The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

5.5 Bootstrap Principle and Bootstrap Confi dence Intervals 67

quite different from most nonparametric approaches that differ from the

bootstrap because these nonparametric tests are based solely on rank-

ings, while the bootstrap uses the actual values. Similar to parametric

procedures, which require pivotal quantities, the bootstrap appears to

function best when an asymptotically pivotal quantity can be used.

Recall that the difference between bootstrap sampling and simple

random sampling is that

1. Instead of sampling from a population the bootstrap samples
   from the original sample.


2. Sampling is done with replacement instead of without
   replacement.

Bootstrap sampling behaves in a similar way to random sampling in
that each sample is a random sample of size n taken from the empirical
distribution function F n , which gives each observation an equal chance
each draw, while simple random sampling is sampling from a popula-
tion distribution F (fi nite in population size N ), but for which, uncon-
ditionally on each draw, each observation has the same chance of
selection, and for the overall sample of size n , every distinct sample
has the same chance 1/CnN, where CnN is the number of ways n objects
can be selected out of N as defi ned in Chapter 1.
The bootstrap principle is very simple. We want to draw inference
about a population based on the sample without make extraneous
unverifi able assumptions. So we consider sampling with replacement
from the empirical distribution F n. It is a way to mimic the sampling
process. Like actors in a play, the empirical distribution acts the part
of the population distribution. Sampling with replacement produces a
bootstrap sample that plays the role of the original sample. Repeating
the process (like performing a play over again) acts like what repeated
sampling of size n from the population would be. Generating bootstrap
samples is like simulating the sampling process.
We now illustrate the simplest bootstrap confi dence interval, called
Efron ’ s percentile method, which is obtained by generating a histogram
of bootstrap estimates of the parameter and using the appropriate per-
centiles to form the confi dence interval. We consider an example taken
from Chernick and Friis ( 2003 ).
In this experiment, a pharmaceutical company wants to market a
new blood - clotting agent that will minimize blood loss during surgery