Sensitivity Analysis for Separable-Effect Assumptions

Why Sensitivity Analysis Is Needed

A separable-effect analysis represents a treatment as two components: \(A_Y\), intended to act on the event of interest, and \(A_D\), intended to act on the competing event. The target contrast can then compare outcomes under interventions that set these components separately. But in many studies the components are delivered together: the observed treatment corresponds to \((A_Y,A_D)=(0,0)\) or \((1,1)\), not to the cross-combinations \((1,0)\) and \((0,1)\). The observed data therefore do not directly reveal what would happen under those cross-combinations.

The dismissible component conditions (DCC) make the cross-regime comparison identifiable under additional causal assumptions. Sensitivity analysis asks a more cautious question: if the DCC are not exactly true, how much might the separable-effect estimate change? It does this by specifying and varying the size of effects that the DCC would set to zero. These assumed effects are not learned from the observed treatment arms alone; their role is to expose how conclusions depend on an untestable assumption.

Definition: Let \(T\) denote time to the first event and \(J\in\{Y,D\}\) its type, where \(Y\) is the event of interest and \(D\) is the competing event. For component assignment \((a_Y,a_D)\), the cause-specific hazard for cause \(k\) is the instantaneous event rate for cause \(k\) at time \(t\), among individuals who have experienced neither event before \(t\), under that assignment. The DCC state that the hazard for \(Y\) does not depend on \(a_D\), and the hazard for \(D\) does not depend on \(a_Y\), conditional on the specified covariates and relevant history.

In a causal analysis, the component interventions must be well-defined, consistency and treatment-version assumptions must be credible, and treatment and censoring assumptions must support estimation of the observed-regime hazards. Positivity and appropriate control of confounding are also needed for those hazards. These conditions do not establish the DCC: the DCC concern how each component affects the other event process. This tutorial focuses on sensitivity to that distinct assumption, not on re-estimating the observed-regime hazards.

A Parameterized Departure From the DCC

Write \(\lambda_Y^{a_Y,a_D}(t)\) and \(\lambda_D^{a_Y,a_D}(t)\) for the cause-specific hazards under the indicated component settings. A convenient sensitivity model uses multiplicative effects on the hazard scale:

\begin{aligned} \lambda_Y^{a_Y,a_D}(t) &= \lambda_Y^{0,0}(t)\exp\{\beta_Y(t)a_Y+\gamma_Y(t)a_D\},\\ \lambda_D^{a_Y,a_D}(t) &= \lambda_D^{0,0}(t)\exp\{\gamma_D(t)a_Y+\beta_D(t)a_D\}. \end{aligned}

Here \(\beta_Y(t)\) is the log hazard ratio for changing \(A_Y\) when \(A_D=0\), while \(\gamma_Y(t)\) is the log hazard ratio for changing \(A_D\) on the \(Y\) hazard. Likewise, \(\beta_D(t)\) is the log hazard ratio for changing \(A_D\) when \(A_Y=0\), and \(\gamma_D(t)\) is the log hazard ratio for changing \(A_Y\) on the \(D\) hazard. The DCC correspond to \(\gamma_Y(t)=\gamma_D(t)=0\) at every time. The model allows departures in either direction: a positive cross-component log hazard ratio increases the relevant hazard, and a negative one decreases it.

Because observed treatment assigns the components together, the observed hazard ratios comparing treatment arm 1 with arm 0 identify the sums \(\beta_Y(t)+\gamma_Y(t)\) and \(\beta_D(t)+\gamma_D(t)\), not their individual terms. Thus, for chosen sensitivity functions \(\gamma_Y(t)\) and \(\gamma_D(t)\), the remaining effects are obtained by subtraction:

\begin{aligned} \beta_Y(t) &= \log\!\left\{\frac{\lambda_Y^{1,1}(t)}{\lambda_Y^{0,0}(t)}\right\}-\gamma_Y(t),\\ \beta_D(t) &= \log\!\left\{\frac{\lambda_D^{1,1}(t)}{\lambda_D^{0,0}(t)}\right\}-\gamma_D(t). \end{aligned}

This is the central sensitivity calculation. The observed diagonal regimes \((0,0)\) and \((1,1)\) anchor the model; the sensitivity parameters allocate each observed joint effect between its intended component and its cross-component effect. With time-varying hazards, these functions may vary over time. A simpler analysis may assume constant cross-component log hazard ratios over follow-up, but that is an additional modeling restriction, not a consequence of the data.

Key distinction: The observed-regime hazards and their joint-treatment hazard ratios can be estimated from data under causal identification assumptions. The decomposition into \(\beta\) and \(\gamma\), and therefore the cross-regime hazards, depends on the chosen sensitivity parameters. A fitted value of \(\gamma=0\) is not evidence that the DCC hold.

From Cross-Regime Hazards to Risks

A separable effect is often reported as a risk contrast, not a hazard contrast. Under a specified pair of cause-specific hazards, the event-free survival probability through time \(t\) is \(S^{a_Y,a_D}(t)=\exp\{-\int_0^t[\lambda_Y^{a_Y,a_D}(u)+\lambda_D^{a_Y,a_D}(u)]\,du\}\). The cumulative incidence of \(Y\) is \(F_Y^{a_Y,a_D}(t)=\int_0^t S^{a_Y,a_D}(u)\lambda_Y^{a_Y,a_D}(u)\,du\). Thus a component-specific risk contrast at a chosen setting of the other component can be calculated from the hazards implied by a sensitivity scenario. It is not generally valid to change only the \(Y\) hazard in the risk formula: if \(\gamma_D\ne0\), changing \(A_Y\) also changes the competing-event hazard.

Worked Example: A Departing Cross-Component Effect

Suppose, hypothetically, that constant hazards per time unit are estimated as \(\lambda_Y^{0,0}=0.08\), \(\lambda_D^{0,0}=0.12\), \(\lambda_Y^{1,1}=0.12\), and \(\lambda_D^{1,1}=0.08\). Consider the separable contrast comparing \((1,0)\) with \((0,0)\) by time 1. Assume constant cross-component hazard ratios of 1.20 for \(A_D\) on the \(Y\) hazard and 0.90 for \(A_Y\) on the \(D\) hazard.

The observed joint-treatment hazard ratio for \(Y\) is \(0.12/0.08=1.50\). The assumed cross-component log hazard ratio is \(\gamma_Y=\log(1.20)\), so the intended-component log hazard ratio is \(\beta_Y=\log(1.50)-\log(1.20)=\log(1.25)\). Therefore \(\lambda_Y^{1,0}=0.08(1.25)=0.10\) per time unit. For \(D\), the observed joint-treatment hazard ratio is \(0.08/0.12=0.6667\). Since \(\gamma_D=\log(0.90)\), \(\beta_D=\log(0.6667)-\log(0.90)=\log(0.7407)\). In the \((1,0)\) regime, \(A_D=0\), so \(\lambda_D^{1,0}=0.12(0.90)=0.108\) per time unit.

For constant hazards \(r_Y,r_D\), the cumulative incidence of \(Y\) by time 1 is \(F_Y(1)=\frac{r_Y}{r_Y+r_D}\{1-\exp[-(r_Y+r_D)]\}\). Under \((0,0)\), this gives \(F_Y^{0,0}(1)=\frac{0.08}{0.20}\{1-\exp(-0.20)\}=0.4(0.181269)=0.07251\), or 7.251%.

Under \((1,0)\), the total hazard is \(0.10+0.108=0.208\), so \(F_Y^{1,0}(1)=\frac{0.10}{0.208}\{1-\exp(-0.208)\}=0.480769(0.187793)=0.09029\), or 9.029%. The risk difference is \(0.09029-0.07251=0.01778\), about 1.778 percentage points by time 1. This conclusion is conditional on the specified sensitivity values and constant-hazard model; the observed treatment arms alone do not identify this cross-regime contrast.

Worked Sensitivity Scenarios

A useful analysis does not present one assumed departure as if it were known. It evaluates a defensible range of values, reports how the target contrast changes, and explains which values materially alter the substantive conclusion. The next examples use the same hypothetical hazards as above to show how the DCC scenario and a tipping point differ.

Worked Example: The DCC Scenario as a Reference Point

Use the observed hypothetical hazards from the preceding example and set both cross-component effects to zero: \(\gamma_Y=\gamma_D=0\). Then the entire observed joint-treatment hazard ratio for each cause is assigned to its intended component. Under \((1,0)\), \(\lambda_Y^{1,0}=0.12\) and \(\lambda_D^{1,0}=0.12\) per time unit.

The risk of \(Y\) by time 1 under \((1,0)\) is \(F_Y^{1,0}(1)=\frac{0.12}{0.24}\{1-\exp(-0.24)\}=0.5(0.213372)=0.1067\), or 10.67%. Under \((0,0)\), the risk remains 7.251%, as calculated above. Using the unrounded risks, the risk difference is \(0.03417837\), which rounds to \(0.03418\), or 3.418 percentage points by time 1.

Compared with the earlier scenario, the estimate is larger because the assumed positive effect of \(A_D\) on the \(Y\) hazard had previously allocated part of the observed 1.50 joint hazard ratio away from \(A_Y\). The difference between these results is a sensitivity finding, not evidence that either assumption is correct.

Worked Example: Finding a Tipping Point

Now set \(\gamma_D=0\) and ask how large the assumed effect of \(A_D\) on the \(Y\) hazard must be for the separable risk difference comparing \((1,0)\) with \((0,0)\) to equal zero. With \(\gamma_D=0\), the competing-event hazard under both regimes is 0.12. The \(Y\) hazard under \((1,0)\) is \(0.08\exp(\beta_Y)\), and the observed joint hazard ratio requires \(\beta_Y+\gamma_Y=\log(1.50)\).

For the risk difference to be zero, the \((1,0)\) and \((0,0)\) risks must match. With the same competing-event hazard, the cumulative incidence formula increases strictly as the \(Y\) hazard increases: for \(r_D>0\), \(F_Y(t)=\int_0^t \exp\{-(r_Y+r_D)u\}r_Y\,du\), and its derivative with respect to \(r_Y\) is \(\int_0^t \exp\{-(r_Y+r_D)u\}(1-r_Yu)\,du\), which is positive for the parameter values here (equivalently, the constant-hazard expression is increasing in \(r_Y\)). Equality therefore occurs when the \(Y\) hazards match, at \(\lambda_Y^{1,0}=0.08\), or \(\beta_Y=0\).

Since \(\beta_Y+\gamma_Y=\log(1.50)\), the tipping value is \(\gamma_Y=\log(1.50)\), corresponding to a cross-component hazard ratio of \(\exp(\gamma_Y)=1.50\). At that value, \(\lambda_Y^{1,0}=0.08\) and \(\lambda_D^{1,0}=0.12\), so \(F_Y^{1,0}(1)=F_Y^{0,0}(1)=0.07251\). Under this model, assigning the entire observed joint \(Y\)-hazard ratio to the cross-component effect removes the separable risk difference. This is a model-based tipping point, not a testable threshold in the data.

Practical Analysis and Diagnostics

A transparent analysis follows a repeatable sequence. First estimate the observed-regime hazards using a method appropriate to the study design and the relevant adjustment variables. Then specify a set or grid of \(\gamma_Y\) and \(\gamma_D\) values, translate each into intended-component effects, construct the cross-regime hazards, and calculate the risk contrast on the target time scale. If hazards are conditional on covariates, calculate risks within the model and standardize over the target covariate distribution; do not treat a conditional hazard ratio as if it were automatically a marginal risk ratio.

Anchor to observed regimes.
Estimate the cause-specific hazards under the observed bundled treatment settings and their joint-treatment hazard ratios.

Choose sensitivity values.
Specify plausible cross-component log hazard ratios, ideally with clinical or mechanistic justification, and state whether they are constant over time or covariates.

Recover cross-regime hazards.
Subtract the assumed cross-component effects from the observed joint log hazard ratios, then construct hazards for the component assignments in the estimand.

Recompute and report risks.
Obtain cumulative incidences and the target contrast for each scenario, including the DCC reference case and meaningful tipping values.

Several checks help reveal when the analysis is going wrong. Confirm that each chosen decomposition reproduces the observed joint-treatment hazard ratios: \(\beta_Y+\gamma_Y\) and \(\beta_D+\gamma_D\) must equal the corresponding observed log hazard ratios. Check that reconstructed hazards remain nonnegative and that the model's time and covariate structure is plausible. Plot the risk contrast across the sensitivity grid; abrupt changes can reflect nonlinear cumulative-incidence behavior or a coding error. Very large or clinically implausible cross-component effects should be identified as such rather than hidden in a single headline estimate.

Sampling uncertainty and sensitivity uncertainty are different. Estimated observed-regime hazards have sampling error, while \(\gamma_Y\) and \(\gamma_D\) represent untestable assumptions. A practical report can calculate confidence intervals conditional on each sensitivity scenario, for example by resampling or propagating model uncertainty while holding the sensitivity values fixed. Those intervals do not account for uncertainty about which sensitivity values are credible. A table or plot across values is needed to show that second dimension.

Comparisons, Limitations, and Common Mistakes

The DCC analysis is the special case \(\gamma_Y=\gamma_D=0\); it gives a single reference estimate but cannot diagnose its own assumption from the observed diagonal treatment arms. A parameterized sensitivity analysis retains the same observed-regime information while showing how departures change the target estimate. This differs from sensitivity analysis for unmeasured confounding, which varies assumptions about residual confounding of treatment and outcome. Here the concern is specifically the unobserved cross-component effects. It also differs from a nonparametric identification strategy: a low-dimensional hazard model supplies a point estimate for each chosen sensitivity value, whereas broader identification approaches address uncertainty about the estimand without fixing a single such decomposition.

The multiplicative hazard parameterization is interpretable and convenient, but it is not universal. It assumes a particular scale for departures, and constant sensitivity parameters may be too restrictive if cross-component effects vary over time or across subgroups. Estimates can also be unstable when observed hazards are poorly estimated, and extrapolating from the observed \((0,0)\) and \((1,1)\) regimes to cross-regimes can be scientifically demanding. A sensitivity analysis does not repair violations of consistency, positivity, censoring assumptions, or confounding control for the observed-regime estimates.

Mistake: treating the DCC as verified because the observed model fits well. The cross-regimes are not observed, so fit to the diagonal arms cannot test whether their component-specific hazards obey the DCC. Correction: label zero cross-effects as an assumption and display results under nonzero alternatives.
Mistake: varying \(\gamma_Y\) without recalibrating \(\beta_Y\). This changes the implied observed joint-treatment hazard ratio and no longer agrees with the observed data. Correction: keep \(\beta_Y+\gamma_Y\) equal to the observed log hazard ratio for every scenario; apply the analogous check for cause \(D\).
Mistake: comparing hazards and calling the result a risk difference. Hazards do not directly equal cumulative incidence, because both event types determine survival without an event. Correction: calculate the cumulative incidence using both cause-specific hazards before forming the risk contrast.
Mistake: interpreting a scenario-specific interval as covering sensitivity uncertainty. An interval conditional on fixed \(\gamma\) values only reflects sampling uncertainty under that scenario. Correction: report how point estimates and intervals vary across the sensitivity range.

Key takeaway: Sensitivity analysis for separable effects makes the untestable DCC explicit. Choose cross-component effects, preserve the observed joint-treatment hazards, reconstruct the cross-regime hazards, and recompute cumulative-incidence contrasts. Conclusions should be stated as conditional on the sensitivity model and its parameter values.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the hazard model, what does \(\gamma_Y=0\) assert, and why can it not generally be tested using only observed bundled treatment arms?
An observed joint-treatment hazard ratio for \(Y\) is 1.80, and a sensitivity scenario sets \(\exp(\gamma_Y)=1.20\). What intended-component hazard ratio follows?
Why must the cumulative incidence calculation use both the \(Y\) and \(D\) hazards?
Suppose the estimated risk difference is 2 percentage points under \(\gamma_Y=\gamma_D=0\), but is close to zero for a larger assumed \(\gamma_Y\). What is the correct interpretation?
Does a confidence interval calculated with \(\gamma_Y,\gamma_D\) fixed represent uncertainty about the validity of those values? Explain.

Solutions. (1) It asserts that changing \(A_D\) has no effect on the \(Y\) hazard, the DCC for that hazard. Bundled arms reveal only the sum \(\beta_Y+\gamma_Y\), so they do not separately identify \(\gamma_Y\). (2) The intended-component hazard ratio is \(1.80/1.20=1.50\), because the log effects add and therefore the hazard ratios multiply. (3) The competing-event hazard changes the probability of remaining event-free and hence the number still at risk of \(Y\); cumulative incidence depends on both hazards. (4) The estimate is sensitive to the assumed cross-component effect. The near-zero contrast is conditional on that scenario, not proof that the true separable effect is zero. (5) No. It reflects sampling uncertainty conditional on the fixed sensitivity values; sensitivity to their validity must be shown by varying them.