Statistical Methods in Medical Research 4ed

652 Statistical methods in epidemiology
and the total sample size is
N ˆ Pk
Ni,
iˆ1
n ˆ Pk
The mean value of x in the whole population is clearly
P
Nimi m ˆ
N :
The appropriate estimate of m is
iˆ1
ni:
P Nixi
^m ˆ , …19:1†
N
which is easily seen to be unbiased. Its variance is
var…^m† ˆ 1
N2 P 2 Ni var…xi†
by (4.2), and this is estimated from the sample as
ˆ 1
N2 P N2 i s2i …1 fi† …19:2†
ni
var…^m† ˆ 1
N2 P N2 i s2i …1 fi†: …19:3†
ni
If all the fi are small, the terms 1 fi may be replaced by unity. In general,
(19.2) will be smaller than the variance of the mean of a random sample because
the terms s2 i measure only the variability within strata. Variability between the mi is irrelevant.
If n is fixed and the fi are small, var…^m† becomes a minimum if the ni are
chosen to be as nearly as possible proportional to Nisi. Thus, if si is constant
from one stratum to another, the ni should be chosen in proportion to Ni, i.e. the
sampling fraction should be constant. Strata with relatively large si have correspondingly
increased sampling fractions. Usually, little will be known about the
si before the survey is carried out and the choice of a constant sampling fraction
will be the most reasonable strategy.
If the survey is designed to estimate a proportion p, the above formulae hold
with xi and mi replaced by pi and pi, the observed and true proportions in the ith
stratum, s2 i by pi…1 pi† and s2 i by nipi…1 pi†=…ni 1† (this being the estimate
of variance in a sample of size ni in which nipi observations are 1 and ni…1 pi)
are 0).