Citation for Statistical test for difference between two odds ratios?

In a comment here, @gung wrote,

I believe they can overlap a little (maybe ~25%) & still be significant at the 5% level. Remember that the 95% CI you see is for the individual OR, but the test of 2 ORs is about the difference between them. However, if they don't overlap at all, then they are definitely significantly different, & if the 95% CI's overlap the other OR point estimate, they definitely don't.

Does anyone have citations for the above statement? A reviewer wants me to calculate if two odds ratios are significantly different to each other.

• Why not just calculate the significance of the difference between two odds ratios directly? Why would you wan to measure the overlap of the 95% CIs & try to get the significance from that? Mar 25 '15 at 1:18
• What is the equation to do this? Mar 25 '15 at 15:30
• To test for the difference of two odds ratios? Do you know the odds ratios & the Ns they are based on? Do you have access to the original data? Mar 25 '15 at 15:38
• Yes, it was a multilevel logistic regression (the bernoulli option using the HLM software). SO I have the ORs and the Ns from that analysis. Mar 25 '15 at 15:46
• The output from the analysis should tell you if they are significantly different, or you should be able to get your software to give that to you by adding some option. Do you have the SEs for the ORs? Are they independent, or do you have an estimate of the covariance of their sampling distributions? Mar 25 '15 at 15:54

From your two logistic regression models, you should have parameter estimates, $\hat\beta_{11}$ and $\hat\beta_{12}$ (where the second subscript refers to the model), and their standard errors. Note that these are on the scale of the log odds and that this is better—there is no need to convert them to odds ratios. If your $N$s are sufficient, these will be normally distributed, as @ssdecontrol explained. The Wald tests that come standard with logistic regression output assume they are normally distributed, for example. In addition, since they came from different models with different data, we can treat them as independent. If you want to test if they are equal, this is simply testing a linear combination of normally distributed parameter estimates, which is a pretty standard thing to do. You can calculate a test statistic as follows:
$$Z = \frac{\hat\beta_{12}-\hat\beta_{11}}{\sqrt{SE(\hat\beta_{12})^2 + SE(\hat\beta_{11})^2}}$$ The resulting $Z$ statistic can be compared to a standard normal distribution to compute the $p$-value.