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Suppose Y are the coordinates given by a PCO of -thatis,
YY = ( I 11 / K ) ( I 11 / K ) ,
(5.26)
where Y Y = . Then the coordinates y of P are (Gower and Hand, 1996, p. 250)
y = 1 Y ( δ 1 / K ) ,
(5.27)
where the final term derives from (5.24) after taking into account that Y 1 = 0 , because
of the centring of Y . Equation (5.27) gives the coordinates of the added point in K 1
or fewer dimensions; in r -dimensional approximations, only the first r columns of Y
and the first r eigenvalues will be needed. A further coordinate in a K th dimension is
necessary for exact representations, but this is rarely needed - see Section 5.4 for further
details. It may be verified that if δ is successively taken to be the n k columns pertaining
to the k th group, then the centroids of the inserted points are at the same position as the
k th group mean in Y ,asisshowninSection5.8.1.
We may write in full matrix form as
1
11 +
11 diag
=
D
2 [diag
(
D
)
(
D
)
],
whence
n
n Dn
n [ n diag
n
=
(
D
)
1 ]
.
(5.28)
From (5.28) we have
n K
n
1 D1
n
=
n k D kk
k =
1
which rearranges to
K
1 D1
n
n
g k Dg k
n k
n
=
+
.
(5.29)
n
k = 1
Recalling that 1 D1 / n is the total sum of squares and g k Dg k / n k is the sum of squares
within the k th group, we see that, apart from sign, the analysis of distance (5.29) is an
expression of the CVA orthogonal analysis of variance:
Total sum of squares = Between-group sum of squares + Within-group sum of squares .
Thus, from (5.29) we may form an AoD table in which the contributions between and
within groups are exhibited. Furthermore, we may break this down into the contribution
arising from different dimensions and sets of dimensions, especially the r fitted dimen-
sions and the remaining residual dimensions. Note that with K groups, the means fit into
K
1 or fewer dimensions so the remaining 'residual' dimensions for the group means
are null.
We have represented within-group variation by choosing d in (5.23) as the successive
columns of D . However, d may refer to a genuine new sample, in which case (5.27)
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