Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples
Construct a prediction interval for the next k observations or next set of k transformed means for a gamma distribution.
1 2 3 4 5 6 7  predIntGamma(x, n.transmean = 1, k = 1, method = "Bonferroni",
pi.type = "twosided", conf.level = 0.95, est.method = "mle",
normal.approx.transform = "kulkarni.powar")
predIntGammaAlt(x, n.transmean = 1, k = 1, method = "Bonferroni",
pi.type = "twosided", conf.level = 0.95, est.method = "mle",
normal.approx.transform = "kulkarni.powar")

x 
numeric vector of nonnegative observations. Missing ( 
n.transmean 
positive integer specifying the sample size associated with the 
k 
positive integer specifying the number of future observations or means
the prediction interval should contain with confidence level 
method 
character string specifying the method to use if the number of future observations
or averages ( 
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 associated with the prediction
interval. 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 
If x
contains any missing (NA
), undefined (NaN
) or
infinite (Inf
, Inf
) values, they will be removed prior to
performing the estimation.
The function predIntGamma
returns a prediction interval as well as
estimates of the shape and scale parameters.
The function predIntGammaAlt
returns a prediction interval as well as
estimates of the mean and coefficient of variation.
Following Krishnamoorthy et al. (2008), the prediction interval is computed by:
using a power transformation on the original data to induce approximate normality,
calling predIntNorm
with the transformed data to
compute the prediction interval, and then
backtransforming the interval to create a 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
predIntGamma
and predIntGammaAlt
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 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 takes values on the positive real line. Special cases of the gamma are the exponential distribution and the chisquare 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.
Prediction intervals have long been applied to quality control and life testing problems (Hahn, 1970b,c; Hahn and Nelson, 1973), and are often discussed in the context of linear regression (Draper and Smith, 1998; Zar, 2010). Prediction intervals are useful for analyzing data from groundwater detection monitoring programs at hazardous and solid waste facilities. References that discuss prediction 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).
Steven P. Millard ([email protected])
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.
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.
Hawkins, D. M., and R.A.J. Wixley. (1986). A Note on the Transformation of ChiSquared 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). NormalBased Methods for a Gamma Distribution: Prediction and Tolerance Intervals and StressStrength 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 SPLUS. 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/R02/084. 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/R07/041, 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/R07/041, 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 ChiSquares. 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/R09007, 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/R09007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division. U.S. Environmental Protection Agency, Washington, D.C.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. PrenticeHall, Upper Saddle River, New Jersey.
GammaDist
, GammaAlt
, estimate.object
,
egamma
, predIntNorm
,
tolIntGamma
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157  # Generate 20 observations from a gamma distribution with parameters
# shape=3 and scale=2, then create a prediciton interval for the
# next observation.
# (Note: the call to set.seed simply allows you to reproduce this
# example.)
set.seed(250)
dat < rgamma(20, shape = 3, scale = 2)
predIntGamma(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
#
#Prediction Interval Method: exact using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on mle of 'shape'
#
#Normal Transform Power: 0.246
#
#Prediction Interval Type: twosided
#
#Confidence Level: 95%
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0.5371569
# UPL = 15.5313783
#
# Using the same data as in the previous example, create an upper
# onesided prediction interval for the next three averages of
# order 2 (i.e., each mean is based on 2 future observations), and
# use the biascorrected estimate of shape.
pred.list < predIntGamma(dat, n.transmean = 2, k = 3,
pi.type = "upper", est.method = "bcmle")
pred.list
#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
#
#Prediction Interval Method: Bonferroni using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on bcmle of 'shape'
#
#Normal Transform Power: 0.246
#
#Prediction Interval Type: upper
#
#Confidence Level: 95%
#
#Number of Future
#Transformed Means: 3
#
#Sample Size for
#Transformed Means: 2
#
#Prediction Interval: LPL = 0.00000
# UPL = 12.17404
#
# Continuing with the above example, assume the distribution shifts
# in the future to a gamma distribution with shape = 5 and scale = 2.
# Create 6 future observations from this distribution, and create 3
# means by pairing the observations sequentially. Note we must first
# transform these observations using the power 0.246, then compute the
# means based on the transformed data, and then transform the means
# back to the original scale and compare them to the upper prediction
# limit of 12.17
set.seed(427)
new.dat < rgamma(6, shape = 5, scale = 2)
p < pred.list$interval$normal.transform.power
p
#[1] 0.246
new.dat.trans < new.dat^p
means.trans < c(mean(new.dat.trans[1:2]), mean(new.dat.trans[3:4]),
mean(new.dat.trans[5:6]))
means < means.trans^(1/p)
means
#[1] 11.74214 17.05299 11.65272
any(means > pred.list$interval$limits["UPL"])
#[1] TRUE
#
# Clean up
rm(dat, pred.list, new.dat, p, new.dat.trans, means.trans, means)
#
# Reproduce part 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 a
# onesided upper 90% prediction limit.
predIntGamma(Gibbons.et.al.09.Alkilinity.vec, pi.type = "upper",
conf.level = 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
#
#Prediction Interval Method: exact using
# Wilson & Hilferty (1931) cuberoot
# transformation to Normality
#
#Normal Transform Power: 0.3333333
#
#Prediction Interval Type: upper
#
#Confidence Level: 90%
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0.0000
# UPL = 85.3495

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