Statistical Methods in Medical Research 4ed

480 Multivariate methods
Allocation is then to the group with the highest score. The method is also
equivalent to allocating an individual to the group with the nearest mean, in
the sense of generalized distance. Computer programs exist for the whole procedure.
This likelihood ratio approach is more appropriate than that of canonical
variates if the main purpose is to form an allocation rule. If the purpose is to gain
insight into the way in which groups differ, using as few dimensions as possible,
then canonical variates are the more appropriate method.
Although the two methods do not usually give identical results, there is a
close relationship between them. Using the first canonical variate gives the same
discrimination as the likelihood method if the group means are collinear in pdimensional
space. Thus, as noted earlier, the two methods are identical for
k ˆ 2 because the two group means may be regarded as collinear. If, for example,
k ˆ 3 and p > 1, there will be k 1 ˆ 2 canonical variates, W1 and W2, and
two independent likelihood ratio discriminators, say Z1 ˆ L1 L2 and
Z2 ˆ L2 L3. (Note that L3 L1 ˆ …Z1 ‡ Z2†.) If the observations are plotted
(i) with W1 and W2 as axes, and (ii) with Z1 and Z2 as axes, it will be found that
the scatter diagrams are essentially similar, differing only in the orientation and
scaling of the axes. Thus, the use of both canonical variates is equivalent to the
likelihood approach. In general, using k 1 canonical variates gives the same
discrimination as the likelihood approach. However, the idea behind using
canonical variates is to reduce the dimensionality, and the only advantage of
this approach is if it is possible to use just a few variates without any material
loss in discriminating ability. For further discussion of this point, see Marriott
(1974).
Scoring systems, using canonical variates
Suppose that a variable is measured on a p-point scale from 1 to p. For example,
a patient's response to treatment may be graded in categories ranging from
`much worse' (scored 1) to `much better' (scored p). This sequence of equidistant
integers may not be the best scale on which to analyse the data. The best method
of scoring will depend on the criterion we wish to optimize. Suppose, for
example, that patients were classified into k treatment groups. It might be
reasonable to choose a system of scoring which maximized differences between
treatments as compared with those within treatments. For this purpose consider
a set of p dummy variables, x1, x2, ..., xp, such that, for a patient in category j,
the value of xj is 1 and that of the other xis is 0. Now choose the first canonical
variate of the xs, say,
g ˆ l1x1 ‡ l2x2 ‡ ...‡ lpxp: