Why Define an Effect in a Specific Population?
A treatment can affect both the event of interest and the chance of experiencing a competing event first. The treatment-specific cumulative incidence of the event of interest is a total causal outcome: it incorporates both pathways. A different question is whether treatment changes the event of interest among people who would avoid the competing event under either treatment. That population is defined by two potential outcomes for each person, not by the group observed to avoid the competing event in one treatment arm.
The survivor average causal effect (SACE) makes this target explicit. Here, “survivor” means a person who would avoid the specified competing event through a fixed follow-up horizon; it does not necessarily mean survival from every cause or absence of the event of interest. The definition is useful for clarifying a scientific question, but the target population is generally not directly observed.
Definition at a Fixed Follow-Up Horizon
Consider a binary treatment \(A\in\{0,1\}\), where 0 denotes control and 1 denotes treatment, and fix a horizon \(\tau\). For each person, let \(C^a\) indicate whether the competing event would occur by \(\tau\) before the event of interest under treatment \(a\). Let \(Y^a\) indicate whether the event of interest would occur by \(\tau\) before the competing event under treatment \(a\). Thus \(Y^a=0\) for a person whose competing event occurs first. Define \(S^a=1-C^a\), the indicator that the person would avoid the competing event under treatment \(a\) through \(\tau\).
The word “survivor” in the name refers to membership in this stratum, not to conditioning on observed follow-up status in a treatment arm. An individual belongs to the stratum only if both potential competing-event indicators equal zero. The two indicators cannot both be observed for the same person: each person receives only one treatment. The SACE is therefore a causal estimand defined through potential outcomes, rather than a direct comparison of two observed groups.
Worked Example: Calculating the Effect in the Target Stratum
Worked Example: Calculating the Effect in the Target Stratum
Suppose a hypothetical population of 100 people has the following potential-outcome pattern over one year. The stratum sizes and risks are stipulated for illustration, so that both potential outcomes can be examined.
| Potential competing-event status \((S^0,S^1)\) | Number of people | Event risk under control among this stratum | Event risk under treatment among this stratum |
|---|---|---|---|
| (1, 1): always avoid | 40 | 20/40 = 0.50 | 12/40 = 0.30 |
| (1, 0): avoid only under control | 20 | 6/20 = 0.30 | Not in the target stratum |
| (0, 1): avoid only under treatment | 10 | Not in the target stratum | 4/10 = 0.40 |
| (0, 0): avoid under neither treatment | 30 | Not in the target stratum | Not in the target stratum |
In the always-avoids stratum, the treatment risk is \(12/40=0.30\), and the control risk is \(20/40=0.50\). Therefore,
In this hypothetical population, treatment reduces the one-year event risk by 20 percentage points among people who would avoid the competing event under either treatment. This is a stratum-specific contrast, not the effect for all 100 people. For comparison, the treatment-arm event risk in the full population is \((12+4)/100=0.16\). The control-arm risk is \((20+6)/100=0.26\), since people with \(S^0=0\) have the competing event first and hence \(Y^0=0\). The total risk difference is \(0.16-0.26=-0.10\), which is different from the SACE of \(-0.20\).
Why the Target Population Matters
The always-avoids stratum is a principal stratum: membership is defined by the pair \((S^0,S^1)\), which remains the same conceptual classification regardless of the treatment actually received. It is not the group of people who happened to avoid the competing event in the treatment arm, nor the group who happened to avoid it in the control arm. Those observed groups can contain different mixtures of principal strata.
This distinction matters because treatment can change who avoids the competing event. A treatment-arm average among people observed to avoid it is therefore an average over people with \(S^1=1\); a control-arm average is over people with \(S^0=1\). Neither group is, in general, exactly the always-avoids stratum. Subtracting those averages does not calculate the SACE.
Worked Example: Arm-Specific Survivor Averages Are Different Mixtures
Use a second hypothetical population. Suppose 50% of people have \((S^0,S^1)=(1,1)\), 20% have \((1,0)\), 10% have \((0,1)\), and 20% have \((0,0)\). Suppose the event risks are 0.40 under control and 0.20 under treatment in the always-avoids stratum. Among people who avoid the competing event only under control, suppose the control event risk is 0.70. Among people who avoid it only under treatment, suppose the treatment event risk is 0.60.
The SACE is computed within the always-avoids group:
The control-arm group that avoids the competing event consists of the 50% always-avoids stratum and the 20% who avoid it only under control. Its event risk is the weighted average
The treatment-arm group that avoids the competing event consists instead of the 50% always-avoids stratum and the 10% who avoid it only under treatment. Its event risk is
The difference between these observed-group risks is approximately \(0.267-0.486=-0.219\), not the SACE of \(-0.20\). More importantly, it compares averages over different populations. The numerical closeness here is incidental; it does not make the arm-specific comparison an estimate of the SACE.
A Short Derivation: The Unobserved Stratum Size
Even in a randomized study, the treatment-specific risks of the competing event reveal only the separate proportions who would avoid it under each treatment. They do not reveal how those statuses pair up within individuals. The size of the always-avoids stratum depends on that pairing.
Let \(p_0=P(S^0=1)\), \(p_1=P(S^1=1)\), and \(q=P(S^0=1,S^1=1)\). Since the overlap of two events cannot exceed either event, \(q\leq \min(p_0,p_1)\). Also, the union of the events cannot have probability greater than 1. By the inclusion-exclusion identity,
Probabilities are nonnegative, so the bounds together give
These bounds concern the target stratum’s size; they do not by themselves identify its treatment and control event risks, and hence do not determine the SACE.
Worked Example: Marginal Risks Do Not Determine the Stratum
Suppose a hypothetical randomized trial estimates that 60% of people would avoid the competing event under control and 70% would avoid it under treatment. Thus \(p_0=0.60\) and \(p_1=0.70\). Applying the bounds gives
so \(0.30\leq q\leq0.60\). The always-avoids stratum could contain between 30% and 60% of the population while preserving these two marginal avoidance probabilities. The trial’s randomization can support comparisons of treatment groups for observed outcomes, but these marginal rates alone cannot tell which individuals belong to both potential-status groups. Nor do they supply the two event risks within that unobserved stratum.
Interpretation, Assumptions, and Limitations
The SACE is an estimand: it defines the population, outcome, follow-up horizon, and contrast of interest. Its definition requires well-defined treatment strategies and potential outcomes under both strategies. For a causal interpretation of treatment effects, the usual requirements include consistency (the outcome under the assigned treatment agrees with that treatment’s potential outcome) and no interference between individuals. These conditions give potential outcomes a coherent treatment interpretation; they do not make the cross-treatment principal stratum observable.
Random assignment helps identify marginal treatment-specific outcome distributions, but generally does not identify the joint distribution of \((S^0,S^1)\) or the outcomes within the always-avoids stratum. Additional assumptions or sensitivity analyses are needed for estimation of the SACE from observed data. Such assumptions should be stated and defended in the scientific setting; the definition alone is not an estimation procedure.
The SACE also answers a narrower question than the total treatment effect on cumulative incidence. It excludes people who would experience the competing event under at least one treatment, even though treatment effects for those people may matter to policy or clinical decisions. It is not automatically a “direct effect” that removes the competing event, and it does not describe all people who remain free of the competing event in either observed arm.
Common Mistake and Practical Comparison
A frequent error is to calculate the event risk among observed competing-event-free participants separately in each arm and call their difference the SACE. The mistaken reasoning is that each arm’s observed survivors must represent the same target population. They need not: each arm selects on its own potential competing-event status, and the selected sets may have different principal-stratum mixtures. The correction is to define the target as \(S^0=S^1=1\) and recognize that membership is not directly observed.
| Contrast | Population represented | What it answers |
|---|---|---|
| Total treatment effect on event-of-interest risk | Everyone in the target population | How treatment changes the overall risk, including its consequences for competing events |
| SACE | People who would avoid the competing event under both treatments | How treatment changes event-of-interest risk in the always-avoids principal stratum |
| Difference in observed survivor averages | Different treatment-specific groups | A comparison of arm-specific selected groups; not generally the SACE |
No one contrast is universally preferable. The total effect is often directly relevant to treatment decisions for the full population. The SACE can address a focused scientific question about people who would avoid the competing event either way, but it depends on an unobserved population and usually requires stronger assumptions for identification.
Check Your Understanding
Answer each question using the fixed-horizon definitions above. Brief solutions follow the questions.
- In one sentence, define the population for the SACE.
- If the event risks in the always-avoids stratum are 0.18 under treatment and 0.25 under control, what is the SACE, and how should it be interpreted?
- If \(P(S^0=1)=0.55\) and \(P(S^1=1)=0.65\), give the bounds on the proportion in the always-avoids stratum.
- Why does randomization alone not generally identify the SACE?
- Is the difference between treatment-arm and control-arm risks among observed competing-event-free participants necessarily the SACE? Explain briefly.
Solutions. (1) It is the population that would avoid the competing event through the horizon under both treatment and control. (2) \(0.18-0.25=-0.07\), a 7-percentage-point lower event risk under treatment in that principal stratum. (3) The lower bound is \(\max(0,0.55+0.65-1)=0.20\), and the upper bound is \(\min(0.55,0.65)=0.55\). (4) Randomization identifies marginal treatment-specific distributions, not each individual’s joint pair \((S^0,S^1)\) or the event risks within the joint stratum. (5) No. The observed groups may contain different mixtures of principal strata, so their risk difference is not generally the SACE.