Kaplan-Meier playground
Censoring is the idea that clicks visually, not from notes. Edit the dataset, flip subjects between death and censored, and watch the product-limit estimate, the Nelson-Aalen overlay and the risk-set table rebuild themselves live.
Estimated survival function (censoring shown as ticks)
Purple: Kaplan-Meier. Dashed gold: exp(−Λ̂) from the Nelson-Aalen cumulative hazard. White ticks: censored subjects leaving the risk set without an event.
Subjects (12)
Risk-set table
| tⱼ | nⱼ | dⱼ | Ŝ(t) KM | Λ̂(t) NA |
|---|---|---|---|---|
| 2 | 12 | 1 | 0.9167 | 0.0833 |
| 4 | 10 | 2 | 0.7333 | 0.2833 |
| 7 | 7 | 1 | 0.6286 | 0.4262 |
| 10 | 5 | 1 | 0.5029 | 0.6262 |
| 12 | 3 | 1 | 0.3352 | 0.9595 |
Kaplan-Meier multiplies (1 − dⱼ/nⱼ) over the event times; Nelson-Aalen sums dⱼ/nⱼ into a cumulative hazard, and exp(−Λ̂) is its survival estimate — always at or above KM, and close to it unless dⱼ/nⱼ is large. Censored subjects leave the risk set without triggering a drop: toggle a few and watch the later steps get bigger as nⱼ shrinks.
The exam question, mechanised
The CS2 question hands you a dozen lives with death and censoring times and asks for the Kaplan-Meier estimate: sort the distinct event times, count the risk set nⱼ and deaths dⱼ at each, multiply up the (1 − dⱼ/nⱼ) factors, and remember censored lives leave the risk set at their censoring time without causing a drop. The table on this page is exactly the working the examiners want written out. Nelson-Aalen sums the same ratios instead of multiplying their complements, estimating the cumulative hazard; exp(−Λ̂) then gives a slightly higher survival estimate, and the gap between the two curves narrows as risk sets grow.
Make it stick. Parametric survival lives in the survival models playground, and sickness processes in the multi-state model simulator. Memori is a flashcard app built by actuarial students, with a ready-made CS2 set in the shop. Join the beta.
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