Why Derive an Influence Function?
The Aalen–Johansen estimator combines estimated cause-specific event increments with estimated event-free survival. An influence function describes, to first order, how one subject’s observed data affect that estimator. It is useful because it identifies which parts of the observation process create variation in a cumulative incidence estimate and supplies the subject-level quantities used in large-sample variance calculations.
The derivation below concerns a fixed cause \(k\) and time horizon \(t\). It uses a continuous-time formulation, so event hazards are continuous and there are no tied event times in the population model. The resulting expression is a first-order, large-sample representation; it is not an exact decomposition of every finite-sample change in the estimator.
Model, Notation, and Assumptions
For subject \(i\), let \(Y_i(u)=I(T_i\geq u,C_i\geq u)\) indicate being event-free and uncensored just before \(u\), and let \(N_{ij}(u)=I(T_i\leq u,J_i=j,T_i\leq C_i)\) count an observed event from cause \(j\). The observed event-process martingale is \(M_{ij}(u)=N_{ij}(u)-\int_0^uY_i(v)\,d\Lambda_j(v)\). Its event jump is offset by the cumulative hazard expected while that subject was at risk.
Assume independent and identically distributed subjects; independent censoring from the event time and cause; and positive probability of remaining uncensored and event-free through the times used in the derivation. In particular, write \(G(u)=P(C\geq u)\) and \(y(u)=P(Y_i(u)=1)=S(u)G(u)\). Require \(y(u)>0\) through \(t\), and assume the hazards are sufficiently regular and integrable for the displayed integrals and first-order approximations to exist. These conditions allow the observed risk set to represent the event-free population, adjusted for independent censoring.
Deriving the First-Order Representation
First consider how a small perturbation to the cause-specific cumulative hazards changes \(F_k(t)\). Since \(S(u)=\exp\{-\sum_{j=1}^K\Lambda_j(u)\}\) in this continuous-time model, perturbing the hazards changes both the target-cause increment and the event-free survival weighting it receives.
The inner integral is the cause-\(k\) incidence still to accrue after time \(u\), namely \(F_k(t)-F_k(u)\) under the continuous-time convention. Thus an error in any cause-specific hazard at \(u\) affects the target estimate by changing survival for the remaining interval. A target-cause hazard error has an additional direct effect through the first integral.
The cause-specific hazard is estimated from events divided by the risk set. To first order, its cumulative-hazard estimation error for the sample is represented by averaging individual observed martingales and dividing by the population risk probability \(y(u)\). In differential form, the subject-level hazard perturbation is \(dM_{ij}(u)/y(u)\). Substituting this into the functional derivative gives the influence function.
Equivalently, the coefficient multiplying the cause-\(j\) martingale is \(a_{kj}(u)=\{I(j=k)S(u)-[F_k(t)-F_k(u)]\}/y(u)\). The estimator has the first-order expansion \(\widehat F_k(t)-F_k(t)=n^{-1}\sum_{i=1}^n\phi_{ik}(t)+o_p(n^{-1/2})\). The influence function has mean zero under the model, and its scale is the change on the estimator’s probability scale due to one subject, before averaging over \(n\) subjects.
The derivation is a functional delta-method argument: differentiate the cumulative incidence functional, then insert the first-order errors in the estimated hazards. The martingale representation accounts for observed events and the time each subject remains in the risk set. With tied event times or sizable finite-sample jumps, the exact product-integral estimator requires care; this continuous-hazard expression describes its first-order limit, rather than replacing the finite-sample tied-time calculation.
What the Contributions Mean
There are two linked effects. A cause-\(k\) event can increase the estimated incidence directly, while events from any cause remove subjects from the event-free risk set and therefore alter the future survival weight. The remaining-incidence term \(F_k(t)-F_k(u)\) represents this latter effect. It is nonnegative, so the indirect contribution of a competing-cause event is negative when considered as an event jump. Censoring does not count as a competing event: it contributes no event jump, but it stops that subject’s risk-set exposure and hence stops the compensator in \(M_{ij}\).
The factor \(1/y(u)=1/\{S(u)G(u)\}\) makes the risk-set condition explicit. When few subjects remain event-free and uncensored, an observed event or a change in risk exposure receives more weight. This is why estimates near the end of follow-up can be sensitive to sparse risk sets, even when the estimated cumulative incidence itself is well defined.
Worked Example: Comparing Local Cause Contributions
Suppose, at a time \(u\) before a fixed horizon, the event-free probability is \(S(u)=0.60\), the probability of remaining uncensored is \(G(u)=0.80\), and hence \(y(u)=0.48\). For the target cause \(k\), suppose \(F_k(t)=0.18\) and \(F_k(u)=0.08\). Calculate the local martingale coefficients for the target cause and for a competing cause \(j\ne k\).
The remaining target-cause incidence after \(u\) is \(0.18-0.08=0.10\). For an event from the target cause, the coefficient is \[ a_{kk}(u)=\frac{0.60-0.10}{0.48}=\frac{0.50}{0.48}=1.042. \] For an event from a competing cause, it is \[ a_{kj}(u)=\frac{0-0.10}{0.48}=-0.2083. \]
Thus, at this time, the target-cause martingale has a positive local coefficient, whereas a competing-cause martingale has a negative one. These are coefficients multiplying \(dM_{ij}(u)\), not stand-alone changes in the estimate: each martingale increment also includes its at-risk hazard compensator.
Worked Example: A Censored Subject’s Risk-Exposure Contribution
Consider a short interval of length \(0.10\) years in which the local coefficients and cause-specific hazards can be treated as constant for a first-order calculation. Suppose \(S=0.80\), \(G=0.625\), so \(y=0.50\), and the remaining target-cause incidence is \(0.10\). Let the target-cause hazard be \(0.10\) per year and a competing-cause hazard be \(0.05\) per year. A subject is event-free and uncensored at the start of the interval, has no event, and is censored at its end.
The coefficients are \(a_{kk}=(0.80-0.10)/0.50=1.4\) and \(a_{kj}=-0.10/0.50=-0.2\). With no event jump, the martingale increments over the interval are their negative compensators: \(dM_{ik}=-0.10(0.10)=-0.010\) and \(dM_{ij}=-0.05(0.10)=-0.005\). The subject’s first-order contribution over this interval is therefore \[ 1.4(-0.010)+(-0.2)(-0.005)=-0.014+0.001=-0.013. \]
This is a local interval contribution, not the subject’s influence over all follow-up. Censoring at the interval end means there is no later risk exposure or martingale contribution from this subject. The example also shows why a censored subject is not simply assigned zero influence: the subject contributes through the at-risk time accumulated before censoring.
Worked Example: Separating a Competing Event Jump From Its Compensator
Use the same local quantities as in the previous example. Now suppose the subject remains at risk throughout the \(0.10\)-year interval and has a competing-cause event at its end. The competing-cause martingale increment is the event jump minus its compensator: \(dM_{ij}=1-0.05(0.10)=0.995\). The target-cause martingale increment, with no target event, is \(dM_{ik}=-0.10(0.10)=-0.010\).
The local contribution is \[ 1.4(-0.010)+(-0.2)(0.995) =-0.014-0.199=-0.213. \] Of this, \(-0.199\) comes from the competing-cause martingale, including its event jump; \(-0.014\) comes from the target-cause compensator during the interval. This calculation uses constant local coefficients as a short-interval approximation. It illustrates the event-jump and exposure pieces separately, rather than claiming an exact finite-sample estimator change.
Practical Checks and Limitations
For an estimated influence function, replace \(S\), \(F_k\), and \(y\) with their estimated counterparts and use the observed subject-specific martingales. In practice, this means checking that the risk-set denominator is positive at the times included and that the censoring and event definitions match those used to construct the Aalen–Johansen estimate. The average of the estimated subject-level influence values should be close to zero, up to numerical and finite-sample error; a substantial discrepancy can signal inconsistent risk-set conventions, event coding, or integration limits.
A common mistake is to treat a competing event as ordinary censoring and conclude that its influence on cause-\(k\) incidence is zero. That reasoning misses the fact that a competing event removes a subject who could otherwise have experienced cause \(k\) before \(t\). The remaining-incidence term gives that event a negative contribution. Censoring is different: under independent censoring it removes observation, not the subject’s underlying event possibility, and its effect enters through the observed risk set and martingale exposure.
Another mistake is to interpret a large coefficient near the end of follow-up as proof that the estimate is biased. The coefficient instead signals that each observed contribution is more influential when \(y(u)\) is small. Independent censoring and adequate positivity remain essential; the influence-function calculation cannot repair informative censoring or create information where almost no one remains at risk. The representation is also a first-order approximation, so it should not be read as an exact finite-sample identity. Unlike a direct resampling analysis, it does not automatically capture every finite-sample irregularity; its main role is to express the leading sampling variation in subject-level form.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(y(u)=0.25\), \(S(u)=0.50\), and \(F_k(t)-F_k(u)=0.05\), what are the local coefficients for a target-cause martingale and a competing-cause martingale?
- Why does the influence function include the future-incidence term \(F_k(t)-F_k(u)\) for causes other than \(k\)?
- What event-process martingale increment does a subject contribute over an interval with no event, while remaining at risk for its full duration \(\Delta\)?
- Explain why independent censoring does not imply that a censored subject has zero influence.
- Name one reason the displayed continuous-time representation should not be treated as an exact finite-sample formula with tied event times.
Solutions. (1) The target coefficient is \((0.50-0.05)/0.25=1.8\); the competing-cause coefficient is \(-0.05/0.25=-0.2\). (2) A hazard error at \(u\) changes survival after \(u\), affecting the target incidence still to accrue by \(t\), even when that hazard belongs to a competing cause. (3) For cause \(j\), it is \(dM_{ij}=-\int_u^{u+\Delta}Y_i(v)\,d\Lambda_j(v)\), which is \(-\lambda_j\Delta\) for constant hazard and continuous at-risk exposure throughout. (4) The subject contributes the compensator accumulated while at risk; censoring then stops further exposure. (5) The expression is a first-order limit based on continuous hazards, whereas tied finite-sample event times involve finite jumps and product-integral details.