Statistical Methods in Medical Research 4ed

espectively. For a 100…1 2a†% interval the limits would be the aBth and
…1 a†Bth largest values. This is known as the percentile method.
It turns out that this kind of interval does not perform all that well: for
example, the coverage probability can stray from the nominal value. A refinement
of the method, known as the bias-corrected and accelerated method, BCa
for short, has better properties. The method still uses the ordered set of the
g 1 , g 2 , ..., g B but chooses the a1Bth and a2Bth largest values for the limits of the
interval. The values a1 and a2 are defined as
w ‡ z
a1 ˆ F w ‡
…a†
1 a…w ‡ z…a† †
…1 a† w ‡ z
a2 ˆ F w ‡
1 a…w ‡ z…1 a† †
…10:15†
…10:16†
and here z…j† is determined by j ˆ F…z…j† †, where F is the standard normal
distribution function. Note that this differs from the definition of a quantile of
the standard normal variable, zj, given in §4.2: the two are related by
z2j ˆ z…j† , j < 1
2 . The quantity w corrects for bias in the distribution of the
statistic. The quantity a, known as the acceleration, is related to the behaviour
of the variance of the statistic: further details can be found in Efron and
Tibshirani (1993, Chapters 14 and 22) and Davison and Hinkley (1997, Chapter
5). If these are both zero, then a1 ˆ a and a2 ˆ 1 a and the BCa method
reduces to the percentile method.
The estimate of w is easily obtained: F…w† is the proportion of the bootstrap
values of g that are less than the observed value. The estimation of a is potentially
more complicated: for a single sample of size n, suppose that g i is the value
of g obtained from the sample with the ith point omitted. If g is the mean of
g , g , ..., g 1 2 n and
Mk ˆ Pn
iˆ1
… g g i†
k
then a ˆ M3=…6M 3=2
2 †. For more complicated data structures, Davison and
Hinkley (1997, Chapter 5) should be consulted.
Example 10.9, continued
10.7 The bootstrap and the jackknife 303
A 95% confidence interval for P…X < Y† can be found by generating 2000 bootstrap
samples and for each one computing the value of D. The 50th and 1950th largest values
are 0 546 and 0 808 and this defines the 95% percentile bootstrap confidence interval for
P…X < Y†.
The BCa interval can also be computed. The proportion of the 2000 values of D that
are less than the observed value, 0 678, is 0 48, giving w ˆ 0 0502. Computation of a is