Canonical Redundancy Analysis

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), ...)

Arguments

object: A "cancor" object
...: Other arguments
x: A "cancor.redundancy" for the print method.
digits: Number of digits to print

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

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.

Functions

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

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/

Author

Michael Friendly

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 
#>  0.173424  0.042113  0.007966  0.223503 
#> 
#> Redundancies for the Ability variables & total Y canonical redundancy
#> 
#>     Ycan1     Ycan2     Ycan3 total Y|X 
#>   0.22491   0.03688   0.01564   0.27743 
## 
## 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