plotTolIntNormDesign.RdCreate plots involving sample size, half-width, estimated standard deviation, coverage, and confidence level for a tolerance interval for a normal distribution.
plotTolIntNormDesign(x.var = "n", y.var = "half.width", range.x.var = NULL,
n = 25, half.width = ifelse(x.var == "sigma.hat", 3 * max.x, 3 * sigma.hat),
sigma.hat = 1, coverage = 0.95, conf.level = 0.95, cov.type = "content",
round.up = FALSE, n.max = 5000, tol = 1e-07, maxiter = 1000, plot.it = TRUE,
add = FALSE, n.points = 100, plot.col = 1, plot.lwd = 3 * par("cex"),
plot.lty = 1, digits = .Options$digits, ..., main = NULL, xlab = NULL,
ylab = NULL, type = "l")character string indicating what variable to use for the x-axis. Possible values
are "n" (sample size; the default), "half.width" (half-width),
"sigma.hat" (estimated standard deviation), "coverage" (the coverage),
and "conf.level" (the confidence level).
character string indicating what variable to use for the y-axis. Possible values
are "half.width" (the half-width; the default), and "n" (sample size).
numeric vector of length 2 indicating the range of the x-variable to use for the plot.
The default value depends on the value of x.var.
When x.var="n" the default value is c(2,50).
When x.var="half.width" the default value is
c(2.5 * sigma.hat, 4 * sigma.hat).
When x.var="sigma.hat", the default value is c(0.1, 2).
When x.var="coverage" or x.var="conf.level", the default value is
c(0.5, 0.99).
positive integer greater than 1 indicating the sample size upon
which the tolerance interval is based. The default value is n=25.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf)
values are not allowed.
positive scalar indicating the half-width of the prediction interval.
The default value depends on the value of x.var. When
x.var="sigma.hat" the default value is 3 times the second value of
range.x.var. When x.var is not equal to "sigma.hat" the
default value is half.width=4*sigma.hat. This argument is ignored
if either x.var="half.width" or y.var="half.width".
numeric scalar specifying the value of the estimated standard deviation.
The default value is sigma.hat=1. This argument is ignored if
x.var="sigma.hat".
numeric scalar between 0 and 1 indicating the desired coverage of the
tolerance interval. The default value is coverage=0.95.
numeric scalar between 0 and 1 indicating the confidence level of the
tolerance interval. The default value is conf.level=0.95.
character string specifying the coverage type for the tolerance interval. The
possible values are "content" (\(\beta\)-content; the default), and
"expectation" (\(\beta\)-expectation).
for the case when y.var="n", logical scalar indicating whether to round
up the values of the computed sample size(s) to the next smallest integer.
The default value is round.up=TRUE.
for the case when y.var="n", positive integer greater than 1 specifying
the maximum possible sample size. The default value is n.max=5000.
for the case when y.var="n", numeric scalar indicating the tolerance
to use in the uniroot search algorithm. The default value is
tol=1e-7.
for the case when y.var="n", positive integer indicating the maximum
number of iterations to use in the uniroot search algorithm.
The default value is maxiter=1000.
a logical scalar indicating whether to create a plot or add to the existing plot
(see explanation of the argument add below) on the current graphics device.
If plot.it=FALSE, no plot is produced, but a list of (x,y) values is returned
(see the section VALUE). The default value is plot.it=TRUE.
a logical scalar indicating whether to add the design plot to the existing plot (add=TRUE),
or to create a plot from scratch (add=FALSE). The default value is add=FALSE.
This argument is ignored if plot.it=FALSE.
a numeric scalar specifying how many (x,y) pairs to use to produce the plot.
There are n.points x-values evenly spaced between range.x.var[1] and range.x.var[2]. The default value is n.points=100.
a numeric scalar or character string determining the color of the plotted line or points. The default value
is plot.col="black". See the entry for col in the help file for par
for more information.
a numeric scalar determining the width of the plotted line. The default value is
3*par("cex"). See the entry for lwd in the help file for par
for more information.
a numeric scalar determining the line type of the plotted line. The default value is
plot.lty=1. See the entry for lty in the help file for par
for more information.
a scalar indicating how many significant digits to print out on the plot. The default
value is the current setting of options("digits").
additional graphical parameters (see par).
See the help files for tolIntNorm, tolIntNormK,
tolIntNormHalfWidth, and tolIntNormN for information
on how to compute a tolerance interval for a normal distribution, how the
half-width is computed when other quantities are fixed, and how the sample size
is computed when other quantities are fixed.
plotTolIntNormDesign invisibly returns a list with components:
x-coordinates of points that have been or would have been plotted.
y-coordinates of points that have been or would have been plotted.
See the help file for tolIntNorm.
See the help file for tolIntNorm.
In the course of designing a sampling program, an environmental scientist may wish
to determine the relationship between sample size, confidence level, and half-width
if one of the objectives of the sampling program is to produce tolerance intervals.
The functions tolIntNormHalfWidth, tolIntNormN, and
plotTolIntNormDesign can be used to investigate these relationships for the
case of normally-distributed observations.
tolIntNorm, tolIntNormK,
tolIntNormN, plotTolIntNormDesign,
Normal.
# Look at the relationship between half-width and sample size for a
# 95% beta-content tolerance interval, assuming an estimated standard
# deviation of 1 and a confidence level of 95%:
dev.new()
plotTolIntNormDesign()
#==========
# Plot half-width vs. coverage for various levels of confidence:
dev.new()
plotTolIntNormDesign(x.var = "coverage", y.var = "half.width",
ylim = c(0, 3.5), main="")
plotTolIntNormDesign(x.var = "coverage", y.var = "half.width",
conf.level = 0.9, add = TRUE, plot.col = "red")
plotTolIntNormDesign(x.var = "coverage", y.var = "half.width",
conf.level = 0.8, add = TRUE, plot.col = "blue")
legend("topleft", c("95%", "90%", "80%"), lty = 1, lwd = 3 * par("cex"),
col = c("black", "red", "blue"), bty = "n")
title(main = paste("Half-Width vs. Coverage for Tolerance Interval",
"with Sigma Hat=1 and Various Confidence Levels", sep = "\n"))
#==========
# Example 17-3 of USEPA (2009, p. 17-17) shows how to construct a
# beta-content upper tolerance limit with 95% coverage and 95%
# confidence using chrysene data and assuming a lognormal distribution.
# The data for this example are stored in EPA.09.Ex.17.3.chrysene.df,
# which contains chrysene concentration data (ppb) found in water
# samples obtained from two background wells (Wells 1 and 2) and
# three compliance wells (Wells 3, 4, and 5). The tolerance limit
# is based on the data from the background wells.
# Here we will first take the log of the data and then estimate the
# standard deviation based on the two background wells. We will use this
# estimate of standard deviation to plot the half-widths of
# future tolerance intervals on the log-scale for various sample sizes.
head(EPA.09.Ex.17.3.chrysene.df)
#> Month Well Well.type Chrysene.ppb
#> 1 1 Well.1 Background 19.7
#> 2 2 Well.1 Background 39.2
#> 3 3 Well.1 Background 7.8
#> 4 4 Well.1 Background 12.8
#> 5 1 Well.2 Background 10.2
#> 6 2 Well.2 Background 7.2
# Month Well Well.type Chrysene.ppb
#1 1 Well.1 Background 19.7
#2 2 Well.1 Background 39.2
#3 3 Well.1 Background 7.8
#4 4 Well.1 Background 12.8
#5 1 Well.2 Background 10.2
#6 2 Well.2 Background 7.2
longToWide(EPA.09.Ex.17.3.chrysene.df, "Chrysene.ppb", "Month", "Well")
#> Well.1 Well.2 Well.3 Well.4 Well.5
#> 1 19.7 10.2 68.0 26.8 47.0
#> 2 39.2 7.2 48.9 17.7 30.5
#> 3 7.8 16.1 30.1 31.9 15.0
#> 4 12.8 5.7 38.1 22.2 23.4
# Well.1 Well.2 Well.3 Well.4 Well.5
#1 19.7 10.2 68.0 26.8 47.0
#2 39.2 7.2 48.9 17.7 30.5
#3 7.8 16.1 30.1 31.9 15.0
#4 12.8 5.7 38.1 22.2 23.4
summary.stats <- summaryStats(log(Chrysene.ppb) ~ Well.type,
data = EPA.09.Ex.17.3.chrysene.df)
summary.stats
#> N Mean SD Median Min Max
#> Background 8 2.5086 0.6279 2.4359 1.7405 3.6687
#> Compliance 12 3.4173 0.4361 3.4111 2.7081 4.2195
#> attr(,"class")
#> [1] "summaryStats"
#> attr(,"stats.in.rows")
#> [1] FALSE
#> attr(,"drop0trailing")
#> [1] TRUE
# N Mean SD Median Min Max
#Background 8 2.5086 0.6279 2.4359 1.7405 3.6687
#Compliance 12 3.4173 0.4361 3.4111 2.7081 4.2195
sigma.hat <- summary.stats["Background", "SD"]
sigma.hat
#> [1] 0.6279
#[1] 0.6279
dev.new()
plotTolIntNormDesign(x.var = "n", y.var = "half.width",
range.x.var = c(5, 40), sigma.hat = sigma.hat, cex.main = 1)
#==========
# Clean up
#---------
rm(summary.stats, sigma.hat)
graphics.off()