# Combining two Odds ratios with the same control group

I am doing a meta-analysis of observational studies. As the disease I am working on has different stages, some of the studies reported ORs (95%CI) for each stage separately in comparison to a single (shared) control population. I wonder if we can combine these two ORs (and their 95%CI) given that they reflect two case groups but a single control group. Some other info:

• The sample size for each case group is different.
• We have the sample size for each of the groups but not the frequency of exposure (otherwise we could use just the raw data).
• Please note that I have no problem combining the ORs, but simply combining two ORs without taking into account that they both have the same control group will cause narrowing of the 95%CI and overweighting of the study in the analysis.

Any ideas?

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## 1 Answer

The chapter on stochastically dependent effect sizes by Gleser and Olkin (2009) in The handbook of research synthesis and meta-analysis (2nd ed.) describes how multiple-treatment and multiple-endpoint studies can be meta-analyzed. Your case can be covered by the "multiple-treatment" part -- that is, you can look at the two disease stages as two different 'treatments' that are both compared against a common control group. In essence, you will have to compute the covariance between the two (log) ORs and take that into consideration when you want to include the two ORs in your meta-analysis.

If you work with R, you may find the metafor package useful. On the metafor package website, you can find code that will reproduce the methods described in the Gleser and Olkin chapter: http://www.metafor-project.org/doku.php/analyses:gleser2009. That should help to get you started.

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