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, ...)

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.

- 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
an optional data frame, list or environment 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.

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.)

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.

```
## IGNORE_RDIFF_BEGIN
## The signs can vary by platform
(ct <- corresp(~ Age + Eth, data = quine))
plot(ct)
corresp(caith)
biplot(corresp(caith, nf = 2))
## IGNORE_RDIFF_END
```

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