Package 'candisc'

Visualizing Generalized Canonical Discriminant and Canonical Correlation Analysis

Description

This package includes functions for computing and visualizing generalized canonical discriminant analyses and canonical correlation analysis for a multivariate linear model. The goal is to provide ways of visualizing such models in a low-dimensional space corresponding to dimensions (linear combinations of the response variables) of maximal relationship to the predictor variables.

Details

Traditional canonical discriminant analysis is restricted to a one-way MANOVA design and is equivalent to canonical correlation analysis between a set of quantitative response variables and a set of dummy variables coded from the factor variable. The candisc package generalizes this to multi-way MANOVA designs for all terms in a multivariate linear model (i.e., an mlm object), computing canonical scores and vectors for each term (giving a candiscList object).

The graphic functions are designed to provide low-rank (1D, 2D, 3D) visualizations of terms in a mlm via the plot.candisc method, and the HE plot heplot.candisc and heplot3d.candisc methods. For mlms with more than a few response variables, these methods often provide a much simpler interpretation of the nature of effects in canonical space than heplots for pairs of responses or an HE plot matrix of all responses in variable space.

Analogously, a multivariate linear (regression) model with quantitative predictors can also be represented in a reduced-rank space by means of a canonical correlation transformation of the Y and X variables to uncorrelated canonical variates, Ycan and Xcan. Computation for this analysis is provided by cancor and related methods. Visualization of these results in canonical space are provided by the plot.cancor, heplot.cancor and heplot3d.cancor methods.

These relations among response variables in linear models can also be useful for “effect ordering” (Friendly & Kwan (2003) for variables in other multivariate data displays to make the displayed relationships more coherent. The function varOrder implements a collection of these methods.

A new vignette, vignette("diabetes", package="candisc"), illustrates some of these methods. A more comprehensive collection of examples is contained in the vignette for the heplots package,

vignette("HE-examples", package="heplots").

The organization of functions in this package and the heplots package may change in a later version.

Author(s)

Michael Friendly and John Fox

Maintainer: Michael Friendly <[email protected]>

References

Friendly, M. (2007). HE plots for Multivariate General Linear Models. Journal of Computational and Graphical Statistics, 16(2) 421–444. http://datavis.ca/papers/jcgs-heplots.pdf, doi:10.1198/106186007X208407.

Friendly, M. & Kwan, E. (2003). Effect Ordering for Data Displays, Computational Statistics and Data Analysis, 43, 509-539. doi:10.1016/S0167-9473(02)00290-6

Friendly, M. & Sigal, M. (2014). Recent Advances in Visualizing Multivariate Linear Models. Revista Colombiana de Estadistica , 37(2), 261-283. doi:10.15446/rce.v37n2spe.47934.

Friendly, M. & Sigal, M. (2017). Graphical Methods for Multivariate Linear Models in Psychological Research: An R Tutorial, The Quantitative Methods for Psychology, 13 (1), 20-45. doi:10.20982/tqmp.13.1.p020.

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

See Also

heplot for details about HE plots.

candisc, cancor for details about canonical discriminant analysis and canonical correlation analysis.

---

Transform a Multivariate Linear model mlm to a Canonical Representation

Description

This function uses candisc to transform the responses in a multivariate linear model to scores on canonical variables for a given term and then uses those scores as responses in a linear (lm) or multivariate linear model (mlm).

The function constructs a model formula of the form Can ~ terms where Can is the canonical score(s) and terms are the terms in the original mlm, then runs lm() with that formula.

Usage

can_lm(mod, term, ...)

Arguments

mod	A mlm object
term	One term in that model
...	Arguments passed to candisc

Value

A lm object if term is a rank 1 hypothesis, otherwise a mlm object

Author(s)

Michael Friendly

See Also

candisc, cancor

Examples

iris.mod <- lm(cbind(Petal.Length, Sepal.Length, Petal.Width, Sepal.Width) ~ Species, data=iris)
iris.can <- can_lm(iris.mod, "Species")
iris.can
car::Anova(iris.mod)
car::Anova(iris.can)

---

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 down-weighted.

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 cancor.default method, this should be a matrix or data.frame whose columns contain the X variables
...	Other arguments, passed to methods
formula	A two-sided formula of the form cbind(y1, y2, y3, ...) ~ x1 + x2 + x3 + ...
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 variance-covariance matrices are computed using cov.wt, and the number of observations is taken as the number of non-zero weights.
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 method = "gensvd" is supported;
y	For the cancor.default method, a matrix or data.frame whose columns contain the Y variables
X.names, Y.names	Character vectors of names for the X and Y variables.
row.names	Observation names in x, y
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 var determining how missing data are handled. Only the default, use="complete" is allowed when observation weights are supplied.
digits	Number of digits passed to print and summary methods
object	A cancor object for related methods.
type	For the coef method, the type of coefficients returned, one of "x", "y", "both". For the scores method, the same list, or "data.frame", which returns a data.frame containing the X and Y canonical scores.
standardize	For the coef method, whether coefficients should be standardized by dividing by the standard deviations of the X and Y variables.

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 low-rank (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 $E = R'R/(n-1)$. Choose a transformation Q that orthogonalizes the error covariance matrix to an identity, that is, $(RQ)'(RQ) = Q' R' R Q = (n-1) I$, 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: X, Y, row.names, set.names
ndim	Number of canonical dimensions extracted, <= min(p,q)
dim	Problem dimensions, a list of 3 components: p (number of X variables), q (number of Y variables), n (sample size)
coef	Canonical coefficients, a list of 2 components: X, Y
scores	Canonical variate scores, a list of 2 components: X, Y
scores	Canonical variate scores, a list of 2 components: X Canonical variate scores for the X variables Y Canonical variate scores for the Y variables
X	The matrix X
Y	The matrix Y
weights	Observation weights, if supplied, else NULL
structure	Structure correlations, a list of 4 components: X.xscores, Y.xscores, X.yscores, Y.yscores
structure	Structure correlations ("loadings"), a list of 4 components: X.xscores Structure correlations of the X variables with the Xcan canonical scores Y.xscores Structure correlations of the Y variables with the Xcan canonical scores X.yscores Structure correlations of the X variables with the Ycan canonical scores Y.yscores Structure correlations of the Y variables with the Ycan canonical scores The formula method also returns components call and terms

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)

---

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.

Usage

candisc(mod, ...)

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

## 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),
  ...
)

## 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,
  ...
)

Arguments

mod	An mlm object, such as computed by lm() with a multivariate response
...	arguments to be passed down. In particular, type="n" can be used with the plot method to suppress the display of canonical scores.
term	the name of one term from mod for which the canonical analysis is performed.
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 hypothesis term.
digits	significant digits to print.
LRtests	logical; should likelihood ratio tests for the canonical dimensions be printed?
object, x	A candisc object
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"
which	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.
conf	Confidence coefficient for the confidence circles around canonical means plotted in the plot method
col	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
pch	A vector of the unique 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
var.labels	Optional vector of variable labels to replace variable names in the plots
var.cex	Character expansion size for variable labels in the plots
var.pos	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.
rev.axes	Logical, a vector of length(which). TRUE causes the orientation of the canonical scores and structure coefficients to be reversed along a given axis.
ellipse	Draw data ellipses for canonical scores?
ellipse.prob	Coverage probability for the data ellipses
fill.alpha	Transparency value for the color used to fill the ellipses. Use fill.alpha to draw the ellipses unfilled.
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.
points.1d	Logical value for plot.candisc when only one canonical dimension.

Details

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.

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, initially 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/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#statug_candisc_sect012.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.

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.

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.

Methods (by class)

• candisc(mlm): "mlm" method.

Methods (by generic)

• print(candisc): print() method for "candisc" objects.

• summary(candisc): summary() method for "candisc" objects.

• coef(candisc): coef() method for "candisc" objects.

• plot(candisc): "plot" method.

Author(s)

Michael Friendly and John Fox

References

Bartlett, M. S. (1938). Further aspects of the theory of multiple regression. Proc. Cambridge Philosophical Society 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)
car::Anova(grass.mod, test="Wilks")

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

# 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)

heplot(iris.can)

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

---

Canonical discriminant analyses

Description

candiscList performs a generalized canonical discriminant analysis for all terms in a multivariate linear model (i.e., an mlm object), computing canonical scores and vectors.

Usage

candiscList(mod, ...)

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

## S3 method for class 'candiscList'
print(x, ...)

## S3 method for class 'candiscList'
summary(object, ...)

## S3 method for class 'candiscList'
plot(x, term, ask = interactive(), graphics = TRUE, ...)

Arguments

mod	An mlm object, such as computed by lm() with a multivariate response
...	arguments to be passed down.
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 the means, structure, scores and coeffs.* components. The default is the rank of the H matrix for the hypothesis term.
object, x	A candiscList object
term	The name of one term to be plotted for the plot method. If not specified, one candisc plot is produced for each term in the mlm object.
ask	If TRUE (the default, when running interactively), a menu of terms is presented; if ask is FALSE, canonical plots for all terms are produced.
graphics	if TRUE (the default, when running interactively), then the menu of terms to plot is presented in a dialog box rather than as a text menu.

Value

An object of class candiscList which is a list of "candisc" objects for the terms in the mlm.

Methods (by class)

• candiscList(mlm): "mlm" method.

Methods (by generic)

• print(candiscList): print() method for "candiscList" objects.

• summary(candiscList): summary() method for "candiscList" objects.

• plot(candiscList): plot() method for "candiscList" objects.

Author(s)

Michael Friendly and John Fox

See Also

candisc, heplot, heplot3d

Examples

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

grass.canL <-candiscList(grass.mod)
names(grass.canL)
names(grass.canL$Species)

## Not run: 
print(grass.canL)

## End(Not run)
plot(grass.canL, type="n", ask=FALSE)
heplot(grass.canL$Species, scale=6)
heplot(grass.canL$Block, scale=2)

---

Indices of observations in a model data frame

Description

Find sequential indices for observations in a data frame corresponding to the unique combinations of the levels of a given model term from a model object or a data frame

Usage

dataIndex(x, term)

Arguments

x	Either a data frame or a model object
term	The name of one term in the model, consisting only of factors

Value

A vector of indices.

Author(s)

Michael Friendly

Examples

factors <- expand.grid(A=factor(1:3),B=factor(1:2),C=factor(1:2))
n <- nrow(factors)
responses <-data.frame(Y1=10+round(10*rnorm(n)),Y2=10+round(10*rnorm(n)))

test <- data.frame(factors, responses)
mod <- lm(cbind(Y1,Y2) ~ A*B, data=test)

dataIndex(mod, "A")
dataIndex(mod, "A:B")

---

Yields from Nitrogen nutrition of grass species

Description

The data frame Grass gives the yield (10 * log10 dry-weight (g)) of eight grass Species in five replicates (Block) grown in sand culture at five levels of nitrogen.

Format

A data frame with 40 observations on the following 7 variables.

Species

a factor with levels B.media D.glomerata F.ovina F.rubra H.pubesens K.cristata L.perenne P.bertolonii

Block

a factor with levels 1 2 3 4 5

N1

species yield at 1 ppm Nitrogen

N9

species yield at 9 ppm Nitrogen

N27

species yield at 27 ppm Nitrogen

N81

species yield at 81 ppm Nitrogen

N243

species yield at 243 ppm Nitrogen

Details

Nitrogen (NaNO3) levels were chosen to vary from what was expected to be from critically low to almost toxic. The amount of Nitrogen can be considered on a log3 scale, with levels 0, 2, 3, 4, 5. Gittins (1985, Ch. 11) treats these as equally spaced for the purpose of testing polynomial trends in Nitrogen level.

The data are also not truly multivariate, but rather a split-plot experimental design. For the purpose of exposition, he regards Species as the experimental unit, so that correlations among the responses refer to a composite representative of a species rather than to an individual exemplar.

Source

Gittins, R. (1985), Canonical Analysis: A Review with Applications in Ecology, Berlin: Springer-Verlag, Table A-5.

Examples

str(Grass)
grass.mod <- lm(cbind(N1,N9,N27,N81,N243) ~ Block + Species, data=Grass)
car::Anova(grass.mod)

grass.canL <-candiscList(grass.mod)
names(grass.canL)
names(grass.canL$Species)

---

Canonical Correlation HE plots

Description

Hypothesis - Error (HE) plots for canonical correlation analysis provide a new graphical method for understanding the relations between two sets of variables, $\mathbf{X}$ and $\mathbf{Y}$. They are similar to HE plots for multivariate multiple regression (MMRA) problems, except that ...

These functions plot ellipses (or ellipsoids in 3D) in canonical space representing the hypothesis and error sums-of-squares-and-products matrices for terms in a multivariate linear model representing the result of a canonical correlation analysis. They provide a low-rank 2D (or 3D) view of the effects in the space of maximum canonical correlations, together with variable vectors representing the correlations of Y variables with the canonical dimensions.

For consistency with heplot.candisc, the plots show effects in the space of the canonical Y variables selected by which.

The interpretation of variable vectors in these plots is different from that of the terms plotted as H "ellipses," which appear as degenerate lines in the plot (because they correspond to 1 df tests of rank(H)=1).

In canonical space, the interpretation of the H ellipses for the terms is the same as in ordinary HE plots: a term is significant iff its H ellipse projects outside the (orthogonalized) E ellipsoid somewhere in the space of the Y canonical dimensions. The orientation of each H ellipse with respect to the Y canonical dimensions indicates which dimensions that X variate contributes to.

On the other hand, the variable vectors shown in these plots are intended only to show the correlations of Y variables with the canonical dimensions. Only their relative lengths and angles with respect to the Y canonical dimensions have meaning. Relative lengths correspond to proportions of variance accounted for in the Y canonical dimensions plotted; angles between the variable vectors and the canonical axes correspond to the structure correlations. The absolute lengths of these vectors are typically manipulated by the scale argument to provide better visual resolution and labeling for the variables.

Setting the aspect ratio of these plots is important for the proper interpretation of angles between the variable vectors and the coordinate axes. However, this then makes it impossible to change the aspect ratio of the plot by re-sizing manually.

Usage

## S3 method for class 'cancor'
heplot(
  mod,
  which = 1:2,
  scale,
  asp = 1,
  var.vectors = "Y",
  var.col = c("blue", "darkgreen"),
  var.lwd = par("lwd"),
  var.cex = par("cex"),
  var.xpd = TRUE,
  prefix = "Ycan",
  suffix = TRUE,
  terms = TRUE,
  ...
)

Arguments

mod	A cancor object
which	A numeric vector containing the indices of the Y canonical dimensions to plot.
scale	Scale factor for the variable vectors in canonical space. If not specified, the function calculates one to make the variable vectors approximately fill the plot window.
asp	aspect ratio setting. Use asp=1 in 2D plots and asp="iso" in 3D plots to ensure equal units on the axes. Use asp=NA in 2D plots and asp=NULL in 3D plots to allow separate scaling for the axes. See Details below.
var.vectors	Which variable vectors to plot? A character vector containing one or more of "X" and "Y".
var.col	Color(s) for variable vectors and labels, a vector of length 1 or 2. The first color is used for Y vectors and the second for X vectors, if these are plotted.
var.lwd	Line width for variable vectors
var.cex	Text size for variable vector labels
var.xpd	logical. Allow variable labels outside the plot box? Does not apply to 3D plots.
prefix	Prefix for labels of the Y canonical dimensions.
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.
terms	Terms for the X variables to be plotted in canonical space. The default, terms=TRUE or terms="X" plots H ellipses for all of the X variables. terms="Xcan" plots H ellipses for all of the X canonical variables, Xcan1, Xcan2, ....
...	Other arguments passed to link[heplots]{heplot}. In particular, you can pass linear hypotheses among the term variables via hypotheses.

Value

Returns invisibly an object of class "heplot", with coordinates for the various hypothesis ellipses and the error ellipse, and the limits of the horizontal and vertical axes.

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

cancor for details on canonical correlation as implemented here; plot.cancor for scatterplots of canonical variable scores. heplot.candisc, heplot, linearHypothesis

Examples

data(Rohwer, package="heplots")
X <- as.matrix(Rohwer[,6:10])
Y <- as.matrix(Rohwer[,3:5])
cc <- cancor(X, Y, set.names=c("PA", "Ability"))

# basic plot
heplot(cc)

# note relationship of joint hypothesis to individual ones
heplot(cc, scale=1.25, hypotheses=list("na+ns"=c("na", "ns")))

# more options
heplot(cc, hypotheses=list("All X"=colnames(X)),
	fill=c(TRUE,FALSE), fill.alpha=0.2,
	var.cex=1.5, var.col="red", var.lwd=3,
	prefix="Y canonical dimension"
	)

# 3D version
## Not run: 
heplot3d(cc, var.lwd=3, var.col="red")

## End(Not run)

---

Canonical Discriminant HE plots

Description

These functions plot ellipses (or ellipsoids in 3D) in canonical discriminant space representing the hypothesis and error sums-of-squares-and-products matrices for terms in a multivariate linear model. They provide a low-rank 2D (or 3D) view of the effects for that term in the space of maximum discrimination.

Usage

## S3 method for class 'candisc'
heplot(
  mod,
  which = 1:2,
  scale,
  asp = 1,
  var.col = "blue",
  var.lwd = par("lwd"),
  var.cex = par("cex"),
  var.pos,
  rev.axes = c(FALSE, FALSE),
  prefix = "Can",
  suffix = TRUE,
  terms = mod$term,
  ...
)

Arguments

mod	A candisc object for one term in a mlm
which	A numeric vector containing the indices of the canonical dimensions to plot.
scale	Scale factor for the variable vectors in canonical space. If not specified, the function calculates one to make the variable vectors approximately fill the plot window.
asp	Aspect ratio for the horizontal and vertical dimensions. The defaults, asp=1 for heplot.candisc and asp="iso" for heplot3d.candisc ensure equal units on all axes, so that angles and lengths of variable vectors are interpretable. As well, the standardized canonical scores are uncorrelated, so the Error ellipse (ellipsoid) should plot as a circle (sphere) in canonical space. For heplot3d.candisc, use asp=NULL to suppress this transformation to iso-scaled axes.
var.col	Color for variable vectors and labels
var.lwd	Line width for variable vectors
var.cex	Text size for variable vector labels
var.pos	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.
rev.axes	Logical, a vector of length(which). TRUE causes the orientation of the canonical scores and structure coefficients to be reversed along a given axis.
prefix	Prefix for labels of canonical dimensions.
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.
terms	Terms from the original mlm whose H ellipses are to be plotted in canonical space. The default is the one term for which the canonical scores were computed. If terms=TRUE, all terms are plotted.
...	Arguments to be passed down to heplot or heplot3d

Details

The generalized canonical discriminant analysis for one term in a mlm is based on the eigenvalues, $\lambda_i$, and eigenvectors, V, of the H and E matrices for that term. This produces uncorrelated canonical scores which give the maximum univariate F statistics. The canonical HE plot is then just the HE plot of the canonical scores for the given term.

For heplot3d.candisc, the default asp="iso" now gives a geometrically correct plot, but the third dimension, CAN3, is often small. Passing an expanded range in zlim to heplot3d usually helps.

Value

heplot.candisc returns invisibly an object of class "heplot", with coordinates for the various hypothesis ellipses and the error ellipse, and the limits of the horizontal and vertical axes.

Similarly, heploted.candisc returns an object of class "heplot3d".

Author(s)

Michael Friendly and John Fox

References

Friendly, M. (2006). Data Ellipses, HE Plots and Reduced-Rank Displays for Multivariate Linear Models: SAS Software and Examples Journal of Statistical Software, 17(6), 1-42. https://www.jstatsoft.org/v17/i06/ doi:10.18637/jss.v017.i06

Friendly, M. (2007). HE plots for Multivariate General Linear Models. Journal of Computational and Graphical Statistics, 16(2) 421–444. http://datavis.ca/papers/jcgs-heplots.pdf, doi:10.1198/106186007X208407.

See Also

candisc, candiscList, heplot, heplot3d, aspect3d

Examples

## Pottery data, from car package
data(Pottery, package = "carData")
pottery.mod <- lm(cbind(Al, Fe, Mg, Ca, Na) ~ Site, data=Pottery)
pottery.can <-candisc(pottery.mod)

heplot(pottery.can, var.lwd=3)
if(requireNamespace("rgl")){
heplot3d(pottery.can, var.lwd=3, scale=10, zlim=c(-3,3), wire=FALSE)
}


# reduce example for CRAN checks time

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

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

heplot(grass.can1, scale=6)
heplot(grass.can1, scale=6, terms=TRUE)
heplot(grass.canL, terms=TRUE, ask=FALSE)

heplot3d(grass.can1, wire=FALSE)
# compare with non-iso scaling
rgl::aspect3d(x=1,y=1,z=1)
# or,
# heplot3d(grass.can1, asp=NULL)


## Can't run this in example
# rgl::play3d(rgl::spin3d(axis = c(1, 0, 0), rpm = 5), duration=12)

# reduce example for CRAN checks time

## FootHead data, from heplots package
library(heplots)
data(FootHead)

# use Helmert contrasts for group
contrasts(FootHead$group) <- contr.helmert

foot.mod <- lm(cbind(width, circum,front.back,eye.top,ear.top,jaw)~group, data=FootHead)
foot.can <- candisc(foot.mod)
heplot(foot.can, main="Candisc HE plot", 
 hypotheses=list("group.1"="group1","group.2"="group2"),
 col=c("red", "blue", "green3", "green3" ), var.col="red")

---

Canonical Discriminant HE plots

Description

These functions plot ellipses (or ellipsoids in 3D) in canonical discriminant space representing the hypothesis and error sums-of-squares-and-products matrices for terms in a multivariate linear model. They provide a low-rank 2D (or 3D) view of the effects for that term in the space of maximum discrimination.

Usage

## S3 method for class 'candiscList'
heplot(mod, term, ask = interactive(), graphics = TRUE, ...)

Arguments

mod	A candiscList object for terms in a mlm
term	The name of one term to be plotted for the heplot and heplot3d methods. If not specified, one plot is produced for each term in the mlm object.
ask	If TRUE (the default), a menu of terms is presented; if ask is FALSE, canonical HE plots for all terms are produced.
graphics	if TRUE (the default, when running interactively), then the menu of terms to plot is presented in a dialog box rather than as a text menu.
...	Arguments to be passed down

Value

No useful value; used for the side-effect of producing canonical HE plots.

Author(s)

Michael Friendly and John Fox

References

Friendly, M. (2006). Data Ellipses, HE Plots and Reduced-Rank Displays for Multivariate Linear Models: SAS Software and Examples Journal of Statistical Software, 17(6), 1-42. https://www.jstatsoft.org/v17/i06/ doi:10.18637/jss.v017.i06.

Friendly, M. (2007). HE plots for Multivariate General Linear Models. Journal of Computational and Graphical Statistics, 16(2) 421–444. http://datavis.ca/papers/jcgs-heplots.pdf, doi:10.1198/106186007X208407.

See Also

candisc, candiscList, heplot, heplot3d

---

High School and Beyond Data

Description

The High School and Beyond Project was a longitudinal study of students in the U.S. carried out in 1980 by the National Center for Education Statistics. Data were collected from 58,270 high school students (28,240 seniors and 30,030 sophomores) and 1,015 secondary schools. The HSB data frame is sample of 600 observations, of unknown characteristics, originally taken from Tatsuoka (1988).

Format

A data frame with 600 observations on the following 15 variables. There is no missing data.

id

Observation id: a numeric vector

gender

a factor with levels male female

race

Race or ethnicity: a factor with levels hispanic asian african-amer white

ses

Socioeconomic status: a factor with levels low middle high

sch

School type: a factor with levels public private

prog

High school program: a factor with levels general academic vocation

locus

Locus of control: a numeric vector

concept

Self-concept: a numeric vector

mot

Motivation: a numeric vector

career

Career plan: a factor with levels clerical craftsman farmer homemaker laborer manager military operative prof1 prof2 proprietor protective sales school service technical not working

read

Standardized reading score: a numeric vector

write

Standardized writing score: a numeric vector

math

Standardized math score: a numeric vector

sci

Standardized science score: a numeric vector

ss

Standardized social science (civics) score: a numeric vector

Source

Tatsuoka, M. M. (1988). Multivariate Analysis: Techniques for Educational and Psychological Research (2nd ed.). New York: Macmillan, Appendix F, 430-442.

References

High School and Beyond data files: http://www.icpsr.umich.edu/icpsrweb/ICPSR/studies/7896

Examples

str(HSB)
# main effects model
hsb.mod <- lm( cbind(read, write, math, sci, ss) ~
		gender + race + ses + sch + prog, data=HSB)
car::Anova(hsb.mod)

# Add some interactions
hsb.mod1 <- update(hsb.mod, . ~ . + gender:race + ses:prog)
heplot(hsb.mod1, col=palette()[c(2,1,3:6)], variables=c("read","math"))

hsb.can1 <- candisc(hsb.mod1, term="race")
heplot(hsb.can1, col=c("red", "black"))

# show canonical results for all terms
## Not run: 
hsb.can <- candiscList(hsb.mod)
hsb.can

## End(Not run)

---

Canonical Correlation Plots

Description

This function produces plots to help visualize X, Y data in canonical space.

The present implementation plots the canonical scores for the Y variables against those for the X variables on given dimensions. We treat this as a view of the data in canonical space, and so offer additional annotations to a standard scatterplot.

Canonical correlation analysis assumes that the all correlations between the X and Y variables can be expressed in terms of correlations the canonical variate pairs, (Xcan1, Ycan1), (Xcan2, Ycan2), ..., and that the relations between these pairs are indeed linear.

Data ellipses, and smoothed (loess) curves, together with the linear regression line for each canonical dimension help to assess whether there are peculiarities in the data that might threaten the validity of CCA. Point identification methods can be useful to determine influential cases.

Usage

## S3 method for class 'cancor'
plot(
  x,
  which = 1,
  xlim,
  ylim,
  xlab,
  ylab,
  points = TRUE,
  add = FALSE,
  col = palette()[1],
  ellipse = TRUE,
  ellipse.args = list(),
  smooth = FALSE,
  smoother.args = list(),
  col.smooth = palette()[3],
  abline = TRUE,
  col.lines = palette()[2],
  lwd = 2,
  labels = rownames(xy),
  id.method = "mahal",
  id.n = 0,
  id.cex = 1,
  id.col = palette()[1],
  ...
)

Arguments

x	A "cancor" object
which	Which dimension to plot? An integer in 1:x$ndim.
xlim, ylim	Limits for x and y axes
xlab, ylab	Labels for x and y axes. If not specified, these are constructed from the set.names component of x.
points	logical. Display the points?
add	logical. Add to an existing plot?
col	Color for points.
ellipse	logical. Draw a data ellipse for the canonical scores?
ellipse.args	A list of arguments passed to dataEllipse. Internally, the function sets the default value for levels to 0.68.
smooth	logical. Draw a (loess) smoothed curve?
smoother.args	Arguments passed to loessLine, which should be consulted for details and defaults.
col.smooth	Color for the smoothed curve.
abline	logical. Draw the linear regression line for Ycan[,which] on Xcan[,which]?
col.lines	Color for the linear regression line
lwd	Line widths
labels	Point labels for point identification via the id.method argument.
id.method	Method used to identify individual points. See showLabels for details. The default, id.method = "mahal" identifies the id.n points furthest from the centroid.
id.n	Number of points to identify
id.cex, id.col	Character size and color for labeled points
...	Other arguments passed down to plot(...) and points(...)

Value

None. Used for its side effect of producing a plot. the value returned

Author(s)

Michael Friendly

References

Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. London: Academic Press.

See Also

cancor,

dataEllipse, loessLine, showLabels

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"))

plot(cc)
# exercise some options
plot(cc, which=1,
     smooth=TRUE, 
     pch = 16,
     id.n=3, ellipse.args=list(fill=TRUE, fill.alpha = 0.2))
plot(cc, which=2, smooth=TRUE)
plot(cc, which=3, smooth=TRUE)


# plot vectors showing structure correlations of Xcan and Ycan with their own variables
plot(cc)
struc <- cc$structure
Xstruc <- struc$X.xscores[,1]
Ystruc <- struc$Y.yscores[,1]
scale <- 2

# place vectors in the margins of the plot
usr <- matrix(par("usr"), nrow=2, dimnames=list(c("min", "max"), c("x", "y")))
ypos <- usr[2,2] - (1:5)/10 
arrows(0, ypos, scale*Xstruc, ypos, angle=10, len=0.1, col="blue")
text(scale*Xstruc, ypos, names(Xstruc), pos=2, col="blue")

xpos <- usr[2,1] - ( 1 + 1:3)/10
arrows(xpos, 0, xpos, scale*Ystruc, angle=10, len=0.1, col="darkgreen")
text(xpos, scale*Ystruc, names(Ystruc), pos=1, col="darkgreen")

---

Get predictor names from a lm-like model

Description

Get predictor names from a lm-like model

Usage

predictor.names(model, ...)

## Default S3 method:
predictor.names(model, ...)

Arguments

model	Model object
...	other arguments (ignored)

Value

A character vector of variable names

Methods (by class)

• predictor.names(default): "default" method.

Examples

#none

---

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 class '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

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.

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

Functions

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

Author(s)

Michael Friendly

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

---

Order variables according to canonical structure or other criteria

Description

The varOrder function implements some features of “effect ordering” (Friendly & Kwan (2003) for variables in a multivariate data display to make the displayed relationships more coherent.

This can be used in pairwise HE plots, scatterplot matrices, parallel coordinate plots, plots of multivariate means, and so forth.

For a numeric data frame, the most useful displays often order variables according to the angles of variable vectors in a 2D principal component analysis or biplot. For a multivariate linear model, the analog is to use the angles of the variable vectors in a 2D canonical discriminant biplot.

Usage

varOrder(x, ...)

## S3 method for class 'mlm'
varOrder(
  x,
  term,
  variables,
  type = c("can", "pc"),
  method = c("angles", "dim1", "dim2", "alphabet", "data", "colmean"),
  names = FALSE,
  descending = FALSE,
  ...
)

## S3 method for class 'data.frame'
varOrder(
  x,
  variables,
  method = c("angles", "dim1", "dim2", "alphabet", "data", "colmean"),
  names = FALSE,
  descending = FALSE,
  ...
)

## Default S3 method:
varOrder(x, ...)

Arguments

x	A multivariate linear model or a numeric data frame
...	Arguments passed to methods
term	For the mlm method, one term in the model for which the canonical structure coefficients are found.
variables	indices or names of the variables to be ordered; defaults to all response variables an MLM or all numeric variables in a data frame.
type	For an MLM, type="can" uses the canonical structure coefficients for the given term; type="pc" uses the principal component variable eigenvectors.
method	One of c("angles", "dim1", "dim2", "alphabet", "data", "colmean") giving the effect ordering method. "angles" Orders variables according to the angles their vectors make with dimensions 1 and 2, counter-clockwise starting from the lower-left quadrant in a 2D biplot or candisc display. "dim1" Orders variables in increasing order of their coordinates on dimension 1 "dim2" Orders variables in increasing order of their coordinates on dimension 2 "alphabet" Orders variables alphabetically "data" Uses the order of the variables in the data frame or the list of responses in the MLM "colmean" Uses the order of the column means of the variables in the data frame or the list of responses in the MLM
names	logical; if TRUE the effect ordered names of the variables are returned; otherwise, their indices in variables are returned.
descending	If TRUE, the ordered result is reversed to a descending order.

Value

A vector of integer indices of the variables or a character vector of their names.

Methods (by class)

• varOrder(mlm): "mlm" method.

• varOrder(data.frame): "data.frame" method.

• varOrder(default): "default" method.

Author(s)

Michael Friendly

References

Friendly, M. & Kwan, E. (2003). Effect Ordering for Data Displays, Computational Statistics and Data Analysis, 43, 509-539. doi:10.1016/S0167-9473(02)00290-6

Examples

data(Wine, package="candisc")
Wine.mod <- lm(as.matrix(Wine[, -1]) ~ Cultivar, data=Wine)
Wine.can <- candisc(Wine.mod)
plot(Wine.can, ellipse=TRUE)

# pairs.mlm HE plot, variables in given order
pairs(Wine.mod, fill=TRUE, fill.alpha=.1, var.cex=1.5)

order <- varOrder(Wine.mod)
pairs(Wine.mod, variables=order, fill=TRUE, fill.alpha=.1, var.cex=1.5)

---

Scale vectors to fill the current plot

Description

Calculates a scale factor so that a collection of vectors nearly fills the current plot, that is, the longest vector does not extend beyond the plot region.

Usage

vecscale(
  vectors,
  bbox = matrix(par("usr"), 2, 2),
  origin = c(0, 0),
  factor = 0.95
)

Arguments

vectors	a two-column matrix giving the end points of a collection of vectors
bbox	the bounding box of the containing plot region within which the vectors are to be plotted
origin	origin of the vectors
factor	maximum length of the rescaled vectors relative to the maximum possible

Value

scale factor, the multiplier of the vectors

Author(s)

Michael Friendly

See Also

vectors

Examples

bbox <- matrix(c(-3, 3, -2, 2), 2, 2)
colnames(bbox) <- c("x","y")
rownames(bbox) <- c("min", "max")
bbox

vecs <- matrix( runif(10, -1, 1), 5, 2)

plot(bbox)
arrows(0, 0, vecs[,1], vecs[,2], angle=10, col="red")
(s <- vecscale(vecs))
arrows(0, 0, s*vecs[,1], s*vecs[,2], angle=10)

---

Draw Labeled Vectors in 2D or 3D

Description

Graphics utility functions to draw vectors from an origin to a collection of points (using arrows in 2D or lines3d in 3D) with labels for each (using text or texts3d).

Usage

vectors(
  x,
  origin = c(0, 0),
  labels = rownames(x),
  scale = 1,
  col = "blue",
  lwd = 1,
  cex = 1,
  length = 0.1,
  angle = 13,
  pos = NULL,
  ...
)

Arguments

x	A two-column matrix or a three-column matrix containing the end points of the vectors
origin	Starting point(s) for the vectors
labels	Labels for the vectors
scale	A multiplier for the length of each vector
col	color(s) for the vectors.
lwd	line width(s) for the vectors.
cex	color(s) for the vectors.
length	For vectors, length of the edges of the arrow head (in inches).
angle	For vectors, angle from the shaft of the arrow to the edge of the arrow head.
pos	For vectors, position of the text label relative to the vector head. If pos==NULL, labels are positioned labels outside, relative to arrow ends.
...	other graphical parameters, such as lty, xpd, ...

Details

The graphical parameters col, lty and lwd can be vectors of length greater than one and will be recycled if necessary

Value

None

Author(s)

Michael Friendly

See Also

arrows, text, segments

lines3d, texts3d

Examples

plot(c(-3, 3), c(-3,3), type="n")
X <- matrix(rnorm(10), ncol=2)
rownames(X) <- LETTERS[1:5]
vectors(X, scale=2, col=palette())

---

Wilks Lambda Tests for Canonical Correlations

Description

Tests the sequential hypotheses that the $i$th canonical correlation and all that follow it are zero,

$\rho_i = \rho_{i+1} = \cdots = 0$

Usage

Wilks(object, ...)

## S3 method for class 'cancor'
Wilks(object, ...)

## S3 method for class 'candisc'
Wilks(object, ...)

Arguments

object	An object of class "cancor""} or \code{"candisc""
...	Other arguments passed to methods (not used)

Details

Wilks' Lambda values are calculated from the eigenvalues and converted to F statistics using Rao's approximation.

Value

A data.frame (of class "anova") containing the test statistics

Methods (by class)

• Wilks(cancor): "cancor" method.

• Wilks(candisc): print() method for "candisc" objects.

Author(s)

Michael Friendly

References

Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. London: Academic Press.

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"))
Wilks(cc)

iris.mod <- lm(cbind(Petal.Length, Sepal.Length, Petal.Width, Sepal.Width) ~ Species, data=iris)
iris.can <- candisc(iris.mod, data=iris)
Wilks(iris.can)

---

Chemical composition of three cultivars of wine

Description

These data are the results of a chemical analysis of wines grown in the same region in Italy but derived from three different cultivars. The analysis determined the quantities of 13 constituents found in each of the three types of wines.

Format

A data frame with 178 observations on the following 14 variables.

Cultivar

a factor with levels barolo grignolino barbera

Alcohol

a numeric vector

MalicAcid

a numeric vector

Ash

a numeric vector

AlcAsh

a numeric vector, Alkalinity of ash

Mg

a numeric vector, Magnesium

Phenols

a numeric vector, Total phenols

Flav

a numeric vector, Flavanoids

NonFlavPhenols

a numeric vector

Proa

a numeric vector, Proanthocyanins

Color

a numeric vector, color intensity

Hue

a numeric vector

OD

a numeric vector, OD280/OD315 of diluted wines

Proline

a numeric vector

Details

This data set is a classic in the machine learning literature as an easy high-D classification problem, but is also of interest for examples of MANOVA and discriminant analysis.

The precise definitions of these variables is unknown: units, how they were measured, etc.

Source

This data set was obtained from the UCI Machine Learning Repository, http://archive.ics.uci.edu/ml/datasets/Wine. This page references a large number of papers that use this data set to compare different methods.

References

In R, a comparable data set is contained in the ggbiplot package.

Examples

data(Wine)
str(Wine)
#summary(Wine)

Wine.mlm <- lm(as.matrix(Wine[, -1]) ~ Cultivar, data=Wine)
Wine.can <- candisc(Wine.mlm)
Wine.can


plot(Wine.can, ellipse=TRUE)
plot(Wine.can, which=1)

---

Wolf skulls

Description

Skull morphometric data on Rocky Mountain and Arctic wolves (Canis Lupus L.) taken from Morrison (1990), originally from Jolicoeur (1959).

Format

A data frame with 25 observations on the following 11 variables.

group

a factor with levels ar:f ar:m rm:f rm:m, comprising the combinations of location and sex

location

a factor with levels ar=Arctic, rm=Rocky Mountain

sex

a factor with levels f=female, m=male

x1

palatal length, a numeric vector

x2

postpalatal length, a numeric vector

x3

zygomatic width, a numeric vector

x4

palatal width outside first upper molars, a numeric vector

x5

palatal width inside second upper molars, a numeric vector

x6

postglenoid foramina width, a numeric vector

x7

interorbital width, a numeric vector

x8

braincase width, a numeric vector

x9

crown length, a numeric vector

Details

All variables are expressed in millimeters.

The goal was to determine how geographic and sex differences among the wolf populations are determined by these skull measurements. For MANOVA or (canonical) discriminant analysis, the factors group or location and sex provide alternative parameterizations.

Source

Morrison, D. F. Multivariate Statistical Methods, (3rd ed.), 1990. New York: McGraw-Hill, p. 288-289.

References

Jolicoeur, P. “Multivariate geographical variation in the wolf Canis lupis L.”, Evolution, XIII, 283–299.

Examples

data(Wolves)

# using group
wolf.mod <-lm(cbind(x1,x2,x3,x4,x5,x6,x7,x8,x9) ~ group, data=Wolves)
car::Anova(wolf.mod)

wolf.can <-candisc(wolf.mod)
plot(wolf.can)
heplot(wolf.can)

# using location, sex
wolf.mod2 <-lm(cbind(x1,x2,x3,x4,x5,x6,x7,x8,x9) ~ location*sex, data=Wolves)
car::Anova(wolf.mod2)

wolf.can2 <-candiscList(wolf.mod2)
plot(wolf.can2)

---