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I have a large dataset that is weighted due to:

  1. probability proportional to size sampling, &
  2. disproportionate stratification.

I would like to compare the means for two different domains within the same sample, using the weighted means and variance estimates. I believe I could proceed using the Delta approximation of variance of a function of random variables: $$ {\rm V}(\mu_1 - \mu_2) = {\rm V}(\mu_1) + {\rm V}(\mu_2) - 2{\rm Cov}(\mu_1, \mu_2) $$ but I cannot find any precedent, nor a different solution, to this problem in the literature. Nor could I find any other questions on this forum that address this specific problem.

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The formula you gave is not an approximation. It is the exact formula for the variance of the difference between two random variables. – Michael Chernick Aug 22 '12 at 19:04
Do the domains overlap, or are they mutually exclusive? What software do you use? People tend to give R answers by default; if you are using something else, you'd want to specify that explicitly. – StasK Aug 22 '12 at 23:33

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