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.