Information Technology Reference
In-Depth Information
Fair
Clerical
Dark
Myfanwy
Scotland
Alisdair
Wales
F
University
School
Jane
Harriet
Brown
Ivor
George
Manual
M
Professional
England
Jeremy
Grey
Postgrad
Figure 8.7
MCA biplot of the data in Table 8.1 based on the associated normalized Burt
matrix
L
−
1
/
2
G
GL
−
1
/
2
2
V
. Category-level points plotted as
Z
=
L
−
1
/
2
V
2
J
2
=
p
V
and samples as
Z
0
=
GZ
/
p
.
Including the unit diagonal blocks implies that when
p
=
2, CA and MCA are not pre-
cisely the same, though the two are closely related; see Gower and Hand (1996) for the
precise relationship. In general, the approximation of the unit diagonal blocks is of no
interest and is detrimental to the good approximation of the contingency tables them-
selves. Greenacre (1988, 2007) omits these superfluous blocks from the approximation,
terming this form of MCA 'joint correspondence analysis' (JCA). Note that, when
p
=
2,
JCA retains only the two-way contingency table (twice), so then JCA and CA coincide,
giving the kind of compatibility desirable in any generalization. JCA requires an iterative
algorithm, at each stage of which the diagonal blocks are updated in a similar way to
that in which simple factor analysis algorithms of a correlation matrix iteratively replace
the unit diagonals by communalities.
Different estimates of goodness of fit accompany all these variants of MCA. Basically,
there are four main types: (i) fits based on the least-squares approximation of the whole
Burt matrix; (ii) fits based on the least-squares approximation of the Burt matrix but
excluding the diagonal blocks; (iii) fits based on the least-squares approximation of the
Burt matrix excluding the diagonal blocks but with a simple adjustment to reduce the
