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<li><a href="#time-to-event-outcome" id="toc-time-to-event-outcome" class="nav-link active" data-scroll-target="#time-to-event-outcome">Time-to-event outcome</a></li>
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<h1 class="title">Extensions in Survival and Longitudinal Analyses</h1>
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<section id="time-to-event-outcome" class="level2 page-columns page-full">
<h2 class="anchored" data-anchor-id="time-to-event-outcome">Time-to-event outcome</h2>
<p>There are two components to a time-to-event (survival) outcome: (1) whether an event occurs and (2) the timing of the event.</p>
<ul>
<li>Example: time-to-CVD, where we are interested both in CVD (cardiovascular disease) status and time from cohort entry to CVD development. The Cox proportional hazards (PH) model is widely used for modeling a time-to-event outcome.</li>
</ul>
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<p><strong>Acknowledgment</strong>: <a href="https://scholar.google.ca/citations?hl=en&user=a8nMkCYAAAAJ&view_op=list_works&sortby=pubdate">Md Belal Hossain</a> contributed to drafting this chapter; some of the ideas presented here stem from his PhD thesis and subsequent publications.</p>
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Time-to-event outcome
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<li>Bross formula requires the exposure, outcome, and proxy covariates to be binary</li>
<li>Ignoring the time aspect in a time-to-event outcome leads to a loss of information, which can significantly impact the selection of the proxy covariates and can impact the effect estimate of interest</li>
<li>ML survival models could be useful in prioritizing and selecting recurrence covariates</li>
</ul>
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<p><span class="citation" data-cites="bross1966spurious schneeweiss2009high">(<a href="references.html#ref-bross1966spurious" role="doc-biblioref">Bross 1966</a>; <a href="references.html#ref-schneeweiss2009high" role="doc-biblioref">Schneeweiss et al. 2009</a>)</span></p>
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<section id="time-dependent-exposure" class="level2 page-columns page-full">
<h2 class="anchored" data-anchor-id="time-dependent-exposure">Time-dependent exposure</h2>
<p>Unlike a time-fixed exposure, whose values are known at study entry (time zero), values for a time-dependent exposure can change over the course of follow-up. Consider an example of a multiple sclerosis (MS) cohort, where we are interested in the relationship between disease-modifying drugs (DMDs) and long-term mortality. Not every patient is exposed to DMDs at their MS diagnosis. Instead, some patients are never exposed to DMDs, some may be exposed to DMDs at the time of their MS diagnosis, while others may be exposed many years later. In this case, the exposure status of patients is not fixed at time zero, but rather depends on time. Simultaneously dealing with time-dependent exposure and residual confounding bias can be challenging.</p>
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Time-dependent exposure
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<ul>
<li>Immortal time bias occurs when a period during follow-up, in which a patient cannot experience the outcome (e.g., death), is misclassified as time under observation, often leading to an overestimation of the treatment’s effectiveness</li>
<li>In the MS example, immortal time can occur if patients must survive long enough to receive a DMD, which could falsely enhance the perceived survival benefit of these drugs</li>
<li>Employing Cox proportional hazards models with time-varying exposure to DMDs can help mitigate immortal time bias. However, this time-dependent Cox regression cannot deal with residual confounding bias.</li>
</ul>
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<p><span class="citation" data-cites="jones2016immortal beyersmann2008impact karim2014marginal">(<a href="references.html#ref-jones2016immortal" role="doc-biblioref">Jones and Fowler 2016</a>; <a href="references.html#ref-beyersmann2008impact" role="doc-biblioref">Beyersmann, Wolkewitz, and Schumacher 2008</a>; <a href="references.html#ref-karim2014marginal" role="doc-biblioref">Karim et al. 2014</a>)</span></p>
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<h2 class="anchored" data-anchor-id="high-dimensional-disease-risk-score-hddrs">High-dimensional disease risk score (hdDRS)</h2>
<p>The hdPS technique might not reduce significant bias due to an overfitted exposure model, particularly with a rare exposure. An alternative confounding adjustment method to hdPS is hdDRS. In contrast, hdPS separates the exposure modelling from the outcome modelling, ultimately giving the end-user more flexibility in adjusting for confounding effects (e.g., via inverse-probability-weighting). On the other hand, the hdDRS achieves the balancing of the confounders by modelling the outcome.</p>
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<li>hdDRS can be an alternative to hdPS for dealing with residual confounding bias</li>
<li>hdDRS could be particularly helpful in situations where the exposure is rare or the outcome is a repeated measure</li>
</ul>
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<p><span class="citation" data-cites="kumamaru2016dimension hossain2025harnessing2">(<a href="references.html#ref-kumamaru2016dimension" role="doc-biblioref">Kumamaru et al. 2016</a>; <a href="references.html#ref-hossain2025harnessing2" role="doc-biblioref">Hossain 2025</a>)</span></p>
</div></div><div id="refs" class="references csl-bib-body hanging-indent" data-entry-spacing="0" role="list" style="display: none">
<div id="ref-beyersmann2008impact" class="csl-entry" role="listitem">
Beyersmann, Jan, Martin Wolkewitz, and Martin Schumacher. 2008. <span>“The Impact of Time-Dependent Bias in Proportional Hazards Modelling.”</span> <em>Statistics in Medicine</em> 27 (30): 6439–54.
</div>
<div id="ref-bross1966spurious" class="csl-entry" role="listitem">
Bross, Irwin DJ. 1966. <span>“Spurious Effects from an Extraneous Variable.”</span> <em>Journal of Chronic Diseases</em> 19 (6): 637–47.
</div>
<div id="ref-hossain2025harnessing2" class="csl-entry" role="listitem">
Hossain, Md Belal. 2025. <span>“Chapter 2: High-Dimensional Disease Risk Score for Dealing with Residual Confounding in Estimating Treatment Effects with a Survival Outcome.”</span> In. Harnessing the power of causal inference; predictive analytics for survival outcomes with health administrative data: applications to tuberculosis research.
</div>
<div id="ref-jones2016immortal" class="csl-entry" role="listitem">
Jones, Mark, and Robert Fowler. 2016. <span>“Immortal Time Bias in Observational Studies of Time-to-Event Outcomes.”</span> <em>Journal of Critical Care</em> 36: 195–99.
</div>
<div id="ref-karim2014marginal" class="csl-entry" role="listitem">
Karim, Mohammad Ehsanul, Paul Gustafson, John Petkau, Yinshan Zhao, Afsaneh Shirani, Elaine Kingwell, Charity Evans, Mia Van Der Kop, Joel Oger, and Helen Tremlett. 2014. <span>“Marginal Structural Cox Models for Estimating the Association Between <span class="math inline">\(\beta\)</span>-Interferon Exposure and Disease Progression in a Multiple Sclerosis Cohort.”</span> <em>American Journal of Epidemiology</em> 180 (2): 160–71.
</div>
<div id="ref-kumamaru2016dimension" class="csl-entry" role="listitem">
Kumamaru, Hiraku, Sebastian Schneeweiss, Robert J Glynn, Soko Setoguchi, and Joshua J Gagne. 2016. <span>“Dimension Reduction and Shrinkage Methods for High Dimensional Disease Risk Scores in Historical Data.”</span> <em>Emerging Themes in Epidemiology</em> 13: 1–10.
</div>
<div id="ref-schneeweiss2009high" class="csl-entry" role="listitem">
Schneeweiss, Sebastian, Jeremy A Rassen, Robert J Glynn, Jerry Avorn, Helen Mogun, and M Alan Brookhart. 2009. <span>“High-Dimensional Propensity Score Adjustment in Studies of Treatment Effects Using Health Care Claims Data.”</span> <em>Epidemiology (Cambridge, Mass.)</em> 20 (4): 512.
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// 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,