Morning Rounds \u00b7 PSM Numericals \u00b7 Round 04<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,600;0,700;1,400;1,600&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n#nps04 *,#nps04 *::before,#nps04 *::after{box-sizing:border-box;margin:0;padding:0}\n#nps04{\n --ob:#1A5F7A;--ob-light:#236E8C;--ob-pale:#E6F3F7;--ob-dark:#114455;\n --correct:#2D6B47;--correct-bg:#EAF6EF;--correct-border:#3A9960;\n --wrong:#B83232;--wrong-bg:#FDF0F0;--wrong-border:#E53935;\n --ink:#0F2A35;--ink-mid:#2A4D5A;--ink-soft:#5A7D8A;\n --line:#B8D8E3;--cream:#F2F8FA;--warm:#F8FCFD;\n font-family:'Source Serif 4',Georgia,serif;\n font-size:16px;color:var(--ink);background:var(--cream);\n line-height:1.7;padding:0 0 64px;\n}\n#nps04 .mr-header{background:linear-gradient(135deg,var(--ob-dark) 0%,var(--ob) 60%,var(--ob-light) 100%);color:#E4F4F9;padding:36px 24px 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.mr-band-c{background:var(--correct-bg);color:var(--correct)}\n#nps04 .mr-band-w{background:var(--wrong-bg);color:var(--wrong)}\n#nps04 .mr-band-s{background:var(--ob-pale);color:var(--ob)}\n#nps04 .mr-retry{display:block;margin:18px auto 4px;background:transparent;border:2px solid var(--ob);color:var(--ob);border-radius:8px;padding:9px 28px;font-family:'Playfair Display',serif;font-size:0.92rem;font-weight:700;cursor:pointer;}\n#nps04 .mr-retry:hover{background:var(--ob);color:#E4F4F9}\n@media(max-width:480px){\n #nps04 .mr-title{font-size:1.4rem}\n #nps04 .mr-num{font-size:1.7rem}\n #nps04 .mr-stem{font-size:0.9rem}\n #nps04 .mr-opt-text{font-size:0.86rem}\n}\n<\/style>\n\n<div id=\"nps04\">\n\n <div class=\"mr-header\">\n <div class=\"mr-series-tag\">Numerical Series · PSM Biostatistics<\/div>\n <div class=\"mr-eyebrow\">Morning Rounds · Round 04 of 07<\/div>\n <div class=\"mr-title\">NNT, NNH, ARR, RRR<br><em>& the Risk Ratios<\/em><\/div>\n <div class=\"mr-subtitle\">Five questions · Treatment effect measures, how they relate & why RRR without ARR misleads<\/div>\n <div class=\"mr-chips\">\n <span class=\"mr-chip\">5 Questions<\/span>\n <span class=\"mr-chip\">+4 \/ −1 scoring<\/span>\n <span class=\"mr-chip\">Options reshuffled<\/span>\n <\/div>\n <\/div>\n\n <div class=\"mr-sentinel\" id=\"nps04-sentinel\"><\/div>\n\n <div class=\"mr-progress\" id=\"nps04-progress\">\n <div class=\"mr-prog-inner\">\n <div class=\"mr-pips\" id=\"nps04-pips\"><\/div>\n <\/div>\n <\/div>\n\n <div class=\"mr-body\">\n <div id=\"nps04-cases\"><\/div>\n <div class=\"mr-submit-wrap\">\n <button class=\"mr-btn\" id=\"nps04-submit\">Submit for Debrief<\/button>\n <\/div>\n <div class=\"mr-score\" id=\"nps04-score\">\n <div class=\"mr-score-in\">\n <div class=\"mr-score-ey\">Round Complete<\/div>\n <div class=\"mr-ring\" id=\"nps04-ring\">\n <div class=\"mr-ring-in\">\n <span class=\"mr-ring-pct\" id=\"nps04-pct\">0%<\/span>\n <span class=\"mr-ring-sub\">net<\/span>\n <\/div>\n <\/div>\n <div class=\"mr-score-title\">Your Debrief<\/div>\n <div class=\"mr-score-net\" id=\"nps04-net\"><\/div>\n <div class=\"mr-verdict\" id=\"nps04-verdict\"><\/div>\n <div class=\"mr-bands\">\n <span class=\"mr-band mr-band-c\" id=\"nps04-ct-c\"><\/span>\n <span class=\"mr-band mr-band-w\" id=\"nps04-ct-w\"><\/span>\n <span class=\"mr-band mr-band-s\" id=\"nps04-ct-s\"><\/span>\n <\/div>\n <button class=\"mr-retry\" id=\"nps04-retry\">↻ New Round<\/button>\n <\/div>\n <\/div>\n <\/div>\n\n<\/div>\n\n<script>\n(function () {\n 'use strict';\n\n var NS = 'nps04';\n var TOTAL = 5;\n var MAX = 20;\n var LTRS = ['A','B','C','D'];\n\n var QS = [\n\n {\n id: 1,\n tag: 'ARR & NNT — Core Calculation',\n stem: 'A trial of a new antihypertensive tests its effect on stroke. In the <strong>control group<\/strong>, 8% have a stroke over 5 years. In the <strong>treatment group<\/strong>, 5% have a stroke. What is the <strong>ARR<\/strong> and <strong>NNT<\/strong>?',\n correct: 'ARR = 3%; NNT = 34',\n opts: [\n 'ARR = 3%; NNT = 34',\n 'ARR = 5%; NNT = 20',\n 'ARR = 3%; NNT = 20',\n 'ARR = 8%; NNT = 13'\n ],\n exp: '<strong>Absolute Risk Reduction (ARR)<\/strong> = Risk in control − Risk in treatment.<span class=\"calc\">ARR = 8% − 5% = <strong>3%<\/strong><br><br>NNT = 1 ÷ ARR = 1 ÷ 0.03 = <strong>33.3 ≈ 34<\/strong><\/span>NNT is the number of patients who must be treated for 5 years to prevent one stroke. Here, 34 patients need treatment to prevent one additional stroke over 5 years \u2014 that is the clinical weight of NNT.<br><br>Trap B uses the control event rate (8%) as ARR. Trap C gets ARR right but calculates NNT as 1÷0.05 = 20, using the wrong denominator. Trap D uses the treatment event rate as ARR. The NNT formula is always <strong>1 ÷ ARR (as a decimal)<\/strong>. Always round up \u2014 you cannot treat a fraction of a patient.'\n },\n\n {\n id: 2,\n tag: 'RRR — Calculation and the Misleading Headline',\n stem: 'Using the same stroke trial (control risk 8%, treatment risk 5%), a pharmaceutical company advertises the drug as reducing stroke risk by <strong>37.5%<\/strong>. This figure is the RRR. Is this claim accurate, and does it give a complete picture?',\n correct: 'Accurate but incomplete; RRR = 37.5% but ARR = only 3%, meaning the absolute benefit is modest despite the impressive relative figure',\n opts: [\n 'Accurate but incomplete; RRR = 37.5% but ARR = only 3%, meaning the absolute benefit is modest despite the impressive relative figure',\n 'Inaccurate; the correct RRR is 62.5%, calculated as treatment risk divided by control risk',\n 'Accurate and complete; RRR of 37.5% fully characterises the treatment benefit without needing ARR',\n 'Inaccurate; RRR should be calculated from absolute numbers of events, not percentages'\n ],\n exp: '<strong>Relative Risk Reduction (RRR)<\/strong> = ARR ÷ Control event rate.<span class=\"calc\">RRR = (8% − 5%) ÷ 8% = 3% ÷ 8% = <strong>0.375 = 37.5%<\/strong><\/span>The figure is arithmetically correct. But notice: a drug that reduces risk from <strong>0.2% to 0.125%<\/strong> also has an RRR of 37.5% \u2014 yet its ARR is only 0.075%, and its NNT is 1333. The RRR is identical; the clinical significance is entirely different.<br><br>This is why <strong>RRR without ARR is incomplete and potentially misleading<\/strong>. Drug advertisements routinely headline RRR because it sounds more impressive. Examiners test this asymmetry directly. The full picture always requires: control event rate, ARR, RRR, and NNT together. Trap C claims RRR alone is sufficient \u2014 that is the pharmaceutical marketing position, not the evidence-based one.'\n },\n\n {\n id: 3,\n tag: 'Relative Risk — Cohort Study Calculation',\n stem: 'In a cohort study of 1000 smokers and 1000 non-smokers followed for 10 years, <strong>80 smokers<\/strong> and <strong>20 non-smokers<\/strong> develop lung cancer. What is the <strong>Relative Risk (RR)<\/strong> and how is it interpreted?',\n correct: 'RR = 4.0; smokers are 4 times as likely to develop lung cancer as non-smokers',\n opts: [\n 'RR = 4.0; smokers are 4 times as likely to develop lung cancer as non-smokers',\n 'RR = 0.25; non-smokers are 75% less likely to develop lung cancer than smokers',\n 'RR = 4.0; the absolute excess risk attributable to smoking is 4 cases per 1000',\n 'RR = 60; calculated as the difference in event numbers between the two groups'\n ],\n exp: '<strong>Relative Risk (RR)<\/strong> = Incidence in exposed ÷ Incidence in unexposed. Used in <strong>cohort studies<\/strong>.<span class=\"calc\">Incidence in smokers = 80 ÷ 1000 = 0.08 (8%)<br>Incidence in non-smokers = 20 ÷ 1000 = 0.02 (2%)<br>RR = 0.08 ÷ 0.02 = <strong>4.0<\/strong><\/span>Interpretation: smokers are <strong>4 times more likely<\/strong> to develop lung cancer than non-smokers over 10 years. RR = 1 means no association. RR >1 means increased risk in the exposed. RR <1 means protective effect.<br><br>Trap B calculates the reciprocal (1÷4 = 0.25) and reframes it from the non-smoker perspective \u2014 mathematically valid but not what was asked. Trap C correctly states RR = 4.0 but then misdefines it as absolute excess risk \u2014 RR is a ratio, not an absolute figure. Trap D subtracts instead of divides. Key exam point: <strong>RR is used in cohort studies; OR is used in case-control studies<\/strong>.'\n },\n\n {\n id: 4,\n tag: 'Odds Ratio — Case-Control Calculation',\n stem: 'In a case-control study of myocardial infarction, <strong>60 of 100 cases<\/strong> were smokers versus <strong>30 of 100 controls<\/strong>. What is the <strong>Odds Ratio (OR)<\/strong> for smoking and MI?',\n correct: 'OR = 3.5; smokers have 3.5 times the odds of MI compared to non-smokers',\n opts: [\n 'OR = 3.5; smokers have 3.5 times the odds of MI compared to non-smokers',\n 'OR = 2.0; calculated as the ratio of smoking prevalence in cases versus controls',\n 'OR = 3.5; this equals the relative risk since case and control numbers are equal',\n 'OR = 0.29; calculated as the odds of smoking in controls divided by odds in cases'\n ],\n exp: 'In a case-control study, we calculate <strong>odds of exposure<\/strong> in cases versus controls \u2014 not incidence rates (those are unavailable).<span class=\"calc\">Odds of smoking in cases = 60 ÷ 40 = 1.5<br>Odds of smoking in controls = 30 ÷ 70 = 0.4286<br>OR = 1.5 ÷ 0.4286 = <strong>3.5<\/strong><\/span>Or using the cross-product formula directly:<span class=\"calc\">OR = (a × d) ÷ (b × c) = (60 × 70) ÷ (40 × 30) = 4200 ÷ 1200 = <strong>3.5<\/strong><\/span>Interpretation: smokers have 3.5 times the <em>odds<\/em> of having had an MI compared to non-smokers.<br><br>Trap B calculates 60÷30 = 2 \u2014 this is the ratio of smoking prevalences, not the odds ratio. Trap C correctly states OR = 3.5 but claims it equals RR \u2014 OR approximates RR only when disease prevalence is <strong>low<\/strong> (the rare disease assumption); with 60% case exposure this assumption fails. Trap D inverts the ratio. The cross-product formula (ad÷bc) is the fastest route.'\n },\n\n {\n id: 5,\n tag: 'NNH — Harm from Adverse Events',\n stem: 'A statin trial shows that <strong>3%<\/strong> of treated patients develop myopathy versus <strong>1%<\/strong> of placebo patients over 3 years. What is the <strong>NNH<\/strong>, and how should it be communicated to a patient?',\n correct: 'NNH = 50; one additional case of myopathy occurs for every 50 patients treated for 3 years',\n opts: [\n 'NNH = 50; one additional case of myopathy occurs for every 50 patients treated for 3 years',\n 'NNH = 33; calculated as 1 divided by the treated group event rate of 3%',\n 'NNH = 50; since NNH exceeds NNT, the drug is definitively net beneficial',\n 'NNH = 100; calculated as 1 divided by the absolute difference expressed as a whole number'\n ],\n exp: '<strong>Number Needed to Harm (NNH)<\/strong> = 1 ÷ Absolute Risk Increase (ARI).<span class=\"calc\">ARI = Risk in treated − Risk in control = 3% − 1% = 2%<br>NNH = 1 ÷ 0.02 = <strong>50<\/strong><\/span>Communication: \"For every 50 patients I treat with this statin for 3 years, one additional patient will develop myopathy who would not have developed it on placebo.\"<br><br>NNH must always be considered alongside NNT. If NNT = 34 (one stroke prevented per 34 patients) and NNH = 50 (one myopathy per 50 patients), the balance favours treatment \u2014 but this is a clinical judgement, not an automatic conclusion. Trap B divides by 3% (treated rate alone, ignoring baseline). Trap C correctly calculates NNH = 50 but then makes the logical error that NNH > NNT automatically means net benefit \u2014 severity of the harm matters equally; myopathy and stroke are not equivalent outcomes. Trap D divides by 2 (the whole number difference) instead of 0.02.'\n }\n\n ];\n\n var answers = {}, answered = 0, shuffled = {}, done = false;\n\n function gid(s) { return document.getElementById(NS + '-' + s); }\n function byId(s) { return document.getElementById(s); }\n\n function shuffleArr(arr) {\n var a = arr.slice(), i, j, t;\n for (i = a.length - 1; i > 0; i--) {\n j = Math.floor(Math.random() * (i + 1));\n t = a[i]; a[i] = a[j]; a[j] = t;\n }\n return a;\n }\n\n function countVal(v) {\n var n = 0, k;\n for (k in answers) { if (answers[k] === v) n++; }\n return n;\n }\n\n function buildPips() {\n var cont = gid('pips'), i, q, wLine, wPip, line, pip;\n cont.innerHTML = '';\n for (i = 0; i < QS.length; i++) {\n q = QS[i];\n if (i > 0) {\n wLine = document.createElement('div');\n wLine.className = 'mr-pip-wrap';\n line = document.createElement('div');\n line.className = 'mr-pip-line';\n line.id = NS + '-pl' + q.id;\n wLine.appendChild(line);\n cont.appendChild(wLine);\n }\n wPip = document.createElement('div');\n wPip.className = 'mr-pip-wrap';\n pip = document.createElement('div');\n pip.className = 'mr-pip';\n pip.id = NS + '-pip' + q.id;\n pip.textContent = String(q.id);\n wPip.appendChild(pip);\n cont.appendChild(wPip);\n }\n }\n\n function build() {\n var cont, i, q, opts, card, top, numDiv, meta, tag, stem,\n rule, optsDiv, expDiv, lbl, txt, j, optEl, ltrSpan, txtSpan;\n\n cont = gid('cases');\n cont.innerHTML = '';\n answers = {}; answered = 0; shuffled = {}; done = false;\n gid('score').style.display = 'none';\n buildPips();\n\n for (i = 0; i < QS.length; i++) {\n q = QS[i];\n opts = shuffleArr(q.opts);\n shuffled[q.id] = opts;\n\n card = document.createElement('div');\n card.className = 'mr-case';\n\n top = document.createElement('div');\n top.className = 'mr-case-top';\n\n numDiv = document.createElement('div');\n numDiv.className = 'mr-num';\n numDiv.textContent = q.id < 10 ? '0' + q.id : String(q.id);\n\n meta = document.createElement('div');\n meta.className = 'mr-meta';\n\n tag = document.createElement('div');\n tag.className = 'mr-tag';\n tag.innerHTML = q.tag;\n\n stem = document.createElement('div');\n stem.className = 'mr-stem';\n stem.innerHTML = q.stem;\n\n meta.appendChild(tag);\n meta.appendChild(stem);\n top.appendChild(numDiv);\n top.appendChild(meta);\n card.appendChild(top);\n\n rule = document.createElement('div');\n rule.className = 'mr-rule';\n card.appendChild(rule);\n\n optsDiv = document.createElement('div');\n optsDiv.className = 'mr-opts';\n\n for (j = 0; j < opts.length; j++) {\n optEl = document.createElement('div');\n optEl.className = 'mr-opt';\n optEl.id = NS + '-o' + q.id + '-' + j;\n optEl.setAttribute('role', 'button');\n optEl.setAttribute('tabindex', '0');\n\n ltrSpan = document.createElement('span');\n ltrSpan.className = 'mr-ltr';\n ltrSpan.textContent = LTRS[j];\n\n txtSpan = document.createElement('span');\n txtSpan.className = 'mr-opt-text';\n txtSpan.innerHTML = opts[j];\n\n optEl.appendChild(ltrSpan);\n optEl.appendChild(txtSpan);\n optsDiv.appendChild(optEl);\n\n (function (qid, oi) {\n optEl.addEventListener('click', function () { pick(qid, oi); });\n }(q.id, j));\n }\n card.appendChild(optsDiv);\n\n expDiv = document.createElement('div');\n expDiv.className = 'mr-exp';\n expDiv.id = NS + '-exp' + q.id;\n\n lbl = document.createElement('div');\n lbl.className = 'mr-exp-lbl';\n lbl.textContent = 'Debrief';\n\n txt = document.createElement('div');\n txt.className = 'mr-exp-text';\n txt.innerHTML = q.exp;\n\n expDiv.appendChild(lbl);\n expDiv.appendChild(txt);\n card.appendChild(expDiv);\n cont.appendChild(card);\n }\n }\n\n function pick(qid, oi) {\n var q, opts, i, el, correct;\n if (answers[qid] !== undefined || done) return;\n\n q = null;\n for (i = 0; i < QS.length; i++) {\n if (QS[i].id === qid) { q = QS[i]; break; }\n }\n if (!q) return;\n\n opts = shuffled[qid];\n correct = (opts[oi] === q.correct);\n answers[qid] = correct ? 'c' : 'w';\n answered++;\n\n for (i = 0; i < opts.length; i++) {\n el = byId(NS + '-o' + qid + '-' + i);\n if (opts[i] === q.correct) {\n el.className = 'mr-opt correct locked';\n } else if (i === oi) {\n el.className = 'mr-opt wrong locked';\n } else {\n el.className = 'mr-opt dimmed locked';\n }\n }\n\n byId(NS + '-exp' + qid).style.display = 'block';\n byId(NS + '-pip' + qid).className = 'mr-pip ' + (correct ? 'correct' : 'wrong');\n\n if (qid > 1) {\n var pl = byId(NS + '-pl' + qid);\n if (pl) { pl.className = 'mr-pip-line done'; }\n }\n }\n\n function showScore() {\n var c, w, s, net, pct, disp, verdicts, vi, sc;\n if (done) return;\n done = true;\n\n c = countVal('c');\n w = countVal('w');\n s = TOTAL - answered;\n net = (c * 4) - w;\n pct = Math.max(0, Math.round((net \/ MAX) * 100));\n disp = Math.min(100, Math.max(0, pct));\n\n gid('ring').style.background =\n 'conic-gradient(#1A5F7A ' + disp + '%, #B8D8E3 0%)';\n\n gid('pct').textContent = pct + '%';\n gid('net').textContent = 'Net Score: ' + net + ' \/ ' + MAX;\n\n verdicts = [\n [5, 'Flawless. ARR, NNT, RRR, RR, OR, NNH \\u2014 all formulae and their traps owned cold.'],\n [4, 'Strong round. One concept to consolidate \\u2014 the debrief has it.'],\n [3, 'Good base. Q2 (RRR without ARR) and Q4 (OR cross-product formula) are the high-yield misses.'],\n [2, 'Two anchors to lock in: NNT = 1\\u00f7ARR; OR = ad\\u00f7bc. Everything else in this round derives from those two.'],\n [0, 'Start with Q1 \\u2014 ARR and NNT are the foundation of this entire round. 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{"id":37374,"date":"2026-08-23T13:32:48","date_gmt":"2026-08-23T08:02:48","guid":{"rendered":"https:\/\/atsixty.com\/?p=37374"},"modified":"2026-08-23T19:26:36","modified_gmt":"2026-08-23T13:56:36","slug":"biostatistics-nnt-nnh-arr-rrr-the-risk-ratios","status":"publish","type":"post","link":"https:\/\/atsixty.com\/index.php\/morning-rounds\/biostatistics-nnt-nnh-arr-rrr-the-risk-ratios\/","title":{"rendered":"Biostatistics – NNT, NNH, ARR, RRR & the Risk Ratios"},"content":{"rendered":"\n\n\n\n\n\n\n\nMorning Rounds \u00b7 PSM Numericals \u00b7 Round 04<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,600;0,700;1,400;1,600&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n#nps04 *,#nps04 *::before,#nps04 *::after{box-sizing:border-box;margin:0;padding:0}\n#nps04{\n --ob:#1A5F7A;--ob-light:#236E8C;--ob-pale:#E6F3F7;--ob-dark:#114455;\n 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#nps04 .mr-opt-text{font-size:0.86rem}\n}\n<\/style>\n\n<div id=\"nps04\">\n\n <div class=\"mr-header\">\n <div class=\"mr-series-tag\">Numerical Series · PSM Biostatistics<\/div>\n <div class=\"mr-eyebrow\">Morning Rounds · Round 04 of 07<\/div>\n <div class=\"mr-title\">NNT, NNH, ARR, RRR<br><em>& the Risk Ratios<\/em><\/div>\n <div class=\"mr-subtitle\">Five questions · Treatment effect measures, how they relate & why RRR without ARR misleads<\/div>\n <div class=\"mr-chips\">\n <span class=\"mr-chip\">5 Questions<\/span>\n <span class=\"mr-chip\">+4 \/ −1 scoring<\/span>\n <span class=\"mr-chip\">Options reshuffled<\/span>\n <\/div>\n <\/div>\n\n <div class=\"mr-sentinel\" id=\"nps04-sentinel\"><\/div>\n\n <div class=\"mr-progress\" id=\"nps04-progress\">\n <div class=\"mr-prog-inner\">\n <div class=\"mr-pips\" id=\"nps04-pips\"><\/div>\n <\/div>\n <\/div>\n\n <div class=\"mr-body\">\n <div id=\"nps04-cases\"><\/div>\n <div class=\"mr-submit-wrap\">\n <button class=\"mr-btn\" id=\"nps04-submit\">Submit for Debrief<\/button>\n <\/div>\n <div class=\"mr-score\" id=\"nps04-score\">\n <div class=\"mr-score-in\">\n <div class=\"mr-score-ey\">Round Complete<\/div>\n <div class=\"mr-ring\" id=\"nps04-ring\">\n <div class=\"mr-ring-in\">\n <span class=\"mr-ring-pct\" id=\"nps04-pct\">0%<\/span>\n <span class=\"mr-ring-sub\">net<\/span>\n <\/div>\n <\/div>\n <div class=\"mr-score-title\">Your Debrief<\/div>\n <div class=\"mr-score-net\" id=\"nps04-net\"><\/div>\n <div class=\"mr-verdict\" id=\"nps04-verdict\"><\/div>\n <div class=\"mr-bands\">\n <span class=\"mr-band mr-band-c\" id=\"nps04-ct-c\"><\/span>\n <span class=\"mr-band mr-band-w\" id=\"nps04-ct-w\"><\/span>\n <span class=\"mr-band mr-band-s\" id=\"nps04-ct-s\"><\/span>\n <\/div>\n <button class=\"mr-retry\" id=\"nps04-retry\">↻ New Round<\/button>\n <\/div>\n <\/div>\n <\/div>\n\n<\/div>\n\n<script>\n(function () {\n 'use strict';\n\n var NS = 'nps04';\n var TOTAL = 5;\n var MAX = 20;\n var LTRS = ['A','B','C','D'];\n\n var QS = [\n\n {\n id: 1,\n tag: 'ARR & NNT — Core Calculation',\n stem: 'A trial of a new antihypertensive tests its effect on stroke. In the <strong>control group<\/strong>, 8% have a stroke over 5 years. In the <strong>treatment group<\/strong>, 5% have a stroke. What is the <strong>ARR<\/strong> and <strong>NNT<\/strong>?',\n correct: 'ARR = 3%; NNT = 34',\n opts: [\n 'ARR = 3%; NNT = 34',\n 'ARR = 5%; NNT = 20',\n 'ARR = 3%; NNT = 20',\n 'ARR = 8%; NNT = 13'\n ],\n exp: '<strong>Absolute Risk Reduction (ARR)<\/strong> = Risk in control − Risk in treatment.<span class=\"calc\">ARR = 8% − 5% = <strong>3%<\/strong><br><br>NNT = 1 ÷ ARR = 1 ÷ 0.03 = <strong>33.3 ≈ 34<\/strong><\/span>NNT is the number of patients who must be treated for 5 years to prevent one stroke. Here, 34 patients need treatment to prevent one additional stroke over 5 years \u2014 that is the clinical weight of NNT.<br><br>Trap B uses the control event rate (8%) as ARR. Trap C gets ARR right but calculates NNT as 1÷0.05 = 20, using the wrong denominator. Trap D uses the treatment event rate as ARR. The NNT formula is always <strong>1 ÷ ARR (as a decimal)<\/strong>. Always round up \u2014 you cannot treat a fraction of a patient.'\n },\n\n {\n id: 2,\n tag: 'RRR — Calculation and the Misleading Headline',\n stem: 'Using the same stroke trial (control risk 8%, treatment risk 5%), a pharmaceutical company advertises the drug as reducing stroke risk by <strong>37.5%<\/strong>. This figure is the RRR. Is this claim accurate, and does it give a complete picture?',\n correct: 'Accurate but incomplete; RRR = 37.5% but ARR = only 3%, meaning the absolute benefit is modest despite the impressive relative figure',\n opts: [\n 'Accurate but incomplete; RRR = 37.5% but ARR = only 3%, meaning the absolute benefit is modest despite the impressive relative figure',\n 'Inaccurate; the correct RRR is 62.5%, calculated as treatment risk divided by control risk',\n 'Accurate and complete; RRR of 37.5% fully characterises the treatment benefit without needing ARR',\n 'Inaccurate; RRR should be calculated from absolute numbers of events, not percentages'\n ],\n exp: '<strong>Relative Risk Reduction (RRR)<\/strong> = ARR ÷ Control event rate.<span class=\"calc\">RRR = (8% − 5%) ÷ 8% = 3% ÷ 8% = <strong>0.375 = 37.5%<\/strong><\/span>The figure is arithmetically correct. But notice: a drug that reduces risk from <strong>0.2% to 0.125%<\/strong> also has an RRR of 37.5% \u2014 yet its ARR is only 0.075%, and its NNT is 1333. The RRR is identical; the clinical significance is entirely different.<br><br>This is why <strong>RRR without ARR is incomplete and potentially misleading<\/strong>. Drug advertisements routinely headline RRR because it sounds more impressive. Examiners test this asymmetry directly. The full picture always requires: control event rate, ARR, RRR, and NNT together. Trap C claims RRR alone is sufficient \u2014 that is the pharmaceutical marketing position, not the evidence-based one.'\n },\n\n {\n id: 3,\n tag: 'Relative Risk — Cohort Study Calculation',\n stem: 'In a cohort study of 1000 smokers and 1000 non-smokers followed for 10 years, <strong>80 smokers<\/strong> and <strong>20 non-smokers<\/strong> develop lung cancer. What is the <strong>Relative Risk (RR)<\/strong> and how is it interpreted?',\n correct: 'RR = 4.0; smokers are 4 times as likely to develop lung cancer as non-smokers',\n opts: [\n 'RR = 4.0; smokers are 4 times as likely to develop lung cancer as non-smokers',\n 'RR = 0.25; non-smokers are 75% less likely to develop lung cancer than smokers',\n 'RR = 4.0; the absolute excess risk attributable to smoking is 4 cases per 1000',\n 'RR = 60; calculated as the difference in event numbers between the two groups'\n ],\n exp: '<strong>Relative Risk (RR)<\/strong> = Incidence in exposed ÷ Incidence in unexposed. Used in <strong>cohort studies<\/strong>.<span class=\"calc\">Incidence in smokers = 80 ÷ 1000 = 0.08 (8%)<br>Incidence in non-smokers = 20 ÷ 1000 = 0.02 (2%)<br>RR = 0.08 ÷ 0.02 = <strong>4.0<\/strong><\/span>Interpretation: smokers are <strong>4 times more likely<\/strong> to develop lung cancer than non-smokers over 10 years. RR = 1 means no association. RR >1 means increased risk in the exposed. RR <1 means protective effect.<br><br>Trap B calculates the reciprocal (1÷4 = 0.25) and reframes it from the non-smoker perspective \u2014 mathematically valid but not what was asked. Trap C correctly states RR = 4.0 but then misdefines it as absolute excess risk \u2014 RR is a ratio, not an absolute figure. Trap D subtracts instead of divides. Key exam point: <strong>RR is used in cohort studies; OR is used in case-control studies<\/strong>.'\n },\n\n {\n id: 4,\n tag: 'Odds Ratio — Case-Control Calculation',\n stem: 'In a case-control study of myocardial infarction, <strong>60 of 100 cases<\/strong> were smokers versus <strong>30 of 100 controls<\/strong>. What is the <strong>Odds Ratio (OR)<\/strong> for smoking and MI?',\n correct: 'OR = 3.5; smokers have 3.5 times the odds of MI compared to non-smokers',\n opts: [\n 'OR = 3.5; smokers have 3.5 times the odds of MI compared to non-smokers',\n 'OR = 2.0; calculated as the ratio of smoking prevalence in cases versus controls',\n 'OR = 3.5; this equals the relative risk since case and control numbers are equal',\n 'OR = 0.29; calculated as the odds of smoking in controls divided by odds in cases'\n ],\n exp: 'In a case-control study, we calculate <strong>odds of exposure<\/strong> in cases versus controls \u2014 not incidence rates (those are unavailable).<span class=\"calc\">Odds of smoking in cases = 60 ÷ 40 = 1.5<br>Odds of smoking in controls = 30 ÷ 70 = 0.4286<br>OR = 1.5 ÷ 0.4286 = <strong>3.5<\/strong><\/span>Or using the cross-product formula directly:<span class=\"calc\">OR = (a × d) ÷ (b × c) = (60 × 70) ÷ (40 × 30) = 4200 ÷ 1200 = <strong>3.5<\/strong><\/span>Interpretation: smokers have 3.5 times the <em>odds<\/em> of having had an MI compared to non-smokers.<br><br>Trap B calculates 60÷30 = 2 \u2014 this is the ratio of smoking prevalences, not the odds ratio. Trap C correctly states OR = 3.5 but claims it equals RR \u2014 OR approximates RR only when disease prevalence is <strong>low<\/strong> (the rare disease assumption); with 60% case exposure this assumption fails. Trap D inverts the ratio. The cross-product formula (ad÷bc) is the fastest route.'\n },\n\n {\n id: 5,\n tag: 'NNH — Harm from Adverse Events',\n stem: 'A statin trial shows that <strong>3%<\/strong> of treated patients develop myopathy versus <strong>1%<\/strong> of placebo patients over 3 years. What is the <strong>NNH<\/strong>, and how should it be communicated to a patient?',\n correct: 'NNH = 50; one additional case of myopathy occurs for every 50 patients treated for 3 years',\n opts: [\n 'NNH = 50; one additional case of myopathy occurs for every 50 patients treated for 3 years',\n 'NNH = 33; calculated as 1 divided by the treated group event rate of 3%',\n 'NNH = 50; since NNH exceeds NNT, the drug is definitively net beneficial',\n 'NNH = 100; calculated as 1 divided by the absolute difference expressed as a whole number'\n ],\n exp: '<strong>Number Needed to Harm (NNH)<\/strong> = 1 ÷ Absolute Risk Increase (ARI).<span class=\"calc\">ARI = Risk in treated − Risk in control = 3% − 1% = 2%<br>NNH = 1 ÷ 0.02 = <strong>50<\/strong><\/span>Communication: \"For every 50 patients I treat with this statin for 3 years, one additional patient will develop myopathy who would not have developed it on placebo.\"<br><br>NNH must always be considered alongside NNT. If NNT = 34 (one stroke prevented per 34 patients) and NNH = 50 (one myopathy per 50 patients), the balance favours treatment \u2014 but this is a clinical judgement, not an automatic conclusion. Trap B divides by 3% (treated rate alone, ignoring baseline). Trap C correctly calculates NNH = 50 but then makes the logical error that NNH > NNT automatically means net benefit \u2014 severity of the harm matters equally; myopathy and stroke are not equivalent outcomes. Trap D divides by 2 (the whole number difference) instead of 0.02.'\n }\n\n ];\n\n var answers = {}, answered = 0, shuffled = {}, done = false;\n\n function gid(s) { return document.getElementById(NS + '-' + s); }\n function byId(s) { return document.getElementById(s); }\n\n function shuffleArr(arr) {\n var a = arr.slice(), i, j, t;\n for (i = a.length - 1; i > 0; i--) {\n j = Math.floor(Math.random() * (i + 1));\n t = a[i]; a[i] = a[j]; a[j] = t;\n }\n return a;\n }\n\n function countVal(v) {\n var n = 0, k;\n for (k in answers) { if (answers[k] === v) n++; }\n return n;\n }\n\n function buildPips() {\n var cont = gid('pips'), i, q, wLine, wPip, line, pip;\n cont.innerHTML = '';\n for (i = 0; i < QS.length; i++) {\n q = QS[i];\n if (i > 0) {\n wLine = document.createElement('div');\n wLine.className = 'mr-pip-wrap';\n line = document.createElement('div');\n line.className = 'mr-pip-line';\n line.id = NS + '-pl' + q.id;\n wLine.appendChild(line);\n cont.appendChild(wLine);\n }\n wPip = document.createElement('div');\n wPip.className = 'mr-pip-wrap';\n pip = document.createElement('div');\n pip.className = 'mr-pip';\n pip.id = NS + '-pip' + q.id;\n pip.textContent = String(q.id);\n wPip.appendChild(pip);\n cont.appendChild(wPip);\n }\n }\n\n function build() {\n var cont, i, q, opts, card, top, numDiv, meta, tag, stem,\n rule, optsDiv, expDiv, lbl, txt, j, optEl, ltrSpan, txtSpan;\n\n cont = gid('cases');\n cont.innerHTML = '';\n answers = {}; answered = 0; shuffled = {}; done = false;\n gid('score').style.display = 'none';\n buildPips();\n\n for (i = 0; i < QS.length; i++) {\n q = QS[i];\n opts = shuffleArr(q.opts);\n shuffled[q.id] = opts;\n\n card = document.createElement('div');\n card.className = 'mr-case';\n\n top = document.createElement('div');\n top.className = 'mr-case-top';\n\n numDiv = document.createElement('div');\n numDiv.className = 'mr-num';\n numDiv.textContent = q.id < 10 ? '0' + q.id : String(q.id);\n\n meta = document.createElement('div');\n meta.className = 'mr-meta';\n\n tag = document.createElement('div');\n tag.className = 'mr-tag';\n tag.innerHTML = q.tag;\n\n stem = document.createElement('div');\n stem.className = 'mr-stem';\n stem.innerHTML = q.stem;\n\n meta.appendChild(tag);\n meta.appendChild(stem);\n top.appendChild(numDiv);\n top.appendChild(meta);\n card.appendChild(top);\n\n rule = document.createElement('div');\n rule.className = 'mr-rule';\n card.appendChild(rule);\n\n optsDiv = document.createElement('div');\n optsDiv.className = 'mr-opts';\n\n for (j = 0; j < opts.length; j++) {\n optEl = document.createElement('div');\n optEl.className = 'mr-opt';\n optEl.id = NS + '-o' + q.id + '-' + j;\n optEl.setAttribute('role', 'button');\n optEl.setAttribute('tabindex', '0');\n\n ltrSpan = document.createElement('span');\n ltrSpan.className = 'mr-ltr';\n ltrSpan.textContent = LTRS[j];\n\n txtSpan = document.createElement('span');\n txtSpan.className = 'mr-opt-text';\n txtSpan.innerHTML = opts[j];\n\n optEl.appendChild(ltrSpan);\n optEl.appendChild(txtSpan);\n optsDiv.appendChild(optEl);\n\n (function (qid, oi) {\n optEl.addEventListener('click', function () { pick(qid, oi); });\n }(q.id, j));\n }\n card.appendChild(optsDiv);\n\n expDiv = document.createElement('div');\n expDiv.className = 'mr-exp';\n expDiv.id = NS + '-exp' + q.id;\n\n lbl = document.createElement('div');\n lbl.className = 'mr-exp-lbl';\n lbl.textContent = 'Debrief';\n\n txt = document.createElement('div');\n txt.className = 'mr-exp-text';\n txt.innerHTML = q.exp;\n\n expDiv.appendChild(lbl);\n expDiv.appendChild(txt);\n card.appendChild(expDiv);\n cont.appendChild(card);\n }\n }\n\n function pick(qid, oi) {\n var q, opts, i, el, correct;\n if (answers[qid] !== undefined || done) return;\n\n q = null;\n for (i = 0; i < QS.length; i++) {\n if (QS[i].id === qid) { q = QS[i]; break; }\n }\n if (!q) return;\n\n opts = shuffled[qid];\n correct = (opts[oi] === q.correct);\n answers[qid] = correct ? 'c' : 'w';\n answered++;\n\n for (i = 0; i < opts.length; i++) {\n el = byId(NS + '-o' + qid + '-' + i);\n if (opts[i] === q.correct) {\n el.className = 'mr-opt correct locked';\n } else if (i === oi) {\n el.className = 'mr-opt wrong locked';\n } else {\n el.className = 'mr-opt dimmed locked';\n }\n }\n\n byId(NS + '-exp' + qid).style.display = 'block';\n byId(NS + '-pip' + qid).className = 'mr-pip ' + (correct ? 'correct' : 'wrong');\n\n if (qid > 1) {\n var pl = byId(NS + '-pl' + qid);\n if (pl) { pl.className = 'mr-pip-line done'; }\n }\n }\n\n function showScore() {\n var c, w, s, net, pct, disp, verdicts, vi, sc;\n if (done) return;\n done = true;\n\n c = countVal('c');\n w = countVal('w');\n s = TOTAL - answered;\n net = (c * 4) - w;\n pct = Math.max(0, Math.round((net \/ MAX) * 100));\n disp = Math.min(100, Math.max(0, pct));\n\n gid('ring').style.background =\n 'conic-gradient(#1A5F7A ' + disp + '%, #B8D8E3 0%)';\n\n gid('pct').textContent = pct + '%';\n gid('net').textContent = 'Net Score: ' + net + ' \/ ' + MAX;\n\n verdicts = [\n [5, 'Flawless. ARR, NNT, RRR, RR, OR, NNH \\u2014 all formulae and their traps owned cold.'],\n [4, 'Strong round. One concept to consolidate \\u2014 the debrief has it.'],\n [3, 'Good base. Q2 (RRR without ARR) and Q4 (OR cross-product formula) are the high-yield misses.'],\n [2, 'Two anchors to lock in: NNT = 1\\u00f7ARR; OR = ad\\u00f7bc. 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