ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Random-intercept covariance and intraclass correlation

Example

For a subject $j$ measured at two distinct visits, let $Y_{j i} = μ + b_{j} + ε_{j i},$ where $E b_{j} = 0$ , $Var (b_{j}) = τ^{2} \geq 0$ , the residuals have mean zero and variance $σ^{2} \geq 0$ , the residuals are independent of each other, and $b_{j}$ is independent of the residual vector $(ε_{j 1}, ε_{j 2})$ . If $τ^{2} + σ^{2} > 0$ , the marginal correlation between the two observations is $Corr (Y_{j 1}, Y_{j 2}) = \frac{τ^{2}}{τ^{2} + σ^{2}} .$ This is the random-intercept intraclass correlation (ICC).

Facts & Assumptions

Given: The two-visit random-intercept special case of Finite-dimensional linear mixed model with explicit longitudinal roles with the moment and independence assumptions in the Example.

[F1]

The mixed-model marginal covariance is $Z G Z^{T} + R$ and does not require Gaussianity. Marginal covariance of the linear mixed model.

[F2]

Variance and covariance are centered second moments. Moments, variance, and covariance on a probability space.

[A1]

$τ^{2}, σ^{2} \geq 0$ and $τ^{2} + σ^{2} > 0$ .

[A2]

For square-integrable real random variables $U, W$ with positive variances, use the convention $Corr (U, W) : = Cov (U, W) / \sqrt{Var (U) Var (W)}$ . Correlation is undefined if either variance is zero.

Verification

Given: The model in the Example.

1.1F1F2A1

For either visit, the random part is $b_{j} + ε_{j i}$ , so its variance is $τ^{2} + σ^{2}$ because the two terms are independent and centered. Equivalently, in the marginal-covariance formula use random-intercept design row $1$ , $G = [τ^{2}]$ , and residual diagonal entry $σ^{2}$ .

1.2F1A1

For the two distinct visits, both observations contain the same $b_{j}$ . The covariance formula gives $Cov (Y_{j 1}, Y_{j 2}) = τ^{2}$ ; the residual cross-covariance is zero by the assumed independence.

2.1A2step 1.1step 1.2A1algebra∎

By [A1], both marginal variances are positive. Dividing the covariance from step 1.2 by the square root of the product of the two equal variances from step 1.1 gives $τ^{2} / (τ^{2} + σ^{2})$ by [A2]. If $τ^{2} = 0 < σ^{2}$ the ICC is $0$ ; if $σ^{2} = 0 < τ^{2}$ it is $1$ ; if both are zero the correlation is undefined.

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Sources

Fan Li, STA610 Lecture 1: The Problem of Correlated Data, PDF pp. 4–8 (standard reference, not scraped)
Duchateau, Janssen, and Rowlands, Linear Mixed Models, Chapter 2 §§2.3.1–2.4.1 (standard reference, not scraped)