Estimate Quantiles of a Normal Distribution Based on Type I Censored Data
Estimate 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)x |
a 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 |
p |
numeric vector of probabilities for which quantiles will be estimated.
All values of |
method |
character string specifying the method of estimating the mean and standard deviation. For singly censored data, the possible values are: For multiply censored data, the possible values are: See the DETAILS section for more information. |
ci |
logical scalar indicating whether to compute a confidence interval for the quantile.
The default value is |
ci.method |
character string indicating what method to use to construct the confidence interval
for the quantile. The possible values are: See the DETAILS section for more information. This argument is ignored if |
ci.type |
character string indicating what kind of confidence interval for the quantile to compute.
The possible values are |
conf.level |
a scalar between 0 and 1 indicating the confidence level of the confidence interval.
The default value is |
digits |
an integer indicating the number of decimal places to round to when printing out
the value of |
nmc |
numeric scalar indicating the number of Monte Carlo simulations to run when
|
seed |
integer supplied to the function |
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-α)\% is equivalent to
an upper β-content tolerance interval with coverage 100p\% and
confidence level 100(1-α)\%. Also, a lower confidence interval for
the p'th quantile with confidence level 100(1-α)\% is equivalent
to a lower β-content tolerance interval with coverage 100(1-p)\% and
confidence level 100(1-α)\%.
eqnormCensored returns a list of class "estimateCensored"
containing the estimated quantile(s) and other information.
See estimateCensored.object for details.
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).
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York.
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.
Owen, D.B. (1962). Handbook of Statistical Tables. Addison-Wesley, Reading, MA.
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.
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.
Stedinger, J. (1983). Confidence Intervals for Design Events. Journal of Hydraulic Engineering 109(1), 13-27.
Stedinger, J.R., R.M. Vogel, and E. Foufoula-Georgiou. (1993). Frequency Analysis of Extreme Events. In: Maidment, D.R., ed. Handbook of Hydrology. McGraw-Hill, New York, Chapter 18, pp.29-30.
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.
Wald, A., and J. Wolfowitz. (1946). Tolerance Limits for a Normal Distribution. Annals of Mathematical Statistics 17, 208-215.
# 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
# [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
# [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 = 9.390624
# 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] 12.56310
#----------
# 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
#...
#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
# 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
# To estimate the 90th percentile on the original scale,
# we need to exponentiate the results
#-------------------------------------------------------
exp(est.list$quantiles)
#90'th %ile
# 52.14674
exp(est.list$interval$limits)
# LCL UCL
# 0.0000 110.9305
#----------
# Clean up
#---------
rm(est.list)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.