Two-stage cluster sampling; unbiased estimation of a population
mean and total
Two-stage cluster sampling: In this chapter we examine two-stage
cluster sampling, a very simple kind of multi-stage sampling. In real-life
surveys, it is common to have several levels of sampling, including use of
stratification and ratio estimators. Here we will consider two-stage cluster
sampling where a simple random sample of clusters is taken, then a simple
random sample of elements in the selected clusters is taken. One example
would be to estimate the proportion of U.S. college students who like Korean
food by taking a SRS of U.S. colleges, then taking a SRS of students in each
selected college.
Notation: The extra stage of sampling here gives us notation that is
slightly changed from that used for single-stage cluster sampling:
N= the number of clusters in the population
n= the number of clusters selected in a SRS
Mi= the number of elements in cluster i
mi= the number of elements selected in a SRS from cluster i
M=
N
P
i=1
Mi= the number of elements in the population
M/N = the average cluster size for the population
yij = the jth observation in the sample from the ith cluster
yi=1
mi
mi
P
j=1
yij = the sample mean for the ith cluster
Unbiased estimation of the population mean µ:In single-stage
cluster sampling the unbiased estimator of τwas N
n
n
P
i=1
yi, where the yiterm
was the total of observations in the ith cluster. We are now sampling from
each cluster, so we do not know these totals. However, we can estimate
the cluster total by multiplying the cluster average (yi) by the number of
elements in the cluster (Mi). We can divide by Mto estimate µ. Then we
have:
N
Mn
n
X
i=1
yiis estimated by N
Mn
n
X
i=1
Miyi, so bµ=N
Mn
n
X
i=1
Miyi=1
M
n
P
i=1
Miyi
n,
and the estimated variance is:
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