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Suppose one wants to compute the odds ratio of a disease for a person in group $A$ versus that of a person in group $B$. Suppose we consider the following: (i) include age in the odds ratio calculation, (ii) include age and height in the odds ratio calculation. Is it possible for the odds ratio for (ii) to be much greater than the odds ratio of (i)?

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Yes, it is possible for group (ii) to be greater. (whether it's MUCH greater would depend on your definition)

For an intuitive point of view, (i) averages out the effect of height among the people in the groups, but (ii) allows the effect to vary.

If you are to look at the calculation (taken from wikipedia on logistic regression), the model is $1/(1+e^{-z})$; so as $z$ gets larger, the odds would increase as well.

Hope this helps.

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Yes, this is entirely possible. What you're talking about is the confounding of the association between Group and Disease by in this case height. It's entirely possible for confounding to push the observed association downward, and when corrected for, to have the odds ratio rise. Indeed, if the confounder is negatively associated with the exposure and the disease, this should be what happens.

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