AW: st: Resampling and compare full sample with subsamples

[Johannes Thrul <Thrul@ift.de>]

Thank you Steve and sorry for the delayed response. 

Could you do me a favor and explain briefly, why you would prefer confidence intervals over hypothesis testing in this case?

Thanks, Johannes





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Ah, you left out most of the detail; your explanation makes sense. To
answer your original question. You want to compare a part A to a whole C
But C = A U B, where B is the observations in C that are not in A. Let
pA and pB be the prevalnce rates in A and B and pW be the prevalence in
the whole. Then if nA, nB, and n are the sample sizes of A,B, and

(*) pC = W pA + (1 - W) pB where W = nA/n.

(**) pC - pA = (1-W)(pB - pA).

A one-sample test comparing A to C is not correct, C is itself a random
sample and pC and pA are correlated. as A is a SRS random sample of C
without replacement, B is also a SRS, pA and pB are slightly negatively
correlated becaus because of (*)

If pA and pB are different, then pA and pC are different (and
vice-versa). Looking at (**) you can see that the proper test is a
*two-sample* test that compares pA and pB. The standard error is
computed under the null hypothesis, and without a finite population
correction. (Cochran, 1977, problem 2.16, p. 48). I myself think that
confidence intervals are preferable to hypothesis tests here.

Reference: Cochran, W. G. (1977). Sampling techniques (3rd ed.). New
York: Wiley.


Steve
sjsamuels@gmail.com

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