DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Nondifferential measurement error relative to named variables

Definition

Let $X, W, Y, Z$ be real random variables on a common probability space. Let $G$ be the smallest sub-sigma-algebra of the ambient sigma-algebra that makes both $X$ and $Z$ measurable; when no $Z$ is named, take $G = σ (X)$ . The error in measuring $X$ by $W$ is nondifferential with respect to $Y$ relative to $(X, Z)$ exactly when $W ⊥ ⁣ ⁣ ⁣ ⊥ Y ∣ G,$ using the library definition Conditional independence given a sigma-algebra. It is differential with respect to $Y$ relative to $(X, Z)$ when this conditional independence fails.

The named variables are part of the definition: changing $Y$ or $Z$ can change the classification. This condition alone fixes no direction of bias for a specified estimator or target. Such a direction requires additional model and target assumptions.

Axiom: This definition assumes the Axiom of Choice (The Axiom of Choice) because the cited conditional-independence definition is stated using the library's conditional-expectation classes.

Depends on

Used by

Classical and Berkson slopes, misclassification, and validation Example

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hernán and Robins, Causal Inference: What If, Chapter 9 (standard reference, not scraped)
Keogh et al., STRATOS guidance on measurement error and misclassification, Part 1 (standard reference, not scraped)