2.4 Other Sampling Methods 27accomplished easily by using the approach of the rejection method
without the need to reject, since repeats are acceptable.
The basic bootstrap idea is to let the original sample data serve as
the population, and then you sample with replacement from the sample
data. The original sample is size n , and bootstrapping is usually done
by taking m samples with replacement with m = n.
However, in recent years, it has been discovered that although
m = n usually works best, there are situations where the choice of m = n
leads to a particular type of incorrect solution which statisticians call
inconsistency. In several of these cases, a consistent bootstrap approach
can be obtained by making m < < n. * This is called the m - out - of - n
bootstrap. For the statistical theory of consistency to hold, both n and
m tend to infi nity, but with m going at a slower rate. So we will see
that bootstrap sampling is conceptually very similar to simple random
sampling. The only difference is that replacement is used for the boot-
strap. For the bootstrap estimates, we will also look at the ages of the
six patients.
2.4 OTHER SAMPLING METHODS
Stratifi ed random sampling is just a little more complicated; the simple
random sampling in a set of m strata are defi ned indexed by k where
k = 1, 2,... , m , and each strata gets a simple random sample of size
n k. An example of stratifi cation might be age group, with k = 1 for ages
1 – 12, k = 2 for ages 13 – 20, k = 3 for ages 21 – 35, k = 4 for ages 36 – 55,
k = 5 for ages 56 – 75, and k = 6 for anyone over 75. A stratifi ed random
sample can work better than a simple random sample if each stratum
has a relatively homogeneous group, but there are marked differences
between strata.
Other forms of sampling are convenience sampling, cluster sam-
pling, and systematic sampling. Cluster sampling is a random sampling
approach that is used when it is easier to randomly select a group of
elements for a sample rather than the individual elements themselves.
Examples could be lists of districts or counties within a state. In cluster
sampling, the item being sampled is a cluster. A cluster is a group of
objects generally found in the same location. For example, in sampling
* By “ m << n , ” we mean that m is much less than n.