tolIntNormCensored {EnvStats}  R Documentation 
Construct a \beta
content or \beta
expectation tolerance
interval for a normal distribution based on Type I or Type II
censored data.
tolIntNormCensored(x, censored, censoring.side = "left", coverage = 0.95,
cov.type = "content", ti.type = "twosided", conf.level = 0.95,
method = "mle", ti.method = "exact.for.complete", seed = NULL,
nmc = 1000)
x 
numeric vector of observations. Missing ( 
censored 
numeric or logical vector indicating which values of 
censoring.side 
character string indicating on which side the censoring occurs. The possible values are

coverage 
a scalar between 0 and 1 indicating the desired coverage of the tolerance interval.
The default value is 
cov.type 
character string specifying the coverage type for the tolerance interval.
The possible values are 
ti.type 
character string indicating what kind of tolerance interval to compute.
The possible values are 
conf.level 
a scalar between 0 and 1 indicating the confidence level associated with the tolerance
interval. The default value is 
method 
character string indicating the method to use for parameter estimation. 
ti.method 
character string specifying the method for constructing the tolerance
interval. Possible values are: 
seed 
for the case when 
nmc 
for the case when 
See the help file for tolIntNorm
for an explanation of
tolerance intervals. When ti.method="gpq"
, the tolerance interval
is constructed using the method of Generalized Pivotal Quantities as
explained in Krishnamoorthy and Mathew (2009, p. 327). When
ti.method="exact.for.complete"
or
ti.method="wald.wolfowitz.for.complete"
, the tolerance interval
is constructed by first computing the maximum likelihood estimates of
the mean and standard deviation by calling
enormCensored
,
then passing these values to the function tolIntNorm
to
produce the tolerance interval as if the estimates were based on
complete rather than censored data. These last two methods are purely
adhoc and their properties need to be studied.
A list of class "estimateCensored"
containing the estimated
parameters, the tolerance interval, and other information.
See estimateCensored.object
for details.
Tolerance intervals have long been applied to quality control and life testing problems (Hahn, 1970b,c; Hahn and Meeker, 1991; Krishnamoorthy and Mathew, 2009). 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).
Steven P. Millard (EnvStats@ProbStatInfo.com)
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 TwoSided Tolerance Intervals for a Normal Distribution. Annals of Mathematical Statistics 35, 762772.
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), 115125.
Hahn, G.J. (1970c). Statistical Intervals for a Normal Population, Part II: Formulas, Assumptions, Some Derivations. Journal of Quality Technology 2(4), 195206.
Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, New York.
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with SPLUS. CRC Press, Boca Raton.
Odeh, R.E., and D.B. Owen. (1980). Tables for Normal Tolerance Limits, Sampling Plans, and Screening. Marcel Dekker, New York.
Owen, D.B. (1962). Handbook of Statistical Tables. AddisonWesley, Reading, MA.
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.
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.
Wald, A., and J. Wolfowitz. (1946). Tolerance Limits for a Normal Distribution. Annals of Mathematical Statistics 17, 208215.
gpqTolIntNormSinglyCensored
, eqnormCensored
,
enormCensored
, estimateCensored.object
.
# Generate 20 observations from a normal distribution with parameters
# mean=10 and sd=3, censor the observations less than 9,
# then create a onesided upper tolerance interval with 90%
# coverage and 95% confidence based on these Type I left, singly
# censored data.
# (Note: the call to set.seed allows you to reproduce this example.
set.seed(250)
dat < sort(rnorm(20, mean = 10, sd = 3))
dat
# [1] 6.406313 7.126621 8.119660 8.277216 8.426941 8.847961
# [7] 8.899098 9.357509 9.525756 9.534858 9.558567 9.847663
#[13] 10.001989 10.014964 10.841384 11.386264 11.721850 12.524300
#[19] 12.602469 12.813429
censored < dat < 9
dat[censored] < 9
tolIntNormCensored(dat, censored, coverage = 0.9, ti.type="upper")
#Results of Distribution Parameter Estimation
#Based on Type I Censored Data
#
#
#Assumed Distribution: Normal
#
#Censoring Side: left
#
#Censoring Level(s): 9
#
#Estimated Parameter(s): mean = 9.700962
# sd = 1.845067
#
#Estimation Method: MLE
#
#Data: dat
#
#Censoring Variable: censored
#
#Sample Size: 20
#
#Percent Censored: 35%
#
#Assumed Sample Size: 20
#
#Tolerance Interval Coverage: 90%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact for
# Complete Data
#
#Tolerance Interval Type: upper
#
#Confidence Level: 95%
#
#Tolerance Interval: LTL = Inf
# UTL = 13.25454
## Not run:
# Note: The true 90'th percentile is 13.84465
#
qnorm(0.9, mean = 10, sd = 3)
# [1] 13.84465
# Compare the result using the method "gpq"
tolIntNormCensored(dat, censored, coverage = 0.9, ti.type="upper",
ti.method = "gpq", seed = 432)$interval$limits
# LTL UTL
# Inf 13.56826
# Clean Up
#
rm(dat, censored)
#==========
# Example 151 of USEPA (2009, p. 1510) shows how to estimate
# the mean and standard deviation using logtransformed multiply
# leftcensored manganese concentration data. Here we'll construct a
# 95
EPA.09.Ex.15.1.manganese.df
# Sample Well Manganese.Orig.ppb Manganese.ppb Censored
# 1 1 Well.1 <5 5.0 TRUE
# 2 2 Well.1 12.1 12.1 FALSE
# 3 3 Well.1 16.9 16.9 FALSE
# ...
# 23 3 Well.5 3.3 3.3 FALSE
# 24 4 Well.5 8.4 8.4 FALSE
# 25 5 Well.5 <2 2.0 TRUE
with(EPA.09.Ex.15.1.manganese.df,
tolIntNormCensored(log(Manganese.ppb), Censored, coverage = 0.9,
ti.type = "upper"))
# Results of Distribution Parameter Estimation
# Based on Type I Censored Data
# 
#
# Assumed Distribution: Normal
#
# Censoring Side: left
#
# Censoring Level(s): 0.6931472 1.6094379
#
# Estimated Parameter(s): mean = 2.215905
# sd = 1.356291
#
# Estimation Method: MLE
#
# Data: log(Manganese.ppb)
#
# Censoring Variable: censored
#
# Sample Size: 25
#
# Percent Censored: 24
#
# Assumed Sample Size: 25
#
# Tolerance Interval Coverage: 90
#
# Coverage Type: content
#
# Tolerance Interval Method: Exact for
# Complete Data
#
# Tolerance Interval Type: upper
#
# Confidence Level: 95
#
# Tolerance Interval: LTL = Inf
# UTL = 4.708904
## End(Not run)