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

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Equal-size cluster design effect under exchangeable within-cluster covariance

Statement

Let $m_{c} \geq 2$ and $k \geq 1$ . In a no-finite-population-correction superpopulation comparison, suppose the sampled PSU outcome vectors are independent and have a common distribution with $E [Y_{i j}] = μ, Var (Y_{i j}) = σ^{2} > 0, Cov (Y_{i j}, Y_{i ℓ}) = ρ σ^{2} (j \neq ℓ),$ for each PSU $i$ , while distinct PSUs are independent. The cluster estimator is the mean of $k$ complete cluster means, ${\overset{ˉ}{Y}}_{C} = \frac{1}{k m_{c}} \sum_{i = 1}^{k} \sum_{j = 1}^{m_{c}} Y_{i j},$ and the reference element SRS estimator is the mean of $k m_{c}$ independent draws from the same one-element marginal distribution. Then $DEFF : = \frac{Var ({\overset{ˉ}{Y}}_{C})}{Var ({\overset{ˉ}{Y}}_{E})} = 1 + (m_{c} - 1) ρ .$ The covariance model's validity entails $- 1 / (m_{c} - 1) \leq ρ \leq 1$ . In particular, $ρ = 0$ means no ICC inflation: both variances equal $σ^{2} / (k m_{c}) > 0$ . This formula makes no claim for unequal cluster sizes or weights, nonexchangeable within-cluster covariance, dependent PSUs, or non-negligible finite-population corrections.

Facts & Assumptions

Given: $m_{c} \geq 2$ , $k \geq 1$ , $σ^{2} > 0$ , independent PSU vectors with equal marginal means and variances, common within-PSU covariance $ρ σ^{2}$ , and an element-SRS reference of $k m_{c}$ independent marginal draws. This is the no-FPC superpopulation model, not a fixed-population design-variance identity.

[F1]

Cluster sampling units, cluster totals, and one-stage SRSWOR: A PSU is a cluster of SSUs; one-stage cluster sampling selects PSUs and observes every SSU in each selected PSU.

[F2]

Moments, variance, and covariance on a probability space: Variance and covariance are the expectations of the centered square and centered product.

[F3]

Covariance is symmetric and bilinear in finite linear combinations: Covariance is symmetric and bilinear in finite linear combinations of square-integrable variables.

[F4]

Variance of a finite sum as the sum of all variances and covariances: The variance of a finite sum of square-integrable real random variables is the sum of all pairwise covariances.

[A1]

Independent PSU outcome vectors share the stated exchangeable within-PSU covariance, and the element-SRS reference uses the same marginal law with its finite-population correction neglected.

Proof

1.1givenF2algebra

Nonnegativity of $Var (Y_{i 1} - Y_{i 2}) = 2 σ^{2} (1 - ρ)$ gives $ρ \leq 1$ ; nonnegativity of $Var (\sum_{j = 1}^{m_{c}} (Y_{i j} - μ)) = m_{c} σ^{2} [1 + (m_{c} - 1) ρ]$ gives $ρ \geq - 1 / (m_{c} - 1)$ . Thus the stated range is necessary.

1.2givenF1F2F3F4A1algebra

By [F1], one cluster mean averages its $m_{c}$ SSU outcomes; expanding its variance with [F4] and [F3] gives $Var (m_{c}^{- 1} \sum_{j = 1}^{m_{c}} Y_{i j}) = (m_{c} σ^{2} + m_{c} (m_{c} - 1) ρ σ^{2}) / m_{c}^{2} = (σ^{2} / m_{c}) [1 + (m_{c} - 1) ρ]$ .

2.1givenF2F3F4A1step 1.2algebra

The $k$ cluster means are independent by [A1], so [F4] and [F3] give $Var ({\overset{ˉ}{Y}}_{C}) = σ^{2} [1 + (m_{c} - 1) ρ] / (k m_{c})$ from step 1.2; the $k m_{c}$ independent element draws each have variance $σ^{2}$ , hence $Var ({\overset{ˉ}{Y}}_{E}) = σ^{2} / (k m_{c})$ .

3.1step 2.1algebra∎

Dividing by the positive reference variance in step 2.1 gives $DEFF = 1 + (m_{c} - 1) ρ$ ; if $ρ = 0$ , this ratio is $1$ while each variance remains $σ^{2} / (k m_{c}) > 0$ .

Remarks

Shalabh derives the equal-cluster relation after neglecting finite-population corrections. On PDF p. 7 the source states that $ρ = 0$ gives relative efficiency $E = 1$ and describes this as “there is no error.” In context this means there is no efficiency loss relative to element SRS due to clustering, not zero sampling variance: both variances remain $σ^{2} / (k m_{c}) > 0$ when $σ^{2} > 0$ .

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Sources

Shalabh, Sampling Theory, Chapter 9: Cluster Sampling, PDF pp. 6–7, IIT Kanpur (standard reference, not scraped)
Sharon L. Lohr, Sampling: Design and Analysis, 2nd ed., §5.2.2, printed pp. 139–141 (standard reference, not scraped)