elnormCensored.RdEstimate the mean and standard deviation parameters of the logarithm of a lognormal distribution given a sample of data that has been subjected to Type I censoring, and optionally construct a confidence interval for the mean.
elnormCensored(x, censored, method = "mle", censoring.side = "left",
ci = FALSE, ci.method = "profile.likelihood", ci.type = "two-sided",
conf.level = 0.95, n.bootstraps = 1000, pivot.statistic = "z",
nmc = 1000, seed = NULL, ...)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 specifying the method of estimation.
For singly censored data, the possible values are: "mle" (maximum likelihood; the default), "bcmle" (bias-corrected maximum likelihood), "ROS" or "qq.reg" (quantile-quantile regression; also called
regression on order statistics and abbreviated ROS), "qq.reg.w.cen.level" (quantile-quantile regression including the
censoring level), "rROS" or "impute.w.qq.reg" (moment estimation based on imputation using
quantile-quantile regression; also called robust regression on order statistics
and abbreviated rROS), "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), "ROS" or "qq.reg" (quantile-quantile regression; also called
regression on order statistics and abbreviated ROS), "rROS" or "impute.w.qq.reg" (moment estimation based on imputation using
quantile-quantile regression; also called robust regression on order statistics
and abbreviated rROS), and "half.cen.level" (moment estimation based on setting the censored
observations to half the censoring level).
See the DETAILS section for more information.
character string indicating on which side the censoring occurs. The possible
values are "left" (the default) and "right".
logical scalar indicating whether to compute a confidence interval for the
mean or variance. The default value is ci=FALSE.
character string indicating what method to use to construct the confidence interval
for the mean. The possible values are: "profile.likelihood" (profile likelihood; the default), "normal.approx" (normal approximation), "normal.approx.w.cov" (normal approximation taking into account the
covariance between the estimated mean and standard deviation; only available for
singly censored data), "gpq" (generalized pivotal quantity), and "bootstrap" (based on bootstrapping).
See the DETAILS section for more information.
This argument is ignored if ci=FALSE.
character string indicating what kind of confidence interval 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.
numeric scalar indicating how many bootstraps to use to construct the
confidence interval for the mean when ci.type="bootstrap". This
argument is ignored if ci=FALSE and/or ci.method does not
equal "bootstrap".
character string indicating which pivot statistic to use in the construction
of the confidence interval for the mean when ci.method="normal.approx" or
ci.method="normal.approx.w.cov" (see the DETAILS section). The possible
values are pivot.statistic="z" (the default) and pivot.statistic="t".
When pivot.statistic="t" you may supply the argument
ci.sample size (see below). The argument pivot.statistic is
ignored if ci=FALSE.
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="bootstrap" or 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.
additional arguments to pass to other functions.
prob.method. Character string indicating what method to use to
compute the plotting positions (empirical probabilities) when method
is one of "ROS", "qq.reg", "qq.reg.w.cen.level",
"rROS", "impute.w.qq.reg",
"impute.w.qq.reg.w.cen.level", "impute.w.mle", or "iterative.impute.w.qq.reg". Possible values are: "kaplan-meier" (product-limit method of Kaplan and Meier (1958)), "nelson" (hazard plotting method of Nelson (1972)), "michael-schucany" (generalization of the product-limit method due to Michael and Schucany (1986)), and "hirsch-stedinger" (generalization of the product-limit method due to Hirsch and Stedinger (1987)).
The default value is prob.method="hirsch-stedinger". The "nelson"
method is only available for censoring.side="right".
See the DETAILS section and the help file for ppointsCensored
for more information.
plot.pos.con. Numeric scalar between 0 and 1 containing the
value of the plotting position constant to use when method is one of
"qq.reg", "qq.reg.w.cen.level", "impute.w.qq.reg", "impute.w.qq.reg.w.cen.level", "impute.w.mle", or "iterative.impute.w.qq.reg". The default value is plot.pos.con=0.375.
See the DETAILS section and the help file for ppointsCensored
for more information.
ci.sample.size. Numeric scalar indicating what sample size to
assume to construct the confidence interval for the mean if
pivot.statistic="t" and ci.method="normal.approx" or
ci.method="normal.approx.w.cov". When method equals
"mle" or "bcmle", the default value is the expected number of
uncensored observations, otherwise it is the observed number of
uncensored observations.
lb.impute. Numeric scalar indicating the lower bound for imputed
observations when method is one of "impute.w.qq.reg", "impute.w.qq.reg.w.cen.level", "impute.w.mle", or "iterative.impute.w.qq.reg". Imputed values smaller than this
value will be set to this value. The default is lb.impute=-Inf.
ub.impute. Numeric scalar indicating the upper bound for imputed
observations when method is one of "impute.w.qq.reg", "impute.w.qq.reg.w.cen.level", "impute.w.mle", or "iterative.impute.w.qq.reg". Imputed values larger than this value
will be set to this value. The default is ub.impute=Inf.
convergence. Character string indicating the kind of convergence
criterion when method="iterative.impute.w.qq.reg". The possible values
are "relative" (the default) and "absolute". See the DETAILS
section for more information.
tol. Numeric scalar indicating the convergence tolerance when method="iterative.impute.w.qq.reg". The default value is tol=1e-6.
If convergence="relative", then the relative difference in the old and
new estimates of the mean and the relative difference in the old and new estimates
of the standard deviation must be less than tol for convergence to be
achieved. If convergence="absolute", then the absolute difference in the
old and new estimates of the mean and the absolute difference in the old and new
estimates of the standard deviation must be less than tol for convergence
to be achieved.
max.iter. Numeric scalar indicating the maximum number of iterations
when method="iterative.impute.w.qq.reg".
t.df. Numeric scalar greater than or equal to 1 that determines the
robustness and efficiency properties of the estimator when method="m.est".
The default value is t.df=3.
If x or censored contain any missing (NA), undefined (NaN) or
infinite (Inf, -Inf) values, they will be removed prior to
performing the estimation.
Let \(X\) denote a random variable with a
lognormal distribution with
parameters meanlog=\(\mu\) and sdlog=\(\sigma\). Then
\(Y = log(X)\) has a normal (Gaussian) distribution with
parameters mean=\(\mu\) and sd=\(\sigma\). Thus, the function
elnormCensored simply calls the function enormCensored using the
log-transformed values of x.
a list of class "estimateCensored" containing the estimated parameters
and other information. See estimateCensored.object for details.
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Helsel, D.R. (1990). Less than Obvious: Statistical Treatment of Data Below the Detection Limit. Environmental Science and Technology 24(12), 1766–1774.
Helsel, D.R. (2012). Statistics for Censored Environmental Data Using Minitab and R, Second Edition. John Wiley & Sons, Hoboken, New Jersey.
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Hirsch, R.M., and J.R. Stedinger. (1987). Plotting Positions for Historical Floods and Their Precision. Water Resources Research 23(4), 715–727.
Korn, L.R., and D.E. Tyler. (2001). Robust Estimation for Chemical Concentration Data Subject to Detection Limits. In Fernholz, L., S. Morgenthaler, and W. Stahel, eds. Statistics in Genetics and in the Environmental Sciences. Birkhauser Verlag, Basel, pp.41–63.
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
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Nelson, W. (1982). Applied Life Data Analysis. John Wiley and Sons, New York, 634pp.
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Pettitt, A. N. (1983). Re-Weighted Least Squares Estimation with Censored and Grouped Data: An Application of the EM Algorithm. Journal of the Royal Statistical Society, Series B 47, 253–260.
Regal, R. (1982). Applying Order Statistic Censored Normal Confidence Intervals to Time Censored Data. Unpublished manuscript, University of Minnesota, Duluth, Department of Mathematical Sciences.
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Saw, J.G. (1961b). The Bias of the Maximum Likelihood Estimators of Location and Scale Parameters Given a Type II Censored Normal Sample. Biometrika 48, 448–451.
Schmee, J., D.Gladstein, and W. Nelson. (1985). Confidence Limits for Parameters of a Normal Distribution from Singly Censored Samples, Using Maximum Likelihood. Technometrics 27(2) 119–128.
Schneider, H. (1986). Truncated and Censored Samples from Normal Populations. Marcel Dekker, New York, New York, 273pp.
Shumway, R.H., A.S. Azari, and P. Johnson. (1989). Estimating Mean Concentrations Under Transformations for Environmental Data With Detection Limits. Technometrics 31(3), 347–356.
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.
Stryhn, H., and J. Christensen. (2003). Confidence Intervals by the Profile Likelihood Method, with Applications in Veterinary Epidemiology. Contributed paper at ISVEE X (November 2003, Chile). https://gilvanguedes.com/wp-content/uploads/2019/05/Profile-Likelihood-CI.pdf.
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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 the mean or a joint confidence region for the mean and standard deviation, rather than rely on a single point-estimate of the mean. Since confidence intervals and regions depend on the properties of the estimators for both the mean and standard deviation, the results of studies that simply evaluated the performance of the mean and standard deviation separately cannot be readily extrapolated to predict the performance of various methods of constructing confidence intervals and regions. Furthermore, for several of the methods that have been proposed to estimate the mean based on type I left-censored data, standard errors of the estimates are not available, hence it is not possible to construct confidence intervals (El-Shaarawi and Dolan, 1989).
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).
Schmee et al. (1985) studied Type II censoring for a normal distribution and
noted that the bias and variances of the maximum likelihood estimators are of the
order \(1/N\), and that the bias is negligible for \(N=100\) and as much as
90% censoring. (If the proportion of censored observations is less than 90%,
the bias becomes negligible for smaller sample sizes.) For small samples with
moderate to high censoring, however, the bias of the mle's causes confidence
intervals based on them using a normal approximation (e.g., method="mle"
and ci.method="normal.approx") to be too short. Schmee et al. (1985)
provide tables for exact confidence intervals for sample sizes up to \(N=100\)
that were created based on Monte Carlo simulation. Schmee et al. (1985) state
that these tables should work well for Type I censored data as well.
Shumway et al. (1989) evaluated the coverage of 90% confidence intervals for the mean based on using a Box-Cox transformation to induce normality, computing the mle's based on the normal distribution, then computing the mean in the original scale. They considered three methods of constructing confidence intervals: the delta method, the bootstrap, and the bias-corrected bootstrap. Shumway et al. (1989) used three parent distributions in their study: Normal(3,1), the square of this distribuiton, and the exponentiation of this distribution (i.e., a lognormal distribution). Based on sample sizes of 10 and 50 with a censoring level at the 10'th or 20'th percentile, Shumway et al. (1989) found that the delta method performed quite well and was superior to the bootstrap method.
Millard et al. (2014; in preparation) show that the coverage of profile likelihood method is excellent.
# 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 or rROS).
# 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 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%
#>
#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 (ROS) and imputation with Q-Q regression (rROS)
#---------------------------------------------------------------
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored))$parameters
#> meanlog sdlog
#> 2.215905 1.356291
# meanlog sdlog
#2.215905 1.356291
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored,
method = "ROS"))$parameters
#> meanlog sdlog
#> 2.293742 1.283635
# meanlog sdlog
#2.293742 1.283635
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored,
method = "rROS"))$parameters
#> meanlog sdlog
#> 2.298656 1.238104
# meanlog sdlog
#2.298656 1.238104
#----------
# The method used to estimate quantiles for a Q-Q plot is
# determined by the argument prob.method. For the functions
# enormCensored and elnormCensored, for any estimation
# method that involves Q-Q regression, the default value of
# prob.method is "hirsch-stedinger" and the default value for the
# plotting position constant is plot.pos.con=0.375.
# Both Helsel (2012) and USEPA (2009) also use the Hirsch-Stedinger
# probability method but set the plotting position constant to 0.
with(EPA.09.Ex.15.1.manganese.df,
elnormCensored(Manganese.ppb, Censored,
method = "rROS", plot.pos.con = 0))$parameters
#> meanlog sdlog
#> 2.277175 1.261431
# 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 = 2.771197
#>
#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 = 2.771197