Ratio between two values with confidence intervals I have two integer values, each with confidence intervals, and I want to calculate the ratio between the two values.
My strategy would be to find all the pairs of integer values within the CIs and then calculate the ratio between them, so as to find the distribution of the ratio between the two values.
I can then calculate the average value, which I can interpret as the average value of the ratio between the two values with their CIs.
My questions are:

*

*Is this strategy sounding or are there any issues?

*Can I interpret the minimum and maximum ratio value I identified with the previous procedure as the Confidence Intervals of my ratio?

*In the case the previous one is true, and considering that the starting confidence levels are 95%, is 95% also the confidence level of the ratio?

R code for a simple simulation:
# simulate data
# x = 250 ± 100
# y = 200 ± 50
x <- seq((250-100), (250+100), by=1)
y <- seq((200-50), (200+50), by=1)

# ratio between all combinations of values
comb <- expand.grid(x, y)
ratio <- comb$Var1/comb$Var2

# summary and histogram
summary(ratio)
hist(ratio)


 A: I do not think that equidistant sampling of values from the confidence intervals produces representative samples, because the probability distribution inside the intervals is not taken into account.
There are, however, two possible sampling methods from measured (empirical) data that take the distribution into account:

*

*The Bootstrap, which draws samples with replacement. Instead of merely counting quantiles of the bootstrap samples ("percentile bootstrap"), it is generally more advisable to use the $BC_a$ bootstrap ("bias corrected accelerated bootstrap").

*The Jackknife, which cyclically omits one sample and then estimates the variance from the $n$ obtained values $\theta_{(i)}$ as
$$\sigma_{JK}(\hat{\theta}) = \sqrt{\frac{n-1}{n}\sum_{i=1}^n (\theta_{(i)}-\theta_{(.)})^2} \quad
 \mbox{ with }\quad \theta_{(.)}=\frac{1}{n}\sum_{i=1}^n\theta_{(i)}$$
The Bootstrap directly yields a non-parametric confidence interval, whilst the Jackknife yields a parametric confidence interval as $z_{1-\alpha/2}\cdot\sigma_{JK}$.
Note that for the specific problem of a ratio of two variables, there is yet another method recently developed by Donner & Zhou named MOVER-R (published in 2012). There is even a readily available R implementation in the package pairwiseCI:
https://rdrr.io/cran/pairwiseCI/man/MOVERR.html
