Delta method for ratio metrics I have the following issue: I would like to do a power analysis (find the right sample size) for a ratio metric ($Z = \frac{X}{Y}$). The in-house statistical software I inherited uses a delta correction or delta method for that and before blindly using that I would rather like to understand what this method does. I understand it is somehow related to the variable $Z$ not being normally distributed and/or $X$ and $Y$ being correlated.
More concretely I would like to understand what happens if I don't use the delta correction, in what way the sample size estimation may be off if I don't use it and in how far it compensates for the (presumed) violation of the prerequisites. Also, I would be interested in whether there is something I can test the distribution against which would allow me to omit the delta correction.

The $\delta$-method is an elementary statistical result for calculating the asymptotic standard error of a smooth function of random variables. A ratio such as $X/Y$ is smooth provided 0 is not in the sample space of $Y$ ($Y$ must be strictly positive or strictly negative). The $\delta$-method has nothing to do with the actual normality of the variables since it is an asymptotic result. If $\bar{X}$ is asymptotically normal and $\bar{Y}$ is asymptotically normal (with a very low probability of achieving 0) then their ratio $\overline{X/Y}$ is asymptotically normal with a variance given by the quadratic form defined by the covariance matrix of $X,Y$ and the Jacobian.
