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This is not the first time I have performed stratified analyses based on logistic regressions, but I have never before been confronted with this situation.

When I test my relationship in the overall population, my OR is 3.98 (3.60-4.40). When I stratify my population according to their obesity, I have these two ORs: 2.05 (1.75-2.41) and 3.52 (3.08-4.04).

I wonder how it is possible that the OR in the overall population is larger than either of the two and whether this could indicate a problem in the model (e. g. too high collinearity between obesity and my outcome).

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    $\begingroup$ Absolutely; this is a classic example of the noncollapsibility of the odds ratio. There is no solution to this problem except to use a collapsible measure of effect size like a risk ratio or risk difference. If you want to retain odds ratios (and I don't know why you would), you need to decide whether you want a conditional estimate of the effect (i.e., given obesity) or a marginal estimate (overall in the population). $\endgroup$
    – Noah
    Commented Apr 24, 2019 at 17:56

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This is not that much unusual and in fact it can be even the case that im the overall sample the odds ratio is >1, let's say OR= 3.98 as in your case, but in both subgroups the odds ratio is even less than 1! This sounds impossible at first glance but the following picture from wikipedia visualizes this effect which is called the simpson's paradox:

simpson's paradox

In this picture the black dotted line shows for the whole sample a negative effect while the two subgroups have positive effects as indicated by the blue and the red line. The same paradox effect can occure with odds ratios.

So what I am saying is that there are situations where the results from the subgroups seem not to be compatible with the whole sample as it is in your data or the subgroups even seem to contradict the whole sample as in the examples in the links I provided. To my knowledge there is no simple universal rule how to interpret such findings but you have to consider theoretical plausible mechanisms (as you can see in the literature). I can also suggest reading this paper that shows the logic behind th simpson's, the lord's paradox and the suppression effect which the authors all together call the reverse effect.

EDIT

Thanks to comments from Noah who mentioned non-collapsibility. This paper reviews "previous explanations of Simpson's paradox that attributed it to two distinct phenomena: confounding and non-collapsibility.". To my understanding this means that the problem you describe is called the simpson's paradaxon in both cases, confounding and non-collapsibility. maybe the paper claryfies which of the two mechanisms is going on with your data.

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    $\begingroup$ Note that even in the absence of confounding, the problem can arise simply due to noncollapsibility. This can happen even in a randomized experiment. $\endgroup$
    – Noah
    Commented Apr 24, 2019 at 18:33
  • $\begingroup$ @Noah: Yes, I just didn't want to write that again since you already did so. Thus I wrote one mechanisms that I spontaneously thought of. $\endgroup$
    – user244721
    Commented Apr 24, 2019 at 18:37
  • $\begingroup$ Thank you stats.and.r @Noah. If I have understood correctly, my results may therefore come either because of non-collapsibility or Simpson's paradox. If I used RRs, the only reason would be the Simpson's paradox. Can the interaction test be false because of any of these reasons or is it unaffected? $\endgroup$
    – Emmanuel.W
    Commented Apr 25, 2019 at 10:50
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Obesity is a confounder and that is why you see the marginal > stratum specific effects. The noncollapsibility effect is always away from the null for prognostic covariates and it could be present too but perhaps masked by the confounding. The unequal effects per stratum is likely an artifact of the sample but could also be an interaction as that can coexist with confounding.

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