Correspondence Regression
Description
This is the basic function for correspondence regression, i.e. the correspondence analysis of a contingency table formed by the categorical variables Y and X, where X can be in turn made up of the combinations of various categorical variables.
Usage
1 2 
Arguments
formula 
A 
data 
The data frame containing the variables specified in the 
part 
The name of a factor partitioning the levels of the lefthand side 
b 
Number of the bootstrap replications. If 
xep 
Logical specifying whether to output the separate terms in the righthand side ( 
std 
Logical specifying whether to output the standardized coordinates. Defaults to 
rel 
Logical specifying whether to divide the coordinates by the 
phi 
Logical specifying whether to compute the output on the scale of the Chisquared value of the contingency table or of the Phisquared value
(which is Chisquared divided by N). Reminiscent of 
chr 
Character specifying the separator string for constructing the interaction terms. 
Details
Correspondence regression rests on the idea, described by Van der Heijden et al. (1989) and quoted in Greenacre (2007: 272), of using correspondence analysis to inspect
the interactions in a loglinear analysis. More specifically, as loglinear analysis or Poisson regression is sometimes used to model a polytomous or multinomial response
variable (in a GLM), correspondence regression enables the analysis of a categorical factor (Y
) in terms of other (possibly interacting) factors (X
). These
are specified in the argument formula
, which can be constructed in all the usual ways of specifying a model formula: e.g. Y ~ X1 * X2
as a shorthand for
Y ~ X1 + X2 + X1 : X2
, or Y ~ X1 * X2  X1 : X2
, Y ~ (X1 + X2 + X3) ^ 2
, etc. Correspondence regression then crosstabulates the Y
factor with all the
combinations in X
, thus producing a typical contingency table, on which a simple correspondence analysis is performed (see Greenacre 2007: 121128 for the outline of
this approach). The more general effects in X
are obtained by aggregating the combinations.
Correspondence regression also allows for inferential validation of the effects, which is done by means of the bootstrap. Setting the argument b
to a number > 0, b
replicates of the contingency table are generated with multinomial sampling. From these, b
new values are derived for the coordinates in both Y
and
X
as well as for the eigenvalues (also called the "principal inertias"). On the basis of the replicate values, confidence intervals, ellipses or ellipsoids can
be computed. CAUTION: bootstrapping is computationally quite intensive, so it can take a while to reach results, especially with a large b
.
The argument parm
can be used when the levels of Y
are grouped/partitioned/nested into clusters and one wants to exclude the heterogeneity between the clusters. Thus,
parm
is equivalent to a random factor, although corregp
currently allows for only one such factor. The use of parm
can be relevant for socalled
lectometric analyses in linguistics.
Value
An object of class "corregp", i.e. a list with components:

A vector of eigenvalues of the correpondence regression. 

The coordinates (matrix) of the Y levels. 

The coordinates of the X levels. If 

A list of the frequencies of every Y and X level. 

If b>0. A list of bootstrap replicates for the eigenvalues, the Y levels and the X levels. 

A list of auxiliary information (such as the U and V matrices of the SVD, the specified values for all the arguments) to be passed to other functions and methods. 
References
Greenacre, M. (2007) Correspondence analysis in practice, Second edition. Boca Raton: Chapman and Hall/CRC.
Van der Heijden, P.G.M., A. de Falguerolles and J. de Leeuw (1989) A combined approach to contingency table analysis using correspondence analysis and loglinear analysis. Applied Statistics 38 (2), 249–292.
See Also
print.corregp
, summary.corregp
, screeplot.corregp
, plot.corregp
.
Examples
1 2 3 