candisc (version 0.9.0)

redundancy: Canonical Redundancy Analysis

Description

Calculates indices of redundancy (Stewart & Love, 1968) from a canonical correlation analysis. These give the proportion of variances of the variables in each set (X and Y) which are accounted for by the variables in the other set through the canonical variates.

Usage

redundancy(object, ...)
# S3 method for cancor.redundancy
print(x, digits = max(getOption("digits") - 3, 3), ...)

Value

An object of class "cancor.redundancy", a list with the following 5 components:

Xcan.redun

Canonical redundancies for the X variables, i.e., the total fraction of X variance accounted for by the Y variables through each canonical variate.

Ycan.redun

Canonical redundancies for the Y variables

X.redun

Total canonical redundancy for the X variables, i.e., the sum of Xcan.redun.

Y.redun

Total canonical redundancy for the Y variables

set.names

names for the X and Y sets of variables

Arguments

object

A "cancor" object

...

Other arguments

x

A "cancor.redundancy" for the print method.

digits

Number of digits to print

Functions

• print(cancor.redundancy): print() method for "cancor.redundancy" objects.

Author

Michael Friendly

Details

The term "redundancy analysis" has a different interpretation and implementation in the environmental ecology literature, such as the vegan. In that context, each \(Y_i\) variable is regressed separately on the predictors in \(X\), to give fitted values \(\widehat{Y} = [\widehat{Y}_1, \widehat{Y}_2, \dots\). Then a PCA of \(\widehat{Y}\) is carried out to determine a reduced-rank structure of the predictions.

References

Muller K. E. (1981). Relationships between redundancy analysis, canonical correlation, and multivariate regression. Psychometrika, 46(2), 139-42.

Stewart, D. and Love, W. (1968). A general canonical correlation index. Psychological Bulletin, 70, 160-163.

Brainder, "Redundancy in canonical correlation analysis", https://brainder.org/2019/12/27/redundancy-in-canonical-correlation-analysis/

See Also

cancor

Examples

	data(Rohwer, package="heplots")
X <- as.matrix(Rohwer[,6:10])  # the PA tests
Y <- as.matrix(Rohwer[,3:5])   # the aptitude/ability variables

cc <- cancor(X, Y, set.names=c("PA", "Ability"))

redundancy(cc)
## 
## Redundancies for the PA variables & total X canonical redundancy
## 
##     Xcan1     Xcan2     Xcan3 total X|Y 
##   0.17342   0.04211   0.00797   0.22350 
## 
## Redundancies for the Ability variables & total Y canonical redundancy
## 
##     Ycan1     Ycan2     Ycan3 total Y|X 
##    0.2249    0.0369    0.0156    0.2774