Statistical Methods in Medical Research 4ed

The bootstrapÐbasic ideas and standard errors
Most statistical analyses attempt to make inferences about a population on the
basis of a sample drawn from that population. The distribution function F of a
variable x in the population is almost always unknown. Analyses usually proceed
by focusing attention on specific summaries of F, such as the mean mF, which are, of course, also unknown (the subscript F is used to emphasize the
dependence of the mean on the underlying distribution). Most analyses estimate
parameters such as mF from random samples drawn from the population, using
estimators that have usually been constructed in a sensible way and which have
been found to have good properties (e.g. minimum variance, unbiased, etc.).
Naturally the values of the estimates will vary from sample to sample, i.e. the
estimates are affected by sampling variation (see §4.1). This source of variability
is usually quantified through the relevant standard error, which for the mean is
simply sF= n
p , or more precisely through the estimate of the standard error,
s= n
p .
The bootstrap approaches this problem from a rather different perspective.
For the problem of estimating a mean and its associated standard error, it
essentially reproduces the usual estimators. However, the new perspective permits
estimation in a very wide variety of situations, including many that have
proved intractable for conventional methods. Rather than trying to make inferences
about F by estimating summaries such as the mean and variance, the
bootstrap approach estimates F directly and then estimates any relevant summaries
through the estimated F. Various estimates, ^F,ofFare possible and some
of these will be discussed briefly below. However, the most general form and the
only one that we consider in detail is the empirical distribution function (EDF),
which is a discrete distribution that gives probability 1=n to each sample point
xi…i ˆ 1, ..., n†. Using the formula for the expectation of a probability distribution
(3.6) the mean of the EDF is m^F , which is
m^F
ˆ 1
n
x1 ‡ 1
n
10.7 The bootstrap and the jackknife 299
x2 ‡ ...‡ 1
n
i.e. m^F ˆ x, the sample mean, which is the usual estimator of mF. The same type
of argument gives the variance of ^F as s2 ^F ˆ n 1 P …xi x† 2 , which again is the
usual estimator, apart from the denominator being n rather than n 1. This
discrepancy arises because xisthe mean of ^F, rather than merely an estimator of
it. The bootstrap estimate of the standard error of the mean is then s^F = n
p ,
which is s= n
p p
but for the factor ‰n=…n 1†Š.
The bootstrap has provided estimators of the mean and its standard error
that are virtually the same as the conventional ones but has done so by first
estimating the distribution function. Because ^F is a known function, properties
such as its mean and variance can be computed and these used as estimators.
xn