Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-20 CHAPTER 14 Bootstrap Methods and Permutation Tests
23 requests from customers of a CLEC during the same time period. The distributions
are both far from normal. Here are some summary statistics:
Service provider n ¯x s
Verizon 1664 8.4 14.7
CLEC 23 16.5 19.5
Difference −8.1
The data suggest that repair times may be longer for CLEC customers. The mean
repair time, for example, is almost twice as long for CLEC customers as for Verizon
customers.
In the setting of Example 14.6 we want to estimate the difference of population
means, µ 1 − µ 2 . We are reluctant to use the two-sample t confidence
interval because one of the samples is both small and very skewed. To compute
the bootstrap standard error for the difference in sample means x 1 − x 2 ,
resample separately from the two samples. Each of our 1000 resamples consists
of two group resamples, one of size 1664 drawn with replacement from
the Verizon data and one of size 23 drawn with replacement from the CLEC
data. For each combined resample, compute the statistic x 1 − x 2 . The 1000
differences form the bootstrap distribution. The bootstrap standard error is
the standard deviation of the bootstrap distribution.
S-PLUS automates the proper bootstrap procedure. Here is some of the
S-PLUS output:
Number of Replications: 1000
Summary Statistics:
Observed Mean Bias SE
meanDiff -8.098 -8.251 -0.1534 4.052
CAUTION
Figure 14.9 shows that the bootstrap distribution is not close to normal. It
has a short right tail and a long left tail, so that it is skewed to the left. Because
the bootstrap distribution is nonnormal, we can’t trust the bootstrap t confidence
interval. When the sampling distribution is nonnormal, no method based on
normality is safe. Fortunately, there are more general ways of using the bootstrap
to get confidence intervals that can be safely applied when the bootstrap
distribution is not normal. These methods, which we discuss in Section 14.4,
are the next step in practical use of the bootstrap.
BEYOND THE BASICS
The bootstrap for a
scatterplot smoother
The bootstrap idea can be applied to quite complicated statistical methods,
such as the scatterplot smoother illustrated in Chapter 2 (page 110).