Power Curves for Sampling Design for Test Based on Nonparametric Simultaneous Prediction Interval
Plot power vs. Δ/σ (scaled minimal detectable difference) for a sampling design for a test based on a nonparametric simultaneous prediction interval. The power is based on assuming the true distribution of the observations is normal.
plotPredIntNparSimultaneousTestPowerCurve(n = 8, n.median = 1, k = 1, m = 2,
r = 1, rule = "k.of.m", lpl.rank = ifelse(pi.type == "upper", 0, 1),
n.plus.one.minus.upl.rank = ifelse(pi.type == "lower", 0, 1), pi.type = "upper",
r.shifted = r, integrate.args.list = NULL, method = "approx", NMC = 100,
range.delta.over.sigma = c(0, 5), plot.it = TRUE, add = FALSE, n.points = 20,
plot.col = "black", plot.lwd = 3 * par("cex"), plot.lty = 1,
digits = .Options$digits, cex.main = par("cex"), ..., main = NULL,
xlab = NULL, ylab = NULL, type = "l")n |
positive integer specifying the sample sizes. |
n.median |
positive odd integer specifying the sample size associated with the
future medians. The default value is |
k |
for the k-of-m rule ( |
m |
positive integer specifying the maximum number of future observations (or
medians) on one future sampling “occasion”.
The default value is |
r |
positive integer specifying the number of future sampling
“occasions”. The default value is |
rule |
character string specifying which rule to use. The possible values are
|
lpl.rank |
non-negative integer indicating the rank of the order statistic to use for
the lower bound of the prediction interval. When |
n.plus.one.minus.upl.rank |
non-negative integer related to the rank of the order statistic to use for
the upper
bound of the prediction interval. A value of |
pi.type |
character string indicating what kind of prediction interval to compute.
The possible values are |
r.shifted |
integer between |
integrate.args.list |
list of arguments to supply to the |
method |
character string indicating what method to use to compute the power. The possible
values are |
NMC |
positive integer indicating the number of Monte Carlo trials to run when |
range.delta.over.sigma |
numeric vector of length 2 indicating the range of the x-variable to use for the
plot. The default value is |
plot.it |
a logical scalar indicating whether to create a plot or add to the existing plot
(see explanation of the argument |
add |
a logical scalar indicating whether to add the design plot to the existing plot ( |
n.points |
a numeric scalar specifying how many (x,y) pairs to use to produce the plot.
There are |
plot.col |
a numeric scalar or character string determining the color of the plotted line or points. The default value
is |
plot.lwd |
a numeric scalar determining the width of the plotted line. The default value is
|
plot.lty |
a numeric scalar determining the line type of the plotted line. The default value is
|
digits |
a scalar indicating how many significant digits to print out on the plot. The default
value is the current setting of |
cex.main, main, xlab, ylab, type, ... |
additional graphical parameters (see |
See the help file for predIntNparSimultaneousTestPower for
information on how to compute the power of a hypothesis test for the difference
between two means of normal distributions based on a nonparametric simultaneous
prediction interval.
plotPredIntNparSimultaneousTestPowerCurve invisibly returns a list with
components:
x.var |
x-coordinates of points that have been or would have been plotted. |
y.var |
y-coordinates of points that have been or would have been plotted. |
See the help file for predIntNparSimultaneous.
In the course of designing a sampling program, an environmental scientist may wish
to determine the relationship between sample size, significance level, power, and
scaled difference if one of the objectives of the sampling program is to determine
whether two distributions differ from each other. The functions
predIntNparSimultaneousTestPower and plotPredIntNparSimultaneousTestPowerCurve can be
used to investigate these relationships for the case of normally-distributed
observations.
Steven P. Millard (EnvStats@ProbStatInfo.com)
See the help file for predIntNparSimultaneous.
Gansecki, M. (2009). Using the Optimal Rank Values Calculator. US Environmental Protection Agency, Region 8, March 10, 2009.
# Example 19-5 of USEPA (2009, p. 19-33) shows how to compute nonparametric upper
# simultaneous prediction limits for various rules based on trace mercury data (ppb)
# collected in the past year from a site with four background wells and 10 compliance
# wells (data for two of the compliance wells are shown in the guidance document).
# The facility must monitor the 10 compliance wells for five constituents
# (including mercury) annually.
# We will pool data from 4 background wells that were sampled on
# a number of different occasions, giving us a sample size of
# n = 20 to use to construct the prediction limit.
# There are 10 compliance wells and we will monitor 5 different
# constituents at each well annually. For this example, USEPA (2009)
# recommends setting r to the product of the number of compliance wells and
# the number of evaluations per year (i.e., r = 10 * 1 = 10).
# Here we will reproduce Figure 19-2 on page 19-35. This figure plots the
# power of the nonparametric simultaneous prediction interval for 6 different
# plans:
# Rule Median.n k m Order.Statistic Achieved.alpha BG.Limit
#1) k.of.m 1 1 3 Max 0.0055 0.28
#2) k.of.m 1 1 4 Max 0.0009 0.28
#3) Modified.CA 1 1 4 Max 0.0140 0.28
#4) k.of.m 3 1 2 Max 0.0060 0.28
#5) k.of.m 1 1 4 2nd 0.0046 0.25
#6) k.of.m 1 1 4 3rd 0.0135 0.24
# Here is the power curve for the 1-of-4 sampling strategy.
dev.new()
plotPredIntNparSimultaneousTestPowerCurve(n = 20, k = 1, m = 4, r = 10,
rule = "k.of.m", n.plus.one.minus.upl.rank = 3, pi.type = "upper",
r.shifted = 1, method = "approx", range.delta.over.sigma = c(0, 5), main = "")
title(main = paste(
"Power Curve for Nonparametric 1-of-4 Sampling Strategy Based on",
"25 Background Samples, SWFPR=10%, and 2 Future Sampling Periods",
sep = "\n"), cex.main = 1.1)
#----------
# Here are the power curves for all 6 sampling strategies.
# Because these take several seconds to create, here we have commented out
# the R commands. To run this example, just remove the pound signs (#) from
# in front of the R commands.
#dev.new()
#plotPredIntNparSimultaneousTestPowerCurve(n = 20, k = 1, m = 4, r = 10,
# rule = "k.of.m", n.plus.one.minus.upl.rank = 3, pi.type = "upper",
# r.shifted = 1, method = "approx", range.delta.over.sigma = c(0, 5), main = "")
#plotPredIntNparSimultaneousTestPowerCurve(n = 20, n.median = 3, k = 1, m = 2,
# r = 10, rule = "k.of.m", n.plus.one.minus.upl.rank = 1, pi.type = "upper",
# r.shifted = 1, method = "approx", range.delta.over.sigma = c(0, 5),
# add = TRUE, plot.col = 2, plot.lty = 2)
#plotPredIntNparSimultaneousTestPowerCurve(n = 20, r = 10, rule = "Modified.CA",
# n.plus.one.minus.upl.rank = 1, pi.type = "upper", r.shifted = 1,
# method = "approx", range.delta.over.sigma = c(0, 5), add = TRUE,
# plot.col = 3, plot.lty = 3)
#plotPredIntNparSimultaneousTestPowerCurve(n = 20, k = 1, m = 4, r = 10,
# rule = "k.of.m", n.plus.one.minus.upl.rank = 2, pi.type = "upper",
# r.shifted = 1, method = "approx", range.delta.over.sigma = c(0, 5),
# add = TRUE, plot.col = 4, plot.lty = 4)
#plotPredIntNparSimultaneousTestPowerCurve(n = 20, k = 1, m = 3, r = 10,
# rule = "k.of.m", n.plus.one.minus.upl.rank = 1, pi.type = "upper",
# r.shifted = 1, method = "approx", range.delta.over.sigma = c(0, 5),
# add = TRUE, plot.col = 5, plot.lty = 5)
#plotPredIntNparSimultaneousTestPowerCurve(n = 20, k = 1, m = 4, r = 10,
# rule = "k.of.m", n.plus.one.minus.upl.rank = 1, pi.type = "upper",
# r.shifted = 1, method = "approx", range.delta.over.sigma = c(0, 5),
# add = TRUE, plot.col = 6, plot.lty = 6)
#legend("topleft", legend = c("1-of-4, 3rd", "1-of-2, Max, Median", "Mod CA",
# "1-of-4, 2nd", "1-of-3, Max", "1-of-4, Max"), lwd = 3 * par("cex"),
# col = 1:6, lty = 1:6, bty = "n")
#title(main = "Figure 19-2. Comparison of Full Power Curves")
#==========
# Clean up
#---------
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