Test for a Difference in Two Odds Ratios

Testing for a statistical difference between two odds ratio estimates can be useful in determining if an association has statistically different effects in different groups or strata of a variable.  For example, maybe an association is stronger for older individuals than younger individuals.  To test for such a difference we need the odds ratio estimate (or more precisely the natural log of the odds ratio estimate, aka the beta estimate from a logistic regression) and the standard error of the log odds ratio.

If you don't have access to the primary data and need to estimate the standard error from a 95% confidence interval (95% CI), see this blog entry.  If you have forgotten how to calculate the standard error of the log odds ratio use this formula:
SE(logOR)=1n1+1n2+1n3+1n4−−−−−−−−−−−−−−−√

To test if two odds ratios are significantly different and get a p-value for the difference follow these steps:
(1) Take the absolute value of the difference between the two log odds ratios. We will call this value δ.
(2) Calculate the standard error for δ, SE(δ), using the formula:
SE21+SE22−−−−−−−−−
√

(3) Calculate the Z score for the test: z=δ/SE(δ)
(4) Calculate the p-value from the z score. The p-value can be easily calculated in R or Microsoft Excel using the below formulas.

R: P-value=2*(1-pnorm(Z))
MS Excel: P-value=2*(1-(NORMDIST(Z,0,1,TRUE)))