Produces a plot of standard deviations versus means with a confidence region based on either robust or classical estimates of location and scale.
xs.plot(x, …)# S3 method for default
xs.plot(x,g,s, degfree, labels.arg=NA, mu, sigma,
probs=c(0.5, 0.95, 0.99), basis=c("robust","classical"),
method=c("chisq","density"), main=paste("X-S plot -", basis, "basis"),
xlab=deparse(substitute(x)), ylab,
contours=TRUE, col.contours="lightgrey", lty.contours=par("lty"),
lwd.contours=par("lwd"),
label.contours=contours, format.clab="p=%3.2f",
pos.clab="bottomright", col.clab=col.contours, cex.clab=0.7,
cex.label=0.7, pos=3, adj=NULL,
pch=par("pch"), col=par("col"), bg=par("bg"), cex=par("cex"),
add=FALSE, …)
XSplot(x, …)
Numeric vector of values to be plotted.
grouping factor of length length(x).
numeric vector of standard deviations of length length(x)
associated with x.
A single value for degrees of freedom associated with all
the standard deviations s. Ignored if s is not supplied.
An optional vector of point labels, coerced to character on use.
A single location used to centre the confidence region. The default is
specified by "basis"; see Details.
A measure of dispersion against which deviations x-mu can be compared.
A vector of probabilities for confidence region contours.
Controls the nature of the location and scale estimators used to produce the confidence contours drawn on the plot. See Details for specification.
The method used to calculate the confidence region. See Details.
Main title for the plot.
x- and y-axis labels,
logical, specifying whether confidence contours should be drawn.
Colour, line type and line width for contour lines.
Logical, controlling whether contour lines are labelled with approximate probabilities.
Format string for contour labels, passed to sprintf.
Specification for location of contour labels. A vector can be provided to give multiple labels. See Details for further description.
Colour and expansion for contour labels.
Graphical parameters passed to points.
Expansion factor for point labels, passed to text.
Specifies position/adjustment of point labels. Passed to text;
see text for details.
If TRUE the plot region is not cleared before plotting; points and contours
are added to the present plot. Use pch=NA to suppress symbols if only added
contours are required.
Other parameters passed to plot.
A list with components:
respectively, the plotted locations and standard deviations.
(the names allow a simple call to plot())
The location and pooled SD estimates \(\hat{\mu}\) and \(\hat{\sigma}\) used to construct the confidence ellipsoids.
A list of sets of coordinates for each confidence region.
A plot of standard deviations against locations is produced, together with optional confidence region(s) calculated (by default) by a method suggested in ISO 13528:2005.
If s is supplied, x is taken as a vector of locations and
s a vector of standard deviations. degfree must be supplied
in this case.
If g is supplied and s is not, the locations and standard deviations
used are the means and standard deviations for each group. degfree is calculated
from the median group size. Groups should, of course, be of the same size for accurate
inference; however, using the median group size allows for some groups with missing values.
If s and g are both supplied, g is ignored with a warning
If requested by contours=TRUE, confidence regions are drawn for each value
of probs. Contour location and shape are controlled by basis
which specifies the location and scale estimators used, and method, which
specifies the method of calculation for the contours. Two methods are supported;
one using the chi-squared distribution (method="chisq") and one based on
equal density countours (method="density"). The default, and the method
recommended by the cited Standard, is method="chisq" and basis="robust".
Both calculations for confidence regions require estimation of a location \(\hat{\mu}\)
and an estimate \(\hat{\sigma}\) of the pooled within-group standard deviation or pooled
estimate from s. If basis="robust", \(\hat{\mu}\) and \(\hat{\sigma}\)
are calculated using algA and algS respectively. If basis="classical",
\(\hat{\mu}\) and \(\hat{\sigma}\) are the mean of the group means and the classical pooled
standard deviation respectively. If mu or sigma are given, these are used
in place of the calculated \(\hat{\mu}\) and \(\hat{\sigma}\) respectively.
If method="chisq", contours for probability \(p\) are calculated as
$$s=\hat{\sigma}\exp\left ( \pm\frac{1}{\sqrt{2(n-1)}}\sqrt{\chi_{2,p}^2-n \left ( \frac{x-\hat{\mu}}{\hat{\sigma}}\right ) ^2}\right )$$
for \(x\) from \( \hat{\mu}-\hat{\sigma}\sqrt{\frac{\chi_{2,p}^2}{n}}\) to \(\hat{\mu}+\hat{\sigma}\sqrt{\frac{\chi_{2,p}^2}{n}}\).
If method="density", contours for probability \(p\) are calculated using
Helmert's distribution to provide constant likelihood contours round the chosen mean
and standard deviation. In the present implementation, these are found using
uniroot to find the mean corresponding to the required density at
given standard deviations. The density chosen is \(d_{max}(1-p)\) where \(p\) is
the probability and \(d_{max}\) the maximum density for Helmert's
distribution for the requisite nunber of degrees of freedom. (See Kruskal
(1946) for a description of Helmert's distribution and, for example,
Pawitan (2001) for the rationale behind the choice of density
contour level.) This seems to give reasonably good results for \(n \ge 3 \)
but is anticonservative (particularly to high \(s\)) for \(n = 2 \).
Contours are by default labelled. Label locations can be specified using pos.clab.
Options are code"top", code"topright", code"right", code"bottomright",
code"bottom", code"bottomleft", code"left" and code"topleft". A vector can be
specified to give labels at more than one such location.
Contour labels are usually placed approximately at the location(s) indicated and adjusted outward
appropriately. For the special case of method="density" and degfree=1 (or where group
sizes \(n=1\)), for which the region is a maximu width at s=0,
"bottomright" and "bottomleft" place labels immediately below the countour boundary
at \(s=0\) and, if specified, "bottom" is replaced with c("bottomright", "bottomleft").
XSplot is an alias for xs.plot.
ISO 13528:2005, Statistical methods for use in proficiency testing by interlaboratory comparisons, International Organization for Standardization, Geneva (2005)
Y Pawitan, (2001) In all likelihood: Statistical Modelling and Inference Using Likelihood,Clarendon Press, Oxford, pp258-9
W Kruskal, American Mathematical Monthly 53, 435-438, (1946)
axis for axis control; points, text for
plotting parameters; sprintf for contour label formatting.
duewer.plot for an alternative plot for locations and associated
standard errors or standard uncertainties;
# NOT RUN {
require(metRology)
set.seed(1017)
x <- rnorm(80)
g <- gl(20,4)
xs.plot(x,g)
#Identical plot with precalculated s:
X <- tapply(x,g,mean)
S <- tapply(x,g,sd)
xs.plot(X, s=S, degfree=3)
#Specify different location and within-group SD estimates:
xs.plot(X, s=S, degfree=3, mu=median(X), sigma=median(S))
#Illustrate multiple contour labelling, point labels and further embellishment
rv <- xs.plot(x,g, pos.clab=c("bottomleft", "bottomright"), labels=TRUE)
abline(v=rv$mu, h=rv$s, col=2)
# }
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