I have a couple things to add together- two different pools of material, and I need to know the total material. Because of various things related to the sampling, I would like to bootstrap the sampling data from each pool. So I generate a list of 100 masses for one pool, and 50 masses for the second pool. Plus, I get the confidence intervals built off the bootstrap. I can just add the mean, to get the mean total mass, but what about the CIs?

I'm still learning bootstrapping, obviously, so sorry if it's a simple question. At first, I figured I could combine them similar to SD's, like sqrt(x^2 + y^2), but that doesn't work. Does simple addition work? Or do I need to build the CI's from scratch using the standard errors from each pool?


1 Answer 1


Correct me if i am wrong but it sounds to me like your problem is simply to get a confidence interval for the sum of two random variables $X$ and $Y$ with $X$ is the total mass from the sample for the first pool and Y is the total mass for the sample from the second pool. I think for this problem you can use either the sum or the average. To get a confidence interval for X+Y you would need to get the sampling distribution of $Z=X+Y$. For a normal distribution a $95\%$ confidence interval for the population mean equates to the

$$\overline{x} \pm 1.96 \cdot s$$

where $\overline{x}$ denotes the sample mean and $s$ is the standard deviation. But this relationship between confidence level does not hold for other distributions. If you assume that $X$ and $Y$ are independent with a known parametric distribution then you can solve the problem without bootstrapping. If you don't you can generate a bootstrap confidence interval for $Z$. Get a bootstrap sample from the sample that generated $X$ and do the same for $Y$. Take the bootstrap estimate for $X$ and add it to the bootstrap estimate for Y to get a bootstrap estimate for $Z$. Repeat this say 10000 times and you will be able to construct a histogram of the $Z$s. The percentile method bootstrap is one simple way to generate approximate confidence intervals. Let $Z_1$ be the value at which $25$ values are at it or below and let $Z_2$ be the values at which $25$ values are at it or above. Then $[Z_1, Z_2]$ is an approximate $95\%$ bootstrap confidence interval for the mean of the distribution for $Z$.

The interval is approximate and there are other types of bootstrap confidence intervals that can be more accurate for the given sample size.

  • 1
    $\begingroup$ So bootstrapping the additive process, rather than bootstrap the individual pools and them add them? That makes sense, I should have thought of that. Lack of sleep must be getting to me... jeez! Thanks. I was using the percentile method for CI's, also. $\endgroup$ May 3, 2012 at 18:25
  • $\begingroup$ I have another question. I built this in R, and it seems to work well. I take a random subsample of the original data for each pool (with replacement, same number of samples as in the original data), add them, and store the result 1000x (or whatever). I then bootstrap those values and generate the final estimate of the mean, CI's, etc. Main question- is this an example of bootstrap aggregation/bagging? $\endgroup$ May 3, 2012 at 22:29
  • $\begingroup$ Nevermind, figured it out. $\endgroup$ May 3, 2012 at 22:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.