Morning Rounds \u00b7 PSM Numericals \u00b7 Round 01<\/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#nps01 *,#nps01 *::before,#nps01 *::after{box-sizing:border-box;margin:0;padding:0}\n#nps01{\n --ob:#1A5F7A;\n --ob-light:#236E8C;\n --ob-pale:#E6F3F7;\n --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#nps01 .mr-header{\n background:linear-gradient(135deg,var(--ob-dark) 0%,var(--ob) 60%,var(--ob-light) 100%);\n color:#E4F4F9;padding:36px 24px 30px;text-align:center;\n 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class=\"mr-eyebrow\">Morning Rounds · Round 01 of 07<\/div>\n <div class=\"mr-title\">The 2×2 Table &<br><em>Its Four Cells<\/em><\/div>\n <div class=\"mr-subtitle\">Five questions · Building the table, filling cells & reading what each number actually means<\/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=\"nps01-sentinel\"><\/div>\n\n <div class=\"mr-progress\" id=\"nps01-progress\">\n <div class=\"mr-prog-inner\">\n <div class=\"mr-pips\" id=\"nps01-pips\"><\/div>\n <\/div>\n <\/div>\n\n <div class=\"mr-body\">\n <div id=\"nps01-cases\"><\/div>\n <div class=\"mr-submit-wrap\">\n <button class=\"mr-btn\" id=\"nps01-submit\">Submit for Debrief<\/button>\n <\/div>\n <div class=\"mr-score\" id=\"nps01-score\">\n <div class=\"mr-score-in\">\n <div class=\"mr-score-ey\">Round Complete<\/div>\n <div class=\"mr-ring\" id=\"nps01-ring\">\n <div class=\"mr-ring-in\">\n <span class=\"mr-ring-pct\" id=\"nps01-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=\"nps01-net\"><\/div>\n <div class=\"mr-verdict\" id=\"nps01-verdict\"><\/div>\n <div class=\"mr-bands\">\n <span class=\"mr-band mr-band-c\" id=\"nps01-ct-c\"><\/span>\n <span class=\"mr-band mr-band-w\" id=\"nps01-ct-w\"><\/span>\n <span class=\"mr-band mr-band-s\" id=\"nps01-ct-s\"><\/span>\n <\/div>\n <button class=\"mr-retry\" id=\"nps01-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 = 'nps01';\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: '2×2 Table — Cell Identification',\n stem: 'A screening test is applied to 1000 people. <strong>200 have the disease<\/strong>. Of these, the test is positive in <strong>180<\/strong>. Of the <strong>800 without disease<\/strong>, the test is positive in <strong>80<\/strong>. Which set of values correctly fills the four cells of the 2×2 table?',\n correct: 'TP=180, FN=20, FP=80, TN=720',\n opts: [\n 'TP=180, FN=20, FP=80, TN=720',\n 'TP=180, FN=80, FP=20, TN=720',\n 'TP=160, FN=40, FP=80, TN=720',\n 'TP=180, FN=20, FP=80, TN=640'\n ],\n exp: 'The 2×2 table places <strong>disease status in columns<\/strong> and <strong>test result in rows<\/strong>.<span class=\"calc\">Disease present (n=200): test positive = 180 (TP), test negative = 20 (FN)<br>Disease absent (n=800): test positive = 80 (FP), test negative = 720 (TN)<\/span>Check: TP+FN = 200 \u2713 FP+TN = 800 \u2713 Total = 1000 \u2713<br><br>Trap B swaps FN and FP \u2014 the most common cell-assignment error. Trap C invents a wrong TP. Trap D miscalculates TN as 800−80−40 = 680 (wrong) by using a number from nowhere. Always verify: each disease column must sum to its known total.'\n },\n\n {\n id: 2,\n tag: '2×2 Table — What True Negative Means',\n stem: 'In the same study (TP=180, FN=20, FP=80, TN=720), a colleague says the <strong>720 true negatives<\/strong> are the most important cell for ruling out disease. A second colleague disagrees, saying <strong>FN=20<\/strong> is the most critical cell clinically. Who is correct, and why?',\n correct: 'The second colleague; false negatives are missed cases \u2014 disease present but test negative \u2014 causing delayed diagnosis and continued transmission or progression',\n opts: [\n 'The second colleague; false negatives are missed cases \u2014 disease present but test negative \u2014 causing delayed diagnosis and continued transmission or progression',\n 'The first colleague; true negatives confirm the absence of disease and allow safe discharge, which is the primary goal of most screening programmes',\n 'Both are correct; TN and FN are mirror images of the same cell and carry equal clinical weight in any screening context',\n 'Neither; the FP cell determines screening programme cost and is the most policy-relevant figure in public health decisions'\n ],\n exp: '<strong>False Negatives (FN)<\/strong> = disease present, test negative. These are the missed cases. In a screening programme, a missed case:<br>• Continues to be untreated (progression risk)<br>• May continue spreading infection (TB, HIV)<br>• Provides false reassurance<br><br><strong>True Negatives (TN)<\/strong> are correctly cleared individuals \u2014 important for programme efficiency, but missing one is not a clinical harm in the same way. The asymmetry is why <strong>sensitivity<\/strong> (which directly measures how few FNs occur) is the priority metric for screening tests \u2014 you want to miss as few diseased people as possible.<br><br>Trap C is wrong: TN and FN are not mirror images. TN = correct negative; FN = incorrect negative. Trap D is a legitimate policy concern but not the answer to \"most critical clinically.\"'\n },\n\n {\n id: 3,\n tag: '2×2 Table — Filling from Partial Data',\n stem: 'A diagnostic test is evaluated in 500 patients. Prevalence of disease is <strong>40%<\/strong>. The test has <strong>sensitivity 90%<\/strong> and <strong>specificity 85%<\/strong>. How many patients fall in each cell?',\n correct: 'TP=180, FN=20, FP=45, TN=255',\n opts: [\n 'TP=180, FN=20, FP=45, TN=255',\n 'TP=180, FN=20, FP=30, TN=270',\n 'TP=160, FN=40, FP=45, TN=255',\n 'TP=180, FN=20, FP=55, TN=245'\n ],\n exp: 'Work step by step from what you know:<span class=\"calc\">Disease present = 40% × 500 = <strong>200<\/strong><br>Disease absent = 60% × 500 = <strong>300<\/strong><br><br>Sensitivity = TP ÷ (TP+FN) = 0.90<br>TP = 0.90 × 200 = <strong>180<\/strong> FN = 200−180 = <strong>20<\/strong><br><br>Specificity = TN ÷ (FP+TN) = 0.85<br>TN = 0.85 × 300 = <strong>255<\/strong> FP = 300−255 = <strong>45<\/strong><\/span>Check: 180+20+45+255 = 500 \u2713<br><br>Trap B uses specificity 90% by mistake for the TN calculation. Trap C applies sensitivity to the wrong denominator (300 instead of 200). Trap D miscalculates FP. The method is always: find disease totals from prevalence first, then apply sensitivity to diseased column and specificity to non-diseased column.'\n },\n\n {\n id: 4,\n tag: '2×2 Table — Identifying False Positive Rate',\n stem: 'From the table in Q3 (TP=180, FN=20, FP=45, TN=255), a health administrator asks for the <strong>false positive rate<\/strong> of the test. Which value is correct, and from which cells is it derived?',\n correct: '15%; calculated as FP ÷ (FP+TN) = 45 ÷ 300',\n opts: [\n '15%; calculated as FP ÷ (FP+TN) = 45 ÷ 300',\n '20%; calculated as FP ÷ (TP+FP) = 45 ÷ 225',\n '15%; calculated as FP ÷ (TP+FP+FP+TN) = 45 ÷ 300',\n '10%; calculated as FN ÷ (TP+FN) = 20 ÷ 200'\n ],\n exp: '<strong>False Positive Rate (FPR) = FP ÷ (FP+TN)<\/strong> \u2014 the proportion of disease-free people who test positive.<span class=\"calc\">FPR = 45 ÷ (45+255) = 45 ÷ 300 = <strong>0.15 = 15%<\/strong><\/span>Note: FPR = <strong>1 − Specificity<\/strong>. Here specificity = 85%, so FPR = 15%. This is a quick cross-check.<br><br>The four rates to know and their denominators:<br>• Sensitivity = TP ÷ (TP+FN) \u2014 denominator: all diseased<br>• Specificity = TN ÷ (FP+TN) \u2014 denominator: all non-diseased<br>• FPR = FP ÷ (FP+TN) = 1−Specificity<br>• FNR = FN ÷ (TP+FN) = 1−Sensitivity<br><br>Trap B calculates positive predictive value, not FPR. Trap D calculates the false negative rate. These are the two most common confusions on this topic.'\n },\n\n {\n id: 5,\n tag: '2×2 Table — Effect of Changing Cut-off',\n stem: 'A laboratory lowers the cut-off value for a blood test, making it easier to test positive. Compared to the original cut-off, what happens to the cells of the 2×2 table?',\n correct: 'TP and FP both increase; FN and TN both decrease \u2014 sensitivity rises, specificity falls',\n opts: [\n 'TP and FP both increase; FN and TN both decrease \u2014 sensitivity rises, specificity falls',\n 'TP increases and FN decreases; FP and TN remain unchanged \u2014 only sensitivity is affected',\n 'FP increases and TN decreases; TP and FN remain unchanged \u2014 only specificity is affected',\n 'All four cells change unpredictably; the direction depends on disease prevalence, not cut-off alone'\n ],\n exp: 'Lowering the cut-off means <strong>more people test positive<\/strong> \u2014 both diseased and non-diseased.<span class=\"calc\">Among diseased: more now test positive → TP ↑, FN ↓<br>Among non-diseased: more now test positive → FP ↑, TN ↓<\/span>Effect on metrics:<br>• Sensitivity = TP÷(TP+FN) → numerator rises, denominator fixed → <strong>sensitivity rises<\/strong><br>• Specificity = TN÷(FP+TN) → numerator falls, denominator fixed → <strong>specificity falls<\/strong><br><br>This is the fundamental <strong>sensitivity-specificity trade-off<\/strong>: they move in opposite directions when the cut-off changes. The total number of diseased and non-diseased people does not change \u2014 only how the test classifies them shifts. Prevalence plays no role in this cut-off direction question. Trap B and C each describe only half the story. Trap D incorrectly invokes prevalence as the determining variable.'\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 ? 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{"id":37365,"date":"2026-08-23T13:21:38","date_gmt":"2026-08-23T07:51:38","guid":{"rendered":"https:\/\/atsixty.com\/?p=37365"},"modified":"2026-08-23T19:25:02","modified_gmt":"2026-08-23T13:55:02","slug":"biostatistics","status":"publish","type":"post","link":"https:\/\/atsixty.com\/index.php\/morning-rounds\/biostatistics\/","title":{"rendered":"Biostatistics"},"content":{"rendered":"\n\n\n\n\n\n\n\nMorning Rounds \u00b7 PSM Numericals \u00b7 Round 01<\/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#nps01 *,#nps01 *::before,#nps01 *::after{box-sizing:border-box;margin:0;padding:0}\n#nps01{\n --ob:#1A5F7A;\n --ob-light:#236E8C;\n --ob-pale:#E6F3F7;\n --ob-dark:#114455;\n --correct:#2D6B47;--correct-bg:#EAF6EF;--correct-border:#3A9960;\n --wrong:#B83232;--wrong-bg:#FDF0F0;--wrong-border:#E53935;\n 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.mr-btn:hover{background:var(--ob-dark)}\n#nps01 .mr-score{display:none;background:var(--warm);border:1px solid var(--line);border-top:4px solid var(--ob);border-radius:10px;margin:24px 0 0;box-shadow:0 2px 12px rgba(15,42,53,0.08);overflow:hidden;}\n#nps01 .mr-score-in{padding:28px 24px;text-align:center}\n#nps01 .mr-score-ey{font-size:0.66rem;letter-spacing:0.14em;text-transform:uppercase;color:var(--ink-soft);margin-bottom:12px;font-weight:600;}\n#nps01 .mr-ring{width:98px;height:98px;border-radius:50%;background:conic-gradient(var(--ob) 0%,var(--line) 0%);display:flex;align-items:center;justify-content:center;margin:0 auto 16px;position:relative;}\n#nps01 .mr-ring::before{content:'';position:absolute;width:76px;height:76px;border-radius:50%;background:var(--warm);}\n#nps01 .mr-ring-in{position:relative;display:flex;flex-direction:column;align-items:center;line-height:1.2}\n#nps01 .mr-ring-pct{font-family:'Playfair Display',serif;font-size:1.3rem;font-weight:700;color:var(--ob)}\n#nps01 .mr-ring-sub{font-size:0.54rem;color:var(--ink-soft);text-transform:uppercase;letter-spacing:0.06em}\n#nps01 .mr-score-title{font-family:'Playfair Display',serif;font-size:1.15rem;font-weight:700;color:var(--ink);margin-bottom:4px}\n#nps01 .mr-score-net{font-size:0.9rem;color:var(--ob);font-weight:600;margin-bottom:4px}\n#nps01 .mr-verdict{font-size:0.83rem;color:var(--ink-soft);font-style:italic;margin-bottom:18px;padding:0 12px}\n#nps01 .mr-bands{display:flex;justify-content:center;gap:10px;flex-wrap:wrap}\n#nps01 .mr-band{padding:5px 13px;border-radius:16px;font-size:0.78rem;font-weight:600}\n#nps01 .mr-band-c{background:var(--correct-bg);color:var(--correct)}\n#nps01 .mr-band-w{background:var(--wrong-bg);color:var(--wrong)}\n#nps01 .mr-band-s{background:var(--ob-pale);color:var(--ob)}\n#nps01 .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#nps01 .mr-retry:hover{background:var(--ob);color:#E4F4F9}\n@media(max-width:480px){\n #nps01 .mr-title{font-size:1.4rem}\n #nps01 .mr-num{font-size:1.7rem}\n #nps01 .mr-stem{font-size:0.9rem}\n #nps01 .mr-opt-text{font-size:0.86rem}\n}\n<\/style>\n\n<div id=\"nps01\">\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 01 of 07<\/div>\n <div class=\"mr-title\">The 2×2 Table &<br><em>Its Four Cells<\/em><\/div>\n <div class=\"mr-subtitle\">Five questions · Building the table, filling cells & reading what each number actually means<\/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=\"nps01-sentinel\"><\/div>\n\n <div class=\"mr-progress\" id=\"nps01-progress\">\n <div class=\"mr-prog-inner\">\n <div class=\"mr-pips\" id=\"nps01-pips\"><\/div>\n <\/div>\n <\/div>\n\n <div class=\"mr-body\">\n <div id=\"nps01-cases\"><\/div>\n <div class=\"mr-submit-wrap\">\n <button class=\"mr-btn\" id=\"nps01-submit\">Submit for Debrief<\/button>\n <\/div>\n <div class=\"mr-score\" id=\"nps01-score\">\n <div class=\"mr-score-in\">\n <div class=\"mr-score-ey\">Round Complete<\/div>\n <div class=\"mr-ring\" id=\"nps01-ring\">\n <div class=\"mr-ring-in\">\n <span class=\"mr-ring-pct\" id=\"nps01-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=\"nps01-net\"><\/div>\n <div class=\"mr-verdict\" id=\"nps01-verdict\"><\/div>\n <div class=\"mr-bands\">\n <span class=\"mr-band mr-band-c\" id=\"nps01-ct-c\"><\/span>\n <span class=\"mr-band mr-band-w\" id=\"nps01-ct-w\"><\/span>\n <span class=\"mr-band mr-band-s\" id=\"nps01-ct-s\"><\/span>\n <\/div>\n <button class=\"mr-retry\" id=\"nps01-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 = 'nps01';\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: '2×2 Table — Cell Identification',\n stem: 'A screening test is applied to 1000 people. <strong>200 have the disease<\/strong>. Of these, the test is positive in <strong>180<\/strong>. Of the <strong>800 without disease<\/strong>, the test is positive in <strong>80<\/strong>. Which set of values correctly fills the four cells of the 2×2 table?',\n correct: 'TP=180, FN=20, FP=80, TN=720',\n opts: [\n 'TP=180, FN=20, FP=80, TN=720',\n 'TP=180, FN=80, FP=20, TN=720',\n 'TP=160, FN=40, FP=80, TN=720',\n 'TP=180, FN=20, FP=80, TN=640'\n ],\n exp: 'The 2×2 table places <strong>disease status in columns<\/strong> and <strong>test result in rows<\/strong>.<span class=\"calc\">Disease present (n=200): test positive = 180 (TP), test negative = 20 (FN)<br>Disease absent (n=800): test positive = 80 (FP), test negative = 720 (TN)<\/span>Check: TP+FN = 200 \u2713 FP+TN = 800 \u2713 Total = 1000 \u2713<br><br>Trap B swaps FN and FP \u2014 the most common cell-assignment error. Trap C invents a wrong TP. Trap D miscalculates TN as 800−80−40 = 680 (wrong) by using a number from nowhere. Always verify: each disease column must sum to its known total.'\n },\n\n {\n id: 2,\n tag: '2×2 Table — What True Negative Means',\n stem: 'In the same study (TP=180, FN=20, FP=80, TN=720), a colleague says the <strong>720 true negatives<\/strong> are the most important cell for ruling out disease. A second colleague disagrees, saying <strong>FN=20<\/strong> is the most critical cell clinically. Who is correct, and why?',\n correct: 'The second colleague; false negatives are missed cases \u2014 disease present but test negative \u2014 causing delayed diagnosis and continued transmission or progression',\n opts: [\n 'The second colleague; false negatives are missed cases \u2014 disease present but test negative \u2014 causing delayed diagnosis and continued transmission or progression',\n 'The first colleague; true negatives confirm the absence of disease and allow safe discharge, which is the primary goal of most screening programmes',\n 'Both are correct; TN and FN are mirror images of the same cell and carry equal clinical weight in any screening context',\n 'Neither; the FP cell determines screening programme cost and is the most policy-relevant figure in public health decisions'\n ],\n exp: '<strong>False Negatives (FN)<\/strong> = disease present, test negative. These are the missed cases. In a screening programme, a missed case:<br>• Continues to be untreated (progression risk)<br>• May continue spreading infection (TB, HIV)<br>• Provides false reassurance<br><br><strong>True Negatives (TN)<\/strong> are correctly cleared individuals \u2014 important for programme efficiency, but missing one is not a clinical harm in the same way. The asymmetry is why <strong>sensitivity<\/strong> (which directly measures how few FNs occur) is the priority metric for screening tests \u2014 you want to miss as few diseased people as possible.<br><br>Trap C is wrong: TN and FN are not mirror images. TN = correct negative; FN = incorrect negative. Trap D is a legitimate policy concern but not the answer to \"most critical clinically.\"'\n },\n\n {\n id: 3,\n tag: '2×2 Table — Filling from Partial Data',\n stem: 'A diagnostic test is evaluated in 500 patients. Prevalence of disease is <strong>40%<\/strong>. The test has <strong>sensitivity 90%<\/strong> and <strong>specificity 85%<\/strong>. How many patients fall in each cell?',\n correct: 'TP=180, FN=20, FP=45, TN=255',\n opts: [\n 'TP=180, FN=20, FP=45, TN=255',\n 'TP=180, FN=20, FP=30, TN=270',\n 'TP=160, FN=40, FP=45, TN=255',\n 'TP=180, FN=20, FP=55, TN=245'\n ],\n exp: 'Work step by step from what you know:<span class=\"calc\">Disease present = 40% × 500 = <strong>200<\/strong><br>Disease absent = 60% × 500 = <strong>300<\/strong><br><br>Sensitivity = TP ÷ (TP+FN) = 0.90<br>TP = 0.90 × 200 = <strong>180<\/strong> FN = 200−180 = <strong>20<\/strong><br><br>Specificity = TN ÷ (FP+TN) = 0.85<br>TN = 0.85 × 300 = <strong>255<\/strong> FP = 300−255 = <strong>45<\/strong><\/span>Check: 180+20+45+255 = 500 \u2713<br><br>Trap B uses specificity 90% by mistake for the TN calculation. Trap C applies sensitivity to the wrong denominator (300 instead of 200). Trap D miscalculates FP. The method is always: find disease totals from prevalence first, then apply sensitivity to diseased column and specificity to non-diseased column.'\n },\n\n {\n id: 4,\n tag: '2×2 Table — Identifying False Positive Rate',\n stem: 'From the table in Q3 (TP=180, FN=20, FP=45, TN=255), a health administrator asks for the <strong>false positive rate<\/strong> of the test. Which value is correct, and from which cells is it derived?',\n correct: '15%; calculated as FP ÷ (FP+TN) = 45 ÷ 300',\n opts: [\n '15%; calculated as FP ÷ (FP+TN) = 45 ÷ 300',\n '20%; calculated as FP ÷ (TP+FP) = 45 ÷ 225',\n '15%; calculated as FP ÷ (TP+FP+FP+TN) = 45 ÷ 300',\n '10%; calculated as FN ÷ (TP+FN) = 20 ÷ 200'\n ],\n exp: '<strong>False Positive Rate (FPR) = FP ÷ (FP+TN)<\/strong> \u2014 the proportion of disease-free people who test positive.<span class=\"calc\">FPR = 45 ÷ (45+255) = 45 ÷ 300 = <strong>0.15 = 15%<\/strong><\/span>Note: FPR = <strong>1 − Specificity<\/strong>. Here specificity = 85%, so FPR = 15%. This is a quick cross-check.<br><br>The four rates to know and their denominators:<br>• Sensitivity = TP ÷ (TP+FN) \u2014 denominator: all diseased<br>• Specificity = TN ÷ (FP+TN) \u2014 denominator: all non-diseased<br>• FPR = FP ÷ (FP+TN) = 1−Specificity<br>• FNR = FN ÷ (TP+FN) = 1−Sensitivity<br><br>Trap B calculates positive predictive value, not FPR. Trap D calculates the false negative rate. These are the two most common confusions on this topic.'\n },\n\n {\n id: 5,\n tag: '2×2 Table — Effect of Changing Cut-off',\n stem: 'A laboratory lowers the cut-off value for a blood test, making it easier to test positive. Compared to the original cut-off, what happens to the cells of the 2×2 table?',\n correct: 'TP and FP both increase; FN and TN both decrease \u2014 sensitivity rises, specificity falls',\n opts: [\n 'TP and FP both increase; FN and TN both decrease \u2014 sensitivity rises, specificity falls',\n 'TP increases and FN decreases; FP and TN remain unchanged \u2014 only sensitivity is affected',\n 'FP increases and TN decreases; TP and FN remain unchanged \u2014 only specificity is affected',\n 'All four cells change unpredictably; the direction depends on disease prevalence, not cut-off alone'\n ],\n exp: 'Lowering the cut-off means <strong>more people test positive<\/strong> \u2014 both diseased and non-diseased.<span class=\"calc\">Among diseased: more now test positive → TP ↑, FN ↓<br>Among non-diseased: more now test positive → FP ↑, TN ↓<\/span>Effect on metrics:<br>• Sensitivity = TP÷(TP+FN) → numerator rises, denominator fixed → <strong>sensitivity rises<\/strong><br>• Specificity = TN÷(FP+TN) → numerator falls, denominator fixed → <strong>specificity falls<\/strong><br><br>This is the fundamental <strong>sensitivity-specificity trade-off<\/strong>: they move in opposite directions when the cut-off changes. The total number of diseased and non-diseased people does not change \u2014 only how the test classifies them shifts. Prevalence plays no role in this cut-off direction question. Trap B and C each describe only half the story. Trap D incorrectly invokes prevalence as the determining variable.'\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 ? 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