LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Moments of the effect for the randomly selected treated unit

Statement

In the two-unit, two-treatment paired randomization, let $δ_{ℓ} = Y_{ℓ} (1) - Y_{ℓ} (0)$ and $τ = (δ_{1} + δ_{2}) / 2$ . Define $S$ to be the fixed treatment effect $δ_{ℓ}$ of whichever unit $ℓ$ is assigned treatment $1$ . Then $E_{d} [S] = τ, {Var}_{d} (S) = \frac{(δ_{1} - δ_{2})^{2}}{4} .$ The random variable $S$ uses both potential outcomes of the selected unit; it is not the observed treated-minus-control contrast and is not a statistic of the observed assignment and response data.

Facts & Assumptions

Given: Two fixed binary-treatment potential-outcome schedules, no interference, and uniform randomization of exactly one unit to treatment $1$ .

[F1]

The finite assignment model holds the potential-outcome schedule fixed and supplies only the declared assignment randomness (Finite randomized treatment-assignment models).

[F2]

The pair target is the average of the two unit-level effects (Finite-population average treatment effect and weak null).

[F3]

The two assignments in a two-treatment, one-block RCBD are equally likely (Randomized complete-block design).

[F4]

Finite-space expectation is the weighted sum of the two values (Expectation of a real random variable on a finite probability space).

[F5]

Variance is the expectation of the squared centered value (Moments, variance, and covariance on a probability space).

[F6]

The observed response vector is $Y_{θ} (Z)$ ; potential responses under unassigned vectors are fixed but latent (Finite randomized treatment-assignment models).

[F7]

A statistic is a measurable function of the observed data (Statistics and estimators).

Proof

1.1F1F3F6F7givenconstruct

Let $S = δ_{1}$ when unit $1$ receives treatment $1$ and $S = δ_{2}$ when unit $2$ receives it. These outcomes have weights $1 / 2$ by [F3]. Under assignment $(1, 0)$ , fix $Y_{1} (1)$ and $Y_{2} (0)$ but change $Y_{1} (0)$ ; the observed assignment and response data stay the same while $S = Y_{1} (1) - Y_{1} (0)$ changes. By [F7], no statistic of the observed data can equal $S$ over the model.

2.1F2F3F4step 1.1algebra

By [F4], $E_{d} [S] = (δ_{1} + δ_{2}) / 2 = τ$ , agreeing with the pair's finite-population average effect [F2].

3.1F3F5step 1.1step 2.1algebra∎

By [F5], ${Var}_{d} (S) = \frac{1}{2} (δ_{1} - τ)^{2} + \frac{1}{2} (δ_{2} - τ)^{2} = (δ_{1} - δ_{2})^{2} / 4$ .

Depends on

Used by

Observed paired-contrast variance is not determined by effect heterogeneity Counterexample

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Sources

Gary W. Oehlert, A First Course in Design and Analysis of Experiments, Chapter 13 §13.2 (standard reference, not scraped)