tolIntNormN.RdCompute the sample size necessary to achieve a specified half-width of a tolerance interval for a normal distribution, given the estimated standard deviation, coverage, and confidence level.
tolIntNormN(half.width, sigma.hat = 1, coverage = 0.95, cov.type = "content",
conf.level = 0.95, method = "wald.wolfowitz", round.up = TRUE, n.max = 5000,
tol = 1e-07, maxiter = 1000)numeric vector of (positive) half-widths.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.
numeric vector specifying the value(s) of the estimated standard deviation(s).
The default value is sigma.hat=1.
numeric vector of values between 0 and 1 indicating the desired coverage of the
tolerance interval. The default value is coverage=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).
numeric vector of values between 0 and 1 indicating the confidence level of the
prediction interval. The default value is conf.level=0.95.
character string specifying the method for constructing the tolerance interval.
The possible values are "exact" (the default) and "wald.wolfowitz"
(the Wald-Wolfowitz approximation).
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.
positive integer greater than 1 specifying the maximum possible sample size.
The default value is n.max=5000.
numeric scalar indicating the tolerance to use in the uniroot
search algorithm. The default value is tol=1e-7.
positive integer indicating the maximum number of iterations to use in the
uniroot search algorithm. The default value is
maxiter=1000.
If the arguments half.width, sigma.hat, coverage, and
conf.level are not all the same length, they are replicated to be the same
length as the length of the longest argument.
The help files for tolIntNorm and tolIntNormK
give formulas for a two-sided tolerance interval based on the sample size, the
observed sample mean and sample standard deviation, and specified confidence level
and coverage. Specifically, the two-sided tolerance interval is given by:
$$[\bar{x} - Ks, \bar{x} + Ks] \;\;\;\;\;\; (1)$$
where \(\bar{x}\) denotes the sample mean:
$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \;\;\;\;\;\; (2)$$
\(s\) denotes the sample standard deviation:
$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (3)$$
and \(K\) denotes a constant that depends on the sample size \(n\), the
confidence level, and the coverage (see the help file for
tolIntNormK). Thus, the half-width of the tolerance interval is
given by:
$$HW = Ks \;\;\;\;\;\; (4)$$
The function tolIntNormN uses the uniroot search algorithm to
determine the sample size for specified values of the half-width, sample
standard deviation, coverage, and confidence level. Note that unlike a
confidence interval, the half-width of a tolerance interval does not
approach 0 as the sample size increases.
numeric vector of sample sizes.
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.
# Look at how the required sample size for a tolerance interval increases
# with increasing coverage:
seq(0.5, 0.9, by = 0.1)
#> [1] 0.5 0.6 0.7 0.8 0.9
#[1] 0.5 0.6 0.7 0.8 0.9
tolIntNormN(half.width = 3, coverage = seq(0.5, 0.9, by = 0.1))
#> [1] 4 4 5 6 9
#[1] 4 4 5 6 9
#----------
# Look at how the required sample size for a tolerance interval decreases
# with increasing half-width:
3:6
#> [1] 3 4 5 6
#[1] 3 4 5 6
tolIntNormN(half.width = 3:6)
#> [1] 15 8 6 5
#[1] 15 8 6 5
tolIntNormN(3:6, round = FALSE)
#> [1] 14.199735 7.022572 5.092374 4.214371
#[1] 14.199735 7.022572 5.092374 4.214371
#----------
# Look at how the required sample size for a tolerance interval increases
# with increasing estimated standard deviation for a fixed half-width:
seq(0.5, 2, by = 0.5)
#> [1] 0.5 1.0 1.5 2.0
#[1] 0.5 1.0 1.5 2.0
tolIntNormN(half.width = 4, sigma.hat = seq(0.5, 2, by = 0.5))
#> [1] 4 8 24 3437
#[1] 4 8 24 3437
#----------
# Look at how the required sample size for a tolerance interval increases
# with increasing confidence level for a fixed half-width:
seq(0.5, 0.9, by = 0.1)
#> [1] 0.5 0.6 0.7 0.8 0.9
#[1] 0.5 0.6 0.7 0.8 0.9
tolIntNormN(half.width = 3, conf.level = seq(0.5, 0.9, by = 0.1))
#> [1] 3 4 5 7 11
#[1] 3 4 5 7 11
#==========
# 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 compute required sample sizes for
# various half-widths on the log-scale.
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
tolIntNormN(half.width = c(4, 2, 1), sigma.hat = sigma.hat)
#> Warning: Value of 'half.width' is too smallfor element3. Try increasing the value of 'n.max'.
#> [1] 4 12 NA
#[1] 4 12 NA
#Warning message:
#In tolIntNormN(half.width = c(4, 2, 1), sigma.hat = sigma.hat) :
# Value of 'half.width' is too smallfor element3.
# Try increasing the value of 'n.max'.
# NOTE: We cannot achieve a half-width of 1 for the given value of
# sigma.hat for a tolerance interval with 95% coverage and
# 95% confidence. The default value of n.max is 5000, but in fact,
# even with a million observations the half width is greater than 1.
tolIntNormHalfWidth(n = 1e6, sigma.hat = sigma.hat)
#> [1] 1.232095
#[1] 1.232095
#==========
# Clean up
#---------
rm(summary.stats, sigma.hat)