Canonical Correlation HE plots

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 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.

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.

Author

Michael Friendly

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)

#> Vector scale factor set to  1 

# 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"
  )

#> Vector scale factor set to  2 

# 3D version
if (FALSE) {
heplot3d(cc, var.lwd=3, var.col="red")
}