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Suppose I have around 20 exposures that potentially affect an outcome and I want to see which exposures have bigger impacts on the outcome. So I want to calculate each exposures' odds ratios by exponentiating the coefficients obtained from logistic regression. So I have the following input set and output set where 1 means it (exposure or outcome) is present and 0=not present:

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So, for example, the first row represents a sample where exposure 1 wasn't present, exposure 2 was present,...exposure 20 was present and the outcome was present. I fit a logistic regression model to this data and exponentiate the coefficients to get odds ratios. The potential problem is that I am going to be working with a VERY sparse data set with many samples. There are many instances where almost all exposures except one or maybe two is going to be present in a sample. My question is if this sparsity is something to be concerned about and if this will make my method of comparing exposures using odds ratios a bad idea.

Page 6 of this paper Greenland 1987 seems to imply that sparsity won't matter too much but I want to see what the statisticians here say. Any links to papers would be appreciated.

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Check out the following paper which directly addresses your query and ways to potentially adjust for the issues that the sparsity is likely to cause (i.e., very large ORs etc.).

http://www.bmj.com/content/352/bmj.i1981

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    $\begingroup$ Note that if you use likelihood profile confidence intervals for ORs, these confidence intervals work correctly even with an OR estimate is $\infty$ or 0. $\endgroup$ Jul 14, 2019 at 10:51

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