I am not sure whether I have interpreted the statement correctly here as it is not written in form of the formula. This is from Rubin's Statistical Analysis 3rd Edition with Missing Data page 75.
"It yields consistent estimates of the regression of Y on X and Z under the assumption that measurement error is nondifferential, which means that W is independent of X given Y and Z because in that case $E(Y ∣ X, Z) = E(Y ∣ X, Z, W)$."
- "W is independent of X given Y and Z" means $P(X|Y,Z,W)=P(X|Y,Z)$ to me. Is this correct here? It could also be $P(W|XYZ)=P(W)$ as W independent from $X|Y,Z$. I could not see how to derive $E(Y|X,Z)=E(Y|X,Z,W)$ here. I would expect this follows from cancellation in conditional probability density.
We consider the following.
$P(Y|X,Z,W)=\frac{P(Y,X,Z,W)}{P(X,Z,W)}=\frac{P(X|Y,Z,W)P(Y,Z,W)}{P(X,Z,W)}=\frac{P(X|Y,Z)P(Y,Z,W)}{P(X,Z,W)}=\frac{P(Y|X,Z)P(X,Z)P(X,Z,W)}{P(Y,Z)P(Y,Z,W)}$
Somehow $P(X,Z)P(X,Z,W)=P(Y,Z)P(Y,Z,W)$ should happen here.
- How do I see $E(Y|X,Z)=E(Y|X,Z,W)$ here?