cancor {candisc}  R Documentation 
Canonical Correlation Analysis
Description
The function cancor
generalizes and regularizes computation for
canonical correlation analysis in a way conducive to visualization using
methods in the heplots
package.
The package provides the following display, extractor and plotting methods for "cancor"
objects
print(), summary()
Print and summarise the CCA
coef()
Extract coefficients for X, Y, or both
scores()
Extract observation scores on the canonical variables
redundancy()
Redundancy analysis: proportion of variances of the variables in each set (X and Y) accounted for by the variables in the other set through the canonical variates
plot()
Plot pairs of canonical scores with a data ellipse and regression line
heplot()
HE plot of the Y canonical variables showing effects of the X variables and projections of the Y variables in this space.
As well, the function provides for observation weights, which may be useful in some situations, as well as providing a basis for robust methods in which potential outliers can be downweighted.
Usage
cancor(x, ...)
## S3 method for class 'formula'
cancor(formula, data, subset, weights, na.rm = TRUE, method = "gensvd", ...)
## Default S3 method:
cancor(
x,
y,
weights,
X.names = colnames(x),
Y.names = colnames(y),
row.names = rownames(x),
xcenter = TRUE,
ycenter = TRUE,
xscale = FALSE,
yscale = FALSE,
ndim = min(p, q),
set.names = c("X", "Y"),
prefix = c("Xcan", "Ycan"),
na.rm = TRUE,
use = if (na.rm) "complete" else "pairwise",
method = "gensvd",
...
)
## S3 method for class 'cancor'
print(x, digits = max(getOption("digits")  2, 3), ...)
## S3 method for class 'cancor'
summary(object, digits = max(getOption("digits")  2, 3), ...)
## S3 method for class 'cancor'
scores(x, type = c("x", "y", "both", "list", "data.frame"), ...)
## S3 method for class 'cancor'
coef(object, type = c("x", "y", "both", "list"), standardize = FALSE, ...)
Arguments
x 
Varies depending on method. For the 
... 
Other arguments, passed to methods 
formula 
A twosided formula of the form 
data 
The data.frame within which the formula is evaluated 
subset 
an optional vector specifying a subset of observations to be used in the calculations. 
weights 
Observation weights. If supplied, this must be a vector of
length equal to the number of observations in X and Y, typically within
[0,1]. In that case, the variancecovariance matrices are computed using

na.rm 
logical, determining whether observations with missing cases are excluded in the computation of the variance matrix of (X,Y). See Notes for details on missing data. 
method 
the method to be used for calculation; currently only

y 
For the 
X.names , Y.names 
Character vectors of names for the X and Y variables. 
row.names 
Observation names in 
xcenter , ycenter 
logical. Center the X, Y variables? [not yet implemented] 
xscale , yscale 
logical. Scale the X, Y variables to unit variance? [not yet implemented] 
ndim 
Number of canonical dimensions to retain in the result, for scores, coefficients, etc. 
set.names 
A vector of two character strings, giving names for the collections of the X, Y variables. 
prefix 
A vector of two character strings, giving prefixes used to name the X and Y canonical variables, respectively. 
use 
argument passed to 
digits 
Number of digits passed to 
object 
A 
type 
For the 
standardize 
For the 
Details
Canonical correlation analysis (CCA), as traditionally presented is used to identify and measure the associations between two sets of quantitative variables, X and Y. It is often used in the same situations for which a multivariate multiple regression analysis (MMRA) would be used.
However, CCA is is “symmetric” in that the sets X and Y have equivalent status, and the goal is to find orthogonal linear combinations of each having maximal (canonical) correlations. On the other hand, MMRA is “asymmetric”, in that the Y set is considered as responses, each one to be explained by separate linear combinations of the Xs.
Let \mathbf{Y}_{n \times p}
and \mathbf{X}_{n \times q}
be two sets of variables over which
CCA is computed. We find canonical coefficients \mathbf{A}_{p \times k}
and
\mathbf{B}_{q \times k}, k=\min(p,q)
such that the canonical variables
\mathbf{U}_{n \times k} = \mathbf{Y} \mathbf{A} \quad \text{and} \quad
\mathbf{V}_{n \times k} = \mathbf{X} \mathbf{B}
have maximal, diagonal correlation structure.
That is, the coefficients \mathbf{A}
and \mathbf{B}
are chosen such that the (canonical)
correlations between
each pair r_i = \text{cor}(\mathbf{u}_i, \mathbf{v}_i), i=1, 2, \dots , k
are maximized and all other pairs are uncorrelated,
r_{ij} = \text{cor}(\mathbf{u}_i, \mathbf{v}_j) = 0, i \ne j
.
Thus, all correlations between the X and Y variables are channeled through the correlations between
the pairs of canonical variates.
For visualization using HE plots, it is most natural to consider
plots representing the relations among the canonical variables for the Y
variables in terms of a multivariate linear model predicting the Y canonical
scores, using either the X variables or the X canonical scores as
predictors. Such plots, using heplot.cancor
provide a
lowrank (1D, 2D, 3D) visualization of the relations between the two sets,
and so are useful in cases when there are more than 2 or 3 variables in each
of X and Y.
The connection between CCA and HE plots for MMRA models can be developed as follows. CCA can also be viewed as a principal component transformation of the predicted values of one set of variables from a regression on the other set of variables, in the metric of the error covariance matrix.
For example, regress the Y variables on the X variables, giving predicted
values \hat{Y} = X (X'X)^{1} X' Y
and residuals R = Y 
\hat{Y}
. The error covariance matrix is E = R'R/(n1)
. Choose a
transformation Q that orthogonalizes the error covariance matrix to an
identity, that is, (RQ)'(RQ) = Q' R' R Q = (n1) I
, and apply the same
transformation to the predicted values to yield, say, Z = \hat{Y} Q
.
Then, a principal component analysis on the covariance matrix of Z gives
eigenvalues of E^{1} H
, and so is equivalent to the MMRA analysis of
lm(Y ~ X)
statistically, but visualized here in canonical space.
Value
An object of class cancorr
, a list with the following
components:
cancor 
Canonical correlations, i.e., the correlations between each canonical variate for the Y variables with the corresponding canonical variate for the X variables. 
names 
Names for various
items, a list of 4 components: 
ndim 
Number of canonical dimensions extracted, 
dim 
Problem dimensions, a list of 3 components:

coef 
Canonical coefficients, a list of 2 components: 
scores 
Canonical variate scores, a list of 2 components: 
scores 
Canonical variate scores, a list of 2 components:

X 
The matrix X 
Y 
The matrix Y 
weights 
Observation weights, if supplied, else 
structure 
Structure correlations, a list of 4 components:

structure 
Structure correlations ("loadings"), a list of 4 components:
The formula method also returns components 
Methods (by class)

cancor(formula)
:"formula"
method. 
cancor(default)
:"default"
method.
Methods (by generic)

print(cancor)
:print()
method for"cancor"
objects. 
summary(cancor)
:summary()
method for"cancor"
objects. 
scores(cancor)
:scores()
method for"cancor"
objects. 
coef(cancor)
:coef()
method for"cancor"
objects.
Note
Not all features of CCA are presently implemented: standardized vs. raw scores, more flexible handling of missing data, other plot methods, ...
Author(s)
Michael Friendly
References
Gittins, R. (1985). Canonical Analysis: A Review with Applications in Ecology, Berlin: Springer.
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. London: Academic Press.
See Also
Other implementations of CCA: cancor
(very
basic), cca
in the yacca (fairly complete, but
very messy return structure), cc
in CCA (fairly
complete, very messy return structure, no longer maintained).
redundancy
, for redundancy analysis;
plot.cancor
, for enhanced scatterplots of the canonical
variates.
heplot.cancor
for CCA HE plots and
heplots
for generic heplot methods.
candisc
for related methods focused on multivariate linear
models with one or more factors among the X variables.
Examples
data(Rohwer, package="heplots")
X < as.matrix(Rohwer[,6:10]) # the PA tests
Y < as.matrix(Rohwer[,3:5]) # the aptitude/ability variables
# visualize the correlation matrix using corrplot()
if (require(corrplot)) {
M < cor(cbind(X,Y))
corrplot(M, method="ellipse", order="hclust", addrect=2, addCoef.col="black")
}
(cc < cancor(X, Y, set.names=c("PA", "Ability")))
## Canonical correlation analysis of:
## 5 PA variables: n, s, ns, na, ss
## with 3 Ability variables: SAT, PPVT, Raven
##
## CanR CanRSQ Eigen percent cum scree
## 1 0.6703 0.44934 0.81599 77.30 77.30 ******************************
## 2 0.3837 0.14719 0.17260 16.35 93.65 ******
## 3 0.2506 0.06282 0.06704 6.35 100.00 **
##
## Test of H0: The canonical correlations in the
## current row and all that follow are zero
##
## CanR WilksL F df1 df2 p.value
## 1 0.67033 0.44011 3.8961 15 168.8 0.000006
## 2 0.38366 0.79923 1.8379 8 124.0 0.076076
## 3 0.25065 0.93718 1.4078 3 63.0 0.248814
# formula method
cc < cancor(cbind(SAT, PPVT, Raven) ~ n + s + ns + na + ss, data=Rohwer,
set.names=c("PA", "Ability"))
# using observation weights
set.seed(12345)
wts < sample(0:1, size=nrow(Rohwer), replace=TRUE, prob=c(.05, .95))
(ccw < cancor(X, Y, set.names=c("PA", "Ability"), weights=wts) )
# show correlations of the canonical scores
zapsmall(cor(scores(cc, type="x"), scores(cc, type="y")))
# standardized coefficients
coef(cc, type="both", standardize=TRUE)
# plot canonical scores
plot(cc,
smooth=TRUE, pch=16, id.n = 3)
text(2, 1.5, paste("Can R =", round(cc$cancor[1], 3)), pos = 4)
plot(cc, which = 2,
smooth=TRUE, pch=16, id.n = 3)
text(2.2, 2.5, paste("Can R =", round(cc$cancor[2], 3)), pos = 4)
##################
data(schooldata)
##################
#fit the MMreg model
school.mod < lm(cbind(reading, mathematics, selfesteem) ~
education + occupation + visit + counseling + teacher, data=schooldata)
car::Anova(school.mod)
pairs(school.mod)
# canonical correlation analysis
school.cc < cancor(cbind(reading, mathematics, selfesteem) ~
education + occupation + visit + counseling + teacher, data=schooldata)
school.cc
heplot(school.cc, xpd=TRUE, scale=0.3)