Ellipses, Data Ellipses, and Confidence Ellipses

These functions draw ellipses, including data ellipses, and confidence ellipses for linear, generalized linear, and possibly other models.

Usage

ellipse(center, shape, radius, log="", center.pch=19, center.cex=1.5, 
  segments=51, draw=TRUE, add=draw, xlab="", ylab="", 
  col=carPalette()[2], lwd=2, fill=FALSE, fill.alpha=0.3, grid=TRUE, ...)

dataEllipse(x, ...)

# S3 method for class 'formula'
dataEllipse(formula, data, subset, weights, xlab, ylab, id=FALSE, ...)

# Default S3 method
dataEllipse(x, y, groups, 
  group.labels=group.levels, ellipse.label, 
  weights, log="", levels=c(0.5, 0.95), center.pch=19, 
  center.cex=1.5, draw=TRUE, plot.points=draw, add=!plot.points, segments=51, 
  robust=FALSE, 
  xlab=deparse(substitute(x)), 
  ylab=deparse(substitute(y)), 
  col=if (missing(groups)) carPalette()[1:2] else carPalette()[1:length(group.levels)],
  pch=if (missing(groups)) 1 else seq(group.levels),
  lwd=2, fill=FALSE, fill.alpha=0.3, grid=TRUE, id=FALSE, 
  label.pos=NULL, label.cex=1.25, label.xpd=FALSE, ...)

confidenceEllipse(model, ...)

# S3 method for class 'lm'
confidenceEllipse(model, which.coef, vcov.=vcov, 
  L, levels=0.95, Scheffe=FALSE, dfn,
  center.pch=19, center.cex=1.5, segments=51, xlab, ylab, 
  col=carPalette()[2], lwd=2, fill=FALSE, fill.alpha=0.3, draw=TRUE, add=!draw,
  grid=TRUE, ...)
  
# S3 method for class 'glm'
confidenceEllipse(model, chisq, ...)

# S3 method for class 'mlm'
confidenceEllipse(model, xlab, ylab, which.coef=1:2, ...)

# Default S3 method
confidenceEllipse(model, which.coef, vcov.=vcov,
  L, levels=0.95, Scheffe=FALSE,  dfn,
  center.pch=19, center.cex=1.5, segments=51, xlab, ylab, 
  col=carPalette()[2], lwd=2, fill=FALSE, fill.alpha=0.3, draw=TRUE, add=!draw, 
  grid=TRUE, ...)
  
confidenceEllipses(model, ...)

# Default S3 method
confidenceEllipses(model, coefnames,  main, grid=TRUE, ...)

# S3 method for class 'mlm'
confidenceEllipses(model, coefnames, main, ...)

Arguments

center: 2-element vector with coordinates of center of ellipse.
shape: $2\times 2$ shape (or covariance) matrix.
radius: radius of circle generating the ellipse.
log: when an ellipse is to be added to an existing plot, indicates whether computations were on logged values and to be plotted on logged axes; "x" if the x-axis is logged, "y" if the y-axis is logged, and "xy" or "yx" if both axes are logged. The default is "", indicating that neither axis is logged.
center.pch: character for plotting ellipse center; if FALSE or NULL the center point isn't plotted.
center.cex: relative size of character for plotting ellipse center.
segments: number of line-segments used to draw ellipse.
draw: if TRUE produce graphical output; if FALSE, only invisibly return coordinates of ellipse(s).
add: if TRUE add ellipse(s) to current plot.
xlab: label for horizontal axis.
ylab: label for vertical axis.
x: a numeric vector, or (if y is missing) a 2-column numeric matrix.
y: a numeric vector, of the same length as x.
groups: optional: a factor to divide the data into groups; a separate ellipse will be plotted for each group (level of the factor).
formula: for the formula method, a model formula, of the form y ~ x or, if plotting by groups, y ~ x | z, where z evaluates to a factor or other variable dividing the data into groups.
data: for the formula method, a data frame within which to evaluate the formula.
subset: for the formula method, expression defining a subset of observations.
group.labels: labels to be plotted for the groups; by default, the levels of the groups factor.
ellipse.label: a label for the ellipse(s) or a vector of labels; if several ellipses are drawn and just one label is given, then that label will be repeated. The default is not to label the ellipses.
label.pos: position(s) of labels for the ellipses, a vector of integers (in 0:4) or character strings (in c("center", "bottom", "left", "top", "right"), or in c("C", "S", "W", "N", "E") used in labeling ellipses, recycled as necessary. Values of 1, 2, 3 and 4, respectively indicate positions below, to the left of, above and to the right of the max/min coordinates of the ellipse; the value 0 specifies the centroid of the ellipse object. The default, label.pos=NULL uses the correlation of the ellipse to determine "top" (r>=0) or "bottom" (r<0)

label.cex

character size for ellipse labels.

label.xpd

allow labels to be drawn outside of plotting region (default FALSE).

weights

a numeric vector of weights, of the same length as x and y to be used by cov.wt or cov.trob in computing a weighted covariance matrix; if absent, weights of 1 are used.

plot.points

if FALSE data ellipses are drawn, but points are not plotted.

levels

draw elliptical contours at these (normal) probability or confidence levels.

robust

if TRUE use the cov.trob function in the MASS package to calculate the center and covariance matrix for the data ellipse.

model

a model object produced by lm or glm.

which.coef

2-element vector giving indices of coefficients to plot; if missing, the first two coefficients (disregarding the regression constant) will be selected.

vcov.

a coefficient-covariance matrix or a function (such as hccm) to compute the coefficent-covariance matrix from model; the default is the vcov function.

Note that arguments supplied to ... are not passed to vcov. when it's a function; in this case either use an anonymous function in which the additional arguments are set, or supply the coefficient covariance matrix directly.

L

As an alternative to selecting coefficients to plot, a transformation matrix can be specified to compute two linear combinations of the coefficients; if the L matrix is given, it takes precedence over the which.coef argument. L should have two rows and as many columns as there are coefficients. It can be given directly as a numeric matrix, or specified by a pair of character-valued expressions, in the same manner as for the link{linearHypothesis} function, but with no right-hand side.

Scheffe

if TRUE scale the ellipse so that its projections onto the axes give Scheffe confidence intervals for the coefficients.

dfn

“numerator” degrees of freedom (or just degrees of freedom for a GLM) for drawing the confidence ellipse. Defaults to the number of coefficients in the model (disregarding the constant) if Scheffe is TRUE, or to 2 otherwise; selecting dfn = 1 will draw the “confidence-interval generating” ellipse, with projections on the axes corresponding to individual confidence intervals with the stated level of coverage.

chisq

if TRUE, the confidence ellipse for the coefficients in a generalized linear model are based on the chisquare statistic, if FALSE on the $F$-statistic. This corresponds to using the default and linear-model methods respectively.

col

color for lines and ellipse center; the default is the second entry in the current car palette (see carPalette and par). For dataEllipse, two colors can be given, in which case the first is for plotted points and the second for lines and the ellipse center; if ellipses are plotted for groups, then this is a vector of colors for the groups.

pch

for dataEllipse this is the plotting character (default, symbol 1, a hollow circle) to use for the points; if ellipses are plotted by groups, then this a vector of plotting characters, with consecutive symbols starting with 1 as the default.

lwd

line width; default is 2 (see par).

fill

fill the ellipse with translucent color col (default, FALSE)?

fill.alpha

transparency of fill (default = 0.3).

...

other plotting parameters to be passed to plot and line.

id

controls point identification; if FALSE (the default), no points are identified; can be a list of named arguments to the showLabels function; TRUE is equivalent to list(method="mahal", n=2, cex=1, col=carPalette()[1], location="lr") (with the default col actually dependent on the number of groups), which identifies the 2 points with the largest Mahalanobis distances from the center of the data.

grid

If TRUE, the default, a light-gray background grid is put on the graph

coefnames

character vector of coefficient names to use to label the diagonal of the pairwise confidence ellipse matrix plotted by confidenceEllipses; defaults to the names of the coefficients in the model.

main

title for matrix of pairwise confidence ellipses.

Details

The ellipse is computed by suitably transforming a unit circle.

dataEllipse superimposes the normal-probability contours over a scatterplot of the data.

confidenceEllipses plots a matrix of all pairwise confidence ellipses; each panel of the matrix is created by confidenceEllipse.

Value

These functions are mainly used for their side effect of producing plots. For greater flexibility (e.g., adding plot annotations), however, ellipse returns invisibly the (x, y) coordinates of the calculated ellipse. dataEllipse and confidenceEllipse return invisibly the coordinates of one or more ellipses, in the latter instance a list named by levels; confidenceEllipses invisibly returns NULL.

References

Fox, J. (2016) Applied Regression Analysis and Generalized Linear Models, Third Edition. Sage.

Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression, Third Edition, Sage.

Monette, G. (1990) Geometry of multiple regression and 3D graphics. In Fox, J. and Long, J. S. (Eds.) Modern Methods of Data Analysis. Sage.

Author

Georges Monette, John Fox jfox@mcmaster.ca, and Michael Friendly.

Examples

dataEllipse(Duncan$income, Duncan$education, levels=0.1*1:9, 
    ellipse.label=0.1*1:9, lty=2, fill=TRUE, fill.alpha=0.1)


# formula method
dataEllipse(education ~ income | type, data=Duncan,
            levels = 0.68)


dataEllipse(education ~ income | type, data=Duncan, 
            levels=0.68, label.xpd=TRUE)


# vary label positions
dataEllipse(education ~ income | type, data=Duncan,
            levels = 0.68,
            fill = TRUE, fill.alpha = 0.1,
            label.pos = c(3, 2, 4),
            xlim = c(0, 100),
            ylim = c(2, 120))


    
confidenceEllipse(lm(prestige ~ income + education, data=Duncan), Scheffe=TRUE)


confidenceEllipse(lm(prestige ~ income + education, data=Duncan), vcov.=hccm)


confidenceEllipse(lm(prestige ~ income + education, data=Duncan), 
  L=c("income + education", "income - education"))

  
confidenceEllipses(lm(prestige ~ income + education + type, data=Duncan),
  fill=TRUE)

cov2cor(vcov(lm(prestige ~ income + education + type, 
  data=Duncan))) # correlations among coefficients
#>             (Intercept)     income  education   typeprof     typewc
#> (Intercept)   1.0000000 -0.3484661 -0.6102747  0.5427211  0.3483203
#> income       -0.3484661  1.0000000 -0.2880828 -0.2016613 -0.1997770
#> education    -0.6102747 -0.2880828  1.0000000 -0.7761809 -0.5592417
#> typeprof      0.5427211 -0.2016613 -0.7761809  1.0000000  0.7072750
#> typewc        0.3483203 -0.1997770 -0.5592417  0.7072750  1.0000000

wts <- rep(1, nrow(Duncan))
wts[c(6, 16)] <- 0 # delete Minister, Conductor
with(Duncan, {
  dataEllipse(income, prestige, levels=0.68)
  dataEllipse(income, prestige, levels=0.68, robust=TRUE, 
      plot.points=FALSE, col="green3")
  dataEllipse(income, prestige, weights=wts, levels=0.68, 
      plot.points=FALSE, col="brown")
  dataEllipse(income, prestige, weights=wts, robust=TRUE, levels=0.68, 
    plot.points=FALSE, col="blue")
  })

    
with(Prestige, dataEllipse(income, education, type, 
    id=list(n=2, labels=rownames(Prestige)), pch=15:17,
    xlim=c(0, 25000), center.pch="+",
    group.labels=c("Blue Collar", "Professional", "White Collar"),
    ylim=c(5, 20), level=.95, fill=TRUE, fill.alpha=0.1))