Average Treatment Effect for Always-Reporters

• AR-ATE is a principal-stratification estimand that isolates treatment effects for always-reporters, a latent subpopulation observable under both treatment and control.
• It employs worst-case randomization tests and studentized estimators to achieve finite-sample exactness and asymptotic validity under the weak null.
• The framework compares design-based inference with Lee trimming, addressing ambiguities from partial always-reporter identification in trials.

Searching arXiv for the cited papers to ground the article in current records. The Average Treatment Effect for Always-Reporters (AR-ATE) is a principal-stratification estimand for randomized controlled trials with attrition, defined on the subpopulation whose outcomes would be observed under both treatment and control. In the notation of potential reporting indicators, it is

$$\tau_{AR}=E\!\left[Y_i(1)-Y_i(0)\mid R_i(1)=R_i(0)=1\right],$$

and in finite-population form, for the realized set of always-reporters $A=\{i:R_i(1)=R_i(0)=1\}$,

$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$

The estimand isolates treatment effects for units whose outcomes are observable in both arms, thereby separating treatment response from post-assignment reporting behavior. A related usage appears in work on weakly positive outcomes with zeros, where the analogous principal stratum is the set of units with strictly positive outcomes under both treatment and control, and AR-ATE corresponds to the intensive-margin effect within that stratum (Chang et al., 26 Mar 2026, Chen et al., 2022).

1. Estimand, setup, and principal-stratum interpretation

The AR-ATE arises in a finite-population randomized controlled trial with attrition under a design-based setup. There are $n$ units, assignment is complete randomization with exactly $n_1$ assigned to treatment and $n_0=n-n_1$ to control, and the assignment vector satisfies $D\sim CR(n,n_1)$. Each unit has potential outcomes $Y_i(1),Y_i(0)$ and potential reporting indicators $R_i(1),R_i(0)$, where $R_i(d)$ indicates whether the outcome would be observed under treatment status $A=\{i:R_i(1)=R_i(0)=1\}$0. Observed reporting is $A=\{i:R_i(1)=R_i(0)=1\}$1, and the observed outcome $A=\{i:R_i(1)=R_i(0)=1\}$2 is available only when $A=\{i:R_i(1)=R_i(0)=1\}$3 (Chang et al., 26 Mar 2026).

The defining feature of AR-ATE is that it targets a latent principal stratum rather than the full randomized population. Under monotonicity in reporting behavior, $A=\{i:R_i(1)=R_i(0)=1\}$4 for all $A=\{i:R_i(1)=R_i(0)=1\}$5, the principal strata are Always-Reporters, Compliers (If-Reporters), Never-Reporters, and Defiers, with Defiers ruled out by monotonicity. The estimand therefore belongs to the same conceptual family as principal-stratification parameters used to study noncompliance, truncation, and post-treatment selection, but here the post-treatment variable is outcome observability (Chang et al., 26 Mar 2026).

This target is motivated by the fact that, under attrition, outcomes are not generally observed for all units in both arms. Principal stratification focuses inference on the subpopulation for which the outcome would be observed regardless of assignment. The data also state that, under monotonicity, the share of Always-Reporters $A=\{i:R_i(1)=R_i(0)=1\}$6 is identified by the control response rate, while Lee’s trimming bounds can partially identify $A=\{i:R_i(1)=R_i(0)=1\}$7 (Chang et al., 26 Mar 2026).

2. Monotonicity, observability, and partial revelation of always-reporter status

The framework assumes SUTVA, complete randomization, and monotonicity in reporting behavior. SUTVA requires that $A=\{i:R_i(1)=R_i(0)=1\}$8 and $A=\{i:R_i(1)=R_i(0)=1\}$9 do not depend on others’ assignments and that there are no hidden versions of treatment. Complete randomization makes the assignment vector the sole source of randomness. Monotonicity, $$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$0, rules out the possibility that treatment reduces reporting for any unit (Chang et al., 26 Mar 2026).

Always-reporter status is only partially observed. Some observed assignment-response patterns reveal principal-stratum membership exactly, while others do not.

Observed pattern $$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$1	Principal-stratum implication	AR status
$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$2	Control reporter	$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$3
$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$4	Treated non-reporter	$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$5
$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$6	Treated reporter; either Always-Reporter or Complier	$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$7
$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$8	Control non-reporter; either Complier or Never-Reporter	$$\tau_{AR}=|A|^{-1}\sum_{i\in A}\big[Y_i(1)-Y_i(0)\big].$$9

This classification is central because the inferential problem is not merely missing outcomes; it is latent membership in the target subpopulation. If $n$0 were known, standard randomization inference under a null hypothesis on always-reporters would be straightforward. The difficulty is that $n$1 is known only up to a feasible set of configurations compatible with the observed assignment and response patterns (Chang et al., 26 Mar 2026).

A plausible implication is that ambiguity is concentrated in treated reporters. Under monotonicity, control reporters are definitively always-reporters, and treated non-reporters are definitively not always-reporters. This induces an asymmetric identification geometry that later drives both worst-case testing and computational simplifications.

3. Worst-case randomization inference

The central inferential proposal is a worst-case randomization test that maximizes the randomization $n$2-value over all always-reporter configurations consistent with the data. Let $n$3 denote the set of feasible always-reporter tables. For any candidate $n$4, one computes a statistic $n$5 and defines the randomization $n$6-value

$n$7

for a right-tailed test. The worst-case $n$8-value is then

$n$9

The decision rule rejects when $n_1$0, or at $n_1$1 when an optional pretest is used (Chang et al., 26 Mar 2026).

Two null hypotheses are distinguished. The sharp null is

$n_1$2

which imposes zero individual treatment effects for always-reporters. The weak null is

$n_1$3

equivalently $n_1$4, while allowing heterogeneous effects. The distinction matters because the sharp null fully specifies the relevant potential outcomes and therefore supports an exact randomization distribution, whereas the weak null supports asymptotic rather than finite-sample validity (Chang et al., 26 Mar 2026).

The worst-case construction is conservative by design. For any fixed $n_1$5, the randomization test is exact under the sharp null by a design-based argument; maximizing over feasible $n_1$6 can only enlarge the $n_1$7-value. This is the mechanism behind finite-sample validity for the sharp null. The same logic also underlies the paper’s asymptotic results under the weak null when studentization is used (Chang et al., 26 Mar 2026).

4. Test statistics, validity theory, and computation

The baseline statistic is a studentized Hájek difference-in-means estimator restricted to a candidate always-reporter set:

$n_1$8

With group means $n_1$9 and $n_0=n-n_1$0 defined analogously, the Hájek-type variance estimator is

$n_0=n-n_1$1

and the studentized statistic is

$n_0=n-n_1$2

The stated rationale for studentization is asymptotic validity under the weak null, because it stabilizes the statistic with an estimated variance and is robust to heterogeneous effects (Chang et al., 26 Mar 2026).

The framework also introduces chi-square-type statistics that incorporate balance in the number of always-reporters across treatment arms. Writing $n_0=n-n_1$3 and $n_0=n-n_1$4, define

$n_0=n-n_1$5

and

$n_0=n-n_1$6

Then

$n_0=n-n_1$7

is two-sided in AR-count balance, while

$n_0=n-n_1$8

is one-sided, penalizing deficits of always-reporters in treatment relative to control. The data state that this AR-count balance component is absent in standard outcome-only randomization tests and can add power and robustness when ambiguous AR membership is consequential (Chang et al., 26 Mar 2026).

The main validity results are twofold. First, for any $n_0=n-n_1$9, the worst-case randomization test using $D\sim CR(n,n_1)$0, $D\sim CR(n,n_1)$1, or $D\sim CR(n,n_1)$2 is exact level-$D\sim CR(n,n_1)$3 for the sharp null. Second, under regularity conditions including non-negligible always-reporters, bounded correlation of potential outcomes among always-reporters, bounded fourth moments, and design balance, the test is asymptotically level-$D\sim CR(n,n_1)$4 for the weak null. The stated intuition is that studentization plus a Berry–Esseen bound for the combinatorial CLT under complete randomization yield asymptotic normality of the outcome-balance component, while the AR-count component may have nonstandard limits and is handled conservatively (Chang et al., 26 Mar 2026).

Implementation depends on the outcome support. Exact enumeration over assignments is typically infeasible for moderate $D\sim CR(n,n_1)$5, so Monte Carlo randomization with draws $D\sim CR(n,n_1)$6 is used in general. For discrete outcomes with finite support, the problem can be reduced to count vectors, and the randomization pmf of treated always-reporters across outcome categories is available in closed form; the resulting complexity is polynomial in $D\sim CR(n,n_1)$7 for fixed support size $D\sim CR(n,n_1)$8, and is practical for small $D\sim CR(n,n_1)$9, including binary outcomes. For continuous outcomes, the paper describes integer-programming-based bounds via an MIQCP-friendly decomposition, partitioning the test-statistic range into intervals and solving subproblems over binary matching variables $Y_i(1),Y_i(0)$0 that encode feasible always-reporter identities. A heuristic stage can provide a fast lower bound for each possible number of always-reporters, and more precise MIQCP subproblems are then solved only when early stopping fails (Chang et al., 26 Mar 2026).

An optional Berger–Boos-style pretest prunes implausible always-reporter configurations before the main test. It uses $Y_i(1),Y_i(0)$1 to test balance in always-reporter counts, keeps only treated-AR counts whose randomization $Y_i(1),Y_i(0)$2-values satisfy $Y_i(1),Y_i(0)$3, and then runs the worst-case test on the pruned set with $Y_i(1),Y_i(0)$4 replaced by $Y_i(1),Y_i(0)$5. The data state that this often drastically reduces the search space, especially when the observed data exhibit obvious balance or imbalance signals (Chang et al., 26 Mar 2026).

5. Confidence sets, Lee bounds, and relation to alternative approaches

Confidence sets for $Y_i(1),Y_i(0)$6 are obtained by test inversion. For a null hypothesis $Y_i(1),Y_i(0)$7, treated outcomes among candidate always-reporters are shifted by $Y_i(1),Y_i(0)$8:

$Y_i(1),Y_i(0)$9

and the worst-case randomization $R_i(1),R_i(0)$0-value is recomputed using $R_i(1),R_i(0)$1 in place of $R_i(1),R_i(0)$2. The confidence set is

$R_i(1),R_i(0)$3

or $R_i(1),R_i(0)$4 depending on convention. Because always-reporter membership remains partially latent, the maximization over feasible $R_i(1),R_i(0)$5 is retained throughout, with Monte Carlo or the integer-programming decomposition used as in hypothesis testing (Chang et al., 26 Mar 2026).

The same estimand can also be partially identified through Lee-style trimming under monotonicity. Under $R_i(1),R_i(0)$6, Lee trimming identifies $R_i(1),R_i(0)$7 as the control response rate, identifies $R_i(1),R_i(0)$8 as the difference in response rates between treated and control, identifies $R_i(1),R_i(0)$9 by the average among control reporters, and bounds $R_i(d)$0 by trimming the treated-reporter distribution by $R_i(d)$1 from the appropriate tail. These are bounds for the same AR-ATE, but they address identification rather than randomization inference for a sharp or weak null (Chang et al., 26 Mar 2026).

The relation to alternative approaches is explicit. IPW under conditional ignorability of attrition requires strong assumptions, and principal-stratification modeling imposes parametric assumptions. The randomization-based framework avoids those assumptions by relying only on assignment randomization and monotonicity, while delivering finite-sample guarantees under the sharp null and uniform asymptotic guarantees under the weak null. This suggests a methodological division: Lee-style trimming provides partial identification of the estimand, whereas worst-case randomization inference provides design-based testing and confidence procedures once the latent always-reporter ambiguity is handled directly (Chang et al., 26 Mar 2026).

6. Intensive-margin analogues, unit dependence with zeros, and limitations

A related but distinct AR-ATE appears when outcomes are weakly positive and may equal zero. In that setting, the relevant principal stratum is

$R_i(d)$2

the set of units with strictly positive outcomes under both treatment and control. The corresponding estimands include

$R_i(d)$3

$R_i(d)$4

and

$R_i(d)$5

The data describe $R_i(d)$6 as “the ATE in logs for individuals with positive outcomes under both treatments,” and identify it as the canonical AR-ATE for the intensive margin (Chen et al., 2022).

This intensive-margin formulation is closely connected to the critique of log-like transformations with zeros. For transformations $R_i(d)$7 that are defined at zero and behave like $R_i(d)$8 for large $R_i(d)$9, the ATE

$A=\{i:R_i(1)=R_i(0)=1\}$00

depends on the units of measurement when treatment affects the extensive margin. Under continuity and monotonicity of $A=\{i:R_i(1)=R_i(0)=1\}$01, the asymptotic relation

$A=\{i:R_i(1)=R_i(0)=1\}$02

shows that rescaling changes the weight placed on zero-to-positive transitions. The data therefore state that ATEs for transformations such as $A=\{i:R_i(1)=R_i(0)=1\}$03 and $A=\{i:R_i(1)=R_i(0)=1\}$04 should not be interpreted as approximating percentage effects when zeros are possible, because they are arbitrarily unit-dependent if the treatment shifts mass at zero (Chen et al., 2022).

In this setting, AR-ATE isolates the intensive margin by conditioning on $A=\{i:R_i(1)=R_i(0)=1\}$05 and thereby excluding extensive-margin contributions. However, the same principal-stratification difficulty reappears: because $A=\{i:R_i(1)=R_i(0)=1\}$06 is latent, $A=\{i:R_i(1)=R_i(0)=1\}$07 and $A=\{i:R_i(1)=R_i(0)=1\}$08 are generally not point-identified from marginals alone. Under the monotonicity condition $A=\{i:R_i(1)=R_i(0)=1\}$09, Lee bounds can partially identify these estimands by trimming the treated-positive distribution. The data report empirical illustrations: in the RCT application of Carranza et al. (2022), $A=\{i:R_i(1)=R_i(0)=1\}$10 and Lee bounds for the intensive margin are $A=\{i:R_i(1)=R_i(0)=1\}$11 in log hours and $A=\{i:R_i(1)=R_i(0)=1\}$12 in weekly hours; in the IV application of Berkouwer and Dean (2022), with extensive margin $A=\{i:R_i(1)=R_i(0)=1\}$13, the complier analogue of $A=\{i:R_i(1)=R_i(0)=1\}$14 is bounded by $A=\{i:R_i(1)=R_i(0)=1\}$15 (Chen et al., 2022).

Several limitations and extensions are explicit. Worst-case $A=\{i:R_i(1)=R_i(0)=1\}$16-values are conservative by construction, and power may be limited when always-reporter membership is highly ambiguous. Continuous outcomes with many ambiguous treated reporters can generate substantial computational burden for MIQCP subproblems. Monotonicity is critical; if it is violated, guarantees weaken. The data state that the framework can be adapted to reverse monotonicity $A=\{i:R_i(1)=R_i(0)=1\}$17 by flipping the roles of the arms, and that sensitivity analysis could relax the hard feasibility constraints to allow a limited number of defiers. Covariate-conditional monotonicity, stratified randomization, regression adjustment, multi-arm designs, and confidence sets for identified sets under monotonicity are all described as open extensions or promising directions (Chang et al., 26 Mar 2026, Chen et al., 2022).