Daniel,
That is an interesting question that crops up with any form of
correlated tests, and here the tests are definitely correlated. My crude
approach is to think -> these are estimates of the same thing, namely a
sigma^2*Chi2 under Ho, so why not average them to get a representative value
and use that for calculating the test. Because of the correlation one cannot
assume a gain in precision by the averaging process, at least, not by a factor
of 1/#imputations as if they were independent. I suppose one could average, or
take the median p-value as, again, being representative.
I've thought about Fisher's note that under Ho (so p~uniform) the
distribution of -2ln(p) is distributed as chi2(df=2), but that does not really
help as we are again back to the correlations between the chi2(2).
One could work with the p-values, but putting them together requires
the conditional Pr(Data1|Ho,imputation 1)=p1, and Pr(Data2|Ho,imputation2)=p2,
but Pr(Data1 & Data2|H0,imputations 1 and 2) is NOT p1*p2 but the conditional
combination p1*Pr(Data2|Ho,imputation2,imputation1).
So I end up with using a representative value for the Chi2.
Peter
------------------------------
Peter J Hannan
Senior Research Fellow
Division of Epidemiology, SPH
University of Minnesota
1300 South Second St. #300
Minneapolis MN 55454-1015
(612)-624-6542 voice
(612)-624-0315 FAX
[email protected]
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