Statistical Methods in Medical Research 4ed

472 Multivariate methods
In the discussion so far it has been assumed that it was known which
variables should be included in the discriminant function (13.4). Usually a set
of possible discriminatory variables is available, and it is required to find the
minimum number of such variables which contribute to the discrimination.
Many computer programs include a stepwise procedure, similar to that in multiple
regression (see p. 357), to achieve this.
The likelihood rule was derived assuming that the xs had a multivariate
normal distribution. Although the method may prove adequate in some cases,
when this assumption breaks down an alternative approach may be preferable.
In particular, if the xs contain categorical variables, then the use of logistic
regression (see §14.2) would be appropriate. The data of Example 13.2 are
analysed using this approach on p. 492.
In most examples of discriminant analysis, there will be little point in testing
the null hypothesis that the two samples are drawn from identical populations.
The important question is how the populations differ. There are, though, some
studies, particularly those in which two treatments are compared by multivariate
data, in which a preliminary significance test is useful in indicating whether there
is much point in further exploration of the data.
Multiple testing
Before discussing multivariate significance tests, we consider the problem in
terms of multiple testing. Suppose each of the p variables is tested by a univariate
method, such as a two-sample t test. Then some variables may differ significantly
between groups whilst others may not. A difficulty in interpretation is that p tests
have been carried out so that, even if the overall null hypothesis that none of the
variables differ between groups is true, the probability of finding at least one
significant difference is higher than the nominal significance level adopted. There
are similarities with the multiple comparisons problem discussed in §8.4.
The Bonferroni procedure is sometimes used to correct for this. If p independent
comparisons are carried out at a significance level of a0 , then the probability
that one or more are significant is 1 …1 a0 † p , and when a0 is small this is
approximately pa0 . Therefore setting a0 ˆ a=p ensures that the probability of
finding one or more significant effects does not exceed a. One problem with this
method is that it takes no account of the multivariate correlations and it is
conservative in the usual multivariate situation when the comparisons are not
independent. A modification due to Simes (1986a) is to order the p values such
that P…1† P…2† ... P…p†. Then the null hypothesis is rejected at level a if any
P…j† ja=p. This modification is more powerful than the Bonferroni procedure
whilst still ensuring that the overall significance level does not exceed a for
positively correlated variables (Sarkar & Chang, 1997)Ðthat is, the test remains
conservative.