Prediction Interval for a Lognormal Distribution
Estimate the mean and standard deviation on the log-scale for a lognormal distribution, or estimate the mean and coefficient of variation for a lognormal distribution (alternative parameterization), and construct a prediction interval for the next k observations or next set of k geometric means.
predIntLnorm(x, n.geomean = 1, k = 1, method = "Bonferroni",
pi.type = "two-sided", conf.level = 0.95)
predIntLnormAlt(x, n.geomean = 1, k = 1, method = "Bonferroni",
pi.type = "two-sided", conf.level = 0.95, est.arg.list = NULL)x |
For For If |
n.geomean |
positive integer specifying the sample size associated with the k future
geometric means. The default value is |
k |
positive integer specifying the number of future observations or geometric means the
prediction interval should contain with confidence level |
method |
character string specifying the method to use if the number of future observations
( |
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 |
est.arg.list |
for |
The function predIntLnorm returns a prediction interval as well as
estimates of the meanlog and sdlog parameters.
The function predIntLnormAlt returns a prediction interval as well as
estimates of the mean and coefficient of variation.
A prediction interval for a lognormal distribution is constructed by taking the
natural logarithm of the observations and constructing a prediction interval
based on the normal (Gaussian) distribution by calling predIntNorm.
These prediction limits are then exponentiated to produce a prediction interval on
the original scale of the data.
If x is a numeric vector, a list of class
"estimate" containing the estimated parameters, the prediction interval,
and other information. See the help file for estimate.object for details.
If x is the result of calling an estimation function,
predIntLnorm returns a list whose class is the same as x.
The list contains the same components as x, as well as a component called
interval containing the prediction interval information.
If x already has a component called interval, this component is
replaced with the prediction interval information.
Prediction and tolerance intervals have long been applied to quality control and life testing problems (Hahn, 1970b,c; Hahn and Nelson, 1973; Krishnamoorthy and Mathew, 2009). In the context of environmental statistics, prediction intervals are useful for analyzing data from groundwater detection monitoring programs at hazardous and solid waste facilities (e.g., Gibbons et al., 2009; Millard and Neerchal, 2001; USEPA, 2009).
Steven P. Millard (EnvStats@ProbStatInfo.com)
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Lewis Publishers, Boca Raton.
Dunnett, C.W. (1955). A Multiple Comparisons Procedure for Comparing Several Treatments with a Control. Journal of the American Statistical Association 50, 1096-1121.
Dunnett, C.W. (1964). New Tables for Multiple Comparisons with a Control. Biometrics 20, 482-491.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.
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.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York.
Helsel, D.R., and R.M. Hirsch. (2002). Statistical Methods in Water Resources. Techniques of Water Resources Investigations, Book 4, chapter A3. U.S. Geological Survey. (available on-line at: http://pubs.usgs.gov/twri/twri4a3/).
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
Millard, S.P., and Neerchal, N.K. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, Florida.
Miller, R.G. (1981a). Simultaneous Statistical Inference. McGraw-Hill, New York.
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 20 observations from a lognormal distribution with parameters
# meanlog=0 and sdlog=1. The exact two-sided 90% prediction interval for
# k=1 future observation is given by: [exp(-1.645), exp(1.645)] = [0.1930, 5.181].
# Use predIntLnorm to estimate the distribution parameters, and construct a
# two-sided 90% prediction interval.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(47)
dat <- rlnorm(20, meanlog = 0, sdlog = 1)
predIntLnorm(dat, conf = 0.9)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Lognormal
#
#Estimated Parameter(s): meanlog = -0.1035722
# sdlog = 0.9106429
#
#Estimation Method: mvue
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: exact
#
#Prediction Interval Type: two-sided
#
#Confidence Level: 90%
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0.1795898
# UPL = 4.5264399
#----------
# Repeat the above example, but do it in two steps.
# First create a list called est.list containing information about the
# estimated parameters, then create the prediction interval.
est.list <- elnorm(dat)
est.list
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Lognormal
#
#Estimated Parameter(s): meanlog = -0.1035722
# sdlog = 0.9106429
#
#Estimation Method: mvue
#
#Data: dat
#
#Sample Size: 20
predIntLnorm(est.list, conf = 0.9)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Lognormal
#
#Estimated Parameter(s): meanlog = -0.1035722
# sdlog = 0.9106429
#
#Estimation Method: mvue
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: exact
#
#Prediction Interval Type: two-sided
#
#Confidence Level: 90%
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0.1795898
# UPL = 4.5264399
#----------
# Using the same data from the first example, create a one-sided
# upper 99% prediction limit for the next 3 geometric means of order 2
# (i.e., each of the 3 future geometric means is based on a sample size
# of 2 future observations).
predIntLnorm(dat, n.geomean = 2, k = 3, conf.level = 0.99,
pi.type = "upper")
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Lognormal
#
#Estimated Parameter(s): meanlog = -0.1035722
# sdlog = 0.9106429
#
#Estimation Method: mvue
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: Bonferroni
#
#Prediction Interval Type: upper
#
#Confidence Level: 99%
#
#Number of Future
#Geometric Means: 3
#
#Sample Size for
#Geometric Means: 2
#
#Prediction Interval: LPL = 0.000000
# UPL = 7.047571
#----------
# Compare the result above that is based on the Bonferroni method
# with the exact method
predIntLnorm(dat, n.geomean = 2, k = 3, conf.level = 0.99,
pi.type = "upper", method = "exact")$interval$limits["UPL"]
# UPL
#7.00316
#----------
# Clean up
rm(dat, est.list)
#--------------------------------------------------------------------
# Example 18-2 of USEPA (2009, p.18-15) shows how to construct a 99%
# upper prediction interval for the log-scale mean of 4 future observations
# (future mean of order 4) assuming a lognormal distribution based on
# chrysene concentrations (ppb) in groundwater at 2 background wells.
# Data were collected once per month over 4 months at the 2 background
# wells, and also at a compliance well.
# The question to be answered is whether there is evidence of
# contamination at the compliance well.
# Here we will follow the example, but look at the geometric mean
# instead of the log-scale mean.
#----------
# The data for this example are stored in EPA.09.Ex.18.2.chrysene.df.
EPA.09.Ex.18.2.chrysene.df
# Month Well Well.type Chrysene.ppb
#1 1 Well.1 Background 6.9
#2 2 Well.1 Background 27.3
#3 3 Well.1 Background 10.8
#4 4 Well.1 Background 8.9
#5 1 Well.2 Background 15.1
#6 2 Well.2 Background 7.2
#7 3 Well.2 Background 48.4
#8 4 Well.2 Background 7.8
#9 1 Well.3 Compliance 68.0
#10 2 Well.3 Compliance 48.9
#11 3 Well.3 Compliance 30.1
#12 4 Well.3 Compliance 38.1
Chrysene.bkgd <- with(EPA.09.Ex.18.2.chrysene.df,
Chrysene.ppb[Well.type == "Background"])
Chrysene.cmpl <- with(EPA.09.Ex.18.2.chrysene.df,
Chrysene.ppb[Well.type == "Compliance"])
#----------
# A Shapiro-Wilks goodness-of-fit test for normality indicates
# we should reject the assumption of normality and assume a
# lognormal distribution for the background well data:
gofTest(Chrysene.bkgd)
#Results of Goodness-of-Fit Test
#-------------------------------
#
#Test Method: Shapiro-Wilk GOF
#
#Hypothesized Distribution: Normal
#
#Estimated Parameter(s): mean = 16.55000
# sd = 14.54441
#
#Estimation Method: mvue
#
#Data: Chrysene.bkgd
#
#Sample Size: 8
#
#Test Statistic: W = 0.7289006
#
#Test Statistic Parameter: n = 8
#
#P-value: 0.004759859
#
#Alternative Hypothesis: True cdf does not equal the
# Normal Distribution.
gofTest(Chrysene.bkgd, dist = "lnorm")
#Results of Goodness-of-Fit Test
#-------------------------------
#
#Test Method: Shapiro-Wilk GOF
#
#Hypothesized Distribution: Lognormal
#
#Estimated Parameter(s): meanlog = 2.5533006
# sdlog = 0.7060038
#
#Estimation Method: mvue
#
#Data: Chrysene.bkgd
#
#Sample Size: 8
#
#Test Statistic: W = 0.8546352
#
#Test Statistic Parameter: n = 8
#
#P-value: 0.1061057
#
#Alternative Hypothesis: True cdf does not equal the
# Lognormal Distribution.
#----------
# Here is the one-sided 99% upper prediction limit for
# a geometric mean based on 4 future observations:
predIntLnorm(Chrysene.bkgd, n.geomean = 4, k = 1,
conf.level = 0.99, pi.type = "upper")
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Lognormal
#
#Estimated Parameter(s): meanlog = 2.5533006
# sdlog = 0.7060038
#
#Estimation Method: mvue
#
#Data: Chrysene.bkgd
#
#Sample Size: 8
#
#Prediction Interval Method: exact
#
#Prediction Interval Type: upper
#
#Confidence Level: 99%
#
#Number of Future
#Geometric Means: 1
#
#Sample Size for
#Geometric Means: 4
#
#Prediction Interval: LPL = 0.00000
# UPL = 46.96613
UPL <- predIntLnorm(Chrysene.bkgd, n.geomean = 4, k = 1,
conf.level = 0.99, pi.type = "upper")$interval$limits["UPL"]
UPL
# UPL
#46.96613
# Is there evidence of contamination at the compliance well?
geoMean(Chrysene.cmpl)
#[1] 44.19034
# Since the geometric mean at the compliance well is less than
# the upper prediction limit, there is no evidence of contamination.
#----------
# Cleanup
#--------
rm(Chrysene.bkgd, Chrysene.cmpl, UPL)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.