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Berkson error preserves the simple linear regression slope

Statement

Let $W, U, ϵ, X, Y$ be square-integrable real random variables on one probability space with $E [W] = E [U] = E [ϵ] = 0$ , $U$ independent of $W$ , and $ϵ$ independent of $W$ . Put $X = W + U, Y = α + β X + ϵ, α, β \in R,$ so $X$ has Berkson additive error relative to $W$ . If $Var (W) > 0$ , the coefficient of the best affine $L^{2}$ predictor of $Y$ from $W$ is $β$ . Independence between $U$ and $ϵ$ is not assumed or needed.

Facts & Assumptions

Given: The square-integrable centered variables and separate independence hypotheses in the Statement.

[F1]

The Berkson model has $X = W + U$ , with centered $U$ independent of $W$ (Additive measurement error and categorical misclassification models).

[F2]

Variance and covariance are the centered-square and centered-product expectations (Moments, variance, and covariance on a probability space).

[F3]

Covariance is bilinear on finite linear combinations, including constants (Covariance is symmetric and bilinear in finite linear combinations).

[F4]

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

[F5]

For independent integrable real random variables, the expectation of their product factors (Expectation factors over a finite product of mutually independent random variables).

[F6]

The one-predictor normal equation for the best affine $L^{2}$ predictor is $γ Var (W) = Cov (Y, W)$ (Normal equations for best affine $L^{2}$ prediction).

[F7]

Square-integrable real variables are integrable on a probability space by Cauchy-Schwarz with the constant-one variable (Cauchy-Schwarz for random variables).

Proof

1.1givenF2F4F5F7

The variables are integrable by [F7]. By [F4] and [F5], independence and centering give $Cov (U, W) = Cov (ϵ, W) = 0$ .

2.1givenF1F2F3step 1.1algebra

Since $X = W + U$ , bilinearity [F3] and step 1.1 give $Cov (Y, W) = Cov (α + β (W + U) + ϵ, W) = β Var (W)$ .

3.1givenF6step 2.1algebra∎

The variance is positive by hypothesis. The normal equation [F6] therefore gives the unique slope $γ = Cov (Y, W) / Var (W) = β$ . No covariance between $U$ and $ϵ$ occurs in this calculation.

Depends on

Used by

Classical and Berkson slopes, misclassification, and validation Example

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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)