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<h1 class="title">d-seperation</h1>
</div>
<div class="quarto-title-meta">
</div>
</header>
<section id="d-separation" class="level1">
<h1>D-separation</h1>
<section id="definition" class="level2">
<h2 class="anchored" data-anchor-id="definition">Definition</h2>
<p>Two variables are said to be d-separated if all paths between them are blocked (otherwise they are d-connected). Two sets of variables are said to be d-separated if each variable in the first set is d-separated from every variable in the second set.</p>
</section>
<section id="rules-when-is-a-path-open-or-blocked" class="level2">
<h2 class="anchored" data-anchor-id="rules-when-is-a-path-open-or-blocked">Rules: When is a path open or blocked?</h2>
<p><strong>1. If there are no variables being conditioned on, a path is blocked if and only if two arrowheads on the path collide at some variable on the path.</strong></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggdag)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stderr">
<pre><code>
Attaching package: 'ggdag'</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>The following object is masked from 'package:stats':
filter</code></pre>
</div>
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>dag1_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> C <span class="sc">~</span> A <span class="sc">+</span> B</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag1_confound_ggdag) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="d-seperation_files/figure-html/unnamed-chunk-2-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<section id="beispiel" class="level3">
<h3 class="anchored" data-anchor-id="beispiel">Beispiel:</h3>
<p><strong>D=Drivingskills</strong><br>
<strong>A=Alkohol</strong><br>
<strong>U=Unfall</strong></p>
<p>Der Pfad zwischen den Drivingskills und dem Alkoholkonsum ist durch den Collider Unfall blockiert.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>dag2_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a> U <span class="sc">~</span> D <span class="sc">+</span> A</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag2_confound_ggdag, <span class="at">layout =</span> <span class="st">"circle"</span>) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="d-seperation_files/figure-html/unnamed-chunk-4-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
<section id="regression" class="level3">
<h3 class="anchored" data-anchor-id="regression">Regression:</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># set number of observations</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>n <span class="ot"><-</span> <span class="dv">10000</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a>D <span class="ot"><-</span> <span class="fu">rnorm</span>(n, <span class="dv">0</span>, <span class="dv">1</span>)</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a>A <span class="ot"><-</span> <span class="fu">rbinom</span>(n, <span class="dv">1</span>, <span class="fl">0.05</span>)</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a>U <span class="ot"><-</span> <span class="fl">0.01</span> <span class="sc">+</span> <span class="fl">0.3</span><span class="sc">*</span>A <span class="sc">-</span> <span class="fl">0.1</span> <span class="sc">*</span> D <span class="sc">+</span> <span class="fu">rnorm</span>(n, <span class="dv">0</span>, <span class="dv">1</span>)</span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a><span class="fu">lm</span>(U <span class="sc">~</span> A <span class="sc">+</span> D)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>
Call:
lm(formula = U ~ A + D)
Coefficients:
(Intercept) A D
0.02153 0.29753 -0.09725 </code></pre>
</div>
</div>
<p><strong>2. Any path that contains a noncollider that has been conditioned on is blocked.</strong><br>
The path between A and C is blocked, conditioning on B.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>dag3_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>C <span class="sc">~</span> B,</span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>B <span class="sc">~</span> A,</span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a><span class="at">coords =</span> <span class="fu">list</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="at">A =</span> <span class="dv">1</span>, <span class="at">B =</span> <span class="dv">2</span>, <span class="at">C =</span> <span class="dv">3</span>),</span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a> <span class="at">y =</span> <span class="fu">c</span>(<span class="at">A =</span> <span class="dv">1</span>, <span class="at">B =</span> <span class="dv">1</span>, <span class="at">C =</span> <span class="dv">1</span>))</span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag3_confound_ggdag) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="d-seperation_files/figure-html/unnamed-chunk-8-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
<section id="beispiel-1" class="level3">
<h3 class="anchored" data-anchor-id="beispiel-1">Beispiel:</h3>
<p><strong>F=Feuer</strong><br>
<strong>R=Rauch</strong><br>
<strong>A=Alarm</strong></p>
<p>Feuer und Alarm sind unabhängig, bedingt auf Rauch.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>dag4_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a> A <span class="sc">~</span> R,</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a> R <span class="sc">~</span> F,</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a> <span class="at">coords =</span> <span class="fu">list</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="at">F =</span> <span class="dv">1</span>, <span class="at">R =</span> <span class="dv">2</span>, <span class="at">A =</span> <span class="dv">3</span>),</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a> <span class="at">y =</span> <span class="fu">c</span>(<span class="at">F =</span> <span class="dv">1</span>, <span class="at">R =</span> <span class="dv">1</span>, <span class="at">A =</span> <span class="dv">1</span>))</span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag4_confound_ggdag) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="d-seperation_files/figure-html/unnamed-chunk-10-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p><strong>3. A collider that has been conditioned on does not block a path.</strong> If we condition on C, the path between A and B is not blocked. They are d-connected.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>dag5_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a> C <span class="sc">~</span> A <span class="sc">+</span> B</span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag5_confound_ggdag, <span class="at">layout =</span> <span class="st">"circle"</span>) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="d-seperation_files/figure-html/unnamed-chunk-12-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Durch die Konditionierung auf “Unfall” ist der Pfad zwischen Drivingskills und Alkohol nicht mehr blockiert.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>dag6_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a> U <span class="sc">~</span> D <span class="sc">+</span> A</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag6_confound_ggdag, <span class="at">layout =</span> <span class="st">"circle"</span>) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="d-seperation_files/figure-html/unnamed-chunk-14-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p><strong>4. A collider that has a descendant that has been conditioned on does not block a path.</strong></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggdag)</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>dag7_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a> C <span class="sc">~</span> A <span class="sc">+</span> B,</span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a> D <span class="sc">~</span>C,</span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a> <span class="at">coords =</span> <span class="fu">list</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="at">B =</span> <span class="dv">1</span>, <span class="at">C =</span> <span class="dv">2</span>, <span class="at">D =</span> <span class="dv">2</span>, <span class="at">A =</span> <span class="dv">3</span>),</span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a> <span class="at">y =</span> <span class="fu">c</span>(<span class="at">B =</span> <span class="dv">3</span>, <span class="at">C =</span> <span class="dv">2</span>, <span class="at">D =</span> <span class="dv">1</span>, <span class="at">A =</span> <span class="dv">3</span>))</span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb12-9"><a href="#cb12-9" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag7_confound_ggdag) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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</section>
<section id="beispiel-2" class="level3">
<h3 class="anchored" data-anchor-id="beispiel-2">Beispiel</h3>
<p><strong>Neue Variable S=Schäden am Auto</strong><br>
Die Variable “Schäden am Auto” ist ein Nachfolger des Colliders “Unfall”. D und A waren d-seperated durch Collider U, aber sind d-connected, sobald wir auf S bedingen.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggdag)</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>dag8_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a> U <span class="sc">~</span> D <span class="sc">+</span> A,</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a> S <span class="sc">~</span> U,</span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a> <span class="at">coords =</span> <span class="fu">list</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="at">D =</span> <span class="dv">1</span>, <span class="at">U =</span> <span class="dv">2</span>, <span class="at">S =</span> <span class="dv">2</span>, <span class="at">A =</span> <span class="dv">3</span>),</span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a> <span class="at">y =</span> <span class="fu">c</span>(<span class="at">D =</span> <span class="dv">3</span>, <span class="at">U =</span> <span class="dv">2</span>, <span class="at">S =</span> <span class="dv">1</span>, <span class="at">A =</span> <span class="dv">3</span>))</span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag8_confound_ggdag) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<section id="excercise" class="level2">
<h2 class="anchored" data-anchor-id="excercise">Excercise</h2>
<p>What sets of variables d-seperate the nonadjacent nodes in this DAG?</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>dag9_confound_ggdag <span class="ot"><-</span> <span class="fu">dagify</span>(</span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a> X <span class="sc">~</span> Z1 <span class="sc">+</span> Z3,</span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a> Z3 <span class="sc">~</span> Z1 <span class="sc">+</span> Z2,</span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a> W <span class="sc">~</span> X,</span>
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a> Y <span class="sc">~</span> W <span class="sc">+</span> Z2 <span class="sc">+</span> Z3,</span>
<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a> <span class="at">coords =</span> <span class="fu">list</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="at">X =</span> <span class="dv">1</span>, <span class="at">Z1 =</span> <span class="dv">1</span>, <span class="at">Z3 =</span> <span class="dv">2</span>, <span class="at">W =</span> <span class="dv">2</span>, <span class="at">Z2 =</span> <span class="dv">3</span>, <span class="at">Y =</span> <span class="dv">3</span>),</span>
<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a> <span class="at">y =</span> <span class="fu">c</span>(<span class="at">X =</span> <span class="dv">1</span>, <span class="at">Z1 =</span> <span class="dv">3</span>, <span class="at">Z3 =</span> <span class="dv">2</span>, <span class="at">W =</span> <span class="dv">1</span>, <span class="at">Z2 =</span> <span class="dv">3</span>, <span class="at">Y =</span> <span class="dv">1</span>))</span>
<span id="cb14-8"><a href="#cb14-8" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb14-9"><a href="#cb14-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot the DAG</span></span>
<span id="cb14-10"><a href="#cb14-10" aria-hidden="true" tabindex="-1"></a><span class="fu">ggdag</span>(dag9_confound_ggdag) <span class="sc">+</span> <span class="fu">theme_dag_blank</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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</div>
</div>
<p>Z1 ⊥⊥ Z2<br>
X ⊥⊥ Y | W, Z3, Z1<br>
oder<br>
X ⊥⊥ Y | W, Z2, Z3<br>
X ⊥⊥ Y | Z1, Z3<br>
W ⊥⊥ Z2 | Z1, Z3<br>
W ⊥⊥ Z2 | X<br>
W ⊥⊥ Z3 | X<br>
W ⊥⊥ Z1 | X<br>
Z1 ⊥⊥ Y |Z3, X, Z2<br>
oder<br>
Z1 ⊥⊥ Y | Z3, W, Z2</p>
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