Statistical Methods in Medical Research 4ed

662 Statistical methods in epidemiology
Direct method
In the direct method the death rate is standardized to the age structure of the
standard population. The directly standardized death rate for the special population
is, therefore,
p 0 P
Nipi
ˆ P : …19:5†
Ni
It is obtained by applying the special death rates, pi, to the standard population
sizes, Ni. Alternatively, p0 can be regarded as a weighted mean of the pi,
using the Ni as weights. The variance of p0 may be estimated as
var…p 0 P 2 …Ni piqi=ni†
†ˆ
… P Ni† 2 , …19:6†
where qi ˆ 1 pi; if, as is often the case, the pi are all small, the binomial
variance of pi, piqi=ni, may be replaced by the Poisson term pi=ni …ˆ ri=n2 i †, giving
var…p 0 P 2 …Ni pi=ni†
†'
… P Ni† 2 : …19:7†
To compare two special populations, A and B, we could calculate a standardized
rate for each (p0 A and p0B ), and consider
d ˆ p 0 A
From (19.5),
P
Ni…pAi pBi†
d ˆ P ,
Ni
which has exactly the same form as (15.15), with wi ˆ Ni, and di ˆ pAi pBi as in
(15.14). The method differs from that of Cochran's test only in using a different
system of weights. The variance is given by
P 2 Ni var…di†
var…d† ˆ
… P Ni† 2 , …19:8†
with var…di† given by (15.17). Again, when the p0i are small, q0i can be put
approximately equal to 1 in (15.17).
If it is required to compare two special populations using the ratio of the
standardized rates, p0 A =p0B , then the variance of the ratio may be obtained using
(19.6) and (5.12).
The variance given by (19.7) may be unsatisfactory for the construction of
confidence limits if the numbers of deaths in the separate age groups are small,
since the normal approximation is then unsatisfactory and the Poisson limits are
p 0 B :