I wonder if case-control matching will bring a new confounding bias into the matched design. In the following figure, $L$ is a confounder, $E$ is the exposure, D is the disease outcome. In the matched design, $L$ can be correlated with $D$ through 1) $L\rightarrow E\rightarrow D$; 2) $L\rightarrow D$ (initial confounding); 3) $L\leftarrow S\rightarrow D$ (selection bias). Since L and D are marginally unassociated, so the overall effect over 1)+2)+3) must be 0. However, when we assess the effect of $E$ on $D$, $E$ is controlled for relative to $L$. So 1) is blocked for $L$. 2)+3) is not zero. Thus, $L$ and $D$ are correlated via paths 2)+3) in the matched design. Not adjusting for $L$ will lead to biased estimate of the exposure effect. (Mansournia et al IJE, 2013).

I understand 2) is the initial confounding effect of $L$; 3) is the selection effect by $L$. Why 3) can not be seen as a new confounding effect also? $L$ is a confounder and it is connected to $Y$ via path 3), which is a criteria for being a confounder.

So the general question is: can we see the backdoor selection bias path by matched confounders as a confounding bias path also? matched confounders are still correlated with $Y$ and $E$ in the matched design and the resulting bias is a confounding bias, right?

case-control design



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