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<h1 class="title toc-ignore">Estimate Null Correlation Problem</h1>
<h4 class="author"><em>Yuxin Zou</em></h4>
<h4 class="date"><em>2018-07-09</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-10-08</p>
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<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/zouyuxin/mash_application/tree/c12be7697b6c89fdc963b01a0585fc722dfc53a1" target="_blank">c12be76</a> </summary></p>
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:
<pre><code>
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</code></pre>
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<a href="https://cdn.rawgit.com/zouyuxin/mash_application/3bfa4f502083ab1d6ba500fc0572aaf7200d6c8d/docs/EstimateCor.html" target="_blank">3bfa4f5</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-13
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
Rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/49d53fb5534caea4e69f8065ad3429c2ec1552e1/analysis/EstimateCor.Rmd" target="_blank">49d53fb</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-13
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<td style="text-align:left;">
wflow_publish(“analysis/EstimateCor.Rmd”)
</td>
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<td style="text-align:left;">
html
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<a href="https://cdn.rawgit.com/zouyuxin/mash_application/6e4f0a1b7a5da19a1dd1ad4747bf8dd0f97de8a5/docs/EstimateCor.html" target="_blank">6e4f0a1</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-03
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
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<td style="text-align:left;">
Rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/c330f07db7d367bdb817136d0bd817a3a505a17e/analysis/EstimateCor.Rmd" target="_blank">c330f07</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-03
</td>
<td style="text-align:left;">
wflow_publish(c(“analysis/EstimateCor.Rmd”))
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<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/zouyuxin/mash_application/298546692568623165a25b72aac7faf332377c70/docs/EstimateCor.html" target="_blank">2985466</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-07-26
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
Rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/e2c5ebd1931ea676f6db7f4526914788f44a4844/analysis/EstimateCor.Rmd" target="_blank">e2c5ebd</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-07-26
</td>
<td style="text-align:left;">
wflow_publish(“analysis/EstimateCor.Rmd”)
</td>
</tr>
</tbody>
</table>
</ul>
<p></details></p>
<hr />
<pre class="r"><code>library(mashr)</code></pre>
<pre><code>Loading required package: ashr</code></pre>
<pre class="r"><code>library(knitr)
library(kableExtra)
source('../code/generateDataV.R')
source('../code/summary.R')</code></pre>
<p>We illustrate the problem about estimating the correlation matrix in <code>mashr</code>.</p>
<p>In my simple simulation, the current approach underestimates the null correlation. We want to find better positive definite estimator. We could try to estimate the pairwise correlation, ie. mle of <span class="math inline">\(\sum_{l,k} \pi_{lk} N_{2}(0, V + w_{l}U_{k})\)</span> for any pair of conditions.</p>
<div id="problem" class="section level2">
<h2>Problem</h2>
<p>Simple simulation in <span class="math inline">\(R^2\)</span> to illustrate the problem: <span class="math display">\[
\hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\
0.5 & 1 \end{matrix}\right))
\]</span></p>
<p><span class="math display">\[
\beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\
0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\
0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\
1 & 1 \end{matrix}\right))
\]</span></p>
<p><span class="math inline">\(\Rightarrow\)</span> <span class="math display">\[
\hat{\beta} \sim \frac{1}{4}N_{2}(0, \left( \begin{matrix} 1 & 0.5 \\
0.5 & 1 \end{matrix} \right)) + \frac{1}{4}N_{2}(0, \left( \begin{matrix} 2 & 0.5 \\
0.5 & 1 \end{matrix} \right)) + \frac{1}{4}N_{2}(0, \left( \begin{matrix} 1 & 0.5 \\
0.5 & 2 \end{matrix} \right)) + \frac{1}{4}N_{2}(0, \left( \begin{matrix} 2 & 1.5 \\
1.5 & 2 \end{matrix} \right))
\]</span></p>
<p>n = 4000</p>
<pre class="r"><code>set.seed(1)
n = 4000; p = 2
Sigma = matrix(c(1,0.5,0.5,1),p,p)
U0 = matrix(0,2,2)
U1 = U0; U1[1,1] = 1
U2 = U0; U2[2,2] = 1
U3 = matrix(1,2,2)
Utrue = list(U0=U0, U1=U1, U2=U2, U3=U3)
data = generate_data(n, p, Sigma, Utrue)</code></pre>
<p>Let’s check the result of <code>mash</code> under different correlation matrix:</p>
<ol style="list-style-type: decimal">
<li>Identity <span class="math display">\[
V.I = I_{2}
\]</span></li>
</ol>
<pre class="r"><code>m.data = mash_set_data(data$Bhat, data$Shat)
U.c = cov_canonical(m.data)
m.I = mash(m.data, U.c, verbose= FALSE)</code></pre>
<ol start="2" style="list-style-type: decimal">
<li>The current approach: truncated empirical correlation, <span class="math inline">\(V.trun\)</span></li>
</ol>
<p>The correlation of samples with maximum |z| scores less than 2.</p>
<pre class="r"><code>Vhat = estimate_null_correlation(m.data, apply_lower_bound = FALSE)
Vhat</code></pre>
<pre><code> [,1] [,2]
[1,] 1.0000000 0.3439205
[2,] 0.3439205 1.0000000</code></pre>
<p>It underestimates the correlation.</p>
<pre class="r"><code># Use underestimate cor
m.data.V = mash_set_data(data$Bhat, data$Shat, V=Vhat)
m.V = mash(m.data.V, U.c, verbose = FALSE)</code></pre>
<ol start="3" style="list-style-type: decimal">
<li>Overestimate correlation <span class="math display">\[
V.o = \left( \begin{matrix} 1 & 0.65 \\ 0.65 & 1\end{matrix} \right)
\]</span></li>
</ol>
<pre class="r"><code># If we overestimate cor
V.o = matrix(c(1,0.65,0.65,1),2,2)
m.data.Vo = mash_set_data(data$Bhat, data$Shat, V=V.o)
m.Vo = mash(m.data.Vo, U.c, verbose=FALSE)</code></pre>
<ol start="4" style="list-style-type: decimal">
<li>mash.1by1</li>
</ol>
<p>We run ash for each condition, and estimate correlation matrix based on the non-significant genes. The estimated cor is closer to the truth.</p>
<pre class="r"><code>m.1by1 = mash_1by1(m.data)
strong = get_significant_results(m.1by1)
V.mash = cor(data$Bhat[-strong,])
V.mash</code></pre>
<pre><code> [,1] [,2]
[1,] 1.0000000 0.4597745
[2,] 0.4597745 1.0000000</code></pre>
<pre class="r"><code>m.data.1by1 = mash_set_data(data$Bhat, data$Shat, V=V.mash)
m.V1by1 = mash(m.data.1by1, U.c, verbose = FALSE)</code></pre>
<ol start="5" style="list-style-type: decimal">
<li>True correlation</li>
</ol>
<pre class="r"><code># With correct cor
m.data.correct = mash_set_data(data$Bhat, data$Shat, V=Sigma)
m.correct = mash(m.data.correct, U.c, verbose = FALSE)</code></pre>
<p>The results are summarized in table:</p>
<pre class="r"><code>null.ind = which(apply(data$B,1,sum) == 0)
V.trun = c(get_loglik(m.V), length(get_significant_results(m.V)), sum(get_significant_results(m.V) %in% null.ind))
V.I = c(get_loglik(m.I), length(get_significant_results(m.I)), sum(get_significant_results(m.I) %in% null.ind))
V.over = c(get_loglik(m.Vo), length(get_significant_results(m.Vo)), sum(get_significant_results(m.Vo) %in% null.ind))
V.1by1 = c(get_loglik(m.V1by1), length(get_significant_results(m.V1by1)), sum(get_significant_results(m.V1by1) %in% null.ind))
V.correct = c(get_loglik(m.correct), length(get_significant_results(m.correct)), sum(get_significant_results(m.correct) %in% null.ind))
temp = cbind(V.I, V.trun, V.1by1, V.correct, V.over)
colnames(temp) = c('Identity','truncate', 'm.1by1', 'true', 'overestimate')
row.names(temp) = c('log likelihood', '# significance', '# False positive')
temp %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
Identity
</th>
<th style="text-align:right;">
truncate
</th>
<th style="text-align:right;">
m.1by1
</th>
<th style="text-align:right;">
true
</th>
<th style="text-align:right;">
overestimate
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
log likelihood
</td>
<td style="text-align:right;">
-12390.14
</td>
<td style="text-align:right;">
-12307.65
</td>
<td style="text-align:right;">
-12304.13
</td>
<td style="text-align:right;">
-12302.62
</td>
<td style="text-align:right;">
-12301.81
</td>
</tr>
<tr>
<td style="text-align:left;">
# significance
</td>
<td style="text-align:right;">
166.00
</td>
<td style="text-align:right;">
30.00
</td>
<td style="text-align:right;">
25.00
</td>
<td style="text-align:right;">
25.00
</td>
<td style="text-align:right;">
70.00
</td>
</tr>
<tr>
<td style="text-align:left;">
# False positive
</td>
<td style="text-align:right;">
14.00
</td>
<td style="text-align:right;">
1.00
</td>
<td style="text-align:right;">
0.00
</td>
<td style="text-align:right;">
0.00
</td>
<td style="text-align:right;">
4.00
</td>
</tr>
</tbody>
</table>
<!--
The estimated `pi` is
```r
par(mfrow=c(2,3))
barplot(get_estimated_pi(m.I), las=2, cex.names = 0.7, main='Identity', ylim=c(0,0.8))
barplot(get_estimated_pi(m.V), las=2, cex.names = 0.7, main='Truncate', ylim=c(0,0.8))
barplot(get_estimated_pi(m.V1by1), las=2, cex.names = 0.7, main='m.1by1', ylim=c(0,0.8))
barplot(get_estimated_pi(m.correct), las=2, cex.names = 0.7, main='True', ylim=c(0,0.8))
barplot(get_estimated_pi(m.Vo), las=2, cex.names = 0.7, main='OverEst', ylim=c(0,0.8))
```
<img src="figure/EstimateCor.Rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" />
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-10-1.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/zouyuxin/mash_application/blob/10d4174c6f044161bf17e623f1b474618cae2114/docs/figure/EstimateCor.Rmd/unnamed-chunk-10-1.png" target="_blank">10d4174</a> </td>
<td style="text-align:left;"> zouyuxin </td>
<td style="text-align:left;"> 2018-08-13 </td>
</tr>
<tr>
<td style="text-align:left;"> <a href="https://github.com/zouyuxin/mash_application/blob/3bfa4f502083ab1d6ba500fc0572aaf7200d6c8d/docs/figure/EstimateCor.Rmd/unnamed-chunk-10-1.png" target="_blank">3bfa4f5</a> </td>
<td style="text-align:left;"> zouyuxin </td>
<td style="text-align:left;"> 2018-08-13 </td>
</tr>
<tr>
<td style="text-align:left;"> <a href="https://github.com/zouyuxin/mash_application/blob/298546692568623165a25b72aac7faf332377c70/docs/figure/EstimateCor.Rmd/unnamed-chunk-10-1.png" target="_blank">2985466</a> </td>
<td style="text-align:left;"> zouyuxin </td>
<td style="text-align:left;"> 2018-07-26 </td>
</tr>
</tbody>
</table>
</details>
-->
<p>The ROC curve:</p>
<pre class="r"><code>m.I.seq = ROC.table(data$B, m.I)
m.V.seq = ROC.table(data$B, m.V)
m.Vo.seq = ROC.table(data$B, m.Vo)
m.V1by1.seq = ROC.table(data$B, m.V1by1)
m.correct.seq = ROC.table(data$B, m.correct)</code></pre>
<p><img src="figure/EstimateCor.Rmd/unnamed-chunk-12-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of unnamed-chunk-12-1.png:</em></summary>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/10d4174c6f044161bf17e623f1b474618cae2114/docs/figure/EstimateCor.Rmd/unnamed-chunk-12-1.png" target="_blank">10d4174</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-13
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/3bfa4f502083ab1d6ba500fc0572aaf7200d6c8d/docs/figure/EstimateCor.Rmd/unnamed-chunk-12-1.png" target="_blank">3bfa4f5</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-13
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/298546692568623165a25b72aac7faf332377c70/docs/figure/EstimateCor.Rmd/unnamed-chunk-12-1.png" target="_blank">2985466</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-07-26
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>Comparing accuracy</p>
<pre class="r"><code>rrmse = rbind(RRMSE(data$B, data$Bhat, list(m.I = m.I, m.V = m.V, m.1by1 = m.V1by1, m.true = m.correct, m.over = m.Vo)))
colnames(rrmse) = c('Identity','V.trun','V.1by1','V.true','V.over')
row.names(rrmse) = 'RRMSE'
rrmse %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
Identity
</th>
<th style="text-align:right;">
V.trun
</th>
<th style="text-align:right;">
V.1by1
</th>
<th style="text-align:right;">
V.true
</th>
<th style="text-align:right;">
V.over
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
RRMSE
</td>
<td style="text-align:right;">
0.6522463
</td>
<td style="text-align:right;">
0.5925754
</td>
<td style="text-align:right;">
0.5811472
</td>
<td style="text-align:right;">
0.5817699
</td>
<td style="text-align:right;">
0.6052702
</td>
</tr>
</tbody>
</table>
<pre class="r"><code>barplot(rrmse, ylim=c(0,(1+max(rrmse))/2), las=2, cex.names = 0.7, main='RRMSE')</code></pre>
<p><img src="figure/EstimateCor.Rmd/unnamed-chunk-14-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of unnamed-chunk-14-1.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;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/568fbe632239ed4a2ec52f4dd5f001b71d615530/docs/figure/EstimateCor.Rmd/unnamed-chunk-14-1.png" target="_blank">568fbe6</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-13
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/3bfa4f502083ab1d6ba500fc0572aaf7200d6c8d/docs/figure/EstimateCor.Rmd/unnamed-chunk-14-1.png" target="_blank">3bfa4f5</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-13
</td>
</tr>
</tbody>
</table>
<p></details></p>
</div>
<div id="solution-mle" class="section level2">
<h2>Solution: MLE</h2>
<div id="k1" class="section level3">
<h3>K=1</h3>
<p>Suppose a simple extreme case <span class="math display">\[
\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right)| \left(\begin{matrix} x \\ y \end{matrix} \right) \sim N_{2}(\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); \left(\begin{matrix} x \\ y \end{matrix} \right), \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix}\right))
\]</span> <span class="math display">\[
\left(\begin{matrix} x \\ y \end{matrix} \right) \sim \delta_{0}
\]</span> <span class="math inline">\(\Rightarrow\)</span> <span class="math display">\[
\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) \sim N_{2}(\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); \left(\begin{matrix} 0 \\ 0 \end{matrix} \right), \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix}\right))
\]</span></p>
<p><span class="math display">\[
f(\hat{x},\hat{y}) = \prod_{i=1}^{n} \frac{1}{2\pi\sqrt{1-\rho^2}} \exp \{-\frac{1}{2(1-\rho^2)}\left[ \hat{x}_{i}^2 + \hat{y}_{i}^2 - 2\rho \hat{x}_{i}\hat{y}_{i}\right] \}
\]</span> The MLE of <span class="math inline">\(\rho\)</span>: <span class="math display">\[
\begin{align*}
l(\rho) &= -\frac{n}{2}\log(1-\rho^2) - \frac{1}{2(1-\rho^2)}\left( \sum_{i=1}^{n} x_{i}^2 + y_{i}^2 - 2\rho x_{i}y_{i} \right) \\
l(\rho)' &= \frac{n\rho}{1-\rho^2} - \frac{\rho}{(1-\rho^2)^2} \sum_{i=1}^{n} (x_{i}^2 + y_{i}^2) + \frac{\rho^2 + 1}{(1-\rho^2)^2} \sum_{i=1}^{n} x_{i}y_{i} = 0 \\
&= \rho^{3} - \rho^{2}\frac{1}{n}\sum_{i=1}^{n} x_{i}y_{i} - \left( 1- \frac{1}{n} \sum_{i=1}^{n} x_{i}^{2} + y_{i}^{2} \right) \rho - \frac{1}{n}\sum_{i=1}^{n} x_{i}y_{i} = 0 \\
l(\rho)'' &= \frac{n(\rho^2+1)}{(1-\rho^2)^2} - \frac{1}{2}\left( \frac{8\rho^2}{(1-\rho^2)^{3}} + \frac{2}{(1-\rho^2)^2} \right)\sum_{i=1}^{n}(x_{i}^2 + y_{i}^2) + \{ \left( \frac{8\rho^2}{(1-\rho^2)^{3}} + \frac{2}{(1-\rho^2)^2} \right)\rho + \frac{4\rho}{(1-\rho^2)^2} \}\sum_{i=1}^{n}x_{i}y_{i}
\end{align*}
\]</span></p>
<p><strong>The log likelihood is not a concave function in general.</strong> The score function has either 1 or 3 real solutions.</p>
<p>Kendall and Stuart (1979) noted that at least one of the roots is real and lies in the interval [−1, 1]. However, it is possible that all three roots are real and in the admissible interval, in which case the likelihood can be evaluated at each root to determine the true maximum likelihood estimate.</p>
<p>I simulate the data with <span class="math inline">\(\rho=0.6\)</span> and plot the loglikelihood function:</p>
<p><img src="figure/EstimateCor.Rmd/unnamed-chunk-15-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of unnamed-chunk-15-1.png:</em></summary>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
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<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/568fbe632239ed4a2ec52f4dd5f001b71d615530/docs/figure/EstimateCor.Rmd/unnamed-chunk-15-1.png" target="_blank">568fbe6</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-08-13
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p><span class="math inline">\(l(\rho)'\)</span> has one real solution</p>
<pre class="r"><code>polyroot(c(- sum(data$Bhat[,1]*data$Bhat[,2]), - (n - sum(data$Bhat[,1]^2 + data$Bhat[,2]^2)), - sum(data$Bhat[,1]*data$Bhat[,2]), n))</code></pre>
<pre><code>[1] 0.6193031+0.000000i 0.0058209+1.009339i 0.0058209-1.009339i</code></pre>
</div>
<div id="in-general" class="section level3">
<h3>In general</h3>
<p>The general derivation is in <a href="EstimateCorOptim.html">estimate correlation mle</a></p>
</div>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] kableExtra_0.9.0 knitr_1.20 mashr_0.2-15 ashr_2.2-14
loaded via a namespace (and not attached):
[1] Rcpp_0.12.19 highr_0.7 pillar_1.3.0
[4] compiler_3.5.1 git2r_0.23.0 plyr_1.8.4
[7] workflowr_1.1.1 R.methodsS3_1.7.1 R.utils_2.6.0
[10] iterators_1.0.10 tools_3.5.1 digest_0.6.15
[13] viridisLite_0.3.0 tibble_1.4.2 evaluate_0.11
[16] lattice_0.20-35 pkgconfig_2.0.2 rlang_0.2.2
[19] Matrix_1.2-14 foreach_1.4.4 rstudioapi_0.7
[22] yaml_2.2.0 parallel_3.5.1 mvtnorm_1.0-8
[25] xml2_1.2.0 httr_1.3.1 stringr_1.3.1
[28] REBayes_1.3 hms_0.4.2 rprojroot_1.3-2
[31] grid_3.5.1 R6_2.2.2 rmarkdown_1.10
[34] rmeta_3.0 readr_1.1.1 magrittr_1.5
[37] whisker_0.3-2 scales_1.0.0 backports_1.1.2
[40] codetools_0.2-15 htmltools_0.3.6 MASS_7.3-50
[43] rvest_0.3.2 abind_1.4-5 assertthat_0.2.0
[46] colorspace_1.3-2 stringi_1.2.4 Rmosek_8.0.69
[49] munsell_0.5.0 doParallel_1.0.14 pscl_1.5.2
[52] truncnorm_1.0-8 SQUAREM_2017.10-1 crayon_1.3.4
[55] R.oo_1.22.0 </code></pre>
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