Solved – Calculate of standard deviation for an Odds Ratio by standard deviations of probabilities

Question

Given the formula $OR=\frac{P_1(1-P_2)}{P_2(1-P_1)}$ for calculating the odds ratio ($OR$). How can I calculate the standard error of $OR$ based on the standard errors of $P_1$ and $P_2$?

Answer 1

If you have the cell counts you can get the standard error for the $\ln(OR)$ (see for exampel here) :

$\sqrt{\frac{1}{n_{11}}+\frac{1}{n_{10}}+\frac{1}{n_{01}}+\frac{1}{n_{00}}}$

Using the delta method, you can approximate the standard error of the $OR$ as:

$OR \times \sqrt{\frac{1}{n_{11}}+\frac{1}{n_{10}}+\frac{1}{n_{01}}+\frac{1}{n_{00}}}$