Sensitivity Analysis for Unmeasured Confounding in Competing Risks

Why examine unmeasured confounding?

A competing-risk causal contrast may be carefully standardized for measured covariates and still depend on an untestable assumption: that no unmeasured variable jointly predicts treatment and the event of interest. Sensitivity analysis makes this dependence quantitative. Rather than claiming that an unmeasured confounder has been removed, it asks how the estimated risk difference would change under specified, plausible assumptions about that confounder.

Fix a follow-up horizon \(\tau\) and an event type \(k\). The outcome here is whether event \(k\) has occurred by \(\tau\) and before any competing event. A competing event is therefore part of the definition of the outcome: it prevents the event of interest from being counted by \(\tau\). The target is a difference in cumulative incidence under treatment strategies, not a hypothetical risk in a world where competing events are eliminated.

The event-specific risk already reflects the competing-event process under each treatment. If \(U\) affects both the event of interest and competing events, its association with \(Y_k^a(\tau)\) incorporates both pathways into the cumulative incidence. This analysis does not estimate an effect among people who would avoid the competing event under either treatment.

Assumptions and sensitivity parameters

The observed risks may be estimated using standardization, weighting, or another suitable method for the measured covariates. The correction below is described after conditioning on \(L=l\). It requires consistency, no interference, adequate positivity for treatment within covariate and confounder strata, and appropriate handling of censoring. Most importantly, it assumes conditional exchangeability after including \(U\): within levels of \(L\) and \(U\), treatment assignment is independent of the potential event outcomes. Because \(U\) is unmeasured, that last assumption cannot be verified from the observed data.

For each treatment \(a\) and covariate stratum \(l\), let \(r_{al}\) be the observed event-\(k\) risk by \(\tau\) among people with \(A=a,L=l\). Let \(p_{al}=P(U=1\mid A=a,L=l)\), and let \(p_l\) be the prevalence of \(U=1\) in the target population within stratum \(L=l\). The prevalence values are either supplied by external information or varied across plausible scenarios. Define \(\gamma_{al}\) as the difference in event risks between \(U=1\) and \(U=0\), under treatment \(a\) and within \(L=l\).

The parameter can vary by treatment and measured-covariate stratum. A positive \(\gamma_{al}\) means that, within that stratum and treatment, the event risk is higher for \(U=1\). A negative value means it is lower. This parameterization accommodates a confounder whose association with the event differs across treatment groups.

Deriving the risk correction

Within a treatment and covariate stratum, the observed risk is a mixture of the risks in the two \(U\) groups. Write those two risks as \(q_{al0}\) and \(q_{al1}\). Then \(r_{al}=(1-p_{al})q_{al0}+p_{al}q_{al1}\), while \(\gamma_{al}=q_{al1}-q_{al0}\). Solving these equations gives \(q_{al0}=r_{al}-p_{al}\gamma_{al}\) and \(q_{al1}=r_{al}+(1-p_{al})\gamma_{al}\).

To estimate the risk under treatment \(a\) in the target population, average those two risks using the target prevalence \(p_l\), rather than the treatment-group prevalence \(p_{al}\). Substitution yields the correction:

Let \(w_l=P(L=l)\) be the target-population proportion in stratum \(l\), with \(\sum_l w_l=1\). The corrected risks and risk difference are therefore

The correction has a direct interpretation. If the treated group has more \(U=1\) people than the target population, then \(p_l-p_{1l}<0\); when \(\gamma_{1l}>0\), reweighting to the target population reduces the treated risk. The control risk is corrected in the same way. The net effect on the risk difference depends on both groups’ prevalence imbalances and outcome associations.

Assessing how strong confounding must be

A useful analysis varies the inputs rather than selecting one pair of values and treating it as known. For example, one can tabulate the corrected risk difference over a range of plausible prevalences and \(\gamma\) values. A tipping point is a combination at which the corrected contrast reaches zero or crosses another clinically important threshold. It is a description of the assumptions needed to change the conclusion, not evidence that those assumptions are true.

In practice, assumptions should be considered jointly and transparently. A strong association between \(U\) and the outcome does not necessarily create large bias if its prevalence is similar across treatment groups. Conversely, even a moderate outcome association may matter when prevalence differs substantially. The direction of the correction is determined by both components, not by the association parameter alone.

Stratification, diagnostics, and alternatives

Several checks help reveal when a scenario is going wrong:

• Check implied probabilities. For each \(a,l\), both \(r_{al}-p_{al}\gamma_{al}\) and \(r_{al}+(1-p_{al})\gamma_{al}\) must lie between 0 and 1. Otherwise the proposed mixture cannot produce the observed risk.
• Check the target population. The weights \(w_l\) and prevalences \(p_l\) must describe the same target population. Mixing a prevalence from one population with stratum weights from another changes the estimand.
• Check positivity and sparse strata. If treatment or \(U\) is nearly absent in a stratum, the observed risks and prevalences may be unstable, and a correction based on them can be highly sensitive to small changes.
• Check the horizon and outcome definition. The \(\gamma\) values must refer to the same event type and follow-up horizon as the observed risks. A risk difference for another event or time point is not interchangeable.
• Check sampling uncertainty separately. The displayed correction treats sensitivity inputs as fixed. Sampling uncertainty in observed risks and prevalences can be assessed, for example, by bootstrap methods; uncertainty about the sensitivity inputs should be conveyed with scenario ranges or distributions, not hidden inside a conventional confidence interval.

This approach complements, rather than replaces, adjustment for measured confounders. Standardization or weighting addresses measured covariates under their assumptions; the sensitivity correction asks how an additional unmeasured variable could alter the result. Probabilistic bias analysis can assign distributions to sensitivity inputs and propagate that uncertainty, while partial-identification bounds answer a different question by reporting a range under weaker restrictions. Neither alternative removes the need to state what is assumed. A single-confounder risk-difference correction is also not an E-value calculation: that measure uses a different effect scale and parameterization.

The method has important limitations. It relies on a chosen binary representation of \(U\), and one confounder may not capture several omitted causes or their dependence. The prevalence and outcome association inputs may be difficult to justify, and a coarse sensitivity grid can miss important interactions or heterogeneity. Most fundamentally, the corrected contrast is causal only if the conditional exchangeability, consistency, positivity, and censoring assumptions hold for the specified model. Sensitivity analysis describes the consequences of assumptions; it does not verify them.

Common mistake

A common error is to adjust only the treated risk because the unmeasured confounder is more prevalent among treated participants. The correction applies to both treatment-specific risks: each is transported from its own \(U\) distribution to the target distribution. In the single-stratum example, the treated risk fell by 1.5 percentage points, but the control risk rose by 1.2 points. Ignoring the latter would give the wrong corrected contrast. Another error is to say that the corrected risk difference is “the true effect.” It is a scenario-specific estimate conditional on the assumed prevalences, associations, and causal assumptions.