Statistical Methods in Medical Research 4ed

D 2 …zA zB†
ˆ
2
…13:5†
variance of z within groups
is as large as possible. The estimated variance of z will, in general, be different in
the two groups, but a pooled estimate could be calculated as in the two-sample t
test.
The solution of this problem is as follows:
1 Calculate the pooled sums of squares and products of the xs within groups,
divide by nA ‡ nB 2 to obtain the matrix of within-group variances and
covariances, and obtain the inverse matrix with general term cij. Here, nA and
nB are the numbers of individuals in groups A and B, respectively.
2 Calculate the bs as follows:
b1 ˆ c11…xA1 xB1†‡c12…xA2 xB2†‡...‡ c1p…xAp xBp†
b2 ˆ c21…xA1 xB1†‡c22…xA2 xB2†‡...‡ c2p…xAp xBp†
bp ˆ cp1…xA1 xB1†‡cp2…xA2 xB2†‡...‡ cpp…xAp xBp†:
…13:6†
Here, xAi is the mean of xi in group A and so on.
To use (13.4) for allocating future individuals to one of the two groups we
need an end-point, or cut-point, to discriminate between A and B. A completely
symmetrical rule would be to use the mean, z0, ofzA and zB. From (13.4),
and
whence
z0 ˆ b1
xA1 ‡ xB1
2
zA ˆ b1xA1 ‡ b2xA2 ‡ ...‡ bpxAp
zB ˆ b1xB1 ‡ b2xB2 ‡ ...‡ bpxBp,
‡ b2
xA2 ‡ xB2
2
13.3 Discriminant analysis 465
‡ ...‡ bp
xAp ‡ xBp
2
…13:7†
: …13:8†
If zA > zB the allocation rule is: allocate an individual to A if z > z0 and to B if
z < z0. Many computer programs calculate the value of z for each individual in
the two original samples. It is thus possible to count how many individuals in the
two groups would have been wrongly classified by the allocation rule. This
unfortunately gives an overoptimistic picture, because the allocation rule has
been determined to be the best (in a certain sense) for these two particular
samples, and it is likely to perform rather less well on the average with subsequent
observations from the two groups (see Hills, 1966).
Various methods have been proposed to overcome this problem (Krzanowski
& Marriott, 1995, §§9.32±9.44). One method is to split the sample randomly into
two subsamples. The first subsample, the training sample, is used to estimate the