Statistical Methods in Medical Research 4ed

476 Multivariate methods
Mean SE (mean) t P
d1 0 53 0 46 1 15 0 28
d2 0 03 0 39 0 08 0 94
d3 0 59 0 49 1 20 0 26
d4 3 10 1 95 1 59 0 15
None of the mean differences is individually significant, but three exceed their standard
errors, and in view of possible correlations between the four differences it seems worth
pursuing a multivariate test.
The matrix of covariances between the di is
and its inverse is
0
B
@
2 1401
0 1188
0 8941
0 1188
1 4868
0 7914
0 8941
0 7914
2 4099
1
3 5922
1 8811 C
4 6011 A
3 5922 1 8811 4 6011 37 8778
0
B
@
0 59389
0 03210
0 12918
0 03210
1 20462
0 65584
0 12918
0 65584
0 93359
1
0 04223
0 14254 C
0 13373 A
0 04223 0 14254 0 13373 0 05373
:
Both matrices are obtained by standard computer programs.
The coefficients bi in the discriminant function are obtained by cross-multiplying the
rows of the inverse matrix by the column of means, giving ( 0 5228, 0 1081,
0 1851, 0 1143). From (13.14), with n ˆ 10,
T 2 ˆ 7 439,
and, with p ˆ 4 in (13.15), the test statistic is
6 7 439=36 ˆ 1 24:
For F ˆ 1 24 on 4 and 6 DF, P ˆ 0 39. There is thus no strong evidence of departures
from zero in the set of four mean differences.
Note that the main contributions to the test statistic come from the variables d1 and d4,
which are positively correlated but have means with opposite signs. If the analysis is
performed with these two variables alone, the test statistic (as F on 2 and 8 DF) becomes
2 80, with P ˆ 0 12. However, the enhanced significance is misleading, since these variables
were selected, only after the first analysis had been performed, as being likely to be
the most sensitive pair.
Classification trees
The discussion of discrimination, or classification, so far has been in terms of
finding a function and allocating individuals into one or other of the two groups