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Re: st: RE: statalist-digest V4 #965
At 11:23 04/08/02 -0400, Stephen Soldz wrote:
I don't think there is any need for a reference, as the point is so
trivial. If you are estimating the difference between 2 population
proportions from 2 different sample proportions on the same sample, then
you are estimating the mean of Z=X-Y, where X and Y are Bernoulli
variables. You are therefore simply estimating the population mean Z from
the sample mean Z. The large sample theory applies, courtesy of the central
limit theorem for ordinary sample means, whether Z is normal (as with the
usual 2-sample t-test) or a discrete distribution with possible values -1,
0 and 1 (as here).
thanks to Nick Cox and Roger Newson for their responses to my question about
robust tests of dependent proportions. Nick gave several references I'll
look up. Roger thinks I wouldn't do to bad with paired t-tests as they are:
"a special case of the Huber variance for clustered data (where the clusters
are the pairs of responses and the observations are the individual
responses)". I wonder if you have a refernce for this I could cite?
The bit about clustered Huber variances is probably not strictly necessary,
but is justified as follows. The conventional SE of the sample mean happens
also to be the Huber SE for estimating the population mean, if you are
using any likelihood function which uses the sample mean as the
maximum-likelihood estimator for the population mean (which includes the
normal likelihood function, and includes also the discrete-distribution
likelihood function with possible values -1, 0 and 1). This is because the
Huber variance is, by definition, the sample mean square of the sample
influence function divided by the number of sampling units. The sample
influence function of the mean, for the i'th sampling unit, is Z_i-Zbar,
where Z_i as the i'th Z-value and Zbar is the sample mean Z-value. A good
reference on influence functions in general is Hampel (1974).
I hope this helps.
Hampel FR. The influence curve and its role in robust estimation. Journal
of the American Statistical Association 1974; 69: 383-397.
Lecturer in Medical Statistics
Department of Public Health Sciences
King's College London
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Opinions expressed are those of the author, not the institution.
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