From Component Hazards to Separable Risk Contrasts
A treatment may affect both the event of interest and a competing event. A separable-effects analysis represents these effects through two treatment components: \(A_Y\), which acts on the cause-specific hazard for the event of interest, and \(A_D\), which acts on the competing-event hazard. The practical calculation is not just a comparison of hazards. It is to combine the component-specific hazards into the probability of experiencing the event of interest by a chosen time, then compare that risk across component settings.
This tutorial works through that calculation for two causes and constant hazards. All numerical values are hypothetical. The examples assume that the component settings correspond to well-defined interventions and that the relevant hazard functions have been identified under the assumptions needed for separable-effect interpretation. The examples do not establish those assumptions from data; they show what follows once the assumptions and hazards are specified.
For the component interpretation to be meaningful, the dismissible component conditions must be plausible: the \(Y\)-component does not affect the competing-event hazard, and the \(D\)-component does not affect the event-of-interest hazard, conditional on the relevant observed history. In the simple examples below there is no covariate history, so this means that the hazard for \(Y\) depends on \(a_Y\), not \(a_D\), and the hazard for \(D\) depends on \(a_D\), not \(a_Y\). These restrictions let us write the hazards as \(\lambda_Y(a_Y)\) and \(\lambda_D(a_D)\). They are modeling and causal assumptions, not consequences of calculating a risk contrast.
Risk Formula Under Constant Hazards
Assume that, while a person is event-free, the two cause-specific hazards are constant over time under each component setting. In a short interval of length \(dt\), the probabilities of the two causes are approximately \(\lambda_Y(a_Y)dt\) and \(\lambda_D(a_D)dt\), respectively. The event-free survival probability therefore declines at the combined rate \(\lambda_Y(a_Y)+\lambda_D(a_D)\). For continuous event times, the two causes are mutually exclusive first events; this calculation does not require assuming independent latent event times.
The formula has two pieces: the fraction of the combined hazard attributed to \(Y\), and the probability of having any event by time \(t\). Consequently, increasing the competing-event hazard can lower the cumulative incidence of \(Y\), even when the cause-specific hazard for \(Y\) is unchanged. That feature is central to interpreting the indirect separable contrast.
Worked Example: Direct and Indirect Contrasts at Five Years
Suppose the hypothetical constant hazards, in events per person-year, are \(\lambda_Y(0)=0.04\), \(\lambda_Y(1)=0.06\), \(\lambda_D(0)=0.03\), and \(\lambda_D(1)=0.05\). These values satisfy the component-specific form: changing \(a_D\) does not change the stated \(Y\)-hazard, and changing \(a_Y\) does not change the stated \(D\)-hazard. Calculate the four \(Y\)-risks by \(t=5\) years.
At \((a_Y,a_D)=(0,0)\), the total hazard is \(0.04+0.03=0.07\) per year: \(R(0,0;5)=(0.04/0.07)[1-\exp(-0.07\times5)]=0.16875\), or about 16.9%.
At \((0,1)\), the total hazard is \(0.04+0.05=0.09\): \(R(0,1;5)=(0.04/0.09)[1-\exp(-0.09\times5)]=0.16105\). At \((1,0)\), it is also \(0.09\), but the target-cause fraction is different: \(R(1,0;5)=(0.06/0.09)[1-\exp(-0.09\times5)]=0.24158\). Finally, \(R(1,1;5)=(0.06/0.11)[1-\exp(-0.11\times5)]=0.23048\). Each risk is a probability by five years, not a hazard rate.
Holding the competing-event component at zero, the direct risk difference is \(R(1,0;5)-R(0,0;5)=0.24158-0.16875=0.07283\), about 7.28 percentage points. Holding it at one gives \(R(1,1;5)-R(0,1;5)=0.23048-0.16105=0.06943\), about 6.94 percentage points.
Holding the \(Y\)-component at zero, the indirect contrast is \(R(0,1;5)-R(0,0;5)=0.16105-0.16875=-0.00770\), about -0.770 percentage points. At \(a_Y=1\), it is \(R(1,1;5)-R(1,0;5)=0.23048-0.24158=-0.01110\), about -1.11 percentage points. Here, increasing the component that raises the competing-event hazard reduces the risk of \(Y\) by five years. These displayed subtractions use the displayed rounded risks and give the contrasts rounded to the same precision.
Why the Risk Contrasts Depend on Both Components
Although the \(Y\)-hazard in the example depends only on \(a_Y\), the direct risk difference is not the same at each value of \(a_D\). The competing hazard changes how many people remain event-free and able to experience \(Y\) as time passes. This changes the risk contrast even though it does not change the target cause-specific hazard. Likewise, the indirect risk difference depends on \(a_Y\), because a different \(Y\)-hazard changes the event-free population exposed to the competing event.
There is a useful additive identity for risk differences. It is a telescoping identity, not a claim that direct and indirect effects are biologically or statistically independent:
The right-hand side is the direct contrast at \(a_D=1\) plus the indirect contrast at \(a_Y=0\). One can instead take the direct contrast at \(a_D=0\) plus the indirect contrast at \(a_Y=1\); the sum is still the same total contrast. The two decompositions can assign different amounts to the direct and indirect parts when there is interaction on the risk-difference scale. Always state which fixed component setting defines each contrast.
Worked Example: A Second Hazard Pattern and the Additive Decomposition
Consider another hypothetical setting, with \(\lambda_Y(0)=0.08\), \(\lambda_Y(1)=0.10\), \(\lambda_D(0)=0.02\), and \(\lambda_D(1)=0.06\) per year. The component-specific assumption is built into these four hazards. Find the risks by three years and compare the total risk difference with one direct-plus-indirect decomposition.
Substitution into the formula gives \(R(0,0;3)=(0.08/0.10)[1-\exp(-0.10\times3)]=0.207345\); \(R(0,1;3)=(0.08/0.14)[1-\exp(-0.14\times3)]=0.195973\); \(R(1,0;3)=(0.10/0.12)[1-\exp(-0.12\times3)]=0.251936\); and \(R(1,1;3)=(0.10/0.16)[1-\exp(-0.16\times3)]=0.238260\).
The direct contrast at \(a_D=0\) is \(0.251936-0.207345=0.044591\), or about 4.46 percentage points. The indirect contrast at \(a_Y=0\) is \(0.195973-0.207345=-0.011372\), or about -1.14 percentage points. Adding these gives \(0.044591-0.011372=0.033219\). This is not the total contrast because these two particular terms do not form the same path through the four risks: the direct contrast at \(a_D=0\) must be paired with the indirect contrast at \(a_Y=1\), not \(a_Y=0\).
For the stated decomposition, use the direct contrast at \(a_D=1\): \(0.238260-0.195973=0.042287\). Pair it with the indirect contrast at \(a_Y=0\): \(0.195973-0.207345=-0.011372\). Their sum is \(0.030915\), equal to the total contrast \(R(1,1;3)-R(0,0;3)=0.238260-0.207345=0.030915\), or about 3.09 percentage points. Values are rounded consistently to six decimal places.
This example also illustrates why labels such as “direct” and “indirect” require their fixed settings to be specified. A direct contrast at \(a_D=0\) is a different contrast from a direct contrast at \(a_D=1\). Neither is automatically the unique direct contribution to the total effect.
Worked Example: The Time Horizon Changes the Risk Contrast
Return to the hazards in the five-year example, but calculate the direct and indirect contrasts by two years. The component-specific hazards remain the same; only the time horizon changes. This isolates the effect of choosing a different horizon.
The four risks are \(R(0,0;2)=(0.04/0.07)[1-\exp(-0.14)]=0.07465244\), \(R(0,1;2)=(0.04/0.09)[1-\exp(-0.18)]=0.07321324\), \(R(1,0;2)=(0.06/0.09)[1-\exp(-0.18)]=0.10981986\), and \(R(1,1;2)=(0.06/0.11)[1-\exp(-0.22)]=0.10771702\).
At \(a_D=0\), the direct risk difference is \(0.10981986-0.07465244=0.03516742\), about 3.52 percentage points. At \(a_Y=0\), the indirect contrast is \(0.07321324-0.07465244=-0.00143920\), about -0.144 percentage points. The total contrast is \(0.10771702-0.07465244=0.03306458\). These two-year values differ from the five-year values because cumulative incidence accumulates over time and the two causes compete to occur first.
Checks, Alternatives, and Limitations
Before interpreting a calculation, check the structure of the model as well as the arithmetic. The total hazard in each setting must be the sum of the two nonnegative cause-specific hazards. The target risk must lie between zero and one. In a two-cause model, the two cumulative incidences together equal \(1-S(t)\), the probability of having either first event by time \(t\). If a fitted or calculated risk violates these checks, revisit the formula, the cause labels, or the component-specific hazard assumptions.
The constant-hazard assumption is a substantive simplification. A poor fit may show up as time-varying hazards in plots or diagnostics, or as systematic disagreement between observed and model-predicted cumulative incidence over time. If hazards vary over time but remain component-specific, the corresponding risk is obtained by integrating the event-free survival multiplied by the time-varying hazard for \(Y\); the constant-hazard closed form no longer applies. A violation of the dismissible component conditions is a more fundamental problem: for example, if \(A_D\) also changes the \(Y\)-hazard, the four hazards cannot be represented by \(\lambda_Y(a_Y)\) alone. The separable component interpretation then does not follow from these calculations.
These risk contrasts answer a different question from a cause-specific hazard ratio, which compares instantaneous event rates among those still event-free. They also differ from contrasts under hypothetical elimination of the competing event, which change the event process being considered rather than assigning the competing-event component to one of its treatment settings. Natural direct and indirect effects use a different decomposition. The choice among these estimands depends on the causal question; a numerical risk contrast does not by itself resolve that choice.
A common mistake is to say that because the \(Y\)-hazard depends only on \(a_Y\), the risk of \(Y\) must also be unaffected by \(a_D\). That reasoning ignores the changing event-free survival: \(R\) includes the factor \(S(u\mid a_Y,a_D)\), which depends on both hazards. Another mistake is to report a direct or indirect contrast without giving its fixed component setting or time horizon. State both, and identify the outcome and the risk-difference scale.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For constant hazards \(\lambda_Y=0.05\) and \(\lambda_D=0.05\) per year, calculate the risk of \(Y\) by two years.
- If only \(\lambda_D(a_D)\) changes while \(\lambda_Y(a_Y)\) stays fixed, must the cumulative incidence of \(Y\) remain unchanged? Explain.
- Write a direct risk difference at \(a_D=0\) and an indirect risk difference at \(a_Y=1\), using \(R(a_Y,a_D;t)\).
- Suppose \(R(0,0;t)=0.10,\ R(1,0;t)=0.16,\ R(0,1;t)=0.08,\) and \(R(1,1;t)=0.13\). Verify the total risk difference using a direct contrast at \(a_D=1\) and an indirect contrast at \(a_Y=0\).
- Give one diagnostic check for a two-cause cumulative-incidence calculation and one assumption whose failure undermines the separable component interpretation.
Solutions. (1) The total hazard is \(0.10\) per year, so \(R=(0.05/0.10)[1-\exp(-0.10\times2)]=0.09063\), or about 9.06% by two years. (2) No. The changed competing hazard changes event-free survival and can therefore change the risk of \(Y\), despite an unchanged \(Y\)-hazard. (3) The direct contrast is \(R(1,0;t)-R(0,0;t)\); the indirect contrast is \(R(1,1;t)-R(1,0;t)\). (4) The total is \(0.13-0.10=0.03\). The stated path gives direct \(0.13-0.08=0.05\) plus indirect \(0.08-0.10=-0.02\), totaling \(0.03\). (5) Check that both cause-specific cumulative incidences sum to \(1-S(t)\) and remain in the range zero to one. A failure of the dismissible component condition—for example, the \(D\)-component also changing the \(Y\)-hazard—undermines the component-specific interpretation used here.