Bounds Under Partial Identification

Why a causal effect may be only partially identified

A competing event can make a target outcome undefined for some people under a treatment. For example, a person who experiences a competing event before the end of follow-up cannot also experience the target event first by that horizon. A comparison of observed target-event risks is still meaningful, but it does not generally identify the causal effect among people who would avoid the competing event under either treatment. Membership in that group depends on two counterfactual competing-event outcomes, only one of which is observed for each person.

This tutorial derives bounds for that group-specific effect. The bounds use the observed treatment-specific risks and, where needed, a principal-stratum proportion: the fraction of the population that would avoid the competing event under both treatment choices. They describe the range of effects compatible with the observed data and assumptions, rather than selecting a single effect from that range.

Potential outcomes and the target estimand

Let treatment be binary, $A\in\{0,1\}$, and fix a follow-up horizon $\tau$. For each treatment choice $a$, let $D(a)=1$ indicate that the person would experience the competing event by $\tau$, before the target event; otherwise $D(a)=0$. For someone with $D(a)=0$, let $Y(a)=1$ indicate a target event by $\tau$, and $Y(a)=0$ indicate no target event by $\tau$. Thus $Y(a)$ is defined for the people with $D(a)=0$; it is not defined for those with $D(a)=1$.

Definition: The always-free principal stratum consists of people with $D(0)=D(1)=0$, who would avoid the competing event under either treatment. Its population proportion is $\pi_{00}=P\{D(0)=0,D(1)=0\}$. The principal-stratum causal risk difference is $$ \Delta_{00}=E\{Y(1)-Y(0)\mid D(0)=D(1)=0\}. $$

The stratum is defined by the pair of counterfactual competing-event statuses, not by the observed status under the treatment actually received. Even in a randomized study, an individual’s pair $(D(0),D(1))$ is not observed. Consequently, randomization identifies treatment-specific marginal distributions but does not, by itself, reveal the stratum or the target risks within it.

The bounds below assume consistency, treatment exchangeability, and positivity: the observed outcomes under the received treatment equal the corresponding potential outcomes; treatment assignment is independent of the relevant potential outcomes (possibly conditional on measured baseline covariates); and each treatment has positive probability within covariate strata used in the analysis. Censoring must also be handled under suitable assumptions so that the treatment-specific event risks are identified. In a randomized study, exchangeability follows from the design; in an observational study, adjustment must adequately address confounding. These assumptions identify the observed marginal risks used below, not the unobserved joint distribution of $D(0)$ and $D(1)$.

What the observed risks say about the stratum size

Write $r_a=P\{D(a)=1\}$ for the competing-event risk under treatment $a$, and $s_a=1-r_a=P\{D(a)=0\}$ for the probability of avoiding that event. The observed data can identify $r_0$ and $r_1$ under the assumptions just stated. They identify the marginal probabilities of avoiding the competing event, but not their overlap.

The Fréchet bounds for two events give the possible range of the overlap:

\max(0,s_0+s_1-1)\ \leq\ \pi_{00}\ \leq\ \min(s_0,s_1).

The lower bound follows because the union of the two avoidance events cannot have probability greater than one: $P\{D(0)=0,D(1)=0\}\geq s_0+s_1-1$. The upper bound follows because their intersection cannot be larger than either event individually. Without additional assumptions or information, every value in this range is compatible with the two observed marginals.

Worked Example: Bounding the Always-Free Proportion

Suppose a hypothetical randomized study estimates competing-event risks by $\tau$ of $r_0=0.20$ under control and $r_1=0.30$ under treatment. The risks of avoiding the competing event are $s_0=1-0.20=0.80$ and $s_1=1-0.30=0.70$.

The possible always-free proportion is therefore $$ \max(0,0.80+0.70-1)=0.50 \ \leq\ \pi_{00}\ \leq\ \min(0.80,0.70)=0.70. $$

The data do not establish that exactly 60% (or any other single fraction in this range) would avoid the competing event under both treatments. They establish only that the proportion is between 50% and 70%, absent further restrictions.

Bounding each treatment-specific risk in the stratum

Let $q_a=P\{D(a)=0,Y(a)=1\}$, the population risk of the target event as the first event by $\tau$ under treatment $a$. This is the target-event cumulative incidence under that treatment; it is a risk in the full population, not a conditional risk among people who avoid the competing event. The observed data identify $q_a$ under the stated assumptions. Necessarily $0\leq q_a\leq s_a$.

Suppose for the moment that the always-free proportion is known to be $p>0$. Under treatment $a$, the always-free group is a subset of size $p$ among the $s_a$ people who would avoid the competing event under $a$. The total population mass with a target event under $a$ is $q_a$. The number of target events in the always-free group can be as small as zero, or as large as $p$, but it must also be consistent with the event count among the other $s_a-p$ people eligible under treatment $a$.

\max\{0,\ q_a-(s_a-p)\} \ \leq\ P\{D(0)=D(1)=0,\ Y(a)=1\} \ \leq\ \min(q_a,p).

For the lower limit, at most $s_a-p$ target events can be assigned to eligible people outside the always-free group, so any excess $q_a-(s_a-p)$ must occur within it. For the upper limit, the number of events cannot exceed either the total $q_a$ or the group size $p$. Dividing by $p$ gives bounds on the stratum-specific risk:

L_a(p)=\max\left\{0,\frac{q_a-(s_a-p)}{p}\right\}, \qquad U_a(p)=\min\left\{1,\frac{q_a}{p}\right\}.

These bounds require no assumption about how target outcomes under one treatment relate to target outcomes under the other. They are sharp given the marginals and $p$: the event outcomes can be allocated across the always-free group and the other eligible people to attain each endpoint.

Worked Example: Risk-Difference Bounds When the Stratum Proportion Is Specified

Continue with the hypothetical risks $s_0=0.80$, $s_1=0.70$, $q_0=0.24$, and $q_1=0.21$. Suppose an additional, explicitly imposed sensitivity specification sets $p=\pi_{00}=0.60$. This value is within the Fréchet range from the preceding example. It is not identified by the observed risks alone.

Under control, the eligible population outside the always-free stratum has size $s_0-p=0.80-0.60=0.20$. Thus $$ L_0(0.60)=\max\left\{0,\frac{0.24-0.20}{0.60}\right\} =\frac{0.04}{0.60}=0.0667, \qquad U_0(0.60)=\min\left\{1,\frac{0.24}{0.60}\right\}=0.400. $$

Under treatment, the outside eligible population has size $s_1-p=0.70-0.60=0.10$, giving $$ L_1(0.60)=\frac{0.21-0.10}{0.60}=0.1833, \qquad U_1(0.60)=\frac{0.21}{0.60}=0.350. $$

The risk difference is treatment risk minus control risk within the same stratum. Subtracting the largest compatible control risk from the smallest compatible treatment risk gives the lower bound; reversing those endpoints gives the upper bound: $$ 0.1833-0.400\ \leq\ \Delta_{00}\ \leq\ 0.350-0.0667, $$ so $-0.2167\leq\Delta_{00}\leq0.2833$. The data and the specified stratum proportion are compatible with effects from about 21.7 percentage points lower to 28.3 percentage points higher target-event risk under treatment. Since the interval includes zero, these bounds alone do not establish the direction of the effect.

When the stratum proportion is not known

Usually, $p$ is not known. Its feasible range is $p_{\min}=\max(0,s_0+s_1-1)$ to $p_{\max}=\min(s_0,s_1)$. For any feasible $p>0$, the risk-difference interval is $$ L_1(p)-U_0(p)\ \leq\ \Delta_{00}\ \leq\ U_1(p)-L_0(p). $$ The bounds allowing $p$ to vary are obtained by taking the smallest lower endpoint and the largest upper endpoint over its feasible range.

For fixed $a$, $L_a(p)$ is nondecreasing in $p$, while $U_a(p)$ is nonincreasing. This follows because $(q_a-s_a+p)/p=1-(s_a-q_a)/p$ increases with $p$, and $q_a/p$ decreases with $p$; taking the maximum or minimum with a constant preserves those directions. Therefore $L_1(p)-U_0(p)$ is nondecreasing, and $U_1(p)-L_0(p)$ is nonincreasing. If $p_{\min}>0$, the widest interval over the feasible range occurs at $p_{\min}$, so evaluating both endpoints there gives the bounds when the stratum proportion is unrestricted within its Fréchet range.

Worked Example: Allowing the Stratum Proportion to Vary

Use the same hypothetical risks: $s_0=0.80$, $s_1=0.70$, $q_0=0.24$, and $q_1=0.21$. The feasible range for $p$ is $[0.50,0.70]$, so the minimum feasible value is positive. By the monotonicity result, evaluate the widest risk-difference bounds at $p=0.50$.

For control, $s_0-p=0.30$, so $$ L_0(0.50)=\max\left\{0,\frac{0.24-0.30}{0.50}\right\}=0, \qquad U_0(0.50)=\frac{0.24}{0.50}=0.480. $$ For treatment, $s_1-p=0.20$, so $$ L_1(0.50)=\frac{0.21-0.20}{0.50}=0.020, \qquad U_1(0.50)=\frac{0.21}{0.50}=0.420. $$

Consequently, $$ 0.020-0.480\ \leq\ \Delta_{00}\ \leq\ 0.420-0, $$ or $-0.460\leq\Delta_{00}\leq0.420$. When the always-free proportion is left unrestricted by the data, the compatible effects span a substantially wider range than under the specification $p=0.60$. This widening reflects uncertainty about both who belongs to the stratum and their treatment-specific risks.

Interpretation, diagnostics, and alternatives

A bound is not a confidence interval. The endpoints describe identification uncertainty: the range of causal effects compatible with the specified assumptions and observed population risks. Sampling uncertainty is separate. In an empirical analysis, the risks and the bound endpoints are estimated from a finite sample, so their estimates also have sampling variability. A report should distinguish the identification region from any confidence interval for its endpoints.

Several checks help detect errors in a calculation or an analysis:

Verify that each target-event risk satisfies $0\leq q_a\leq s_a$. A target-event risk exceeding the probability of avoiding the competing event signals inconsistent definitions or computation.
Check that the chosen $p$ is in the Fréchet range and that $p>0$. If $p=0$, the conditional estimand $\Delta_{00}$ is undefined.
Check that each risk bound lies between zero and one, and that the lower endpoint does not exceed the upper endpoint.
Confirm that both treatment-specific risks refer to the same horizon, event ordering, target event, and principal stratum.
In adjusted observational analyses, check treatment overlap and whether the assumptions used to identify $r_a$ and $q_a$ are credible. Bounds cannot repair confounding or unsupported extrapolation.

If $p_{\min}=0$, the observed marginal risks permit the always-free stratum to be empty. The conditional effect is then not defined at that value. If the scientific target requires a nonempty stratum but the data provide no positive lower bound for its size, the analyst should state that limitation and, if appropriate, show results over prespecified positive values of $p$. Silently dividing by zero or treating a near-zero stratum as if its effect were precisely estimated is not justified.

These bounds answer a different question from a treatment-policy risk contrast, which compares full-population risks under the assigned treatments and is directly about what happens with the competing event included. They also differ from a hypothetical contrast that asks what would happen if the competing event were eliminated. Neither contrast is automatically a substitute for the always-free principal-stratum effect. Stronger assumptions, such as restrictions on the joint counterfactual competing-event statuses or on the allocation of target outcomes, can narrow the bounds, but those restrictions are not established by the observed marginal risks alone.

Common mistake: treating observed survivors as the principal stratum

A tempting but incorrect argument is: “Among people who avoided the competing event in the control arm, compare those in control with people who avoided it in the treatment arm; these are the people who would avoid it under either treatment.” The first group is selected by $D(0)=0$, while the second is selected by $D(1)=0$. Each can include people who would experience the competing event under the other treatment. The observed groups therefore need not be the same people, and comparing their risks does not identify $\Delta_{00}$.

The correction is to keep the counterfactual joint membership explicit. Use the observed marginals to bound the size of the overlap, and then use the event-count constraints to bound each treatment-specific risk within that overlap. A point estimate requires additional identifying assumptions; it cannot be obtained merely by relabelling observed competing-event-free participants as always-free.

Key takeaway: The observed treatment-specific competing-event and target-event risks identify marginal quantities. They generally do not identify the principal-stratum effect. Fréchet bounds on the stratum proportion, combined with feasible allocations of target events within that stratum, yield an interpretable range of causal risk differences.

Check Your Understanding

Use the definitions in this tutorial. The questions concern hypothetical risks, not real study results.

If $s_0=0.75$ and $s_1=0.65$, what is the Fréchet range for the always-free proportion?
Suppose $s_a=0.70$, $q_a=0.18$, and $p=0.50$. Calculate the lower and upper bounds for the target-event risk within the always-free stratum under treatment $a$.
Why does randomization not, by itself, identify the always-free principal-stratum effect?
If the feasible range for $p$ includes zero, what is the issue with reporting a conditional effect for the always-free stratum?

Solutions. (1) The range is $[\max(0,0.75+0.65-1),\min(0.75,0.65)]=[0.40,0.65]$. (2) Here $s_a-p=0.20$, so the lower bound is $\max\{0,(0.18-0.20)/0.50\}=0$, and the upper bound is $\min\{1,0.18/0.50\}=0.36$. (3) Randomization identifies treatment-specific marginal distributions, but each person reveals only one of $D(0)$ and $D(1)$; it does not identify their joint distribution or stratum membership. (4) At $p=0$, the conditioning stratum is empty and its conditional effect is undefined. If the stratum might be arbitrarily small, the data may not support informative bounds without an additional positive lower bound or other assumptions.