tolIntGamma.RdConstruct a \(\beta\)-content or \(\beta\)-expectation tolerance interval for a gamma distribution.
tolIntGamma(x, coverage = 0.95, cov.type = "content",
ti.type = "two-sided", conf.level = 0.95, method = "exact",
est.method = "mle", normal.approx.transform = "kulkarni.powar")
tolIntGammaAlt(x, coverage = 0.95, cov.type = "content",
ti.type = "two-sided", conf.level = 0.95, method = "exact",
est.method = "mle", normal.approx.transform = "kulkarni.powar")numeric vector of non-negative observations. Missing (NA), undefined (NaN), and
infinite (Inf, -Inf) values are allowed but will be removed.
a scalar between 0 and 1 indicating the desired coverage of the tolerance interval.
The default value is coverage=0.95. If cov.type="expectation",
this argument is ignored.
character string specifying the coverage type for the tolerance interval.
The possible values are "content" (\(\beta\)-content; the default), and
"expectation" (\(\beta\)-expectation). See the DETAILS section for more
information.
character string indicating what kind of tolerance interval to compute.
The possible values are "two-sided" (the default), "lower", and
"upper".
a scalar between 0 and 1 indicating the confidence level associated with the tolerance
interval. The default value is conf.level=0.95.
for the case of a two-sided tolerance interval, a character string specifying the
method for constructing the two-sided normal distribution tolerance interval using
the transformed data. This argument is ignored if ti.type="lower" or
ti.type="upper". The possible values are "exact" (the default) and "wald.wolfowitz" (the Wald-Wolfowitz approximation). See the DETAILS section
of the help file for tolIntNorm for more information.
character string specifying the method of estimation for the shape and scale
distribution parameters. The possible values are
"mle" (maximum likelihood; the default),
"bcmle" (bias-corrected mle), "mme" (method of moments), and
"mmue" (method of moments based on the unbiased estimator of variance).
See the DETAILS section of the help file for egamma for more information.
character string indicating which power transformation to use.
Possible values are "kulkarni.powar" (the default), "cube.root", and "fourth.root". See the DETAILS section for more informaiton.
The function tolIntGamma returns a tolerance interval as well as
estimates of the shape and scale parameters.
The function tolIntGammaAlt returns a tolerance interval as well as
estimates of the mean and coefficient of variation.
The tolerance interval is computed by 1) using a power transformation on the original
data to induce approximate normality, 2) using tolIntNorm to compute
the tolerance interval, and then 3) back-transforming the interval to create a tolerance
interval on the original scale. (Krishnamoorthy et al., 2008).
The value normal.approx.transform="cube.root" uses
the cube root transformation suggested by Wilson and Hilferty (1931) and used by
Krishnamoorthy et al. (2008) and Singh et al. (2010b), and the value
normal.approx.transform="fourth.root" uses the fourth root transformation suggested
by Hawkins and Wixley (1986) and used by Singh et al. (2010b).
The default value normal.approx.transform="kulkarni.powar"
uses the "Optimum Power Normal Approximation Method" of Kulkarni and Powar (2010).
The "optimum" power \(p\) is determined by:
| \(p = -0.0705 - 0.178 \, shape + 0.475 \, \sqrt{shape}\) | if \(shape \le 1.5\) |
| \(p = 0.246\) | if \(shape > 1.5\) |
where \(shape\) denotes the estimate of the shape parameter. Although
Kulkarni and Powar (2010) use the maximum likelihood estimate of shape to
determine the power \(p\), for the functions tolIntGamma and tolIntGammaAlt the power \(p\) is based on
whatever estimate of shape is used (e.g., est.method="mle", est.method="bcmle", etc.).
A list of class "estimate" containing the estimated parameters,
the tolerance interval, and other information. See estimate.object
for details.
In addition to the usual components contained in an object of class
"estimate", the returned value also includes an additional
component within the "interval" component:
the value of the power used to transform the original data to approximate normality.
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Lewis Publishers, Boca Raton.
Draper, N., and H. Smith. (1998). Applied Regression Analysis. Third Edition. John Wiley and Sons, New York.
Ellison, B.E. (1964). On Two-Sided Tolerance Intervals for a Normal Distribution. Annals of Mathematical Statistics 35, 762-772.
Evans, M., N. Hastings, and B. Peacock. (1993). Statistical Distributions. Second Edition. John Wiley and Sons, New York, Chapter 18.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.
Guttman, I. (1970). Statistical Tolerance Regions: Classical and Bayesian. Hafner Publishing Co., Darien, CT.
Hahn, G.J. (1970b). Statistical Intervals for a Normal Population, Part I: Tables, Examples and Applications. Journal of Quality Technology 2(3), 115-125.
Hahn, G.J. (1970c). Statistical Intervals for a Normal Population, Part II: Formulas, Assumptions, Some Derivations. Journal of Quality Technology 2(4), 195-206.
Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, New York.
Hawkins, D. M., and R.A.J. Wixley. (1986). A Note on the Transformation of Chi-Squared Variables to Normality. The American Statistician, 40, 296–298.
Johnson, N.L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York, Chapter 17.
Krishnamoorthy K., T. Mathew, and S. Mukherjee. (2008). Normal-Based Methods for a Gamma Distribution: Prediction and Tolerance Intervals and Stress-Strength Reliability. Technometrics, 50(1), 69–78.
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
Kulkarni, H.V., and S.K. Powar. (2010). A New Method for Interval Estimation of the Mean of the Gamma Distribution. Lifetime Data Analysis, 16, 431–447.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton.
Singh, A., A.K. Singh, and R.J. Iaci. (2002). Estimation of the Exposure Point Concentration Term Using a Gamma Distribution. EPA/600/R-02/084. October 2002. Technology Support Center for Monitoring and Site Characterization, Office of Research and Development, Office of Solid Waste and Emergency Response, U.S. Environmental Protection Agency, Washington, D.C.
Singh, A., R. Maichle, and N. Armbya. (2010a). ProUCL Version 4.1.00 User Guide (Draft). EPA/600/R-07/041, May 2010. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.
Singh, A., N. Armbya, and A. Singh. (2010b). ProUCL Version 4.1.00 Technical Guide (Draft). EPA/600/R-07/041, May 2010. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.
Wilson, E.B., and M.M. Hilferty. (1931). The Distribution of Chi-Squares. Proceedings of the National Academy of Sciences, 17, 684–688.
USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.
USEPA. (2010). Errata Sheet - March 2009 Unified Guidance. EPA 530/R-09-007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division. U.S. Environmental Protection Agency, Washington, D.C.
The gamma distribution takes values on the positive real line. Special cases of the gamma are the exponential distribution and the chi-square distributions. Applications of the gamma include life testing, statistical ecology, queuing theory, inventory control, and precipitation processes. A gamma distribution starts to resemble a normal distribution as the shape parameter a tends to infinity.
Some EPA guidance documents (e.g., Singh et al., 2002; Singh et al., 2010a,b) strongly recommend against using a lognormal model for environmental data and recommend trying a gamma distribuiton instead.
Tolerance intervals have long been applied to quality control and life testing problems (Hahn, 1970b,c; Hahn and Meeker, 1991). References that discuss tolerance intervals in the context of environmental monitoring include: Berthouex and Brown (2002, Chapter 21), Gibbons et al. (2009), Millard and Neerchal (2001, Chapter 6), Singh et al. (2010b), and USEPA (2009).
It is possible for the lower tolerance limit based on the transformed data to be less than 0. In this case, the lower tolerance limit on the original scale is set to 0 and a warning is issued stating that the normal approximation is not accurate in this case.
# Generate 20 observations from a gamma distribution with parameters
# shape=3 and scale=2, then create a tolerance interval.
# (Note: the call to set.seed simply allows you to reproduce this
# example.)
set.seed(250)
dat <- rgamma(20, shape = 3, scale = 2)
tolIntGamma(dat)
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Gamma
#>
#> Estimated Parameter(s): shape = 2.203862
#> scale = 2.174928
#>
#> Estimation Method: MLE
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Tolerance Interval Coverage: 95%
#>
#> Coverage Type: content
#>
#> Tolerance Interval Method: Exact using
#> Kulkarni & Powar (2010)
#> transformation to Normality
#> based on MLE of 'shape'
#>
#> Normal Transform Power: 0.246
#>
#> Tolerance Interval Type: two-sided
#>
#> Confidence Level: 95%
#>
#> Tolerance Interval: LTL = 0.2340438
#> UTL = 21.2996464
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 2.203862
# scale = 2.174928
#
#Estimation Method: mle
#
#Data: dat
#
#Sample Size: 20
#
#Tolerance Interval Coverage: 95%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on mle of 'shape'
#
#Tolerance Interval Type: two-sided
#
#Confidence Level: 95%
#
#Number of Future Observations: 1
#
#Tolerance Interval: LTL = 0.2340438
# UTL = 21.2996464
#--------------------------------------------------------------------
# Using the same data as in the previous example, create an upper
# one-sided tolerance interval and use the bias-corrected estimate of
# shape.
tolIntGamma(dat, ti.type = "upper", est.method = "bcmle")
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Gamma
#>
#> Estimated Parameter(s): shape = 1.906616
#> scale = 2.514005
#>
#> Estimation Method: Bias-Corrected MLE
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Tolerance Interval Coverage: 95%
#>
#> Coverage Type: content
#>
#> Tolerance Interval Method: Exact using
#> Kulkarni & Powar (2010)
#> transformation to Normality
#> based on Bias-Corrected MLE of 'shape'
#>
#> Normal Transform Power: 0.246
#>
#> Tolerance Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Tolerance Interval: LTL = 0.00000
#> UTL = 17.72107
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 1.906616
# scale = 2.514005
#
#Estimation Method: bcmle
#
#Data: dat
#
#Sample Size: 20
#
#Tolerance Interval Coverage: 95%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on bcmle of 'shape'
#
#Tolerance Interval Type: upper
#
#Confidence Level: 95%
#
#Tolerance Interval: LTL = 0.00000
# UTL = 17.72107
#----------
# Clean up
rm(dat)
#--------------------------------------------------------------------
# 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. Here we will use the same chrysene data but assume a
# gamma distribution.
attach(EPA.09.Ex.17.3.chrysene.df)
Chrysene <- Chrysene.ppb[Well.type == "Background"]
#----------
# First perform a goodness-of-fit test for a gamma distribution
gofTest(Chrysene, dist = "gamma")
#>
#> Results of Goodness-of-Fit Test
#> -------------------------------
#>
#> Test Method: Shapiro-Wilk GOF Based on
#> Chen & Balakrisnan (1995)
#>
#> Hypothesized Distribution: Gamma
#>
#> Estimated Parameter(s): shape = 2.806929
#> scale = 5.286026
#>
#> Estimation Method: MLE
#>
#> Data: Chrysene
#>
#> Sample Size: 8
#>
#> Test Statistic: W = 0.9156306
#>
#> Test Statistic Parameter: n = 8
#>
#> P-value: 0.3954223
#>
#> Alternative Hypothesis: True cdf does not equal the
#> Gamma Distribution.
#Results of Goodness-of-Fit Test
#-------------------------------
#
#Test Method: Shapiro-Wilk GOF Based on
# Chen & Balakrisnan (1995)
#
#Hypothesized Distribution: Gamma
#
#Estimated Parameter(s): shape = 2.806929
# scale = 5.286026
#
#Estimation Method: mle
#
#Data: Chrysene
#
#Sample Size: 8
#
#Test Statistic: W = 0.9156306
#
#Test Statistic Parameter: n = 8
#
#P-value: 0.3954223
#
#Alternative Hypothesis: True cdf does not equal the
# Gamma Distribution.
#----------
# Now compute the upper tolerance limit
tolIntGamma(Chrysene, ti.type = "upper", coverage = 0.95,
conf.level = 0.95)
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Gamma
#>
#> Estimated Parameter(s): shape = 2.806929
#> scale = 5.286026
#>
#> Estimation Method: MLE
#>
#> Data: Chrysene
#>
#> Sample Size: 8
#>
#> Tolerance Interval Coverage: 95%
#>
#> Coverage Type: content
#>
#> Tolerance Interval Method: Exact using
#> Kulkarni & Powar (2010)
#> transformation to Normality
#> based on MLE of 'shape'
#>
#> Normal Transform Power: 0.246
#>
#> Tolerance Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Tolerance Interval: LTL = 0.00000
#> UTL = 69.32425
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 2.806929
# scale = 5.286026
#
#Estimation Method: mle
#
#Data: Chrysene
#
#Sample Size: 8
#
#Tolerance Interval Coverage: 95%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on mle of 'shape'
#
#Tolerance Interval Type: upper
#
#Confidence Level: 95%
#
#Tolerance Interval: LTL = 0.00000
# UTL = 69.32425
#----------
# Compare this upper tolerance limit of 69 ppb to the upper tolerance limit
# assuming a lognormal distribution.
tolIntLnorm(Chrysene, ti.type = "upper", coverage = 0.95,
conf.level = 0.95)$interval$limits["UTL"]
#> UTL
#> 90.9247
# UTL
#90.9247
#----------
# Clean up
rm(Chrysene)
detach("EPA.09.Ex.17.3.chrysene.df")
#--------------------------------------------------------------------
# Reproduce some of the example on page 73 of
# Krishnamoorthy et al. (2008), which uses alkalinity concentrations
# reported in Gibbons (1994) and Gibbons et al. (2009) to construct
# two-sided and one-sided upper tolerance limits for various values
# of coverage using a 95% confidence level.
tolIntGamma(Gibbons.et.al.09.Alkilinity.vec, ti.type = "upper",
coverage = 0.9, normal.approx.transform = "cube.root")
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Gamma
#>
#> Estimated Parameter(s): shape = 9.375013
#> scale = 6.202461
#>
#> Estimation Method: MLE
#>
#> Data: Gibbons.et.al.09.Alkilinity.vec
#>
#> Sample Size: 27
#>
#> Tolerance Interval Coverage: 90%
#>
#> Coverage Type: content
#>
#> Tolerance Interval Method: Exact using
#> Wilson & Hilferty (1931) cube-root
#> transformation to Normality
#>
#> Normal Transform Power: 0.3333333
#>
#> Tolerance Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Tolerance Interval: LTL = 0.00000
#> UTL = 97.70502
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 9.375013
# scale = 6.202461
#
#Estimation Method: mle
#
#Data: Gibbons.et.al.09.Alkilinity.vec
#
#Sample Size: 27
#
#Tolerance Interval Coverage: 90%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact using
# Wilson & Hilferty (1931) cube-root
# transformation to Normality
#
#Tolerance Interval Type: upper
#
#Confidence Level: 95%
#
#Tolerance Interval: LTL = 0.00000
# UTL = 97.70502
tolIntGamma(Gibbons.et.al.09.Alkilinity.vec,
coverage = 0.99, normal.approx.transform = "cube.root")
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Gamma
#>
#> Estimated Parameter(s): shape = 9.375013
#> scale = 6.202461
#>
#> Estimation Method: MLE
#>
#> Data: Gibbons.et.al.09.Alkilinity.vec
#>
#> Sample Size: 27
#>
#> Tolerance Interval Coverage: 99%
#>
#> Coverage Type: content
#>
#> Tolerance Interval Method: Exact using
#> Wilson & Hilferty (1931) cube-root
#> transformation to Normality
#>
#> Normal Transform Power: 0.3333333
#>
#> Tolerance Interval Type: two-sided
#>
#> Confidence Level: 95%
#>
#> Tolerance Interval: LTL = 13.14318
#> UTL = 148.43876
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 9.375013
# scale = 6.202461
#
#Estimation Method: mle
#
#Data: Gibbons.et.al.09.Alkilinity.vec
#
#Sample Size: 27
#
#Tolerance Interval Coverage: 99%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact using
# Wilson & Hilferty (1931) cube-root
# transformation to Normality
#
#Tolerance Interval Type: two-sided
#
#Confidence Level: 95%
#
#Tolerance Interval: LTL = 13.14318
# UTL = 148.43876