Restricting Eligibility Changes the Estimand

Why Eligibility Defines the Causal Question

A treatment effect is always an effect for some population. Eligibility criteria make that population explicit: for example, adults meeting a baseline risk threshold, patients without a contraindication, or students enrolled before a program begins. Restricting the target population can be substantively sensible, but it changes which units contribute to the average effect. The resulting estimand need not equal the effect for everyone in the source population.

The essential distinction is between defining who the effect concerns and estimating that effect from observed data. Eligibility defines the target. Additional design or identification assumptions are needed to learn its causal effect from observed outcomes. The introductory material on defining target populations establishes why an estimand must name its population; here we make the eligibility restriction explicit and calculate its consequences.

The eligibility rule may be represented as \(R=r(X)\), where \(X\) contains baseline characteristics and \(r\) is a stated rule. For instance, \(R=1\) might mean age is between 50 and 69 years and a baseline risk score exceeds a specified threshold. The rule, its measurement time, and the population to which it is applied are part of the estimand. A vague description such as “high-risk patients” is not enough to determine a unique target.

Eligibility is not synonymous with treatment receipt. The eligible population can contain treated and untreated units, and the restricted ATE averages their potential treatment effects without weighting specifically toward either group. Nor is eligibility the same as a post-treatment response. Defining the target by a variable affected by treatment can select units differently under treatment and control, so it generally does not define this baseline restricted effect.

How Restriction Changes the Average

Suppose \(X\) divides the eligible population into mutually exclusive baseline groups. Within group \(x\), write the conditional average treatment effect as \(\tau(x)=E[Y(1)-Y(0)\mid X=x,R=1]\). The restricted effect averages these group effects using the covariate distribution among eligible units:

for discrete \(X\); for continuous \(X\), the sum is an integral over the eligible covariate distribution. Thus, restriction can change the effect even if every subgroup-specific effect stays the same: the eligible population may assign different weights to subgroups. If treatment effects are constant across all groups, changing those weights does not change the average effect. This is a direct consequence of averaging, not a contradiction between analyses.

For an empirical finite population, the corresponding target is the average of the eligible units’ individual effects. If the eligible units are indexed by \(i\), then \(\tau_R=N_R^{-1}\sum_{i:R_i=1}\{Y_i(1)-Y_i(0)\}\), where \(N_R\) is the number meeting the rule. In real data, both potential outcomes are never observed for the same unit. A calculation using both is therefore a transparent hypothetical illustration; estimating the target from observed data requires an identification strategy.

Subgroup Weights and Eligibility Rules

When only subgroup averages are available, the eligible effect is calculated using the subgroup proportions among eligible units, not their proportions in the entire source population. In a sample, these proportions estimate the eligible population weights only if the sample represents the intended eligible population or the analysis otherwise accounts for the sampling design.

An eligibility threshold can also change the weights and therefore the target effect. For example, “age at least 60” and “age 50 through 69” are distinct rules, even if the same study data are used. A useful analysis specifies the threshold before examining outcomes, reports how many units meet it, and states the population to which the resulting average is intended to apply.

Identification From Observed Outcomes

The estimand \(\tau_R\) is defined by potential outcomes whether or not it can be identified from a particular data set. To identify it from observational data, a common sufficient set of conditions is consistency, no interference between units, conditional exchangeability within the eligible population, and positivity there. Consistency means the observed outcome equals the potential outcome under the treatment actually received. No interference means one unit’s treatment does not change another unit’s potential outcome. Conditional exchangeability requires \(\{Y(1),Y(0)\}\perp A\mid X,R=1\), and positivity requires \(0<P(A=a\mid X=x,R=1)<1\) for \(a=0,1\) in covariate strata represented in the eligible target population. In a randomized study, random assignment can supply exchangeability by design, subject to implementation and follow-up issues.

Under these conditions, and with eligibility determined from baseline information, the effect can be written as an average of conditional observed-outcome contrasts over the eligible covariate distribution:

This expression makes two requirements visible. First, treatment and control outcomes must be comparable within the relevant covariate strata; eligibility alone does not remove confounding. Second, the covariate distribution used for the outer average must be that of the target eligible population. If some eligible covariate strata have no treated units or no controls, positivity fails there, and the effect for the full stated eligible population is not identified by this formula without further assumptions.

Practical Checks and Common Pitfalls

Before calculating or reporting an eligibility-restricted effect, verify that the rule is operational and measured before treatment; that the restricted population is the one relevant to the scientific or clinical question; and that there is treatment and control information across the eligible covariate strata needed for identification. Report the number or proportion eligible, the rule’s substantive meaning, and the outcome units. If eligibility is based on several criteria, state how they are combined.

• Changing the rule after seeing outcomes. Choosing a threshold because it gives a preferred estimate makes the target data-dependent and can undermine interpretation. Prespecify the rule when possible and label exploratory restrictions honestly.
• Using a post-treatment criterion. Restricting to people who responded, adhered, or remained event-free after treatment can compare selected groups whose membership may depend on treatment. That is not the baseline eligibility estimand defined here.
• Assuming restriction solves confounding. A carefully defined eligible population does not make treatment assignment exchangeable. Confounding control and eligibility definition are separate tasks.
• Using the wrong weights. Averaging subgroup effects equally, or using the source-population weights when the target is restricted, generally produces a different estimand.
• Overgeneralizing the result. An effect among eligible units does not, without additional transport assumptions, establish the effect among excluded units or in a different population.

The restricted ATE differs from the ATT, which averages effects among treated units, and from the ATC, which averages among control units: eligibility alone does not condition the target on treatment status. It also differs from the ATO, whose target is defined by treatment-overlap weighting. These are alternative population questions, not interchangeable labels. Matching may later alter which units contribute to an analysis, but the eligibility rule itself should be stated independently of a particular estimation procedure.