# Generate 15 observations from a lognormal distribution with
# parameters meanlog=3 and sdlog=0.5, and censor observations less than 10.
# Then generate 15 more observations from this distribution and censor
# observations less than 9.
# 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 <- rlnorm(15, meanlog = 3, sdlog = 0.5)
sort(x.1)
# [1] 8.051717 9.651611 11.671282 12.271247 12.664108 17.446124
# [7] 17.707301 20.238069 20.487219 21.025510 21.208197 22.036554
#[13] 25.710773 28.661973 54.453557
censored.1 <- x.1 < 10
x.1[censored.1] <- 10
x.2 <- rlnorm(15, meanlog = 3, sdlog = 0.5)
sort(x.2)
# [1] 6.289074 7.511164 8.988267 9.179006 12.869408 14.130081
# [7] 16.941937 17.060513 19.287572 19.682126 20.363893 22.750203
#[13] 24.744306 28.089325 37.792873
censored.2 <- x.2 < 9
x.2[censored.2] <- 9
x <- c(x.1, x.2)
censored <- c(censored.1, censored.2)
eqlnormCensored(x, censored, p = 0.9, ci = TRUE, ci.type = "upper")
#Results of Distribution Parameter Estimation
#Based on Type I Censored Data
#--------------------------------------------
#
#Assumed Distribution: Lognormal
#
#Censoring Side: left
#
#Censoring Level(s): 9 10
#
#Estimated Parameter(s): meanlog = 2.8099300
# sdlog = 0.5137151
#
#Estimation Method: MLE
#
#Estimated Quantile(s): 90'th %ile = 32.08159
#
#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 = 0.00000
# UCL = 41.38716
#----------
# Compare these results with the true 90'th percentile:
qlnorm(p = 0.9, meanlog = 3, sd = 0.5)
#[1] 38.1214
#----------
# 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.
# 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:
#---------------------------------------------------------------
with(EPA.09.Ex.15.1.manganese.df,
eqlnormCensored(Manganese.ppb, Censored,
p = 0.9, ci = TRUE, ci.type = "upper"))
#Results of Distribution Parameter Estimation
#Based on Type I Censored Data
#--------------------------------------------
#
#Assumed Distribution: Lognormal
#
#Censoring Side: left
#
#Censoring Level(s): 2 5
#
#Estimated Parameter(s): meanlog = 2.215905
# sdlog = 1.356291
#
#Estimation Method: MLE
#
#Estimated Quantile(s): 90'th %ile = 52.14674
#
#Quantile Estimation Method: Quantile(s) Based on
# MLE Estimators
#
#Data: 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 = 0.0000
# UCL = 110.9305Run the code above in your browser using DataLab