Nelson-Aalen Cumulative Hazard H(t), Aalen Variance & Fleming-Harrington Survival

Why Nelson-Aalen avoids zero-survival collapse at the final event time

When the last remaining subject at risk fails (d_k = n_k), the Kaplan-Meier product-limit estimator multiplies by (1 - d_k / n_k) = 0 and collapses estimated survival strictly to 0%. By contrast, Wayne Nelson (1972) and Odd Aalen (1978) accumulated discrete hazard increments directly: H(t) = SUM_{t_i <= t} (d_i / n_i), with martingale variance estimator Var(H(t)) = SUM_{t_i <= t} (d_i / n_i^2).

Exponentiating negative cumulative hazard yields the Fleming-Harrington (Breslow) survival estimator S_FH(t) = EXP(-H(t)), which stays strictly positive even when d_k = n_k and exhibits smaller small-sample bias in competing-risks and reliability engineering.

Nelson-Aalen Hazard Increment & Cumulative H(t) Formula

Divide events C2 by risk-set size B2 per interval and accumulate SUM($D$2:D2) down the table.

=IF(B2 <= 0, 0, C2 / B2)

Aalen Standard Error & Fleming-Harrington Survival Formula

Sum C_i / (B_i ^ 2) for Aalen variance, take SQRT for SE(H), and evaluate ROUND(EXP(-H_t), 4) for survival probability.

=ROUND(EXP(-SUM(D2:D5)), 4)

Further reading

• Aalen, O. (1978) — Nonparametric Inference for a Family of Counting Processes, Annals of Statistics