Statistical Methods in Medical Research 4ed

1993; Proskin, 1993). There is thus scope for overviews of related meta-analyses
(M 2 -analyses?), with the looming possibility of M 3 -analyses, and so on!
Analysis
A useful starting-point is the method of weighting described in §8.2. The difference
in efficacy between two treatments will be measured by a suitable parameter,
typically chosen as the difference in mean responses or (for binary
responses) the difference in proportions or the log odds ratio. Suppose there
are k trials, and that for the ith trial the true value of the difference parameter is
ui, estimated by the statistic yi, with estimated sampling variance vi. Define the
weight wi ˆ 1=vi. Then, if all the ui are equal to some common value u, a suitable
estimate of u is the weighted mean
P
wiyi
y ˆ P , …18:6†
wi
as in (8.13), and, approximately, var…y† ˆ1= P wi. The assumption that the
ui are constant may be tested, as in (8.14) and (8.15), by the heterogeneity
statistic
G ˆ P wi…yi y† 2 ˆ P wiy 2 P
i … wiyi†
2 = P wi, …18:7†
distributed approximately as x 2 …k 1† .
18.10 Meta-analysis 643
For trials with a binary outcome, these formulae may be applied to provide
p
combined estimates of log relative risk (using (4.24) for vi)
or log odds ratio
(using (4.26). In the latter case, an alternative approach is to use the method of
Yusuf et al. (1985) given in (4.33). If Oi and Ei are the observed and expected
numbers of critical events for one of the treatments, the log odds ratio is
estimated by yi ˆ…Oi Ei†=Vi, with var…yi† ˆ1=Vi, where Vi is calculated, as
in (4.32), from the 2 2 table for the trial. Then the weighted mean (18.6)
becomes
P
…Oi Ei†
y ˆ P ,
Vi
with var…y† ˆ1= P Vi. The heterogeneity statistic (18.7), distributed approximately
as x2 …k 1† , becomes
G 0 ˆ P …Oi Ei† 2
‰
Vi
P …Oi Ei†Š 2
P :
Vi
As noted in §4.5, this method is biased when the treatment effect is large (i.e.
when the odds ratio departs greatly from unity). The report by the Early Breast
Cancer Trialists' Collaborative Group (1990) suggests that the bias is not serious