eqnormCensored.RdEstimate quantiles of a normal distribution given a sample of data that has been subjected to Type I censoring, and optionally construct a confidence interval for a quantile.
eqnormCensored(x, censored, censoring.side = "left", p = 0.5, method = "mle",
ci = FALSE, ci.method = "exact.for.complete", ci.type = "two-sided",
conf.level = 0.95, digits = 0, nmc = 1000, seed = NULL)a numeric vector of observations.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf)
values are allowed but will be removed.
numeric or logical vector indicating which values of x are censored.
This must be the same length as x. If the mode of censored is
"logical", TRUE values correspond to elements of x that
are censored, and FALSE values correspond to elements of x that
are not censored. If the mode of censored is "numeric",
it must contain only 1's and 0's; 1 corresponds to
TRUE and 0 corresponds to FALSE. Missing (NA)
values are allowed but will be removed.
character string indicating on which side the censoring occurs. The possible
values are "left" (the default) and "right".
numeric vector of probabilities for which quantiles will be estimated.
All values of p must be between 0 and 1. When ci=TRUE, p
must be a scalar. The default value is p=0.5.
character string specifying the method of estimating the mean and standard deviation.
For singly censored data, the possible values are: "mle" (maximum likelihood; the default), "bcmle" (bias-corrected maximum likelihood), "qq.reg" (quantile-quantile regression), "qq.reg.w.cen.level" (quantile-quantile regression including the
censoring level), "impute.w.qq.reg" (moment estimation based on imputation using the
qq.reg method), "impute.w.qq.reg.w.cen.level" (moment estimation based on imputation
using the qq.reg.w.cen.level method), "impute.w.mle" (moment estimation based on imputation using the mle), "iterative.impute.w.qq.reg" (moment estimation based on iterative
imputation using the qq.reg method), "m.est" (robust M-estimation), and "half.cen.level" (moment estimation based on setting the censored
observations to half the censoring level).
For multiply censored data, the possible values are: "mle" (maximum likelihood; the default), "qq.reg" (quantile-quantile regression), "impute.w.qq.reg" (moment estimation based on imputation using the
qq.reg method), and "half.cen.level" (moment estimation based on setting the censored
observations to half the censoring level).
See the DETAILS section for more information.
logical scalar indicating whether to compute a confidence interval for the quantile.
The default value is ci=FALSE.
character string indicating what method to use to construct the confidence interval
for the quantile. The possible values are: "exact.for.complete" (exact method for complete (i.e., uncensored) data; the default), "gpq" (method based on generalized pivotal quantities), and "normal.approx" (normal approximation).
See the DETAILS section for more information. This argument is ignored if ci=FALSE.
character string indicating what kind of confidence interval for the quantile to compute.
The possible values are "two-sided" (the default), "lower", and
"upper". This argument is ignored if ci=FALSE.
a scalar between 0 and 1 indicating the confidence level of the confidence interval.
The default value is conf.level=0.95. This argument is ignored if ci=FALSE.
an integer indicating the number of decimal places to round to when printing out
the value of 100*p. The default value is digits=0.
numeric scalar indicating the number of Monte Carlo simulations to run when
ci.method="gpq". The default is nmc=1000. This argument is ignored
if ci=FALSE.
integer supplied to the function set.seed and used when
ci.method="gpq". The default value is
seed=NULL, in which case the current value of .Random.seed is used.
This argument is ignored when ci=FALSE.
Estimating Quantiles
Quantiles are estimated by:
estimating the mean and standard deviation parameters by
calling enormCensored, and then
calling the function qnorm and using the estimated
values for the mean and standard deviation.
The estimated quantile thus depends on the method of estimating the mean and
standard deviation.
Confidence Intervals for Quantiles
Exact Method When Data are Complete (ci.method="exact.for.complete")
When ci.method="exact.for.complete", the function eqnormCensored
calls the function eqnorm, supplying it with the estimated mean
and standard deviation, and setting the argument ci.method="exact". Thus, this
is the exact method for computing a confidence interval for a quantile had the data
been complete. Because the data have been subjected to Type I censoring, this method
of constructing a confidence interval for the quantile is an approximation.
Normal Approximation (ci.method="normal.approx")
When ci.method="normal.approx", the function eqnormCensored
calls the function eqnorm, supplying it with the estimated mean
and standard deviation, and setting the argument ci.method="normal.approx".
Thus, this is the normal approximation method for computing a confidence interval
for a quantile had the data been complete. Because the data have been subjected
to Type I censoring, this method of constructing a confidence interval for the
quantile is an approximation both because of the normal approximation and because
the estimates of the mean and standard devation are based on censored, instead of
complete, data.
Generalized Pivotal Quantity (ci.method="gpq")
When ci.method="gpq", the function eqnormCensored uses the
relationship between confidence intervals for quantiles and tolerance intervals
and calls the function tolIntNormCensored with the argument
ti.method="gpq" to construct the confidence interval.
Specifically, it can be shown
(e.g., Conover, 1980, pp.119-121) that an upper confidence interval for the
\(p\)'th quantile with confidence level \(100(1-\alpha)\%\) is equivalent to
an upper \(\beta\)-content tolerance interval with coverage \(100p\%\) and
confidence level \(100(1-\alpha)\%\). Also, a lower confidence interval for
the \(p\)'th quantile with confidence level \(100(1-\alpha)\%\) is equivalent
to a lower \(\beta\)-content tolerance interval with coverage \(100(1-p)\%\) and
confidence level \(100(1-\alpha)\%\).
eqnormCensored returns a list of class "estimateCensored"
containing the estimated quantile(s) and other information.
See estimateCensored.object for details.
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Lewis Publishers, Boca Raton.
Caudill, S.P., L.-Y. Wong, W.E. Turner, R. Lee, A. Henderson, D. G. Patterson Jr. (2007). Percentile Estimation Using Variable Censored Data. Chemosphere 68, 169–180.
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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.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.
Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York, NY, pp.132-136.
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.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY, pp.88-90.
Hewett, P., and G.H. Ganser. (2007). A Comparison of Several Methods for Analyzing Censored Data. Annals of Occupational Hygiene 51(7), 611–632.
Johnson, N.L., and B.L. Welch. (1940). Applications of the Non-Central t-Distribution. Biometrika 31, 362-389.
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
Kroll, C.N., and J.R. Stedinger. (1996). Estimation of Moments and Quantiles Using Censored Data. Water Resources Research 32(4), 1005–1012.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. 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.
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Serasinghe, S.K. (2010). A Simulation Comparison of Parametric and Nonparametric Estimators of Quantiles from Right Censored Data. A Report submitted in partial fulfillment of the requirements for the degree Master of Science, Department of Statistics, College of Arts and Sciences, Kansas State University, Manhattan, Kansas.
Singh, A., R. Maichle, and S. Lee. (2006). On the Computation of a 95% Upper Confidence Limit of the Unknown Population Mean Based Upon Data Sets with Below Detection Limit Observations. EPA/600/R-06/022, March 2006. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.
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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.
Stedinger, J. (1983). Confidence Intervals for Design Events. Journal of Hydraulic Engineering 109(1), 13-27.
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Percentiles are sometimes used in environmental standards and regulations. For example, Berthouex and Brown (2002, p.71) note that England has water quality limits based on the 90th and 95th percentiles of monitoring data not exceeding specified levels. They also note that the U.S. EPA has specifications for air quality monitoring, aquatic standards on toxic chemicals, and maximum daily limits for industrial effluents that are all based on percentiles. Given the importance of these quantities, it is essential to characterize the amount of uncertainty associated with the estimates of these quantities. This is done with confidence intervals.
A sample of data contains censored observations if some of the observations are reported only as being below or above some censoring level. In environmental data analysis, Type I left-censored data sets are common, with values being reported as “less than the detection limit” (e.g., Helsel, 2012). Data sets with only one censoring level are called singly censored; data sets with multiple censoring levels are called multiply or progressively censored.
Statistical methods for dealing with censored data sets have a long history in the field of survival analysis and life testing. More recently, researchers in the environmental field have proposed alternative methods of computing estimates and confidence intervals in addition to the classical ones such as maximum likelihood estimation.
Helsel (2012, Chapter 6) gives an excellent review of past studies of the properties of various estimators based on censored environmental data.
In practice, it is better to use a confidence interval for a percentile, rather than rely on a single point-estimate of percentile. Confidence intervals for percentiles of a normal distribution depend on the properties of the estimators for both the mean and standard deviation.
Few studies have been done to evaluate the performance of methods for constructing confidence intervals for the mean or joint confidence regions for the mean and standard deviation when data are subjected to single or multiple censoring (see, for example, Singh et al., 2006). Studies to evaluate the performance of a confidence interval for a percentile include: Caudill et al. (2007), Hewett and Ganner (2007), Kroll and Stedinger (1996), and Serasinghe (2010).
# Generate 15 observations from a normal distribution with
# parameters mean=10 and sd=2, and censor observations less than 8.
# Then generate 15 more observations from this distribution and censor
# observations less than 7.
# Then estimate the 90th percentile and create a one-sided upper 95%
# confidence interval for that percentile.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(47)
x.1 <- rnorm(15, mean = 10, sd = 2)
sort(x.1)
#> [1] 6.343542 7.068499 7.828525 8.029036 8.155088 9.436470 9.495908
#> [8] 10.030262 10.079205 10.182946 10.217551 10.370811 10.987640 11.422285
#> [15] 13.989393
# [1] 6.343542 7.068499 7.828525 8.029036 8.155088 9.436470
# [7] 9.495908 10.030262 10.079205 10.182946 10.217551 10.370811
#[13] 10.987640 11.422285 13.989393
censored.1 <- x.1 < 8
x.1[censored.1] <- 8
x.2 <- rnorm(15, mean = 10, sd = 2)
sort(x.2)
#> [1] 5.355255 6.065562 6.783680 6.867676 8.219412 8.593224 9.319168
#> [8] 9.347066 9.837844 9.918844 10.055054 10.498296 10.834382 11.341558
#> [15] 12.528482
# [1] 5.355255 6.065562 6.783680 6.867676 8.219412 8.593224
# [7] 9.319168 9.347066 9.837844 9.918844 10.055054 10.498296
#[13] 10.834382 11.341558 12.528482
censored.2 <- x.2 < 7
x.2[censored.2] <- 7
x <- c(x.1, x.2)
censored <- c(censored.1, censored.2)
eqnormCensored(x, censored, p = 0.9, ci = TRUE, ci.type = "upper")
#>
#> Results of Distribution Parameter Estimation
#> Based on Type I Censored Data
#> --------------------------------------------
#>
#> Assumed Distribution: Normal
#>
#> Censoring Side: left
#>
#> Censoring Level(s): 7 8
#>
#> Estimated Parameter(s): mean = 9.227116
#> sd = 2.073889
#>
#> Estimation Method: MLE
#>
#> Estimated Quantile(s): 90'th %ile = 11.88491
#>
#> Quantile Estimation Method: Quantile(s) Based on
#> MLE Estimators
#>
#> Data: x
#>
#> Censoring Variable: censored
#>
#> Sample Size: 30
#>
#> Percent Censored: 23.33333%
#>
#> Confidence Interval for: 90'th %ile
#>
#> Assumed Sample Size: 30
#>
#> Confidence Interval Method: Exact for
#> Complete Data
#>
#> Confidence Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Confidence Interval: LCL = -Inf
#> UCL = 12.9131
#>
#Results of Distribution Parameter Estimation
#Based on Type I Censored Data
#--------------------------------------------
#
#Assumed Distribution: Normal
#
#Censoring Side: left
#
#Censoring Level(s): 7 8
#
#Estimated Parameter(s): mean = 9.390624
# sd = 1.827156
#
#Estimation Method: MLE
#
#Estimated Quantile(s): 90'th %ile = 11.73222
#
#Quantile Estimation Method: Quantile(s) Based on
# MLE Estimators
#
#Data: x
#
#Censoring Variable: censored
#
#Sample Size: 30
#
#Percent Censored: 16.66667%
#
#Confidence Interval for: 90'th %ile
#
#Assumed Sample Size: 30
#
#Confidence Interval Method: Exact for
# Complete Data
#
#Confidence Interval Type: upper
#
#Confidence Level: 95%
#
#Confidence Interval: LCL = -Inf
# UCL = 12.63808
#----------
# Compare these results with the true 90'th percentile:
qnorm(p = 0.9, mean = 10, sd = 2)
#> [1] 12.5631
#[1] 12.56310
#----------
# Clean up
rm(x.1, censored.1, x.2, censored.2, x, censored)
#==========
# Chapter 15 of USEPA (2009) gives several examples of estimating the mean
# and standard deviation of a lognormal distribution on the log-scale using
# manganese concentrations (ppb) in groundwater at five background wells.
# In EnvStats these data are stored in the data frame
# EPA.09.Ex.15.1.manganese.df.
# Here we will estimate the mean and standard deviation using the MLE,
# and then construct an upper 95% confidence limit for the 90th percentile.
# We will log-transform the original observations and then call
# eqnormCensored. Alternatively, we could have more simply called
# eqlnormCensored.
# First look at the data:
#-----------------------
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
#> 4 4 Well.1 21.6 21.6 FALSE
#> 5 5 Well.1 <2 2.0 TRUE
#> 6 1 Well.2 <5 5.0 TRUE
#> 7 2 Well.2 7.7 7.7 FALSE
#> 8 3 Well.2 53.6 53.6 FALSE
#> 9 4 Well.2 9.5 9.5 FALSE
#> 10 5 Well.2 45.9 45.9 FALSE
#> 11 1 Well.3 <5 5.0 TRUE
#> 12 2 Well.3 5.3 5.3 FALSE
#> 13 3 Well.3 12.6 12.6 FALSE
#> 14 4 Well.3 106.3 106.3 FALSE
#> 15 5 Well.3 34.5 34.5 FALSE
#> 16 1 Well.4 6.3 6.3 FALSE
#> 17 2 Well.4 11.9 11.9 FALSE
#> 18 3 Well.4 10 10.0 FALSE
#> 19 4 Well.4 <2 2.0 TRUE
#> 20 5 Well.4 77.2 77.2 FALSE
#> 21 1 Well.5 17.9 17.9 FALSE
#> 22 2 Well.5 22.7 22.7 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
# 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
longToWide(EPA.09.Ex.15.1.manganese.df,
"Manganese.Orig.ppb", "Sample", "Well",
paste.row.name = TRUE)
#> Well.1 Well.2 Well.3 Well.4 Well.5
#> Sample.1 <5 <5 <5 6.3 17.9
#> Sample.2 12.1 7.7 5.3 11.9 22.7
#> Sample.3 16.9 53.6 12.6 10 3.3
#> Sample.4 21.6 9.5 106.3 <2 8.4
#> Sample.5 <2 45.9 34.5 77.2 <2
# Well.1 Well.2 Well.3 Well.4 Well.5
#Sample.1 <5 <5 <5 6.3 17.9
#Sample.2 12.1 7.7 5.3 11.9 22.7
#Sample.3 16.9 53.6 12.6 10 3.3
#Sample.4 21.6 9.5 106.3 <2 8.4
#Sample.5 <2 45.9 34.5 77.2 <2
# Now estimate the mean, standard deviation, and 90th percentile
# on the log-scale using the MLE, and construct an upper 95%
# confidence limit for the 90th percentile on the log-scale:
#---------------------------------------------------------------
est.list <- with(EPA.09.Ex.15.1.manganese.df,
eqnormCensored(log(Manganese.ppb), Censored,
p = 0.9, ci = TRUE, ci.type = "upper"))
est.list
#>
#> 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
#>
#> Estimated Quantile(s): 90'th %ile = 3.954062
#>
#> Quantile Estimation Method: Quantile(s) Based on
#> MLE Estimators
#>
#> Data: log(Manganese.ppb)
#>
#> Censoring Variable: censored
#>
#> Sample Size: 25
#>
#> Percent Censored: 24%
#>
#> Confidence Interval for: 90'th %ile
#>
#> Assumed Sample Size: 25
#>
#> Confidence Interval Method: Exact for
#> Complete Data
#>
#> Confidence Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Confidence Interval: LCL = -Inf
#> UCL = 4.708904
#>
#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
#
#Estimated Quantile(s): 90'th %ile = 3.954062
#
#Quantile Estimation Method: Quantile(s) Based on
# MLE Estimators
#
#Data: log(Manganese.ppb)
#
#Censoring Variable: censored
#
#Sample Size: 25
#
#Percent Censored: 24%
#
#Confidence Interval for: 90'th %ile
#
#Assumed Sample Size: 25
#
#Confidence Interval Method: Exact for
# Complete Data
#
#Confidence Interval Type: upper
#
#Confidence Level: 95%
#
#Confidence Interval: LCL = -Inf
# UCL = 4.708904
# To estimate the 90th percentile on the original scale,
# we need to exponentiate the results
#-------------------------------------------------------
exp(est.list$quantiles)
#> 90'th %ile
#> 52.14674
#90'th %ile
# 52.14674
exp(est.list$interval$limits)
#> LCL UCL
#> 0.0000 110.9305
# LCL UCL
# 0.0000 110.9305
#----------
# Clean up
#---------
rm(est.list)