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1406 lines (1231 loc) · 106 KB
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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta charset="utf-8">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1">
<style type="text/css">
@font-face {
font-family: octicons-link;
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}
body {
-webkit-text-size-adjust: 100%;
text-size-adjust: 100%;
color: #333;
font-family: "Helvetica Neue", Helvetica, "Segoe UI", Arial, freesans, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";
font-size: 16px;
line-height: 1.6;
word-wrap: break-word;
}
a {
background-color: transparent;
}
a:active,
a:hover {
outline: 0;
}
strong {
font-weight: bold;
}
h1 {
font-size: 2em;
margin: 0.67em 0;
}
img {
border: 0;
}
hr {
box-sizing: content-box;
height: 0;
}
pre {
overflow: auto;
}
code,
kbd,
pre {
font-family: monospace, monospace;
font-size: 1em;
}
input {
color: inherit;
font: inherit;
margin: 0;
}
html input[disabled] {
cursor: default;
}
input {
line-height: normal;
}
input[type="checkbox"] {
box-sizing: border-box;
padding: 0;
}
table {
border-collapse: collapse;
border-spacing: 0;
}
td,
th {
padding: 0;
}
* {
box-sizing: border-box;
}
input {
font: 13px / 1.4 Helvetica, arial, nimbussansl, liberationsans, freesans, clean, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol";
}
a {
color: #4078c0;
text-decoration: none;
}
a:hover,
a:active {
text-decoration: underline;
}
hr {
height: 0;
margin: 15px 0;
overflow: hidden;
background: transparent;
border: 0;
border-bottom: 1px solid #ddd;
}
hr:before {
display: table;
content: "";
}
hr:after {
display: table;
clear: both;
content: "";
}
h1,
h2,
h3,
h4,
h5,
h6 {
margin-top: 15px;
margin-bottom: 15px;
line-height: 1.1;
}
h1 {
font-size: 30px;
}
h2 {
font-size: 21px;
}
h3 {
font-size: 16px;
}
h4 {
font-size: 14px;
}
h5 {
font-size: 12px;
}
h6 {
font-size: 11px;
}
blockquote {
margin: 0;
}
ul,
ol {
padding: 0;
margin-top: 0;
margin-bottom: 0;
}
ol ol,
ul ol {
list-style-type: lower-roman;
}
ul ul ol,
ul ol ol,
ol ul ol,
ol ol ol {
list-style-type: lower-alpha;
}
dd {
margin-left: 0;
}
code {
font-family: Consolas, "Liberation Mono", Menlo, Courier, monospace;
font-size: 12px;
}
pre {
margin-top: 0;
margin-bottom: 0;
font: 12px Consolas, "Liberation Mono", Menlo, Courier, monospace;
}
.select::-ms-expand {
opacity: 0;
}
.octicon {
font: normal normal normal 16px/1 octicons-link;
display: inline-block;
text-decoration: none;
text-rendering: auto;
-webkit-font-smoothing: antialiased;
-moz-osx-font-smoothing: grayscale;
-webkit-user-select: none;
-moz-user-select: none;
-ms-user-select: none;
user-select: none;
}
.octicon-link:before {
content: '\f05c';
}
.markdown-body:before {
display: table;
content: "";
}
.markdown-body:after {
display: table;
clear: both;
content: "";
}
.markdown-body>*:first-child {
margin-top: 0 !important;
}
.markdown-body>*:last-child {
margin-bottom: 0 !important;
}
a:not([href]) {
color: inherit;
text-decoration: none;
}
.anchor {
display: inline-block;
padding-right: 2px;
margin-left: -18px;
}
.anchor:focus {
outline: none;
}
h1,
h2,
h3,
h4,
h5,
h6 {
margin-top: 1em;
margin-bottom: 16px;
font-weight: bold;
line-height: 1.4;
}
h1 .octicon-link,
h2 .octicon-link,
h3 .octicon-link,
h4 .octicon-link,
h5 .octicon-link,
h6 .octicon-link {
color: #000;
vertical-align: middle;
visibility: hidden;
}
h1:hover .anchor,
h2:hover .anchor,
h3:hover .anchor,
h4:hover .anchor,
h5:hover .anchor,
h6:hover .anchor {
text-decoration: none;
}
h1:hover .anchor .octicon-link,
h2:hover .anchor .octicon-link,
h3:hover .anchor .octicon-link,
h4:hover .anchor .octicon-link,
h5:hover .anchor .octicon-link,
h6:hover .anchor .octicon-link {
visibility: visible;
}
h1 {
padding-bottom: 0.3em;
font-size: 2.25em;
line-height: 1.2;
border-bottom: 1px solid #eee;
}
h1 .anchor {
line-height: 1;
}
h2 {
padding-bottom: 0.3em;
font-size: 1.75em;
line-height: 1.225;
border-bottom: 1px solid #eee;
}
h2 .anchor {
line-height: 1;
}
h3 {
font-size: 1.5em;
line-height: 1.43;
}
h3 .anchor {
line-height: 1.2;
}
h4 {
font-size: 1.25em;
}
h4 .anchor {
line-height: 1.2;
}
h5 {
font-size: 1em;
}
h5 .anchor {
line-height: 1.1;
}
h6 {
font-size: 1em;
color: #777;
}
h6 .anchor {
line-height: 1.1;
}
p,
blockquote,
ul,
ol,
dl,
table,
pre {
margin-top: 0;
margin-bottom: 16px;
}
hr {
height: 4px;
padding: 0;
margin: 16px 0;
background-color: #e7e7e7;
border: 0 none;
}
ul,
ol {
padding-left: 2em;
}
ul ul,
ul ol,
ol ol,
ol ul {
margin-top: 0;
margin-bottom: 0;
}
li>p {
margin-top: 16px;
}
dl {
padding: 0;
}
dl dt {
padding: 0;
margin-top: 16px;
font-size: 1em;
font-style: italic;
font-weight: bold;
}
dl dd {
padding: 0 16px;
margin-bottom: 16px;
}
blockquote {
padding: 0 15px;
color: #777;
border-left: 4px solid #ddd;
}
blockquote>:first-child {
margin-top: 0;
}
blockquote>:last-child {
margin-bottom: 0;
}
table {
display: block;
width: 100%;
overflow: auto;
word-break: normal;
word-break: keep-all;
}
table th {
font-weight: bold;
}
table th,
table td {
padding: 6px 13px;
border: 1px solid #ddd;
}
table tr {
background-color: #fff;
border-top: 1px solid #ccc;
}
table tr:nth-child(2n) {
background-color: #f8f8f8;
}
img {
max-width: 100%;
box-sizing: content-box;
background-color: #fff;
}
code {
padding: 0;
padding-top: 0.2em;
padding-bottom: 0.2em;
margin: 0;
font-size: 85%;
background-color: rgba(0,0,0,0.04);
border-radius: 3px;
}
code:before,
code:after {
letter-spacing: -0.2em;
content: "\00a0";
}
pre>code {
padding: 0;
margin: 0;
font-size: 100%;
word-break: normal;
white-space: pre;
background: transparent;
border: 0;
}
.highlight {
margin-bottom: 16px;
}
.highlight pre,
pre {
padding: 16px;
overflow: auto;
font-size: 85%;
line-height: 1.45;
background-color: #f7f7f7;
border-radius: 3px;
}
.highlight pre {
margin-bottom: 0;
word-break: normal;
}
pre {
word-wrap: normal;
}
pre code {
display: inline;
max-width: initial;
padding: 0;
margin: 0;
overflow: initial;
line-height: inherit;
word-wrap: normal;
background-color: transparent;
border: 0;
}
pre code:before,
pre code:after {
content: normal;
}
kbd {
display: inline-block;
padding: 3px 5px;
font-size: 11px;
line-height: 10px;
color: #555;
vertical-align: middle;
background-color: #fcfcfc;
border: solid 1px #ccc;
border-bottom-color: #bbb;
border-radius: 3px;
box-shadow: inset 0 -1px 0 #bbb;
}
.pl-c {
color: #969896;
}
.pl-c1,
.pl-s .pl-v {
color: #0086b3;
}
.pl-e,
.pl-en {
color: #795da3;
}
.pl-s .pl-s1,
.pl-smi {
color: #333;
}
.pl-ent {
color: #63a35c;
}
.pl-k {
color: #a71d5d;
}
.pl-pds,
.pl-s,
.pl-s .pl-pse .pl-s1,
.pl-sr,
.pl-sr .pl-cce,
.pl-sr .pl-sra,
.pl-sr .pl-sre {
color: #183691;
}
.pl-v {
color: #ed6a43;
}
.pl-id {
color: #b52a1d;
}
.pl-ii {
background-color: #b52a1d;
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<body>
<h1 id="a-quick-introduction-to-the-ergmito-r-package">A quick introduction to the ergmito R package</h1>
<p>George G Vega Yon SONIC Speaker Series<br>Northwestern University, IL<br>March 20, 2019</p>
<h2 id="contents">Contents</h2>
<ol>
<li>Setup</li>
</ol>
<p>Key parts of <code>ergmito</code></p>
<ol start="2">
<li><p>Likelihood function</p></li>
<li><p>Estimation function</p></li>
<li><p>Simulation</p></li>
</ol>
<p>Using gnet</p>
<ol start="5">
<li>A preview example</li>
</ol>
<h2 id="setup">Setup</h2>
<ul>
<li><p>Install the <code>ergmito</code> R package: Follow the instructions available on github <a href="https://github.com/muriteams/ergmito">https://github.com/muriteams/ergmito</a></p></li>
<li><p>Install the <code>gnet</code> R package: Follow the instructions available on github <a href="https://github.com/muriteams/ergmito">https://github.com/muriteams/ergmito</a></p></li>
<li><p>Install the <code>ergm</code> R package: <code>install.packages("ergm")</code></p></li>
</ul>
<h2 id="likelihood">Likelihood</h2>
<ul>
<li><p>The <code>ergmito</code> package’s core function consists on the likelihood function</p></li>
<li><p>The function itself is a wrapper of <code>ergm::ergm.allstats</code> which returns exactly that: all the statistics of a graph of a given size.</p></li>
<li><p>One important detail: We are not calculating exactly the same function as the one observed in the formula for ERGMs, we are using the fact that <strong>isomorphism</strong> occurs farily often, so instead we used weights:</p></li>
</ul>
<h2 id="optimization">Optimization</h2>
<ul>
<li><p>The <code>ergmito</code> function works in the following way:</p>
<ol>
<li><p>Reads in the data and creates a log-likelihood function together with the corresponding gradient function.</p></li>
<li><p>Uses the <code>stats::optim</code> function in R to maximize the log-likelihood. Some details:</p>
<ol>
<li><p>Uses the “BFGS” algorithm for optimization (standard in R)</p></li>
<li><p>Instead of approximating the gradient, we pass the true analytical gradient (since log-likelihood can be computed exactly, so this can be).</p></li>
<li><p>Solves the same optimization problem 5 times (see the <code>ntries</code> parameter) just to make sure that we are getting a maximum.</p></li>
</ol></li>
</ol>
<p>The last point after experiencing some convergence issues in the optmization algorithms which were not yielding consistent results. Earlier we were using the limited memory bounded version of BFGS which turned out not to be very appropiate for us.</p></li>
</ul>
<h2 id="example-1-the-fivenets-network">Example 1: The fivenets network</h2>
<ul>
<li><p>This network was generated using the sampler function available in the package (<code>ergmito::new_rergmito</code>)</p></li>
<li><p>The population parameters in this dataset are <code>par[edges] = -2.0</code> and <code>par[nodematch.female] = 0.2</code>.</p></li>
<li><p>Before fitting the true model, let’s take a look at the simplest example, estimating the probability of a bernoulli graph:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb1-1" title="1"><span class="kw">library</span>(ergmito)</a>
<a class="sourceLine" id="cb1-2" title="2"></a>
<a class="sourceLine" id="cb1-3" title="3"><span class="kw">data</span>(<span class="st">"fivenets"</span>, <span class="dt">package=</span><span class="st">"ergmito"</span>)</a>
<a class="sourceLine" id="cb1-4" title="4"></a>
<a class="sourceLine" id="cb1-5" title="5">(model0 <-<span class="st"> </span><span class="kw">ergmito</span>(fivenets <span class="op">~</span><span class="st"> </span>edges))</a></code></pre></div>
<pre><code>##
## ERGMito estimates
##
## Coefficients:
## edges
## -0.6931
</code></pre></li>
<li><p>Since this is a very simple model (no markovian dependency, i.e. edges are independent), we could have just estimated this by treating the total number of observed ties as a binomal random variable. In such case, the MLE estimate is simply the total number of edges over the possible number of edges. We can compare the odds of this model with the binomial MLE estimate (should obtain the same):</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb3-1" title="1"><span class="co"># Calculating the probability of observing a tie</span></a>
<a class="sourceLine" id="cb3-2" title="2">odds <-<span class="st"> </span><span class="kw">exp</span>(<span class="kw">coef</span>(model0))</a>
<a class="sourceLine" id="cb3-3" title="3">odds<span class="op">/</span>(<span class="dv">1</span> <span class="op">+</span><span class="st"> </span>odds)</a></code></pre></div>
<pre><code>## edges
## 0.3333333
</code></pre>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb5-1" title="1"><span class="co"># Figuring out MLE estimate</span></a>
<a class="sourceLine" id="cb5-2" title="2">n <-<span class="st"> </span><span class="kw">nvertex</span>(model0) <span class="co"># This function returns the number of vertices</span></a>
<a class="sourceLine" id="cb5-3" title="3">m <-<span class="st"> </span><span class="kw">nedges</span>(model0) <span class="co"># this other function, the number of ties (see ?nvertex)</span></a>
<a class="sourceLine" id="cb5-4" title="4"></a>
<a class="sourceLine" id="cb5-5" title="5"><span class="co"># Since model0 has multiple networks, both m and n are vectors</span></a>
<a class="sourceLine" id="cb5-6" title="6">n; m</a></code></pre></div>
<pre><code>## [1] 4 4 4 4 4
## [1] 2 7 4 5 2
</code></pre>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb7-1" title="1"><span class="co"># MLE for the probability of a tie (this SHOULD match the above)</span></a>
<a class="sourceLine" id="cb7-2" title="2"><span class="kw">mean</span>(m<span class="op">/</span>(n<span class="op">*</span>(n<span class="dv">-1</span>)))</a></code></pre></div>
<pre><code>## [1] 0.3333333
</code></pre></li>
<li><p>Just like <code>ergm</code> objects, <code>ergmito</code> objects have a lot of components. Let’s take a look at what the object includes:</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb9-1" title="1"><span class="kw">str</span>(model0, <span class="dt">max.level =</span> <span class="dv">1</span>)</a></code></pre></div>
<pre><code>## List of 12
## $ call : language ergmito(model = fivenets ~ edges)
## $ coef : Named num -0.693
## ..- attr(*, "names")= chr "edges"
## $ iterations: Named int 12
## ..- attr(*, "names")= chr "function"
## $ mle.lik :Class 'logLik' : -38 (df=1)
## $ null.lik :Class 'logLik' : -42 (df=0)
## $ covar : num [1, 1] 0.075
## ..- attr(*, "dimnames")=List of 2
## $ coef.init : num [1:5, 1] -0.464 1.347 -0.848 -1.254 -0.294
## $ formulae :List of 7
## ..- attr(*, "class")= chr "ergmito_loglik"
## $ nobs : num 60
## $ network :List of 5
## $ init : num [1:5, 1] -0.464 1.347 -0.848 -1.254 -0.294
## $ optim.out :List of 6
## - attr(*, "class")= chr [1:2] "ergmito" "ergm"
</code></pre>
<p>The <code>optim.out</code> component is what the <code>stats::optim</code> function returns</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb11-1" title="1"><span class="kw">str</span>(model0<span class="op">$</span>optim.out)</a></code></pre></div>
<pre><code>## List of 6
## $ par : Named num -0.693
## ..- attr(*, "names")= chr "edges"
## $ value : num -38.2
## $ counts : Named int [1:2] 12 6
## ..- attr(*, "names")= chr [1:2] "function" "gradient"
## $ convergence: int 0
## $ message : NULL
## $ hessian : num [1, 1] -13.3
</code></pre>
<p>The <code>formulae</code> object is what actually holds the log-likelihood function and the gradient. It has its own print method:</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb13-1" title="1">model0<span class="op">$</span>formulae</a></code></pre></div>
<pre><code>## ergmito log-likelihood function
## Number of networks: 5
## Model: fivenets ~ edges
## Available elements by using the $ operator:
## loglik: function (params, stats = NULL) grad : function (params, stats = NULL)
</code></pre></li>
</ul>
<hr />
<ul>
<li><p>We can use the different methods available for <code>ergmito</code> class object:</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb15-1" title="1"><span class="kw">confint</span>(model0)</a></code></pre></div>
<pre><code>## 2.5 % 97.5 %
## edges -1.229905 -0.1563889
</code></pre>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb17-1" title="1"><span class="kw">vcov</span>(model0)</a></code></pre></div>
<pre><code>## edges
## edges 0.075
</code></pre>
<div class="sourceCode" id="cb19"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb19-1" title="1"><span class="kw">logLik</span>(model0)</a></code></pre></div>
<pre><code>## 'log Lik.' -38.19085 (df=1)
</code></pre>
<div class="sourceCode" id="cb21"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb21-1" title="1"><span class="kw">AIC</span>(model0)</a></code></pre></div>
<pre><code>## [1] 78.3817
</code></pre>
<div class="sourceCode" id="cb23"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb23-1" title="1"><span class="kw">BIC</span>(model0)</a></code></pre></div>
<pre><code>## [1] 80.47604
</code></pre>
<div class="sourceCode" id="cb25"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb25-1" title="1"><span class="kw">summary</span>(model0)</a></code></pre></div>
<pre><code>## $coefs
## Estimate Std. Error z value Pr(>|z|)
## edges -0.6931472 0.2738613 -2.531016 0.01137328
##
## $aic
## [1] 78.3817
##
## $bic
## [1] 80.47604
##
## $model
## [1] "fivenets ~ edges"
##
## attr(,"class")
## [1] "ergmito_summary"
</code></pre>
<div class="sourceCode" id="cb27"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb27-1" title="1"><span class="kw">nobs</span>(model0)</a></code></pre></div>
<pre><code>## [1] 60
</code></pre></li>
<li><p>Now, let’s try to fit other models. We will store the results and try to add them up together in a single output table</p>
<div class="sourceCode" id="cb29"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb29-1" title="1">model1 <-<span class="st"> </span><span class="kw">ergmito</span>(fivenets <span class="op">~</span><span class="st"> </span>edges <span class="op">+</span><span class="st"> </span>ttriad)</a>
<a class="sourceLine" id="cb29-2" title="2">model2 <-<span class="st"> </span><span class="kw">ergmito</span>(fivenets <span class="op">~</span><span class="st"> </span>edges <span class="op">+</span><span class="st"> </span><span class="kw">nodeicov</span>(<span class="st">"female"</span>))</a>
<a class="sourceLine" id="cb29-3" title="3">model3 <-<span class="st"> </span><span class="kw">ergmito</span>(fivenets <span class="op">~</span><span class="st"> </span>edges <span class="op">+</span><span class="st"> </span><span class="kw">nodematch</span>(<span class="st">"female"</span>))</a></code></pre></div></li>
</ul>
<hr />
<ul>
<li><p>Furthermore, the package includes methods for the <a href="http://github.com/leifeld/texreg/"><strong>texreg</strong></a> R package, so we can export lists of fitted models directly</p>
<div class="sourceCode" id="cb30"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb30-1" title="1"><span class="kw">library</span>(texreg)</a></code></pre></div>
<pre><code>## Version: 1.36.23
## Date: 2017-03-03
## Author: Philip Leifeld (University of Glasgow)
##
## Please cite the JSS article in your publications -- see citation("texreg").
</code></pre>
<div class="sourceCode" id="cb32"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb32-1" title="1"><span class="kw">htmlreg</span>(</a>
<a class="sourceLine" id="cb32-2" title="2"> <span class="dt">l =</span> <span class="kw">list</span>(<span class="dt">Baseline =</span> model0, <span class="dt">Balance =</span> model1, <span class="dt">icovFemale =</span> model2, <span class="dt">Homophily =</span> model3),</a>
<a class="sourceLine" id="cb32-3" title="3"> <span class="dt">doctype =</span> <span class="ot">FALSE</span>,</a>
<a class="sourceLine" id="cb32-4" title="4"> <span class="dt">caption =</span> <span class="st">"My first ergmito table"</span></a>
<a class="sourceLine" id="cb32-5" title="5"> )</a></code></pre></div>
<table cellspacing="0" align="center" style="border: none;">
<caption align="bottom" style="margin-top:0.3em;">
<p>My first ergmito table</p>
</caption>
<tr>
<th style="text-align: left; border-top: 2px solid black; border-bottom: 1px solid black; padding-right: 12px;">
<p><b></b></p>
</th>
<th style="text-align: left; border-top: 2px solid black; border-bottom: 1px solid black; padding-right: 12px;">
<p><b>Baseline</b></p>
</th>
<th style="text-align: left; border-top: 2px solid black; border-bottom: 1px solid black; padding-right: 12px;">
<p><b>Balance</b></p>
</th>
<th style="text-align: left; border-top: 2px solid black; border-bottom: 1px solid black; padding-right: 12px;">
<p><b>icovFemale</b></p>
</th>
<th style="text-align: left; border-top: 2px solid black; border-bottom: 1px solid black; padding-right: 12px;">
<p><b>Homophily</b></p>
</th>
</tr>
<tr>
<td style="padding-right: 12px; border: none;">
<p>edges</p>
</td>
<td style="padding-right: 12px; border: none;">
<p>-0.69<sup style="vertical-align: 0px;">*</sup></p>
</td>
<td style="padding-right: 12px; border: none;">
<p>-0.90<sup style="vertical-align: 0px;">*</sup></p>
</td>
<td style="padding-right: 12px; border: none;">
<p>-0.31</p>
</td>
<td style="padding-right: 12px; border: none;">
<p>-1.70<sup style="vertical-align: 0px;">**</sup></p>
</td>
</tr>
<tr>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
<p>(0.27)</p>
</td>
<td style="padding-right: 12px; border: none;">
<p>(0.39)</p>
</td>
<td style="padding-right: 12px; border: none;">
<p>(0.35)</p>
</td>
<td style="padding-right: 12px; border: none;">
<p>(0.54)</p>
</td>
</tr>
<tr>
<td style="padding-right: 12px; border: none;">
<p>ttriple</p>
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
<p>0.27</p>
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
</tr>
<tr>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
<p>(0.36)</p>
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
</tr>
<tr>
<td style="padding-right: 12px; border: none;">
<p>nodeicov.female</p>
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
<p>-0.95</p>
</td>
<td style="padding-right: 12px; border: none;">
</td>
</tr>
<tr>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
<p>(0.58)</p>
</td>
<td style="padding-right: 12px; border: none;">
</td>
</tr>
<tr>
<td style="padding-right: 12px; border: none;">
<p>nodematch.female</p>
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">
</td>
<td style="padding-right: 12px; border: none;">