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<h1 class="title toc-ignore">Intervalles de confiance</h1>
</div>
<div id="TOC">
<ul>
<li><a href="#ic_moyenne">Intervalle de confiance d’une moyenne</a></li>
<li><a href="#ic_proportion">Intervalle de confiance d’une proportion</a></li>
<li><a href="#survey">Données pondérées et l’extension survey</a></li>
</ul>
</div>
<p>Nous utiliserons dans ce chapitre les données de l’enquête <em>Histoire de vie 2003</em> fournies avec l’extension <code class="pkg">questionr</code>.</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>(questionr)</a>
<a class="sourceLine" id="cb1-2" title="2"><span class="kw">data</span>(<span class="st">"hdv2003"</span>)</a>
<a class="sourceLine" id="cb1-3" title="3">d <-<span class="st"> </span>hdv2003</a></code></pre></div>
<div id="ic_moyenne" class="section level2">
<h2>Intervalle de confiance d’une moyenne</h2>
<p>L’<dfn>intervalle de confiance d’une moyenne</dfn><dfn data-index="moyenne, intervalle de confiance"></dfn><dfn data-index="intervalle de confiance"></dfn> peut être calculé avec la fonction <code data-pkg="stats">t.test</code> (fonction qui permet également de réaliser un test <em>t</em> de Student comme nous le verrons dans le chapitre dédié aux <a href="comparaisons-moyennes-et-proportions.html">comparaisons de moyennes</a>) :</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb2-1" title="1"><span class="kw">t.test</span>(d<span class="op">$</span>heures.tv)</a></code></pre></div>
<pre><code>
One Sample t-test
data: d$heures.tv
t = 56.505, df = 1994, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
2.168593 2.324540
sample estimates:
mean of x
2.246566 </code></pre>
<p>Le niveau de confiance peut être précisé via l’argument <code>conf.level</code> :</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb4-1" title="1"><span class="kw">t.test</span>(d<span class="op">$</span>heures.tv, <span class="dt">conf.level =</span> <span class="fl">0.9</span>)</a></code></pre></div>
<pre><code>
One Sample t-test
data: d$heures.tv
t = 56.505, df = 1994, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
90 percent confidence interval:
2.181138 2.311995
sample estimates:
mean of x
2.246566 </code></pre>
<p>Le nombre d’heures moyennes à regarder la télévision parmi les enquêtés s’avère être de 2,2 heures, avec un intervalle de confiance à 95 % de [2,17 - 2,33] et un intervalle de confiance à 90 % de [2,18 - 2,31].</p>
</div>
<div id="ic_proportion" class="section level2">
<h2>Intervalle de confiance d’une proportion</h2>
<p>La fonction <code data-pkg="stats">prop.test</code> permet de calculer l’<dfn>intervalle de confiance d’une proportion</dfn><dfn data-index="proportion, intervalle de confiance"></dfn><dfn data-index="intervalle de confiance"></dfn>. Une première possibilité consiste à lui transmettre une table à une dimension et deux entrées. Par exemple, si l’on s’intéresse à la proportion de personnes ayant pratiqué une activité physique au cours des douze derniers mois :</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb6-1" title="1"><span class="kw">freq</span>(d<span class="op">$</span>sport)</a></code></pre></div>
<div data-pagedtable="false">
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<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb7-1" title="1"><span class="kw">prop.test</span>(<span class="kw">table</span>(d<span class="op">$</span>sport))</a></code></pre></div>
<pre><code>
1-sample proportions test with continuity correction
data: table(d$sport), null probability 0.5
X-squared = 152.9, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
0.6169447 0.6595179
sample estimates:
p
0.6385 </code></pre>
<p>On remarquera que la fonction a calculé l’intervalle de confiance correspondant à la première entrée du tableau, autrement dit celui de la proportion d’enquêtés n’ayant pas pratiqué une activité sportive. Or, nous sommes intéressé par la proportion complémentaire, à savoir celle d’enquêtés ayant pratiqué une activité sportive. On peut dès lors modifier l’ordre de la table en indiquant notre modalité d’intérêt avec la fonction <code data-pkg="stats">relevel</code> ou bien indiquer à <code data-pkg="stats">prop.test</code> d’abord le nombre de succès puis l’effectif total :</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb9-1" title="1"><span class="kw">prop.test</span>(<span class="kw">table</span>(<span class="kw">relevel</span>(d<span class="op">$</span>sport, <span class="st">"Oui"</span>)))</a></code></pre></div>
<pre><code>
1-sample proportions test with continuity correction
data: table(relevel(d$sport, "Oui")), null probability 0.5
X-squared = 152.9, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
0.3404821 0.3830553
sample estimates:
p
0.3615 </code></pre>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb11-1" title="1"><span class="kw">prop.test</span>(<span class="kw">sum</span>(d<span class="op">$</span>sport <span class="op">==</span><span class="st"> "Oui"</span>), <span class="kw">length</span>(d<span class="op">$</span>sport))</a></code></pre></div>
<pre><code>
1-sample proportions test with continuity correction
data: sum(d$sport == "Oui") out of length(d$sport), null probability 0.5
X-squared = 152.9, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
0.3404821 0.3830553
sample estimates:
p
0.3615 </code></pre>
<p>Enfin, le niveau de confiance peut être modifié via l’argument <code>conf.level</code> :</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb13-1" title="1"><span class="kw">prop.test</span>(<span class="kw">table</span>(<span class="kw">relevel</span>(d<span class="op">$</span>sport, <span class="st">"Oui"</span>)), <span class="dt">conf.level =</span> <span class="fl">0.9</span>)</a></code></pre></div>
<pre><code>
1-sample proportions test with continuity correction
data: table(relevel(d$sport, "Oui")), null probability 0.5
X-squared = 152.9, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is not equal to 0.5
90 percent confidence interval:
0.3437806 0.3795989
sample estimates:
p
0.3615 </code></pre>
<div class="note">
<p>Il existe de nombreuses manières de calculer un intervalle de confiance pour une proportion. En l’occurence, l’intervalle calculé par <code data-pkg="stats">prop.test</code> correspond dans le cas présent à un intervalle bilatéral selon la méthode des scores de Wilson avec correction de continuité. Pour plus d’information, on pourra lire <a href="http://joseph.larmarange.net/?Intervalle-de-confiance-bilateral" class="uri">http://joseph.larmarange.net/?Intervalle-de-confiance-bilateral</a>.</p>
</div>
<div class="note">
<p>Pour se simplifier un peu la vie, le package <code class="pkg">JLutils</code> propose une fonction <code data-pkg="JLutils">prop.ci</code> (et ses deux variantes <code data-pkg="JLutils">prop.ci.lower</code> et <code data-pkg="JLutils">prop.ci.upper</code>) permettant d’appeler plus facilement <code data-pkg="stats">prop.test</code> et renvoyant directement l’intervalle de confiance.</p>
<p><code class="pkg">JLutils</code> n’étant disponible que sur <a href="https://github.com/larmarange/JLutils">GitHub</a>, on aura recours au package <code class="pkg">devtools</code> et à sa fonction <code data-pkg="devtools">install_github</code> pour l’installer :</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">library</span>(devtools)</a>
<a class="sourceLine" id="cb15-2" title="2"><span class="kw">install_github</span>(<span class="st">"larmarange/JLutils"</span>)</a></code></pre></div>
<p><code data-pkg="JLutils">prop.ci</code> fonction accepte directement un tri à plat obtenu avec <code data-pkg="base">table</code>, un vecteur de données, un vecteur logique (issu d’une condition), ou bien le nombre de succès et le nombre total d’essais. Voir les exemples ci-après :</p>
<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb16-1" title="1"><span class="kw">library</span>(JLutils)</a>
<a class="sourceLine" id="cb16-2" title="2"><span class="kw">freq</span>(d<span class="op">$</span>sport)</a></code></pre></div>
<div data-pagedtable="false">
<script data-pagedtable-source type="application/json">
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</script>
</div>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb17-1" title="1"><span class="kw">prop.ci</span>(d<span class="op">$</span>sport)</a></code></pre></div>
<pre><code>[1] 0.6169447 0.6595179</code></pre>
<div class="sourceCode" id="cb19"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb19-1" title="1"><span class="kw">prop.ci.lower</span>(d<span class="op">$</span>sport)</a></code></pre></div>
<pre><code>[1] 0.6169447</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">prop.ci.upper</span>(d<span class="op">$</span>sport)</a></code></pre></div>
<pre><code>[1] 0.6595179</code></pre>
<div class="sourceCode" id="cb23"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb23-1" title="1"><span class="kw">prop.ci</span>(d<span class="op">$</span>sport, <span class="dt">conf.level =</span> <span class="fl">0.9</span>)</a></code></pre></div>
<pre><code>[1] 0.6204011 0.6562194</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">prop.ci</span>(<span class="kw">table</span>(d<span class="op">$</span>sport))</a></code></pre></div>
<pre><code>[1] 0.6169447 0.6595179</code></pre>
<div class="sourceCode" id="cb27"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb27-1" title="1"><span class="kw">prop.ci</span>(d<span class="op">$</span>sport <span class="op">==</span><span class="st"> "Non"</span>)</a></code></pre></div>
<pre><code>[1] 0.6169447 0.6595179</code></pre>
<div class="sourceCode" id="cb29"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb29-1" title="1"><span class="kw">prop.ci</span>(d<span class="op">$</span>sport <span class="op">==</span><span class="st"> "Oui"</span>)</a></code></pre></div>
<pre><code>[1] 0.3404821 0.3830553</code></pre>
<div class="sourceCode" id="cb31"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb31-1" title="1"><span class="kw">prop.ci.lower</span>(<span class="kw">c</span>(<span class="dv">1277</span>, <span class="dv">723</span>), <span class="dt">n =</span> <span class="dv">2000</span>)</a></code></pre></div>
<pre><code>[1] 0.6169447 0.3404821</code></pre>
<div class="sourceCode" id="cb33"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb33-1" title="1"><span class="kw">prop.ci.upper</span>(<span class="kw">c</span>(<span class="dv">1277</span>, <span class="dv">723</span>), <span class="dt">n =</span> <span class="dv">2000</span>)</a></code></pre></div>
<pre><code>[1] 0.6595179 0.3830553</code></pre>
</div>
</div>
<div id="survey" class="section level2">
<h2>Données pondérées et l’extension survey</h2>
<p>Lorsque l’on utilise des données pondérées définies à l’aide de l’extension <code class="pkg">survey</code><a href="#fn1" class="footnote-ref" id="fnref1"><sup>1</sup></a>, l’<dfn>intervalle de confiance d’une moyenne</dfn><dfn data-index="moyenne, intervalle de confiance"></dfn> s’obtient avec <code data-pkg="survey" data-rdoc="surveysummary">confint</code> et celui d’une <dfn data-index="intervalle de confiance d'une proportion">proportion</dfn><dfn data-index="proportion, intervalle de confiance"></dfn> avec <code data-pkg="survey">svyciprop</code>.</p>
<p>Quelques exemples :</p>
<div class="sourceCode" id="cb35"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb35-1" title="1"><span class="kw">library</span>(survey)</a>
<a class="sourceLine" id="cb35-2" title="2">dw <-<span class="st"> </span><span class="kw">svydesign</span>(<span class="dt">ids =</span> <span class="op">~</span><span class="dv">1</span>, <span class="dt">data =</span> d, <span class="dt">weights =</span> <span class="op">~</span>poids)</a>
<a class="sourceLine" id="cb35-3" title="3"><span class="kw">svymean</span>(<span class="op">~</span>age, dw)</a></code></pre></div>
<pre><code> mean SE
age 46.347 0.5284</code></pre>
<div class="sourceCode" id="cb37"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb37-1" title="1"><span class="kw">confint</span>(<span class="kw">svymean</span>(<span class="op">~</span>age, dw)) <span class="co"># Intervalle de confiance d'une moyenne</span></a></code></pre></div>
<pre><code> 2.5 % 97.5 %
age 45.3117 47.38282</code></pre>
<div class="sourceCode" id="cb39"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb39-1" title="1"><span class="kw">confint</span>(<span class="kw">svyby</span>(<span class="op">~</span>age, <span class="op">~</span>sexe, dw, svymean)) <span class="co"># Intervalles de confiance pour chaque sexe</span></a></code></pre></div>
<pre><code> 2.5 % 97.5 %
Homme 43.74781 46.65618
Femme 45.88867 48.79758</code></pre>
<div class="sourceCode" id="cb41"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb41-1" title="1"><span class="kw">freq</span>(<span class="kw">svytable</span>(<span class="op">~</span>sexe, dw))</a></code></pre></div>
<div data-pagedtable="false">
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<div class="sourceCode" id="cb42"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb42-1" title="1"><span class="kw">svyciprop</span>(<span class="op">~</span>sexe, dw) <span class="co"># Intervalle de confiance d'une proportion</span></a></code></pre></div>
<pre><code> 2.5% 97.5%
sexe 0.535 0.507 0.56</code></pre>
</div>
<div class="footnotes">
<hr />
<ol>
<li id="fn1"><p>Voir le chapitre dédié aux <a href="donnees-ponderees.html#survey">données pondérées</a>.<a href="#fnref1" class="footnote-back">↩</a></p></li>
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}); // Si <pre> est le dernier enfant de son parent
// Ajout liens rdocumentation et tooltip
$("code[data-pkg]").each(function( index ) {
pkg = $(this).attr('data-pkg');
if ($(this).attr('data-rdoc') !== undefined) {
rdocumentation = $(this).attr('data-rdoc');
} else {
rdocumentation = $(this).text();
}
fonction = $(this).text();
//$(this).wrap('<a href="http://www.rdocumentation.org/packages/'+pkg+'/functions/'+rdocumentation+'">');
$(this).wrap('<a href="http://rdrr.io/pkg/'+pkg+'/sym/'+rdocumentation+'">');
$(this).attr('data-toggle','tooltip');
$(this).attr('data-placement','top');
$(this).attr('title','package : ' + pkg);
$('[data-toggle="tooltip"]').tooltip();
});
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//$(this).wrap('<a href="http://www.rdocumentation.org/packages/'+$(this).text()+'">');
$(this).wrap('<a href="http://rdrr.io/pkg/'+$(this).text()+'">');
});
// Figures
$("figure").each(function( index ) {
if ($(this).children("figcaption").length > 0)
$(this).children("figcaption:first").prepend('<span class="figure-number">Figure '+(index+1)+'.</span> ');
else
$(this).append($("<figcaption>").append('<span class="figure-number">Figure '+(index+1)+'</span>'));
});
// Colorbox
/*----
jQuery('article div img').colorbox({
maxWidth: '90%',
maxHeight: '90%',
rel: 'figures',
current: "",
href: function(){
return $(this).attr('src');
},
title: function(){
return $(this).attr('alt');
}
});
jQuery('article div img').css('cursor', 'pointer');
jQuery('figure img').colorbox({
maxWidth: '90%',
maxHeight: '90%',
rel: 'figures',
current: "",
href: function(){
return $(this).attr('src');
},
title: function(){
return $(this).parent().children("figcaption").text();
}
});
jQuery('figure img').css('cursor', 'pointer');
-----*/
/* Clipboard --------------------------*/
function changeTooltipMessage(element, msg) {
var tooltipOriginalTitle=element.getAttribute('data-original-title');
element.setAttribute('data-original-title', msg);
$(element).tooltip('show');
element.setAttribute('data-original-title', tooltipOriginalTitle);
}
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var copyButton = "<button type='button' class='btn btn-primary btn-copy-ex' type = 'submit' title='Copier dans le presse-papier' aria-label='Copier dans le presse-papier' data-toggle='tooltip' data-placement='left auto' data-trigger='hover' data-clipboard-copy><i class='fa fa-copy'></i></button>";
$(".examples, div.sourceCode").addClass("hasCopyButton");
// Insert copy buttons:
$(copyButton).prependTo(".hasCopyButton");
// Initialize tooltips:
$('.btn-copy-ex').tooltip({container: 'body'});
// Initialize clipboard:
var clipboardBtnCopies = new ClipboardJS('[data-clipboard-copy]', {
text: function(trigger) {
return trigger.parentNode.textContent;
}
});
clipboardBtnCopies.on('success', function(e) {
changeTooltipMessage(e.trigger, 'Copié !');
e.clearSelection();
});
clipboardBtnCopies.on('error', function() {
changeTooltipMessage(e.trigger,'Appuyez sur Ctrl+C ou Command+C pour copier');
});
});
}
// Bigfoot
$(".footnotes > li").addClass("footnote");
$.bigfoot({
activateOnHover: true,
actionOriginalFN: "ignore"
});
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