Suppose that each unit in the population can be divided
into a number of smaller units, or subunits. A sample on n units has been selected.
If subunits within a selected unit give similar results, we can select and
measure a sample of the subunits in any chosen unit to avoid uneconomically
measure them all. This technique is called subsampling, since the unit
is measured by its samples, or multi-stage sampling, since the
sample is taken in more than one step. The firstly selected sample units are
called the primary units, and then select a sample of subunits from
these primary units.
The prime stimulus for multi-stage sampling is
administrative convenience. It is more flexible than one-stage sampling. It
reduces to one-stage sampling, unless this is the best choice of sample size of
subsample. We have chance of selecting smaller value which appears more efficient.
How to estimate
means and variances in Multi-stage Sampling (with units of equal sizes)?
In two-stage sampling the sampling plan gives fir a method
for selecting n units. Then for each selected unit, a method is given for
selecting the specified number of subunits from it. In finding the mean and
variance of an estimate, averages must be taken over all samples that can be
generated by this two-stage process.
One way of calculating this average is first to average the
estimate over all selects, then average over all possible selections of n units
by the plan.
=value obtained for the jth subunit in the ith
=sample mean per subunit in the ith primary unit
=over-all sample mean per subunit
=variance among primary unit means
=variance among subunits within primary units
Theorem 10.1. if the n units and the m subunits from each chosen unit are selected
by simple random sampling is an unbiased estimate of with variance
Theorem 10.2. Under the conditions of theorem 10.1, an unbiased estimate of is
Corollary. A useful result is
It follows that an unbiased estimate of is
If m=M that is, the formula becomes to simple random sampling of the units.
If n-N, the formula is that for proportional stratified
random sampling, since primary units may them be regarded as strata, all of
which are sampled.
When is negligible, we have the simple result
Thus the estimated variance can be computed from knowledge
of the unit means only, which is also helpful when subsampling is systematic.
It depends on the type of cost function. If travel costs
between units are unimportant, one form of cost function is
The first component of cost, , is proportional to the number of primary units in the
sample; the second, , is proportional to the total number of second-stage units
or elements. We got
The last term on the right didnít depend on the choice of n
and m. Minimizing V for fixed C, or C for fixed V, is equivalent to minimizing
By the Cauchy-Schwarz inequality