AMR/docs/articles/PCA.html

339 lines
21 KiB
HTML
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE html>
<!-- Generated by pkgdown: do not edit by hand --><html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>How to conduct principal component analysis (PCA) for AMR • AMR (for R)</title>
<!-- favicons --><link rel="icon" type="image/png" sizes="16x16" href="../favicon-16x16.png">
<link rel="icon" type="image/png" sizes="32x32" href="../favicon-32x32.png">
<link rel="apple-touch-icon" type="image/png" sizes="180x180" href="../apple-touch-icon.png">
<link rel="apple-touch-icon" type="image/png" sizes="120x120" href="../apple-touch-icon-120x120.png">
<link rel="apple-touch-icon" type="image/png" sizes="76x76" href="../apple-touch-icon-76x76.png">
<link rel="apple-touch-icon" type="image/png" sizes="60x60" href="../apple-touch-icon-60x60.png">
<!-- jquery --><script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script><!-- Bootstrap --><link href="https://cdnjs.cloudflare.com/ajax/libs/bootswatch/3.4.0/flatly/bootstrap.min.css" rel="stylesheet" crossorigin="anonymous">
<script src="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/js/bootstrap.min.js" integrity="sha256-nuL8/2cJ5NDSSwnKD8VqreErSWHtnEP9E7AySL+1ev4=" crossorigin="anonymous"></script><!-- bootstrap-toc --><link rel="stylesheet" href="../bootstrap-toc.css">
<script src="../bootstrap-toc.js"></script><!-- Font Awesome icons --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" integrity="sha256-mmgLkCYLUQbXn0B1SRqzHar6dCnv9oZFPEC1g1cwlkk=" crossorigin="anonymous">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/v4-shims.min.css" integrity="sha256-wZjR52fzng1pJHwx4aV2AO3yyTOXrcDW7jBpJtTwVxw=" crossorigin="anonymous">
<!-- clipboard.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js" integrity="sha256-inc5kl9MA1hkeYUt+EC3BhlIgyp/2jDIyBLS6k3UxPI=" crossorigin="anonymous"></script><!-- headroom.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/headroom.min.js" integrity="sha256-AsUX4SJE1+yuDu5+mAVzJbuYNPHj/WroHuZ8Ir/CkE0=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script><!-- pkgdown --><link href="../pkgdown.css" rel="stylesheet">
<script src="../pkgdown.js"></script><link href="../extra.css" rel="stylesheet">
<script src="../extra.js"></script><meta property="og:title" content="How to conduct principal component analysis (PCA) for AMR">
<meta property="og:description" content="AMR">
<meta property="og:image" content="https://msberends.gitlab.io/AMR/logo.svg">
<!-- mathjax --><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js" integrity="sha256-nvJJv9wWKEm88qvoQl9ekL2J+k/RWIsaSScxxlsrv8k=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/config/TeX-AMS-MML_HTMLorMML.js" integrity="sha256-84DKXVJXs0/F8OTMzX4UR909+jtl4G7SPypPavF+GfA=" crossorigin="anonymous"></script><!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script>
<script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script>
<![endif]-->
</head>
<body data-spy="scroll" data-target="#toc">
<div class="container template-article">
<header><div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<span class="navbar-brand">
<a class="navbar-link" href="../index.html">AMR (for R)</a>
<span class="version label label-default" data-toggle="tooltip" data-placement="bottom" title="Latest development version">1.1.0</span>
</span>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li>
<a href="../index.html">
<span class="fa fa-home"></span>
Home
</a>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false">
<span class="fa fa-question-circle"></span>
How to
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li>
<a href="../articles/AMR.html">
<span class="fa fa-directions"></span>
Conduct AMR analysis
</a>
</li>
<li>
<a href="../articles/resistance_predict.html">
<span class="fa fa-dice"></span>
Predict antimicrobial resistance
</a>
</li>
<li>
<a href="../articles/PCA.html">
<span class="fa fa-compress"></span>
Conduct principal component analysis for AMR
</a>
</li>
<li>
<a href="../articles/MDR.html">
<span class="fa fa-skull-crossbones"></span>
Determine multi-drug resistance (MDR)
</a>
</li>
<li>
<a href="../articles/WHONET.html">
<span class="fa fa-globe-americas"></span>
Work with WHONET data
</a>
</li>
<li>
<a href="../articles/SPSS.html">
<span class="fa fa-file-upload"></span>
Import data from SPSS/SAS/Stata
</a>
</li>
<li>
<a href="../articles/EUCAST.html">
<span class="fa fa-exchange-alt"></span>
Apply EUCAST rules
</a>
</li>
<li>
<a href="../reference/mo_property.html">
<span class="fa fa-bug"></span>
Get properties of a microorganism
</a>
</li>
<li>
<a href="../reference/ab_property.html">
<span class="fa fa-capsules"></span>
Get properties of an antibiotic
</a>
</li>
<li>
<a href="../articles/benchmarks.html">
<span class="fa fa-shipping-fast"></span>
Other: benchmarks
</a>
</li>
</ul>
</li>
<li>
<a href="../reference/">
<span class="fa fa-book-open"></span>
Manual
</a>
</li>
<li>
<a href="../authors.html">
<span class="fa fa-users"></span>
Authors
</a>
</li>
<li>
<a href="../news/">
<span class="far fa far fa-newspaper"></span>
Changelog
</a>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li>
<a href="https://gitlab.com/msberends/AMR">
<span class="fab fa fab fa-gitlab"></span>
Source Code
</a>
</li>
<li>
<a href="../LICENSE-text.html">
<span class="fa fa-book"></span>
Licence
</a>
</li>
</ul>
</div>
<!--/.nav-collapse -->
</div>
<!--/.container -->
</div>
<!--/.navbar -->
</header><div class="row">
<div class="col-md-9 contents">
<div class="page-header toc-ignore">
<h1 data-toc-skip>How to conduct principal component analysis (PCA) for AMR</h1>
<h4 class="author">Matthijs S. Berends</h4>
<h4 class="date">15 April 2020</h4>
<small class="dont-index">Source: <a href="https://gitlab.com/msberends/AMR/blob/master/vignettes/PCA.Rmd"><code>vignettes/PCA.Rmd</code></a></small>
<div class="hidden name"><code>PCA.Rmd</code></div>
</div>
<p><strong>NOTE: This page will be updated soon, as the pca() function is currently being developed.</strong></p>
<div id="introduction" class="section level1">
<h1 class="hasAnchor">
<a href="#introduction" class="anchor"></a>Introduction</h1>
</div>
<div id="transforming" class="section level1">
<h1 class="hasAnchor">
<a href="#transforming" class="anchor"></a>Transforming</h1>
<p>For PCA, we need to transform our AMR data first. This is what the <code>example_isolates</code> data set in this package looks like:</p>
<div class="sourceCode" id="cb1"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/library.html">library</a></span>(<span class="no">AMR</span>)
<span class="fu"><a href="https://rdrr.io/r/base/library.html">library</a></span>(<span class="no">dplyr</span>)
<span class="fu"><a href="https://dplyr.tidyverse.org/reference/reexports.html">glimpse</a></span>(<span class="no">example_isolates</span>)
<span class="co"># Rows: 2,000</span>
<span class="co"># Columns: 49</span>
<span class="co"># $ date &lt;date&gt; 2002-01-02, 2002-01-03, 2002-01-07, 2002-01-07, 2002…</span>
<span class="co"># $ hospital_id &lt;fct&gt; D, D, B, B, B, B, D, D, B, B, D, D, D, D, D, B, B, B,…</span>
<span class="co"># $ ward_icu &lt;lgl&gt; FALSE, FALSE, TRUE, TRUE, TRUE, TRUE, FALSE, FALSE, T…</span>
<span class="co"># $ ward_clinical &lt;lgl&gt; TRUE, TRUE, FALSE, FALSE, FALSE, FALSE, TRUE, TRUE, F…</span>
<span class="co"># $ ward_outpatient &lt;lgl&gt; FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALS…</span>
<span class="co"># $ age &lt;dbl&gt; 65, 65, 45, 45, 45, 45, 78, 78, 45, 79, 67, 67, 71, 7…</span>
<span class="co"># $ gender &lt;chr&gt; "F", "F", "F", "F", "F", "F", "M", "M", "F", "F", "M"…</span>
<span class="co"># $ patient_id &lt;chr&gt; "A77334", "A77334", "067927", "067927", "067927", "06…</span>
<span class="co"># $ mo &lt;mo&gt; B_ESCHR_COLI, B_ESCHR_COLI, B_STPHY_EPDR, B_STPHY_EPDR…</span>
<span class="co"># $ PEN &lt;rsi&gt; R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R,…</span>
<span class="co"># $ OXA &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ FLC &lt;rsi&gt; NA, NA, R, R, R, R, S, S, R, S, S, S, NA, NA, NA, NA,…</span>
<span class="co"># $ AMX &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ AMC &lt;rsi&gt; I, I, NA, NA, NA, NA, S, S, NA, NA, S, S, I, I, R, I,…</span>
<span class="co"># $ AMP &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ TZP &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ CZO &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ FEP &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ CXM &lt;rsi&gt; I, I, R, R, R, R, S, S, R, S, S, S, S, S, NA, S, S, R…</span>
<span class="co"># $ FOX &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ CTX &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, S, S,…</span>
<span class="co"># $ CAZ &lt;rsi&gt; NA, NA, R, R, R, R, R, R, R, R, R, R, NA, NA, NA, S, …</span>
<span class="co"># $ CRO &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, S, S,…</span>
<span class="co"># $ GEN &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ TOB &lt;rsi&gt; NA, NA, NA, NA, NA, NA, S, S, NA, NA, NA, NA, S, S, N…</span>
<span class="co"># $ AMK &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ KAN &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ TMP &lt;rsi&gt; R, R, S, S, R, R, R, R, S, S, NA, NA, S, S, S, S, S, …</span>
<span class="co"># $ SXT &lt;rsi&gt; R, R, S, S, NA, NA, NA, NA, S, S, NA, NA, S, S, S, S,…</span>
<span class="co"># $ NIT &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ FOS &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ LNZ &lt;rsi&gt; R, R, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, R, R, R…</span>
<span class="co"># $ CIP &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, S, S, NA, NA, NA, NA,…</span>
<span class="co"># $ MFX &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ VAN &lt;rsi&gt; R, R, S, S, S, S, S, S, S, S, NA, NA, R, R, R, R, R, …</span>
<span class="co"># $ TEC &lt;rsi&gt; R, R, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, R, R, R…</span>
<span class="co"># $ TCY &lt;rsi&gt; R, R, S, S, S, S, S, S, S, I, S, S, NA, NA, I, R, R, …</span>
<span class="co"># $ TGC &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ DOX &lt;rsi&gt; NA, NA, S, S, S, S, S, S, S, NA, S, S, NA, NA, NA, R,…</span>
<span class="co"># $ ERY &lt;rsi&gt; R, R, R, R, R, R, S, S, R, S, S, S, R, R, R, R, R, R,…</span>
<span class="co"># $ CLI &lt;rsi&gt; NA, NA, NA, NA, NA, R, NA, NA, NA, NA, NA, NA, NA, NA…</span>
<span class="co"># $ AZM &lt;rsi&gt; R, R, R, R, R, R, S, S, R, S, S, S, R, R, R, R, R, R,…</span>
<span class="co"># $ IPM &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, S, S,…</span>
<span class="co"># $ MEM &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ MTR &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ CHL &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ COL &lt;rsi&gt; NA, NA, R, R, R, R, R, R, R, R, R, R, NA, NA, NA, R, …</span>
<span class="co"># $ MUP &lt;rsi&gt; NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…</span>
<span class="co"># $ RIF &lt;rsi&gt; R, R, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, R, R, R…</span></pre></body></html></div>
<p>Now to transform this to a data set with only resistance percentages per taxonomic order and genus:</p>
<div class="sourceCode" id="cb2"><html><body><pre class="r"><span class="no">resistance_data</span> <span class="kw">&lt;-</span> <span class="no">example_isolates</span> <span class="kw">%&gt;%</span>
<span class="fu"><a href="https://dplyr.tidyverse.org/reference/group_by.html">group_by</a></span>(<span class="kw">order</span> <span class="kw">=</span> <span class="fu"><a href="../reference/mo_property.html">mo_order</a></span>(<span class="no">mo</span>), <span class="co"># group on anything, like order</span>
<span class="kw">genus</span> <span class="kw">=</span> <span class="fu"><a href="../reference/mo_property.html">mo_genus</a></span>(<span class="no">mo</span>)) <span class="kw">%&gt;%</span> <span class="co"># and genus as we do here</span>
<span class="fu"><a href="https://dplyr.tidyverse.org/reference/summarise_all.html">summarise_if</a></span>(<span class="no">is.rsi</span>, <span class="no">resistance</span>) <span class="kw">%&gt;%</span> <span class="co"># then get resistance of all drugs</span>
<span class="fu"><a href="https://dplyr.tidyverse.org/reference/select.html">select</a></span>(<span class="no">order</span>, <span class="no">genus</span>, <span class="no">AMC</span>, <span class="no">CXM</span>, <span class="no">CTX</span>,
<span class="no">CAZ</span>, <span class="no">GEN</span>, <span class="no">TOB</span>, <span class="no">TMP</span>, <span class="no">SXT</span>) <span class="co"># and select only relevant columns</span>
<span class="fu"><a href="https://rdrr.io/r/utils/head.html">head</a></span>(<span class="no">resistance_data</span>)
<span class="co"># # A tibble: 6 x 10</span>
<span class="co"># # Groups: order [2]</span>
<span class="co"># order genus AMC CXM CTX CAZ GEN TOB TMP SXT</span>
<span class="co"># &lt;chr&gt; &lt;chr&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;</span>
<span class="co"># 1 (unknown orde… Micrococcoides NA NA NA NA NA NA NA NA</span>
<span class="co"># 2 Actinomycetal… Actinomyces NA NA NA NA NA NA NA NA</span>
<span class="co"># 3 Actinomycetal… Corynebacterium NA NA NA NA NA NA NA NA</span>
<span class="co"># 4 Actinomycetal… Dermabacter NA NA NA NA NA NA NA NA</span>
<span class="co"># 5 Actinomycetal… Micrococcus NA NA NA NA NA NA NA NA</span>
<span class="co"># 6 Actinomycetal… Propionibacter… NA NA NA NA NA NA NA NA</span></pre></body></html></div>
</div>
<div id="perform-principal-component-analysis" class="section level1">
<h1 class="hasAnchor">
<a href="#perform-principal-component-analysis" class="anchor"></a>Perform principal component analysis</h1>
<p>The new <code><a href="../reference/pca.html">pca()</a></code> function will automatically filter on rows that contain numeric values in all selected variables, so we now only need to do:</p>
<div class="sourceCode" id="cb3"><html><body><pre class="r"><span class="no">pca_result</span> <span class="kw">&lt;-</span> <span class="fu"><a href="../reference/pca.html">pca</a></span>(<span class="no">resistance_data</span>)
<span class="co"># NOTE: Columns selected for PCA: AMC/CXM/CTX/CAZ/GEN/TOB/TMP/SXT.</span>
<span class="co"># Total observations available: 7.</span></pre></body></html></div>
<p>The result can be reviewed with the good old <code><a href="https://rdrr.io/r/base/summary.html">summary()</a></code> function:</p>
<div class="sourceCode" id="cb4"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(<span class="no">pca_result</span>)
<span class="co"># Importance of components:</span>
<span class="co"># PC1 PC2 PC3 PC4 PC5 PC6 PC7</span>
<span class="co"># Standard deviation 2.1580 1.6783 0.61282 0.33017 0.20150 0.03190 2.123e-16</span>
<span class="co"># Proportion of Variance 0.5821 0.3521 0.04694 0.01363 0.00508 0.00013 0.000e+00</span>
<span class="co"># Cumulative Proportion 0.5821 0.9342 0.98117 0.99480 0.99987 1.00000 1.000e+00</span></pre></body></html></div>
<p>Good news. The first two components explain a total of 93.4% of the variance (see the PC1 and PC2 values of the <em>Proportion of Variance</em>. We can create a so-called biplot with the base R <code><a href="https://rdrr.io/r/stats/biplot.html">biplot()</a></code> function, to see which antimicrobial resistance per drug explain the difference per microorganism.</p>
</div>
<div id="plotting-the-results" class="section level1">
<h1 class="hasAnchor">
<a href="#plotting-the-results" class="anchor"></a>Plotting the results</h1>
<div class="sourceCode" id="cb5"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/stats/biplot.html">biplot</a></span>(<span class="no">pca_result</span>)</pre></body></html></div>
<p><img src="PCA_files/figure-html/unnamed-chunk-5-1.png" width="750"></p>
<p>But we cant see the explanation of the points. Perhaps this works better with our new <code><a href="../reference/ggplot_pca.html">ggplot_pca()</a></code> function, that automatically adds the right labels and even groups:</p>
<div class="sourceCode" id="cb6"><html><body><pre class="r"><span class="fu"><a href="../reference/ggplot_pca.html">ggplot_pca</a></span>(<span class="no">pca_result</span>)</pre></body></html></div>
<p><img src="PCA_files/figure-html/unnamed-chunk-6-1.png" width="750"></p>
<p>You can also print an ellipse per group, and edit the appearance:</p>
<div class="sourceCode" id="cb7"><html><body><pre class="r">
<span class="fu"><a href="../reference/ggplot_pca.html">ggplot_pca</a></span>(<span class="no">pca_result</span>, <span class="kw">ellipse</span> <span class="kw">=</span> <span class="fl">TRUE</span>) +
<span class="kw pkg">ggplot2</span><span class="kw ns">::</span><span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">title</span> <span class="kw">=</span> <span class="st">"An AMR/PCA biplot!"</span>)</pre></body></html></div>
<p><img src="PCA_files/figure-html/unnamed-chunk-7-1.png" width="750"></p>
</div>
</div>
<div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar">
<nav id="toc" data-toggle="toc"><h2 data-toc-skip>Contents</h2>
</nav>
</div>
</div>
<footer><div class="copyright">
<p>Developed by <a href="https://www.rug.nl/staff/m.s.berends/">Matthijs S. Berends</a>, <a href="https://www.rug.nl/staff/c.f.luz/">Christian F. Luz</a>, <a href="https://www.rug.nl/staff/a.w.friedrich/">Alexander W. Friedrich</a>, <a href="https://www.rug.nl/staff/b.sinha/">Bhanu N. M. Sinha</a>, <a href="https://www.rug.nl/staff/c.j.albers/">Casper J. Albers</a>, <a href="https://www.rug.nl/staff/c.glasner/">Corinna Glasner</a>.</p>
</div>
<div class="pkgdown">
<p>Site built with <a href="https://pkgdown.r-lib.org/">pkgdown</a> 1.5.0.</p>
</div>
</footer>
</div>
</body>
</html>