[R] p-values from bootstrap - what am I not understanding?

[Peter Dalgaard]

Johan Jackson wrote:
> Dear stats experts:
> Me and my little brain must be missing something regarding bootstrapping. I
> understand how to get a 95%CI and how to hypothesis test using bootstrapping
> (e.g., reject or not the null). However, I'd also like to get a p-value from
> it, and to me this seems simple, but it seems no-one does what I would like
> to do to get a p-value, which suggests I'm not understanding something.
> Rather, it seems that when people want a p-value using resampling methods,
> they immediately jump to permutation testing (e.g., destroying dependencies
> so as to create a null distribution). SO - here's my thought on getting a
> p-value by bootstrapping. Could someone tell me what is wrong with my
> approach? Thanks:
> 
> STEPS TO GETTING P-VALUES FROM BOOTSTRAPPING - PROBABLY WRONG:
> 
> 1) sample B times with replacement, figure out theta* (your statistic of
> interest). B is large (> 1000)
> 
> 2) get the distribution of theta*
> 
> 3) the mean of theta* is generally near your observed theta. In the same way
> that we use non-centrality parameters in other situations, move the
> distribution of theta* such that the distribution is centered around the
> value corresponding to your null hypothesis (e.g., make the distribution
> have a mean theta = 0)
> 
> 4) Two methods for finding 2-tailed p-values (assuming here that your
> observed theta is above the null value):
> Method 1: find the percent of recentered theta*'s that are above your
> observed theta. p-value = 2 * this percent
> Method 2: find the percent of recentered theta*'s that are above the
> absolute value of your observed value. This is your p-value.
> 
> So this seems simple. But I can't find people discussing this. So I'm
> thinking I'm wrong. Could someone explain where I've gone wrong?


There's nothing particularly wrong about this line of reasoning, or at 
least not (much) worse than the calculation of CI. After all, one 
definition of a CI at level 1-alpha is that it contains values of theta0 
for which the hypothesis theta=theta0 is accepted at level alpha. (Not 
the only possible definition, though.)

The crucial bit in both cases is the assumption of approximate 
translation invariance, which holds asymptotically, but maybe not well 
enough in small samples.

There are some braintwisters connected with the bootstrap; e.g., if the 
bootstrap distribution is skewed to the right, should the CI be skewed 
to the right or to the left? The answer is that it cannot be decided 
based on the distribution of theta* alone since that depends only on the 
true theta, and we need to know what the distribution would have been 
had a different theta been the true one.

The point is that these things get tricky, so most people head for the 
safe haven of permutation testing, where it is rather more easy to feel 
that you know what you are doing.

For a rather different approach, you might want to look into the theory 
of empirical likelihood (book by Art Owen, or just Google it).

-- 
    O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
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