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The most important bit of theory about nonparametric bootstrap hypothesis tests is that, in general, there ain't any!
The reason is that the empirical F hat
is close to the
true (population) distribution, and may or may not satisfy
the null hypothesis. Thus the bootstrap samples are generally not
simulations from a distribution satisfying the null hypothesis. Hence they
are, in general, completely useless for doing a hypothesis test, because,
whether parametric or nonparametric and whether done analytically or by
simulation, the reference distribution for a hypothesis test must
be a distribution satisfying the null hypothesis. (How else can it be relevant
to rejecting
the null hypothesis?)
A naive person attempting to do a bootstrap test just calculates a P-value as something like
mean(tstat.star > tstat.hat)
where tstat.hat
is the value of the test statistic calculated
for the actual data and tstat.star
is a vector of values of
the test statistic calculated for bootstrap samples.
The resulting test is a valid hypothesis test in the sense that a test with nominal significance level α actually has that significance level.
But (a big but!) this test typically has no power. It rejects the null
hypothesis with probability α regardless of what the alternative is.
No matter how large the deviation of the true parameter value from the
null hypothesis, the naive bootstrap test typically doesn't find
any statistical significance.
Hence the naive bootstrap test proves nothing except
a little knowledge is a dangerous thing.
There are a variety of special situations in which something that makes sense as a nonparametric bootstrap hypothesis test can be done. Efron and Tibshirani, Chapter 16 describe a few. But if you don't see a general principle in their explanation, don't worry. There isn't any.
There is one limited general principle mentioned in passing in their Section 16.4, which we would like to amplify.
Any confidence interval has a hypothesis test dual to it.
Thus if you know how to do a bootstrap confidence interval for some parameter, then you also know how to do hypothesis tests concerning that single parameter.
The decision theoretic view of inverting confidence intervals to get tests is the simplest. Just reject H_{0} if the interval doesn't contain the hypothesized value of the parameter. More precisely, reject H_{0} at level α if the interval with coverage probability 1 - α doesn't cover the hypothesized value of the parameter.
The only tricky part is doing one-tailed tests, which involves the use of one-tailed intervals. But once one sees that, the rest is obvious.
P-values are a bit trickier. Here is one way to describe the P-value corresponding to a confidence interval or, more precisely, to a recipe for creating confidence intervals of any specified coverage probability. The P-value that goes with a confidence interval recipe is the α such that the confidence interval with coverage probability 1 - α has the value of the parameter hypothesized under H_{0} exactly on top of one of the endpoints of the interval.
It is actually a little easier to see this with one-sided confidence intervals and one-tailed tests, because there is only one endpoint of such an interval (the other endpoint is infinity or minus infinity) that we need to adjust to get the endpoint exactly on top of the hypothesized parameter value.
If one likes equal-tailed two-sided intervals, as Efron and Tibshirani do, then the usual rule holds
The two-tailed P-value is twice the lower of the two one-tailed P-values.
So just do both one-tailed tests and double the P-value of the one that is less than one-half.
A side note for the curious, which will not be mentioned again: if one does not have any particular fondness for equal-tailed two-sided intervals, then there is no unambiguous way to calculate P-values for two-tailed tests. One gets a different P-value for each different confidence interval recipe.
The data here are just two variables x
and y
which may or may not be correlated. That's what we test.
for
loop should be familiar
by now. Because there are two variables to sample, we use the usual
trick of sampling indices k
rather than the data x
and y
.
theta.star
is equal to the hypothesized value of the
parameter under the null hypothesis (here zero).
The α such that F^{-1}(α) = 0 is
just F(0), that is, we calculate the probability that
theta.star
is less than or equal to zero, which is how
ltpv
is defined.
1 / nboot
.
So if the P-value should be really, really small (highly, highly
statistically significant) we need a really, really big nboot
.
The bcanon
doesn't produce a theta.star
vector
for us. Nor, for that matter, are its intervals based on percentiles of
the theta.star
distribution, so we can't do the test the
same way as in the preceding example.
However, bcanon
does purport to produce for any alpha
the corresponding confpoint
. So we just ask for a whole lot of
confpoints
corresponding to
the whole range of possible alpha
values and interpolate as before.
bca
should be more
or less obvious. The kcor
function is defined the way
it has to be for bootstrapping two variables. The only example I find
of this isn't for BC_{a} but
for variance stabilized bootstrap t
but the principle is the same.
seq(along = x)
is just a short way to say 1:n
when n
has previously been assigned
the value length(x)
. This just avoids the assignment.
alpha
argument to bcanon
asks for
all confpointson a grid of values ranging from zero to one.
bca.out
. The component
we want is bca.out$confpoints
which is a matrix with two columns
bca.out$confpoints[ , 1]
is the supplied alpha
values.
bca.out$confpoints[ , 2]
is the corresponding BC_{a}
confpoints.
confpointvaries as a function of
alpha
. We want to find the alpha
where the
curve crosses zero, which is the horizontal line in the plot.
approx
command defining ltpv
does this
(approximately).
The same general principle of inverting confidence intervals to find tests applies to any confidence interval whatsoever (nonparametric bootstrap or not).
Annoyingly, if you really want to know the details of inverting some other interval (say ABC), we leave these as an exercise for the reader.
(Of course, change annoyingly
to mercifully
if you have had enough of this subject.)