Estimation and graduation of survival models
CS2 tests nonparametric estimation from censored lifetimes and the graduation of crude rates into smooth, defensible tables.
Estimation
The Kaplan–Meier estimator is Ŝ(t)=∏_{t_j≤t}(1−d_j/n_j), where d_j is the number of deaths at t_j and n_j the risk set just before t_j. An individual censored at time c stays at risk until c and is removed after it. Greenwood's formula gives Var(Ŝ(t))=Ŝ(t)²∑ d_j/[n_j(n_j−d_j)]. Nelson–Aalen estimates H(t) by ∑ d_j/n_j. The log-rank test compares observed and expected deaths across groups.
Graduation
Graduation fits a smooth curve - parametric (Gompertz–Makeham, Weibull), spline or Whittaker–Henderson - to crude rates. Compare graduated and crude values with chi-square, sign, runs or serial-correlation tests, and inspect cumulative deviations. A good graduation balances smoothness against fidelity; it must not hide real cohort or duration effects.
Worked example
Ten lives start at time 0. Deaths occur at t=2 and t=5; one life is censored at t=3; another death occurs at t=6. At t=2, n=10, d=1, so Ŝ=0.9. At t=5 the censored life has left, so n=9, d=1 and Ŝ=0.9×8/9=0.8. At t=6, n=8, d=1 and Ŝ=0.8×7/8=0.7. The censoring at t=3 affects later risk sets but contributes no factor of its own.
Traps
Do not include a death in the risk set after it occurs. Fix the convention when censoring and death share a time. Greenwood gives the variance of Ŝ, not of H. A graduation test is not a goodness-of-fit test for the underlying model. Do not over-smooth a table that shows genuine changes.