Delta Method to calculate the standard error of ratio in an AB testing context In AB testing context, if we have a control group and test group (2 groups), and I'd like to calculate the relative difference (Mean test/ Mean control -1) and the confidence interval of this ratio difference, I would need to calculate the standard error for the ratio difference.
I've seen a calculation of the standard error to be:
$$
\sqrt{\left [ SE^{2}_{test} + \left ( Mean_{test}/Mean_{control} \right )^{2} * SE_{control}^{2} \right ]/Mean_{control}^{2}}
$$
But I can't seem to find information on how this is derived. Does any one have pointers?

The image from this question might clarify the Delta method for a ratio of distributions

In this case the variables have $\mu_X = \mu_Y = 1$ and you get the approximation
$$\frac{X}{Y} \approx 1 + X - Y$$
We could use this result and generalize it by using the parameterization $X = U/\mu_U$ and $Y = V/\mu_V$,then we get
$$\frac{U/\mu_U}{V/\mu_V} \approx 1 + U/\mu_U - V/\mu_V$$
or
$$\frac{U}{V} \approx \mu_U/\mu_V +  \frac{1}{\mu_V} (U - V\frac{\mu_U}{\mu_V}) $$
then
$$\sigma_{U/V} \approx   \frac{1}{\mu_V} \sqrt {\sigma_U^2 + \frac{\mu_U^2}{\mu_V^2}\sigma_V^2}$$
