Statistical Methods in Medical Research 4ed

678 Statistical methods in epidemiology
Example 19.6
Table 19.5 summarizes results from 10 retrospective surveys in which patients with lung
cancer and control subjects were classified as smokers or non-smokers. In most or all of
these surveys, cases and controls would have been matched, but the original data are
usually not presented in sufficient detail to enable relative risks to be estimated from
(19.26) and matching is ignored in the present analysis. (The effect of ignoring matching
when it is present is, if anything, to underestimate the departure of the relative risk from
unity.) The data were compiled by Cornfield (1956) and have been referred to also by Gart
(1962).
Defining wi as the reciprocal of var(ln ^c i) from (4.26), the weighted mean is
P wi ln ^c i
P wi
ˆ
161 36
ˆ 1 531,
105 4
and the pooled estimate of c is exp(1 531) ˆ 4 62.
For the heterogeneity test (8.15), the x2 …9† statistic is
P
wi…ln ^c i† 2 … P wi ln ^c i† 2
P ˆ 253 678 247 061 ˆ 6 62 …P ˆ 0 68†:
wi
There is no strong evidence of heterogeneity between separate estimates. It is, of course,
likely that the relative risk varies to some extent from study to study, particularly as the
factor `smoking' covers such a wide range of activity. However, the sampling variation of
the separate estimates is evidently too large to enable such real variation to emerge. If we
assume that all the variation is due to sampling error, the variance of the weighted mean
of ln ^c i can be obtained as
1
P ˆ 0 00949:
wi
Approximate 95% confidence limits for ln c are
p
1 531 …1 96† 0 00949 ˆ 1 340 and 1 722:
The corresponding limits for c are obtained by exponentials as 3 82 and 5 60.
The Mantel±Haenszel estimator of c is
302 840
RMH ˆ ˆ 4 68:
64 687
The statistic for testing that this estimate differs from unity is
X 2 MH ˆ…3793 3554 85†2 =193 17
ˆ 293 60 …P < 0 001†:
The test-based method of calculating confidence limits from (19.23) gives
p
SE…ln RMH† ˆln…4 68†= 293 60
ˆ 1 543=17 13
ˆ 0 0901,
and approximate 95% confidence limits for ln c are