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Higher competing mortality lowers incidence in a two-time table

Example

Cause 1 is cancer death and cause 2 is cardiovascular death. Consider two exact groups of 16 right-censored records, with follow-up through time $2$ . Remaining event-free subjects are censored at $2$ . At time $1$ , the low-CV group has 4 cancer deaths and 4 cardiovascular deaths; the high-CV group has 4 cancer deaths and 8 cardiovascular deaths. At time $2$ , the corresponding risk sets and cancer deaths are 8 and 2 in the low-CV group, and 4 and 1 in the high-CV group. Events from different causes at the same time are grouped in the common risk-set update.

Facts & Assumptions

Given: Both groups have 16 subjects; events are observed at times $1$ and $2$ with the risk and cause counts stated above. Compute the cause-1 Aalen–Johansen incidence and compare it with the complement of the cause-specific Kaplan–Meier product that treats cause 2 as censored.

[F1]

Competing-risk data, cause-specific hazard measures, and cumulative incidence: the observed cause- $k$ cumulative incidence is $F_{k} (t) = P (T \leq t, D = k)$ .

[F2]

Aalen–Johansen cumulative-incidence estimator: at an event time $u$ , the cause- $k$ increment is $\hat{S} (u -) Δ N_{k} (u) / Y (u)$ and all-cause survival is multiplied by $1 - Δ N (u) / Y (u)$ .

[F3]

Event-time observation, censoring, and truncation schemes: under right censoring, $U = \min (T, C)$ and $δ = 1 {T \leq C}$ ; an event tied with censoring is counted as an event.

[F4]

Nelson–Aalen, Kaplan–Meier, and Greenwood estimators: the single-event Kaplan–Meier product multiplies $1 - Δ N (u) / Y (u)$ over event times, with tied event counts grouped in the risk set.

Verification

1.1F2givenalgebra

At time $1$ , both groups have $Y (1) = 16$ and $Δ N_{1} (1) = 4$ . The low-CV group has $Δ N (1) = 8$ , so $\hat{S} (1) = 1 - 8 / 16 = 1 / 2$ and ${\hat{F}}_{1} (1) = 4 / 16 = 1 / 4$ . The high-CV group has $Δ N (1) = 12$ , so $\hat{S} (1) = 1 - 12 / 16 = 1 / 4$ and ${\hat{F}}_{1} (1) = 4 / 16 = 1 / 4$ .

2.1F1step 1.1F2givenalgebra

At time $2$ , the low-CV risk count is $8$ and $Δ N_{1} (2) = 2$ , giving increment $(1 / 2) (2 / 8) = 1 / 8$ and ${\hat{F}}_{1} (2) = 1 / 4 + 1 / 8 = 3 / 8$ . The high-CV risk count is $4$ and $Δ N_{1} (2) = 1$ , giving increment $(1 / 4) (1 / 4) = 1 / 16$ and ${\hat{F}}_{1} (2) = 1 / 4 + 1 / 16 = 5 / 16$ . Thus higher cardiovascular mortality lowers the observed cancer cumulative incidence even though both groups have cause-1 event fractions $1 / 4$ at each event time.

3.1F1F3F4step 2.1givenalgebra∎

If cause-2 deaths are instead recoded as censorings for the cause-1 Kaplan–Meier calculation, the low-CV product is $(1 - 4 / 16) (1 - 2 / 8) = (3 / 4)^{2} = 9 / 16$ , and the high-CV product is $(1 - 4 / 16) (1 - 1 / 4) = (3 / 4)^{2} = 9 / 16$ . Both complements are $7 / 16$ , which differs from both observed cause-1 incidences, $3 / 8$ and $5 / 16$ . The cause-specific product-limit complement therefore does not equal the observed cause-specific cumulative incidence in these records.

Depends on

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Sources

Lu Tian and Richard Olshen, Competing Risk and Alternatives to Cox Model, Stanford University STAT331, slides 6–8, 12 (standard reference, not scraped)
Janne Pitkäniemi and Esa Läärä, Survival analysis with competing risks, Statistical Practice in Epidemiology using R, University of Tartu, slides 13–24 (standard reference, not scraped)
Peter C. Austin, D. S. Lee, and Jason P. Fine, Introduction to the Analysis of Survival Data in the Presence of Competing Risks, Circulation 133(6) (2016), pp. 601–609 (standard reference, not scraped)