Statistical Methods in Medical Research 4ed

216 Comparison of several groups
should be regarded as heterogeneous, in the sense that sampling variation is
unlikely to explain the differences among them.
We now consider a method of wide generality for tackling other problems of
this sort. It depends on certain assumptions, which may or may not be precisely
true in any particular application, but it is nevertheless very useful in providing
approximate solutions to many problems. The general situation is that data are
available in k groups, each providing an estimate of some parameter. We wish
first to test whether there is evidence of heterogeneity between the estimates, and
then in the absence of heterogeneity to obtain a single estimate of the parameter
from the whole data set.
Suppose we observe k quantities, Y1, Y2, ..., Yk, and we know that Yi is
N…mi, Vi†, where the means mi are unknown but the variances Vi are known. To
test the hypothesis that all the mi are equal to some specified value m, we could
calculate a weighted sum of squares
G0 ˆ P
‰…Yi m† 2 =ViŠ ˆ P
wi…Yi m† 2 , …8:12†
i
where wi ˆ 1=Vi. The quantity G0 is the sum of squares of k standardized normal
deviates and, from (5.2), it follows the x 2 …k† distribution.
However, to test the hypothesis of homogeneity we do not usually wish to
specify the value m. Let us replace m by the weighted mean
and calculate
P
Y ˆ
i wiYi
P
i wi
i
, …8:13†
G ˆ P
wi…Yi Y† 2 : …8:14†
i
The quantity wi is called a weight: note that it is the reciprocal of the variance Vi,
so if, for example, Y1 is more precise than Y2, then V1 < V2 and w1 > w2 and Y1
will be given the higher weight.
A little algebra gives the alternative formula:
G ˆ P
i
wiY 2
i
… P
i wiYi† 2 = P
i wi: …8:15†
Note that if all the wi ˆ 1, (8.14) is the usual sum of squares about the mean, Y
becomes the usual unweighted mean and (8.15) becomes the usual short-cut
formula (2.3).
It can be shown that, on the null hypothesis of homogeneity, G is distributed
as x2 …k 1† . Replacing m by Y has resulted in the loss of one degree of freedom.
High values of G indicate evidence against homogeneity.