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Does anyone know a formula for pooling odds ratios across studies? I'm trying to do a quick pen and paper pooling of different numbers of odds ratios from different studies where I don't have the original data, only the odds ratios and sometimes confidence intervals.

I'm trying to determine whether four things (race, socioeconomic status, smoking, and drug use) are affecting a certain response (disease incidence).

A paper I've read about it says that each of the four things is positively associated with the response and then lists several odds ratios greater than 1.00 found in the literature, as well as 95% confidence intervals for the odds ratios. For example, for race, the odds ratios with 95% confidence intervals are 1.45 (1.10, 1.80), 3.5 (1.0, 6.0), 2.33 (2.03, 2.66), and 1.98 (1.5, 2.48).

I'm aware that multiple comparisons - not multiple odds ratios for each of the four independent variables but the comparisons of the four independent variables here separately and not all at once - being a problem. They being compared for their effects on the disease incidence, there are four effects, and four comparisons will inflate type I error. The problem is that I'm really not sure how to clean up this information that I have. I keep wondering if there is any way to, say, pool odds ratios? For example, I calculated new odds ratios since I can calculate the n's using the confidence intervals, but I still have four now. What about an overall ratio? And I don't even think my cleaned up odds are Kosher since I'm using a bunch of different studies. Therefore, I'm having a lot of trouble getting anywhere with this. Anyone have any ideas?

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    $\begingroup$ I'm not clear on what you're trying to pool. You have multiple studies for each of the 4 predictors, yes? Are trying to pool them to get 4 different odds ratios, one for each predictor? Or, are you trying to combine the 4 predictors into one overall odds ratio for a person who exhibits all 4 of the traits? A useful search term for the former is "meta-analysis". The latter is probably not feasible, although it's outside my area of expertise. See also stats.stackexchange.com/questions/371917/… $\endgroup$
    – David Luke Thiessen
    Jun 11, 2022 at 13:47
  • $\begingroup$ I'm trying to do a meta-analysis on the four odds ratios, for race, socioeconomic status, drug use, and smoking. I'm having trouble with it because I'm starting with several odds ratios and confidence intervals (from several studies) for each of those 4 explanatory variables. I'm addressing a report from another author who's flooding me with subgroup analyses and I'm trying to pool a response, if that makes any sense. $\endgroup$
    – Jamie Watts
    Jun 11, 2022 at 15:20
  • $\begingroup$ I'm aware of strong associations between the explanatory variables. The opposing author is implying that four factors are contributing to disease incidence but it's clear that confounding is present and could explain the high odds ratios in all four factors. $\endgroup$
    – Jamie Watts
    Jun 11, 2022 at 18:34

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