

if (file.exists(skew_file)) {
    res <- readRDS(skew_file)
} else {
    res <- mclapply(X = 1:B, FUN = simIteration, bench = bdplus, m = m,
                    pi0 = pi0, es_dist = es_dist, icovariate = icovariate,
                    ts_dist = ts_dist, null_dist = null_dist,
                    seed = seed, mc.cores = cores)
    saveRDS(res, file = skew_file)
}
res_i <- lapply(res, `[[`, "informative")
res_u <- lapply(res, `[[`, "uninformative")

5.2 Covariate Diagnostics

Here, we show the relationship between the independent covariate and p-values for a single replication of the experiment.

onerun <- simIteration(bdplus, m = m, pi0 = pi0, es_dist = es_dist, ts_dist = ts_dist,
                       icovariate = icovariate, null_dist = null_dist, execute = FALSE)
rank_scatter(onerun, pvalue = "pval", covariate = "ind_covariate")

strat_hist(onerun, pvalue = "pval", covariate = "ind_covariate", maxy = 10, numQ = 3)

5.3 Benchmark Metrics

We plot the averaged results across 100 replications.

resdf <- plotsim_standardize(res_i, alpha = seq(0.01, 0.10, 0.01))

plotsim_average(resdf, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE)

We also take a look at the distribution of rejects for each method as a function of the effect size and independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "effect_size")

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate")

Finally, (if enough methods produce rejections at 0.05) we take a look at the overlap of rejections between methods.

if (numberMethodsReject(resdf, alphacutoff = ualpha, filterSet = excludeSet) >= 3) {
    aggupset(res_i, alpha = ualpha, supplementary = FALSE, return_list = FALSE)
} else {
    message("Not enough methods found rejections at alpha ", ualpha, 
            "; skipping upset plot")
}

We also compare the simulation results with and without an informative covariate.

resdfu <- plotsim_standardize(res_u, alpha = seq(0.01, 0.10, 0.01))

resdfiu <- dplyr::full_join(select(resdf, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            select(resdfu, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            by = c("rep", "blabel", "param.alpha", "key",
                                   "performanceMetric", "alpha"),
                            suffix = c(".info", ".uninfo"))
resdfiu <- dplyr::mutate(resdfiu, value = value.info - value.uninfo)

plotsim_average(resdfiu, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

6 Bimodal Setting

Finally, we consider the setting where the effect sizes under the alternative are distributed according to a bimodal distribution (equal mixture of two normal distributions centered at -2, 2, with variance 1), again, as defined in the ASH simulations.

6.1 Data Simulation

es_dist <- function(n) { 2*sampler_bimodal(n) }       # functional: dist of alternative test stats
seed <- 913

We next run the simulations.

if (file.exists(bimodal_file)) {
    res <- readRDS(bimodal_file)
} else {
    res <- mclapply(X = 1:B, FUN = simIteration, bench = bdplus, m = m,
                    pi0 = pi0, es_dist = es_dist, icovariate = icovariate,
                    ts_dist = ts_dist, null_dist = null_dist,
                    seed = seed, mc.cores = cores)
    saveRDS(res, file = bimodal_file)
}
res_i <- lapply(res, `[[`, "informative")
res_u <- lapply(res, `[[`, "uninformative")

6.2 Covariate Diagnostics

Here, we show the relationship between the independent covariate and p-values for a single replication of the experiment.

onerun <- simIteration(bdplus, m = m, pi0 = pi0, es_dist = es_dist, ts_dist = ts_dist,
                       icovariate = icovariate, null_dist = null_dist, execute = FALSE)
rank_scatter(onerun, pvalue = "pval", covariate = "ind_covariate")

strat_hist(onerun, pvalue = "pval", covariate = "ind_covariate", maxy = 10, numQ = 3)

6.3 Benchmark Metrics

We plot the averaged results across 100 replications.

resdf <- plotsim_standardize(res_i, alpha = seq(0.01, 0.10, 0.01))

plotsim_average(resdf, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE) 

plotsim_average(resdf, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE)

We also take a look at the distribution of rejects for each method as a function of the effect size and independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "effect_size")

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate")

Finally, (if enough methods produce rejections at 0.05) we take a look at the overlap of rejections between methods.

if (numberMethodsReject(resdf, alphacutoff = ualpha, filterSet = excludeSet) >= 3) {
    aggupset(res_i, alpha = ualpha, supplementary = FALSE, return_list = FALSE)
} else {
    message("Not enough methods found rejections at alpha ", ualpha, 
            "; skipping upset plot")
}

We also compare the simulation results with and without an informative covariate.

resdfu <- plotsim_standardize(res_u, alpha = seq(0.01, 0.10, 0.01))

resdfiu <- dplyr::full_join(select(resdf, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            select(resdfu, rep, blabel, param.alpha, key,
                                   performanceMetric, alpha, value),
                            by = c("rep", "blabel", "param.alpha", "key",
                                   "performanceMetric", "alpha"),
                            suffix = c(".info", ".uninfo"))
resdfiu <- dplyr::mutate(resdfiu, value = value.info - value.uninfo)

plotsim_average(resdfiu, met="rejections", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="FDR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TPR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

plotsim_average(resdfiu, met="TNR", filter_set = excludeSet,
                merge_ihw = TRUE, errorBars = TRUE, diffplot = TRUE)

7 Session Info

sessionInfo()
## R version 3.5.0 (2018-04-23)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: CentOS Linux 7 (Core)
## 
## Matrix products: default
## BLAS: /usr/lib64/libblas.so.3.4.2
## LAPACK: /usr/lib64/liblapack.so.3.4.2
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
##  [1] splines   parallel  stats4    stats     graphics  grDevices utils    
##  [8] datasets  methods   base     
## 
## other attached packages:
##  [1] bindrcpp_0.2.2              cowplot_0.9.3              
##  [3] hexbin_1.27.2               adaptMT_1.0.0              
##  [5] FDRreg_0.1                  fda_2.4.8                  
##  [7] Matrix_1.2-14               fdrtool_1.2.15             
##  [9] swfdr_1.6.0                 qvalue_2.12.0              
## [11] ashr_2.2-7                  IHW_1.8.0                  
## [13] magrittr_1.5                SummarizedBenchmark_0.99.2 
## [15] mclust_5.4.1                stringr_1.3.1              
## [17] rlang_0.2.2                 UpSetR_1.3.3               
## [19] SummarizedExperiment_1.10.1 DelayedArray_0.6.6         
## [21] BiocParallel_1.14.2         matrixStats_0.54.0         
## [23] Biobase_2.40.0              GenomicRanges_1.32.7       
## [25] GenomeInfoDb_1.16.0         IRanges_2.14.12            
## [27] S4Vectors_0.18.3            BiocGenerics_0.26.0        
## [29] tidyr_0.8.1                 ggplot2_3.0.0              
## [31] dplyr_0.7.7                
## 
## loaded via a namespace (and not attached):
##  [1] ggdendro_0.1-20        foreach_1.4.4          assertthat_0.2.0      
##  [4] ggstance_0.3.1         GenomeInfoDbData_1.1.0 ggrepel_0.8.0         
##  [7] mosaic_1.4.0           yaml_2.2.0             slam_0.1-43           
## [10] pillar_1.3.0           backports_1.1.2        lattice_0.20-35       
## [13] glue_1.3.0             digest_0.6.18          XVector_0.20.0        
## [16] colorspace_1.3-2       htmltools_0.3.6        plyr_1.8.4            
## [19] pkgconfig_2.0.2        broom_0.5.0            zlibbioc_1.26.0       
## [22] purrr_0.2.5            scales_1.0.0           mosaicData_0.17.0     
## [25] tibble_1.4.2           withr_2.1.2            lazyeval_0.2.1        
## [28] crayon_1.3.4           evaluate_0.12          nlme_3.1-137          
## [31] doParallel_1.0.14      MASS_7.3-51            truncnorm_1.0-8       
## [34] tools_3.5.0            munsell_0.5.0          compiler_3.5.0        
## [37] grid_3.5.0             RCurl_1.95-4.11        iterators_1.0.10      
## [40] labeling_0.3           mosaicCore_0.6.0       bitops_1.0-6          
## [43] rmarkdown_1.10         gtable_0.2.0           codetools_0.2-15      
## [46] reshape2_1.4.3         R6_2.3.0               gridExtra_2.3         
## [49] knitr_1.20             ggformula_0.9.0        bindr_0.1.1           
## [52] rprojroot_1.3-2        lpsymphony_1.8.0       stringi_1.2.4         
## [55] pscl_1.5.2             SQUAREM_2017.10-1      Rcpp_0.12.19          
## [58] tidyselect_0.2.5