Simple Correspondence Analysis

Find the principal canonical correlation and corresponding row- and column-scores from a correspondence analysis of a two-way contingency table.

corresp(x, ...)

# S3 method for matrix
corresp(x, nf = 1, ...)

# S3 method for factor
corresp(x, y, ...)

# S3 method for data.frame
corresp(x, ...)

# S3 method for xtabs
corresp(x, ...)

# S3 method for formula
corresp(formula, data, ...)

Arguments

x, formula	The function is generic, accepting various forms of the principal argument for specifying a two-way frequency table. Currently accepted forms are matrices, data frames (coerced to frequency tables), objects of class `"xtabs"` and formulae of the form `~ F1 + F2`, where `F1` and `F2` are factors.
nf	The number of factors to be computed. Note that although 1 is the most usual, one school of thought takes the first two singular vectors for a sort of biplot.
y	a second factor for a cross-classification.
data	a data frame against which to preferentially resolve variables in the formula.
...	If the principal argument is a formula, a data frame may be specified as well from which variables in the formula are preferentially satisfied.

Value

An list object of class "correspondence" for which print, plot and biplot methods are supplied. The main components are the canonical correlation(s) and the row and column scores.

Details

See Venables & Ripley (2002). The plot method produces a graphical representation of the table if nf=1, with the areas of circles representing the numbers of points. If nf is two or more the biplot method is called, which plots the second and third columns of the matrices A = Dr^(-1/2) U L and B = Dc^(-1/2) V L where the singular value decomposition is U L V. Thus the x-axis is the canonical correlation times the row and column scores. Although this is called a biplot, it does not have any useful inner product relationship between the row and column scores. Think of this as an equally-scaled plot with two unrelated sets of labels. The origin is marked on the plot with a cross. (For other versions of this plot see the book.)

References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

Gower, J. C. and Hand, D. J. (1996) Biplots. Chapman & Hall.

Examples

(ct <- corresp(~ Age + Eth, data = quine))
#> First canonical correlation(s): 0.05317534 
#> 
#>  Age scores:
#>         F0         F1         F2         F3 
#> -0.3344445  1.4246090 -1.0320002 -0.4612728 
#> 
#>  Eth scores:
#>          A          N 
#> -1.0563816  0.9466276 
plot(ct)

corresp(caith)
#> First canonical correlation(s): 0.4463684 
#> 
#>  Row scores:
#>        blue       light      medium        dark 
#> -0.89679252 -0.98731818  0.07530627  1.57434710 
#> 
#>  Column scores:
#>        fair         red      medium        dark       black 
#> -1.21871379 -0.52257500 -0.09414671  1.31888486  2.45176017 
biplot(corresp(caith, nf = 2))