>I need to perform the two-sample paired t-test where the samples to
>be compared are the mixtures of normals sharing the same mean but
>differing by their variances.
>
>More specifically, suppose that the sample X contains n1 terms drawn
>from N(mu, sigma1), n2 terms drawn from N(mu, sigma2), etc. The
>sample Y has the same structure. I need to test the null hypothesis
>of equality of means, mu.X=mu.Y, and assign a p-value. Obviously, the
>X & Y are not normally distributed, and t-statistic is not
>t-distributed. Is there a general approach for performing something
>like t-test in this setting? Note: it is not a Behrens-Fisher
>problem, it is something different. Any ideas/references/software
>implementations are highly appreciated.
Are n1, n2, ... fixed?
If the null hypothesis is correct, then do the terms in Y have
the same variances as the corresponding terms in X?
If the answer to both questions is yes, then I suggest using a
a stratified permutation test. This does not require that the
strata sizes in X and Y be identical.
If you have a data frame with variables "values", "treatment", "strata"
where values contains the numerical values, treatment has two levels
indicating sample X or Y, and strata = j for the observations
with variance sigmaj, then
permutationTestMeans(data, treatment = treatment, group = strata)
would do it. Example using artificial data is below.
This requires S+Resample; see the bottom of this message.
You could also use a different statistic; rather than a simple
average of the values within each treatment, take a weighted
mean of the strata means, with weights on the strata means
proportional to stratum size / estimated stratum variance.
The permutationTestMeans function doesn't support this, but the
more general permutationTest2 funtion would.
> set.seed(0)
> data _ data.frame(runif(60), treatment=rep(1:2, 30),
+ strata = sample(1:3, size=60, replace=T))
> permutationTestMeans(data, treatment = treatment, group = strata)
Call:
permutationTestMeans(data = data, treatment = treatment, group = strata)
Number of Replications: 999
Summary Statistics:
Observed Mean SE alternative p.value
X1 0.04311 0.0004573 0.076 two.sided 0.578
P-values for each variable and stratum:
alternative 1 2 3
X1 two.sided 0.08 0.396 0.818
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