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Classical and Berkson slopes, misclassification, and validation

Example

Each construction below is a separate probability model.

Classical error. Let $(X, U)$ be uniform on ${- 1, 1}^{2}$ , put $W = X + U$ and $Y = 2 X$ . The latent-value slope is $2$ , while the measured-value slope is $1$ .
Berkson error. Let $(W, U)$ instead be uniform on ${- 1, 1}^{2}$ , put $X = W + U$ and $Y = 2 X$ . Under this different error direction and independence condition, the slope of $Y$ on $W$ is $2$ .
Nondifferential binary misclassification. Let $X$ be fair binary and $Y = X$ . Given either value of $X$ , set $W = X$ with probability $1 / 5$ and $W = 1 - X$ with probability $4 / 5$ . This mechanism is nondifferential relative to $Y$ given $X$ , but it reverses the sign of the observed mean contrast: the true contrast is $1$ and the contrast by $W$ is $- 3 / 5$ .
Differential binary misclassification. Give $(X, Y)$ probabilities $3 / 8, 1 / 8, 1 / 8, 3 / 8$ at $(0, 0), (0, 1), (1, 0), (1, 1)$ , respectively, and set $W = 1 - Y$ . The true contrast of $Y$ across $X$ is $1 / 2$ , while the contrast across $W$ is $- 1$ ; here $W$ is not conditionally independent of $Y$ given $X$ .
Unidentified variance split and validation. Let $A, B, C, D$ be the coordinate signs on the uniform space ${- 1, 1}^{4}$ . The decompositions $X_{0} = A + B, U_{0} = C + D$ and $X_{1} = A + B + C, U_{1} = D$ both give $W = A + B + C + D$ . Their latent-error variance pairs are $(2, 2)$ and $(3, 1)$ . The joint validation laws of $(X, W)$ differ: $P (X_{0} = 0) = 1 / 2$ but $P (X_{1} = 0) = 0$ . Thus observing $X$ adds information that separates these two candidate laws.

The fifth construction shows what additional validation information can do for this candidate pair; it does not assert a universal correction rule.

Facts & Assumptions

Given: The finite probability models and their cell probabilities specified in the Example.

[F1]

Classical and Berkson errors have the opposite additive equations and their own independence conditions (Additive measurement error and categorical misclassification models).

[F2]

Variance and covariance are centered expectations, and covariance is bilinear (Moments, variance, and covariance on a probability space, Covariance is symmetric and bilinear in finite linear combinations).

[F3]

Independence of real random variables is independence of their generated sigma-algebras (Independent random elements).

[F4]

Expectations of products of finite independent integrable families factor (Expectation factors over a finite product of mutually independent random variables).

[F5]

Under the stated classical hypotheses, the measured-value slope is the variance-ratio formula in the classical error theorem (Classical error attenuates the simple linear regression slope).

[F6]

Under the stated Berkson hypotheses, the measured-value slope equals the latent-value slope (Berkson error preserves the simple linear regression slope).

[F7]

Nondifferentiality is conditional independence relative to the named variables (Nondifferential measurement error relative to named variables).

[F8]

The general conditional-independence definition explicitly assumes the Axiom of Choice so that its conditional-expectation classes are available (Conditional independence given a sigma-algebra, The Axiom of Choice).

[F9]

The observed law need not identify the latent-error variance pair (The observed law need not identify the latent-error variance split).

[F10]

Conditional independence is the conditional-expectation factorization; it holds automatically when one side is measurable with respect to the conditioning sigma-algebra (Conditional independence given a sigma-algebra).

Verification

1.1givenF1F2F3F4F5algebra

In the first construction, each of the four sign pairs has probability $1 / 4$ and each coordinate sign has probability $1 / 2$ , so $X$ and $U$ are independent. Each has $E [X] = 0$ and $Var (X) = 1$ , and likewise for $U$ . Their covariance is $0$ by [F2]–[F4]. The error is classical, $ϵ = 0$ , and the response coefficient is $β = 2$ . Applying [F5] gives measured slope $2 (1) / (1 + 1) = 1$ , while $Y = 2 X$ gives latent slope $2$ .

1.2givenF1F2F3F4F6

In the second construction, $W$ and $U$ are independent centered signs with variance $1$ , $X = W + U$ , and $ϵ = 0$ . Thus the distinct Berkson hypotheses hold, and [F6] gives measured slope $2$ .

1.3givenF7F8F10algebra

In the third construction $Y = X$ . On each positive-probability stratum $X = x$ , the variable $Y$ is constant. The conditional-independence definition [F10], applied under the explicit Axiom of Choice premise [F8], therefore gives for all bounded measurable $f, g$ , $E [f (W) g (Y) ∣ σ (X)] = g (X) E [f (W) ∣ σ (X)] = E [f (W) ∣ σ (X)] E [g (Y) ∣ σ (X)] .$ By [F7], the error is nondifferential relative to $Y$ given $X$ .

1.4givenF7F8F10algebra

In the fourth construction, $E [Y ∣ X = 1] - E [Y ∣ X = 0] = 3 / 4 - 1 / 4 = 1 / 2$ . Since $W = 1 - Y$ , $E [Y ∣ W = 1] - E [Y ∣ W = 0] = 0 - 1 = - 1$ . On the positive stratum $X = 1$ , the conditional expectation of $1_{{W = 1}} 1_{{Y = 1}}$ is $0$ , whereas the product of the two conditional expectations is $(1 / 4) (3 / 4) = 3 / 16$ . The factorization in [F10] fails, so the error is differential relative to $Y$ given $X$ .

1.5givenF1F3F4algebra

On the uniform four-sign space, every specified pattern of $k$ distinct signs has probability $2^{- k}$ ; thus the coordinates are mutually independent, centered, and have variance $1$ . For sign-pattern sets $E, F$ on disjoint blocks of sizes $2, 2$ or $3, 1$ , the joint probability is $∣ E ∣ ∣ F ∣ / 16$ , the product of the two block probabilities; hence $X_{i}$ is independent of $U_{i}$ for $i = 0, 1$ . Their distinct-coordinate covariances are zero by [F4]. They satisfy $W_{0} = W_{1} = A + B + C + D$ pointwise.

2.1givenF2step 1.3algebra

The true contrast is $E [Y ∣ X = 1] - E [Y ∣ X = 0] = 1$ . The table gives $P (W = 1) = 1 / 2$ , $E [Y ∣ W = 1] = 1 / 5$ , and $E [Y ∣ W = 0] = 4 / 5$ , so the measured contrast is $1 / 5 - 4 / 5 = - 3 / 5$ . Thus this nondifferential mechanism reverses the contrast direction.

2.2F2F3F4F9step 1.5algebra

Bilinearity [F2] and the zero coordinate covariances from step 1.5 give $Var (X_{0}) = 1 + 1 = 2$ , $Var (U_{0}) = 1 + 1 = 2$ , $Var (X_{1}) = 1 + 1 + 1 = 3$ , and $Var (U_{1}) = 1$ . Thus the same observed $W$ law is compatible with distinct variance pairs, as asserted in [F9].

3.1givenstep 1.5step 2.2algebra∎

The event $X_{0} = 0$ requires opposite signs among $A, B$ , so has probability $2 / 4 = 1 / 2$ . Since $X_{1} = A + B + C$ is a sum of three signs, it takes values only in ${- 3, - 1, 1, 3}$ , and therefore $P (X_{1} = 0) = 0$ . These unequal marginals show that the joint validation laws of $(X, W)$ differ, so observing $X$ adds information that separates these two candidate laws.

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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)