I have an Odds Ratio of a:b ($OR_{ab}$) and I can switch it to the Odds Ratio of b:a by taking the inverse.
$$OR_{ba}=1/OR_{ab}$$
I know the standard error of the original odds ratio ($SE_{ab}$).
How can I calculate the standard error of this new odds ratio ($OR_{ba}$)?
Best Answer
If you just have the odds ratio $\mathrm{OR}_{ab}$ and its corresponding standard error, you can use the delta method to calculate the approximative standard error of the inverse $\mathrm{OR}_{ba}=1/\mathrm{OR}_{ab}$. Let $T$ be a random variable with mean $\theta$. Further, let $g(T)$ be a differentiable function for which we want an estimate of variance. The approximate the variance of $g(T)$ is: $$ \mathrm{Var}(g(T)) \approx g'(\theta)^2\mathrm{Var}(T) $$ In your case we have $T=\mathrm{OR}_{ab}$ and $g(\mathrm{OR}_{ab})= \frac{1}{\mathrm{OR}_{ab}}$. The first derivative is $g'(\mathrm{OR}_{ab}) = -\frac{1}{\mathrm{OR}_{ab}^{2}}$. So the approximative variance is: $$ \mathrm{Var}\left(\dfrac{1}{\mathrm{OR_{ab}}}\right) \approx \left(-\dfrac{1}{\widehat{\mathrm{OR_{ab}}}^{2}}\right)^{2}\cdot\mathrm{Var}({\mathrm{OR_{ab}}}) = \left(\dfrac{1}{\widehat{\mathrm{OR_{ab}}}}\right)^{4}\cdot\mathrm{Var}({\mathrm{OR_{ab}}}) $$ The variance on the right hand side is unknown. But the squared standard error of $\mathrm{OR}_{ab}$ is an estimate of it. So to get an estimate of the standard error $\mathrm{OR}_{ba}$, plug in the squared standard error $\mathrm{SE}(\mathrm{OR}_{ab})^2$ as well as the estimate of the odds ratio $\widehat{\mathrm{OR}}_{ab}$. Then take the square root to get an estimate of the standard error of the inverse odds ratio $\mathrm{OR}_{ba}$.