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<h1 class="title toc-ignore">Example Analysis with CAUSE: LDL -> CAD</h1>
<h4 class="author"><em>Jean Morrison</em></h4>
<h4 class="date"><em>2018-10-16</em></h4>
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<p><strong>Last updated:</strong> 2019-06-25</p>
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Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated. <br><br> Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use <code>wflow_publish</code> or <code>wflow_git_commit</code>). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:
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<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>This document will walk through a real genome-sized example of how to use CAUSE. Some of the steps will take 5-10 minutes. The LD pruning steps will benefit from access to a cluster or multiple cores. For steps that require long computation we also provide output files that can be downloaded to make it easier to run through the example.</p>
<p>We will be analyzing GWAS data for LDL cholesterol and for coronary artery disease to test for a causal relationship of LDL on CAD. The analysis will have the following steps:</p>
<ol style="list-style-type: decimal">
<li>Format the data for use with CAUSE</li>
<li>Calculate nuisance parameters</li>
<li>LD pruning</li>
<li>Fit CAUSE</li>
<li>Look at results</li>
</ol>
<p>Step 3 will require LD information estimated from the 1000 Genomes CEU population using LDshrink <a href="https://zenodo.org/record/1464357#.W8a-fxROmV4">here</a>. LD data are about 11 Gb. The GWAS data we will use are about 320 Gb. However, in this tutorial you will be able to skip the large data steps and simply download the results.</p>
</div>
<div id="step-0-install-cause" class="section level2">
<h2>Step 0: Install CAUSE</h2>
<p>In R</p>
<pre class="r"><code>devtools::install_github("jean997/cause")</code></pre>
<p>Please be sure you are using <code>mixsqp-0-97</code> which is currently the version available in CRAN *not the latest version available on GitHub.</p>
</div>
<div id="step-1-format-data-for-cause" class="section level2">
<h2>Step 1: Format Data for CAUSE</h2>
<p>We will use <code>read_tsv</code> to read in summary statistics for a GWAS of LDL cholesterol and a GWAS of coronary artery disease from the internet. We will then combine these and format them for use with CAUSE. First read in the data. For LDL Cholesterol, we use summary statistics from Willer et al (2013) [PMID: 24097068]. For CAD we use summary statistics from van der Harst et al. (2017) [PMID: 29212778]</p>
<pre class="r"><code>library(readr)
library(dplyr)</code></pre>
<pre><code>
Attaching package: 'dplyr'</code></pre>
<pre><code>The following objects are masked from 'package:stats':
filter, lag</code></pre>
<pre><code>The following objects are masked from 'package:base':
intersect, setdiff, setequal, union</code></pre>
<pre class="r"><code>library(cause)</code></pre>
<pre class="r"><code>X1 <- read_tsv("http://csg.sph.umich.edu/abecasis/public/lipids2013/jointGwasMc_LDL.txt.gz")
X2 <- read_tsv("ftp://ftp.ebi.ac.uk/pub/databases/gwas/summary_statistics/vanderHarstP_29212778_GCST005194/CAD_META.gz")</code></pre>
<p>CAUSE needs the following information from each data set: SNP or variant ID, effect size, and standard error, effect allele and other allele. For convenience, we provide a simple function that will merge data sets and produce a <code>cause_data</code> object that can be used with later functions. This step and the rest of the analysis are done in R.</p>
<p>The function <code>gwas_format_cause</code> will try to merge two data sets and and align effect sizes to correspond to the same allele. It will remove variants with ambiguous alleles (G/C or A/T) or with alleles that do not match between data sets (e.g A/G in one data set and A/C in the other). It will not remove variants that are simply strand flipped between the two data sets (e. g. A/C in one data set, T/G in the other).</p>
<p>LDL column headers:</p>
<ul>
<li>SNP: rsid</li>
<li>Effect: beta</li>
<li>Standard Error: se</li>
<li>Effect Allele: A1</li>
<li>Other Allele: A2</li>
</ul>
<p>CAD column headers:</p>
<ul>
<li>SNP: oldID</li>
<li>Effect: Effect</li>
<li>Standard Error: StdErr</li>
<li>Effect Allele: Allele1</li>
<li>Other Allele: Allele2</li>
</ul>
<pre class="r"><code>X <- gwas_format_cause(X1, X2, snp_name_cols = c("rsid", "oldID"),
beta_hat_cols = c("beta", "Effect"),
se_cols = c("se", "StdErr"),
A1_cols = c("A1", "Allele1"),
A2_cols = c("A2", "Allele2"))
head(X)</code></pre>
<p>Alternatively, you can download already formatted data <a href="https://github.com/jean997/cause/blob/master/example_data/LDL_CAD_merged.RDS">here</a> and read it in using <code>readRDS</code>.</p>
<pre class="r"><code>X <- readRDS("example_data/LDL_CAD_merged.RDS")</code></pre>
<p>There are likely more efficient ways to do this merge. If you would like to process the data yourself, you can construct a <code>cause_data</code> object from a data frame using the constructor <code>new_cause_data(X)</code> where <code>X</code> is any data frame that includes the columns <code>snp</code>, <code>beta_hat_1</code>, <code>seb1</code>, <code>beta_hat_2</code>, and <code>seb2</code>.</p>
</div>
<div id="step-2-calculate-nuisance-parameters" class="section level2">
<h2>Step 2: Calculate nuisance parameters</h2>
<p>The next step is to estimate the parameters that define the prior distribution of <span class="math inline">\(\beta_{M}\)</span> and <span class="math inline">\(\theta\)</span> and to estimate <span class="math inline">\(\rho\)</span>, the correlation between summary statistics that is due to sample overlap or population structure. We will do this with a random subset of 1,000,000 variants since our data set is large. <code>est_cause_params</code> estimates the nuisance parameters by finding the maximum a posteriori estimate of <span class="math inline">\(\rho\)</span> and the mixing parameters when <span class="math inline">\(\gamma = \eta = 0\)</span>. This step takes a several minutes.</p>
<pre class="r"><code>set.seed(100)
varlist <- with(X, sample(snp, size=1000000, replace=FALSE))
params <- est_cause_params(X, varlist)</code></pre>
<pre><code>Estimating CAUSE parameters with 1000000 variants.
1 0.1530926
2 0.0006548855
3 8.291495e-06
4 1.06693e-07
5 5.555824e-10 </code></pre>
<p>The object <code>params</code> is of class <code>cause_params</code> and contains information about the fit as well as the maximum a posteriori estimates of the mixing parameters (<span class="math inline">\(\pi\)</span>) and <span class="math inline">\(\rho\)</span>. The object <code>params$mix_grid</code> is a data frame defining the distribution of summary statistics. The column <code>S1</code> is the variance for trait 1 (<span class="math inline">\(M\)</span>), the column <code>S2</code> is the variance for trait 2 (<span class="math inline">\(Y\)</span>) and the column <code>pi</code> is the mixture proportion assigned to that variance combination.</p>
<pre class="r"><code>class(params)</code></pre>
<pre><code>[1] "cause_params"</code></pre>
<pre class="r"><code>names(params)</code></pre>
<pre><code> [1] "rho" "pi" "mix_grid" "loglik" "PIS"
[6] "RHO" "LLS" "converged" "prior" "var" </code></pre>
<pre class="r"><code>params$rho</code></pre>
<pre><code>[1] 0.06465867</code></pre>
<pre class="r"><code>head(params$mix_grid)</code></pre>
<pre><code> S1 S2 pi
1 0.000000000 0.000000000 0.349613038
2 0.000000000 0.003408693 0.156938837
3 0.003257738 0.003408693 0.002752562
4 0.000000000 0.004820620 0.176297684
5 0.003257738 0.004820620 0.143141741
6 0.004607137 0.006817386 0.049773176</code></pre>
<p>So, for example, in this case, we have estimated that 35% of variants have trait 1 variance and trait 2 equal to 0 meaning that they have no association with either trait.</p>
<p>Tip: Do not try to estimate the nuisance parameters with substantially fewer than 100,000 variants. This can lead to poor estimates of the mixing parameters whih leads to bad model comparisons.</p>
</div>
<div id="step-3-ld-pruning" class="section level2">
<h2>Step 3: LD Pruning</h2>
<p>We estimate CAUSE posterior distributions using an LD pruned set of variants, prioritizing variants with low trait <span class="math inline">\(M\)</span> (LDL) <span class="math inline">\(p\)</span>-values. The CAUSE R package contains a function to help do this. The function <code>ld_prune</code> uses a greedy algorithm that selects the variant wtih the lowest LDL p-value and removes all variants in LD with the selected variant and then repeats until no variants are left. This step requires LD estimates. You can download estimates made in the 1000 Genomes CEU population <a href="https://zenodo.org/record/1464357#.W8a-fxROmV4">here</a>. We first demonstrate use of the function for one chromosome and then show an example of how to parallelize the analysis over all 22 autosomes.</p>
<pre class="r"><code>ld <- readRDS("example_data/ld_data/chr22_AF0.05_0.1.RDS")
snp_info <- readRDS("example_data/ld_data/chr22_AF0.05_snpdata.RDS")
head(ld)</code></pre>
<pre><code> rowsnp colsnp r2
1 rs62224609 rs376238049 0.9012642
2 rs62224609 rs62224614 0.9907366
3 rs62224609 rs7286962 0.9907366
4 rs62224609 rs55926024 0.1103000
5 rs62224609 rs117246541 0.9012642
6 rs62224609 rs62224618 0.9907366</code></pre>
<pre class="r"><code>head(snp_info)</code></pre>
<pre><code># A tibble: 6 x 9
AF SNP allele chr pos snp_id region_id map ld_snp_id
<dbl> <chr> <chr> <int> <int> <int> <int> <dbl> <int>
1 0.884 rs62224609 T,C 22 16051249 7.98e7 22 0 79758556
2 0.904 rs4965031 G,A 22 16052080 7.98e7 22 0 79758578
3 0.646 rs3756846… A,AAAAC 22 16052167 7.98e7 22 0 79758584
4 0.894 rs3762380… C,T 22 16052962 7.98e7 22 0 79758602
5 0.934 rs2007775… C,A 22 16052986 7.98e7 22 0 79758604
6 0.934 rs80167676 A,T 22 16053444 7.98e7 22 0 79758627</code></pre>
<p>The <code>ld</code> data frame should contain the column names <code>rowsnp</code>, <code>colsnp</code>, and <code>r2</code>. The <code>snp_info</code> data frame contains information about all of the chromosome 22 variants with allele frequency greater than 0.05. The only piece of information we need from this data frame is the list of variants <code>snp_info$SNP</code> which provides the total list of variants used in the LD calculations.</p>
<div id="ld-pruning-for-one-chromosome" class="section level3">
<h3>LD pruning for one chromosome</h3>
<p>The <code>ld_prune</code> function is somewhat flexible in its arguments, see <code>help(ld_prune)</code>.</p>
<pre class="r"><code>variants <- X %>% mutate(pval1 = 2*pnorm(abs(beta_hat_1/seb1), lower.tail=FALSE))
pruned <- ld_prune(variants = variants,
ld = ld, total_ld_variants = snp_info$SNP,
pval_cols = c("pval1"),
pval_thresh = c(1e-3))</code></pre>
<pre><code>You have suppplied information for 2023362 variants.
Of these, 22835 have LD information.</code></pre>
<pre class="r"><code>length(pruned)</code></pre>
<pre><code>[1] 15</code></pre>
<p><code>ld_prune</code> retunrs a list of vectors of length equal to the length of the <code>pval_cols</code> argument. In this case <code>pval_cols= c(NA, "pval1")</code> meaning that the first element of <code>pruned</code> will be a randomly pruned list and the second will be pruned preferentially choosing variants with low values of <code>pval1</code>. We also apply a threshold specified by the <code>pval_thresh</code> argument. For the first list there is no threshold. For the second, the threshold is 0.001.</p>
</div>
<div id="parallelizing-over-chromosomes" class="section level3">
<h3>Parallelizing over chromosomes</h3>
<p>We highly recommend parallelizing for whole genome LD pruning. One way to do this is with the parallel pacakge, though this option uses a lot of memory.</p>
<pre class="r"><code>library(parallel)
cores <- parallel::detectCores()-1
ld_files <- paste0("example_data/ld_data/chr", 1:22, "_AF0.05_0.1.RDS")
snp_info_files <- paste0("example_data/ld_data/chr", 1:22, "_AF0.05_snpdata.RDS")
cl <- makeCluster(cores, type="PSOCK")
clusterExport(cl, varlist=c("variants", "ld_files", "snp_info_files"))
clusterEvalQ(cl, library(cause))
system.time(
pruned <- parLapply(cl, seq_along(ld_files[20:22]), fun=function(i){
ld <- readRDS(ld_files[i])
snp_info <- readRDS(snp_info_files[i])
ld_prune(variants = variants,
ld = ld, total_ld_variants = snp_info$SNP,
pval_cols = c("pval1"),
pval_thresh = c( 1e-3))
})
)
stopCluster(cl)
top_ldl_pruned_vars <- unlist(pruned)</code></pre>
<p>A better option is to parallelize over the nodes of a compute cluster and then combine results.</p>
<p>Tip: If you are analyzing many phenotypes first obtain a list of variants present in all data sets and then LD prune this list. You can use this single set of variants estimate nuisance parameters for every pair as long as there are enough of them.</p>
<p>Download the resulting variant list: <a href="https://github.com/jean997/cause/blob/master/example_data/top_ldl_pruned_vars.RDS">top LDL list</a></p>
</div>
</div>
<div id="step-4-fit-cause" class="section level2">
<h2>Step 4: Fit CAUSE</h2>
<p>Now that we have formatted data, an LD pruned set of variants, and nuisance parameters estimated, we can fit CAUSE! The function <code>cause::cause</code> will estimate posterior distributions under the confounding and causal models and calculate the elpd for both models as well as for the null model in which there is neither a causal or a confounding effect. This might take 5-10 minutes.</p>
<p>Note: To estimate the posterior distributions, we only need the variants that are most associated with the mediator. This is because other variants don’t add any information about the relationship between the traits. When we LD pruned, we used a <span class="math inline">\(p\)</span>-value threshold of 0.001. The exact value of this threshold isn’t important as long as it is fairly lenient. Including additional variants may slow down computation but shouldn’t change the results</p>
<pre class="r"><code>top_vars <- readRDS("example_data/top_ldl_pruned_vars.RDS")
res <- cause(X=X, variants = top_vars, param_ests = params)</code></pre>
<pre><code>Estimating CAUSE posteriors using 1215 variants.</code></pre>
</div>
<div id="step-5-look-at-results" class="section level2">
<h2>Step 5: Look at Results</h2>
<p>The resulting <code>cause</code> object contains an object for the partial sharing model fit (<code>conf</code>), and object for the causal model fit (<code>full</code>) and a table of ELPD results.</p>
<pre class="r"><code>class(res)</code></pre>
<pre><code>[1] "cause"</code></pre>
<pre class="r"><code>names(res)</code></pre>
<pre><code>[1] "conf" "full" "elpd" "loos" "data" "sigma_g" "qalpha"
[8] "qbeta" </code></pre>
<pre class="r"><code>res$elpd</code></pre>
<pre><code> model1 model2 delta_elpd se_delta_elpd z
1 null conf -62.005377 7.783698 -7.966056
2 null full -70.272030 8.940126 -7.860295
3 conf full -8.266653 1.242831 -6.651469</code></pre>
<pre class="r"><code>class(res$conf)</code></pre>
<pre><code>[1] "cause_post"</code></pre>
<pre class="r"><code>class(res$full)</code></pre>
<pre><code>[1] "cause_post"</code></pre>
<p>The <code>elpd</code> table has the following columns:</p>
<ul>
<li>model1, model2: The models being compared</li>
<li>delta_elpd: Estimated difference in elpd. If delta_elpd is negative, model 2 is a better fit</li>
<li>se_delta_elpd: Estimated standard error of delta_elpd</li>
<li>z: delta_elpd/se_delta_elpd. A z-score that can be compared to a normal distribution to test if the difference in model fit is significant.</li>
</ul>
<p>In this case we see that the full (causal) model is significantly better than the confounding model from the thrid line of the table. The <span class="math inline">\(z\)</span>-score is -6.65 corresponding to a p-value of 1.510^{-11}.</p>
<p>For each model (partial sharing and full) we can plot the posterior distributions of the parameters. Dotted lines show the prior distributions.</p>
<pre class="r"><code>plot(res$conf)</code></pre>
<p><img src="figure/ldl_cad.Rmd/plot1-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of plot1-1.png:</em></summary>
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<p></details></p>
<pre class="r"><code>plot(res$full)</code></pre>
<p><img src="figure/ldl_cad.Rmd/plot1-2.png" width="672" style="display: block; margin: auto;" /></p>
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<p></details></p>
<p>The <code>summary</code> method will summarize the posterior medians and credible intervals.</p>
<pre class="r"><code>summary(res, ci_size=0.95)</code></pre>
<pre><code>p-value testing that causal model is a better fit: 1.5e-11
Posterior medians and 95 % credible intervals:
model gamma eta
[1,] "Confounding Only" NA "0.38 (0.31, 0.45)"
[2,] "Causal" "0.34 (0.28, 0.4)" "-0.01 (-0.65, 0.53)"
q
[1,] "0.78 (0.64, 0.88)"
[2,] "0.03 (0, 0.25)" </code></pre>
<p>The <code>plot</code> method applied to a <code>cause</code> object will arrange all of this information on one spread.</p>
<pre class="r"><code>plot(res)</code></pre>
<p><img src="figure/ldl_cad.Rmd/plot2-1.png" width="672" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/jean997/cause/blob/d8d148608a68c9fd853964a999d3ac3e216978cc/docs/figure/ldl_cad.Rmd/plot2-1.png" target="_blank">d8d1486</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2019-06-25
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/33b373230107121d24eb71d1d4a7bdd70488f4c4/docs/figure/ldl_cad.Rmd/plot2-1.png" target="_blank">33b3732</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2019-06-25
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/286f4e9e0a432565b96c38bba3a1b155c02f9a78/docs/figure/ldl_cad.Rmd/plot2-1.png" target="_blank">286f4e9</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2019-06-25
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/4a8f76ccd2554d1f46985507b41899ced9e02a3e/docs/figure/ldl_cad.Rmd/plot2-1.png" target="_blank">4a8f76c</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2018-11-06
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/a34393dedf81051e37783951c9e3b74aba80de65/docs/figure/ldl_cad.Rmd/plot2-1.png" target="_blank">a34393d</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2018-10-24
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/bbe49018f1c79ea1f2bfb1c1c4ac2228722dac60/docs/figure/ldl_cad.Rmd/plot2-1.png" target="_blank">bbe4901</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2018-10-17
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/73690eb831172fcd7130e9b4d2620f85b0c528c1/docs/figure/ldl_cad.Rmd/plot2-1.png" target="_blank">73690eb</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2018-10-17
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>The <code>plot</code> method can also produce scatter plots of the data showing for each model, the probability that each variant is acting through the confounder and the contribution of each variant to the ELPD test statistic.</p>
<pre class="r"><code>plot(res, type="data")</code></pre>
<p><img src="figure/ldl_cad.Rmd/plot3-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of plot3-1.png:</em></summary>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/d8d148608a68c9fd853964a999d3ac3e216978cc/docs/figure/ldl_cad.Rmd/plot3-1.png" target="_blank">d8d1486</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2019-06-25
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/33b373230107121d24eb71d1d4a7bdd70488f4c4/docs/figure/ldl_cad.Rmd/plot3-1.png" target="_blank">33b3732</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2019-06-25
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/286f4e9e0a432565b96c38bba3a1b155c02f9a78/docs/figure/ldl_cad.Rmd/plot3-1.png" target="_blank">286f4e9</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2019-06-25
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/jean997/cause/blob/4a8f76ccd2554d1f46985507b41899ced9e02a3e/docs/figure/ldl_cad.Rmd/plot3-1.png" target="_blank">4a8f76c</a>
</td>
<td style="text-align:left;">
Jean Morrison
</td>
<td style="text-align:left;">
2018-11-06
</td>
</tr>
</tbody>
</table>
<p></details></p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.5.2 (2018-12-20)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.2 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/libopenblasp-r0.2.20.so
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] stats graphics grDevices utils datasets methods base
other attached packages:
[1] bindrcpp_0.2.2 cause_0.2.0.0097 dplyr_0.7.6 readr_1.1.1
loaded via a namespace (and not attached):
[1] tidyselect_0.2.4 ashr_2.2-32 purrr_0.2.5
[4] lattice_0.20-35 colorspace_1.3-2 htmltools_0.3.6
[7] loo_2.0.0 yaml_2.2.0 utf8_1.1.4
[10] rlang_0.3.1 R.oo_1.22.0 mixsqp_0.1-97
[13] pillar_1.3.1 glue_1.3.1 R.utils_2.7.0
[16] matrixStats_0.54.0 foreach_1.4.4 bindr_0.1.1
[19] plyr_1.8.4 stringr_1.3.1 munsell_0.5.0
[22] gtable_0.2.0 workflowr_1.1.1 R.methodsS3_1.7.1
[25] codetools_0.2-15 evaluate_0.11 labeling_0.3
[28] knitr_1.20 doParallel_1.0.14 pscl_1.5.2
[31] parallel_3.5.2 fansi_0.4.0 Rcpp_1.0.0
[34] backports_1.1.2 scales_1.0.0 RcppParallel_4.4.1
[37] truncnorm_1.0-8 gridExtra_2.3 ggplot2_3.1.0
[40] hms_0.4.2 digest_0.6.18 stringi_1.2.4
[43] numDeriv_2016.8-1 grid_3.5.2 rprojroot_1.3-2
[46] cli_1.0.1 tools_3.5.2 magrittr_1.5
[49] lazyeval_0.2.1 tibble_2.0.0 crayon_1.3.4
[52] whisker_0.3-2 tidyr_0.8.1 pkgconfig_2.0.2
[55] MASS_7.3-50 Matrix_1.2-14 SQUAREM_2017.10-1
[58] assertthat_0.2.1 rmarkdown_1.10 iterators_1.0.10
[61] R6_2.4.0 intervals_0.15.1 git2r_0.25.2
[64] compiler_3.5.2 </code></pre>
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