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For a 2x2xk table, where k are strata from repeated cross-sectional studies over the course of 10 years, would it be suitable with a Mantel-Haenszel test of common odds ratio or would a logistic regression analysis be more apropriate? The 2x2 part are purely nominal data.

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With large strata they should be identical. Only subtle differences arise between the two tests, more discussion here.

I would prefer the logistic regression model because it could be extended to use fixed or random effects to account for age / period / cohort effects of study follow-up.

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