Statistical Methods in Medical Research 4ed

8.7 Comparison of several variances
The one-way analysis of variance (§8.1) is a generalization of the two-sample t
test (§4.3). Occasionally one requires a generalization of the F test (used, as in
§5.1, for the comparison of two variances) to the situation where more than two
estimates of variance are to be compared. In a one-way analysis of variance, for
example, the primary purpose is to compare means, but one might wish to test
the significance of differences between variances, both for the intrinsic interest of
this comparison and also because the analysis of variance involves an assumption
that the group variances are equal.
Suppose there are k estimates of variance, s2 i , having possibly different
degrees of freedom, ni. (If the ith group contains ni observations, ni ˆ ni 1.)
On the assumption that the observations are randomly selected from normal
distributions, an approximate significance test due to Bartlett (1937) consists in
calculating
s 2 ˆ P nis2 P
i = ni
and
1
C ˆ 1 ‡
3…k 1†
8.7 Comparison of several variances 233
M ˆ P … ni†ln
s 2 P ni ln s2 i
X 1
ni
1
P ni
and referring M=C to the x2 …k 1† distribution. Here `ln' refers to the natural
logarithm (see p. 126). The quantity C is likely to be near 1 and need be
calculated only in marginal cases. Worked examples are given by Snedecor and
Cochran (1989, §13.10).
Bartlett's test is perhaps less useful than might be thought, for two reasons.
First, like the F test, it is rather sensitive to non-normality. Secondly, with
have to differ very considerably
samples of moderate size, the true variances s2 i
before there is a reasonable chance of obtaining a significant test result. To put
this point another way, even if M=C is non-significant, the estimated s2 i may
differ substantially, and so may the true s2 i . If possible inequality in the s2 i is
important, it may therefore be wise to assume it even if the test result is nonsignificant.
In some situations moderate inequality in the s2 i will not matter very
much, so again the significance test is not relevant.
An alternative test which is less influenced by non-normality is due to Levene
(1960). In this test the deviations of each value from its group mean, or median,
are calculated and negative deviations changed to positive, that is, the absolute
deviations are used. Then a test of the equality of the mean values of the
absolute deviations over the groups using a one-way analysis of variance (§8.1)
is carried out. Since the mean value of the absolute deviation is proportional to