Cluster Sampling

1

• Cluster Sampling
• Basic ideas
• A population can often be grouped in
  clusters. It will be more convenient and less
  expensive to sample in clusters than
  individually.
• Cluster sampling is only practical way to
  sample in many situations.
• Geographic clusters are often used in
  community surveys.
•   Time clusters are often used in surveying
  activities in progress.
• Cluster sampling in general results in larger
  sampling error than simple random sampling. As
  the cluster gets larger, the sampling error
  increases.

2

• Choice of clusters
• Clusters should be well defined, i.e., every
  element of the population must belong to one and
  only one cluster.
• The number of elements (cluster size) or at
  least a reasonable estimate of its size should
  be known.
• Sufficiently small clusters are preferred, so
  that the number of clusters is sufficiently
  large to facilitate sampling.
• Clusters should be formed to minimize the
  intra-class correlation.
• Heterogeneous clusters are preferred over
  homogeneous clusters.
• Homogeneous clusters imply less information
  contained in the cluster and lead to larger
  sampling error.

3

• Types of cluster sampling
• We will study the following five types of cluster
  sampling
•   1. Single stage cluster sampling from equal
  clusters
• 2. Single stage cluster sampling from unequal
  clusters, including PPS sampling
• 3. Two stage cluster sampling from equal clusters
• 4. Two stage cluster sampling from unequal
  clusters
• 5. Multistage sampling
• One way to handle sampling from unequal
  clusters is sampling with selection
  probability proportional to the size of cluster
  

4

• Single-Stage Sampling from Equal Clusters
• Basic features
•   Simple random sampling is generally used in
  sampling from equal clusters.
•   Single stage cluster sampling from equal
  clusters is equivalent to simple random sampling
  described in Chapter 3, if data are aggregated
  to cluster totals or means.
•   The repeated systematic sampling that we
  studied in Chapter 4 is equivalent to single
  stage cluster sampling from equal clusters.

5

• Comparison with simple random sampling
•   If we form every cluster in the population
  taking a simple random sample from the
  population, then cluster sampling is as good as
  simple random sampling.
• Using the notations defined in box 9.1 on page
  241,
• the sampling variance of is

• where ? is the intra-class correlation
  coefficient.
• In this cluster sample, there are
  elements. The sampling variance of simple
  random sample of will be

6

• Then the ratio of sampling variance of cluster
  sampling to that of simple random sampling will
  be

• This ratio is called the design effect
  of cluster sampling.
• For example, using the data on page 246, the
  intra-cluster correlation for the number of
  persons over 65 years of age is 0.051, which
  can be calculated by
• Then the approximate design effect from this
  example is
• which confirms the small design effect shown
  for age 65 (STATA output) on page 248.

7

• Design effect and intra-class correlation
•   The design effect defined above is a function
  of ?, the intra-class correlation coefficient,
  which varies from
• When ? is negative cluster sampling performs
  better than simple random sampling.
• When ? is zero, cluster sampling variance
  would be about the same as simple random
  sampling variance.
• For a given ?, the sampling variance of a
  cluster sample will increase as the size of
  cluster increases.

8

• Examples of intra-class correlation
• The intra-class correlation in household
  clusters is negative with respect to sex ratio,
  since each household contains both male and
  female, i.e. households are heterogeneous with
  respect to sex composition.
• Neighborhood clusters of households are
  homogeneous with respect to demographic
  characteristics but heterogeneous with respect to
  most health characteristics.
•   See the handout (reliability of estimates with
  alternative cluster sizes in the Health
  Interview Survey).

9

•  Estimator of sampling variance
•   The estimators given in Box 9.2 can be
  rewritten
• which measures variance among cluster
  totals

10

• An alternative variance estimator for the
  ratio is
• This formula is easier to use than the
  formula in Box 9.2