= Noncollapsibility of the odds ratio
A conditional <odds ratio> can differ from the marginal <odds ratio> after averaging over a predictor, even without <confounding>. If conditional risks are $p_a(x)$, averaging gives $\bar p_a=\int p_a(x)dF(x)$, and odds of $\bar p_a$ are generally not averages of the conditional odds. Consequently a common conditional odds multiplier need not equal the marginal odds multiplier. However, if both groups use the same distribution $F$ and $p_1(x)>p_0(x)$ everywhere, then $\bar p_1>\bar p_0$ and the marginal odds ratio is still above one. Noncollapsibility alone cannot reverse a uniformly signed effect under an identical background distribution.
Back to article page
