ContingencyTable: To Configuration (ca)...

A command that creates a Configuration object from the selected ContingencyTable object by means of Correspondence analysis.

Settings

Number of dimensions
The dimensionality of the Configuration.
Scaling of the final configuration
determines whether row points are in the centre of gravity of column points, or, column points are in the centre of gravity of row points, or, whether roes and columns are treated symmetrically.

Algorithm

1. We start with the following transformation of the entries fij:

hij = fij / √ (fi+f+j) - √ (fi+f+j) / N,

where hij is the entry for a cell in the matrix H with transformed data, fi+ is the total count for row i, f+j is the total count for column j and N is the grand total. This can be written in matrix form as:

H = R–1/2FC–1/2R1/2uuC1/2 / N,

where R and C are diagonal matrices with the row and column totals, respectively and u a column vector with all elements equal to 1.

2. Next the singular value decomposition of matrix H is performed:

H = K Λ L′,

where KK = I, LL = I, and Λ is a diagonal matrix with singular values.

3. Now the row (X) and column points (Y) can be determined. Three normalizations are possible:

• Scale row points in the centre of gravity of column points
X = √N R–1/2KΛ
Y = √N C–1/2U
• Scale column points in the centre of gravity of row points
X = √N R–1/2K
Y = √N C–1/2LΛ
• Treat row points and column points symmetrically
X = √N R–1/2KΛ–1/2
Y = √N C–1/2LΛ–1/2

For more details see Gifi (1990), chapter 8.


© djmw, April 7, 2004