Aalen-Johansen Competing-Risks Cumulative Incidence vs Naive 1 - KM

Expose why 1 - KM_1 + 1 - KM_2 impossibly exceeds total failure 1 - S_all

In cardiology, transplant, and credit-prepayment cohorts, a subject who dies of Non-CV mortality (Event 2) can NEVER subsequently experience Cardiovascular death (Event 1). Treating competing Event 2 deaths as independent right-censoring inside a cause-specific Kaplan-Meier curve 1 - KM_1(t) implicitly assumes dead patients continue facing the same Event 1 hazard as survivors — systematically overestimating cumulative risk!

Aalen and Johansen (1978) weight each cause-1 hazard jump d_{1j} / n_j by the ALL-CAUSE event-free survival S_all(t_{j-1}) entering interval j: delta_CIF_{1j} = S_all(t_{j-1}) * (d_{1j} / n_j), guaranteeing CIF_1(t) + CIF_2(t) + S_all(t) = 1.0000 identically! Verify on 100 patients across 3 intervals: t_1 (n=100, d_1=10 CV, d_2=10 Non-CV), t_2 (n=80, d_1=16 CV, d_2=8 Non-CV), t_3 (n=56, d_1=14 CV, d_2=14 Non-CV). Here S_all = {0.80, 0.56, 0.28}, true CIF_1 = 0.10 + 0.80*(16/80) + 0.56*(14/56) = 0.10 + 0.16 + 0.14 = 0.4000 (and CIF_2 = 0.3200, summing cleanly to 1 - 0.28 = 0.7200), whereas naive 1 - KM_1 reports 1 - 0.90*0.80*0.75 = 0.4600 (+6.0 percentage points of phantom risk!).

Interval Aalen-Johansen Cause-1 Incidence Jump Formula

Multiply prior-row all-cause survival S_all(t_{j-1}) (starting at 1.0000 at t_0) by the interval cause-specific hazard d_{1j} / n_j.

=IF(B3 <= 0, 0, E2 * (C3 / B3))

Cumulative Incidence CIF_1(t) vs Naive 1 - KM_1 Equivalence Check (Zero Competing Risk)

Verify the fundamental theorem by zeroing out all competing events (set D3 = 0, D4 = 0, D5 = 0): when competing risks vanish, Aalen-Johansen CIF_1 and 1 - KM_1 collapse to the EXACT same number (0.4600)!

=IF(OR(B3 <= 0, C3 + D3 > B3), "Invalid Risk Set", ROUND(SUM(F3:F5), 4))

Further reading

• Arte Sheets Lesson — Greenwood Kaplan-Meier Survival Variance and Log-Log CI
• Arte Sheets Lesson — Restricted Mean Survival Time (RMST) Exact Step-Area Formula