candisc {candisc}R Documentation

Canonical discriminant analysis


candisc performs a generalized canonical discriminant analysis for one term in a multivariate linear model (i.e., an mlm object), computing canonical scores and vectors. It represents a transformation of the original variables into a canonical space of maximal differences for the term, controlling for other model terms.

In typical usage, the term should be a factor or interaction corresponding to a multivariate test with 2 or more degrees of freedom for the null hypothesis.


candisc(mod, ...)

## S3 method for class 'mlm'
candisc(mod, term, type = "2", manova, ndim = rank, ...)

## S3 method for class 'candisc'
coef(object, type = c("std", "raw", "structure"), ...)

## S3 method for class 'candisc'
plot(x, which = 1:2, conf = 0.95, col, pch, scale, asp = 1,
    var.col = "blue", var.lwd = par("lwd"), var.labels, var.cex = 1, var.pos,
    rev.axes=c(FALSE, FALSE),
		ellipse=FALSE, ellipse.prob = 0.68, fill.alpha=0.1,
    prefix = "Can", suffix=TRUE, 
    titles.1d = c("Canonical scores", "Structure"),
    points.1d = FALSE, ...)
## S3 method for class 'candisc'
print(x, digits=max(getOption("digits") - 2, 3), LRtests=TRUE, ...)

## S3 method for class 'candisc'
summary(object, means = TRUE, scores = FALSE, coef = c("std"),
    ndim, digits = max(getOption("digits") - 2, 4), ...)



An mlm object, such as computed by lm() with a multivariate response


the name of one term from mod for which the canonical analysis is performed.


type of test for the model term, one of: "II", "III", "2", or "3"


the Anova.mlm object corresponding to mod. Normally, this is computed internally by Anova(mod)


Number of dimensions to store in (or retrieve from, for the summary method) the means, structure, scores and coeffs.* components. The default is the rank of the H matrix for the hypothesis term.

object, x

A candisc object


A vector of one or two integers, selecting the canonical dimension(s) to plot. If the canonical structure for a term has ndim==1, or length(which)==1, a 1D representation of canonical scores and structure coefficients is produced by the plot method. Otherwise, a 2D plot is produced.


Confidence coefficient for the confidence circles around canonical means plotted in the plot method


A vector of the unique colors to be used for the levels of the term in the plot method, one for each level of the term. In this version, you should assign colors and point symbols explicitly, rather than relying on the somewhat arbitrary defaults, based on palette


A vector of the unique point symbols to be used for the levels of the term in the plot method



Scale factor for the variable vectors in canonical space. If not specified, a scale factor is calculated to make the variable vectors approximately fill the plot space.


Aspect ratio for the plot method. The asp=1 (the default) assures that the units on the horizontal and vertical axes are the same, so that lengths and angles of the variable vectors are interpretable.


Color used to plot variable vectors


Line width used to plot variable vectors


Optional vector of variable labels to replace variable names in the plots


Character expansion size for variable labels in the plots


Position(s) of variable vector labels wrt. the end point. If not specified, the labels are out-justified left and right with respect to the end points.


Logical, a vector of length(which). TRUE causes the orientation of the canonical scores and structure coefficients to be reversed along a given axis.


Draw data ellipses for canonical scores?


Coverage probability for the data ellipses


Transparency value for the color used to fill the ellipses. Use fill.alpha to draw the ellipses unfilled.


Prefix used to label the canonical dimensions plotted


Suffix for labels of canonical dimensions. If suffix=TRUE the percent of hypothesis (H) variance accounted for by each canonical dimension is added to the axis label.


A character vector of length 2, containing titles for the panels used to plot the canonical scores and structure vectors, for the case in which there is only one canonical dimension.


Logical value for plot.candisc when only one canonical dimension.


Logical value used to determine if canonical means are printed


Logical value used to determine if canonical scores are printed


Type of coefficients printed by the summary method. Any one or more of "std", "raw", or "structure"


significant digits to print.


logical; should likelihood ratio tests for the canonical dimensions be printed?


arguments to be passed down. In particular, type="n" can be used with the plot method to suppress the display of canonical scores.


Canonical discriminant analysis is typically carried out in conjunction with a one-way MANOVA design. It represents a linear transformation of the response variables into a canonical space in which (a) each successive canonical variate produces maximal separation among the groups (e.g., maximum univariate F statistics), and (b) all canonical variates are mutually uncorrelated. For a one-way MANOVA with g groups and p responses, there are dfh = min( g-1, p) such canonical dimensions, and tests, initally stated by Bartlett (1938) allow one to determine the number of significant canonical dimensions.

Computational details for the one-way case are described in Cooley & Lohnes (1971), and in the SAS/STAT User's Guide, "The CANDISC procedure: Computational Details,"

A generalized canonical discriminant analysis extends this idea to a general multivariate linear model. Analysis of each term in the mlm produces a rank dfh H matrix sum of squares and crossproducts matrix that is tested against the rank dfe E matrix by the standard multivariate tests (Wilks' Lambda, Hotelling-Lawley trace, Pillai trace, Roy's maximum root test). For any given term in the mlm, the generalized canonical discriminant analysis amounts to a standard discriminant analysis based on the H matrix for that term in relation to the full-model E matrix.

The plot method for candisc objects is typically a 2D plot, similar to a biplot. It shows the canonical scores for the groups defined by the term as points and the canonical structure coefficients as vectors from the origin.

If the canonical structure for a term has ndim==1, or length(which)==1, the 1D representation consists of a boxplot of canonical scores and a vector diagram showing the magnitudes of the structure coefficients.


An object of class candisc with the following components:


hypothesis degrees of freedom for term


error degrees of freedom for the mlm


number of non-zero eigenvalues of HE^{-1}


eigenvalues of HE^{-1}


squared canonical correlations


A vector containing the percentages of the canrsq of their total.


Number of canonical dimensions stored in the means, structure and coeffs.* components


A data.frame containing the class means for the levels of the factor(s) in the term


A data frame containing the levels of the factor(s) in the term


name of the term


A character vector containing the names of the terms in the mlm object


A matrix containing the raw canonical coefficients


A matrix containing the standardized canonical coefficients


A matrix containing the canonical structure coefficients on ndim dimensions, i.e., the correlations between the original variates and the canonical scores. These are sometimes referred to as Total Structure Coefficients.


A data frame containing the predictors in the mlm model and the canonical scores on ndim dimensions. These are calculated as Y %*% coeffs.raw, where Y contains the standardized response variables.


Michael Friendly and John Fox


Bartlett, M. S. (1938). Further aspects of the theory of multiple regression. Proc. Camb. Phil. Soc. 34, 33-34.

Cooley, W.W. & Lohnes, P.R. (1971). Multivariate Data Analysis, New York: Wiley.

Gittins, R. (1985). Canonical Analysis: A Review with Applications in Ecology, Berlin: Springer.

See Also

candiscList, heplot, heplot3d


grass.mod <- lm(cbind(N1,N9,N27,N81,N243) ~ Block + Species, data=Grass)
Anova(grass.mod, test="Wilks")

grass.can1 <-candisc(grass.mod, term="Species")

# library(heplots)
heplot(grass.can1, scale=6, fill=TRUE)

# iris data
iris.mod <- lm(cbind(Petal.Length, Sepal.Length, Petal.Width, Sepal.Width) ~ Species, data=iris)
iris.can <- candisc(iris.mod, data=iris)
#-- assign colors and symbols corresponding to species
col <- c("red", "brown", "green3")
pch <- 1:3
plot(iris.can, col=col, pch=pch)


# 1-dim plot
iris.can1 <- candisc(iris.mod, data=iris, ndim=1)

[Package candisc version 0.8-5 Index]