1 Summary

In this set of simulations, we consider settings with both null and non-null tests with an informative and uninformative covariate. The informative covariate is sampled uniformly from the interval [0, 1], and the conditional probability of a test being non-null is a smooth (cubic) function of the covariate. The uninformative covariate is simply uninformly sampled independently from the interval [0, 1]. The uninformative covariate is included as a baseline to compare the informative covariate against. We include simulation results again with Gaussian, t-distributed, and Chi-Squared 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
gauss_file <- file.path(resdir, "informative-cubic-benchmark-gaussian.rds")
tdist_file <- file.path(resdir, "informative-cubic-benchmark-t5.rds")
tdist11_file <- file.path(resdir, "informative-cubic-benchmark-t11.rds")
chisq_file <- file.path(resdir, "informative-cubic-benchmark-chisq4.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".

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))

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
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 Gaussian Setting

First, we consider the setting with Gaussian test statistics.

3.1 Data Simulation

es_dist <- rnorm_generator(3)     # functional: dist of alternative test stats
ts_dist <- rnorm_perturber(1)     # functional: sampling dist/noise for test stats
null_dist <- rnorm_2pvaluer(1)    # functional: dist to calc p-values
seed <- 608

We next run the simulations (including Scott’s FDR Regression).

if (file.exists(gauss_file)) {
    res <- readRDS(gauss_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 = gauss_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")

We also look at the FDR as a function of the independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate", metric = "FDR")
## Warning: Removed 35 rows containing missing values (geom_path).
## Warning: Removed 37 rows containing missing values (geom_errorbar).

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 Student’s t Setting (df = 5)

Next, we consider the setting with t-distributed test statistics.

4.1 Data Simulation

es_dist <- rnorm_generator(3)  # functional: dist of alternative test stats
ts_dist <- rt_perturber(5)     # functional: sampling dist/noise for test stats
null_dist <- rt_2pvaluer(5)    # functional: dist to calc p-values
seed <- 815

For the t-distributed setting, 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 statistic distribution.

bdplust <- modifyBMethod(bdplus, "ashq", df = 5)

We next run the simulations (including Scott’s FDR Regression and ASH with degrees of freedom specified).

if (file.exists(tdist_file)) {
    res <- readRDS(tdist_file)
} else {
    res <- mclapply(X = 1:B, FUN = simIteration, bench = bdplust, 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 = tdist_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")
## Warning: Removed 25 rows containing missing values (geom_path).
## Warning: Removed 25 rows containing missing values (geom_errorbar).

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate")
## Warning: Removed 25 rows containing missing values (geom_path).

## Warning: Removed 25 rows containing missing values (geom_errorbar).

We also look at the FDR as a function of the independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate", metric = "FDR")
## Warning: Removed 196 rows containing missing values (geom_path).
## Warning: Removed 196 rows containing missing values (geom_errorbar).

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 Student’s t Setting (df = 11)

Next, we consider a second setting with t-distributed test statistics.

5.1 Data Simulation

es_dist <- rnorm_generator(3)  # functional: dist of alternative test stats
ts_dist <- rt_perturber(11)    # functional: sampling dist/noise for test stats
null_dist <- rt_2pvaluer(11)   # functional: dist to calc p-values
seed <- 9158

For the t-distributed setting, 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 statistic distribution.

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

We next run the simulations (including Scott’s FDR Regression and ASH with degrees of freedom specified).

if (file.exists(tdist11_file)) {
    res <- readRDS(tdist11_file)
} else {
    res <- mclapply(X = 1:B, FUN = simIteration, bench = bdplust, 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 = tdist11_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")

We also look at the FDR as a function of the independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate", metric = "FDR")
## Warning: Removed 69 rows containing missing values (geom_path).
## Warning: Removed 72 rows containing missing values (geom_errorbar).

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 Chi-Squared Setting

Finally, we consider the setting with chi-squared distributed test statistics.

6.1 Data Simulation

es_dist <- rnorm_generator(15)  # functional: dist of alternative test stats
ts_dist <- rchisq_perturber(4)  # functional: sampling dist/noise for test stats
null_dist <- rchisq_pvaluer(4)  # functional: dist to calc p-values
seed <- 1023

For the chi-squared distributed setting, we must change the “mode” setting for ASH from the default of 0 to "estimate" since the mode of null and alternative test statistics are no longer centered at 0. While both approximate Normality and unimodality of effects are violated in this simulation setting, by allowing the mode to be estimated, rather than forced to 0, should return more comparable results.

bdchi <- modifyBMethod(bd, "ashq", mode = "empirical")

We next run the simulations. We do not include FDR Regression because the test statistics are not approximately normally distributed.

if (file.exists(chisq_file)) {
    res <- readRDS(chisq_file)
} else {
    res <- mclapply(X = 1:B, FUN = simIteration, bench = bdchi, 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 = chisq_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(bdchi, 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)

Since ashq is not appropriate when the test statistics are not approximately normal or t, we also plot the results without this method.

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

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

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

plotsim_average(resdf, met = "TNR", filter_set = c(excludeSet, "ashq"),
                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. Again, we plot these without ashq.

covariateLinePlot(lapply(res_i, function(x) { x[, -which(colnames(x) == "ashq")] }),
                  alpha = ualpha, covname = "effect_size")

covariateLinePlot(lapply(res_i, function(x) { x[, -which(colnames(x) == "ashq")] }),
                  alpha = ualpha, covname = "ind_covariate")

We also look at the FDR as a function of the independent covariate.

covariateLinePlot(res_i, alpha = ualpha, covname = "ind_covariate", metric = "FDR")
## Warning: Removed 52 rows containing missing values (geom_path).
## Warning: Removed 53 rows containing missing values (geom_errorbar).

Finally, (if enough methods produce rejections at 0.05) we take a look at the overlap of rejections between methods. Again, without ashq.

if (numberMethodsReject(resdf, alphacutoff = ualpha, filterSet = c(excludeSet, "ashq")) >= 3) {
    aggupset(lapply(res_i, function(x) { x[, -which(colnames(x) == "ashq")] }),
             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