Simultaneous Prediction Interval for a Gamma Distribution
Estimate the shape and scale parameters for a gamma distribution, or estimate the mean and coefficient of variation for a gamma distribution (alternative parameterization), and construct a simultaneous prediction interval for the next r sampling occasions, based on one of three possible rules: k-of-m, California, or Modified California.
predIntGammaSimultaneous(x, n.transmean = 1, k = 1, m = 2, r = 1,
rule = "k.of.m", delta.over.sigma = 0, pi.type = "upper", conf.level = 0.95,
K.tol = 1e-07, est.method = "mle", normal.approx.transform = "kulkarni.powar")
predIntGammaAltSimultaneous(x, n.transmean = 1, k = 1, m = 2, r = 1,
rule = "k.of.m", delta.over.sigma = 0, pi.type = "upper", conf.level = 0.95,
K.tol = 1e-07, est.method = "mle", normal.approx.transform = "kulkarni.powar")x |
numeric vector of non-negative observations. Missing ( |
n.transmean |
positive integer specifying the sample size associated with future transformed
means (see the DETAILS section for an explanation of what the transformation is).
The default value is |
k |
for the k-of-m rule ( |
m |
positive integer specifying the maximum number of future observations (or
transformed means) 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
|
delta.over.sigma |
numeric scalar indicating the ratio Δ/σ. The quantity
Δ (delta) denotes the difference between the mean of the population
(on the transformed scale) that was sampled to construct the prediction interval,
and the mean of the population (on the transformed scale) that will be sampled to
produce the future observations. The quantity σ (sigma) denotes the
population standard deviation (on the transformed scale) for both populations.
See the DETAILS section below for more information. The default value is
|
pi.type |
character string indicating what kind of prediction interval to compute.
The possible values are |
conf.level |
a scalar between 0 and 1 indicating the confidence level of the prediction interval.
The default value is |
K.tol |
numeric scalar indicating the tolerance to use in the nonlinear search algorithm to
compute K. The default value is |
est.method |
character string specifying the method of estimation for the shape and scale
distribution parameters. The possible values are
|
normal.approx.transform |
character string indicating which power transformation to use.
Possible values are |
The function predIntGammaSimultaneous returns a simultaneous prediction
interval as well as estimates of the shape and scale parameters.
The function predIntGammaAltSimultaneous returns a simultaneous prediction
interval as well as estimates of the mean and coefficient of variation.
Following Krishnamoorthy et al. (2008), the simultaneous prediction interval is computed by:
using a power transformation on the original data to induce approximate normality,
calling predIntNormSimultaneous with the transformed data to
compute the simultaneous prediction interval, and then
back-transforming the interval to create a simultaneous prediction interval on the original scale.
The argument normal.approx.transform determines which transformation is used.
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 \, √{shape} | if shape ≤ 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 predIntGammaSimultaneous and predIntGammaAltSimultaneous the power
p is based on whatever estimate of shape is used
(e.g., est.method="mle", est.method="bcmle", etc.).
When the argument n.transmean is larger than 1 (i.e., you are
constructing a prediction interval for future means, not just single
observations), in order to properly compare a future mean with the
prediction limits, you must follow these steps:
Take the observations that will be used to compute the mean and
transform them by raising them to the power given by the value in the
component interval$normal.transform.power (see the section VALUE below).
Compute the mean of the transformed observations.
Take the mean computed in step 2 above and raise it to the inverse of the power originally used to transform the observations.
A list of class "estimate" containing the estimated parameters,
the simultaneous prediction 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 two additional
components within the "interval" component:
n.transmean |
the value of |
normal.transform.power |
the value of the power used to transform the original data to approximate normality. |
It is possible for the lower prediction limit based on the transformed data to be less than 0. In this case, the lower prediction 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.
The Gamma Distribution
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.
Motivation
Prediction and tolerance intervals have long been applied to quality control and
life testing problems (Hahn, 1970b,c; Hahn and Nelson, 1973). In the context of
environmental statistics, prediction intervals are useful for analyzing data from
groundwater detection monitoring programs at hazardous and solid waste facilities.
One of the main statistical problems that plague groundwater monitoring programs at hazardous and solid waste facilities is the requirement of testing several wells and several constituents at each well on each sampling occasion. This is an obvious multiple comparisons problem, and the naive approach of using a standard t-test at a conventional α-level (e.g., 0.05 or 0.01) for each test leads to a very high probability of at least one significant result on each sampling occasion, when in fact no contamination has occurred. This problem was pointed out years ago by Millard (1987) and others.
Davis and McNichols (1987) proposed simultaneous prediction intervals as a way of controlling the facility-wide false positive rate (FWFPR) while maintaining adequate power to detect contamination in the groundwater. Because of the ubiquitous presence of spatial variability, it is usually best to use simultaneous prediction intervals at each well (Davis, 1998a). That is, by constructing prediction intervals based on background (pre-landfill) data on each well, and comparing future observations at a well to the prediction interval for that particular well. In each of these cases, the individual α-level at each well is equal to the FWFRP divided by the product of the number of wells and constituents.
Often, observations at downgradient wells are not available prior to the construction and operation of the landfill. In this case, upgradient well data can be combined to create a background prediction interval, and observations at each downgradient well can be compared to this prediction interval. If spatial variability is present and a major source of variation, however, this method is not really valid (Davis, 1994; Davis, 1998a).
Chapter 19 of USEPA (2009) contains an extensive discussion of using the 1-of-m rule and the Modified California rule.
Chapters 1 and 3 of Gibbons et al. (2009) discuss simultaneous prediction intervals
for the normal and lognormal distributions, respectively.
The k-of-m Rule
For the k-of-m rule, Davis and McNichols (1987) give tables with
“optimal” choices of k (in terms of best power for a given overall
confidence level) for selected values of m, r, and n. They found
that the optimal ratios of k to m (i.e., k/m) are generally small,
in the range of 15-50%.
The California Rule
The California rule was mandated in that state for groundwater monitoring at waste
disposal facilities when resampling verification is part of the statistical program
(Barclay's Code of California Regulations, 1991). The California code mandates a
“California” rule with m ≥ 3. The motivation for this rule may have
been a desire to have a majority of the observations in bounds (Davis, 1998a). For
example, for a k-of-m rule with k=1 and m=3, a monitoring
location will pass if the first observation is out of bounds, the second resample
is out of bounds, but the last resample is in bounds, so that 2 out of 3 observations
are out of bounds. For the California rule with m=3, either the first
observation must be in bounds, or the next 2 observations must be in bounds in order
for the monitoring location to pass.
Davis (1998a) states that if the FWFPR is kept constant, then the California rule
offers little increased power compared to the k-of-m rule, and can
actually decrease the power of detecting contamination.
The Modified California Rule
The Modified California Rule was proposed as a compromise between a 1-of-m
rule and the California rule. For a given FWFPR, the Modified California rule
achieves better power than the California rule, and still requires at least as many
observations in bounds as out of bounds, unlike a 1-of-m rule.
Different Notations Between Different References
For the k-of-m rule described in this help file, both
Davis and McNichols (1987) and USEPA (2009, Chapter 19) use the variable
p instead of k to represent the minimum number
of future observations the interval should contain on each of the r sampling
occasions.
Gibbons et al. (2009, Chapter 1) presents extensive lists of the value of K for both k-of-m rules and California rules. Gibbons et al.'s notation reverses the meaning of k and r compared to the notation used in this help file. That is, in Gibbons et al.'s notation, k represents the number of future sampling occasions or monitoring wells, and r represents the minimum number of observations the interval should contain on each sampling occasion.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Barclay's California Code of Regulations. (1991). Title 22, Section 66264.97 [concerning hazardous waste facilities] and Title 23, Section 2550.7(e)(8) [concerning solid waste facilities]. Barclay's Law Publishers, San Francisco, CA.
Davis, C.B. (1998a). Ground-Water Statistics \& Regulations: Principles, Progress and Problems. Second Edition. Environmetrics \& Statistics Limited, Henderson, NV.
Davis, C.B. (1998b). Personal Communication, September 3, 1998.
Davis, C.B., and R.J. McNichols. (1987). One-sided Intervals for at Least p of m Observations from a Lognormal Population on Each of r Future Occasions. Technometrics 29, 359–370.
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.
Fertig, K.W., and N.R. Mann. (1977). One-Sided Prediction Intervals for at Least p Out of m Future Observations From a Lognormal Population. Technometrics 19, 167–177.
Hahn, G.J. (1969). Factors for Calculating Two-Sided Prediction Intervals for Samples from a Lognormal Distribution. Journal of the American Statistical Association 64(327), 878-898.
Hahn, G.J. (1970a). Additional Factors for Calculating Prediction Intervals for Samples from a Lognormal Distribution. Journal of the American Statistical Association 65(332), 1668-1676.
Hahn, G.J. (1970b). Statistical Intervals for a Lognormal Population, Part I: Tables, Examples and Applications. Journal of Quality Technology 2(3), 115-125.
Hahn, G.J. (1970c). Statistical Intervals for a Lognormal 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.
Hahn, G., and W. Nelson. (1973). A Survey of Prediction Intervals and Their Applications. Journal of Quality Technology 5, 178-188.
Hall, I.J., and R.R. Prairie. (1973). One-Sided Prediction Intervals to Contain at Least m Out of k Future Observations. Technometrics 15, 897–914.
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. (1987). Environmental Monitoring, Statistics, and the Law: Room for Improvement (with Comment). The American Statistician 41(4), 249–259.
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.
# Generate 8 observations from a gamma distribution with parameters
# mean=10 and cv=1, then use predIntGammaAltSimultaneous to estimate the
# mean and coefficient of variation of the true distribution and construct an
# upper 95% prediction interval to contain at least 1 out of the next
# 3 observations.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(479)
dat <- rgammaAlt(8, mean = 10, cv = 1)
predIntGammaAltSimultaneous(dat, k = 1, m = 3)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): mean = 13.875825
# cv = 1.049504
#
#Estimation Method: MLE
#
#Data: dat
#
#Sample Size: 8
#
#Prediction Interval Method: exact using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on MLE of 'shape'
#
#Normal Transform Power: 0.2204908
#
#Prediction Interval Type: upper
#
#Confidence Level: 95%
#
#Minimum Number of
#Future Observations
#Interval Should Contain: 1
#
#Total Number of
#Future Observations: 3
#
#Prediction Interval: LPL = 0.00000
# UPL = 15.87101
#----------
# Compare the 95% 1-of-3 upper prediction limit to the California and
# Modified California upper prediction limits. Note that the upper
# prediction limit for the Modified California rule is between the limit
# for the 1-of-3 rule and the limit for the California rule.
predIntGammaAltSimultaneous(dat, k = 1, m = 3)$interval$limits["UPL"]
# UPL
#15.87101
predIntGammaAltSimultaneous(dat, m = 3, rule = "CA")$interval$limits["UPL"]
# UPL
#34.11499
predIntGammaAltSimultaneous(dat, rule = "Modified.CA")$interval$limits["UPL"]
# UPL
#22.58809
#----------
# Show how the upper 95% simultaneous prediction limit increases
# as the number of future sampling occasions r increases.
# Here, we'll use the 1-of-3 rule.
predIntGammaAltSimultaneous(dat, k = 1, m = 3)$interval$limits["UPL"]
# UPL
#15.87101
predIntGammaAltSimultaneous(dat, k = 1, m = 3, r = 10)$interval$limits["UPL"]
# UPL
#37.86825
#----------
# Compare the upper simultaneous prediction limit for the 1-of-3 rule
# based on individual observations versus based on transformed means of
# order 4.
predIntGammaAltSimultaneous(dat, k = 1, m = 3)$interval$limits["UPL"]
# UPL
#15.87101
predIntGammaAltSimultaneous(dat, n.transmean = 4, k = 1,
m = 3)$interval$limits["UPL"]
# UPL
#14.76528
#==========
# Example 19-1 of USEPA (2009, p. 19-17) shows how to compute an
# upper simultaneous prediction limit for the 1-of-3 rule for
# r = 2 future sampling occasions. The data for this example are
# stored in EPA.09.Ex.19.1.sulfate.df.
# We will pool data from 4 background wells that were sampled on
# a number of different occasions, giving us a sample size of
# n = 25 to use to construct the prediction limit.
# There are 50 compliance wells and we will monitor 10 different
# constituents at each well at each of the r=2 future sampling
# occasions. To determine the confidence level we require for
# the simultaneous prediction interval, USEPA (2009) recommends
# setting the individual Type I Error level at each well to
# 1 - (1 - SWFPR)^(1 / (Number of Constituents * Number of Wells))
# which translates to setting the confidence limit to
# (1 - SWFPR)^(1 / (Number of Constituents * Number of Wells))
# where SWFPR = site-wide false positive rate. For this example, we
# will set SWFPR = 0.1. Thus, the confidence level is given by:
nc <- 10
nw <- 50
SWFPR <- 0.1
conf.level <- (1 - SWFPR)^(1 / (nc * nw))
conf.level
#[1] 0.9997893
#----------
# Look at the data:
names(EPA.09.Ex.19.1.sulfate.df)
#[1] "Well" "Month" "Day"
#[4] "Year" "Date" "Sulfate.mg.per.l"
#[7] "log.Sulfate.mg.per.l"
EPA.09.Ex.19.1.sulfate.df[,
c("Well", "Date", "Sulfate.mg.per.l", "log.Sulfate.mg.per.l")]
# Well Date Sulfate.mg.per.l log.Sulfate.mg.per.l
#1 GW-01 1999-07-08 63.0 4.143135
#2 GW-01 1999-09-12 51.0 3.931826
#3 GW-01 1999-10-16 60.0 4.094345
#4 GW-01 1999-11-02 86.0 4.454347
#5 GW-04 1999-07-09 104.0 4.644391
#6 GW-04 1999-09-14 102.0 4.624973
#7 GW-04 1999-10-12 84.0 4.430817
#8 GW-04 1999-11-15 72.0 4.276666
#9 GW-08 1997-10-12 31.0 3.433987
#10 GW-08 1997-11-16 84.0 4.430817
#11 GW-08 1998-01-28 65.0 4.174387
#12 GW-08 1999-04-20 41.0 3.713572
#13 GW-08 2002-06-04 51.8 3.947390
#14 GW-08 2002-09-16 57.5 4.051785
#15 GW-08 2002-12-02 66.8 4.201703
#16 GW-08 2003-03-24 87.1 4.467057
#17 GW-09 1997-10-16 59.0 4.077537
#18 GW-09 1998-01-28 85.0 4.442651
#19 GW-09 1998-04-12 75.0 4.317488
#20 GW-09 1998-07-12 99.0 4.595120
#21 GW-09 2000-01-30 75.8 4.328098
#22 GW-09 2000-04-24 82.5 4.412798
#23 GW-09 2000-10-24 85.5 4.448516
#24 GW-09 2002-12-01 188.0 5.236442
#25 GW-09 2003-03-24 150.0 5.010635
# The EPA guidance document constructs the upper simultaneous
# prediction limit for the 1-of-3 plan assuming a lognormal
# distribution for the sulfate data. Here we will compare
# the value of the limit based on assuming a lognormal distribution
# versus assuming a gamma distribution.
Sulfate <- EPA.09.Ex.19.1.sulfate.df$Sulfate.mg.per.l
pred.int.list.lnorm <-
predIntLnormSimultaneous(x = Sulfate, k = 1, m = 3, r = 2,
rule = "k.of.m", pi.type = "upper", conf.level = conf.level)
pred.int.list.gamma <-
predIntGammaSimultaneous(x = Sulfate, k = 1, m = 3, r = 2,
rule = "k.of.m", pi.type = "upper", conf.level = conf.level)
pred.int.list.lnorm$interval$limits["UPL"]
# UPL
#159.5497
pred.int.list.gamma$interval$limits["UPL"]
# UPL
#153.3232
#==========
# Cleanup
#--------
rm(dat, nc, nw, SWFPR, conf.level, Sulfate, pred.int.list.lnorm,
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