Choosing and Interpreting the Causal Estimand

Why the Estimand Must Be Chosen Explicitly

A causal estimate is interpretable only when the question it answers is clear. “Does treatment work?” leaves several choices unresolved: work for whom, compared with what strategy, on which outcome, by when, and measured on what scale? The estimand makes those choices explicit. It is the target quantity, not the particular statistical method used to estimate it.

The target-trial protocol specifies the eligible population, treatment strategies, time zero, follow-up, and endpoint. The estimand brings these choices together in a quantitative contrast. Even when two analyses use the same data and endpoint, they can answer different causal questions if they target different populations or use different contrast scales.

Definition: Let \(g_1\) and \(g_0\) denote two treatment strategies, and let \(T^g\) be the counterfactual time to the event under strategy \(g\). For a specified target population and time horizon \(\tau\), the causal risk under \(g\) is \(R_g(\tau)=P_\star(T^g\leq\tau)\). The subscript \(\star\) indicates that probability is taken over the target population. A causal estimand is a specified quantity, such as \(R_{g_1}(\tau)-R_{g_0}(\tau)\), together with its population, strategies, outcome, and time horizon.

The counterfactual \(T^g\) describes what the event time would be for an individual if that individual followed strategy \(g\). The two potential event times need not both be observed for any one person. The estimand is defined in terms of these counterfactual outcomes; whether it can be identified and estimated from observed data is a separate question.

Risk, Risk Difference, and Risk Ratio

For a binary outcome by time \(\tau\), risk is the probability that the event has occurred by that horizon. It is an absolute quantity, typically reported as a proportion or percentage. When comparing two strategies, the two most direct risk-scale contrasts are the risk difference and risk ratio.

Definition: For strategies \(g_1\) and \(g_0\), the risk difference is \(RD(\tau)=R_{g_1}(\tau)-R_{g_0}(\tau)\). The risk ratio is \(RR(\tau)=R_{g_1}(\tau)/R_{g_0}(\tau)\), provided \(R_{g_0}(\tau)>0\). The direction of each contrast depends on the order of strategies in its definition.

A negative risk difference means that the event is less common under \(g_1\) than under \(g_0\); its magnitude is an absolute change in probability. A risk ratio below one means the risk under \(g_1\) is a fraction of the risk under \(g_0\). For example, \(RR=0.70\) means the risk is 30% lower relative to the comparator risk, not 30 percentage points lower.

Result: If \(R_{g_0}(\tau)>0\), then \(RD(\tau)=R_{g_0}(\tau)\{RR(\tau)-1\}\). Thus the risk difference and risk ratio encode the same two risks on different scales, but their numerical values and practical interpretations differ.

To verify the identity, substitute the definition of the risk ratio:

R_{g_0}(\tau)\{RR(\tau)-1\} =R_{g_0}(\tau)\left\{\frac{R_{g_1}(\tau)}{R_{g_0}(\tau)}-1\right\} =R_{g_1}(\tau)-R_{g_0}(\tau) =RD(\tau).

The condition \(R_{g_0}(\tau)>0\) is necessary for the ratio and this derivation. If comparator risk is zero, the risk difference remains defined, but the risk ratio does not.

Worked Example: Comparing One-Year Event Risks

Suppose the target population is adults eligible for either of two specified treatment strategies, and the outcome is hospitalization within one year. Under \(g_1\), the one-year risk is 0.12; under \(g_0\), it is 0.18. Calculate and interpret both contrasts.

Risk difference: \(RD(1)=0.12-0.18=-0.06\). The risk is 6 percentage points lower under \(g_1\) than under \(g_0\) over one year.

Risk ratio: \(RR(1)=0.12/0.18=2/3\approx0.667\). The risk under \(g_1\) is about 0.667 times the risk under \(g_0\), or about 33.3% lower relative to \(g_0\), since \(1-0.667\approx0.333\).

The identity checks the calculation: \(0.18(0.667-1)\approx-0.060\). The population, strategies, hospitalization outcome, and one-year horizon are part of what these numbers mean; omitting any of them leaves the causal question underspecified.

Survival Contrasts at a Specified Time

The survival function under strategy \(g\) is \(S_g(t)=P_\star(T^g>t)\), the probability of remaining event-free beyond time \(t\). For an event time without ties at the boundary, this is the complement of the risk by \(t\). A survival contrast therefore describes event-free probability rather than event probability.

Result: For the same event, target population, strategies, and time \(t\), \(S_g(t)=1-R_g(t)\). Consequently, the survival difference \(S_{g_1}(t)-S_{g_0}(t)\) is the negative of the risk difference \(R_{g_1}(t)-R_{g_0}(t)\).

The result follows directly from complementary events:

S_{g_1}(t)-S_{g_0}(t) =\{1-R_{g_1}(t)\}-\{1-R_{g_0}(t)\} =-\{R_{g_1}(t)-R_{g_0}(t)\}.

Thus a positive survival difference means more people remain event-free under \(g_1\); for the same event and time, it corresponds to a negative risk difference. This sign change is easy to overlook when switching outcome descriptions.

A survival ratio, \(S_{g_1}(t)/S_{g_0}(t)\), is another possible fixed-time contrast when \(S_{g_0}(t)>0\). It is not the risk ratio: the numerators and denominators refer to complementary probabilities. A survival contrast also needs a time point, or an explicitly defined summary across time, because the difference between survival curves may vary over follow-up.

Worked Example: Event-Free Survival at Two Years

For the target population of interest, suppose the event-free survival probabilities at two years are \(S_{g_1}(2)=0.82\) and \(S_{g_0}(2)=0.75\). The event is death, and \(g_1\) and \(g_0\) are the two prespecified strategies. Find the survival difference and ratio, then express the comparison on the risk scale.

Survival difference: \(0.82-0.75=0.07\). Event-free survival at two years is 7 percentage points higher under \(g_1\).

Survival ratio: \(0.82/0.75\approx1.093\). The event-free probability under \(g_1\) is about 1.093 times that under \(g_0\).

The corresponding two-year event risks are \(R_{g_1}(2)=1-0.82=0.18\) and \(R_{g_0}(2)=1-0.75=0.25\). Therefore the risk difference is \(0.18-0.25=-0.07\), as required by the result above, and the risk ratio is \(0.18/0.25=0.72\). It would be incorrect to call 1.093 the risk ratio: it is a ratio of survival probabilities.

The Target Population Is Part of the Quantity

A marginal risk averages over the covariate distribution of a particular population. Change that distribution and the marginal risk, and potentially the contrast, can change—even if the strategy-specific risks within each covariate group stay fixed. The target population is therefore not just descriptive background; it is one of the estimand’s defining components.

The following calculation illustrates this point. Suppose \(L\) marks a baseline risk group. For a target population with group proportion \(q=P_\star(L=\text{high})\), the risk under strategy \(g\) is the population-weighted average of the group-specific risks:

R_g(\tau) =qP(T^g\leq\tau\mid L=\text{high}) +(1-q)P(T^g\leq\tau\mid L=\text{low}).

Worked Example: A Contrast Changes With the Target Population

Suppose the one-year event risks under \(g_1\) and \(g_0\) are 0.30 and 0.20, respectively, in the high-risk group, and 0.02 and 0.04 in the low-risk group. Calculate the marginal risks and contrasts when the high-risk group makes up 20% of the target population, then when it makes up 60%.

Target with 20% high-risk members: Under \(g_1\), the risk is \(0.20(0.30)+0.80(0.02)=0.060+0.016=0.076\). Under \(g_0\), it is \(0.20(0.20)+0.80(0.04)=0.040+0.032=0.072\). Thus \(RD=0.076-0.072=0.004\) and \(RR=0.076/0.072\approx1.056\).

Target with 60% high-risk members: Under \(g_1\), the risk is \(0.60(0.30)+0.40(0.02)=0.180+0.008=0.188\). Under \(g_0\), it is \(0.60(0.20)+0.40(0.04)=0.120+0.016=0.136\). Thus \(RD=0.188-0.136=0.052\) and \(RR=0.188/0.136\approx1.382\).

The group-specific risks have not changed, but the population-level contrasts have: the high-risk group, in which \(g_1\) has higher risk, receives more weight in the second target population. The correct interpretation names which population the marginal contrast targets.

Identification Is Separate From Definition

A causal risk is defined using counterfactual outcomes, but observed data do not automatically reveal it. In a simple setting with baseline treatment \(A\in\{0,1\}\), baseline covariates \(L\), and binary outcome \(Y\) by a fixed horizon, the following assumptions identify the risk under treatment level \(a\). Assume the observed study represents the target population, or that the covariate distribution and conditional outcome information have been appropriately transported to it.

Consistency: if \(A=a\), then the observed outcome \(Y\) equals the potential outcome \(Y^a\).
Conditional exchangeability: \(Y^a\) is independent of \(A\) given \(L\).
Positivity: for covariate values occurring in the target population, \(P(A=a\mid L)>0\).

Under these conditions, the g-formula for the risk is \(E_\star[E(Y\mid A=a,L)]\). To see why, consistency gives \(E(Y^a\mid A=a,L)=E(Y\mid A=a,L)\). Conditional exchangeability gives \(E(Y^a\mid L)=E(Y^a\mid A=a,L)\), and hence \(E(Y^a\mid L)=E(Y\mid A=a,L)\). Averaging both sides over the target distribution of \(L\) yields:

R_a=E_\star[Y^a] =E_\star\{E(Y^a\mid L)\} =E_\star\{E(Y\mid A=a,L)\}.

The first equality is the risk definition for a binary event indicator; the second is iterated expectation; the third follows from the assumptions just stated. Positivity ensures that the observed conditional mean on the right is available across the target covariate support. This formula is a standardization identity, not a guarantee that the assumptions are true. More complex longitudinal strategies require assumptions appropriate to their treatment histories and are not reduced to this baseline-treatment formula. \(\square\)

Choosing a Scale and Checking the Interpretation

No single scale is universally best. A risk difference communicates absolute impact and can be directly related to the number of events expected in a population of a given size. A risk ratio communicates relative change and may be useful when the proportional comparison is scientifically meaningful. Fixed-time survival contrasts focus on the probability of remaining event-free at a clinically chosen horizon. Whichever is selected, report the underlying risks when possible, so the scale does not hide the baseline event frequency.

A hazard ratio is a different kind of contrast: it compares instantaneous event rates among those still event-free at each time. It is not a risk ratio at a fixed horizon, and it does not by itself specify the absolute probability of an event by that horizon. A survival curve or fixed-time survival probability answers a different question.

Common warning signs include a contrast with no stated time horizon, a risk ratio whose comparator risk is zero, and a claimed population-level effect that does not specify the population weights. Also check that the two strategies define the same outcome and follow-up horizon. If one reported risk is cumulative incidence of an event and the other is survival from that event, the signs and denominators must be interpreted accordingly. A clear report states the target population, both strategies, outcome definition, time horizon, contrast scale, and direction of subtraction or division.

Key takeaway: A causal estimand is not just a number. It is a number for a named population, under named strategies, for a named outcome and time horizon, expressed on a stated scale.

Check Your Understanding

Answer each question, then compare with the brief solutions.

By 18 months, risks are 0.08 under \(g_1\) and 0.10 under \(g_0\). Give the risk difference and risk ratio, and interpret each.
If event-free survival at a specified time is 0.90 under \(g_1\) and 0.84 under \(g_0\), what are the risks and risk difference at that time?
Why can the same group-specific risks produce different marginal risk differences in two target populations?
State the three assumptions used in the simple baseline-treatment identification formula.

Solutions. (1) \(RD=0.08-0.10=-0.02\), a 2-percentage-point lower risk under \(g_1\); \(RR=0.08/0.10=0.80\), a 20% lower risk relative to \(g_0\). (2) Risks are \(1-0.90=0.10\) and \(1-0.84=0.16\); the risk difference is \(0.10-0.16=-0.06\), the negative of the survival difference. (3) Marginal risks average group-specific risks using the target population’s group proportions, so different proportions give different weighted averages. (4) Consistency, conditional exchangeability given baseline covariates, and positivity over the target covariate support.