| Title: | Support for Metrological Applications |
|---|---|
| Description: | Provides classes and calculation and plotting functions for metrology applications, including measurement uncertainty estimation and inter-laboratory metrology comparison studies. |
| Authors: | Stephen L R Ellison <[email protected]>. |
| Maintainer: | Stephen L R Ellison <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.9-28-1 |
| Built: | 2024-12-21 06:42:22 UTC |
| Source: | CRAN |
Generates a parametric bootstrap for the median of scaled differences from each point in a data set to all other points..
bootMSD(x, ...) ## Default S3 method: bootMSD(x, s = mad, B = 3000, probs = c(0.95, 0.99), method = c("rnorm", "lhs"), keep = FALSE, labels = names(x), ...) ## S3 method for class 'MSD' bootMSD(x, B = 3000, probs = c(0.95, 0.99), method = c("rnorm", "lhs"), keep = FALSE, labels = names(x), ...)bootMSD(x, ...) ## Default S3 method: bootMSD(x, s = mad, B = 3000, probs = c(0.95, 0.99), method = c("rnorm", "lhs"), keep = FALSE, labels = names(x), ...) ## S3 method for class 'MSD' bootMSD(x, B = 3000, probs = c(0.95, 0.99), method = c("rnorm", "lhs"), keep = FALSE, labels = names(x), ...)
x |
An R object. For the default method, a vector of observations. For the |
s |
Either a function returning an estimate of scale for |
B |
Scalar number of bootstrap replicates. |
probs |
Vector of probabilities at which to calculate upper quantiles. Passed to
|
method |
Character value describing the simulation method. |
keep |
If |
labels |
Character vector of labels for individual values. |
... |
Parameters passed to other methods. |
bootMSD calculates a parametric bootstrap simulation (or Monte carlo simulation)
of the results of msd applied to data. This allows individual case-specific
quantiles and p-values to be estimated that allow for different standard errors
(or standard uncertainties) s.
The sampling method is currently either sampling from rnorm or by latin hypercube sampling
using lhs.
Individual upper quantiles for probabilities probs and p-values are estimated
directly from the bootstrap replicates. Quantiles use quantile. p-values
are estimated from the proportion of replicates that exceed the observed MSD calculated by
msd. Note that the print method for the summary object does
not report zero proportions as identically zero.
An object of class "bootMSD", consisting of a vector of length length(x) of median
scaled absolute deviations for each observation, with attributes:
msdvector of raw calculated MSD values calculated by msd
labelscharacter vector of labels, by default taken from x
probsvextor of probabilities supplied and used for quantiles
critical.valuesmatrix of quantiles. Each row corresponds to a probability
in probs and each column to an individual data point.
pvalsp-values estimated as the observed proportion of
simulated values exceeding the MSD value calculated by msd.
BNumber of bootstrap replicates used.
methodThe sampling method used by the parametric bootstrap.
tIf keep == TRUE, the individual bootstrap replicates
generated by bootMSD. Set to NA if keep == FALSE.
Summary, print and plot methods are provided for the class; see bootMSD-class.
S. L. R. Ellison [email protected]
msd, bootMSD-class, print.bootMSD,
plot.bootMSD, summary.bootMSD.
data(Pb) ## Not run: #Default method: set.seed(1023) boot.Pb.default <- bootMSD(Pb$value, Pb$u) # Uses individual standard uncertainties summary(boot.Pb.default) #Method for MSD object: msd.Pb<-msd(Pb$value, Pb$u) # Uses individual standard uncertainties boot.Pb <- bootMSD(msd.Pb, B=5000) #Increased replication compared to default summary(boot.Pb) # NOTE: The default summary gives individual observation p-values. # To correct for multiple comparisons, apply # a suitable p-value adjustment: summary(boot.Pb, p.adjust="holm") ## End(Not run)data(Pb) ## Not run: #Default method: set.seed(1023) boot.Pb.default <- bootMSD(Pb$value, Pb$u) # Uses individual standard uncertainties summary(boot.Pb.default) #Method for MSD object: msd.Pb<-msd(Pb$value, Pb$u) # Uses individual standard uncertainties boot.Pb <- bootMSD(msd.Pb, B=5000) #Increased replication compared to default summary(boot.Pb) # NOTE: The default summary gives individual observation p-values. # To correct for multiple comparisons, apply # a suitable p-value adjustment: summary(boot.Pb, p.adjust="holm") ## End(Not run)
bootMSD and associated methods.The object class returned by bootMSD and associated
print, summary, and plotting classes.
## S3 method for class 'bootMSD' print(x, ...) ## S3 method for class 'bootMSD' plot(x, ...) ## S3 method for class 'bootMSD' barplot(height, ylab="MSD", names.arg=height$labels, crit.vals=TRUE, lty.crit=c(2,1), col.crit=2, lwd.crit=c(1,2), ... ) ## S3 method for class 'bootMSD' summary(object, p.adjust="none", ...) ## S3 method for class 'summary.bootMSD' print(x, digits=3, ..., signif.stars = getOption("show.signif.stars"), signif.legend=signif.stars)## S3 method for class 'bootMSD' print(x, ...) ## S3 method for class 'bootMSD' plot(x, ...) ## S3 method for class 'bootMSD' barplot(height, ylab="MSD", names.arg=height$labels, crit.vals=TRUE, lty.crit=c(2,1), col.crit=2, lwd.crit=c(1,2), ... ) ## S3 method for class 'bootMSD' summary(object, p.adjust="none", ...) ## S3 method for class 'summary.bootMSD' print(x, digits=3, ..., signif.stars = getOption("show.signif.stars"), signif.legend=signif.stars)
x |
An R object. For |
height |
An object of class |
object |
An object of class |
p.adjust |
Multiple correction method for calculated p-values, passed to
|
ylab |
Label for vertical axis, passed to |
names.arg |
Labels for individual bars in bar plot, passed to |
crit.vals |
If |
lty.crit, col.crit, lwd.crit
|
Vectors of line style parameters for plotted critical values, passed to
|
digits |
integer; passed to |
signif.stars |
logical; if |
signif.legend |
logical; if |
... |
Parameters passed to other methods. |
The default plot method is an alias for the barplot method.
For the plot methods, quantiles for each point are taken directly from the quantiles
calulated by bootMSD and retained in the returned object.
For the summary method, p-values are initially calculated as the observed
proportion of simulated values exceeding the MSD value calculated by msd. The
summary method additionally returns p-values after adjustment
for multiple comparisons using the adjustment method specified.
The print method for the summary.bootMSD object prints the summary as a data
frame adjusted with columns for the calculated MSD values, data-specific upper quantiles
(one column for each probability supplied to bootMSD and the p-values
after adjustment for multiple comparisons based on the proportion of simulated values
exceeding the observed MSD. Where that proportion is zero, the summary replaces the
raw zero proportion with 1/B, corrects that proportion using the requested
adjustment method, andreports the p-value as less than ("<") the resulting
adjusted value.
The print method returns the object, invisibly.
The plot and barplot methods return the values at the midpoint of each bar.
The summary method returns an object of class "summary.bootMSD" which
is a list with members:
msdCalculated MSD values from msd
labelscharacter vector of labels for individual data points
probsProbabilities used for quantiles
critical.valuesmatrix of quantiles. Each row corresponds to a probability
in probs and each column to an individual data point.
pvalsp-values estimated as the observed proportion of
simulated values exceeding the MSD value calculated by msd.
p.adjustCharacter value containing the name of the p-value adjustment method used.
p.adj p-values adjusted using the given p-value adjustment method
specified by p.adjust.
BNumber of bootstrap replicates used.
methodThe sampling method used by the parametric bootstrap.
S. L. R. Ellison [email protected]
## Not run: data(Pb) msd.Pb<-msd(Pb$value, Pb$u) # Uses individual standard uncertainties set.seed(1023) boot.Pb <- bootMSD(msd.Pb) summary(boot.Pb) # The default summary gives individual observation p-values. To # avoid over-interpretation for the study as a whole, # apply a sensible p-value adjustment: summary(boot.Pb, p.adjust="holm") plot(boot.Pb, crit=TRUE) ## End(Not run)## Not run: data(Pb) msd.Pb<-msd(Pb$value, Pb$u) # Uses individual standard uncertainties set.seed(1023) boot.Pb <- bootMSD(msd.Pb) summary(boot.Pb) # The default summary gives individual observation p-values. To # avoid over-interpretation for the study as a whole, # apply a sensible p-value adjustment: summary(boot.Pb, p.adjust="holm") plot(boot.Pb, crit=TRUE) ## End(Not run)
Produces a consistency plot for typical metrology comparison data.
cplot(x,u,labels=names(x), p.adjust.method="holm", ordered=TRUE, breaks=c(0,0.001,0.01, 0.05, 0.1,1), col=terrain.colors(length(breaks)-1), log.p=FALSE, main=paste("Consistency map -", deparse(substitute(x))), subtitle=NULL, key=FALSE, key.width=2.54, key.height=0.6,...)cplot(x,u,labels=names(x), p.adjust.method="holm", ordered=TRUE, breaks=c(0,0.001,0.01, 0.05, 0.1,1), col=terrain.colors(length(breaks)-1), log.p=FALSE, main=paste("Consistency map -", deparse(substitute(x))), subtitle=NULL, key=FALSE, key.width=2.54, key.height=0.6,...)
x |
Vector of reported values |
u |
Vector of length |
labels |
Vector of of length |
p.adjust.method |
p-value adjustment method; passed to |
ordered |
If |
breaks |
Vector of breaks; passed to |
col |
Vector of colours of length |
log.p |
If |
main, subtitle
|
Main and subtitle for plot. |
key |
If |
key.width, key.height
|
Width and height of key, if plotted. See details for specification. |
... |
Graphical parameters passed to |
Calculates the (square, symmetric matrix of) optionally adjusted p-values for a
two-tailed z-test of |x[i]-x[j]|/sqrt(u[i]^2+u[j]^2) against zero
and plots the p-values as an image.
p.adjust is called prior to plotting to correct for multiple comparisons.
To suppress adjustment, set p.adjust.method="none".
key.height is a fraction of the figure region height. key.width is the width of the key area in cm, unless under 1, in which case it is interpreted as a fraction of the plot region width.
If log.p is TRUE and subtitle NULL,
a subtitle indicating the use of log.p is added to the plot,
Invisibly returns a matrix of pairwise test p-values or, if log.p==TRUE,
matrix of -log_10(p).
S Ellison [email protected]
data(Pb) cplot(Pb$value, Pb$u, key=TRUE)data(Pb) cplot(Pb$value, Pb$u, key=TRUE)
Produces a dot plot of typical metrology comparison data (value/uncertainty) with error bars, assigned value and uncertainty and optional percentage deviation axis or marginal density
kplot(x, ...) ## Default S3 method: kplot(x,U=NULL, labels=names(x), assigned=NULL, U.assigned=NULL, U.lo=U, U.hi=U, k=2, strata=NULL, do.percent=!is.null(assigned) && !do.pdf, ordered=TRUE, order.strata=levels(strata), xlim=c(0.5, length(x)+0.5), ylim, main=NULL, xlab=NULL, ylab=NULL, axis.main=2, axis.pct=4, at=1:length(x), at.main=NULL, cex.axis=0.8, las=2, las.pct=1, ylab.line=2.5, ylab.line.pct=2.1, ci.width=0.03, col.ci=par("fg"), lty.ci=par("lty"), lwd.ci=par("lwd"), pch=21, col=par("fg"), bg="white", add.outliers=FALSE, outlier.offset=0.2, mar=NULL, box=TRUE, do.pdf=FALSE, do.individual.pdf=do.pdf, col.pdf=par("fg"), lwd.pdf=1, lty.pdf=1, do.total.pdf=TRUE, col.total.pdf=col.pdf[1], lwd.total.pdf=2, lty.total.pdf=1, n.pdf=200, pdf.layout=c(4,1), pdf.scale=0.7, pdf.offset=0.05, xlim.pdf, pdf.axis=FALSE, las.pdf=0, mgp.pdf=c(3,0.5,0), ...) ## S3 method for class 'ilab' kplot(x, ...) kpoints(x,U=NULL, labels=names(x), U.lo=U, U.hi=U, k=2, strata=NULL, ordered=TRUE, order.strata=levels(strata), at=1:length(x), ci.width=0.03, col.ci=par("fg"), lty.ci=par("lty"), lwd.ci=par("lwd"), pch=21, col=par("fg"), bg="white", add.outliers=FALSE, outlier.offset=0.2, ...)kplot(x, ...) ## Default S3 method: kplot(x,U=NULL, labels=names(x), assigned=NULL, U.assigned=NULL, U.lo=U, U.hi=U, k=2, strata=NULL, do.percent=!is.null(assigned) && !do.pdf, ordered=TRUE, order.strata=levels(strata), xlim=c(0.5, length(x)+0.5), ylim, main=NULL, xlab=NULL, ylab=NULL, axis.main=2, axis.pct=4, at=1:length(x), at.main=NULL, cex.axis=0.8, las=2, las.pct=1, ylab.line=2.5, ylab.line.pct=2.1, ci.width=0.03, col.ci=par("fg"), lty.ci=par("lty"), lwd.ci=par("lwd"), pch=21, col=par("fg"), bg="white", add.outliers=FALSE, outlier.offset=0.2, mar=NULL, box=TRUE, do.pdf=FALSE, do.individual.pdf=do.pdf, col.pdf=par("fg"), lwd.pdf=1, lty.pdf=1, do.total.pdf=TRUE, col.total.pdf=col.pdf[1], lwd.total.pdf=2, lty.total.pdf=1, n.pdf=200, pdf.layout=c(4,1), pdf.scale=0.7, pdf.offset=0.05, xlim.pdf, pdf.axis=FALSE, las.pdf=0, mgp.pdf=c(3,0.5,0), ...) ## S3 method for class 'ilab' kplot(x, ...) kpoints(x,U=NULL, labels=names(x), U.lo=U, U.hi=U, k=2, strata=NULL, ordered=TRUE, order.strata=levels(strata), at=1:length(x), ci.width=0.03, col.ci=par("fg"), lty.ci=par("lty"), lwd.ci=par("lwd"), pch=21, col=par("fg"), bg="white", add.outliers=FALSE, outlier.offset=0.2, ...)
x |
an R object. For the default method, a vector of reported values. For the ilab method, an object of class ‘ilab’ |
U |
Vector of length |
labels |
Vector of of length |
assigned |
Assigned value for the comparison. Plotted as a horizontal line on the plot. |
U.assigned |
Expanded uncertainty for the assigned value |
U.lo, U.hi
|
Vectors of of length |
k |
Coverage factor originally used in calculating U. Required only if
|
strata |
A Factor identifying subsets of the data. Currently not implemented. |
do.percent |
Logical indicating whether percentage deviation should be
plotted as a secondary axis. Defaults to |
ordered |
If |
order.strata |
Character vector showing the order of plotting for strata. Currently not implemented. |
xlim, ylim
|
Plot limits as in |
main, xlab, ylab
|
Titles; see |
axis.main, axis.pct
|
Integers specifying on which side of the plot the relevant axis is to
be drawn, passed to |
at |
Vector of x-axis locations for the data points, x-axis tick marks and labels.
Defaults to |
at.main |
The points at which tick-marks are to be drawn on the main (y) axis.
Passed to |
cex.axis |
The magnification to be used for axis annotation
relative to the current setting of 'cex'. Passed to |
las, las.pct
|
Integers defining x- and y axis and percentage axis label
orientation; see |
ylab.line, ylab.line.pct
|
Margin lines for main and percentage axis titles.
Passed to |
ci.width |
Width of error bar terminators, passed to |
col.ci, lty.ci, lwd.ci
|
Graphical parameters for the error bars; passed
to |
pch, col, bg
|
Graphical parameters for data points, passed to |
add.outliers |
If |
outlier.offset |
X-offset (in x-axis units) specifying lateral location of outlier tet labels relative to x-axis location of the outlier indicator. |
mar |
Plot margins as in |
box |
If |
do.pdf |
If |
do.individual.pdf |
Logical controlling whether the individual densities are plotted as well as/instead of the combined density. |
col.pdf, lwd.pdf, lty.pdf
|
Graphical parameters controlling the appearance of the marginal density plot(s). Vectors are permitted, allowing different styles for each individual pdf. |
do.total.pdf |
Logical controlling whether the sum of individual densities is plotted. |
col.total.pdf, lwd.total.pdf, lty.total.pdf
|
Graphical parameters controlling the appearance of the marginal density plot for the combined density. |
n.pdf |
Number of points used to construct the marginal density. |
pdf.layout |
Vector of length 2 specifying the relative sizes of the main plot and
marginal density plot. See |
pdf.scale, pdf.offset
|
Offset and scaling factor used to control the location and height of the marginal density plot(s). |
xlim.pdf |
Controls the x-axis (i.e. the horizontal axis) for the marginal density plotting area. |
pdf.axis |
If |
las.pdf, mgp.pdf
|
Axis control parameters passed to |
... |
Parameters passed to other functions; currently unused. |
If do.pdf=TRUE a marginal density plot is added. This plot is constructed
from a set of (currently) normal densities centred at x with standard
deviation U/k.
If a marginal density is plotted, par("layout") is changed to
pdf.layout; otherwise, par("layout") is set to matrix(1).
Both override any previously set layout. par("layout") is preserved on exit.
The ‘ilab’ method passes all parameters in ‘...’ to the default method
with default values for x, upper and lower bounds U.lo and U.hi,
labels and title taken from the ilab object.
kpoints is a convenience function for adding points with confidence
intervals to an existing plot. kpoints is not a generic function
and requires a vector x. Note that kpoints does not check for
the presence of a marginal density plot.
Invisibly returns a list with components:
order |
The order for plotting the original data, as returned by |
at |
x-axis locations used, in plotting order. |
S Ellison [email protected]
data(Pb) kplot(Pb$value, Pb$U, assigned=2.99, U.assigned=0.06) kplot(Pb$value, Pb$U, assigned=2.99, U.assigned=0.06, do.pdf=TRUE) #Use of return value for annotation kp<-kplot(Pb$value, Pb$U, assigned=2.99, U.assigned=0.06) text(kp$at, Pb$value-Pb$U, Pb$lab, srt=90, pos=4, cex=0.7)data(Pb) kplot(Pb$value, Pb$U, assigned=2.99, U.assigned=0.06) kplot(Pb$value, Pb$U, assigned=2.99, U.assigned=0.06, do.pdf=TRUE) #Use of return value for annotation kp<-kplot(Pb$value, Pb$U, assigned=2.99, U.assigned=0.06) text(kp$at, Pb$value-Pb$U, Pb$lab, srt=90, pos=4, cex=0.7)
Generates median of scaled differences from each point in a data set to all other points..
msd(x, s=mad , ...)msd(x, s=mad , ...)
x |
Vector of observations |
s |
Either a function returning an estimate of scale for |
... |
Parameters passed to |
For each observation x[i], msd calculates the median of |x[i]-x[j]|/sqrt(s[i]^2+s[j]^2), j!=i,
that is, the median of differences divided by the estimated uncertainties of the distance.
If s is a function, it is applied to x and replicated to length length(x); if
a scalar, it is replicated to length length(x).
The median scaled difference is a measure of how ‘far’ an individual observation is from
the majority of the other values in the data set. As a rule of thumb, values above
2 are indicative of a suspect (x[i], s[i]) data pair; that is, a value x[i] that
is remote from a large fraction of the remaining data given its associated standard
error or standard uncertainty s[i].
An object of class "MSD", consisting of a vector of length length(x) of median
scaled absolute deviations for each observation, with attributes:
names |
character vector of names, taken from |
x |
values supplied as |
s |
values supplied as |
Print and plotting methods are currently provided for the "MSD" class;
see MSD-class.
S. L. R. Ellison [email protected]
data(Pb) msd(Pb$value) # Uses mad(Pb$value) as scale estimate msd(Pb$value, Pb$u) # Scales differences using standard uncertaintiesdata(Pb) msd(Pb$value) # Uses mad(Pb$value) as scale estimate msd(Pb$value, Pb$u) # Scales differences using standard uncertainties
msd.Print and plotting methods for the MSD object class returned by msd.
## S3 method for class 'MSD' print(x, ...) ## S3 method for class 'MSD' plot(x, type="h", ylab="MSD", ...) ## S3 method for class 'MSD' barplot(height, ylab="MSD", names.arg=names(height), crit.vals=TRUE, lty.crit=c(2,1), col.crit=2, lwd.crit=c(1,2), probs=c(0.95, 0.99), n=length(height), ... )## S3 method for class 'MSD' print(x, ...) ## S3 method for class 'MSD' plot(x, type="h", ylab="MSD", ...) ## S3 method for class 'MSD' barplot(height, ylab="MSD", names.arg=names(height), crit.vals=TRUE, lty.crit=c(2,1), col.crit=2, lwd.crit=c(1,2), probs=c(0.95, 0.99), n=length(height), ... )
x, height
|
Object of class |
type |
The plot type. See |
ylab |
Label for vertical axis, passed to |
names.arg |
Labels for individual bars in bar plot, passed to |
crit.vals |
If |
lty.crit, col.crit, lwd.crit
|
Vectors of line style parameters for plotted critical values, passed to
|
probs |
vector of probabilities at which critical values are drawn. |
n |
integer number of observations for critical value calculation; passed to
|
... |
Parameters passed to other methods. |
See msd for the object description.
For the barplot method, critical values are ‘single-observation’ quantiles.
For use as an outlier test, use probabilities adjusted for multiple comparison;
for example, for the barplot method, consider raising the default probs
to the power .
The print method returns the object, invisibly.
The plot method returns NULL, invisibly.
The barplot methods return the values at the midpoint of each bar.
S. L. R. Ellison [email protected]
data(Pb) msd.Pb<-msd(Pb$value, Pb$u) # Uses individual standard uncertainties names(msd.Pb) <- as.character(Pb$lab) plot(msd.Pb) barplot(msd.Pb)data(Pb) msd.Pb<-msd(Pb$value, Pb$u) # Uses individual standard uncertainties names(msd.Pb) <- as.character(Pb$lab) plot(msd.Pb) barplot(msd.Pb)
Cumulative lower tail probability and quantile for median of scaled differences.
dmsd(q, n, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199) pmsd(q, n, lower.tail = TRUE, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199) qmsd(p, n, lower.tail = TRUE, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199)dmsd(q, n, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199) pmsd(q, n, lower.tail = TRUE, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199) qmsd(p, n, lower.tail = TRUE, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199)
q |
Vector of quantiles. |
p |
Vector of probabilities. |
n |
Number of observations from which msd was calculated. Unused (and can be missing)
for |
lower.tail |
logical; if TRUE (the default), probabilities are P[X <= x]; otherwise, P[X > x]. |
method |
Calculation method. See details. |
max.odd |
Highest odd |
pmsd, dmsd and qmsd return probabilities, densities and quantiles, respectively,
for the median scaled difference applied to a single observation in a standard normal
distribution, where otehr values are also IID normal.
n is the number of observations in the data set of interest and not the degrees of
freedom or number of differences (msd for a value x[i] in a set of n observations
involves n-1 scaled differences).
n, p and q are recycled to the length of the longest, as necessary.
method determines the method of calculation.
For method="fast", probabilities are calculated using monotonic spline
interpolation on precalculated probabilities. qmsd with method="fast" is obtained
by root-finding on the corresponding spline function using uniroot, and densities are
estimated from the first derivative of the interpolating spline. This provides fast
calculation, and values for most practical probabilities are within 10^-6 of exact calculations.
For high probabilites and for low quantiles (below 0.48) at high n, fast quantile accuracy
is poorer due to the very low function gradients in this regions, but is still guaranteed
monotonic with p.
For method="exact", probabilities and densities are calculated using quadrature
integration for an order statistic. For odd n, this requires a double integral. Values for
odd n accordingly take about an order of magnitude longer to obtain than for even n.
This can be slow (seconds for a vector of several hundred values of q on an Intel x86
machine running at 1-2GHz). qmsd with method="exact" is obtained by root-finding from
pmsd(..., method="excat") using uniroot, and is over an order of magnitude slower than
pmsd pmsd.
For method="exact", asymptotic (large ) probabilities, densities and quantiles are
returned. n is unused and can be missing.
For method="exact", odd n above max.odd are replaced with the next lower
even value. This provides a fair approximation for n above 30 (though the fast method is better)
and a good approximation above the default of 199. Values of max.odd above 199 are not recommended
as integration can become unstable at high odd n; a warning is issued if max.odd > 199.
For method="even", an exact calculation is performed with any odd n replaced with the
next lower even value. This is equivalent to setting method="exact" and max.odd=0.
This is provided for interest only; the method="fast" method provides a substantially better
approximation for odd n than method="even" and is faster.
Note that these functions are appropriate for the distribution of single values. If
seeking an outlier test in a data set of size , either adjust p for
comparisons before applying qmsd to find a critical value, or adjust the returned
p-values using, for example, Holm adjustment.
A vector of length length(p) or length(q) (or, if longer, length(n)) of
cumulative probabilities, densities or quantiles respectively.
S Ellison [email protected]
msd for calculation of MSD values, and bootMSD for
a parametric bootstrap (MCS) method of obtaining p-values and quantiles
for the more general non-IID case.
data(Pb) msd(Pb$value) # Uses mad(Pb$value) as scale estimate msd(Pb$value, Pb$u) # Scales differences using standard uncertaintiesdata(Pb) msd(Pb$value) # Uses mad(Pb$value) as scale estimate msd(Pb$value, Pb$u) # Scales differences using standard uncertainties
Provides classes and calculation and plotting functions for metrology applications, including measurement uncertainty estimation and inter-laboratory metrology comparison studies.
The metRology package includes functions for:
Plotting for Key Comparisons (dot-and-bar, consistency)
Uncertainty evaluation using algebraic or numeric differentiation, with support for correlation
Monte Carlo evaluation of uncertainty (including correlation for normally distributed variables)
Classes and functions for location estimates for metrology comparisons
Mandel's h and k statistics and plots for interlaboratory studies
Support functions for an excel interface
Changes in version 0.9-28-1 from version 0.9-28-0 include:
Fixed blockplot x-axis label, which was incorrect.
Changes in version 0.9-28-0 from version 0.9-27-2 include:
A new plot, blockplot, added. A “block plot” is a histogram variant identifiying
individual data points, which appear as “blocks” in the plot. blockplot provides for grouped
data, which generates vertically separated subplots for each group. Fills and label colours can be specified
for each data point.
Changes in version 0.9-27-2 from version 0.9-26-2 include:
pmsd and related functions will now use fast interpolation by default, and
provide exact values for both odd- and even- data sets up to .
gplot (called by plot.mandel.kh) now has a spacing parameter
which allows finer control of vertical line spacing.
Changes in version 0.9-26-2 from version 0.9-26-1 include:
Fix to a bug in reml.loc which failed to report the standard uncertainty u
correctly.
cplot now respects cex.axis as a plot parameter.
Changes in version 0.9-26-1 from version 0.9-26-0 include:
Added plot and barplot methods for MSD class.
Minor correction to code in msd to prevent over-replication
of estimated s when s is a function and returns a vector.
Changes in version 0.9-26 from version 0.9-25 include:
msd now returns an object of class "MSD" which
includes the original data as attributes to permit subsequent bootstrapping.
A new function, bootMSD that performs parametric
bootstrapping for MSD objects to obtain critical values
and p-values for the general case where standard uncertainties/standard
errors differ appreciably.
Improvements in version 0.9-25 from version 0.9-23 include:
plot.d.ellipse now takes default xlab and ylab from
dimnames in the supplied cov.dellipse.
Improvements in version 0.9-23 from version 0.9-22 include:
A wholly new Youden plot (see yplot), with many options for confidence ellipses
A REML location estimate, reml.loc, in addition to vr.mle;
reml.loc can use means and standard uncertainties/standard errors instead of raw data
and when doing so does not require degrees of freedom.
Incremental improvements in handling for the median scaled difference measure of anomalies.
msd is faster and less memory-intensive, and pmsd now uses a
beta formulation to extend to very high n (at least 1e6 - if you feel very
patient).
Support for log and log.p in dt.scaled.
Corrections and bugfixes include:
amends plot.drop1.uncert to give a plot for each measure of change specified in which
corrects a grep warning appearing in drop1.uncert;
corrects an unnecessary 'missing u' error message in version 0.9-22's uncert()
when cov was specified and u was not.
Stephen L R Ellison <[email protected]>.
Maintainer: Stephen L R Ellison <[email protected]>
Algorithm A is an implementation of Huber's location and scale estimate with iterated scale.
algA(x, k = 1.5, na.rm = FALSE, tol = .Machine$double.eps^0.25, maxiter = 25, verbose = FALSE)algA(x, k = 1.5, na.rm = FALSE, tol = .Machine$double.eps^0.25, maxiter = 25, verbose = FALSE)
x |
numeric vector or array of values. |
k |
Tuning factor; Winsorisation occurs ar k standard deviations. |
na.rm |
a logical value indicating whether |
tol |
Convergence tolerance Iteration continues until the relative
change in estimated sd drops below |
maxiter |
Maximum number of iterations permitted. |
verbose |
Controls information displayed during iteration; see Details. |
Algorithm A is the robust estimate of location described in ISO 5725-5:1998. It proceeds by winsorisation and re-estimation of scale and location.
The argument k controls the point at which values are Winsorised
and hence controls the efficiency. At k=1.5, the value chosen by
ISO 5725, the estimator has asymptotic efficiency at the Normal of 0.964.
With iterated estimate of scale and k=1.5, the estimator has a
breakdown point of about 30
The convergence criterion for Algorithm A is not specified in ISO 5725-5:1998.
The criterion chosen here is reasonably stringent but the results will differ
from those obtained using other choices. Use verbose=2 to check the
effect of different tolerance or maximum iteration count.
If verbose is non-zero, the current iteration number
and estimate are printed; if verbose>1, the current set
of truncated values is also printed.
mu |
Robust estimate of location |
s |
Robust estimate of scale |
Algorithm A uses the corrected median absolute deviation as the initial
estimate of scale; an error is returned if the resulting scale estimate is
zero, which can occur with over 50% of the data set equal. huberM in
the robustbase package uses an alternative scale estimate in these
circumstances.
Algorithm A is identical to Huber's estimate with variable scale.
The implementation here differs from hubers from MASS in:
hubers allows prior specification of fixed scale (which provides higher breakdown if chosen properly) or location
the option of verbose output in algA,
a maximum iteration option in algA
the convergence criterion; hubers converges on changes in mu,
whilst this implementation of Algorithm A converges on changes in s.
Internally, Algorithm A multiplies by a correction factor for
standard deviation whilst hubers divides by a correction factor
applied to the variance; the actual correction to s is identical.
The principal reasons for providing an implementation in the metRology package are i) to ensure a close implementation of the cited Standard irrespective of other package developments (though the MASS implementation has proved very stable) and ii) to make the implementation easy to recognise for users of the ISO standard.
S L R Ellison [email protected]
ISO 5725-5:1998 Accuracy (trueness and precision) of measurement methods and results - Part 5: Alternative methods for the determination of the precision of a standard measurement method
Maronna R A, Martin R D, Yohai V J (2006) Robust statistics - theory and methods. Jhn Wiley and Sons, West Sussex, England.
#Creosote example from ISO 5725-5:1998 #Means for each group are: cm <-c(24.140, 20.155, 19.500, 20.300, 20.705, 17.570, 20.100, 20.940, 21.185) algA(cm, verbose=TRUE) #Iteration 4 corresponds very closely to the ISO 5725 answer#Creosote example from ISO 5725-5:1998 #Means for each group are: cm <-c(24.140, 20.155, 19.500, 20.300, 20.705, 17.570, 20.100, 20.940, 21.185) algA(cm, verbose=TRUE) #Iteration 4 corresponds very closely to the ISO 5725 answer
‘Algorithm S’ calculates a robust estimate of pooled standard deviation from a set of standard deviations
algS(s, degfree, na.rm = FALSE, prob.eta = 0.9, is.range = FALSE, tol = .Machine$double.eps^0.25, maxiter = 25, verbose = FALSE)algS(s, degfree, na.rm = FALSE, prob.eta = 0.9, is.range = FALSE, tol = .Machine$double.eps^0.25, maxiter = 25, verbose = FALSE)
s |
A vector of standard deviations or, if |
degfree |
Scalar number of degrees of freedom associated with all
values in |
na.rm |
a logical value indicating whether 'NA' values should be stripped before the computation proceeds. |
prob.eta |
prob.eta is set to specify the lower tail area of the chi-squared distribution used as a cut-off. |
is.range |
if is.range is TRUE, s is interpreted as a vector of positive differences of duplcate observations and degfree is set to 1 |
tol |
Convergence tolerance Iteration continues until the relative
change in estimated pooled sd drops below |
maxiter |
Maximum number of iterations permitted. |
verbose |
Controls information displayed during iteration; see Details. |
Algorithm S is suggested by ISO 5725-5:1998 as a robust estimator of
pooled standard deviation from standard deviations of
groups of size .
The algorithm calculates a ‘limit factor’, , set to
qchisq(prob.eta, degfree). Following an initial estimate of
as median(s), the standard deviations
are replaced with
and an updated value for calculated as
where is the number of standard deviations and is calculated as
If the are ranges of two values, ISO 5725 recommends
carrying out the above iteration on the ranges and then dividing by
; in the implementation here, this
is done prior to returning the estimate.
If verbose is non-zero, the current iteration number
and estimate are printed; if verbose>1, the current set
of truncated values is also printed.
A scalar estimate of poooled standard deviation.
S L R Ellison [email protected]
ISO 5725-5:1998 Accuracy (trueness and precision) of measurement methods and results - Part 5: Alternative methods for the determination of the precision of a standard measurement method
#example from ISO 5725-5:1998 (cell ranges for percent creosote) cdiff <- c(0.28, 0.49, 0.40, 0.00, 0.35, 1.98, 0.80, 0.32, 0.95) algS(cdiff, is.range=TRUE) #Compare with the sd of the two values (based on the range) c.sd <- cdiff/sqrt(2) algS(c.sd, degfree=1, verbose=TRUE)#example from ISO 5725-5:1998 (cell ranges for percent creosote) cdiff <- c(0.28, 0.49, 0.40, 0.00, 0.35, 1.98, 0.80, 0.32, 0.95) algS(cdiff, is.range=TRUE) #Compare with the sd of the two values (based on the range) c.sd <- cdiff/sqrt(2) algS(c.sd, degfree=1, verbose=TRUE)
A data frame containing reported duplicate results for dietary fibre from a collaborative study.
data(apricot)data(apricot)
A data frame containing duplicate results from 9 laboratories:
Factor giving abbreviated laboratory identifier
The reported fibre content.
Replicate results appear on separate rows.
B. W. Li, M. S. Cardozo (1994) Determination of total dietary fibre in foods and products with little or no starch, non-enzymatic gravimetric method: collaborative study. J. AOAC Int. 77, 687-689, 1994
A. L. Ruhkin, C. J. Biggerstaff and M. G. Vangel (2000) Restricted maximum likelihood estimation of a common mean and the Mandel-Paule algorithm. J. Stat. Planning an Inferences 83, 319-330, 2008
B. W. Li, M. S. Cardozo (1994) Determination of total dietary fibre in foods and products with little or no starch, non-enzymatic gravimetric method: collaborative study. J. AOAC Int. 77, 687-689, 1994
barplot.mandel.kh produces a bar plot of Mandel's statistics, suitably
grouped and with appropriate indicator lines for unusual values.
## S3 method for class 'mandel.kh' barplot(height, probs = c(0.95, 0.99), main, xlab = attr(height, "grouped.by"), ylab = attr(height, "mandel.type"), separators = TRUE, zero.line = TRUE, ylim, p.adjust = "none", frame.plot = TRUE, ..., col.ind = 1, lty.ind = c(2, 1), lwd.ind = 1, col.sep = "lightgrey", lwd.sep = 1, lty.sep = 1, lwd.zero = 1, col.zero = 1, lty.zero = 1)## S3 method for class 'mandel.kh' barplot(height, probs = c(0.95, 0.99), main, xlab = attr(height, "grouped.by"), ylab = attr(height, "mandel.type"), separators = TRUE, zero.line = TRUE, ylim, p.adjust = "none", frame.plot = TRUE, ..., col.ind = 1, lty.ind = c(2, 1), lwd.ind = 1, col.sep = "lightgrey", lwd.sep = 1, lty.sep = 1, lwd.zero = 1, col.zero = 1, lty.zero = 1)
height |
An object of class |
probs |
Indicator lines are drawn for these probabilities. Note that
|
main |
a main title for the plot. If missing, the default is
|
xlab |
a label for the x axis; defaults to the |
ylab |
a label for the x axis; defaults to the |
separators |
Logical; if |
zero.line |
logical; if |
ylim |
the y limits of the plot. For Mandel's k, the default lower limit is zero. |
p.adjust |
Correction method for probabilities. If not |
frame.plot |
Logical; If |
... |
Other (usually graphical) parameters passed to |
col.ind, lty.ind, lwd.ind
|
Graphical parameters used for the indicator lines, recyckled to |
col.sep, lwd.sep, lty.sep
|
Graphical parameters used for the separator lines. |
lwd.zero, col.zero, lty.zero
|
Graphical parameters used for the zero line. |
Mandel's statistics are traditionally plotted for inter-laboratory study data, grouped by laboratory, to give a rapid graphical view of laboratory bias and relative precision. This plot produces a grouped, side-by-side bar plot.
For classical Mandel statistics, indicator lines are drawn based on qmandelh
or qmandelk as appropriate. For robust variants, indicator lines use
qnorm for the statistic and qf(probs, n, Inf) for
the statistic. Note that this corresponds to taking the robust estimates of
location and scale as true values, so will be somewhat anticonservative.
barplot.mandel.kh returns a numeric vector of mid-points of the groups along the x-axis.
S Ellison [email protected]
Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva (1994).
mandel.h, mandel.k, mandel.kh,
pmandelh, pmandelk for probabilities, quantiles etc.
See plot.mandel.kh for the 'classic' Mandel plot.
data(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) barplot(h, las=2) # Lab 4 shows consistent low bias; # Lab 23 several extreme values. #Use colours to identify particular measurands: barplot(h, las=2, col=1:8) legend("bottomleft", legend=names(h), fill=1:8, cex=0.7, bg="white") #Example of Mandel's k: k <- with(RMstudy, mandel.k(RMstudy[2:9], g=Lab)) barplot(k, las=2) # Lab 8 looks unusually variable; # Lab 14 unusually precisedata(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) barplot(h, las=2) # Lab 4 shows consistent low bias; # Lab 23 several extreme values. #Use colours to identify particular measurands: barplot(h, las=2, col=1:8) legend("bottomleft", legend=names(h), fill=1:8, cex=0.7, bg="white") #Example of Mandel's k: k <- with(RMstudy, mandel.k(RMstudy[2:9], g=Lab)) barplot(k, las=2) # Lab 8 looks unusually variable; # Lab 14 unusually precise
bkp draws block plots based on block plot objects generated by blockplot and its methods.
It is normally called from within blockplot but can be invoked directly.
bkp(x, labels = x$labels, xlim = NULL, ylim = NULL, main = NULL, xlab = NULL, ylab = "Frequency", square = FALSE, add = FALSE, offset = 0, grp.spacing = 2, grp.at = NA, fill = NA, border = NULL, density = NULL, angle = 45, lty = 1, lwd = 2, label.col = 1, cex = NA, adj = c(0.5, 0.4), uline = 2, uline.lwd = lwd, uline.lty = 1, uline.col = if (!is.null(border)) border else par("fg"), grp.labs = FALSE, grp.pos = 1, glab.control = list(), axes = c(TRUE, FALSE), asp = NA, frame.plot = any(axes), drop.unused = TRUE, unused.label="[Missing]", ...)bkp(x, labels = x$labels, xlim = NULL, ylim = NULL, main = NULL, xlab = NULL, ylab = "Frequency", square = FALSE, add = FALSE, offset = 0, grp.spacing = 2, grp.at = NA, fill = NA, border = NULL, density = NULL, angle = 45, lty = 1, lwd = 2, label.col = 1, cex = NA, adj = c(0.5, 0.4), uline = 2, uline.lwd = lwd, uline.lty = 1, uline.col = if (!is.null(border)) border else par("fg"), grp.labs = FALSE, grp.pos = 1, glab.control = list(), axes = c(TRUE, FALSE), asp = NA, frame.plot = any(axes), drop.unused = TRUE, unused.label="[Missing]", ...)
x |
An R object. For the default method, a vector of values for which the blockplot is desired.
For the formula method, a valid formula, such as |
labels |
Labels for data points, passed to |
xlim |
Limits |
ylim |
limits for the vertical range of the plot. Will be overridden if |
main |
Main title for the plot, passed to |
xlab, ylab
|
x- and y-axis labels for the plot. As usual, either can be expressions (see |
square |
Logical: If |
add |
Logical: If |
offset |
Numeric scalar value. Vertical offset for the plot, in units of block height.
|
grp.spacing |
Numeric scalar, giving the minimum vertical spacing (in units of block height) between subplots when there is more than one group. |
grp.at |
Optional vector specifying explicit vertical locations for subplot
baselines (including the first group). The default ( |
fill |
Fill colour for the rectangles (“blocks”) making up the plot.
Recycled to length |
border |
Border colour for the rectangles (“blocks”) making up the plot.
Recycled to length |
density |
Shading line density for (“blocks”) making up the plot. Recycled
to length |
angle |
Shading line angle for (“blocks”) making up the plot. Recycled
to length |
lty |
Border line type for (“blocks”) making up the plot. Recycled
to length |
lwd |
Border line width for (“blocks”) making up the plot. Recycled
to length |
label.col |
Colour used for text labels in each (“block”) making up the plot.
Recycled to length |
cex |
Size of text labels in each (“block”) making up the plot.
Recycled to length |
adj |
Vector of two values giving text location adjustment for all block labels.
Passed to |
uline |
Specification for the distance by which the baseline for each subplot extends beyond the data range for the group. See ‘Graphical elements’ for details. The default is two units either side. |
uline.lwd |
Scalar: Line width for the subplot baseline(s). |
uline.lty |
Scalar: Line type for the subplot baseline(s). |
uline.col |
Scalar: Line colour for the subplot baseline(s). |
grp.labs |
Logical, determining whether group labels are plotted, or a vector of labels. See ‘Details’. |
grp.pos |
Specification for the position of group labels. Follows |
glab.control |
List of arguments to be passed to |
axes |
Logical, indicating whether axes are drawn. Can be a vector of two logicals, specifying horizontal and vertical axes respectively. See ‘Graphical elements’ for details. |
asp |
Aspect ratio, passed to |
frame.plot |
Logical, controlling whether a frame (box) is drawn round the plot. |
drop.unused |
Logical specification for the treatment of empty groups. If |
unused.label |
Character string appended to missing group labels. |
... |
Further parameters passed to |
bkp provides considerable control of graphical elements. The main
elements, and the arguments controlling their location and appearance, are:
A block plot of a single group of data has the general appearance of a histogram. However, instead of vertical bars (of possibly variable width) indicating the number of data points within the bin interval, each bin is a stack of rectangles, each corresponding to a single data point and with an optional label identifying the datum.
Block plots of this kind are useful for data sets of modest size; typically 10-100 per group, as individual labels quickly become hard to identify in larger data sets.
By default, blockplot produces one such plot for a set of data. If
a series of such plots is needed, this can be accommodated either by using
blockplot with add=TRUE to build up a plot manually, setting
xlim, ylim and breaks to accommodate all the required
groups. Alternatively, a grouping factor can be provided (via argument
groups) which will produce a series of subplots, laid out automatically.
The use of groups and the corresponding layout options are detailed below
(see “Groups”).
The vertical position of a single block plot within the figure
can be set using offset, which sets the baseline height,
in units of block height, from the figure origin. This is useful for
separating several groups that are added manually; just set offset
appropriately for each separate plot. Note that setting offset
has no effect on the automatic ylim setting, which means that
ylim must be set manually to accommodate the vertical offset.
Each individual rectangle (“block” in the plot corresponds to a single data point. In this implementation, blocks appear in rank order from left to right and from bottom to top; that is, data are placed in vertical bins as in a normal histogram but, in addition, the vertical ordering of blocks corresponds to the data order within each bin, with blocks at the bottom corresponding to lower values.
Blocks are always 1 unit high, so the total vertical height of each bin corresponds
directly to frequency (not density) in a histogram. The block width is the interval
between breaks, which must be equispaced.
By default, the apparent aspect ratio for blocks depends primarily on
xlim and ylim and the height and width of the plotting device.
However, setting square=TRUE will cause the plot aspect ration (asp)
to be set such that the bocks appear square in the current plot window.
Fill colour, border colour and style, fill effects and text colour of individual
blocks can all be controlled using fill, border,
density, angle, lty, lwd and label.col,
as the relevant arguments can be vectors of length length(x). This
allows conditional formatting, for example to identify
a particular data point or some secondary grouping variable.
The baseline for each subplot is controlled by uline,
as follows:
TRUE:The line extends the full width of the plot;
FALSE:No baseline is drawn;
If numeric (as for the default), uline specifies
the distance that the baseline extends beyond each end of the data,
in units of block width. uline can be length 2 numeric vector,
which specifies the baseline extension on the left and right sides
respectively.
Colour, line type, and line width for the subplot baseline(s) can be controlled
with uline.col, uline.lty, and uline.lwd respectively.
Axes can be controlled with the axes argument, which
controls whether or not axes are drawn. If a vector of two logical
values (as for the default), axes specifies drawing for horizontal
and vertical axes respectively.
The horizontal axis is normally continuous for the plot. If a
vertical (frequency) axis is requested (either by axes=TRUE or,
for example, by axes=c(TRUE, TRUE), a vertical axis is drawn
for each subplot, starting at zero at the baseline and terminating
at the highest vertical value in the subplot. Vertical axes,
restarting at 0 at the next subplot baseline, are drawn if
there is more than one group.
blockplot provides a simple grouping mechanism to display
separate subplots for different groups of data on the same figure. The
default method provides for a grouping variable specified via groups.
The formula method provides a somewhat more flexible interface, allowing
specification of more than one grouping variable. Like boxplot,
if there is more than one goruping variable in the formula, subplots are drawn
for each (non-empty) level of the interaction term.
Subplots for different groups are arranged vertically. Vertical position can be
specified explicitly via grp.at or, more simply, by setting
grp.spacing. The latter sets grp.at to equal vertical
spacing such that the smallest vertical gap is grp.spacing.
Both grp.at and grp.spacing are in units of block height;
that is, grpspacing=2 (the default) means that the smallest vertical
gap is equivalent to two blocks.
Labels can be added to each subplot. These are controlled by grp.labs ..
grp.labs provides the specification for group labels, and can be a single logical or
a vector of labels. Effects of grp.labs are as follows:
FALSE (The default): No group labels are drawn.
TRUE Labels are taken as levels(groups),
and set to "1" if there is only one group.
Vector If a character vector (or expression) is provided, these are used as labels for the groups plotted.
WARNING: If missing values in x cause group levels to be
dropped, those groups will not be plotted. grp.labs must
have the same length as the number of groups plotted. An error
is generated if the length of labels differs from the number
of groups actually plotted.
grp.pos specifies the general positioning of group labels
relative to each subplot. grp.pos follows pos in text:
Values of 1, 2, 3 and 4, respectively
indicate positions below, to the left of, above and to the right of the plot.
The detailed positioning of group labels is automatic; the four positions
specified by grp.pos correspond approximately to the midpoints of the
corresponding edge of each plot, where the ‘edges’ are the baseline,
leftmost block, topmost block and rightmost block. Labels are placed a short
distance outward from these edges. Labels are justified according to position;
grp.pos is re-used as the default pos argument to text.
Further control of group label position is available via grp.control,
which is a list (empty by default) of arguments passed to text.
Ths can include arguments such as pos and adj, as well as
appearance elements such as col, cex etc.
bxp returns the original object x with additional elements:
grp.at |
The vertical coordinated for the subplot baselines. |
blockwidth |
The width of the blocks in the plot. |
Stephen L R Ellison [email protected].
ISO 5725-2:1994 Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva, 1994.
For constructing breaks and grouping:
blockplot
For graphical elements:
text, rect
#A simple blockplot set.seed(55) x<-rnorm(48, 15) b <- blockplot(x) #Aspect ratio control for square blocks bkp(b, square=TRUE) # Specifying groups grp <- gl(3, 16) bg <- blockplot(x~grp) # Add vertical axes ( axes=TRUE asks for horizontal and vertical axes) bkp(bg, axes=TRUE ) #Axes both left and right par(mar=c(5,4,4,4)+0.1) bkp(bg, axes=c(TRUE, TRUE, FALSE, TRUE) ) #Note that axes[3] is FALSE to suppress top axis # Vectorised colour specification bkp(bg, square=TRUE, fill=ifelse(1:48 %in% c(15, 23, 24), "gold", "white") ) # Group labelling bkp(bg, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2)#A simple blockplot set.seed(55) x<-rnorm(48, 15) b <- blockplot(x) #Aspect ratio control for square blocks bkp(b, square=TRUE) # Specifying groups grp <- gl(3, 16) bg <- blockplot(x~grp) # Add vertical axes ( axes=TRUE asks for horizontal and vertical axes) bkp(bg, axes=TRUE ) #Axes both left and right par(mar=c(5,4,4,4)+0.1) bkp(bg, axes=c(TRUE, TRUE, FALSE, TRUE) ) #Note that axes[3] is FALSE to suppress top axis # Vectorised colour specification bkp(bg, square=TRUE, fill=ifelse(1:48 %in% c(15, 23, 24), "gold", "white") ) # Group labelling bkp(bg, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2)
A “block plot” is a histogram variant identifiying individual data points. Histogram bars are replaced
by a stack of rectangles (“blocks”, each of which is labelled. blockplot provides for grouped
data, which generates vertically separated subplots for each group. Fills and label colours can be specified
for each data point.
blockplot(x, ...) bplot(x, ...) ## Default S3 method: blockplot(x, breaks = "23", labels = paste(1:length(x)), groups = NA, xlim = NULL, ylim = NULL, main = NULL, xlab = NULL, ylab = "Frequency", grp.labs = FALSE, include.lowest = TRUE, right = TRUE, nclass = NULL, plot = TRUE, add=FALSE, ...) ## S3 method for class 'formula' blockplot(x, data = NULL, ..., subset, main = NULL, xlab = NULL) nclass.23(x)blockplot(x, ...) bplot(x, ...) ## Default S3 method: blockplot(x, breaks = "23", labels = paste(1:length(x)), groups = NA, xlim = NULL, ylim = NULL, main = NULL, xlab = NULL, ylab = "Frequency", grp.labs = FALSE, include.lowest = TRUE, right = TRUE, nclass = NULL, plot = TRUE, add=FALSE, ...) ## S3 method for class 'formula' blockplot(x, data = NULL, ..., subset, main = NULL, xlab = NULL) nclass.23(x)
x |
An R object. For the default method, a vector of values for which the blockplot is desired.
For the formula method, a valid formula, such as |
data |
For the formula method, a data frame or list from which the variables in |
subset |
For the formula method, an optional vector specifying a subset of observations to be used for plotting. |
breaks |
Either a specification for choosing breakpoints for “binning” the data, or a vector giving the breakpoints themselves. The specification can be a single number, a function, or a character string identifying a function. See ‘Details’ for detailed specification. |
labels |
Labels for data points, passed to |
groups |
An optional grouping variable, coerced to factor. If present, one subplot is produced for each non-empty group. |
xlim |
Limits |
ylim |
limits for the vertical range of the plot. Will be overridden if |
main |
Main title for the plot, passed to |
xlab, ylab
|
x- and y-axis labels for the plot. As usual, either can be expressions (see |
grp.labs |
Logical, determining whether group labels are plotted, or a vector of labels. See ‘Details’. |
include.lowest |
Logical, indicating whether a value equal to the lowest
(or highest, for |
right |
Logical, indicating whether the bin intervals should be closed on the right (and open
on the left) or vice versa. Passed to |
nclass |
Suggested number of classes for |
plot |
If |
add |
If |
... |
Further parameters passed to other functions, in particular, |
blockplot produces a block plot - a histogram variant identifying individual
data points. Histogram bars are replaced by a stack of rectangles (“blocks”,
each of which can be (and by default, is) labelled.
bplot is an alias for blockplot.
For the formula method, x is a formula, such as y ~ grp, in which y is a numeric vector
of data values to be split into groups according to the grouping variable grp (usually a factor).
More than one grouping variable can be specified, in which case subplots are produced for each level
of the interaction between grouping factors.
The specification for breakpoints, breaks, is modelled closely on that for
hist. breaks can be one of:
a vector giving the (equally spaced) breakpoints between bins;
a function to compute the vector of breakpoints;
a single number giving the suggested number of bins for the blockplot;
a character string naming an algorithm to compute the
number of cells. Values of "23" (the default),
"Sturges", "Scott", "FD" and
"Freedman-Diaconis" are currently
supported; see below for their effect
a function to compute the number of bins.
In the last three cases the number is a suggestion only, as the breakpoints will be set to “pretty” values.
The different character string specifications correspond to “nclass”
functions, including those used by hist; see nclass.FD
for details of those. In addition, the default "23" corresponds to the
function nclass.23. This is just a wrapper for the one-line expression
ceiling(length(x)^(2/3)),
which appears to provide good results for block plots.
Considerable control of graphical elements is provided by the plotting function
bkp, which is called by blockplot. In particular, arguments
passed through ... to bkp can control:
The general shape of the plot, including the asbect ratio of the “blocks”;
whether a plot should be added to an existing figure (add)
the fill colour and shading, the border width, type and colour, and the font size and colour of individual blocks;
the vertical location of the plot in the figure region offset;
the vertical spacing between multiple plots on the same figure when a grouping
variable is provided (grp.spacing and grp.at;
the presence, location and appearance of labels for individual subplots;
whether axes are plotted on any of the four sides of the plot;
the appearance or omission of empty groups.
See bkp for further details.
Blockplot currently returns an object of class blockplot, which is a list with elements:
x |
The original data |
groups |
If there is more than one group, a factor of groups for each data point,
with additional attribute |
x.left |
Vector of x-coordinates for the left side of each block |
x.height |
Vector of y-coordinates for each box, relative to the group baseline |
x.mid |
Vector of x-coordinates for the middle of each block (the text location) |
x.mid |
Vector of x-coordinates for the middle of each block (the text location) |
The name “block plot” may not be in general use, but the package author has been unable to identify either an alternative designation or an original source for this type of plot. An example - apparently hand drawn - was given in ISO 5725-2:1994 (referenced above).
S Ellison [email protected]
ISO 5725-2:1994 Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva, 1994.
For plotting and control of plot appearance:
link{bkp}
For graphical elements:
text, rect
For specification of breaks:
link{nclass.Sturges}, link{nclass.Scott}, link{nclass.FD}
#A simple blockplot set.seed(55) x<-rnorm(48, 15) blockplot(x) #Aspect ratio control for square blocks blockplot(x, square=TRUE) # Specifying groups grp <- gl(3, 16) blockplot(x, groups=grp) #Formula interface blockplot(x~grp) #Vectorised colour specification blockplot(x~grp, square=TRUE, fill=ifelse(1:48 %in% c(15, 23, 24), "gold", "white") ) #Group labelling blockplot(x~grp, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2) #A missing group xm <- x xm[ grp == "2" ] <- NA blockplot(xm~grp, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2) blockplot(xm~grp, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2, drop.unused=FALSE)#A simple blockplot set.seed(55) x<-rnorm(48, 15) blockplot(x) #Aspect ratio control for square blocks blockplot(x, square=TRUE) # Specifying groups grp <- gl(3, 16) blockplot(x, groups=grp) #Formula interface blockplot(x~grp) #Vectorised colour specification blockplot(x~grp, square=TRUE, fill=ifelse(1:48 %in% c(15, 23, 24), "gold", "white") ) #Group labelling blockplot(x~grp, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2) #A missing group xm <- x xm[ grp == "2" ] <- NA blockplot(xm~grp, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2) blockplot(xm~grp, square=TRUE, grp.labs=paste("Level", 1:3), grp.pos=2, drop.unused=FALSE)
Produces a box plot of Mandel's statistics, with optional outlier labels and indicator lines for unusual values.
## S3 method for class 'mandel.kh' boxplot(x, probs=c(0.95, 0.99), main, xlab=attr(x, "grouped.by"), ylab=attr(x, "mandel.type"), separators=FALSE, zero.line=TRUE, ylim, p.adjust="none", frame.plot = TRUE, horizontal=FALSE, at, ... , col.ind=1, lty.ind=c(2,1), lwd.ind=1, col.sep="lightgrey", lwd.sep=1, lty.sep=1, lwd.zero=1, col.zero=1, lty.zero=1, outlier.labels=row.names(x), cex.lab=0.7, col.lab=1, adj=NULL, pos=NULL, srt=0 )## S3 method for class 'mandel.kh' boxplot(x, probs=c(0.95, 0.99), main, xlab=attr(x, "grouped.by"), ylab=attr(x, "mandel.type"), separators=FALSE, zero.line=TRUE, ylim, p.adjust="none", frame.plot = TRUE, horizontal=FALSE, at, ... , col.ind=1, lty.ind=c(2,1), lwd.ind=1, col.sep="lightgrey", lwd.sep=1, lty.sep=1, lwd.zero=1, col.zero=1, lty.zero=1, outlier.labels=row.names(x), cex.lab=0.7, col.lab=1, adj=NULL, pos=NULL, srt=0 )
x |
An object of class |
probs |
Indicator lines are drawn for these probabilities. Note that
|
main |
a main title for the plot. If missing, the default is
|
xlab |
a label for the x axis; defaults to the |
ylab |
a label for the x axis; defaults to the |
separators |
Logical; if |
zero.line |
logical; if |
ylim |
the y limits of the plot. For Mandel's k, the default lower limit for y is zero. |
p.adjust |
Correction method for probabilities. If not |
frame.plot |
Logical; If |
horizontal |
if |
at |
numeric vector giving the locations where the boxplots should
be drawn; defaults to |
... |
Other (usually graphical) parameters passed to |
col.ind, lty.ind, lwd.ind
|
Graphical parameters used for the indicator lines, recyckled to |
col.sep, lwd.sep, lty.sep
|
Graphical parameters used for the separator lines. |
lwd.zero, col.zero, lty.zero
|
Graphical parameters used for the zero line. |
outlier.labels |
Either a logical indicating whether outliers should be labelled or a character vector of length nrow(x) giving labels. Defaults to row.names(x). |
cex.lab, col.lab
|
Character size and colour for outlier labels, passed to |
adj, pos
|
Position of outlier labels relative to outliers; passed to |
srt |
Label rotation, in degrees, for outlier labels; passed to |
This plot produces a box plot (using boxplot.default) of the variables in an object of
class "mandel.kh".
If labels are specified for outliers (the default), outliers are first located based on the locations given by boxplot.default. WARNING: ties may be mislabelled, as the label allocated will be the _first_ point at that location.
Indicator lines are, if requested, drawn as for plot.mandel.kh.
Vertical separators are drawn at midpoints of at. If
boxplot.mandel.kh returns the box plot statistics returned by
boxplot, invisibly.
S Ellison [email protected]
boxplot for box plot arguments, and text for outlier label
location, colour and rotation.
mandel.h, mandel.k, mandel.kh,
pmandelh, pmandelk for probabilities, quantiles etc.
See plot.mandel.kh for the 'classic' Mandel plot.
data(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) boxplot(h, las=2) #Recall that for normally distributed data mandel's h should #have the same dispersion (and IQR) for all groups. But outliers adversely #affect the estimate of dispersion, so the interquartile ranges differ. #The same effect also accounts for the many boxplot outliers visible #inside the classical Mandel indicator lines; the indicators also #assume normality. #with separators: boxplot(h, las=2, separators=TRUE) #With different labels and label colours: boxplot(h, las=2, outlier.labels=paste(1:nrow(h)), col.lab=1:5) #... and a horizontal variant (note use of pos to change label positions) par(omd=c(0.1,1,0,1)) #to make room for axis labels boxplot(h, las=1, separators=TRUE, horizontal=TRUE, pos=1)data(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) boxplot(h, las=2) #Recall that for normally distributed data mandel's h should #have the same dispersion (and IQR) for all groups. But outliers adversely #affect the estimate of dispersion, so the interquartile ranges differ. #The same effect also accounts for the many boxplot outliers visible #inside the classical Mandel indicator lines; the indicators also #assume normality. #with separators: boxplot(h, las=2, separators=TRUE) #With different labels and label colours: boxplot(h, las=2, outlier.labels=paste(1:nrow(h)), col.lab=1:5) #... and a horizontal variant (note use of pos to change label positions) par(omd=c(0.1,1,0,1)) #to make room for axis labels boxplot(h, las=1, separators=TRUE, horizontal=TRUE, pos=1)
Functions to build and update correlation and covariance matrices using a compact specification of off-diagonal terms.
buildCor(n, cors) updateCor(cor, cors, cor.names) buildCov(s, covs, vars = s^2, cors, cov.names) updateCov(cov, covs, cors, cov.names)buildCor(n, cors) updateCor(cor, cors, cor.names) buildCov(s, covs, vars = s^2, cors, cov.names) updateCov(cov, covs, cors, cov.names)
n |
scalar: number of rows/colums required in correlation matrix. |
cors, covs
|
3-column matrix or data frame specification of individual correlation or covariance terms. Can also be a vector of length 3. See Details. |
s, vars
|
vector of standard deviations or variances respectively. One of
|
cor.names, cov.names
|
vectors of names for the rows and columns of the returned
matrix. |
cor, cov
|
correlation or covariance matrix requiring amendment. |
For buildCor, the size of the returned correlation matrix is set using n;
an n by n correlation matrix is returned. For buildCov the size is
set to length(s) by length(s).
Each row of cors specifies a correlation term in the form
. That is, the first two columns give the row
and column in the desired correlation matrix, and the third gives the relevant
correlation coefficient. On constructing or updating the correlation matrix,
is set equal to , so it is only necessary
to specify one of or .
covs specifies covariance terms in the same way except that the third column of
covs must be a covariance.
If either cors or covs is a vector of length 3, it is coerced
to a matrix of three columns.
If cor.names or cov.names are present, the matrix returned has
dimnames set to the names supplied.
All four functions test for positive definite return values and generate a warning if not positive definite.
A square symmetric correlation or covariance matrix.
S. L. R. Ellison [email protected]
None.
None.
#Duplicate correlation for example for uncert() buildCor(4, cors=c(3, 4, 0.5)) #Multiple correlations r<-buildCor(3, cors=rbind( c(1,2,0.5), c(2,3,0.25) ) ) r updateCor(r, cors=c(1,3,0.13)) #perhaps more realistic buildCov(1:3, cors=rbind( c(1,2,0.5), c(2,3,0.25),c(1,3,0.13) ) )#Duplicate correlation for example for uncert() buildCor(4, cors=c(3, 4, 0.5)) #Multiple correlations r<-buildCor(3, cors=rbind( c(1,2,0.5), c(2,3,0.25) ) ) r updateCor(r, cors=c(1,3,0.13)) #perhaps more realistic buildCov(1:3, cors=rbind( c(1,2,0.5), c(2,3,0.25),c(1,3,0.13) ) )
Chromium data for two different materials included in an interlaboratory study intended to provide data for certification of a reference material.
data("chromium")data("chromium")
A data frame with 28 observations on the following 2 variables.
QCChromium concentrations (ug/kg) reported on a material used as a quality control material
RMChromium concentrations (ug/kg) reported on a candidate reference material material used as a quality control material
Chromium data for two different materials included in an interlaboratory study intended to provide data for certification of a crab tissue reference material. The study included a previously certified reference material (near end of stock) to serve as a quality control (QC) check. Laboratories were asked to report five replicate measurements on the candidate reference material and three for the QC material. Each row in the data set corresponds to the mean of replicate results reported by each laboratory.
Inspection of the data suggests that one laboratory interchanged or mislabelled
the test materials; this is hard to see in univariate plots but relatively easy
to see in a Youden plot (a type of pairwise scatter plot - see youden.plot).
Private communication - Pending publication
data(chromium) yplot(chromium)data(chromium) yplot(chromium)
Extracts the individual nonzero contributions to the combined uncertainty in an 'uncert' object.
contribs(object, scope, as.sd = FALSE, keep.sign = TRUE, simplify = TRUE, expand.dot=TRUE)contribs(object, scope, as.sd = FALSE, keep.sign = TRUE, simplify = TRUE, expand.dot=TRUE)
object |
|
scope |
An expression, one-sided formula or character vector describing the particular variables for which contributions are desired. If missing, contributions for all variables are returned. |
as.sd |
logical; controls whether values are returned in the form of standard uncertainties or variance contributions. See Details. |
keep.sign |
logical; controls whether the sign of the cobntributions is appended to
the return value when |
simplify |
logical. If |
expand.dot |
logical; if |
contribs calculates the contribution matrix where
.
In general, these values are possibly negative (co)variance contributions
to the variance (squared standard uncertainty) in . In GUM notation
(‘the GUM’ is JCGM 100 (2008) - see references), the diagonal elements of
C are squared standard uncertainties in . The form of the
return value depends on simplify, as.sd and keep.sign.
If as.sd is FALSE (the default), contributions
are returned unchanged. For the diagonal elements of (contributions for
individual individual terms), this form corresponds to squared standard uncertainties
in GUM notation.
If as.sd=TRUE, the magnitude of the value returned is .
For the diagonal elements of this corresponds to standard uncertainties
in GUM notation.
If as.sd=TRUE, keep.sign controls whether the values are signed or
returned as absolute values. If keep.sign=TRUE, the value returned is
. If false,
the absolute value is returned. Note that the sign is returned solely to indicate
the direction of the original contribution. keep.sign has no effect if
as.sd=FALSE.
If simplify=FALSE (the default), the requested elements of the contribution matrix
are returned as a matrix. If simplify=FALSE, the return value is a vector
containing only those terms with nonzero values in the associated correlation matrix.
The threshold for deciding a correlation is nonzero is that its magnitude is greater
than 2*.Machine$double.eps.
Off-diagonal terms for the same pair of variables are summed, that is, for
the pair
the (single) value returned is .
The contributions returned can be limited to a chosen subset using scope;
only the terms involving variables included in scope are returned.
scope can be an expression, formula or character vector of variable names.
If an expression or formula, only those contributions involving variables in
the expression or formula are returned.
Any variable names in scope which are not present in
row.names(object$budget) are silently ignored except for
the formula specification which will return an error.
If simplify=FALSE, the matrix returned always contains all contributions
involving individual variables in scope. If simplify=TRUE, however, specifying
scope as a formula provides additional control over the returned contributions:
If a formula, scope accepts the usual model formula operators ‘.’, ‘+’, ‘-’,
‘*’ and ‘^’, but the interpretation is not quite identical to lm.
First, if present, ‘.’ is taken by default as ‘all contributions’, implying
all single terms and all pairwise terms (like ‘.^2)’ in other formula specifications).
This can be disabled by specifying expand.dot=FALSE.
The negation operator ‘-’ removes terms, but removing a single variable also removes any
associated covariance contributions. For example, scope=~.-A is expanded to all single
and pairwise contributions to the uncertanty budget that do not involve A.
Interaction-like terms of the form A:B are interpreted as indicating the total
off-diagonal contribution, that is, A:B is equivalent to B:A and the associated
value returned is based on .
Cross-terms like ~A*B are supported and expand, as usual, to ~A+B+A:B.
Unlike the two other scope specifications, single terms in the formula do not
automatically imply off-diagonal terms; A+B will not return the off-diagonal contribution for
A and B. Use A*B or (A+B)^2 etc. to get off-diagonal contributions.
Cross-terms of order above two are ignored so A*B*C safely returns only the set of
individual and pairwise terms, but it is perhaps more precise to use (A+B+C)^2.
I() and other operators or functions are not supported.
A named vector or matrix of contributions. Names for off-diagonal contributions in the vector format are constructed from the names of the two contributing variables.
S. L. R. Ellison [email protected]
JCGM 100 (2008) Evaluation of measurement data - Guide to the expression of uncertainty in measurement. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf. (JCGM 100:2008 is a public domain copy of ISO/IEC Guide to the expression of uncertainty in measurement (1995) ).
#Example with negative correlation x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<- -0.5 u.form.c<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM", cor=u.cor) contribs(u.form.c, simplify=FALSE) contribs(u.form.c) contribs(u.form.c, as.sd=TRUE) contribs(u.form.c, as.sd=TRUE, keep.sign=FALSE) contribs(u.form.c, scope=c("a", "c", "d") ) #Effects of formula specification for scope: contribs(u.form.c, ~.) #All contributions contribs(u.form.c, ~(a+b+c+d)^2) #same as ~. contribs(u.form.c, ~a+b+c+d ) #single-variable contributions only contribs(u.form.c, ~., expand.dot=FALSE ) # as ~a+b+c+d contribs(u.form.c, ~.-d) #Drops d and c:d contribs(u.form.c, ~.-c:d) contribs(u.form.c, ~c+d) contribs(u.form.c, ~c*d)#Example with negative correlation x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<- -0.5 u.form.c<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM", cor=u.cor) contribs(u.form.c, simplify=FALSE) contribs(u.form.c) contribs(u.form.c, as.sd=TRUE) contribs(u.form.c, as.sd=TRUE, keep.sign=FALSE) contribs(u.form.c, scope=c("a", "c", "d") ) #Effects of formula specification for scope: contribs(u.form.c, ~.) #All contributions contribs(u.form.c, ~(a+b+c+d)^2) #same as ~. contribs(u.form.c, ~a+b+c+d ) #single-variable contributions only contribs(u.form.c, ~., expand.dot=FALSE ) # as ~a+b+c+d contribs(u.form.c, ~.-d) #Drops d and c:d contribs(u.form.c, ~.-c:d) contribs(u.form.c, ~c+d) contribs(u.form.c, ~c*d)
Constructs a covariance matrix and associated location using a variety
of (possibly robust) estimators. The returned object is suitable for use
by plot.d.ellipse.
cov.dellipse(x, y = NULL, cov.method = c("spearman", "kendall", "pearson", "MCD", "OGK", "GK", "gk", "rgk", "mcd", "mve"), scalefn = NULL, locfn = NULL, cov.control = list())cov.dellipse(x, y = NULL, cov.method = c("spearman", "kendall", "pearson", "MCD", "OGK", "GK", "gk", "rgk", "mcd", "mve"), scalefn = NULL, locfn = NULL, cov.control = list())
x |
An R |
y |
A numeric vector of the same length as |
cov.method |
A character value specifying the covariance method used. |
scalefn |
A function that computes univariate scale and (optionally) location estimates from a
numeric vector.
If provided, |
locfn |
A function that computes univariate location estimates from a numeric vector.
If used, |
cov.control |
A named list of arguments passed to the covariance calculation used. Note that this can
override |
cov.dellipse is a wrapper for a range of covariance estimation methods found in
various packages. Its operation and defaults depend on the particular covariance
estimator specified by cov.method. Details for each are as follows.
spearman, kendall
By default, the median and mad are used as location and scale respectively,
and the covariance is calculated from the product of scale estimates and the
Spearman rank correlation or Kendall's tau respectively.
If either scalefn or locfnis supplied, scalefn is used for scale estimation and
locfn for location. For both spearman and kendall, scalefn is
only used as a scale estimator and need not take a mu.too argument.
pearsonBy default, the mean and sd are used as location and scale respectively,
and the covariance is calculated from the product of scale estimates and the
Pearson correlation.
If either scalefn or locfnis supplied, scalefn is used for scale
estimation and locfn for location, making it possible (if not very sensible) to
use a combination of robust scale or location functions with the Pearson correlation coefficient.
For this case, scalefn is only used as a scale estimator and need
not take a mu.too argument.
MCD, mcd
Both compute the Minimum Covariance Determinant (MCD) estimator, a robust multivariate
location and scale estimate with a high breakdown point, via the 'Fast MCD' or 'Deterministic MCD'
("DetMcd") algorithm. "MCD" uses the implementation covMcd in the robustbase package;
"mcd" uses cov.mcd in the MASS package.
Neither require or use scalefn or locfn.
Note that these MCD implementations differ appreciably for small samples (at least to n=60). MCD
includes consistency and finite sample correction whereas mcd apparently does not apply a finite
sample correction. As a result, the mcd scales can be considerably smaller for modest
data set sizes.
OGKComputes the orthogonalized pairwise covariance matrix estimate described by Maronna and Zamar (2002),
as implemented by the covOGK in the robustbase package.
By default, scale and location use scaleTau2 from robustbase. Alternatives
can be specified either by providing both scalefn and locfn or by including
an argument sigmamu in cov.control, which is passed to covOGK. See
covOGK for a description of sigmamu.
If sigmamu is not present in cov.control and both scalefn and locfn
are provided, scale and location are constructed from scalefn and locfn. If only one
of these is provided, a warning is issued and ]{scaleTau2} is used.
GKComputes a simple pairwise covariance estimate suggested by Gnanadesikan and Kettenring (1972),
as implemented by the covGK in the robustbase package.
By default, scale and location use scaleTau2 from robustbase. Alternatives
can be specified either by providing scalefn and locfn or by including
an argument scalefn in cov.control, which is passed to covGK. See
covGK for a description of scalefn.
If scalefn is not present in cov.control, scale and location are constructed from scalefn
and locfn. If locfn is omitted, scalefn is used if it takes an argument mu.too
and the median is used otherwise.
gkAs GK, except that the variables are scaled to unit (robust) sd (using scalefn) before
calculating the covariance (which is then rescaled). This can prevent large scale differences from
masking outliers in a variable with small scale.
rgkImplements Gnanadesikan and Kettenring's second covariance estimate
based on scaled variables and a robust correlation
calculated as
where and are robust variances of
and respectively, calculated using scalefn.
The advantage over "gk" and "GK" is that the correlation
coefficient is guaranteed to be in , making for a positive definite covariance matrix. Scaling also
helps prevent large scale differences from masking outliers in a variable with small scale.
mveUses uses cov.mve in the MASS package, which is based on the location and covariance matrix for
a minimum volume ellipsoid. The method neither requires nor uses scalefn or locfn.
An object of class cov.dellipse, which is a list with (at least) components
Character string describing method; identical to cov.method
2x2 covariance matrix
2x2 correlation matrix
vector (length 2) specifying centre of ellipse
vector, length 2, specifying scale estimates for each variable
number of points (rows) used in the covariance estimate
This list is intended to be consistent with that returned by cov.wt.
Stephen L R Ellison
Maronna, R.A. and Zamar, R.H. (2002) Robust estimates of location and dispersion of high-dimensional datasets; Technometrics 44(4), 307-317.
Gnanadesikan, R. and John R. Kettenring (1972) Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics 28, 81-124
cov.rob in MASS, covMcd, covOGK and
covGK in robustbase.
data(potassium) cov.dellipse(potassium) #Defaults to Spearman rank correlation #With different method cov.dellipse(potassium, cov.method="OGK") #Same as above but specifying control parameters library(robustbase) #For scaleTau2 cov.dellipse(potassium, cov.method="OGK", cov.control=list(sigmamu=scaleTau2)) #With individually specified (mad) scale cov.dellipse(potassium, cov.method="GK", scalefn=mad) #Defaults to median for location because mad() #does not accept a mu.too argument cov.dellipse(potassium, cov.method="GK", scalefn=scaleTau2) #Defaults to specified scalefn for location because scaleTau2 #accepts mu.too=TRUEdata(potassium) cov.dellipse(potassium) #Defaults to Spearman rank correlation #With different method cov.dellipse(potassium, cov.method="OGK") #Same as above but specifying control parameters library(robustbase) #For scaleTau2 cov.dellipse(potassium, cov.method="OGK", cov.control=list(sigmamu=scaleTau2)) #With individually specified (mad) scale cov.dellipse(potassium, cov.method="GK", scalefn=mad) #Defaults to median for location because mad() #does not accept a mu.too argument cov.dellipse(potassium, cov.method="GK", scalefn=scaleTau2) #Defaults to specified scalefn for location because scaleTau2 #accepts mu.too=TRUE
Constructs and optionally plots a set of probability ellipses for a bivariate normal distribution with defined centre and covariance.
data.ellipse(cov, probs = 0.95, plot = TRUE, npoints = 100, ...) ## S3 method for class 'd.ellipse' summary(object, ...) ## S3 method for class 'd.ellipse' print(x, ...)data.ellipse(cov, probs = 0.95, plot = TRUE, npoints = 100, ...) ## S3 method for class 'd.ellipse' summary(object, ...) ## S3 method for class 'd.ellipse' print(x, ...)
cov |
Covariance and location object of class |
probs |
A vector of probabilities at which ellipses will be constructed. |
plot |
Logical specifying whether the ellipses constructed will additionally be plotted.
If |
npoints |
Integer number of points for each quadrant of the ellipses returned. |
object |
Object of class |
x |
Object of class |
... |
Arguments passed to other methods, particularly |
data.ellipse constructs and returns one set of x, y coordinates for each value of
probs, in a form that can be passed directly to polygon.
Ellipses are constructed from the upper probs quantile of the F distribution using
T = sqrt( 2 * (n - 1) * qf(probs, 2, n - 1)/(n - 2))
where n is the number of pairs used in forming the covariance matrix.
If the number of points is missing or NA, Inf is substituted.
Summary and print methods are provided. The summary method returns a list with the
same names as class d.ellipse, each containing a default summary of the respective
member of the d.ellipse object. The print method returns its argument invisibly.
An object of class d.ellipse, consisting of:
ellipses |
A named list of ellipsoids named for each probability in |
probs |
Numeric vector of probabilities as supplied by |
cov |
Covariance object of class |
S L R Ellison
ISO 13528:2005 Statistical methods for use in proficiency testing by interlaboratory comparisons. (2005) International organization for Standardizaton, Geneva
Jackson, J. E. (1956) Quality control methods for two related variables. Industrial Quality Control, Vol. 7, pp. 2-6
Jackson, J. E. (1959) Control Methods for Several Related Variables. Technometrics, Vol. 1, pp. 359-377
cov.dellipse, plot.d.ellipse, polygon
data(chromium) cov.Cr <- cov.dellipse(chromium) dellipse.Cr <- data.ellipse(cov.Cr, plot=FALSE) summary(dellipse.Cr)data(chromium) cov.Cr <- cov.dellipse(chromium) dellipse.Cr <- data.ellipse(cov.Cr, plot=FALSE) summary(dellipse.Cr)
Calculates derSimonian-Laird estimate of location, with standard error, assuming a random-effects model
dsl(x, ..., na.rm = FALSE) ## Default S3 method: dsl(x, s, n = NULL, groups = NULL, ..., na.rm = FALSE)dsl(x, ..., na.rm = FALSE) ## Default S3 method: dsl(x, s, n = NULL, groups = NULL, ..., na.rm = FALSE)
x |
numeric vector of mean values for groups, or (if |
s |
numeric vector of length |
n |
integer giving the number of observations in each group. May be a vector
of length |
groups |
factor, or vetor which can be coerced to factor, of groups. If
present, |
na.rm |
logical: if |
... |
Further parameters passed to other methods. |
dsl implements the derSimonian-Laird random-effects estimate of location,
using the implementation described by Jackson (2010).
The estimator assumes a model of the form
in which is drawn from and
is drawn from .
The estimator forms a direct calculation of , and uses this to
form revised estimates of standard error
in x, calculates weights as the inverse of these and in turn calculates a
weighted mean, allowing for any calculated excess variance .
This implementation permits input in the form of:
means x and standard errors s, in which case neither n nor
groups are supplied;
means x, standard deviations s and group size(s) n,
standard errors then being calculated as s/sqrt(n)
individual observations x with a groupinf factor groups, in
which case standard errors are calculated from the groups using tapply.
A loc.est object; see loc.est for details. In the returned object, individual
values xi are always input means (calculated from groups and n as
necessary); method.details is returned as a list containing:
The estimated location.
The standard error in the location.
The excess variance (as a standard deviation).
S L R Ellison [email protected]
Jackson et al. (2010) J Stat Plan Inf 140, 961-970
#PCB measurements in a sediment from Key Comparison CCQM-K25 #s are reported standard uncertainties pcb105 <- data.frame(x=c(10.21, 10.9, 10.94, 10.58, 10.81, 9.62, 10.8), s=c(0.381, 0.250, 0.130, 0.410, 0.445, 0.196, 0.093)) with( pcb105, dsl(x, s) )#PCB measurements in a sediment from Key Comparison CCQM-K25 #s are reported standard uncertainties pcb105 <- data.frame(x=c(10.21, 10.9, 10.94, 10.58, 10.81, 9.62, 10.8), s=c(0.381, 0.250, 0.130, 0.410, 0.445, 0.196, 0.093)) with( pcb105, dsl(x, s) )
drop1 calculates revised combined uncertainty for single variable deletions from
an object of class 'uncert'.
## S3 method for class 'uncert' drop1(object, scope, simplify = TRUE, which=c("% Change", "var", "u", "var.change", "u.change"), ...) ## S3 method for class 'uncertMC' drop1(object, scope, simplify = TRUE, which=c("% Change", "var", "u", "var.change", "u.change"), ...) #Print and plot methods ## S3 method for class 'drop1.uncert' print(x, ..., digits=2) ## S3 method for class 'drop1.uncert' plot(x, ..., which=c("% Change", "var", "u", "var.change", "u.change"))## S3 method for class 'uncert' drop1(object, scope, simplify = TRUE, which=c("% Change", "var", "u", "var.change", "u.change"), ...) ## S3 method for class 'uncertMC' drop1(object, scope, simplify = TRUE, which=c("% Change", "var", "u", "var.change", "u.change"), ...) #Print and plot methods ## S3 method for class 'drop1.uncert' print(x, ..., digits=2) ## S3 method for class 'drop1.uncert' plot(x, ..., which=c("% Change", "var", "u", "var.change", "u.change"))
object |
An object of class ‘uncert’ or ‘uncertMC’. |
scope |
character vector, expression or formula containing the list of variables to be dropped. If missing, all variables in object$budget are taken as scope. |
simplify |
logical. If |
which |
logical; controls the form of information returned when
|
x |
An object of class ‘drop1.uncert’ returned by |
... |
Further objects passed to other functions. |
digits |
number of digits used to format the output. See the |
By analogy with drop1, drop1.uncert perfoms single variable deletions from
the uncertainty budget in object, calculates the resulting uncertainty and returns the
results in the form requested by simplify and which.
‘Single variable deletion’ of a variable is equivalent to setting the uncertainty
to zero. Note that this also sets covariance terms involving
to zero. drop1.uncert does not support the deletion of single terms such as .
In the case of ‘uncertMC’ objects, drop1 currently requires object$MC$x to be
present (i.e. uncertMC called with keep.x=TRUE). The uncertMC
method does not support correlation.
For which="var.change", which="u.change" and which="% Change" the
change on dropping a variable is negative if the uncertainty reduces on removing the variable.
The print method simply prints the output with a header formed from the expr attribute
and with '%' appended to the "% Change" column.
The plot method produces a barplot of the chosen data column. A plot for each value in which
is produced. Arguments in ‘...’ are passed to barplot. If not already present in ‘...’
a default main title and ylab are used. The expr attribute is shown as marginal text if not NA.
If simplify=FALSE, an object of class ‘drop1.uncert’, consisting of a
data frame with row names corresponding to row.names(object$budget), columns
corresponding to all possible values of which in the order "var", "u", "var.change",
"u.change", "% Change", and an attribute expr containing a copy of the expr
value of the 'uncert' object to which drop1.uncert is applied.
If simplify=TRUE, the column of the above data frame corresponding to which
is returned as a vector with names row.names(object$budget).
S. L. R. Ellison, [email protected]
None.
uncert, uncert-class, format for digits,
barplot for available plot parameters.
#Continuing the example from plot.uncert: require(graphics) d1<-drop1(u.form.c, simplify=FALSE) d1 plot(d1) drop1(u.form.c) #% change only#Continuing the example from plot.uncert: require(graphics) d1<-drop1(u.form.c, simplify=FALSE) d1 plot(d1) drop1(u.form.c) #% change only
Produces a Duewer concordance/apparent precision plot, showing relative precision or uncertainty plotted against (relative) deviation from assigned value.
dplot(x, ...) duewer.plot(x, ...) ## Default S3 method: duewer.plot(x,s,mu=median(x),sigma=mad(x), s0=median(s), labels=NA, radius=1:3, units=c("z","x"), main, xlab, ylab, xlim, ylim, at.xax=NULL, at.yax=NULL, aspect, col.contours="lightgrey", lty.contours=par("lty"), lwd.contours=par("lwd"), label.contours=T, format.clab="p=%4.3f", cex=par("cex"), cex.label=0.7, pos=3, adj=NULL, pos.clab="bottomright", col.clab=col.contours, cex.axis=par("cex.axis"), pch=par("pch"), las=par("las"), col=par("col"), bg=par("bg"), ...)dplot(x, ...) duewer.plot(x, ...) ## Default S3 method: duewer.plot(x,s,mu=median(x),sigma=mad(x), s0=median(s), labels=NA, radius=1:3, units=c("z","x"), main, xlab, ylab, xlim, ylim, at.xax=NULL, at.yax=NULL, aspect, col.contours="lightgrey", lty.contours=par("lty"), lwd.contours=par("lwd"), label.contours=T, format.clab="p=%4.3f", cex=par("cex"), cex.label=0.7, pos=3, adj=NULL, pos.clab="bottomright", col.clab=col.contours, cex.axis=par("cex.axis"), pch=par("pch"), las=par("las"), col=par("col"), bg=par("bg"), ...)
x |
Numeric vector of values to be plotted. |
s |
Numeric vector of standard deviations, standard errors or uncertainties
of length |
mu |
A single location against which to compare x. |
sigma |
A measure of dispersion against which deviations x-mu can be compared. |
s0 |
A typical, expected or reference value for the standard uncertainties s |
labels |
An optional vector of point labels of length |
radius |
A vector of radii for reference lines in the classic Duewer plot. |
units |
Controls scaling of the plot. If set to |
main |
Main title for the plot, passed to |
xlab, ylab
|
x- and y-axis labels, passed to |
xlim, ylim
|
x- and y-limits for the plot. |
at.xax, at.yax
|
Locations for x- and yaxis tick marks, passed to |
aspect |
The aspect ratio for the plot, passed to |
col.contours, lty.contours, lwd.contours
|
Colour, line type and line width for contour lines. |
label.contours |
Logical, controlling whether countour lines are labelled with approximate probabilities. |
format.clab |
format string for contour labels; passed to |
cex |
Expansion factor for plotted symbols. |
cex.label |
Expansion factor for point labels. |
pos, adj
|
Specifies position/adjustment of point labels. Passed to |
pos.clab |
Specification for location of contour labels. Options are '"top"', '"topright"', '"right"',
'"bottomright"', '"bottom"', '"bottomleft"', '"left"', '"topleft"'. A vector can be provided
to give multiple labels. Contour labels for |
col.clab |
Colour for contour labels. |
cex.axis |
Expansion factor for axis labels. |
las |
Axis label orientation, passed to |
pch, col, bg
|
Graphical parameters passed to |
... |
Other parameters passed to plotting functions. Currently unused. |
A Duewer plot is a plot of dispersion against location. Classically, this has been applied to
multiple observations from laboratories. Locations x are mean results of the form
(x-mu)/s and and dispersions s are the associated sd. The principle has also
been applied to multiple results for different measurands per laboratory, by calculating
z-scores for all observations relative to the assigned value and dispersion for each measurand
and then plotting mean and sd of the scores. More recently the plot has been used to summarise
reported values and (usually) standard uncertainties in metrology comparisons to allow
quick assessment of anomalies within data sets.
The traditional plot includes visual guides in the form of semicircular contours at multiples of (x-mu)/sigma for the x-axis and s/s0 for the y-axis, s0 being a median or other estimate of the typical standard deviation.
Contours are, by default, labelled with probabilities corresponding to quantiles of the normal distribution.
dplot is an alias for duewer.plot.
This function is called for its side effect, which is the production of a plot.
S Ellison [email protected]
Duewer, D, Probably in Anal. Chem. in about 1990
axis for axis control, points, text for
plotting parameters; sprintf for contour label format.
xs.plot for a plot of location and scale data with probabilistic
confidence regions.
require(metRology) data(Pb) Pb duewer.plot(Pb$value, Pb$u) duewer.plot(Pb$value, Pb$u, basis="prob", df=5) #Illustrate contour labelling duewer.plot(Pb$value, Pb$u, pos.clab="bottom")require(metRology) data(Pb) Pb duewer.plot(Pb$value, Pb$u) duewer.plot(Pb$value, Pb$u, basis="prob", df=5) #Illustrate contour labelling duewer.plot(Pb$value, Pb$u, pos.clab="bottom")
Functions for manipulating interlaboratory study objects objects of class ‘ilab’.
## S3 method for class 'ilab' subset(x, subset, drop=FALSE, ...) ## S3 method for class 'ilab' x[i, j]## S3 method for class 'ilab' subset(x, subset, drop=FALSE, ...) ## S3 method for class 'ilab' x[i, j]
x |
An object of class ‘ilab’ |
subset |
logical expression indicating elements or rows to keep: missing values are taken as false. |
drop |
passed on to '[' indexing operator. |
... |
Parameters passed to other functions |
i, j
|
elements to extract. May be numeric or logical vectors. |
For the subset method, subset is an expression evaluated in the frame
of ilab$data and in the parent environment if objects are not found in
ilab$data. Note that since ilab$distrib and ilab$distrib.pars are
not in ilab$data, any operation on these must be specified in full.
The indexing method '[' operates on both rows and columns of the object. However,
only the $data element can be addressed with the j; the distrib and
distrib.pars elements are unaffected by j and will always be included
in the returned object.
An object of class ‘ilab’ with fewer rows and (if j is present)
fewer columns.
Removing the standard columns from ‘ilab’ objects using '[' may have unforeseen
consequences for other functions; only the print method is likely to operate successfully.
S. L. R. Ellison [email protected]
None, yet.
data(Pb) il.pb<-construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) subset(il.pb, u < 0.03) il.pb[1:6,]data(Pb) il.pb<-construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) subset(il.pb, u < 0.03) il.pb[1:6,]
gplot is primarily used by plot.mandel.kh to produce the underlying grouped data plot.
gplot(x, main = NULL, xlab = NULL, ylab = deparse(substitute(x)), ylim = NULL, las = 1, axes = TRUE, cex.axis = 1, frame.plot = axes, lwd = 1, lty = 1, col = par("col"), separators = TRUE, col.sep = "lightgrey", lwd.sep = 1, lty.sep = 1, zero.line = TRUE, lwd.zero = 1, col.zero = 1, lty.zero = 1, spacing=NA, ...)gplot(x, main = NULL, xlab = NULL, ylab = deparse(substitute(x)), ylim = NULL, las = 1, axes = TRUE, cex.axis = 1, frame.plot = axes, lwd = 1, lty = 1, col = par("col"), separators = TRUE, col.sep = "lightgrey", lwd.sep = 1, lty.sep = 1, zero.line = TRUE, lwd.zero = 1, col.zero = 1, lty.zero = 1, spacing=NA, ...)
x |
A matrix or data frame to be plotted. |
main |
Main title for the plot. |
xlab, ylab
|
Labels for x and y axes. |
ylim |
the y limits of the plot. |
las |
the style of the axis labels; see |
axes |
a logical value indicating whether axes should be drawn on the plot. |
cex.axis |
The magnification to be used for axis annotation relative to the current setting of 'cex'. |
frame.plot |
Logical; If |
lwd, lty, col
|
Graphical parameters used for the plotted vertical lines corresponding to each value in x. |
separators |
Logical; if |
col.sep, lwd.sep, lty.sep
|
Graphical parameters used for the separator lines. |
zero.line |
logical; if |
lwd.zero, col.zero, lty.zero
|
Graphical parameters used for the zero line. |
... |
Other graphical parameters passed to |
spacing |
Spacing for data within each group, as a fraction of inter-group spacing. Defaults to 0.3 or less. |
gplot produces a plot of type="h", with values in x grouped by row and with
optional vertical separators between groups. The plotting order (left to right) is
in order of stack(as.data.frame(t(x))); each group corresoponds to a row in x.
Because gplot is primarily a supporting function for plot.mandel.kh,
it assumes a suitable object will be provided and does minimal checking
to ensure an appropriate object class. Error messages may not be
very informative.
A numeric vector of mid-points of the groups along the x-axis.
S Ellison [email protected]
Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva (1994).
data(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) gplot(h, las=2) #Note the absence of indicator lines, title etc. #compared to plot(h)data(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) gplot(h, las=2) #Note the absence of indicator lines, title etc. #compared to plot(h)
A function for propagation of measurement uncertainty for typical metrology applications using the methods from the Joint Committee on Guides in Metrology (JCGM) Guide to the Expression of Uncertainty in Measurement (GUM). This approach approximates the uncertainty of a function of random variables that define a measurement result by computing the uncertainty of the first-order Taylor series for the measurement function. This function also serves as the primary computational tool underlying the GUM uncertainty templates found in the metRology for Microsoft Excel user interface.
GUM(var.name, x.i, u.i, nu.i, measurement.fnc, correlation = diag(length(var.name)), shared.u.i = var.name, cl = 0.95, cov.factor = "Student's t", sig.digits.U = 2, ...)GUM(var.name, x.i, u.i, nu.i, measurement.fnc, correlation = diag(length(var.name)), shared.u.i = var.name, cl = 0.95, cov.factor = "Student's t", sig.digits.U = 2, ...)
var.name |
Character vector of input variable names. |
x.i |
Vector of input variable values. |
u.i |
Vector of standard uncertainties (i.e. standard errors) for each input variable value. |
nu.i |
Degrees of freedom associated with each standard uncertainty. |
measurement.fnc |
Character string specifying the functional relationship between input variables that defines the output measurement result. |
correlation |
Matrix giving the correlation between the different input variable values. Default is to assume no correlation between input variable values. |
shared.u.i |
Character vector giving the relative relationship between the standard uncertainties for each variable value. Groups of variables based on a common shared standard uncertainty will all share the same variable name. The default is to assume all standard uncertainties are assessed independently, resulting a value of shared.u.i that is identical to var.name. |
cl |
Nominal confidence level to be used to compute the expanded uncertainty of the output measurement result. Default value is 0.95. |
cov.factor |
Type of coverage factor to be used. The default is to use the value from the Student's t distribution with confidence level specified above and nu.eff effective degrees of freedom. |
sig.digits.U |
Number of significant digits to be reported in the expanded uncertainty of the measurement result. The measurement result will be rounded to the same number of decimal places. |
... |
Arguments passed to other functions. Currently unimplemented. |
Whenever possible, sensitivity coefficients are obtained analytically using the gradient attribute of the
deriv function. In situations where some part of the measurement function is not found in
derivative table, sensitivity coefficients are obtained by numeric partial differentiation using the
grad function from the package numDeriv.
A list containing the 9 components:
y |
Value of the measurement result obtained by evaluating the measurement function at the input variable values. |
uc |
The combined standard uncertainty of the measurement result, y. |
nu.eff |
The effective degrees of freedom associated with uc, computed using the Welch-Satterthwaite formula. |
cl |
The nominal confidence level used to obtain the coverage factor, k. |
k |
The coverage factor used to control the confidence level associated with the expanded uncertainty of the measurement result. |
U |
The expanded uncertainty of the measurement result, computed as U=k*uc. |
contributions |
Relative variance contributed to the standard uncertainty (uc) of the measurement result from each input variable. |
sensitivities |
Sensitivity coefficient associated with each input variable. |
msgs |
Error and warning messages that point out potential problems with the inputs to the |
Hung-kung Liu [email protected] and Will Guthrie [email protected]
Joint Committee on Guides in Metrology (JCGM), Evaluation of Measurement Data Guide to the Expression of Uncertainty in Measurement, http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf, 2008.
GUM.validate a function to assess the statistical performance of GUM uncertainty intervals
for the application of interest in terms of average attained coverage probability.
uncert for a family of functions focused on the study and comparison of different approaches
and numerical options in uncertainty analysis.
## a simple uncertainty analysis for the product of two quantities GUM(c("x1","x2"),c(2.3,1.1),c(0.030,0.015),c(5,9999),"x1*x2") ## example of the difference in the measurements of two standards, each ## with a standard uncertainty based on a common value drawn from a control chart ## representative of the measurement process made using a check standard that ## is comparable to the two individual standards under study GUM(c("s1","s2"),c(45.3,46.0),c(0.26,0.26),c(134,134),"s1-s2",shared.u.i=c("s1","s1")) ## compare with results for equivalent, alternative specification of shared.u.i GUM(c("s1","s2"),c(45.3,46.0),c(0.26,0.26),c(134,134),"s1-s2",shared.u.i=c("s2","s2"))## a simple uncertainty analysis for the product of two quantities GUM(c("x1","x2"),c(2.3,1.1),c(0.030,0.015),c(5,9999),"x1*x2") ## example of the difference in the measurements of two standards, each ## with a standard uncertainty based on a common value drawn from a control chart ## representative of the measurement process made using a check standard that ## is comparable to the two individual standards under study GUM(c("s1","s2"),c(45.3,46.0),c(0.26,0.26),c(134,134),"s1-s2",shared.u.i=c("s1","s1")) ## compare with results for equivalent, alternative specification of shared.u.i GUM(c("s1","s2"),c(45.3,46.0),c(0.26,0.26),c(134,134),"s1-s2",shared.u.i=c("s2","s2"))
Calibration of an end gauge for length measurement. Notation is based on the presentation in chapter 3 of Data Modeling for Metrology and Testing in Measurement Science.
GUM.H.1 data(GUM.H.1)GUM.H.1 data(GUM.H.1)
A list containing 10 components that give the uncertainty budget, measurement function, and other quantities needed to carry out an uncertainty analysis using the methods from the Guide to the Expression of Uncertainty in Measurement:
Character vector giving the name of each input variable included in the analysis.
Expression giving the units of each input variable
Vector giving the reported value for each input variable.
Vector giving the standard uncertainty associated with each reported value.
Vector giving the degrees of freedom associated with each standard uncertainty.
Character vector indicating the method of evaluation (Type A or Type B) for each standard uncertainty.
Character vector listing the probability distribution assumed to describe each variable value.
Character string giving the measurement function for the output variable.
Matrix giving the correlations between input variable values.
Character vector describing which standard uncertainties, if any, are based on a common underlying standard uncertainty. A vector that is the same as var.name indicates that all standard uncertainties are based on independent assessments of standard uncertainty. A vector with fewer names than in var.names indicates that one or more variables are derived from a common uncertainty assessment.
Details can be found in the GUM and in Data Modeling for Metrology and Testing in Measurement Science.
Joint Committee on Guides in Metrology (JCGM), Evaluation of Measurement Data Guide to the Expression of Uncertainty in Measurement, http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf, 2008.
Guthrie, W.F. et. al., "Three Statistical Paradigms for Assessment and Interpretation of Measurement Uncertainty" in Data Modeling for Metrology and Testing in Measurement Science, F. Pavese and A.B. Forbes, eds., Birkhauser, Boston, 2009.
A function for assessing the statistical performance of measurement uncertainty intervals for
particular metrology applications computed using the methods from the Joint Committee on
Guides in Metrology (JCGM) Guide to the Expression of Uncertainty in Measurement (GUM).
The validation is carried out using the input values as true values in a simulation that
directly checks the attained coverage probability of the uncertainty
intervals produced using the GUM function.
GUM.validate(var.name, x.i, u.i, nu.i, type, distribution, measurement.fnc, correlation = diag(length(var.name)), shared.u.i = var.name, cl = 0.95, cov.factor = "Student's t", sig.digits.U = 2)GUM.validate(var.name, x.i, u.i, nu.i, type, distribution, measurement.fnc, correlation = diag(length(var.name)), shared.u.i = var.name, cl = 0.95, cov.factor = "Student's t", sig.digits.U = 2)
var.name |
Character vector of input variable names. |
x.i |
Vector of input variable values. |
u.i |
Vector of standard uncertainties (i.e. standard errors) for each input variable value. |
nu.i |
Degrees of freedom associated with each standard uncertainty. |
type |
Character vector of values "A" and "B" indicating the methods used to evaluate the standard uncertainty of each input value. Standard uncertainties evaluated using statistical methods are denoted Type A in the GUM, while standard uncertainties evaluated using other means are denoted Type B. |
distribution |
Character vector of probability distributions associated with the potential values taken on by each input variable. The current possible choices are "Normal" (i.e. Gaussian), "Triangular", or "Rectangular" (i.e. Uniform). |
measurement.fnc |
Character string specifying the functional relationship between input variables that defines the output measurement result. |
correlation |
Matrix giving the correlation between the different input variable values. Default is to assume no correlation between input variable values. |
shared.u.i |
Character vector giving the relative relationship between the standard uncertainties for each variable value. Groups of variables based on a common shared standard uncertainty share will all share the same variable name. The default is to assume all standard uncertainties are assessed independently, resulting a value of shared.u.i that is identical to var.name. |
cl |
Nominal confidence level to be used to compute the expanded uncertainty of the output measurement result. Default value is 0.95. |
cov.factor |
Type of coverage factor to be used. The default is to use the value from the Student's t distribution with confidence level specified above and nu.eff effective degrees of freedom. |
sig.digits.U |
Number of significant digits to be reported in the expanded uncertainty of the measurement result. The measurement result will be rounded to the same number of decimal places. |
Currently 1000 simulated sets of uncertainty data are used for the computation of the attained confidence level.
A Monte Carlo assessment of the attained coverage of expanded uncertainty intervals like those produced
using the GUM function for the application of interest.
Hung-kung Liu [email protected] and Will Guthrie [email protected]
Joint Committee on Guides in Metrology (JCGM), Evaluation of Measurement Data Guide to the Expression of Uncertainty in Measurement, http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf, 2008.
GUM a function to compute GUM uncertainty intervals for general metrological applications.
## a simple uncertainty analysis for the product of two quantities GUM.validate(c("x1","x2"), c(2.3,1.1), c(0.030,0.015), c(5,9999), c("A","B"),c("Normal","Rectangular"),"x1*x2")## a simple uncertainty analysis for the product of two quantities GUM.validate(c("x1","x2"), c(2.3,1.1), c(0.030,0.015), c(5,9999), c("A","B"),c("Normal","Rectangular"),"x1*x2")
The ‘ilab’ class and its constructor function.
construct.ilab(org, item, measurand, x, u, df, k, U, U.lower, U.upper, distrib=NULL, distrib.pars=NULL, study=NA, title=NA, p=0.95, ...)construct.ilab(org, item, measurand, x, u, df, k, U, U.lower, U.upper, distrib=NULL, distrib.pars=NULL, study=NA, title=NA, p=0.95, ...)
org |
Character vector or factor of organisation names. |
item |
vector or factor of identifiers for test items. Coerced to factor on storage. |
measurand |
vector or factor identifying the measurand(s) involved in the study. |
x |
numeric vector of reported values. |
u |
numeric vector of reported standard uncertainties or standard errors associated with x. |
df |
optional numeric vector of degrees of freedom associated with each reported uncertainty. |
k |
numeric vector of coverage factors. The coverage factor is the factor multiplying u to obtain U. |
U |
numeric or character vector of expanded uncertainties or confidence interval half-widths. Coerced to numeric but may include a character representation of interval limits; see Details. |
U.lower, U.upper
|
numeric vectors of lower and upper limits for the confidence interval around x, allowing asymmetric intervals. Defaults to U or to the limits specified by U. See Details. |
distrib |
A character vector of length |
distrib.pars |
A named list of lists of parameters describing the distributions
associated with |
study |
A character value or vector or a factor identifying different studies
or study populations within the data set. Typically used, for example, for identifying
participants in global and regional components of a combined study. Recycled to length
|
title |
An optional title for the study. May be a character vector, in which case each element is displayed on a separate line when printed. |
p |
Confidence level assumed to apply to |
... |
Other named factors or character vectors used to group observations. |
If U is a character vector, it may contain character representations of range.
Two forms are permitted:
Interpreted as limits of a range from a to b. U.lower
and U.upper are calculated from these limits and x
U.upper is set to a in "+a",
and U.lower is set to b in "-b".
If distrib.pars is missing, an attempt is made to deduce appropriate
distribution parameters from x, u, df and distrib.
In doing so, the following assumptions and values apply for the respective distributions:
mean=x$name, sd=u$name.
min=x-sqrt(3)*u, max=x+sqrt(3)*u.
min=x-sqrt(6)*u, max=x+sqrt(6)*u, mode=x.
df=df, mean=x, sd=u.
In addition, if distrib contains "t" or "t.scaled", and
df is NA, the corresponding degrees of freedom are chosen based on
k and p.
An object of class ‘ilab’ consisting of:
title |
A character value or vector describing the study |
|||||||||||||||||||||||
subset |
A character string describing any subset operation used to form the object. |
|||||||||||||||||||||||
data |
A data frame with columns:
|
|||||||||||||||||||||||
distrib |
An unnamed list of distribution names. |
|||||||||||||||||||||||
distrib.pars |
An unnamed list of lists of parameters describing the distributions
associated with |
S. L. R. Ellison [email protected]
None, yet.
print.ilab, subset.ilab, plot.ilab
data(Pb) construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) #Illustrate default for U and automatic distrib.pars construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, distrib="norm") construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, distrib="t.scaled")data(Pb) construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) #Illustrate default for U and automatic distrib.pars construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, distrib="norm") construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, distrib="t.scaled")
Calculates a 'largest consistent subset' given values and associated uncertainty information.
LCS(x, u, p = 0.05, method = "enum", simplify = FALSE, verbose = FALSE)LCS(x, u, p = 0.05, method = "enum", simplify = FALSE, verbose = FALSE)
x |
Vector of observations. |
u |
Vector of standard errors or standard uncertainties associated with |
p |
Significance level at which consistency is tested. |
method |
Subset identification method. Currently only 'enum' is supported. |
simplify |
If |
verbose |
Logical: Controls the level of reporting during the search. |
LCS obtains the largest subset(s) of x which pass a chi-squared
test for consistency, taking the uncertainties u into account.
method controls the search method used. Method "enum" uses complete enumeration
of all subsets of size n, starting at n==length(x) and decreasing n
until at least one consistent subset is found. No other method is currently supported; if
a different method is specified, LCS provides a warning and continues with "enum".
There may be more than on consistent subset of size n. If so, LCS returns all such
subsets unless simplify is TRUE, in which case LCS prints a short warning
and returns the subset with smallest estimated uncertainty as estimated for the Graybill-Deal
weighted mean assuming large degrees of freedom in u.
verbose controls the level of reporting. If TRUE, LCS prints the progress of
the search.
The general idea of a Largest Consistent Subset as implemented here was suggested by Cox (2006), though at least one other related method has been suggested by Heydorn (2006). It has, however, been criticised as an estimator (Toman and Possolo (2009)) ; see Warning below.
If there is only one subset of maximum size, or if simplify=TRUE, a vector of indices
for x representing the largest consistent subset.
If there is more than one subset of maximum size and simplify=FALSE, a matrix of indices
in which the rows contain the indices of each subset.
LCS methods are essentially equivalent to unsupervised outlier rejection. In general, this results in a possibly extreme low estimated variance for an arbitrarily small subset (in the limit of gross inconsistency, LCS will return subsets of size 1). The estimated uncertainty calculated for the Graybill-Deal weighted mean of the subset(s) does not generally take account of the subset selection process or the dispersion of the complete data set, so is not an estimate of sampling variance.
LCS is therefore not recommended for consensus value estimation. It is however, quite useful for identifying value/uncertainty outliers.
S. Ellison [email protected]
Cox, M. G. (2007) The evaluation of key comparison data: determining the largest consistent subset. Metrologia 44, 187-200 (2007)
Heydorn, K. (2006) The determination of an accepted reference value from proficiency data with stated uncertainties. Accred Qual Assur 10, 479-484 (2006)
Toman, B. and Possolo, A. (2009) Laboratory effects models for interlaboratory comparisons. Accred. Qual. Assur. 14, 553-563 (2009)
None.
data(Pb) with(Pb, LCS(value, U/k))data(Pb) with(Pb, LCS(value, U/k))
The location estimate class contains output from a variety of estimators used in the metRology package.
A print method is provided.
## S3 method for class 'loc.est' print(x, ...)## S3 method for class 'loc.est' print(x, ...)
x |
An object of class ‘loc.est’ |
... |
Parameters passed to other functions. Currently unused. |
An object of class ‘loc.est’ is a list containing
Scalar estimate of location
Standard uncertainty (usually equivalent to standard error) of the location estimate.
Degrees of freedom associated with the location estimate (may be NA)
Numeric vector of individual values contributing to the estimate
Numeric vector of uncertainties initially associated with xi.
Numeric vector of degrees of freedom associated with ui.
Numeric vector of ‘effective uncertainties’ in xi after
any additional terms or adjustments are added (see below).
Numeric vector of weights associated with xi (see below).
Character string describing the method used to obtain the estimate.
An optional list of additional details provided by the particular method used.
The ‘effective uncertainties’ u.eff arise from some estimation methods (for example,
Mandel-Paule). These typically involve either the estimation of an additional variance term,
a scale adjustment to the output value uncertainty or (for example in the case of the
arithmetic mean) replacement of the initial individual uncertainties with some single
estimate based on the dispersion of values. These adjustments are usually equivalent to
replacing the estimator used with a weighted mean using weights .
The weight vector w is not equivalent to . Rather, it
gives the ratio of prior weights to posterior weights, which combine
prior weights with some additional weighting. Posterior weights arise in particular when using
robust estimators, and are generally 1 otherwise. The returned location estimate in such cases
can be calculated as sum(w*x/(u^2))/sum(w/(u^2))).
method.details is an optional list that may contain anything from a short
summary of a scale factor or additional variance to a complete object (e.g. an rlm
object) returned by the function used to calculate the estimate.
The print method is called for its side effect; no value is returned.
S. L. R. Ellison [email protected]
None, yet
## Cd heat of vapourisation example (see ?mpaule) x2<-c(27.044, 26.022, 26.340, 26.787, 26.796) v<-c(3, 76, 464, 3, 14)*1e-3 mp<-mpaule(x2, sqrt(v)) print(mp)## Cd heat of vapourisation example (see ?mpaule) x2<-c(27.044, 26.022, 26.340, 26.787, 26.796) v<-c(3, 76, 464, 3, 14)*1e-3 mp<-mpaule(x2, sqrt(v)) print(mp)
Functions for calculating M- and MM-estimators for location given values and associated standard errors or standard uncertainties.
MM.estimate(x, ...) ## Default S3 method: MM.estimate(x, u, c = 4.685, ...) huber.estimate(x, ...) ## Default S3 method: huber.estimate(x, u, k= 1.345, ...)MM.estimate(x, ...) ## Default S3 method: MM.estimate(x, u, c = 4.685, ...) huber.estimate(x, ...) ## Default S3 method: huber.estimate(x, u, k= 1.345, ...)
x |
numeric vector of mean values for groups |
u |
numeric vector of standard deviations or standard uncertainties associated with the values |
c, k
|
Tuning parameters passed to other functions (see |
... |
Parameters passed to other functions. |
These functions are wrappers for robust estimation using rlm. All simply
call rlm with the formula x~1 and weights 1/u^2.
An object of class ‘loc.est’.
S. L. R. Ellison [email protected]
None, yet.
## Cd heat of vapourisation example ## from Paule, R. C. and Mandel, J. (1982) - see ?mpaule x2<-c(27.044, 26.022, 26.340, 26.787, 26.796) v<-c(3, 76, 464, 3, 14)*1e-3 MM.estimate(x2, sqrt(v)) huber.estimate(x2, sqrt(v))## Cd heat of vapourisation example ## from Paule, R. C. and Mandel, J. (1982) - see ?mpaule x2<-c(27.044, 26.022, 26.340, 26.787, 26.796) v<-c(3, 76, 464, 3, 14)*1e-3 MM.estimate(x2, sqrt(v)) huber.estimate(x2, sqrt(v))
Density, distribution function, quantile function and random generation for Mandel's h statistic, a measure of relative deviation from a common mean.
dmandelh(x, g, log = FALSE) pmandelh(q, g, lower.tail = TRUE, log.p = FALSE) qmandelh(p, g, lower.tail = TRUE, log.p = FALSE) rmandelh(B, g)dmandelh(x, g, log = FALSE) pmandelh(q, g, lower.tail = TRUE, log.p = FALSE) qmandelh(p, g, lower.tail = TRUE, log.p = FALSE) rmandelh(B, g)
x, q
|
vector of quantiles. |
p |
vector of probabilities. |
g |
number of means for which h is calculated. |
B |
Number of observations. If 'length(B) > 1', the length is taken to be the number required. |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x]; otherwise, P[X > x]. |
log, log.p
|
logical; if TRUE, probabilities p are given as log(p). |
Mandel's h is calculated for a particular mean value y[i] in a set of
mean values y as
h[i] = ( y[i] - mean(y) )/sd(y) )
The density, probabilities and quantiles can be derived from the beta distribution: (1+h*sqrt(g)/(g-1))/2 is distributed as Beta((g-2)/2, (g-2)/2).
dmandelh returns the density at x, pmandelh the cumulative probability,
qmandelh the quantiles for probability p and rmandelh returns B
random values drawn from the distribution.
Vector values of x, p, q and g are permitted, in which case the functions return vectors.
Note that rmandelh uses B and not n (as do most R random
number functions) for number of random draws; this is for compatibility with
the relevant functions for Mandel's k, for which n is conventionally
used for the number of replicates per group. Be careful when using named parameters!
S. L. R. Ellison, [email protected]
None.
#Generate the 95% and 99% quantiles for comparison with tables in #ISO 5725:1996 Part 2: n <- 3:30 round(qmandelh(0.975, n), 2) #95% 2-tailed round(qmandelh(0.995, n), 2) #99% 2-tailed#Generate the 95% and 99% quantiles for comparison with tables in #ISO 5725:1996 Part 2: n <- 3:30 round(qmandelh(0.975, n), 2) #95% 2-tailed round(qmandelh(0.995, n), 2) #99% 2-tailed
Density, distribution function, quantile function and random generation for Mandel's k statistic, a measure of relative precision compared to a common variance.
dmandelk(x, g, n, log = FALSE) pmandelk(q, g, n, lower.tail = TRUE, log.p = FALSE) qmandelk(p, g, n, lower.tail = TRUE, log.p = FALSE) rmandelk(B, g, n)dmandelk(x, g, n, log = FALSE) pmandelk(q, g, n, lower.tail = TRUE, log.p = FALSE) qmandelk(p, g, n, lower.tail = TRUE, log.p = FALSE) rmandelk(B, g, n)
x, q
|
vector of quantiles. |
p |
vector of probabilities. |
g |
number of groups for which k is calculated. |
n |
number of observations in each group of data for which k is calculated. |
B |
Number of observations. If 'length(B) > 1', the length is taken to be the number required. |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x]; otherwise, P[X > x]. |
log, log.p
|
logical; if TRUE, probabilities p are given as log(p). |
Mandel's k for one of a set of standard deviations is calculated as
Since the numerator is chi-squared(n-1), or Gamma((n-1)/2, 2), and the denominator
can be written as the sum of the same quantity and a pooled variance with distribution
Gamma((g-1)*(n-1)/2, 2), k is distributed as Beta((n-1)/2, (g-1)(n-1)/2).
Quantiles, probabilities, density and random numbers can therefore be generated
from the Beta distribution. For example, qmandelk is calculated as
sqrt( g * qbeta( (n-1)/2, (g-1)*(n-1)/2)).
dmandelh returns the density at x, pmandelh the cumulative probability,
qmandelh the quantiles for probability p and rmandelh returns B
random values drawn from the distribution.
Vector values of x, p, q and g are permitted, in which case the functions return vectors.
Note that rmandelk uses B and not n (as do most R random
number functions) for number of random draws; this is because n is conventionally
used for the number of replicates per group. Be careful when using named parameters!
S. L. R. Ellison, [email protected]
None.
#Generate the 95% and 99% quantiles for comparison with tables in #ISO 5725:1996 Part 2: round(qmandelk(0.95, g=3:30, n=3), 2) #95% upper tail round(qmandelk(0.99, g=3:30, n=3), 2) #99% upper tail#Generate the 95% and 99% quantiles for comparison with tables in #ISO 5725:1996 Part 2: round(qmandelk(0.95, g=3:30, n=3), 2) #95% upper tail round(qmandelk(0.99, g=3:30, n=3), 2) #99% upper tail
Calculate a weighted mean, between-group standard deviation and standard error on the weighted mean using the Mandel-Paule algorithm.
mpaule(x, ..., tol=.Machine$double.eps^0.25, maxiter=25) ## Default S3 method: mpaule(x, u=NULL, n=NULL, groups=NULL, tol=.Machine$double.eps^0.25, maxiter=25, ...) mandel.paule(x, ..., tol=.Machine$double.eps^0.25, maxiter=25)mpaule(x, ..., tol=.Machine$double.eps^0.25, maxiter=25) ## Default S3 method: mpaule(x, u=NULL, n=NULL, groups=NULL, tol=.Machine$double.eps^0.25, maxiter=25, ...) mandel.paule(x, ..., tol=.Machine$double.eps^0.25, maxiter=25)
x |
numeric vector of mean values for groups, or (if |
u |
numeric vector of standard deviations or standard uncertainties
associated with the values |
n |
integer vector of numbers in each group. If |
groups |
factor, or vetor which can be coerced to factor, of groups. If present, |
... |
Additional parameters passed to other functions. |
tol |
numeric tolerance; iteration stops when the variance step size drops below |
maxiter |
numeric maximum number of iterations |
The Mandel-Paule algorithm finds the between-method variance by iteratively solving the
equation relating the weighted mean to the weighting factor applied. The weighting factor is
the inverse of the sum of the standard error in x and the between-group variance.
If the iterative procedure produces a negative estimate for the between-group variance, the between-group variance is set to zero.
For the default method, if u is present and n=NULL, u is interpreted as
a vector of standard uncertainties or standard errors. If n is not NULL, u
is interpreted as a vector of standard deviations and standard errors are calculated as
u/sqrt(n).
If groups is not NULL, x is interpreted as a vector of individual
observations grouped by groups, and the algorithm is applied to the corresponding
group means and standard errors.
If maxiter is set less than 1, no iterations are performed and the consensus mean
is returned as NA.
mandel.paule is an alias for mpaule retained for backward compatibility.
A loc.est object; see loc.est for details. In the returned object, df is
set to where is the number of non-NA observations or
group means as appropriate, and method.details is returned as :
var.between |
the estimated between-group variance) |
iter |
the number of iterations taken |
converged |
|
S. Cowen [email protected] with amendments by S. L. R. Ellison.
Paule, R. C. and Mandel, J. (1982), J Res Nat Bur Stand, 87, (5) 377-385
## the second example in the paper cited above x <- c(201.533, 216.55) s <- c(0.154, 0.25) n <- c(6, 2) mpaule(x, s/sqrt(n)) ## Cd heat of vapourisation example from the paper cited above x2<-c(27.044, 26.022, 26.340, 26.787, 26.796) v<-c(3, 76, 464, 3, 14)*1e-3 mpaule(x2, sqrt(v))## the second example in the paper cited above x <- c(201.533, 216.55) s <- c(0.154, 0.25) n <- c(6, 2) mpaule(x, s/sqrt(n)) ## Cd heat of vapourisation example from the paper cited above x2<-c(27.044, 26.022, 26.340, 26.787, 26.796) v<-c(3, 76, 464, 3, 14)*1e-3 mpaule(x2, sqrt(v))
mandel.h calculates Mandel's h statistics for replicate observations.
Mandel's h is an indication of relative deviation from the mean value.
mandel.h(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## Default S3 method: mandel.h(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'ilab' mandel.h(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...)mandel.h(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## Default S3 method: mandel.h(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'ilab' mandel.h(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...)
x |
An R object (see Details below), which contains replicate observations or,
if |
g |
A primary grouping factor, usually corresponding to Laboratory in an
inter-laboratory study. If not present, |
m |
A secondary grouping factor, usually corresponding to test item
or measured quantity. |
na.rm |
A logical value indicating whether 'NA' values should be
stripped before the computation proceeds. Passed to functions
such as |
rowname |
A single character label for the primary grouping factor (e.g. "Lab", "Organisation"). |
method |
Character scalar giving the calculation method. |
n |
scalar number of observations per group. Required only if |
... |
Additional parameters passed to |
mandel.h is a convenience wrapper for mandel.kh(..., type="h"). It is generic,
with methods for numeric vectors, arrays, data frames, matrices and objects of class
'ilab'. All parameters are passed to mandel.kh.
Mandel's h is an indicators of relative deviation for grouped
sets of observations. Given a set of observations where
denotes observation , for measurand or test item and group
(usually laboratory) , , Mandel's is given by:
where
If x is a vector, one-dimensional array or single-column matrix, values are aggregated
by g and, if present, by m. If x is a data frame or matrix, each column
is aggregated by g and m silently ignored if present. In all cases, if g
is NULL or missing, each row (or value, if a vector) in x
is taken as a pre-calculated mean (for Mandel's ) or standard deviation (for Mandel's ).
If x is an object of class 'ilab', g defaults to '$org' and
m to $measurand.
The returned object includes a label ('grouped.by') for the primary grouping factor.
For the 'ilab' method, this is "Organisation". For other methods, If rowname is
non-null, rowname is used. If rowname is NULL, the default is deparse(substitute(g));
if g is also NULL or missing, "Row" is used.
If method="robust", Mandel's is replaced by a robust z score calculated by
replacing and with the robust estimates of location and scale
obtained using Huber's estimate with tuning constant k set to 1.5 (unless otherwise
specified in ...).
mandel.h returns an object of class "mandel.kh", which is a data frame consisting
of the required Mandel's statistics and in which each row corresponds to a level of g
and each column to a level of m or (if x was a matrix or data frame) to the
corresponding column in x. In addition to the class, the object has attributes:
"h" or "k"
Character scalar giving the label used for the grouping
factor g; see Details above for the defaults.
Number of observations per group (n if specified
S Ellison [email protected]
Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva (1994).
mandel.k, mandel.kh;
pmandelh, pmandelk for probabilities, quantiles etc.;
plot.mandel.kh, barplot.mandel.kh for plotting methods.
data(RMstudy) #Data frame examples: note no secondary grouping factor h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) plot(h, las=2) #Vector variant RMstk <- stack(RMstudy[,2:9]) names(RMstk) <- c("x", "meas") #names replace 'values' and 'ind' RMstk$Lab <- rep(RMstudy$Lab, 8) h2 <- with(RMstk, mandel.h(x, g=Lab, m=meas, rowname="Laboratory")) #Note use of rowname to override g plot(h2, las=2) #ilab method RM.ilab <- with(RMstk, construct.ilab(org=Lab, x=x, measurand=meas, item=factor(rep("CRM", nrow(RMstk))) ) ) plot(mandel.h(RM.ilab)) #Robust variant hrob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="h", method="robust")) plot(hrob, las=2)data(RMstudy) #Data frame examples: note no secondary grouping factor h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) plot(h, las=2) #Vector variant RMstk <- stack(RMstudy[,2:9]) names(RMstk) <- c("x", "meas") #names replace 'values' and 'ind' RMstk$Lab <- rep(RMstudy$Lab, 8) h2 <- with(RMstk, mandel.h(x, g=Lab, m=meas, rowname="Laboratory")) #Note use of rowname to override g plot(h2, las=2) #ilab method RM.ilab <- with(RMstk, construct.ilab(org=Lab, x=x, measurand=meas, item=factor(rep("CRM", nrow(RMstk))) ) ) plot(mandel.h(RM.ilab)) #Robust variant hrob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="h", method="robust")) plot(hrob, las=2)
mandel.k calculates Mandel's k statistics for replicate observations.
Mandel's k an indicator of precision compared to the pooled standard deviation across
all groups.
mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## Default S3 method: mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'ilab' mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...)mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## Default S3 method: mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'ilab' mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...)
x |
An R object (see Details below), which contains replicate observations or,
if |
g |
A primary grouping factor, usually corresponding to Laboratory in an
inter-laboratory study. If not present, |
m |
A secondary grouping factor, usually corresponding to test item
or measured quantity. |
na.rm |
A logical value indicating whether 'NA' values should be
stripped before the computation proceeds. Passed to functions
such as |
rowname |
A single character label for the primary grouping factor (e.g. "Lab", "Organisation"). |
method |
Character scalar giving the calculation method. |
n |
scalar number of observations per group. Required only if |
... |
Additional parameters passed to other methods. Currently not implemented. |
mandel.k is a convenience wrapper for mandel.kh(..., type="k"). It is generic,
with methods for numeric vectors, arrays, data frames, matrices and objects of
class 'ilab'. All parameters are passed to mandel.kh.
Mandel's is an indicator of relative dispersion for grouped
sets of observations. Given a set of observations where
denotes observation , for measurand or test item and group
(usually laboratory) , , Mandel's is given by:
where is the standard deviation of values over .
If x is a vector, one-dimensional array or single-column matrix, values are aggregated
by g and, if present, by m. If x is a data frame or matrix, each column
is aggregated by g and m silently ignored if present. In all cases, if g
is NULL or missing, each row (or value, if a vector) in x
is taken as a pre-calculated mean (for Mandel's h) or standard deviation (for Mandel's k).
If x is an object of class 'ilab', g defaults to '$org' and
m to $measurand.
The returned object includes a label ('grouped.by') for the primary grouping factor.
For the 'ilab' method, this is "Organisation". For other methods, If rowname is
non-null, rowname is used. If rowname is NULL, the default is deparse(substitute(g));
if g is also NULL or missing, "Row" is used.
If method="robust", Mandel's is calculated by replacing the classical pooled standard
deviation with the robust pooled standard deviation calculated by algorithm S (see algS).
mandel.k returns an object of class "mandel.kh", which is a data frame consisting
of the required Mandel's statistics and in which each row corresponds to a level of g
and each column to a level of m or (if x was a matrix or data frame) to the
corresponding column in x. In addition to the class, the object has attributes:
"h" or "k"
Character scalar giving the label used for the grouping
factor g; see Details above for the defaults.
Number of observations per group (n if specified
S Ellison [email protected]
Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva (1994).
mandel.h, mandel.kh;
pmandelh, pmandelk for probabilities, quantiles etc.;
plot.mandel.kh, barplot.mandel.kh for plotting methods.
data(RMstudy) #Data frame examples: note no secondary grouping factor h <- with(RMstudy, mandel.k(RMstudy[2:9], g=Lab)) plot(h, las=2) #Vector variant RMstk <- stack(RMstudy[,2:9]) names(RMstk) <- c("x", "meas") #names replace 'values' and 'ind' RMstk$Lab <- rep(RMstudy$Lab, 8) h2 <- with(RMstk, mandel.k(x, g=Lab, m=meas, rowname="Laboratory")) #Note use of rowname to override g plot(h2, las=2) #ilab method RM.ilab <- with(RMstk, construct.ilab(org=Lab, x=x, measurand=meas, item=factor(rep("CRM", nrow(RMstk))) ) ) plot(mandel.k(RM.ilab)) #Robust variant krob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="k", method="robust")) plot(krob, las=2)data(RMstudy) #Data frame examples: note no secondary grouping factor h <- with(RMstudy, mandel.k(RMstudy[2:9], g=Lab)) plot(h, las=2) #Vector variant RMstk <- stack(RMstudy[,2:9]) names(RMstk) <- c("x", "meas") #names replace 'values' and 'ind' RMstk$Lab <- rep(RMstudy$Lab, 8) h2 <- with(RMstk, mandel.k(x, g=Lab, m=meas, rowname="Laboratory")) #Note use of rowname to override g plot(h2, las=2) #ilab method RM.ilab <- with(RMstk, construct.ilab(org=Lab, x=x, measurand=meas, item=factor(rep("CRM", nrow(RMstk))) ) ) plot(mandel.k(RM.ilab)) #Robust variant krob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="k", method="robust")) plot(krob, las=2)
mandel.kh calculates Mandel's h and k statistics for replicate observations.
These are traditionally used to provide a rapid graphical summary of results
from an inter-laboratory exercise in which each organisation provides replicate
observations of one or more measurands on one or more test items.
Mandel's h is an indication of relative deviation from the mean value; Mandel's
k is an indicator of precision compared to the pooled standard deviation across
all groups.
mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## Default S3 method: mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'data.frame' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'matrix' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'array' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'ilab' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...)mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## Default S3 method: mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'data.frame' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'matrix' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'array' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'ilab' mandel.kh(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, type = c("h", "k"), method=c("classical", "robust"), n = NA, ...)
x |
An R object (see Details below), which contains replicate observations or,
if |
g |
A primary grouping factor, usually corresponding to Laboratory in an
inter-laboratory study. If not present, |
m |
A secondary grouping factor, usually corresponding to test item
or measured quantity. |
na.rm |
A logical value indicating whether 'NA' values should be
stripped before the computation proceeds. Passed to functions
such as |
rowname |
A single character label for the primary grouping factor (e.g. "Lab", "Organisation"). |
type |
Character denoting the statistic to be calculated; may be "h" or "k". |
method |
Character scalar giving the calculation method. |
n |
scalar number of observations per group. Required only if |
... |
Additional parameters passed to |
mandel.kh can be called directly, but is usually intended to be called via
convenience functions mandel.h or mandel.k.
mandel.kh is a generic, with methods for numeric vectors, arrays, data
frames, matrices and objects of class 'ilab'.
Mandel's statistics are simple indicators of relative deviation or precision for grouped
sets of observations. Given a set of observations where
denotes observation , for measurand or test item and group
(usually laboratory) , , Mandel's and are given by:
where
and
where is the standard deviation of values over .
If x is a vector, one-dimensional array or single-column matrix, values are aggregated
by g and, if present, by m. If x is a data frame or matrix, each column
is aggregated by g and m silently ignored if present. In all cases, if g
is NULL or missing, each row (or value, if a vector) in x
is taken as a pre-calculated mean (for Mandel's ) or standard deviation (for Mandel's ).
If x is an object of class 'ilab', g defaults to '$org' and
m to $measurand.
The returned object includes a label ('grouped.by') for the primary grouping factor.
For the 'ilab' method, this is "Organisation". For other methods, If rowname is
non-null, rowname is used. If rowname is NULL, the default is deparse(substitute(g));
if g is also NULL or missing, "Row" is used.
If method="robust", Mandel's is replaced by a robust z score calculated by replacing
and with the robust estimates of location and scale obtained using Huber's estimate with tuning constant
k set to 1.5 (or as included in ...), and Mandel's is calculated by replacing the
classical pooled standard deviation in the denominator with the robust pooled standard deviation
calculated by algorithm S (see algS).
mandel.kh returns an object of class "mandel.kh", which is a data frame consisting
of the required Mandel's statistics and in which each row corresponds to a level of g
and each column to a level of m or (if x was a matrix or data frame) to the
corresponding column in x. In addition to the class, the object has attributes:
"h" or "k"
Character scalar giving the label used for the grouping
factor g; see Details above for the defaults.
Number of observations per group (n if specified
S Ellison [email protected]
Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva (1994).
mandel.h, mandel.k for convenience functions;
pmandelh, pmandelk for probabilities, quantiles etc.;
plot.mandel.kh, barplot.mandel.kh for plotting methods.
algS and hubers for robust estimates used when method="robust".
data(RMstudy) #Data frame examples: note no secondary grouping factor h <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="h")) plot(h, las=2) k <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="k")) plot(k, las=2) #Vector variant RMstk <- stack(RMstudy[,2:9]) names(RMstk) <- c("x", "meas") #names replace 'values' and 'ind' RMstk$Lab <- rep(RMstudy$Lab, 8) h2 <- with(RMstk, mandel.kh(x, g=Lab, m=meas, rowname="Laboratory")) #Note use of rowname to override g plot(h2, las=2) #ilab method RM.ilab <- with(RMstk, construct.ilab(org=Lab, x=x, measurand=meas, item=factor(rep("CRM", nrow(RMstk))) ) ) plot(mandel.kh(RM.ilab, type="h")) #Robust variants hrob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="h", method="robust")) plot(hrob, las=2) krob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="k", method="robust")) plot(krob, las=2)data(RMstudy) #Data frame examples: note no secondary grouping factor h <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="h")) plot(h, las=2) k <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="k")) plot(k, las=2) #Vector variant RMstk <- stack(RMstudy[,2:9]) names(RMstk) <- c("x", "meas") #names replace 'values' and 'ind' RMstk$Lab <- rep(RMstudy$Lab, 8) h2 <- with(RMstk, mandel.kh(x, g=Lab, m=meas, rowname="Laboratory")) #Note use of rowname to override g plot(h2, las=2) #ilab method RM.ilab <- with(RMstk, construct.ilab(org=Lab, x=x, measurand=meas, item=factor(rep("CRM", nrow(RMstk))) ) ) plot(mandel.kh(RM.ilab, type="h")) #Robust variants hrob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="h", method="robust")) plot(hrob, las=2) krob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="k", method="robust")) plot(krob, las=2)
Functions for printing and plotting interlaboratory study objects objects of class ‘ilab’.
## S3 method for class 'ilab' print(x, ..., digits = NULL, right = FALSE) ## S3 method for class 'ilab' plot(x, ...)## S3 method for class 'ilab' print(x, ..., digits = NULL, right = FALSE) ## S3 method for class 'ilab' plot(x, ...)
x |
An object of class ‘ilab’ |
digits |
Number of digits to display in budget and (if present) distribution parameter lists.
Passed to |
right |
If TRUE, strings in uncertainty budget are right-justified.
This differs from the default in |
... |
Parameters passed to other functions |
The print method uses print.data.frame to display the data after formatting the
distrib and distrib.pars elements.
The plot method passes the object to kplot.
The print and plot methods are called for their side effects.
S. L. R. Ellison [email protected]
None, yet.
ilab-class, subset.ilab kplot.
data(Pb) il.pb<-construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) print(il.pb) plot(il.pb)data(Pb) il.pb<-construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) print(il.pb) plot(il.pb)
Calculate a weighted mean, between-group standard deviation and standard error on the weighted mean using the Maximum likelihood algorithm of Vangel-Rukhin.
mle.1wre(x, s2, n, init.mu = mean(x), init.sigma2 = var(x), labels = c(1:length(x)), max.iter = 200, tol = .Machine$double.eps^0.5, trace = FALSE)mle.1wre(x, s2, n, init.mu = mean(x), init.sigma2 = var(x), labels = c(1:length(x)), max.iter = 200, tol = .Machine$double.eps^0.5, trace = FALSE)
x |
numeric vector of the sample mean values of each group |
s2 |
numeric vector of the sample variances of each group |
n |
integer vector of sample size of each group |
init.mu |
numeric initial value for the mean |
init.sigma2 |
numeric initial value for the between-group component of variance |
labels |
vector of group names. Coerced to character on use. |
max.iter |
numeric maximum number of iterations |
tol |
numeric tolerance; iteration stops when the relative step size drops below 'tol' |
trace |
when TRUE shows the sequence of intermediate results |
The Vangel-Rukhin MLE algorithm finds the between-method variance by iteratively solving the equation relating the weighted mean to the weighting factor applied. The weighting factor is the inverse of the sum of the standard error in 'x' and the between-method variance, scaled by the between-method variance.
For the default method, 's2' is interpreted as a vector of sample variances. 'x' is interpreted as a vector of sample means and the algorithm is applied to the corresponding group means, variances, and sample sizes.
The Vangel-Rukhin MLE algorithm shows an improvement in the number of iterations required to converge over the classical MLE based on the Score equations.
The function mle.1wre implements the MLE for the one way random effects based on the Fisher scoring equations and is provided for comparison purpose only.
mle.1wre returns an object of class "summary.mle.1wre" which contains
the following fields:
mu |
the estimated mean |
var.mu |
the variance associated with the estimated mean |
sigma2 |
the estimated between variance component |
llh |
the log likelihood of the estimates |
tot.iter |
the total number of iterations ran |
cur.rel.abs.error |
the current relative absolute error reached |
sigmai2 |
a vector with the estimates of the within variance components |
H. Gasca-Aragon
Vangel, M. G. and Rukhin, A. L. (1999), Biometrics, Vol 55, No. 1 pp 129-136
Searle, S. R., Cassella, G., and McCulloch, C. E. (1992). Variance Components. New York: Wiley.
##=================================================================== ## the dietary fiber in apples example in the Vangel and Rukhin paper ##=================================================================== m1 <- c(12.46, 13.035, 12.44, 12.87, 13.42, 12.08, 13.18, 14.335, 12.23) s1 <- c(0.028, 0.233, 0.325, 0.071, 0.339, 0.325, 0.099, 0.064, 0.212) n1 <- c(2, 2, 2, 2, 2, 2, 2, 2, 2) mle.1wre(m1, s1^2, n1, tol=1e-6) # output: # 12.90585 # 0.2234490 # 0.4262122 # 12.46790 13.34380 # 6##=================================================================== ## the dietary fiber in apples example in the Vangel and Rukhin paper ##=================================================================== m1 <- c(12.46, 13.035, 12.44, 12.87, 13.42, 12.08, 13.18, 14.335, 12.23) s1 <- c(0.028, 0.233, 0.325, 0.071, 0.339, 0.325, 0.099, 0.064, 0.212) n1 <- c(2, 2, 2, 2, 2, 2, 2, 2, 2) mle.1wre(m1, s1^2, n1, tol=1e-6) # output: # 12.90585 # 0.2234490 # 0.4262122 # 12.46790 13.34380 # 6
A data frame containing reported results for lead (in mg/kg) from CCQM Key Comparison CCQM-K30.
PbPb
A data frame containing 11 reported results with uncertainty data:
Factor giving abbreviated laboratory identifier
The reported value for lead (mg/kg)
Standard uncertainty (mg/kg). The values in Pb were calculated from
the reported expanded uncertainty U and coverage factor k using u=U/k.
Coverage factor. Conventionally, the coverage factor is set to a suitable
quantile of Student's t based on the Welch-Satterthwaite effective degrees of freedom
or simply set to 2 for approximately 95% confidence. In this data set, labs all quoted
k for approximately 95% confidence.
Expanded uncertainty as reported by labs.
Factor indicating general measurement methodology:
Isotope dilution mass spectrometry
Inductively coupled plasma spectrometry
Graphite furnace atomic absorbtion spectrometry
logical; Whether the reported result was included in the calculation of the Key Comparison Reference Value for the study.
The study involved circulation of a homogeneous set of samples of wine for analysis for lead (Pb) content by a number of National Measurement Institutes.
The Key Comparison Reference Value, or assigned value for the lead content, was set at 2.99 mg/kg with expanded uncertainty 0.06 mg/kg.
Hearn, R., Santamaria-Fernandez, R. and Sargent, M. (2008) Final report on key comparison CCQM-K30: Determination of lead in wine. Metrologia 45, 08001, 2008
See source.
Plots a number of data ellipses specified by
## S3 method for class 'd.ellipse' plot(x, col.ellipse = 1, lty.ellipse = 1, lwd.ellipse = 1, fill = NA, density = NULL, angle = 45, add = FALSE, npoints = 100, xlim = NA, ylim = NA, prinax = FALSE, col.prinax = 1, lty.prinax = 1, lwd.prinax = 1, xlab=NULL, ylab=NULL, ...)## S3 method for class 'd.ellipse' plot(x, col.ellipse = 1, lty.ellipse = 1, lwd.ellipse = 1, fill = NA, density = NULL, angle = 45, add = FALSE, npoints = 100, xlim = NA, ylim = NA, prinax = FALSE, col.prinax = 1, lty.prinax = 1, lwd.prinax = 1, xlab=NULL, ylab=NULL, ...)
x |
An object of class |
col.ellipse, lty.ellipse, lwd.ellipse
|
Colour, line type and line width for the ellipse(s). Can be vectors, allowing different
colour, line type etc. Recycled as necessary to length |
fill, density, angle
|
Fill colour, line density and line angle for each ellipse in |
add |
If |
npoints |
Controls the number of points used to form each ellipse. See |
xlim, ylim
|
Plot limits. Ignored if |
prinax |
If |
col.prinax, lty.prinax, lwd.prinax
|
Colour, line type and line width for principal axes. |
xlab, ylab
|
Axis labels passed to |
... |
Additional arguments, passed to |
A series of ellipses specified in x is plotted.
The function is primarily used for adding ellipses to a Youden plot.
The function is called for its side effect, which is the drawing of ellipses.
S L R Ellison
data(chromium) cov.Cr <- cov.dellipse(chromium) dellipse.Cr <- data.ellipse(cov.Cr, plot=FALSE) plot(dellipse.Cr)data(chromium) cov.Cr <- cov.dellipse(chromium) dellipse.Cr <- data.ellipse(cov.Cr, plot=FALSE) plot(dellipse.Cr)
plot.mandel.kh produces classic plots of Mandel's statistics, suitably
grouped and with appropriate indicator lines for unusual values.
## S3 method for class 'mandel.kh' plot(x, probs = c(0.95, 0.99), main, xlab = attr(x, "grouped.by"), ylab = attr(x, "mandel.type"), ylim = NULL, las = 1, axes = TRUE, cex.axis = 1, frame.plot = axes, lwd = 1, lty = 1, col = par("col"), col.ind = 1, lty.ind = c(2, 1), lwd.ind = 1, separators = TRUE, col.sep = "lightgrey", lwd.sep = 1, lty.sep = 1, zero.line = TRUE, lwd.zero = 1, col.zero = 1, lty.zero = 1, p.adjust = "none", ...)## S3 method for class 'mandel.kh' plot(x, probs = c(0.95, 0.99), main, xlab = attr(x, "grouped.by"), ylab = attr(x, "mandel.type"), ylim = NULL, las = 1, axes = TRUE, cex.axis = 1, frame.plot = axes, lwd = 1, lty = 1, col = par("col"), col.ind = 1, lty.ind = c(2, 1), lwd.ind = 1, separators = TRUE, col.sep = "lightgrey", lwd.sep = 1, lty.sep = 1, zero.line = TRUE, lwd.zero = 1, col.zero = 1, lty.zero = 1, p.adjust = "none", ...)
x |
An object of class |
probs |
Indicator lines are drawn for these probabilities. Note that
|
main |
a main title for the plot. If missing, the default is
|
xlab |
a label for the x axis; defaults to the |
ylab |
a label for the x axis; defaults to the |
ylim |
the y limits of the plot. For Mandel's k, the default lower limit is zero. |
las |
the style of the axis labels; see |
axes |
a logical value indicating whether axes should be drawn on the plot. |
cex.axis |
The magnification to be used for axis annotation relative to the current setting of 'cex'. |
frame.plot |
Logical; If |
lwd, lty, col
|
Graphical parameters used for the plotted vertical lines corresponding to each value of Mandel's statistics (the plot is of type "h"). All are recycled across the prinmary grouping factor, allowing different measurands/test items to be identified more clearly. |
col.ind, lty.ind, lwd.ind
|
Graphical parameters used for the indicator lines, recyckled to |
separators |
Logical; if |
col.sep, lwd.sep, lty.sep
|
Graphical parameters used for the separator lines. |
zero.line |
logical; if |
lwd.zero, col.zero, lty.zero
|
Graphical parameters used for the zero line. |
p.adjust |
Correction method for probabilities. If not |
... |
Other (usually graphical) parameters passed to |
Mandel's statistics are traditionally plotted for inter-laboratory study data,
grouped by laboratory, to give a rapid graphical view of laboratory bias and
relative precision. The traditional plot is a plot of type "h", that is,
simple vertical lines from the x-axis.
For classical Mandel statistics, indicator lines are drawn based on qmandelh
or qmandelk as appropriate. For robust variants, indicator lines use
qnorm for the statistic and qf(probs, n, Inf) for
the statistic. Note that this corresponds to taking the robust estimates of
location and scale as true values, so will be somewhat anticonservative.
plot.mandel.kh uses gplot for the main plot.
plot.mandel.kh returns a numeric vector of mid-points of the groups along the x-axis.
S Ellison [email protected]
Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva (1994).
mandel.h, mandel.k, mandel.kh,
pmandelh, pmandelk for probabilities, quantiles etc.
See barplot.mandel.kh for an alternative plotting method.
gplot for the underlying plotting function.
data(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) plot(h, las=2) # Lab 4 shows consistent low bias; # Lab 23 several extreme values. #Use colours to identify particular measurands: plot(h, las=2, col=1:8) legend("bottomleft", legend=names(h), col=1:8, lty=1, cex=0.7, bg="white") #Example of Mandel's k: k <- with(RMstudy, mandel.k(RMstudy[2:9], g=Lab)) plot(k, las=2) # Lab 8 looks unusually variable; # Lab 14 unusually precisedata(RMstudy) h <- with(RMstudy, mandel.h(RMstudy[2:9], g=Lab)) plot(h, las=2) # Lab 4 shows consistent low bias; # Lab 23 several extreme values. #Use colours to identify particular measurands: plot(h, las=2, col=1:8) legend("bottomleft", legend=names(h), col=1:8, lty=1, cex=0.7, bg="white") #Example of Mandel's k: k <- with(RMstudy, mandel.k(RMstudy[2:9], g=Lab)) plot(k, las=2) # Lab 8 looks unusually variable; # Lab 14 unusually precise
Plots for uncertainty budgets produced by uncert calls.
## S3 method for class 'uncert' plot(x, which = c(1,2,4,5), main = paste(deparse(substitute(x))), ask = prod(par("mfcol")) < length(which) && dev.interactive(), caption = list("Variance and covariance contributions", expression(sqrt(group("|", "Variance and covariance contributions", "|"))), expression("Contribution " * u[i](y) == c[i] * u[i]), "Combined contribution", "Correlation (x,y)", "Covariances (x,y)"), cex.caption = 1, ...)## S3 method for class 'uncert' plot(x, which = c(1,2,4,5), main = paste(deparse(substitute(x))), ask = prod(par("mfcol")) < length(which) && dev.interactive(), caption = list("Variance and covariance contributions", expression(sqrt(group("|", "Variance and covariance contributions", "|"))), expression("Contribution " * u[i](y) == c[i] * u[i]), "Combined contribution", "Correlation (x,y)", "Covariances (x,y)"), cex.caption = 1, ...)
x |
An object of class |
which |
Integer in 1:6; the particular variant(s) of plot required. A vector is permitted, in which case plots are produced in ascending order. |
main |
Main title for the plot |
ask |
logical; if 'TRUE', the user is _ask_ed before each plot, see 'par("ask=")' |
caption |
A list of captions for all the different plots. |
cex.caption |
Text size for captions. Note that if the number of figures per page is over 2, captions are further scaled by 0.8 |
... |
Further parameters passed to |
For uncert objects created with methods other than MC, the plot types are:
which=1A barplot of all non-zero contributions to the combined
uncertainty. These are derived from the covariance matrix and the coefficients .
For terms on the diagonal of the covariance matrix, these are ; for
off-diagonal terms (the correlation terms), .
The threshold for deciding an off-diagonal term is nonzero is that its magnitude
is greater than 2*.Machine$double.eps. Note that off-diagonal contributions
may be negative.
which=2As for which=1 except that the square root of the
absolute value is plotted. For the 'diagonal' terms, these are just eqnu_i(y)
in the nomenclature used by the GUM.
which=3A barplot of , without the correlation terms.
which=4A barplot of the sum of all (co)variance contributions
associated with each , that is,
.
which=5A barplot of the theoretical correlations
.
which=6A barplot of the theoretical covariances .
Values of which outside this range are silently ignored.
For the X-Y correlation and covariance plots, the covariances are calculated from the
covariance matrix (supplied to uncert() as cov
or calculated as outer(u,u,"*")*cor) and sensitivity coefficients
as .
In fact the calculation used is simpler: cov.xy <- V %*% ci. The correlations
are calculated in turn from these using .
Perhaps the most informative plots are for which=1, which=2,
which=4 and which=5. The first of these includes all nonzero signed contributions,
making the negative contributions visible; the second (which=2) makes direct
comparison of magnitudes easier. The combined contribution plot is the effect on
the total variance of removing all terms associated with a particular variable; it
shows how much would reduce if the uncertainty for were
reduced to zero. Note that in some cases with negative correlation the combined uncertainty can increase,
on dropping a variable, shown by a negative reduction in the plot. (which=5) is among the most
direct indications of the relative importance of individual parameters.
Objects created with the MC method are passed to plot.uncertMC.
Invisibly returns the default return value for the last plot produced.
S. L. R. Ellison, [email protected]
None.
uncert, barplot, plot.uncertMC.
#An example with negative correlation x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<- -0.5 u.form.c<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM", cor=u.cor) par(mfrow=c(3,2)) plot(u.form.c, which=1:6, las=1, horiz=TRUE) #Note use of barplot parameters#An example with negative correlation x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<- -0.5 u.form.c<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM", cor=u.cor) par(mfrow=c(3,2)) plot(u.form.c, which=1:6, las=1, horiz=TRUE) #Note use of barplot parameters
Plots for uncertainty evaluations produced by uncertMC or calls
to uncert with method=MC.
## S3 method for class 'uncertMC' plot(x, which = 1:2, main=paste("Monte Carlo evaluation -",deparse(substitute(x))), ask = prod(par("mfcol")) < length(which) && dev.interactive(), caption = list("Histogram", "Q-Q plot", "Density", "Correlation x-y", "Covariance x-y"), xlab = paste(deparse(substitute(x)), "$y", sep = ""), ..., cex.caption = 1, cex.main = 1.25, lwd.y = 2, col.y = 2, lty.y, col.qqline = NULL, lty.qqline = NULL, lwd.qqline = NULL)## S3 method for class 'uncertMC' plot(x, which = 1:2, main=paste("Monte Carlo evaluation -",deparse(substitute(x))), ask = prod(par("mfcol")) < length(which) && dev.interactive(), caption = list("Histogram", "Q-Q plot", "Density", "Correlation x-y", "Covariance x-y"), xlab = paste(deparse(substitute(x)), "$y", sep = ""), ..., cex.caption = 1, cex.main = 1.25, lwd.y = 2, col.y = 2, lty.y, col.qqline = NULL, lty.qqline = NULL, lwd.qqline = NULL)
x |
An object of class |
which |
Integer in 1:5; the particular variant(s) of plot required.
A vector is permitted, in which case plots are produced in ascending order
of |
main |
Main title for the plot |
ask |
logical; if 'TRUE', the user is _ask_ed before each plot, see 'par("ask=")' |
caption |
A list of captions for all the different plots. |
xlab |
x-axis label, currently passed only to the histogram plot. |
... |
Additional parameters passed to other functions. See details for which parameters are passed. |
cex.caption |
Expansion factor for individual plot captions; as |
cex.main |
Expansion factor for main title; as |
lwd.y, col.y, lty.y
|
Line width and colour for the location line in the histogram and density plots. Setting lwd.y=0 suppresses the location line. |
col.qqline, lty.qqline, lwd.qqline
|
Graphical parameters for the Q-Q line in the Q-Q plot. |
For uncert objects created with methods other than MC, the plot types are:
which=1A histogram of the MC replicates in x$MC$y,
with optional line for x$MC$y. The histogram is produced using
hist.default
which=2A Q-Q plot of the MC replicates in x$MC$y,
with Q-Q line. The plot uses qqnorm.default. If datax is not
present (in sQuote...), it is set to TRUE.
which=3A density plot of the MC replicates in x$MC$y.
The plot calls density.default to calculate the density and
plot.density to produce the plot.
which=4A bar plot of eqncor(x_i,y) if x$y
is present. Any correlation method supported by stats::cor may
be included in ‘...’ (e.g as method="pearson".
which=5A bar plot of eqncov(x_i,y) if x$y
is present. Any correlation method supported by stats::cov may
be included in ‘...’ (e.g as method="pearson".
Values outside 1:5 are silently ignored.
Parameters in ‘...’ are passed to the various plot methods or calculations called. Only those parameters relevant to a given plot are passed to each calculation or plotting function, so ‘...’ can include any parameter accepted by any of the functions called.
For the x-y correlation and x-y covariance plot, values in x$cor.xy are
used if available. If not, stats::cor or stats::cov is called on values
in x$MC$y and x$MC$x if the latter is available
(i.e. uncertMC was called with keep.x=TRUE). If neither
x$cor.xy nor x$MC$x is present, or if method is
unknown, the plot is skipped with a warning.
plot.uncertMC invisibly returns NULL.
S. L. R. Ellison, [email protected]
None.
uncertMC-class, hist,
qqnorm, qqline,
density, plot.density
expr <- expression(a/(b-c)) x <- list(a=1, b=3, c=2) u <- lapply(x, function(x) x/20) set.seed(403) u.invexpr<-uncertMC(expr, x, u, distrib=rep("norm", 3), B=999, keep.x=TRUE ) par(mfrow=c(2,2)) plot(u.invexpr, which=1:4, pch=20, method="k") # method="k" gives Kendall correlationexpr <- expression(a/(b-c)) x <- list(a=1, b=3, c=2) u <- lapply(x, function(x) x/20) set.seed(403) u.invexpr<-uncertMC(expr, x, u, distrib=rep("norm", 3), B=999, keep.x=TRUE ) par(mfrow=c(2,2)) plot(u.invexpr, which=1:4, pch=20, method="k") # method="k" gives Kendall correlation
Potassium data for two different materials included in an interlaboratory study intended to provide data for certification of a reference material.
data("potassium")data("potassium")
A data frame with 25 observations on the following 2 variables.
QCPotassium concentrations (mg/kg) reported on a material used as a quality control material
RMPotassium concentrations (mg/kg) reported on a candidate reference material material used as a quality control material
Potassium data for two different materials included in an interlaboratory study intended to provide data for certification of a crab tissue reference material. The study included a previously certified reference material (near end of stock) to serve as a quality control (QC) check. Laboratories were asked to report five replicate measurements on the candidate reference material and three for the QC material. Each row in the data set corresponds to the mean of replicate results reported by each laboratory.
Inspection of the data suggests that one laboratory interchanged or mislabelled
the test materials. The anomalous results appear as an outlier for both QC and RM, as well as
being visible as an off-diagonal outlier in a Youden plot - see youden.plot).
Private communication - Pending publication
data(potassium) yplot(potassium)data(potassium) yplot(potassium)
Functions to combine ilab objects.
rbind(..., deparse.level = 1) ## Default S3 method: rbind(..., deparse.level = 1) ## S3 method for class 'ilab' rbind(..., deparse.level = 1) ## S3 method for class 'ilab' c(..., recursive=FALSE) cbind(..., deparse.level = 1) ## Default S3 method: cbind(..., deparse.level = 1) ## S3 method for class 'ilab' cbind(..., deparse.level = 1)rbind(..., deparse.level = 1) ## Default S3 method: rbind(..., deparse.level = 1) ## S3 method for class 'ilab' rbind(..., deparse.level = 1) ## S3 method for class 'ilab' c(..., recursive=FALSE) cbind(..., deparse.level = 1) ## Default S3 method: cbind(..., deparse.level = 1) ## S3 method for class 'ilab' cbind(..., deparse.level = 1)
... |
For |
deparse.level |
integer controlling the construction of labels. Passed
to |
recursive |
logical, controlling recursion in lists. Included solely for
consistency with |
ilab |
object of class ‘ilab’. |
The generic and default cbind and rbind functions defined by metRology use the first
object in '...' to decide which method to apply. This differs from the behaviour
of these functions in the base package, which dispatch based on inspection of
all objects in '...' (see rbind in the base package for details).
Control is, however, passed to the base package functions if an ilab object is not first
in the list.
The rbind method for class 'ilab' combines objects by applying
rbind to the $data elements in turn and then concatenating the
$distrib and $distrib.pars elements using the default c
method.
Combination of the $data elements follows the rules of rbind.data.frame;
in particular, names must match, but need not be in the same order and the return value
column classes will be coerced to match the first if necessary.
c.ilab simply passes the objects to rbind.ilab, using the default value
for deparse.level. recursive is ignored. An error is returned if any
objects in '...' are not of class ‘ilab’.
The cbind method for ‘ilab’ objects combines objects of class ‘ilab’
with atomic objects or data frames by applying base::cbind to $data in the
(single) supplied ilab object and the items listed in '...'. base::cbind
will extend scalars, vectors or columns in data frames to length nrow(ilab$data) if
their length is an integer fraction of nrow(ilab$data). Unlike base::cbind,
cbind.ilab does not permit vectors or data frames longer than nrow(ilab$data)
and will return an error in such cases. cbind.ilab will also return an error if any objects in
'...' are not one of atomic, data frame or class ‘ilab’, if more than one
‘ilab’ object is supplied or if none are.
An object of class ‘ilab’. The title for the returned object is the title for the first ilab object in the list.
For the cbind method, the returned object will have additional columns in the
$data element, and the title will be unchanged.
Because of the unusual method dispatch behaviour of base::cbind and
base::rbind, which use the data frame method if any objects in
'...' are data frames, metRology masks the base package functions
in order to guarantee correct dispatch when data frames are included in '...'.
No adverse effects are currently known, but please report any to the package maintainer.
Calling base::cbind directly will provide a work-around if necessary.
S. L. R. Ellison [email protected]
None, yet.
base package functions cbind, c.
data(Pb) il1<-construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) rbind(il1, il1)data(Pb) il1<-construct.ilab(org=Pb$lab, x=Pb$value, measurand="Pb", item="none", u=Pb$u, k=Pb$k, U=Pb$U, title=c("CCQM K30", "Lead in wine"), method=Pb$method) rbind(il1, il1)
Calculates REML estimate of location, with standard error, assuming a random-effects model
reml.loc(x, ..., na.rm = FALSE) ## Default S3 method: reml.loc(x, s, n = NULL, groups = NULL, na.rm = FALSE, tol=.Machine$double.eps^0.5, REML=TRUE, ...)reml.loc(x, ..., na.rm = FALSE) ## Default S3 method: reml.loc(x, s, n = NULL, groups = NULL, na.rm = FALSE, tol=.Machine$double.eps^0.5, REML=TRUE, ...)
x |
numeric vector of mean values for groups, or (if |
s |
numeric vector of length |
n |
integer giving the number of observations in each group. May be a vector
of length |
groups |
factor, or vetor which can be coerced to factor, of groups. If
present, |
na.rm |
logical: if |
tol |
numeric tolerance for convergence, used by |
REML |
logical: if |
... |
Further parameters passed to |
reml.loc finds an excess variance and location that maximise the
restricted maximum likelihood criterion.
The estimator assumes a model of the form
in which is drawn from and
is drawn from .
By default the function maximises the data-dependent part of the negative log restricted likelihood:
where and is the number of mean values.
If REML=FALSE, the final term is omitted to give the maximum likelihood criterion.
This implementation permits input in the form of:
means x and standard errors s, in which case neither n nor
groups are supplied;
means x, standard deviations s and group size(s) n,
standard errors then being calculated as s/sqrt(n)
individual observations x with a groupinf factor groups, in
which case standard errors are calculated from the groups using tapply.
A loc.est object; see loc.est for details. In the returned object, individual
values xi are always input means (calculated from groups and n as
necessary); method.details is returned as a list containing:
The estimated location.
The standard error in the location.
The excess variance (as a standard deviation).
Logical, giving the value of REML used.
S L R Ellison [email protected]
None, but see documentation for the metafor package for a more general implementation of REML.
#PCB measurements in a sediment from Key Comparison CCQM-K25 #s are reported standard uncertainties pcb105 <- data.frame(x=c(10.21, 10.9, 10.94, 10.58, 10.81, 9.62, 10.8), s=c(0.381, 0.250, 0.130, 0.410, 0.445, 0.196, 0.093)) with( pcb105, reml.loc(x, s) )#PCB measurements in a sediment from Key Comparison CCQM-K25 #s are reported standard uncertainties pcb105 <- data.frame(x=c(10.21, 10.9, 10.94, 10.58, 10.81, 9.62, 10.8), s=c(0.381, 0.250, 0.130, 0.410, 0.445, 0.196, 0.093)) with( pcb105, reml.loc(x, s) )
A data frame containing reported replicate results, in mg/L, for elements reported in an inter-laboratory certification study for a candiate drinking reference material.
data(RMstudy)data(RMstudy)
A data frame with 145 observations on the following 9 variables.
Laba factor with levels Lab1 - Lab29
Arsenica numeric vector
Cadmiuma numeric vector
Chromiuma numeric vector
Coppera numeric vector
Leada numeric vector
Manganesea numeric vector
Nickela numeric vector
Zinca numeric vector
The data set includes results for eight of a total of 23 elements studied in an
inter-laboratory study of a candidate reference material. All observations are
reported in ug/l.
Laboratories were asked to report 5 replicate observations for each element. Replicate
observations appear on separate rows. Most but not all laboratories reported five
replicates, and some laboratories reported no results for some elements. The eight
elements included in the data set are those for which no more than three laboratories
failed to report any results. Missing observations are coded NA.
Laboratories were coded anonymously in order of receipt of results.
LGC limited, Teddington, UK (Private communication).
Student's t distribution for 'df' degrees of freedom, shifted by 'mean' and scaled by 'sd'.
dt.scaled(x, df, mean = 0, sd = 1, ncp, log = FALSE) pt.scaled(q, df, mean = 0, sd = 1, ncp, lower.tail = TRUE, log.p = FALSE) qt.scaled(p, df, mean = 0, sd = 1, ncp, lower.tail = TRUE, log.p = FALSE) rt.scaled(n, df, mean = 0, sd = 1, ncp)dt.scaled(x, df, mean = 0, sd = 1, ncp, log = FALSE) pt.scaled(q, df, mean = 0, sd = 1, ncp, lower.tail = TRUE, log.p = FALSE) qt.scaled(p, df, mean = 0, sd = 1, ncp, lower.tail = TRUE, log.p = FALSE) rt.scaled(n, df, mean = 0, sd = 1, ncp)
x, q
|
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
df |
degrees of freedom (> 0, maybe non-integer). |
mean |
mean value for the shifted, scaled distribution. |
sd |
Scale factor for the shifted, scaled distribution. |
ncp |
non-centrality parameter delta; currently except for |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x]; otherwise, P[X > x]. |
log, log.p
|
logical; if TRUE, probabilities p are given as log(p). |
These are wrappers for the corresponding t distribution functions in package stats.
The scaled, shifted t distribution has mean mean and variance sd^2 * df/(df-2)
The scaled, shifted t distribution is used for Monte Carlo evaluation when a value x has
been assigned a standard uncertainty u associated with with df degrees of freedom;
the corresponding distribution function for that is then t.scaled with
mean=x, sd=u and df=df.
dt.scaled gives the density, pt.scaled gives the distribution function,
qt.scaled gives the quantile function, and rt.scaled generates random deviates.
Invalid arguments will result in return value NaN, with a
warning.
S. L. R. Ellison [email protected]
u<-rt.scaled(20, df=5, mean=11, sd=0.7) qt.scaled(c(0.025,0.975), Inf, mean=10, sd=1) #10 +- 1.96*sd require(graphics) hist(rt.scaled(10000, df=4, mean=11, sd=0.7), breaks=50, probability=TRUE) x<-seq(0,25, 0.05) lines(x,dnorm(x,mean=11, sd=0.7), col=2)u<-rt.scaled(20, df=5, mean=11, sd=0.7) qt.scaled(c(0.025,0.975), Inf, mean=10, sd=1) #10 +- 1.96*sd require(graphics) hist(rt.scaled(10000, df=4, mean=11, sd=0.7), breaks=50, probability=TRUE) x<-seq(0,25, 0.05) lines(x,dnorm(x,mean=11, sd=0.7), col=2)
Density, distribution function, quantile function and random generation for the triangular distribution with range 'min' to 'max' and mode equal to 'mode'.
dtri(x, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, log = FALSE) ptri(q, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, lower.tail = TRUE, log.p = FALSE) qtri(p, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, lower.tail = TRUE, log.p = FALSE) rtri(n, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2)dtri(x, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, log = FALSE) ptri(q, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, lower.tail = TRUE, log.p = FALSE) qtri(p, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, lower.tail = TRUE, log.p = FALSE) rtri(n, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2)
x, q
|
Vector of quantiles. |
p |
Vector of quantiles. |
n |
Number of observations. If 'length(n) > 1', the length is taken to be the number required. |
min |
Vector of lower limits of distribution. |
max |
Vector of upper limits of distribution. |
mode |
Vector of modes |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x]; otherwise, P[X > x]. |
log, log.p
|
logical; if TRUE, probabilities p are given as log(p). |
The triangular distribution has density
and 0 elsewhere.
The mean is
and the variance is
The default values of min, max and mode give a distribution with
mean 0 and unit variance.
If min>max, min amd max will be silently interchanged. If
mode is not within [min, max], the functions return NA, with a warning.
rtri calls runif(n, 0, 1) to generate probabilities which are passed to
qtri.
A vector of densities, probabilities, quantiles or random deviates.
dtri gives the density, ptri gives the distribution function,
qtri gives the quantile function, and rtri generates random deviates.
S. L. R. Ellison [email protected]
require(graphics) x<-seq(-3,3,0.02) par(mfrow=c(2,1)) plot(x, dtri(x), type="l", main="Density") plot(x, ptri(x), type="l", main="p(X<x)") u <- rtri(5000) var(rtri(10000,-1,1)) # ~ = 1/6 = 0.167require(graphics) x<-seq(-3,3,0.02) par(mfrow=c(2,1)) plot(x, dtri(x), type="l", main="Density") plot(x, ptri(x), type="l", main="p(X<x)") u <- rtri(5000) var(rtri(10000,-1,1)) # ~ = 1/6 = 0.167
Functions for estimating measurement uncertainty from standard uncertainties and either sensitivity coefficients or (for some methods) expressions or functions. Correlation is supported via either a correlation or covariance matrix.
uncert(obj, ...) ## Default S3 method: uncert(obj, c, method = c("GUM", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, x=NULL, keep.x = TRUE, u=obj, ...) ## S3 method for class 'expression' uncert(obj, x, u, method=c("GUM", "NUM", "kragten", "k2", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, delta=0.01, keep.x = TRUE, ...) ## S3 method for class 'function' uncert(obj, x, u, method=c("NUM", "kragten", "k2", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, delta=0.01, keep.x = TRUE, ...) ## S3 method for class 'formula' uncert(obj, x, u, method=c("GUM", "NUM", "kragten", "k2", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, delta=0.01, keep.x = TRUE, ...)uncert(obj, ...) ## Default S3 method: uncert(obj, c, method = c("GUM", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, x=NULL, keep.x = TRUE, u=obj, ...) ## S3 method for class 'expression' uncert(obj, x, u, method=c("GUM", "NUM", "kragten", "k2", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, delta=0.01, keep.x = TRUE, ...) ## S3 method for class 'function' uncert(obj, x, u, method=c("NUM", "kragten", "k2", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, delta=0.01, keep.x = TRUE, ...) ## S3 method for class 'formula' uncert(obj, x, u, method=c("GUM", "NUM", "kragten", "k2", "MC"), cor, cov, distrib=NULL, distrib.pars=NULL, B=200, delta=0.01, keep.x = TRUE, ...)
obj |
An R object used for method dispatch; see below. Methods currently exist for numeric vector, expression, function, or formula |
u |
For the default method, a numeric vector of standard uncertainties. For
the formula or expression methods, a named list of standard uncertainties.Note that
for the default method, |
c |
A numeric vector of senstivity coefficients. |
x |
For the expression or formula methods, an R object which can be used
as an environment by |
method |
Method of uncertainty evaluation. The current list of methods is:
|
cor, cov
|
A (square, symmetric) correlation or covariance matrix, respectively.
If neither is specified, |
distrib |
For |
distrib.pars |
For |
B |
Number of Monte Carlo replicates. |
delta |
Step size for numerical differentiation. |
keep.x |
For |
... |
Additional parameters to be passed to a function (for the function method) or used in an expression (for expression or formula method). |
The default “GUM” method applies first-order error propagation principles to estimate
a combined standard uncertainty from a set of sensitivity coefficients and either a
set of standard uncertainties and a correlation matrix (which defaults to an identity matrix)
or a covariance matrix. Both options use the same calculation, which is simply
(t(c) %*% cov) %*% c ; standard uncertainties are first combined with
the correlation matrix provided to form the covariance matrix. Since the correlation
matrix defaults to the identity matrix, the default is combination without
correlation.
The default method takes obj as a vector of uncertainty contributions unless u
is specified, in which case u is used. It is not necessary to specify both.
The expression method requires obj to be a differentiable R expression which can
be evaluated in the environment x to provide a numeric value.
For the function method, obj must be an R function which takes parameters from x and
returns a numeric value.
For the formula method, obj must be a formula with no left-hand side (e.g. ~a*x+b*x^2)
which can be evaluated in the environment x to provide a numeric value.
The formula and expression methods first calculate derivatives for the expression or formula,
evaluate them using the supplied values of x and then pass the resulting sensitivity
coefficients, with supplied u, cor or cov to uncert.default.
The derivatives for the “GUM” method (formula and expression methods only) are algorithmic
derivatives (that is, algebraic or analytical derivatives) obtained using deriv
applied to expr and formula.
Numerical derivatives are computed in different ways depending on the method specified:
- For method="NUM", the derivatives are calculated as
.
- For method="kragten", derivatives are calculated as
.
- For method="k2", derivatives are calculated as
.
"NUM" is likely to give a close approximation to analytical differentiation provided that
delta is appreciably less than 1 but not so small as to give step sizes near machine
precision. "k2" is equivalent to "NUM" with delta=1.0. Both will give zero coefficients
at stationary points (e.g minima), leading to under-estimation of uncertainty if
the curvature is large. "kragten" uses a deliberately one-sided (and large) step to
avoid this problem; as a result, "kragten" is a poorer (sometimes much poorer) estimate of
the analytical differential but likely a better approximation to the truth.
Since these methods rely on u, if u is unspecified and cov is
provided, u is extracted from cov (using sqrt(diag(cov))). It is
assumed that the row and column order in cov is identical to the order of named
parameters in x.
Derivatives (and uncertainty contributions) are computed for all parameters in
x. Additional parameters used in FUN, expr or formula
may be included in ...; these will be treated as constants in the
uncertainty calculation.
If distrib is missing, or if it is a list with some members missing, the distribution
is assumed Normal and distrib$name is set to "norm". Similarly, if distrib.pars
or a member of it is missing, the default parameters for x$name are
list(mean=x$name, sd=u$name). If the list is not named, names(x) are used
(so the list must be in order of names(x)).
If method="MC", uncert calls uncertMC. Distributions and
distribution parameters are required and B must be present and >1. See uncertMC
for details of distribution specification.
For other evaluation methods, the distributions are silently ignored.
An object of class ‘uncert’ or, for method="MC" of class ‘uncertMC’.
See uncert-class and uncertMC-class for details.
S. L. R. Ellison [email protected]
JCGM 100 (2008) Evaluation of measurement data - Guide to the expression of uncertainty in measurement. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf. (JCGM 100:2008 is a public domain copy of ISO/IEC Guide to the expression of uncertainty in measurement (1995) ).
Kragten, J. (1994) Calculating standard deviations and confidence intervals with a universally applicable spreadsheet technique, Analyst, 119, 2161-2166.
Ellison, S. L. R. (2005) Including correlation effects in an improved spreadsheet calculation of combined standard uncertainties, Accred. Qual. Assur. 10, 338-343.
For method="MC" see uncertMC and uncertMC-class.
expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncert(expr, x, u, method="NUM") u.expr #Compare with default: uncert(u=c(0.1, 0.3, 0.2, 1.1), c=c(1.0, 2.0, 3.0, 0.5)) #... or with function method f <- function(a,b,c,d) a+b*2+c*3+d/2 u.fun<-uncert(f, x, u, method="NUM") u.fun #.. or with the formula method u.form<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM") u.form #An example with correlation u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<-0.5 u.formc<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM", cor=u.cor) u.formc #A Monte Carlo example #See uncertMC for a less linear example u.formc.MC<-uncert(~a+b*2+c*3+d/2, x, u, method="MC", cor=u.cor, B=200) u.formc.MCexpr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncert(expr, x, u, method="NUM") u.expr #Compare with default: uncert(u=c(0.1, 0.3, 0.2, 1.1), c=c(1.0, 2.0, 3.0, 0.5)) #... or with function method f <- function(a,b,c,d) a+b*2+c*3+d/2 u.fun<-uncert(f, x, u, method="NUM") u.fun #.. or with the formula method u.form<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM") u.form #An example with correlation u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<-0.5 u.formc<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM", cor=u.cor) u.formc #A Monte Carlo example #See uncertMC for a less linear example u.formc.MC<-uncert(~a+b*2+c*3+d/2, x, u, method="MC", cor=u.cor, B=200) u.formc.MC
Object returned by uncert calls.
## S3 method for class 'uncert' print(x, digits=NULL, right=FALSE, ..., simplify=TRUE) ## S3 method for class 'uncert' summary(object, ..., simplify=TRUE)## S3 method for class 'uncert' print(x, digits=NULL, right=FALSE, ..., simplify=TRUE) ## S3 method for class 'uncert' summary(object, ..., simplify=TRUE)
x, object
|
An object of class uncert |
digits |
Number of digits to display in budget and (if present) distribution parameter lists.
Passed to |
right |
If TRUE, strings in uncertainty budget are right-justified.
This differs from the default in |
... |
Other parameters passed to |
simplify |
If |
summary.uncert simply calls print.uncert.
An object of class "uncert" contains:
The matched call
The calculated value (for function, expression or formula methods) or NA
The combined standard uncertainty
The uncertainty evaluation method used.
A data frame consisting of:
x (if supplied; otherwise a vector of NA's).
The standard uncertainties in input quantities (originally provided as u)
The degrees of freedom asscociated with u
Sensitivity coefficients either provided as c or (for the formula, function
and expression methods) as calculated.
The product of u and c. These are the contributions to the
combined uncertainty for uncorrelated quantities.
Any relevant parameters other than those in $budget$x (typically addditional constants passed to function or expression methods)
If available, a named list of the distributions associated with u.
The list contains either root nams of distribution functions (e.g "norm" or
function definitions.
If available, a list of lists of parameters describing the
distributions associated with u.
The covariance matrix used
The correlation matrix used
A data frame of covariances between x and y. Row names correspond to the correlation method used. For all uncertainty evaluation methods but MC, the only correlation calculation is "theoretical"; for MC row names include all methods supported by stats::cor at the time the object was created.
A data frame of correlations between x and y, of the same form as cov.xy
For the formula and expression methods, the result of a call to deriv;
an expression which evaluates to the value with attributes corresponding to the derivatives
(that is, an expression which can be evaluated to give the value and sensitivity coefficients)
print and summary methods invisibly return the original object.
The print method provides a formatted printout of the object. By default,
simplify=TRUE; this displays a shortened listing. Columns in $budget are
suppressed if all NA (typically df when not specified).
summary is currently an alias for the print method.
S. L. R. Ellison [email protected]
None.
uncert, especially for calculation methods;
plot.uncert, uncertMC-class,
print.data.frame, format.
expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncert(expr, x, u, method="NUM") print(u.expr)expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncert(expr, x, u, method="NUM") print(u.expr)
uncertMC estimates measurement uncertainty from a function,
expression or formula by Monte Carlo simulation.
uncertMC(expr, x, u, method = "MC", df, cor, cov, distrib, distrib.pars, B = 200, keep.x = TRUE, vectorized=TRUE, ...)uncertMC(expr, x, u, method = "MC", df, cor, cov, distrib, distrib.pars, B = 200, keep.x = TRUE, vectorized=TRUE, ...)
expr |
An expression, function, or formula with no left-hand side (e.g.
|
x |
A named list or vector of parameters supplied to |
u |
A named list or named vector of length |
method |
Method of uncertainty evaluation. The only method currently supported
by |
df |
A named list or named vector of degrees of freedom. |
cor, cov
|
Optional (square, symmetric) correlation or covariance matrices, respectively.
If neither is specified, |
distrib |
A character vector of length |
distrib.pars |
A named list of lists of parameters describing the distributions
associated with |
B |
Number of Monte Carlo replicates. |
keep.x |
If |
vectorized |
If |
... |
Additional parameters to be passed to a function (for the function method) or used in an expression (for expression or formula method). |
Although most likely to be called by uncert, uncertMC may be called directly.
If any of x, u, df, distrib or distrib.pars are not lists,
they are coerced to lists. If x is not named, arbitrary names of the form 'Xn'
are applied. If u, df, distrib or distrib.pars do not have
names, the names will be set to names(x) if they are of length exactly
length(x); if not, an error is returned.
For Monte Carlo evaluation, distributions and distribution parameters are needed but
defaults are used if some or all are absent. If distrib is missing, or
if it is a list with some members missing, the distribution is assumed Normal
and any missing member of distrib is set to "norm".
Distributions are usually identified by the root of the distribution function name; for example
to specify the Normal, distrib$name="norm". At present, only the random value
generator (e.g. rnorm) is used. Names of user-specified distributions functions can also be
used, provided they have a random value generator named r<dist> where <dist>
is the abbreviated distribution. Parameters are passed to distribution functions using
do.call, so the function must accept the parameters supplied in distrib.pars.
If distrib.pars or members of it are missing, an attempt is made to deduce
appropriate distribution parameters from x, u, df and distrib.
In doing so, the following assumptions and values apply for the respective distributions:
mean=x$name, sd=u$name.
min=x-sqrt(3)*u, max=x+sqrt(3)*u.
min=x-sqrt(6)*u, max=x+sqrt(6)*u, mode=x.
df=df, mean=x, sd=u.
If either cor or cov are present, a test is made to see if off-diagonal
elements are significant. If not, uncertMC treats the values as independent.
The test simply checks whether the sum of off-diagonal elements of cor (calculated
from cov if cov is present) is bigger than
.Machine.double.eps*nrow^2.
Correlation is supported as long as all correlated variables are normally distributed.
If correlation is present, uncertMC follows a two-stage simulation procedure.
First, variables showing correlation are identified. Following a check that
their associated distrib values are all "norm", mvrnorm from
the MASS library is called to generate the simulated x values for those variables.
Second, any remaining (i.e. independent) variables are simulated from their respective
distrib and distrib.pars.
Vectorisation makes a difference to execution speed. If vectorize=TRUE, MC evaluation
uses eval using the simulated data as the evaluation environment; if not, apply
is used row-wise on the simulated input matrix. This makes an appreciable difference to
execution speed (typically eval is faster by a factor of 5 or more) so the default
assumes vectorised expressions. However, not all functions and expressions take vector arguments,
especially user functions involving complicated arithmetic or numerical solutions. Use vectorize=FALSE
for functions or expressions that do not take vector arguments.
Note: One common symptom of an expression that does not take vector arguments is
an R warning indicating that only the first element (typically of a parameter in x) is used.
uncertMC may also return NA for u on attempting to take the sd of a single simulated point.
An object of class uncertMC. See uncertMC-class for details.
S. L. R. Ellison [email protected]
JCGM 100 (2008) Evaluation of measurement data - Guide to the expression of uncertainty in measurement. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf. (JCGM 100:2008 is a public domain copy of ISO/IEC Guide to the expression of uncertainty in measurement (1995) ).
Kragten, J. (1994) Calculating standard deviations and confidence intervals with a universally applicable spreadsheet technique, Analyst, 119, 2161-2166.
Ellison, S. L. R. (2005) Including correlation effects in an improved spreadsheet calculation of combined standard uncertainties, Accred. Qual. Assur. 10, 338-343.
uncert, uncert-class, uncertMC-class
expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.MC<-uncertMC(expr, x, u, distrib=rep("norm", 4), method="MC") print(u.MC, simplify=FALSE) #An example with correlation u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<-0.5 u.formc.MC<-uncertMC(~a+b*2+c*3+d/2, x, u, cor=u.cor, keep.x=TRUE) u.formc.MC #A non-linear example expr <- expression(a/(b-c)) x <- list(a=1, b=3, c=2) u <- lapply(x, function(x) x/20) set.seed(403) u.invexpr<-uncertMC(expr, x, u, distrib=rep("norm", 3), B=999, keep.x=TRUE ) u.invexpr #Look at effect of vectorize system.time(uncertMC(expr, x, u, distrib=rep("norm", 3), B=9999, keep.x=TRUE )) system.time(uncertMC(expr, x, u, distrib=rep("norm", 3), B=9999, keep.x=TRUE, vectorize=FALSE))expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.MC<-uncertMC(expr, x, u, distrib=rep("norm", 4), method="MC") print(u.MC, simplify=FALSE) #An example with correlation u.cor<-diag(1,4) u.cor[3,4]<-u.cor[4,3]<-0.5 u.formc.MC<-uncertMC(~a+b*2+c*3+d/2, x, u, cor=u.cor, keep.x=TRUE) u.formc.MC #A non-linear example expr <- expression(a/(b-c)) x <- list(a=1, b=3, c=2) u <- lapply(x, function(x) x/20) set.seed(403) u.invexpr<-uncertMC(expr, x, u, distrib=rep("norm", 3), B=999, keep.x=TRUE ) u.invexpr #Look at effect of vectorize system.time(uncertMC(expr, x, u, distrib=rep("norm", 3), B=9999, keep.x=TRUE )) system.time(uncertMC(expr, x, u, distrib=rep("norm", 3), B=9999, keep.x=TRUE, vectorize=FALSE))
Object returned by uncertMC calls and by uncertainty with method="MC".
summary.uncertMC is currently an alias for print.uncertMC.
## S3 method for class 'uncertMC' print(x, digits=NULL, right=FALSE, ..., simplify=TRUE, minimise=FALSE) ## S3 method for class 'uncertMC' summary(object, digits=NULL, right=FALSE, ..., simplify=TRUE, minimise=FALSE)## S3 method for class 'uncertMC' print(x, digits=NULL, right=FALSE, ..., simplify=TRUE, minimise=FALSE) ## S3 method for class 'uncertMC' summary(object, digits=NULL, right=FALSE, ..., simplify=TRUE, minimise=FALSE)
x, object
|
An object of class |
digits |
Number of digits to display in budget and (if present) distribution parameter lists.
Passed to |
right |
If TRUE, strings in uncertainty budget are right-justified.
This differs from the default in |
... |
Other parameters passed to |
simplify |
If |
minimise |
If |
An object of class "uncertMC" inherits from class "uncert". In addition to the contents of the "uncert" object, it contains the results from the MC replication as a list MC. The complete description is:
The matched call
The expression, formula or function supplied to uncertMC.
The uncertainty evaluation method used (always 'MC').
The number of Monte Carlo replicates used.
A data frame consisting of:
The starting values x.
The standard uncertainties in input quantities (originally provided as u)
The degrees of freedom asscociated with u
Sensitivity coefficients estimated from the MC output (see uncertMC
for how this is done).
If available, a named list of the distributions associated with u.
The list contains either root nams of distribution functions (e.g "norm" or
function definitions.
If available, a list of lists of parameters describing the
distributions associated with u.
If supplied, any relevant parameters other than those in $budget$x
(typically addditional constants passed to function or expression methods)
The covariance matrix used
The correlation matrix used
A data frame of covariances between x and y. The Row names correspond
to the correlation method used. For uncertMC objects only
"pearson" is currently supported (because "kendall" and "spearman"
take a very long time to compute)
A data frame of correlations between x and y, of the same form as cov.xy
A list containing:
The value of .Random.seed when uncertMC was called.
The B Monte Carlo replicates of the standard uncertainty
calculated as sd(y).
If uncertMC is called with keep.x=TRUE, a data frame
whose columns are the Monte Carlo replicates of the variables in x.
print and summary methods invisibly return the original object.
The print method provides a formatted printout of the object. By default,
simplify=TRUE; this displays a shortened listing. Columns in $data are
suppressed if all NA.
summary is currently an alias for the print method.
S. L. R. Ellison [email protected]
None.
uncert, uncertMC, uncert-class,
print.data.frame, format.
set.seed(13*17) expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncertMC(expr, x, u, distrib=rep("norm", 4), method="MC") print(u.expr)set.seed(13*17) expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncertMC(expr, x, u, distrib=rep("norm", 4), method="MC") print(u.expr)
‘update’ allows modification of components of ‘uncert’ or ‘uncertMC’ objects, including the uncertainty estimation method used, and will recalculate the estimate and return a new ‘uncert’ or ‘uncertMC’ object. Individual elements of most components can be amended.
## S3 method for class 'uncert' update(object, expr = NULL, method = NULL, x = NULL, u = NULL, c=NULL, df = NULL, cov = NULL, cor = NULL, distrib = NULL, distrib.pars = NULL, delta = NULL, B = NULL, keep.x = NULL, ...)## S3 method for class 'uncert' update(object, expr = NULL, method = NULL, x = NULL, u = NULL, c=NULL, df = NULL, cov = NULL, cor = NULL, distrib = NULL, distrib.pars = NULL, delta = NULL, B = NULL, keep.x = NULL, ...)
object |
An object of class ‘uncert’ |
expr |
An expression, formula or function. |
method |
Uncertainty evaluation method. May be any of the methods listed for |
x, u, df
|
Named list, vector or array of values to update elements of
object |
c |
Update to |
cov, cor
|
A covariance or correlation matrix. Only one of |
distrib |
Named list or character vector of updated distribution names. |
distrib.pars |
Named list of updates for distribution names and parameters. |
delta |
Scalar value updating |
B |
Updated number of Monte Carlo iterations for ‘uncertMC’ objects or specification of
|
keep.x |
Update to |
... |
Other values passed to |
Update will use the values provided to update the object given, call the original function with the revised parameters and return the result as an object of class ‘uncert’ or ‘uncert’ depending on the uncertainty evaluation method used.
Note that updating with a different value of method may result in an object of
different class. Updating an ‘uncertMC’ object with a method other than "MC"
will return an object of class ‘uncert’; similarly, updating an ‘uncert’ object
using method="MC" will return an object of class ‘uncertMC’.
Updates to vector or list elements of uncert such as x, u, df
etc. can be specified as named lists, named vectors or arrays, with names corresponding to
names of the input quantities in the uncertainty budget (that is, the names may correspond
to one or more of row.names(uncert$budget)). If names are present, only the
corresponding individual members are updated.
If names are not present, the complete vector or list in uncert is replaced, and names added.
Matrix elements cor and cov must be specified completely; see
buildCor, updateCor and associated functions for
covariance matrices for compact update methods.
An object of class ‘uncert’ or, for method="MC" of class ‘uncertMC’.
See uncert-class and uncertMC-class for details.
S. L. R. Ellison [email protected]
None, yet.
uncert-class, uncert-class, uncertMC, uncertMC-class
#From uncert: expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncert(expr, x, u, method="NUM") u.expr update(u.expr, u=list(a=0.3)) update(u.expr, method="MC")#From uncert: expr <- expression(a+b*2+c*3+d/2) x <- list(a=1, b=3, c=2, d=11) u <- lapply(x, function(x) x/10) u.expr<-uncert(expr, x, u, method="NUM") u.expr update(u.expr, u=list(a=0.3)) update(u.expr, method="MC")
Calculate a weighted mean, between-group standard deviation and standard error on the weighted mean using the Maximum likelihood algorithm of Vangel-Rukhin.
vr.mle(x, s2, n, init.mu = mean(x), init.sigma2 = var(x), labels = c(1:length(x)), max.iter = 1000, tol = .Machine$double.eps^0.5, trace = FALSE) ## S3 method for class 'summary.vr.mle' print(x, ..., digits=3)vr.mle(x, s2, n, init.mu = mean(x), init.sigma2 = var(x), labels = c(1:length(x)), max.iter = 1000, tol = .Machine$double.eps^0.5, trace = FALSE) ## S3 method for class 'summary.vr.mle' print(x, ..., digits=3)
x |
numeric vector of the sample mean values of each group |
s2 |
numeric vector of the sample variances of each group |
n |
integer vector of sample size of each group |
init.mu |
numeric initial value for the mean |
init.sigma2 |
numeric initial value for the between-group component of variance |
labels |
vector of group names. Coerced to character on use. |
max.iter |
numeric maximum number of iterations |
tol |
numeric tolerance; iteration stops when the relative step size drops below 'tol' |
trace |
when TRUE shows the sequence of intermediate results |
..., digits
|
Passed to |
The Vangel-Rukhin MLE algorithm finds the between-method variance by iteratively solving the equation relating the weighted mean to the weighting factor applied. The weighting factor is the inverse of the sum of the standard error in 'x' and the between-method variance, scaled by the between-method variance.
For the default method, 's2' is interpreted as a vector of sample variances. 'x' is interpreted as a vector of sample means and the algorithm is applied to the corresponding group means, variances, and sample sizes.
The Vangel-Rukhin MLE algorithm shows an improvement in the number of iterations required to converge over the classical MLE based on the Score equations.
The function mle.1wre implements the MLE for the one way random effects based on the Fisher scoring equations and is provided for comparison purpose only.
vr.mle returns an object of class "summary.vr.mle" which contains
the following fields:
mu |
the estimated mean |
var.mu |
the variance associated with the estimated mean |
sigma2 |
the estimated between variance component |
llh |
the log likelihood of the estimates |
tot.iter |
the total number of iterations ran |
cur.rel.abs.error |
the current relative absolute error reached |
gammai |
a vector with the estimates of the weights |
converged |
TRUE is convergence criteria was met, FALSE otherwise |
reduced.model |
TRUE implies that a reduced model, with no
between-group effect, is suggested, based on |
H. Gasca-Aragon
Vangel, M. G. and Rukhin, A. L. (1999), Biometrics, Vol 55, No. 1 pp 129-136
Searle, S. R., Cassella, G., and McCulloch, C. E. (1992). Variance Components. New York: Wiley.
##=================================================================== ## the dietary fiber in apples example in the Vangel and Rukhin paper ##=================================================================== m1 <- c(12.46, 13.035, 12.44, 12.87, 13.42, 12.08, 13.18, 14.335, 12.23) s1 <- c(0.028, 0.233, 0.325, 0.071, 0.339, 0.325, 0.099, 0.064, 0.212) n1 <- c(2, 2, 2, 2, 2, 2, 2, 2, 2) res<- vr.mle(m1, s1^2, n1, tol=1e-6) res$mu sqrt(res$var.mu) res$sigma2 res$mu+c(-1,1)*qnorm(0.975)*sqrt(res$var.mu) res$tot.iter res$converged res$reduced.model # output # 12.90585 # 0.2234490 # 0.4262122 # 12.46790 13.34380 # 6 # converged = TRUE # reduced.model = FALSE##=================================================================== ## the dietary fiber in apples example in the Vangel and Rukhin paper ##=================================================================== m1 <- c(12.46, 13.035, 12.44, 12.87, 13.42, 12.08, 13.18, 14.335, 12.23) s1 <- c(0.028, 0.233, 0.325, 0.071, 0.339, 0.325, 0.099, 0.064, 0.212) n1 <- c(2, 2, 2, 2, 2, 2, 2, 2, 2) res<- vr.mle(m1, s1^2, n1, tol=1e-6) res$mu sqrt(res$var.mu) res$sigma2 res$mu+c(-1,1)*qnorm(0.975)*sqrt(res$var.mu) res$tot.iter res$converged res$reduced.model # output # 12.90585 # 0.2234490 # 0.4262122 # 12.46790 13.34380 # 6 # converged = TRUE # reduced.model = FALSE
Provides the Welch-Satterthwaite effective degrees of freedom given standard uncertainties and associated degrees of freedom.
w.s is an alias for welch.satterthwaite.
w.s(ui, df, ci = rep(1, length(ui)), uc=sqrt(sum((ci*ui)^2))) welch.satterthwaite(ui, df, ci = rep(1, length(ui)), uc=sqrt(sum((ci*ui)^2)))w.s(ui, df, ci = rep(1, length(ui)), uc=sqrt(sum((ci*ui)^2))) welch.satterthwaite(ui, df, ci = rep(1, length(ui)), uc=sqrt(sum((ci*ui)^2)))
ui |
Standard uncertainties |
df |
Degrees of freedom |
ci |
Sensitivity coefficients |
uc |
Combined standard uncertainty |
Implements the Welch-Satterthwaite equation as provided in the ISO Guide to the expression of
uncertainty in measurement (1995) (See JCGM 100:2008). This assumes that uc is the
uncertainty in a measurement result , where , ci are
the partial derivatives and ui is the standard uncertainty associated with xi.
The implementation assumes that the combined uncertainty uc is equal to
sqrt(sum((ci*ui)^2). An independent estimate of uc can be provided.
The ci are 'sensitivity coefficients'; the default is 1, so that the ui
can be given either as standard uncertainties in the values of influence quantities ,
together with the associated ci, or as contributions ci*ui to the uncertainty in .
Correlation is not supported, because the Welch-Satterthwaite equation is only valid for independent variances.
The calculated effective degrees of freedom associated with uc.
S. L. R. Ellison [email protected]
JCGM 100 (2008) Evaluation of measurement data - Guide to the expression of uncertainty in measurement. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf. (JCGM 100:2008 is a public domain copy of ISO/IEC Guide to the expression of uncertainty in measurement (1995) ).
Satterthwaite, F. E. (1946), An Approximate Distribution of Estimates of Variance Components., Biometrics Bulletin 2, 110-114, doi:10.2307/3002019
Welch, B. L. (1947), The generalization of "Student's" problem when several different population variances are involved., Biometrika 34 28-35
None, yet.
u <- c(0.1, 0.3, 0.2, 1.1) ci <- c(1.0, 2.0, 3.0, 0.5) degfree <- c(Inf,6,8,3) w.s(ui=u,df=degfree, ci=ci)u <- c(0.1, 0.3, 0.2, 1.1) ci <- c(1.0, 2.0, 3.0, 0.5) degfree <- c(Inf,6,8,3) w.s(ui=u,df=degfree, ci=ci)
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, ...) ## Default S3 method: 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, ...)xs.plot(x, ...) ## Default S3 method: 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, ...)
x |
Numeric vector of values to be plotted. |
g |
grouping factor of length |
s |
numeric vector of standard deviations of length |
degfree |
A single value for degrees of freedom associated with all
the standard deviations |
labels.arg |
An optional vector of point labels, coerced to character on use. |
mu |
A single location used to centre the confidence region. The default is
specified by |
sigma |
A measure of dispersion against which deviations x-mu can be compared. |
probs |
A vector of probabilities for confidence region contours. |
basis |
Controls the nature of the location and scale estimators used to produce the confidence contours drawn on the plot. See Details for specification. |
method |
The method used to calculate the confidence region. See Details. |
main |
Main title for the plot. |
xlab, ylab
|
x- and y-axis labels, |
contours |
logical, specifying whether confidence contours should be drawn. |
col.contours, lty.contours, lwd.contours
|
Colour, line type and line width for contour lines. |
label.contours |
Logical, controlling whether contour lines are labelled with approximate probabilities. |
format.clab |
Format string for contour labels, passed to |
pos.clab |
Specification for location of contour labels. A vector can be provided to give multiple labels. See Details for further description. |
col.clab, cex.clab
|
Colour and expansion for contour labels. |
pch, col, bg, cex
|
Graphical parameters passed to |
cex.label |
Expansion factor for point labels, passed to |
pos, adj
|
Specifies position/adjustment of point labels. Passed to |
add |
If TRUE the plot region is not cleared before plotting; points and contours
are added to the present plot. Use |
... |
Other parameters passed to |
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
and an estimate of the pooled within-group standard deviation or pooled
estimate from s. If basis="robust", and
are calculated using algA and algS respectively. If basis="classical",
and 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 and respectively.
If method="chisq", contours for probability are calculated as
for from
to
.
If method="density", contours for probability 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 where is
the probability and 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
but is anticonservative (particularly to high ) for .
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 ), for which the region is a maximu width at s=0,
"bottomright" and "bottomleft" place labels immediately below the countour boundary
at and, if specified, "bottom" is replaced with c("bottomright", "bottomleft").
XSplot is an alias for xs.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 and
used to construct the confidence ellipsoids.
A list of sets of coordinates for each confidence region.
S Ellison [email protected]
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;
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)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)
A Youden plot is a bivariate scatter plot, named for its use by W. M Youden in interlaboratory studies. This implementation includes data ellipses based on Pearson, Spearman, Kendall or several robust covariance measures.
youden.plot(x, ...) yplot(x, ...) ## Default S3 method: youden.plot(x, y = NULL, type = c("points", "labels", "both", "outliers"), labels, probs = c(0.95, 0.99), x0, y0, pch = par("pch"), cex = par("cex"), col = par("col"), bg = par("bg"), main, xlab, ylab, xlim = c("data", "ellipse", "all"), ylim = c("data", "ellipse", "all"), col.axes = 2, lwd.axes = 1, lty.axes = 1, cex.lab = 0.7, pos = 3, out.method = c("F", "chisq", "n"), n.out, p.out = 0.99, add = FALSE, ...)youden.plot(x, ...) yplot(x, ...) ## Default S3 method: youden.plot(x, y = NULL, type = c("points", "labels", "both", "outliers"), labels, probs = c(0.95, 0.99), x0, y0, pch = par("pch"), cex = par("cex"), col = par("col"), bg = par("bg"), main, xlab, ylab, xlim = c("data", "ellipse", "all"), ylim = c("data", "ellipse", "all"), col.axes = 2, lwd.axes = 1, lty.axes = 1, cex.lab = 0.7, pos = 3, out.method = c("F", "chisq", "n"), n.out, p.out = 0.99, add = FALSE, ...)
x |
An R |
y |
A numeric vector of the same length as |
type |
The type of plot produced. See Details. |
labels |
Character vector of text labels for data points. Defaults to |
probs |
Numeric vector of probabilities for data ellipses. |
x0, y0
|
If specified, data ellipses will be centred on |
pch, cex, col, bg
|
passed to |
main, xlab, ylab
|
Plot titles. If missing, titles are based on the names of the objects plotted. |
xlim, ylim
|
Specifications for horizintal and vertical plot limits. Each can be either a length 2
numeric vector (as usual) or a character value matching one of |
col.axes, lwd.axes, lty.axes
|
Colour, line width and line type for vertical and horizontal location markers drawn through the ellipse centre. |
cex.lab |
Size for data point labels; see |
pos |
a position specifier for data point labels; see |
out.method |
Character specifying outlier marking. See Details. |
n.out |
Number of outliers marked if |
p.out |
Confidence level at which points are marked as outliers if |
add |
If |
... |
Named arguments passed to other functions. In particular:
|
type controls the type of plot produced. Allowed types and their effect are:
pointsPoints only are drawn.
labelsPoint labels only are drawn
bothPoints are drawn with labels
outliersPoints are drawn and outlying points are labelled (see below)
Ellipses are constructed based on a location and covariance matrix constructed from the data
by the method specified by cov.method. probs specifies the approximate coverage.
See data.ellipse for details of covariance methods and ellipse specification.
The outlier identification method, if any, is specified by out.method and controlled by
one of n.out or p.out. If out.method is "F" or "chisq",
points with Mahalanobis distance greater than an upper critical value with probability p.out
are considered to be outliers. The critical values used are
"F"Mahalanobis distance greater than
2 * (n-1) * qf(p.out, 2, n-1) / (n-2)
"chisq"Mahalanobis distance greater than qchisq(p.out, 2)
which( md > 2 * (n-1) * qf(p.out, 2, n-1) / (n-2) ) #F dist The Mahalanobis distance is calculated based on the covariance matrix used to consstruct plot ellipses.
If out.method is "n", the outermost n.out points (judged by Mahalanobis distance)
are marked as outliers. Specifying out.method="n" and n.out=0 suppresses outlier
identification.
If outliers are marked, a list of marked outliers is included in the returned list
(see Value, below).
yplot is an alias for youden.plot
Invisibly returns the plotted data ellipses as an object of class d.ellipse.
S L R Ellison ([email protected])
Youden, W.J. and Steiner, E.H. (1975) Statistical Manual of the AOAC. AOAC International, Washington, US. ISBN 0-935584-15-3
ISO 13528:2005, Statistical methods for use in proficiency testing by interlaboratory comparisons, International Organization for Standardization, Geneva (2005)
data(chromium) data(potassium) ( yy <- youden.plot(chromium, main="Chromium") ) #With outlier ID (F based) youden.plot(chromium, main="Chromium", xlim='a', ylim='a', type='o', p.out=0.95) #Note use of xlim="a" etc. to ensure both ellipses and data are included. #Top 5 most distant outliers (5 is also the default) youden.plot(chromium, main="Chromium", xlim='a', ylim='a', type='o', out.method="n", n.out=5) #With ellipse principal axes #(useful to specify asp=1 or the axes will not always appear orthogonal) youden.plot(chromium, main="Chromium", xlim='a', ylim='a', type='o', p.out=0.99, prinax=TRUE, lty.prinax=2, asp=1.0) youden.plot(potassium, main="Potassium") #A different pairs plot ... panel.youden <- function(x, y, ...) youden.plot(x, y, add=TRUE, type="o", cex=1, pos=1, p.out=0.95) pairs(chromium, upper.panel=panel.youden)data(chromium) data(potassium) ( yy <- youden.plot(chromium, main="Chromium") ) #With outlier ID (F based) youden.plot(chromium, main="Chromium", xlim='a', ylim='a', type='o', p.out=0.95) #Note use of xlim="a" etc. to ensure both ellipses and data are included. #Top 5 most distant outliers (5 is also the default) youden.plot(chromium, main="Chromium", xlim='a', ylim='a', type='o', out.method="n", n.out=5) #With ellipse principal axes #(useful to specify asp=1 or the axes will not always appear orthogonal) youden.plot(chromium, main="Chromium", xlim='a', ylim='a', type='o', p.out=0.99, prinax=TRUE, lty.prinax=2, asp=1.0) youden.plot(potassium, main="Potassium") #A different pairs plot ... panel.youden <- function(x, y, ...) youden.plot(x, y, add=TRUE, type="o", cex=1, pos=1, p.out=0.95) pairs(chromium, upper.panel=panel.youden)