Case Study: 16S microbiome data (ENIGMA)

1 Summary

Here we analyze a subset of the ENIGMA data (Smith, M. and Rocha, A. et al 2015). The dataset includes water samples from 67 wells with well-characterized biochemical metadata (i.e. pH, heavy metal concentrations, etc). We’ll download the processed OTU table and associated metadata from Zenodo and put it in the data/enigma_results folder.

Note that I’ve uploaded this to our repo as a gzipped file, make sure to gunzip the OTU table before knitting this.

We did some preliminary correlation analysis and found that the metadata variables show a range of correlation with different OTUs: some metadata variables don’t seem correlated with any OTUs, whereas others seem correlated with basically the entire community. We’ll pick three of these metadata variables to illustrate our FDR correction methods in cases where we expect different numbers of true significant associations.

2 Workspace Setup

library(dplyr)
library(tidyr)
library(ggplot2)
library(magrittr)
library(SummarizedBenchmark)

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

## project data/results folders
datdir <- "data"
resdir <- "results"
sbdir <- "../../results/microbiome"
dir.create(datdir, showWarnings = FALSE)
dir.create(resdir, showWarnings = FALSE)
dir.create(sbdir, showWarnings = FALSE)

enigma_dir <- file.path(datdir, "enigma_results")
dir.create(enigma_dir, showWarnings = FALSE)

# From their paper, we expect SO4 to have little correlation, pH
# to have high correlations, and Al to be in the middle
so4_result_file <- file.path(resdir, "enigma-so4-otus-results.rds")
so4_bench_file <- file.path(sbdir, "enigma-so4-otus-benchmark.rds")
so4_bench_file_abun <- file.path(sbdir, "enigma-so4-otus-abun-benchmark.rds")
so4_bench_file_uninf <- file.path(sbdir, "enigma-so4-otus-uninf-benchmark.rds")

ph_result_file <- file.path(resdir, "enigma-ph-otus-results.rds")
ph_bench_file <- file.path(sbdir, "enigma-ph-otus-benchmark.rds")
ph_bench_file_abun <- file.path(sbdir, "enigma-ph-otus-abun-benchmark.rds")
ph_bench_file_uninf <- file.path(sbdir, "enigma-ph-otus-uninf-benchmark.rds")

al_result_file <- file.path(resdir, "enigma-al-otus-results.rds")
al_bench_file <- file.path(sbdir, "enigma-al-otus-benchmark.rds")
al_bench_file_abun <- file.path(sbdir, "enigma-al-otus-abun-benchmark.rds")
al_bench_file_uninf <- file.path(sbdir, "enigma-al-otus-uninf-benchmark.rds")

3 Data Preparation

We download the data from Zenodo if not already available.

if (!file.exists(file.path(enigma_dir, "enigma.metadata.txt"))) {
    download.file("https://zenodo.org/record/1455793/files/enigma.metadata.txt",
                  destfile = file.path(enigma_dir, "enigma.metadata.txt"))
}

if (!file.exists(file.path(enigma_dir, "enigma.otu_table_resampled_updated_r.txt.gz"))) {
    download.file("https://zenodo.org/record/1455793/files/enigma.otu_table_resampled_updated_r.txt.gz",
                  destfile = file.path(enigma_dir, "enigma.otu_table_resampled_updated_r.txt.gz"))
}

## load OTU table and metadata
otu <- read.table(file.path(enigma_dir, "enigma.otu_table_resampled_updated_r.txt.gz"), 
                  sep='\t', header = TRUE, row.names = 1)
meta <- read.csv(file.path(enigma_dir, "enigma.metadata.txt"), sep='\t')

# Select only the 0.02 um samples
meta <- meta[endsWith(as.character(meta$Sample), "02"), ]
# Make metadata sample ID match OTU table
meta$Sample <- gsub("_", "\\.", as.character(meta$Sample))
meta$Sample <- substr(meta$Sample, 1, nchar(meta$Sample)-3)

## keep only samples with both metadata and 16S data
keep_samples <- intersect(colnames(otu), meta$Sample)
otu <- otu[, keep_samples]
meta <- dplyr::filter(meta, Sample %in% keep_samples)

## Make sure samples in metadata and OTU table match
meta <- meta[match(colnames(otu), meta$Sample), ]

We’ll do a little filtering here before calculating correlations. We’ll apply a minimum threshold of 10 reads per OTU across all samples. After removing these shallow OTUs, we also get rid of any samples with too few reads. We define the minimum number of reads per OTU in min_otu, and the minimum number of reads per sample in min_sample.

After we’ve removed any shallow OTUs and samples, we’ll convert the OTU table to relative abundances.

min_otu <- 10
minp_otu <- 0.01
min_sample <- 100

## Remove OTUs w/ <= min reads, w/ <= min prop, samples w/ <= min reads
otu <- otu[rowSums(otu) > min_otu, ]
otu <- otu[rowSums(otu > 0) / ncol(otu) > minp_otu, ]
otu <- otu[, colSums(otu) > min_sample]

## Update metadata with new samples
meta <- dplyr::filter(meta, Sample %in% colnames(otu))

## Remove empty OTUs
otu <- otu[rowSums(otu) > 0, ]

## Convert to relative abundance
abun_otu <- t(t(otu) / rowSums(t(otu)))

## Add pseudo counts
zeroabun <- 0
abun_otu <- abun_otu + zeroabun

4 Data Analysis

4.1 pH

Next, we need to calculate the pvalues for correlations between abundances of each OTU and the metadata variables of interest. (i.e. we’re correlating each OTU’s abundances across wells with the metadata variable across wells). Let’s start with “pH”. We’ll store the pvalue results in a dataframe.

While we’re at it, we’ll also calculate the mean abundance and ubiquity (detection rate) of each OTU. Later, we can assign their values to a new column called ind_covariate for use in downstream steps.

if (!file.exists(ph_result_file)) {
    res <- test_microbiome_corr(abundance = abun_otu, meta_col = meta$pH,
                                shift = zeroabun)
    saveRDS(res, file = ph_result_file)
} else {
    res <- readRDS(ph_result_file)
}

Add random (uninformative) covariate.

set.seed(44318)
res$rand_covar <- rnorm(nrow(res))

4.1.1 Covariate Diagnostics

Here we look to see if the covariates do indeed look informative.

4.1.1.1 Ubiquity

strat_hist(res, pvalue="pval", covariate="ubiquity", maxy=10, binwidth=0.05)

## 
## Attaching package: 'cowplot'

## The following object is masked from 'package:ggplot2':
## 
##     ggsave

rank_scatter(res, pvalue="pval", covariate="ubiquity")

4.1.1.2 Mean Abundance (across non-zero samples)

strat_hist(res, pvalue="pval", covariate="mean_abun_present", maxy=8, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="mean_abun_present")

4.1.1.3 Random

strat_hist(res, pvalue="pval", covariate="rand_covar", maxy=8, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="rand_covar")

4.1.2 Multiple-Testing Correction - ubiquity

Let’s use ubiquity as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = ubiquity)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(ph_bench_file)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = ph_bench_file)
} else {
    sb <- readRDS(ph_bench_file)
}

4.1.3 Benchmark Metrics - ubiquity

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

4.1.4 Multiple-Testing Correction - mean abundance

Let’s use mean_abun_present as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = mean_abun_present)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(ph_bench_file_abun)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = ph_bench_file_abun)
} else {
    sb <- readRDS(ph_bench_file_abun)
}

4.1.5 Benchmark Metrics - mean abundance

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

4.1.6 Multiple-Testing Correction - random

Let’s use rand_covar as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = rand_covar)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(ph_bench_file_uninf)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = ph_bench_file_uninf)
} else {
    sb <- readRDS(ph_bench_file_uninf)
}

4.1.7 Benchmark Metrics - random

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

4.2 SO4

Next, we’ll look at SO4 levels

if (!file.exists(so4_result_file)) {
    res <- test_microbiome_corr(abundance = abun_otu, meta_col = meta$SO4_mgL,
                                shift = zeroabun)
    saveRDS(res, file = so4_result_file)
} else {
    res <- readRDS(so4_result_file)
}

Add random (uninformative) covariate.

set.seed(27349)
res$rand_covar <- rnorm(nrow(res))

4.2.1 Covariate Diagnostics

Here we look to see if the covariates do indeed look informative.

4.2.1.1 Ubiquity

strat_hist(res, pvalue="pval", covariate="ubiquity", maxy=10, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="ubiquity")

4.2.1.2 Mean Abundance (across non-zero samples)

strat_hist(res, pvalue="pval", covariate="mean_abun_present", maxy=8, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="mean_abun_present")

4.2.1.3 Random

strat_hist(res, pvalue="pval", covariate="rand_covar", maxy=8, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="rand_covar")

4.2.2 Multiple-Testing Correction - ubiquity

Let’s use ubiquity as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = ubiquity)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(so4_bench_file)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = so4_bench_file)
} else {
    sb <- readRDS(so4_bench_file)
}

4.2.3 Benchmark Metrics - ubiquity

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

4.2.4 Multiple-Testing Correction - mean abundance

Let’s use mean_abun_present as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = mean_abun_present)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(so4_bench_file_abun)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = so4_bench_file_abun)
} else {
    sb <- readRDS(so4_bench_file_abun)
}

4.2.5 Benchmark Metrics - mean abundance

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

4.2.6 Multiple-Testing Correction - random

Let’s use rand_covar as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = mean_abun_present)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(so4_bench_file_uninf)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = so4_bench_file_uninf)
} else {
    sb <- readRDS(so4_bench_file_uninf)
}

4.2.7 Benchmark Metrics - random

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

4.3 Al

Finally, we’ll look at Al levels

if (!file.exists(al_result_file)) {
    res <- test_microbiome_corr(abundance = abun_otu, meta_col = meta$Al_mgL,
                                shift = zeroabun)
    saveRDS(res, file = al_result_file)
} else {
    res <- readRDS(al_result_file)
}

# remove OTUs with missing pvals
res <- res %>% filter(!is.na(pval))

Add random (uninformative) covariate.

set.seed(19474)
res$rand_covar <- rnorm(nrow(res))

4.3.1 Covariate Diagnostics

Here we look to see if the covariates do indeed look informative.

4.3.1.1 Ubiquity

strat_hist(res, pvalue="pval", covariate="ubiquity", maxy=10, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="ubiquity")

4.3.1.2 Mean Abundance (across non-zero samples)

strat_hist(res, pvalue="pval", covariate="mean_abun_present", maxy=8, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="mean_abun_present")

4.3.1.3 Random

strat_hist(res, pvalue="pval", covariate="rand_covar", maxy=8, binwidth=0.05)

rank_scatter(res, pvalue="pval", covariate="rand_covar")

4.3.2 Multiple-Testing Correction - ubiquity

Let’s use ubiquity as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = ubiquity)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(al_bench_file)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = al_bench_file)
} else {
    sb <- readRDS(al_bench_file)
}

4.3.3 Benchmark Metrics - ubiquity

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

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

4.3.4 Multiple-Testing Correction - mean abundance

Let’s use mean_abun_present as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = mean_abun_present)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(al_bench_file_abun)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = al_bench_file_abun)
} else {
    sb <- readRDS(al_bench_file_abun)
}

4.3.5 Benchmark Metrics - mean abundance

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

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

4.3.6 Multiple-Testing Correction - random

Let’s use rand_covar as our ind_covariate.

res <- dplyr::mutate(res, ind_covariate = rand_covar)

First, we’ll create an object of BenchDesign class to hold the data and add the benchmark methods to the BenchDesign object. We remove ASH from the default comparison.

Then, we’ll construct the SummarizedBenchmark object, which will run the functions specified in each method (these are actually sourced in from the helper scripts).

if (!file.exists(al_bench_file_uninf)) {
    bd <- initializeBenchDesign()
    bd <- dropBMethod(bd, "ashq")
    sb <- buildBench(bd, data = res, ftCols = "ind_covariate")
    saveRDS(sb, file = al_bench_file_uninf)
} else {
    sb <- readRDS(al_bench_file_uninf)
}

4.3.7 Benchmark Metrics - random

Next, we’ll add the default performance metric for q-value assays. First, we have to rename the assay to ‘qvalue’.

assayNames(sb) <- "qvalue"
sb <- addDefaultMetrics(sb)

Now, we’ll plot the results.

rejections_scatter( sb, as_fraction=FALSE, supplementary=FALSE)

rejection_scatter_bins(sb, covariate="ind_covariate", supplementary=FALSE)

covariateLinePlot(sb, alpha = 0.05, covname = "ind_covariate")

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

5 Response/Covariate comparison

Here we compare the method ranks for the different comparisons at alpha = 0.10.

plotMethodRanks(c(so4_bench_file, so4_bench_file_abun, so4_bench_file_uninf,
                  ph_bench_file, ph_bench_file_abun, ph_bench_file_uninf,
                  al_bench_file, al_bench_file_abun, al_bench_file_uninf), 
                colLabels = c("SO4-ubiquity", "SO4-abun", "SO4-uninf",
                              "pH-ubiquity", "ph-abun", "pH-uninf",
                              "Al-ubiquity", "Al-abun", "Al-uninf"),
                alpha = 0.10, xlab = "Comparison")

6 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] adaptMT_1.0.0               FDRreg_0.2-1               
##  [3] mvtnorm_1.0-8               BayesLogit_0.6             
##  [5] fda_2.4.7                   Matrix_1.2-14              
##  [7] fdrtool_1.2.15              swfdr_1.6.0                
##  [9] qvalue_2.12.0               ashr_2.2-7                 
## [11] IHW_1.8.0                   hexbin_1.27.2              
## [13] cowplot_0.9.2               bindrcpp_0.2.2             
## [15] SummarizedBenchmark_0.99.1  mclust_5.4                 
## [17] stringr_1.3.1               rlang_0.2.1                
## [19] UpSetR_1.3.3                SummarizedExperiment_1.10.1
## [21] DelayedArray_0.6.1          BiocParallel_1.14.2        
## [23] matrixStats_0.53.1          Biobase_2.40.0             
## [25] GenomicRanges_1.32.3        GenomeInfoDb_1.16.0        
## [27] IRanges_2.14.10             S4Vectors_0.18.3           
## [29] BiocGenerics_0.26.0         magrittr_1.5               
## [31] ggplot2_3.0.0               tidyr_0.8.1                
## [33] dplyr_0.7.5                
## 
## loaded via a namespace (and not attached):
##  [1] nlme_3.1-137           bitops_1.0-6           doParallel_1.0.11     
##  [4] RColorBrewer_1.1-2     rprojroot_1.3-2        tools_3.5.0           
##  [7] backports_1.1.2        R6_2.2.2               lazyeval_0.2.1        
## [10] colorspace_1.3-2       withr_2.1.2            tidyselect_0.2.4      
## [13] gridExtra_2.3          mnormt_1.5-5           compiler_3.5.0        
## [16] ggdendro_0.1-20        labeling_0.3           slam_0.1-43           
## [19] mosaicCore_0.5.0       scales_0.5.0           SQUAREM_2017.10-1     
## [22] psych_1.8.4            digest_0.6.15          ggformula_0.7.0       
## [25] foreign_0.8-70         rmarkdown_1.10         XVector_0.20.0        
## [28] pscl_1.5.2             pkgconfig_2.0.1        htmltools_0.3.6       
## [31] lpsymphony_1.8.0       bindr_0.1.1            RCurl_1.95-4.10       
## [34] GenomeInfoDbData_1.1.0 mosaicData_0.16.0      Rcpp_0.12.17          
## [37] munsell_0.4.3          stringi_1.2.2          yaml_2.1.19           
## [40] MASS_7.3-50            zlibbioc_1.26.0        plyr_1.8.4            
## [43] grid_3.5.0             lattice_0.20-35        knitr_1.20            
## [46] pillar_1.2.3           reshape2_1.4.3         codetools_0.2-15      
## [49] glue_1.2.0             evaluate_0.10.1        foreach_1.4.4         
## [52] gtable_0.2.0           purrr_0.2.5            assertthat_0.2.0      
## [55] broom_0.4.4            truncnorm_1.0-8        tibble_1.4.2          
## [58] iterators_1.0.9        mosaic_1.2.0