Policy Transmission and the Actual Unit of Randomization

Conceptual map

IV-13.01Intervention units
IV-13.02Administrative dynamics
IV-13.03Handoff chains
IV-13.04Sample boundaries
IV-13.05Interpretation of effects

1. Assignment, delivery, and exposure are separate variables

A policy experiment identifies contrasts generated by its assignment mechanism. Administrative delivery determines how much treatment those assignments actually produce.

Let $Z\in\{0,1\}$ denote randomized assignment at the declared experimental unit, $D(z)\in\{0,1\}$ the potential receipt of a policy, and $$Y(z)$$ the potential outcome under assignment. Randomization means $$Z$$ is independent of the full potential-outcome vector, possibly conditional on design strata. The intention-to-treat effects are $ITT_{Y}=\mathbb{E}[Y\mid Z=1]-\mathbb{E}[Y\mid Z=0]$ and $ITT_{D}=\mathbb{E}[D\mid Z=1]-\mathbb{E}[D\mid Z=0]$ .

Definition 1 · Actual unit of randomization

The unit is the entity whose assignment is independently drawn by the design: person, household, branch, district, date, or cluster. Repeated records observed below that level do not create additional independent assignments.

2. A handoff chain maps assignment into receipt

Suppose delivery requires ordered stages $R_{1},\ldots,R_{K}$ , and every successful unit must pass all stages. Then the chain rule gives

\[P(D=1|Z=z)=P(R_{1}=1|Z=z)\prod _{k=2}^{K}P(R_{k}=1|R_{1:k-1}=1,Z=z).\](1)

This factorization is a probability identity for a serial all-stages path. A workflow with optional branches, retries, or skipped stages needs a state-transition representation or a sum over valid paths. Multiplying unconditional stage success rates generally gives the wrong delivery probability because the risk sets differ.

Proposition 1 · Random-assignment Wald estimand

If assignment is randomized, exclusion gives $$Y(z)=Y(D(z))$$ , receipt is monotone so $D(1)\ge D(0)$ , and $ITT_{D}>0$ , then $ITT_{Y}/ITT_{D}$ equals the average receipt effect for compliers.

Proof. Exclusion and binary receipt imply $$Y(D(1))-Y(D(0))=[Y(1)-Y(0)][D(1)-D(0)]$$ . Randomization identifies the expectation of this expression and the expectation of $$D(1)-D(0)$$ . Monotonicity makes the latter indicator equal one only for compliers. Dividing by its positive mean conditions the outcome effect on that group. ∎

3. A delivery experiment

Let half the eligible units be assigned to an administrative encouragement. Suppose receipt probabilities are 0.6 under $$Z=1$$ and 0.1 under $$Z=0$$ . If receipt adds two outcome units for every unit and assignment has no direct effect, then $ITT_{D}=0.5$ , $ITT_{Y}=2(0.5)=1$ , and the Wald ratio is two.

The outcome ITT remains a valid assignment effect when exclusion or monotonicity fails. The receipt-effect interpretation then fails. A contrast among delivered units, $\mathbb{E}[Y\mid D=1]-\mathbb{E}[Y\mid D=0]$ , compares groups selected partly by their potential delivery behavior and generally lacks a causal interpretation.

4. Interference, missing stages, and weak delivery

Failure case · Individual standard errors for clustered assignment

When branches are randomized and customers are observed, assignment varies at the branch level. Treating customer records as independent randomizations understates design uncertainty unless the covariance structure justifies that treatment.

Interference requires potential outcomes indexed by relevant assignment vectors or exposure mappings. Attrition after assignment can break identification unless outcome missingness is handled by design, weighting, bounds, or defensible models. A zero first stage makes the Wald ratio undefined; a small first stage makes conventional approximations unreliable. Equality at the monotonicity boundary permits units with no assignment response. Stage-specific rates are unidentified when stage timestamps or risk-set indicators are absent. Calendar windows must include the full delivery latency or explicitly classify right-censored cases.

5. Implementation, exercises, and sources

Record the randomization unit, strata, assignment probability, interference neighborhood, stage order, risk set, retry rule, delivery date, outcome window, and missingness status. Estimate assignment contrasts at the design level. Decompose delivery only after verifying that administrative fields encode the declared states and that every probability uses the correct conditional risk set.

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Exercises

Verify the three effects in Section 3.
Write a path-probability formula when delivery occurs through either stages $A\to B$ or the direct stage C.
Explain what remains identified if some units assigned to control obtain treatment and monotonicity is credible.

Partial solutions

1. Receipt changes by 0.5, so the constant receipt effect generates an outcome change of one and ratio two. 2. Add the probabilities of the two disjoint successful paths; shared or overlapping paths require inclusion-exclusion or a transition model. 3. The assignment ITT is identified, and the Wald ratio identifies the complier average effect when exclusion and the positive-first-stage condition also hold.

Guido W. Imbens and Joshua D. Angrist (1994), “Identification and Estimation of Local Average Treatment Effects,” Econometrica 62, 467–475.Instrumental assignment and complier effects.
Peter M. Aronow and Cyrus Samii (2017), “Estimating Average Causal Effects under General Interference,” Annals of Applied Statistics 11, 1912–1947.Exposure mappings under interference.
Chae-Yeon Xon (2026), “Almost a Policy: Transmission through Administrative Handoffs.”Related administrative-delivery application.

6. Audit checkpoint

Randomization unit, assignment support, strata, cluster count, interference mapping, potential-outcome indexing, stage order, path branching, conditional risk sets, retries, delivery definition, exclusion, monotonicity, first-stage magnitude, outcome window, attrition, censoring, covariance estimator, and estimand population.

7. Scope boundary

The chapter covers randomized encouragement with administrative transmission. Observational instruments, time-varying treatment regimes, and network-optimal designs require additional assumptions and methods.