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<li><a href="#build-model-formula-based-on-all-variables" id="toc-build-model-formula-based-on-all-variables" class="nav-link active" data-scroll-target="#build-model-formula-based-on-all-variables"><span class="header-section-number">15.1</span> Build model formula based on all variables</a></li>
<li><a href="#fit-the-ps-model-with-super-learner" id="toc-fit-the-ps-model-with-super-learner" class="nav-link" data-scroll-target="#fit-the-ps-model-with-super-learner"><span class="header-section-number">15.2</span> Fit the PS model with super learner</a></li>
<li><a href="#obtain-log-or-from-unadjusted-outcome-model" id="toc-obtain-log-or-from-unadjusted-outcome-model" class="nav-link" data-scroll-target="#obtain-log-or-from-unadjusted-outcome-model"><span class="header-section-number">15.3</span> Obtain log-OR from unadjusted outcome model</a></li>
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<p>We show an example using a super learner using 3 candidate learners.</p>
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<p>If you want to know more about Super Learner, look at other <a href="https://ehsanx.github.io/TMLEworkshop/g-computation-using-ml.html#g-comp-using-superlearner">tutorials</a>.</p>
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<p></p><figure class="figure"><p></p>
<div>
<pre class="mermaid mermaid-js">flowchart LR
S(Super Learner) --> l(Logistic regression)
S --> g(LASSO)
S --> m(Multivariate Adaptive Regression Splines MARS)
style S fill:#90EE90;
</pre>
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<p></p></figure><p></p>
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<p>The super learning approach is <mark>fundamentally different from the pure ML</mark> or LASSO approach discussed earlier. Here all of the candidate learners are using <mark><code>exposure</code> as their outcome</mark> while running the model.</p>
</div></div><section id="build-model-formula-based-on-all-variables" class="level2 page-columns page-full" data-number="15.1">
<h2 data-number="15.1" class="anchored" data-anchor-id="build-model-formula-based-on-all-variables"><span class="header-section-number">15.1</span> Build model formula based on all variables</h2>
<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>proxy.list <span class="ot"><-</span> <span class="fu">names</span>(out3<span class="sc">$</span>autoselected_covariate_df[,<span class="sc">-</span><span class="dv">1</span>])</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span>(proxy.list)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="co">#> [1] 100</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>covform <span class="ot"><-</span> <span class="fu">paste0</span>(investigator.specified.covariates, <span class="at">collapse =</span> <span class="st">"+"</span>)</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>proxyform <span class="ot"><-</span> <span class="fu">paste0</span>(proxy.list, <span class="at">collapse =</span> <span class="st">"+"</span>)</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>rhsformula <span class="ot"><-</span> <span class="fu">paste0</span>(<span class="fu">c</span>(covform, proxyform), <span class="at">collapse =</span> <span class="st">"+"</span>)</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>ps.formula <span class="ot"><-</span> <span class="fu">as.formula</span>(<span class="fu">paste0</span>(<span class="st">"exposure"</span>, <span class="st">"~"</span>, rhsformula))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>We work with all proxies</p>
</div></div></section>
<section id="fit-the-ps-model-with-super-learner" class="level2 page-columns page-full" data-number="15.2">
<h2 data-number="15.2" class="anchored" data-anchor-id="fit-the-ps-model-with-super-learner"><span class="header-section-number">15.2</span> Fit the PS model with super learner</h2>
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<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(WeightIt)</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>W.out <span class="ot"><-</span> <span class="fu">weightit</span>(ps.formula, </span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> hdps.data, </span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a> <span class="at">estimand =</span> <span class="st">"ATE"</span>,</span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> <span class="at">method =</span> <span class="st">"super"</span>,</span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a> <span class="at">SL.library =</span> <span class="fu">c</span>(<span class="st">"SL.glm"</span>, </span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a> <span class="st">"SL.glmnet"</span>,</span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> <span class="st">"SL.earth"</span>))</span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a><span class="co">#> Loading required namespace: glmnet</span></span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a><span class="co">#> Loading required namespace: earth</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>Propensity score model fit based on super learning algorithm to be able to calculate the inverse probability weights.</p>
</div></div></section>
<section id="obtain-log-or-from-unadjusted-outcome-model" class="level2 page-columns page-full" data-number="15.3">
<h2 data-number="15.3" class="anchored" data-anchor-id="obtain-log-or-from-unadjusted-outcome-model"><span class="header-section-number">15.3</span> Obtain log-OR from unadjusted outcome model</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(W.out<span class="sc">$</span>ps)</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="co">#> Min. 1st Qu. Median Mean 3rd Qu. Max. </span></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a><span class="co">#> 0.01826 0.22575 0.39324 0.42094 0.59037 0.98809</span></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a>out.formula <span class="ot"><-</span> <span class="fu">as.formula</span>(<span class="fu">paste0</span>(<span class="st">"outcome"</span>, <span class="st">"~"</span>, <span class="st">"exposure"</span>))</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a>fit <span class="ot"><-</span> <span class="fu">glm</span>(out.formula,</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> hdps.data,</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> <span class="at">weights =</span> W.out<span class="sc">$</span>weights,</span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> <span class="at">family=</span> <span class="fu">binomial</span>(<span class="at">link =</span> <span class="st">"logit"</span>))</span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a>fit.summary <span class="ot"><-</span> <span class="fu">summary</span>(fit)<span class="sc">$</span>coef[<span class="st">"exposure"</span>,</span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">c</span>(<span class="st">"Estimate"</span>, </span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a> <span class="st">"Std. Error"</span>, </span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a> <span class="st">"Pr(>|z|)"</span>)]</span>
<span id="cb3-13"><a href="#cb3-13" aria-hidden="true" tabindex="-1"></a>fit.summary[<span class="dv">2</span>] <span class="ot"><-</span> <span class="fu">sqrt</span>(sandwich<span class="sc">::</span><span class="fu">sandwich</span>(fit)[<span class="dv">2</span>,<span class="dv">2</span>])</span>
<span id="cb3-14"><a href="#cb3-14" aria-hidden="true" tabindex="-1"></a><span class="fu">require</span>(lmtest)</span>
<span id="cb3-15"><a href="#cb3-15" aria-hidden="true" tabindex="-1"></a>conf.int <span class="ot"><-</span> <span class="fu">confint</span>(fit, <span class="st">"exposure"</span>, <span class="at">level =</span> <span class="fl">0.95</span>, <span class="at">method =</span> <span class="st">"hc1"</span>)</span>
<span id="cb3-16"><a href="#cb3-16" aria-hidden="true" tabindex="-1"></a>fit.summary_with_ci.sl <span class="ot"><-</span> <span class="fu">c</span>(fit.summary, conf.int)</span>
<span id="cb3-17"><a href="#cb3-17" aria-hidden="true" tabindex="-1"></a>knitr<span class="sc">::</span><span class="fu">kable</span>(<span class="fu">t</span>(<span class="fu">round</span>(fit.summary_with_ci.sl,<span class="dv">2</span>))) </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<table class="caption-top table table-sm table-striped small">
<thead>
<tr class="header">
<th style="text-align: right;">Estimate</th>
<th style="text-align: right;">Std. Error</th>
<th style="text-align: right;">Pr(>|z|)</th>
<th style="text-align: right;">2.5 %</th>
<th style="text-align: right;">97.5 %</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">0.42</td>
<td style="text-align: right;">0.1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0.31</td>
<td style="text-align: right;">0.53</td>
</tr>
</tbody>
</table>
</div>
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<p>Summary of results (log-OR).</p>
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const link = links[i];
if (!isInternal(link.href)) {
// undo the damage that might have been done by quarto-nav.js in the case of
// links that we want to consider external
if (link.dataset.originalHref !== undefined) {
link.href = link.dataset.originalHref;
}
}
}
function tippyHover(el, contentFn, onTriggerFn, onUntriggerFn) {
const config = {
allowHTML: true,
maxWidth: 500,
delay: 100,
arrow: false,
appendTo: function(el) {
return el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start',
};
if (contentFn) {
config.content = contentFn;
}
if (onTriggerFn) {
config.onTrigger = onTriggerFn;
}
if (onUntriggerFn) {
config.onUntrigger = onUntriggerFn;
}
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
if (note) {
return note.innerHTML;
} else {
return "";
}
});
}
const xrefs = window.document.querySelectorAll('a.quarto-xref');
const processXRef = (id, note) => {
// Strip column container classes
const stripColumnClz = (el) => {
el.classList.remove("page-full", "page-columns");
if (el.children) {
for (const child of el.children) {
stripColumnClz(child);
}
}
}
stripColumnClz(note)
if (id === null || id.startsWith('sec-')) {
// Special case sections, only their first couple elements
const container = document.createElement("div");
if (note.children && note.children.length > 2) {
container.appendChild(note.children[0].cloneNode(true));
for (let i = 1; i < note.children.length; i++) {
const child = note.children[i];
if (child.tagName === "P" && child.innerText === "") {
continue;
} else {
container.appendChild(child.cloneNode(true));
break;
}
}
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(container);
}
return container.innerHTML
} else {
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(note);
}
return note.innerHTML;
}
} else {
// Remove any anchor links if they are present
const anchorLink = note.querySelector('a.anchorjs-link');
if (anchorLink) {
anchorLink.remove();
}
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(note);
}
if (note.classList.contains("callout")) {
return note.outerHTML;
} else {
return note.innerHTML;
}
}
}
for (var i=0; i<xrefs.length; i++) {
const xref = xrefs[i];
tippyHover(xref, undefined, function(instance) {
instance.disable();
let url = xref.getAttribute('href');
let hash = undefined;
if (url.startsWith('#')) {
hash = url;
} else {
try { hash = new URL(url).hash; } catch {}
}
if (hash) {
const id = hash.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
if (note !== null) {
try {
const html = processXRef(id, note.cloneNode(true));
instance.setContent(html);
} finally {
instance.enable();
instance.show();
}
} else {
// See if we can fetch this
fetch(url.split('#')[0])
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.getElementById(id);
if (note !== null) {
const html = processXRef(id, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
} else {
// See if we can fetch a full url (with no hash to target)
// This is a special case and we should probably do some content thinning / targeting
fetch(url)
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.querySelector('main.content');
if (note !== null) {
// This should only happen for chapter cross references
// (since there is no id in the URL)
// remove the first header
if (note.children.length > 0 && note.children[0].tagName === "HEADER") {
note.children[0].remove();
}
const html = processXRef(null, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
}, function(instance) {
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
const selectCodeLines = (annoteEl) => {
const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
const lineIds = lines.map((line) => {
return targetCell + "-" + line;
})
let top = null;
let height = null;
let parent = null;
if (lineIds.length > 0) {
//compute the position of the single el (top and bottom and make a div)
const el = window.document.getElementById(lineIds[0]);
top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
if (lineIds.length > 1) {
const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
if (top !== null && height !== null && parent !== null) {
// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
div.style.left = 0;
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
if (gutterDiv === null) {
gutterDiv = window.document.createElement("div");
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;
}
};
const unselectCodeLines = () => {
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Handle positioning of the toggle
window.addEventListener(
"resize",
throttle(() => {
elRect = undefined;
if (selectedAnnoteEl) {
selectCodeLines(selectedAnnoteEl);
}
}, 10)
);
function throttle(fn, ms) {
let throttle = false;
let timer;
return (...args) => {
if(!throttle) { // first call gets through
fn.apply(this, args);
throttle = true;
} else { // all the others get throttled
if(timer) clearTimeout(timer); // cancel #2
timer = setTimeout(() => {
fn.apply(this, args);
timer = throttle = false;
}, ms);
}
};
}
// Attach click handler to the DT
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
for (const annoteDlNode of annoteDls) {
annoteDlNode.addEventListener('click', (event) => {
const clickedEl = event.target;
if (clickedEl !== selectedAnnoteEl) {
unselectCodeLines();
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
if (activeEl) {
activeEl.classList.remove('code-annotation-active');
}
selectCodeLines(clickedEl);
clickedEl.classList.add('code-annotation-active');
} else {
// Unselect the line
unselectCodeLines();
clickedEl.classList.remove('code-annotation-active');
}
});
}
const findCites = (el) => {
const parentEl = el.parentElement;
if (parentEl) {
const cites = parentEl.dataset.cites;
if (cites) {
return {
el,
cites: cites.split(' ')
};
} else {
return findCites(el.parentElement)
}
} else {