Power Curves for Sampling Design for Test Based on Nonparametric Simultaneous Prediction Interval

Description

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.

Usage

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  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") 

Arguments

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 n.median=1 (i.e., individual observations). Note that all future medians must be based on the same sample size.

k

for the k-of-m rule (rule="k.of.m"), a positive integer specifying the minimum number of observations (or medians) out of m observations (or medians) (all obtained on one future sampling “occassion”) the prediction interval should contain. The default value is k=1. This argument is ignored when the argument rule is not equal to "k.of.m".

m

positive integer specifying the maximum number of future observations (or medians) on one future sampling “occasion”. The default value is m=2, except when rule="Modified.CA", in which case this argument is ignored and m is automatically set equal to 4.

r

positive integer specifying the number of future sampling “occasions”. The default value is r=1.

rule

character string specifying which rule to use. The possible values are "k.of.m" (k-of-m rule; the default), "CA" (California rule), and "Modified.CA" (modified California rule).

lpl.rank

non-negative integer indicating the rank of the order statistic to use for the lower bound of the prediction interval. When pi.type="lower", the default value is lpl.rank=1 (implying the minimum value of x is used as the lower bound of the prediction interval). When pi.type="upper", the argument lpl.rank is set equal to 0.

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 n.plus.one.minus.upl.rank=1 means use the first largest value, and in general a value of
n.plus.one.minus.upl.rank=i means use the i'th largest value. When
pi.type="upper", the default value is n.plus.one.minus.upl.rank=1. When pi.type="lower", the argument n.plus.one.minus.upl.rank is set equal to 0.

pi.type

character string indicating what kind of prediction interval to compute. The possible values are "two.sided" (the default), "lower", and "upper".

r.shifted

integer between 1 and r specifying the number of future sampling occasions for which the scaled mean is shifted by Δ/σ. The default value is r.shifted=r.

integrate.args.list

list of arguments to supply to the integrate function. The default value is NULL.

method

character string indicating what method to use to compute the power. The possible values are "approx" (the default) and "simulate" (use Monte Carlo simulation).

NMC

positive integer indicating the number of Monte Carlo trials to run when
method="simulate". The default value is NMC=100.

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 range.delta.over.sigma=c(0,5).

plot.it

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.

add

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.

n.points

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.

plot.col

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.

plot.lwd

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.

plot.lty

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.

digits

a scalar indicating how many significant digits to print out on the plot. The default value is the current setting of options("digits").

cex.main, main, xlab, ylab, type, ...

additional graphical parameters (see par).

Details

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.

Value

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.

Note

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.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

See the help file for predIntNparSimultaneous.

Gansecki, M. (2009). Using the Optimal Rank Values Calculator. US Environmental Protection Agency, Region 8, March 10, 2009.

See Also

predIntNparSimultaneousTestPower, predIntNparSimultaneous, predIntNparSimultaneousN, predIntNparSimultaneousConfLevel, plotPredIntNparSimultaneousDesign, predIntNpar, tolIntNpar.

Examples

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  # 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
  #---------
  graphics.off()

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