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:
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:
(3) Calculate the Z score for the test:
(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.
MS Excel: P-value=2*(1-(NORMDIST(Z,0,1,TRUE)))
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