# 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,
# Q-Q regression (also called parametric regression on order statistics
# or ROS; e.g., USEPA, 2009 and Helsel, 2012), and imputation with Q-Q
# regression (also called robust ROS).
# 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 and standard deviation on the log-scale
# using the MLE:
#---------------------------------------------------------------
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored))
#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
#
#Data: Manganese.ppb
#
#Censoring Variable: Censored
#
#Sample Size: 25
#
#Percent Censored: 24%
# Now compare the MLE with the estimators based on
# Q-Q regression and imputation with Q-Q regression
#--------------------------------------------------
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored))$parameters
# meanlog sdlog
#2.215905 1.356291
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored,
method = "qq.reg"))$parameters
# meanlog sdlog
#2.293742 1.283635
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored,
method = "impute.w.qq.reg"))$parameters
# meanlog sdlog
#2.298656 1.238104
#----------
# The method used to estimate quantiles for a Q-Q plot is
# determined by the argument prob.method. For the functions
# enormCensored and elnormCensored, for any estimation
# method that involves Q-Q regression, the default value of
# prob.method is "hirsch-stedinger" and the default value for the
# plotting position constant is plot.pos.con=0.375.
# Both Helsel (2012) and USEPA (2009) also use the Hirsch-Stedinger
# probability method but set the plotting position constant to 0.
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored,
method = "impute.w.qq.reg", plot.pos.con = 0))$parameters
# meanlog sdlog
#2.277175 1.261431
#----------
# Using the same data as above, compute a confidence interval
# for the mean on the log-scale using the profile-likelihood
# method.
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored, ci = TRUE))
#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
#
#Data: Manganese.ppb
#
#Censoring Variable: Censored
#
#Sample Size: 25
#
#Percent Censored: 24%
#
#Confidence Interval for: meanlog
#
#Confidence Interval Method: Profile Likelihood
#
#Confidence Interval Type: two-sided
#
#Confidence Level: 95%
#
#Confidence Interval: LCL = 1.595062
# UCL = 2.771197Run the code above in your browser using DataLab