Graphics Reference
In-Depth Information
Correspondence Analysis
and Reciprocal Averaging
3.8
Correspondence analysis represents therowsandcolumns ofatwo-waycontingency
table as points in Euclidean spaces of dimension usually chosen as two. It can also be
used on any data matrix that has non-negative entries and can also be extended to
higher way tables, but this aspect is not covered here. he technique will be illus-
trated using the contingency table displayed in Table
.
, which records the number
of papers in a research database which employ specific keywords in their descriptive
fields (title, keywords or abstract) over an
-year period.
First distances are measured between the rows of the table. Ordinary Euclidean
distance, treating the
columns as
dimensions for the rows, is not appropriate,
since doubling the size of the sample for any one rowwill alter the Euclidean distance
dramatically. For contingency tables where only the overall total is fixed, this may be
not be a problem, but it will be for tables where row totals can be chosen arbitrarily.
To overcome this problem, χ
distances are used. he χ
distance, d
ii
′
, between the
ith and i
′
th rows is defined by
p
j
=
x
ij
r
i
x
i
′
j
r
i
′
c
j
d
ii
′
=
−
,
(
.
)
where x
ij
is the entry in the table in the ith row and jth column, r
i
is the ith row
sum and c
j
is the jth column sum. A space is now found where points in the space
represent the rows of the table and where Euclidean distance between points equals
the χ
distance between the corresponding rows of the table. Greenacre (
) gives
a comprehensive account of how this space is found, but only a brief summary can
be given here.
Let
denote the table that has been normalized to have overall total equal to
X
unity. Let
r
bethediagonal matrix oftherowsumsof
, andlet
c
bethediagonal
D
X
D
matrix of column sums of
X
. Let the generalized singular value decomposition of
X
be given by
T
,
X=AD
B
(
.
)
λ
Table
.
.
References employing keywords
Keywords
Total
Reciprocal av.
Correspond. anal.
Ind. diff. scal.
Classical scaling
Procrustes anal.
Mult. scaling
Total
