Statistical Methods in Medical Research 4ed

Apart from the increased precision in the estimation of the population mean,
stratification may be adopted to provide reasonably precise estimates of the
means for each of the strata. This may lead to departures from the optimal
allocation described above, so that none of the sample sizes in the separate strata
become too small.
Example 19.1
It is desired to estimate the prevalence of a certain condition (i.e. the proportion of
affected individuals) in a population of 5000 people by taking a sample of size 100.
Suppose that the prevalence is known to be associated with age and that the population
can be divided into three strata defined by age, with the following numbers of
individuals:
Stratum, i Age (years) Ni
1 0±14 1200
2 15±44 2200
3 45± 1600
5000
Suppose that the true prevalences, pi, in the different strata are as follows:
Stratum, i pi
1 0 02
2 0 08
3 0 15
19.2 The planning of surveys 653
It is easily verified that the overall prevalence, p (ˆ P Nipi= P Ni), is 0 088. A simple
random sample of size n ˆ 100 would therefore give an estimate p with variance
(0 088)(0 912)/100 ˆ 0 000803.
For stratified sampling with optimal allocation the sample sizes in the strata should be
chosen in proportion to Ni ‰
p pi…1 pi†Š. The values of this quantity in the three strata are
168, 597 and 571, which are in the proportions 0 126, 0 447 and 0 427, respectively. The
optimal sample sizes are, therefore, n1 ˆ 12, n2 ˆ 45 and n3 ˆ 43, or numbers very close to
these (there being a little doubt about the effect of rounding to the nearest integer). Let us
call this allocation A. This depends on the unknown pi, and therefore could hardly be
used in practice. If we knew very little about the likely variation in the pi, we might choose
the ni / Ni, ignoring the effect of the changing standard deviation. This would give, for
allocation B, n1 ˆ 24, n2 ˆ 44 and n3 ˆ 32. Thirdly, we might have some idea that the
prevalence (and therefore the standard deviation) increased with age, and therefore adjust
the allocation rather arbitrarily to give, say, n1 ˆ 20, n2 ˆ 40 and n3 ˆ 40 (allocation C).
The estimate of ^p is, in each case, given by the formula equivalent to (19.1),