Sorry everyone, I think Efron & Tibshirani's book covers my
problem, I just hadn't read far enough. A little learning is a dangerous
thing,
or in my case a dangerously nonretentive memory is a constant hindrance.
My first algorithm does give a bootstrap significance value and confidence
interval:
On p88/89 of Efron & Tibshirani, in dealing with the two sample means
problem,
they say to sample the T*'s from the T's and the U*'s separately from the
U's.
This is because
F -> t1,...,tn independently of G -> u1,...,un
where F and G are the unknown pdfs. So P -> x = (t,u) is best estimated
by Fhat -> t* independently of Ghat -> u*.
However, on page 202 they use the same data set as on p89
to demonstrate a permutation test, which is exactly the algorithm suggested
by my permutational colleague.
The text makes clear that which I had overlooked, that the first one is
testing a
hypothesis of location, the second one is testing a more general hypothesis
that F = G.
Also that bootstrap histograms are centred on Rhat, while permutation
histograms
are centered on 1. So the second algorithm is testing how far the observed
ratio is from 1, while the bootstrap test measures how far 1 is from Rhat.
I'd still enjoy hearing from anyone using bootstrap in the context of ratio
estimators.
Thanks very much for your help
Graeme
--
original message:
Hello everyone
This is probably a really silly question, but I would greatly
appreciate anyone's thoughts as input.
I am trying to use the bootstrap to test hypotheses and generate
confidence intervals about a ratio estimator. The scientific
question is to determine whether or not a compound causes
upregulation , or downregulation, of some genetic material.
If T1, ..., Tn are n samples of treated genetic material,
and U1,..., Un are n samples of untreated genetic material,
then in my naive way I thought that calculating
R = Tbar/Ubar
would be a useful test statistic (Tbar = mean(T1,...,Tn), etc).
I also thought that using the bootstrap to estimate the
sampling dn of R would be straightforward. I used the following
algorithm:
for b in 1:B {
sample at random with replacement from T1,...,Tn -> T1*,...,Tn* ->
Tbar*
sample at random with replacement from U1,...,Un -> U1*,...,Un* ->
Ubar*
calculate R*(b) = Tbar*/Ubar*
}
A confidence interval for the true ratio can be estimated using the
quantiles
of R*(b). I then used an estimated 95% CI to carry out a significance test:
if 1 belongs to the interval, then a null hypothesis of "no up or down
regulation" cannot be rejected at the 5% level.
Aha said a colleague, you've been thinking backwards. Start with the null
hypothesis:
H0: there is no up or down reguation, ie R=1.
If H0 is true, then the placement of the n T values and the n U values
are irrelevant. In other words, while the test statistic R*(b) is OK, the
sampling
should have been different:
for b in 1:B {
sample 2n values at random with replacement from T1,...,Tn,U1,...,Un
call the first n T1*,...,Tn*
call the second n U1*,...,Un*
calculate R*(b) as before
}
Then the approximate p-value for the test is given by the frequency with
which R*(b) is smaller than the observed value.
Both arguments seem compelling to me. The second algorithm is undoubtedly
a bootstrap hypothesis test; but could the bootstrap dn be used to make a
CI for
the true R? I don't think so.
The first algorithm seems to me to make such an interval (especially since
T and U are independent random variables). But is the hypothesis test
implied
by the interval wrong headed?
As I say, any thoughts very welcome. Thanks for your help.
Graeme
--
Dr Graeme Archer
Statistical Sciences, Smithkline Beecham Pharmaceuticals,
Harlow, Essex UK.
Tel 01279 622 181
Email [log in to unmask]
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