Why combine treatment and outcome models?
A competing-risk causal question often asks how the probability of a particular event by a specified time would differ if everyone received treatment rather than control. For instance, one may compare the risk of a disease-related event by one year under two treatment strategies, while treating a competing event as an event that prevents the disease-related event from occurring first. The target is a marginal risk in the full population, not a risk conditional on remaining event-free.
Standardization estimates each treatment-specific risk by averaging outcome-model predictions over the covariate distribution. Inverse probability weighting instead reweights observed outcomes using a treatment model. An augmented inverse-probability weighted (AIPW) estimator combines these approaches. Under the identification assumptions, it can remain consistent if either the treatment model or the outcome model is correctly specified, although this protection does not make the assumptions optional or guarantee good finite-sample performance.
The target is specific to cause \(k\), horizon \(\tau\), and the treatment strategies represented by \(A=1\) and \(A=0\). It is a cumulative-incidence contrast under those strategies: it does not ask what the risk would be if competing events were eliminated. The earlier material on competing-risk g-formula and inverse probability weighting provides the standard identification context; here the focus is the augmentation that combines the models.
Identification and the AIPW estimator
Write \(e_a(L)=P(A=a\mid L)\) for the treatment propensity and \(m_a(L)=E\{Y_k(\tau)\mid A=a,L\}\) for the conditional cause-specific risk. Identification of \(\psi_a=E\{m_a(L)\}\) requires consistency, conditional exchangeability \(Y_k^a(\tau)\perp A\mid L\), and treatment positivity \(e_a(L)>0\) wherever the covariate distribution has support. These assumptions concern the causal interpretation; model fit alone cannot establish them.
For this basic formulation, suppose the outcome through \(\tau\) is ascertained for everyone. With right censoring before \(\tau\), an appropriate censoring adjustment is also needed. One simple valid setup treats the outcome as observed only when \(R=I(C\geq\tau)=1\), assumes censoring is independent of the event outcome conditional on \(A,L\), and defines \(G(\tau\mid A,L)=P(C\geq\tau\mid A,L)>0\). Events known before earlier censoring may be deliberately treated as unobserved under this setup. More efficient event-time-specific methods can use additional observed information, but require their own correctly specified censoring adjustment.
This expression applies when every \(Y_{k,i}(\tau)\) is observed. With the simple censoring setup above, replace the residual term by \(I(A_i=a)R_i\{Y_{k,i}(\tau)-\widehat m_a(L_i)\}/\{\widehat e_a(L_i)\widehat G(\tau\mid A_i,L_i)\}\). The outcome regression still predicts the risk, while the weighted residual corrects its average using observed outcomes. The competing event remains a zero for the cause-\(k\) indicator, not a censored observation.
The formula is an augmented estimator because it starts with the outcome prediction and adds a weighted correction. If the outcome regression is right, the conditional residual has mean zero among people assigned treatment \(a\), so the correction has expectation zero even if the fitted propensity is wrong, provided the weights are defined and the regularity conditions hold. If the propensity is right, inverse weighting makes the observed residual average represent the population residual average within covariate strata; that cancels the outcome-model error after averaging. These are two distinct routes to consistency, not a guarantee when both models are wrong.
Worked Example: Calculating Both Risks and Their Difference
Suppose six hypothetical participants have complete follow-up through \(\tau\). For simplicity, the fitted propensities are \(0.5\) for either treatment, and the fitted outcome predictions are \(0.30\) under treatment and \(0.20\) under control for every participant. The observed outcome is the indicator for the event of interest by \(\tau\); a competing event, if it occurred, would be recorded as zero for this outcome.
| Participant | \(A_i\) | Observed \(Y_{k,i}(\tau)\) | Treatment-arm score | Control-arm score |
|---|---|---|---|---|
| 1 | 1 | 1 | 0.30 + 2(1 − 0.30) = 1.70 | 0.20 |
| 2 | 0 | 0 | 0.30 | 0.20 + 2(0 − 0.20) = −0.20 |
| 3 | 1 | 0 | 0.30 + 2(0 − 0.30) = −0.30 | 0.20 |
| 4 | 0 | 1 | 0.30 | 0.20 + 2(1 − 0.20) = 1.80 |
| 5 | 1 | 0 | 0.30 + 2(0 − 0.30) = −0.30 | 0.20 |
| 6 | 0 | 0 | 0.30 | 0.20 + 2(0 − 0.20) = −0.20 |
For a participant not assigned to an arm, that arm’s score is just its outcome prediction. For a participant assigned to the arm, the weighted residual is added. Thus the treatment risk estimate is \(\widehat{\psi}_1=(1.70+0.30-0.30+0.30-0.30+0.30)/6=2.00/6=0.3333\). The control risk estimate is \(\widehat{\psi}_0=(0.20-0.20+0.20+1.80+0.20-0.20)/6=2.00/6=0.3333\).
The estimated risk difference is \(0.3333-0.3333=0\), or zero percentage points by \(\tau\) in this hypothetical sample. This small example illustrates the calculation, not evidence that the population effect is zero. The negative and greater-than-one individual scores are possible; the estimator averages scores and does not constrain each contribution to be a probability.
Why the augmentation is doubly robust
The key calculation can be made conditional on \(L\). Suppress \(L\) in the notation, let the true propensity for treatment \(a\) be \(e_a\), and let the true conditional risk be \(m_a=E(Y_k(\tau)\mid A=a,L)\). Let \(q_a\) be a proposed outcome prediction and \(e_a^*\) a proposed propensity. The conditional expected AIPW score for arm \(a\) is
If \(q_a=m_a\), the right side is \(m_a\) regardless of the proposed propensity, as long as the expression is defined. If \(e_a^*=e_a\), it is \(q_a+(m_a-q_a)=m_a\), regardless of the proposed outcome prediction. Averaging over \(L\) then yields \(\psi_a\). This establishes the double-robust property at the population level under the causal identification assumptions. It does not say that an estimator is unbiased in every finite sample, nor does it rescue violations of exchangeability or positivity.
Worked Example: Checking the Two Robustness Routes
Consider a hypothetical population with two equally common covariate strata. The true cause-\(k\) risks are \(m_1=(0.40,0.20)\) under treatment and \(m_0=(0.10,0.30)\) under control. Therefore the marginal risks are \(\psi_1=(0.40+0.20)/2=0.30\) and \(\psi_0=(0.10+0.30)/2=0.20\), so \(\Delta=0.10\), or 10 percentage points by \(\tau\).
First, suppose the outcome predictions are correct but a fitted treatment propensity is misspecified as \(e_1^*=0.50\) in both strata. In each stratum \(q_1=m_1\), so the conditional expected score is \(q_1+(e_1/e_1^*)(m_1-q_1)=m_1\); the residual term has expectation zero. Averaging gives \(0.30\). The same argument gives \(0.20\) for control and the risk difference remains \(0.10\).
Second, suppose the true treatment propensities are \(e_1=(0.25,0.75)\), but the outcome prediction for treatment is misspecified as \(q_1=0.25\) in both strata. With the correct propensity, the conditional expected scores are \(0.25+(0.40-0.25)=0.40\) in the first stratum and \(0.25+(0.20-0.25)=0.20\) in the second. Their average is \(0.30\). Applying the same calculation to control gives its true marginal risk \(0.20\). Thus the contrast is again \(0.10\). This is an expectation argument under the assumptions, not a claim that a small realized sample must produce exactly those values.
Uncertainty, diagnostics, and alternatives
For independent observations and suitable regularity conditions, the arm-specific AIPW score provides an influence-function-based variance estimate. In the fully observed case, define each estimated arm score \(S_{ia}\) as the quantity inside the brackets in the estimator. The estimated influence contribution for the contrast is \(D_i=(S_{i1}-S_{i0})-\widehat{\Delta}\). A standard error is the sample standard deviation of the \(D_i\)'s divided by \(\sqrt n\); a large-sample Wald interval is \(\widehat{\Delta}\pm1.96\,SE\). With censoring adjustment, the scores must include the censoring weights and any required nuisance-estimation terms or resampling procedure.
Worked Example: Standard Error and an Approximate Interval
Use the six hypothetical records in the first worked example. The individual differences \(S_{i1}-S_{i0}\) are \(1.50, 0.50, -0.50, -1.50, -0.50, 0.50\). Their mean is zero, matching \(\widehat{\Delta}=0\). Their sample variance is \(\{1.50^2+0.50^2+(-0.50)^2+(-1.50)^2+(-0.50)^2+0.50^2\}/(6-1)=5.50/5=1.10\).
Therefore \(SE=\sqrt{1.10/6}=0.4282\). The approximate 95% Wald interval is \(0\pm1.96(0.4282)=(-0.8393,0.8393)\), a risk-difference interval spanning about −83.9 to 83.9 percentage points by \(\tau\). Its great width reflects the very small, illustrative sample and the variability of the weighted scores. In repeated samples under the model and regularity conditions, this construction is intended to cover the target risk difference about 95% of the time; it is not a 95% probability statement about this fixed parameter after observing the data.
Diagnostics should examine both nuisance models and the resulting scores. Check whether fitted treatment probabilities approach zero for either arm, whether treated and control participants overlap across important covariate patterns, and whether estimates are dominated by a few large weights. Inspect outcome-model calibration and predictions within treatment and covariate strata. For censoring-adjusted analyses, also inspect censoring probabilities. Large influence contributions, unstable estimates under modest model changes, and unusually wide intervals signal that positivity, model fit, or sample size may be problematic. Truncating weights can reduce variance but changes the estimator and can introduce bias; report the choice and assess sensitivity rather than treating truncation as a cure for poor overlap.
Standardization relies directly on the outcome model and can be efficient when that model is well specified. Inverse probability weighting relies on the treatment model and can be sensitive to extreme weights. AIPW adds a residual correction and offers consistency if one of those two models is correct, but it still depends on measured-confounder exchangeability, positivity, consistency, correct handling of censoring, and appropriate uncertainty estimation. Flexible nuisance models and cross-fitting can reduce dependence on restrictive functional forms; valid inference still requires adequate nuisance estimation and regularity, including control of the product of relevant nuisance estimation errors. In small samples, asymptotic intervals and apparent double robustness may perform poorly.
Common mistake
Wrong reasoning: “The estimator is doubly robust, so if treatment assignment is confounded or some covariate patterns have no treated participants, the other model will fix the problem.” Correction: double robustness concerns misspecification of one of the two nuisance models under the causal identification assumptions. It does not remove unmeasured confounding, create information where positivity fails, or correct an invalid censoring assumption. A second common error is to code a competing event as censored when defining \(Y_k(\tau)\); for a cause-specific cumulative-incidence risk, that competing event is a zero for the event-of-interest indicator.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For the risk of cause \(k\) by \(\tau\), how is a competing event before \(\tau\) coded in \(Y_k(\tau)\)?
- State the two nuisance-model conditions that give the AIPW estimator its double-robust property.
- If the fitted outcome model is correct but the fitted propensity model is wrong, why can the arm-specific score still have the right expectation?
- What does an estimated propensity close to zero for one treatment arm suggest, and why is it a concern?
- Does double robustness address unmeasured confounding? Explain briefly.
Solutions. (1) It is coded \(0\), because the event of interest did not occur first by \(\tau\); it is not censoring. (2) Either the outcome regression \(m_a(L)\) is correct or the treatment propensity \(e_a(L)\) is correct, subject to the causal assumptions and regularity conditions. (3) When \(q_a=m_a\), the residual \(Y_k(\tau)-q_a\) has conditional mean zero among those assigned arm \(a\), so its weighted expectation is zero when defined; the score expectation is \(m_a\). (4) It signals weak treatment positivity and potentially extreme inverse weights, making estimates unstable and dependent on sparse observations. (5) No. Double robustness addresses nuisance-model misspecification; causal identification still requires exchangeability given measured covariates, among other assumptions.