candisc (version 0.5-16)

candisc: Canonical discriminant analysis

Description

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. To be of any use, the term should be a factor or interaction corresponding to a multivariate test with 2 or more degrees of freedom for the null hypothesis.

Usage

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"), prefix = "Can", suffix=TRUE, 
    titles.1d = c("Canonical scores", "Structure"), ...)
    
## S3 method for class 'candisc':
print(x, digits=max(getOption("digits") - 2, 3), ...)

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

Arguments

mod

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

term

the name of one term from mod

type

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

manova

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

ndim

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 hyp

object, x

A candisc object

which

A vector of two integers, selecting the canonical dimensions to plot

conf

Confidence coefficient for the confidence circles plotted in the plot method

col

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

pch

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

scale

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.

asp

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.

var.col

Color used to plot variable vectors

var.lwd

Line width used to plot variable vectors

prefix

Prefix used to label the canonical dimensions plotted

suffix

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.

titles.1d

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.

means

Logical value used to determine if canonical means are printed

scores

Logical value used to determine if canonical scores are printed

coef

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

digits

significant digits to print.

...

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

Value

• An object of class candisc with the following components:
• dfh
  hypothesis degrees of freedom for term
• dfe
  error degrees of freedom for the mlm
• rank
  number of non-zero eigenvalues of $HE^{-1}$
• eigenvalues
  eigenvalues of $HE^{-1}$
• canrsq
  squared canonical correlations
• pct
  A vector containing the percentages of the canrsq of their total.
• ndim
  Number of canonical dimensions stored in the means, structure and coeffs.* components
• means
  A data.frame containing the class means for the levels of the factor(s) in the term
• factors
  A data frame containing the levels of the factor(s) in the term
• term
  name of the term
• terms
  A character vector containing the names of the terms in the mlm object
• coeffs.raw
  A matrix containing the raw canonical coefficients
• coeffs.std
  A matrix containing the standardized canonical coefficients
• structure
  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.
• scores
  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.

Details

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," http://support.sas.com/onlinedoc/913/getDoc/en/statug.hlp/candisc_sect12.htm. A generalized canonical discriminant analysis extends this idea to a general multivariate linear model. Analysis of each term in the mlm produces a rank $df_h$ H matrix sum of squares and crossproducts matrix that is tested against the rank $df_e$ 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.

References

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

Examples

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")
plot(grass.can1, type="n")

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

# 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 <- rep(c("red", "black", "blue"), each=50)
pch <- rep(1:3, each=50)
plot(iris.can, col=col, pch=pch)

heplot(iris.can)

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