1. ## Question about conditional odds ratio

I have a question about the definition of a conditional odds ratio (COR). Let's assume a very simple scenario of three binary variables Y,X and Z. I'm interested on the COR of Y and X (conditional on Z).

Is the way to approach this to estimate two ORs separately by fixing Z=0 and then Z=1 (an example would be here https://onlinecourses.science.psu.edu/stat504/node/105 under the conditional odds ratio section).

Or is it more appropriate to estimate one OR with a logistic regression model adjusted for Z (an example would be here https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3935453/ in section 3.1 starting with equation #2 for simplicity).

I'm confused at the two different ways to approach the same(?) concept?

2. ## Re: Question about conditional odds ratio

Well, I won't have enough time right now to write out all of what I would like to write. But think about why you would want to report marginal ORs or conditional ORs.

The big thing that comes to mind is that you are trying to address a confounding variable. You have an OR, but when controlling for a third variable the OR changes, they are conditional on another variable, like in the traditional confounding triangle. Traditionally, you would see the CMH approaches like at the Penn State page. Now that logistic regression is so easily accessible, it is a more common approach, since you can get a single estimate of OR while controlling for Z. It allows you to pull out the Z's effect and leave behind the X's effect. While the older approach of stratifying provides two estimates, which can also be helpful in better understanding what may be happening between the variables and their relationships.

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4. ## Re: Question about conditional odds ratio

Would it be ok to assume that we would prefer to use the logistic regression approach to get a single estimate of OR for X unless we are specifically interested in the OR for a specific subgroup of Z (especially if there is more than one confounder variables instead of just Z)?

5. ## Re: Question about conditional odds ratio

Not sure if there is a hard rule but that is typically how I treat it. Also, normally you are not just controlling for one covariate, but multiple. This makes using logistic more advantageous. That and you can request the ORs by confounding variable levels in logistic reg, so you can still get at what CMH conveys.

I put this in some notes I had once, EM = effect modification:

• Effect Modification: If ORs significantly different after stratification.
• Confounding: changes from crude but both the ORs are the same
• No EM or confounding: No difference in stratified ORs compared to crude OR
• Both EM and confounding: stratified ORs different, but both are below or bigger than crude.

• Both EM and confounding: ORs straddle the crude OR
I haven't check into this to much, though.

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