Plots for a Sampling Design Based on a Tolerance Interval for a Normal Distribution

Create 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")

Arguments

x.var

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).

y.var

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).

range.x.var

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).

n

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.

half.width

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".

sigma.hat

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".

coverage

numeric scalar between 0 and 1 indicating the desired coverage of the tolerance interval. The default value is coverage=0.95.

conf.level

numeric scalar between 0 and 1 indicating the confidence level of the tolerance interval. The default value is conf.level=0.95.

cov.type

character string specifying the coverage type for the tolerance interval. The possible values are "content" (\(\beta\)-content; the default), and "expectation" (\(\beta\)-expectation).

round.up

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.

n.max

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.

tol

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.

maxiter

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.

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").

main, xlab, ylab, type, ...

additional graphical parameters (see par).

Details

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.

Value

plotTolIntNormDesign 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.

References

See the help file for tolIntNorm.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

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.

See also

tolIntNorm, tolIntNormK, tolIntNormN, plotTolIntNormDesign, Normal.

Examples

  # 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
  #            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()