30:T474, Vertical drops ⇒ Median survival Where does the curve first cross 0.5? ⇒ Between 6.5 mo and 7 mo; linear interpolation gives ≈ 6.8 mo. What if all observations were censored after a point? The curve would stay flat at its last value indefinitely — we would have no information about later survival. How does censoring affect nini and the variance of S^(t)S^(t)? What happens if several events share the same time? All their deaths are tallied in didi for that single row, producing a bigger step.

What does a horizontal stretch in the KM curve indicate?

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No events occurred during that interval (only censoring or nothing at all).

What causes the step-down at a specific time?

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One or more events; the size of the drop is proportional to di/nidi /ni.

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No events occurred during that interval (only censoring or nothing at all).

31:[] 2e:{"title":"Survival curves","cells":[{"id":0,"value":"# Survival curves","pinned":false,"hidden":false,"mode":"md"},{"id":5,"value":"survivalData = [\n { id: 1, studyEntryAge: 68, studyExitAge: 69.5, time: 1.5, event: 1 },\n { id: 2, studyEntryAge: 53, studyExitAge: 56, time: 3.0, event: 0 },\n { id: 3, studyEntryAge: 61, studyExitAge: 64.5, time: 3.5, event: 1 },\n { id: 4, studyEntryAge: 50, studyExitAge: 54, time: 4.0, event: 1 },\n { id: 5, studyEntryAge: 64, studyExitAge: 70, time: 6.0, event: 0 },\n { id: 6, studyEntryAge: 56, studyExitAge: 62.5, time: 6.5, event: 1 },\n { id: 7, studyEntryAge: 66, studyExitAge: 73, time: 7.0, event: 1 },\n { id: 8, studyEntryAge: 63, studyExitAge: 71.5, time: 8.5, event: 0 },\n { id: 9, studyEntryAge: 55, studyExitAge: 64, time: 9.0, event: 1 },\n { id: 10, studyEntryAge: 50, studyExitAge: 60, time: 10.0, event: 0 }\n]","pinned":true,"hidden":false,"mode":"ojs"},{"id":7,"value":"survivalData","pinned":true,"hidden":false,"mode":"ojs"},{"id":64,"value":"## Survival data by study entry/exit age","pinned":false,"hidden":false,"mode":"md"},{"id":56,"value":"Plot.plot({\n height: 250,\n insetTop: 10,\n insetBottom: 10,\n\n // lock the y-scale domain to the sorted list of ids\n y: {\n type: \"point\",\n domain: survivalData // [1,2,…,10]\n .map((d) => d.id)\n .sort((a, b) => a - b) // ascending\n },\n\n x: { label: \"Age (years)\" },\n\n marks: [\n // horizontal “sticks” from entry age to exit age\n Plot.ruleY(survivalData, {\n y: \"id\",\n x1: \"studyEntryAge\",\n x2: \"studyExitAge\",\n strokeWidth: 3\n }),\n\n // dots at the exit age\n Plot.dot(survivalData, {\n x: \"studyExitAge\",\n y: \"id\",\n fill: \"event\",\n stroke: \"black\",\n r: 6,\n tip: true\n })\n ],\n\n color: { domain: [0, 1], range: [\"white\", \"red\"] }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":66,"value":"## Survival data by survival time","pinned":false,"hidden":false,"mode":"md"},{"id":46,"value":"Plot.plot({\n height: 250,\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY(survivalData, {\n y: \"id\",\n x: \"time\",\n sort: { y: \"x\" },\n strokeWidth: 3\n }),\n Plot.dot(survivalData, {\n x: \"time\",\n y: \"id\",\n fill: \"event\",\n stroke: \"black\",\n r: 6,\n tip: true\n })\n ],\n y: { type: \"point\" },\n color: { domain: [0, 1], range: [\"white\", \"red\"] },\n insetTop: 10,\n insetBottom: 10\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":68,"value":"## Cumulative survival\n\n${tex`\\prod_{t_i < t} \\frac{n_i - d_i}{n_i}`}","pinned":false,"hidden":false,"mode":"md"},{"id":14,"value":"data = manualKMEstimation(survivalData)","pinned":true,"hidden":false,"mode":"ojs"},{"id":18,"value":"data","pinned":true,"hidden":false,"mode":"ojs"},{"id":27,"value":"kmPlot(survivalData)","pinned":true,"hidden":false,"mode":"ojs"},{"id":33,"value":"$30","pinned":false,"hidden":false,"mode":"html"},{"id":30,"value":"Flat segments\nWhat does a horizontal stretch in the KM curve indicate?\n⇒ No events occurred during that interval (only censoring or nothing at all).\n\nVertical drops\nWhat causes the step-down at a specific time?\n⇒ One or more events; the size of the drop is proportional to di/nidi/ni.\n\nMedian survival\nWhere does the curve first cross 0.5?\n⇒ Between 6.5 mo and 7 mo; linear interpolation gives ≈ 6.8 mo.\n\nWhat if all observations were censored after a point?\nThe curve would stay flat at its last value indefinitely — we would have no information about later survival.\n\nHow does censoring affect nini and the variance of S^(t)S^(t)?\n\nWhat happens if several events share the same time?\nAll their deaths are tallied in didi for that single row, producing a bigger step.","pinned":false,"hidden":false,"mode":"md"},{"id":9,"value":"# Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"function manualKMEstimation(data) {\n // Sort by time\n const sorted = data.slice().sort((a, b) => a.time - b.time);\n\n // Get unique time points\n const timePoints = [...new Set(sorted.map(d => d.time))];\n\n let nAtRisk = data.length;\n let cumulativeSurvival = 1;\n const table = [];\n\n for (const t of timePoints) {\n const eventsAtT = sorted.filter(d => d.time === t && d.event === 1).length;\n const censoredAtT = sorted.filter(d => d.time === t && d.event === 0).length;\n\n if (eventsAtT > 0) {\n const conditionalSurvival = (nAtRisk - eventsAtT) / nAtRisk;\n cumulativeSurvival *= conditionalSurvival;\n table.push({\n time: t,\n events: eventsAtT,\n censored: censoredAtT,\n atRisk: nAtRisk,\n conditionalSurvival: conditionalSurvival.toFixed(3),\n cumulativeSurvival: cumulativeSurvival.toFixed(3)\n });\n }\n\n nAtRisk -= eventsAtT + censoredAtT;\n }\n\n return table;\n}\n","pinned":true,"hidden":false,"mode":"ojs"},{"id":23,"value":"function kmEstimate(data) {\n const sorted = data.slice().sort((a, b) => a.time - b.time);\n\n let n = sorted.length;\n let survival = 1;\n const curve = [{time: 0, survival: 1}];\n\n for (const {time, event} of sorted) {\n if (event === 1) {\n const d = 1;\n survival *= (n - d) / n;\n curve.push({time, survival});\n }\n // either event or censored -> one less at risk\n n -= 1; \n }\n return curve;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":25,"value":"function kmPlot(data) {\n const curve = kmEstimate(data);\n return Plot.plot({\n y: {label: \"Survival\"},\n marks: [\n Plot.lineY(curve, {\n x: \"time\",\n y: \"survival\",\n stroke: \"steelblue\",\n curve: \"step-after\"\n }),\n Plot.ruleY([0.5], {strokeDasharray: \"4,4\", strokeOpacity: 0.6}) // median guide\n ]\n });\n}","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}