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I have estimated two values: $X$ & $Y$ each of which have a standard error, $s_X$ and $s_Y$. I have assumed Normality for the construction of confidence intervals.

In an ideal situation, these two values should be the same. I'm running simulations to find out when they are the same or when they are different.

Previously, I have found their difference $W = X-Y$ and $s_W = \sqrt{s_X^2 + s_Y^2}$.

However, these values were inappropriate for the next steps of evaluation and so it would be better to find their ratio so I now have $Z = X/Y$ (I know that $Y \neq 0$).

How do I calculate the standard error of $Z$, $s_Z$? Or how would I find a confidence interval?

Michael Barrowman
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  • This is a special (easy) case of the situation where $X$ and $Y$ are correlated. See https://stats.stackexchange.com/questions/474676/how-to-combine-standard-errors-for-correlated-variables/474685#474685. – whuber Aug 20 '20 at 21:10

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From the Stata documentation of the ratio command:

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A 95% CI can be constructed by $$\hat R\pm 1.96 \cdot \sqrt{\hat Var(\hat R)},$$ where the SE of the ratio is $\sqrt{\hat Var(\hat R)}$.

dimitriy
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  • What if the Covariances are unknown? – Michael Barrowman Aug 20 '20 at 20:50
  • There should only be one covariance here. You can either estimate it, assume it is zero (but only if that makes sense), or do some bounding exercise about how big/small it has to be for your CI to exclude zero (or whatever number you care about). – dimitriy Aug 20 '20 at 20:55
  • You assumed the covariance away for the SE of the difference, so perhaps that applies here as well. But take a look [here](https://en.wikipedia.org/wiki/Variance#Weighted_sum_of_variables) for the general formula, of which yours is a special case. – dimitriy Aug 20 '20 at 21:03
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    The Stata documentation agrees with our thread at https://stats.stackexchange.com/questions/474676/how-to-combine-standard-errors-for-correlated-variables/474685#474685, which also shows how the solution can be obtained. – whuber Aug 20 '20 at 21:12