How can I use a taylor series variance approximation to construct a confidence interval?

Question

I read that the variance of the ratio of counts from a multinomial distribution can be approximated as

$$ var\left[\frac{X_1}{X_2}\right] \approx \frac{1}{N}\left(\frac{p_1}{p_2}\right)^2\left(\frac{1}{p_1} + \frac{1}{p_2}\right) $$

from https://jsdajournal.springeropen.com/articles/10.1186/s40488-018-0083-x.

The method outlined was a taylor series approximation. However, I am wondering if this would be valid for making a confidence interval by by the form:

$$ \frac{X_1}{X_2}\pm z_{1-\alpha/2} \sqrt{\dfrac{Var\left(\frac{X_1}{X_2}\right)}{N}} $$

It seems to result in very narrow widths and so I suspect something is wrong.

Answer 1

1. I suspect that you have mis-applied the formula: you may have accidentally divided by $N$ twice. (The first formula you display already divides by the sample size $N$, and so there should not be a need to divide by $N$ again in your second displayed formula).

2. The authors of the paper you cite discuss in some detail limitations of their approximation and suggest that you run simulations to see if it gives reasonable results in your intended application.