lnorm.ml {STAND} | R Documentation |
ML Estimation for Lognormal Data with Non-detects
Description
When an exposure measurement may be less than a detection limit closed form and exact methods have not been developed for the lognormal model. The maximum likelihood (ML) principle is used to develop an algorithm for parameter estimation, and to obtain large sample equivalents of confidence limits for the mean exposure level, the 100pth percentile, and the exceedance fraction. For a detailed discussion of assumptions, properties, and computational issues related to ML estimation see Cox and Hinkley (1979) and Cohen (1991).
Usage
lnorm.ml(dd)
Arguments
dd |
An n by 2 matrix or data frame with |
Details
For notational convenience the m detected values x[i]
are listed first
followed by the nx[i]
indicating non-detects, so that the data are
x[i], i = 1, \ldots , m, nx[i] i = m + 1, \ldots ,n
. If nx[i]
is the same for each
non-detect, this is referred to as a left singly censored sample (Type I
censoring) and nx
is the limit of detection(LOD). If the nx[i]
are different,
this is known as randomly (or progressively) left-censored data[see
Cohen(1991) and Schmoyer et al (1996)]. In some situations a value of 0 is
recorded when the exposure measurement is less than the LOD. In this
situation, the value of nx[i]
is the LOD indicating that x
is in the interval
(0, nx[i])
. The probability density function for lognormal distribution is
g(x;\mu,\sigma)= exp[-(log(x) - \mu)^2/(2\sigma^2)] /[\sigma x \sqrt(2\Pi )]
where y = log(x)
is normally distributed with mean \mu
and standard
deviation \sigma
[Atkinson and Brown (1969)]. The geometric mean of X is
GM = exp(\mu)
and the geometric standard deviation is GSD = exp(\sigma)
.
Strom and Stansberry (2000) provide a summary of these and other
relationships for lognormal parameters. Assuming the data are a random
sample from a lognormal distribution, the log of the likelihood function for
the unknown parameters \mu
and \sigma
given the data is
L (\mu, \sigma )=\sum log[g(x; \mu, \sigma )] + \sum log[G (nx; \mu, \sigma )],
where G(x; \mu , \sigma)
is the lognormal distribution function, i.e., G(nx; \mu , \sigma)
is the probability that x \le nx
.
The first summation is over i = 1, \ldots , m
, and the second is over i = m +
1, \ldots ,n
.
To test that the mean of X > L
, Ho: E(X) > L
at the
\alpha = 1- \gamma
significance level a one-sided upper 100\gamma\%
confidence limit can be used. One method for calculating this UCL is to use the
censored data equivalent of Cox's direct method; i.e., calculate the ML
estimate of \phi =\mu + [1/2] \sigma ^2
, and var(\phi) = var(\mu + [1/2] \sigma ^2)
where
var(\phi )= var(\mu ) + [1/4] var(\sigma^2)+cov(\mu ,\sigma^2).
The ML estimator of E(X) is exp(\phi)
, the 100\gamma {\%}
LCL for E(X)
is exp[\phi - t var(\phi )
], and the 100\gamma\%
UCL for
E(x) is exp[\phi + t var(\phi )
], where t = t(\gamma , m-1)
. The
resulting confidence interval (LCL, UCL) has confidence level 100(2\gamma
-1)\%
. An equivalent procedure is to estimate \phi = \mu + [1/2] \sigma^2
and its standard error directly, i.e., by maximizing the log-likelihood with
parameters \mu + [1/2]\sigma^2
and \sigma^2
. ML estimates of \mu , \sigma , \phi , \sigma^2
,
estimates of their standard errors, and covariance terms are calculated.
Value
A list with components:
mu |
ML estimate of |
sigma |
ML estimate of |
logEX |
ML estimate of log of E(X) |
SigmaSq |
ML estimate of |
se.mu |
ML estimate of standard error of |
se.sigma |
ML estimate of standard error of |
se.logEX |
ML estimate of standard error of log of E(X) |
se.Sigmasq |
ML estimate of standard error of |
cov.musig |
ML estimate of cov( |
m |
number of detects |
n |
number of observations in the data set |
m2log(L) |
-2 times the log-likelihood function |
convergence |
convergence indicator from |
Note
Local function ndln
is called by optim
for mu
and sigma
and local function ndln2
is called by optim
for logEX
and Sigmasq
.
Author(s)
E. L. Frome
References
Cohen, A. C. (1991), Truncated and Censored Samples, Marcel Decker, New York
Cox, D. R. and D. V. Hinkley (1979), Theoretical Statistics, Chapman and Hall, New York.
Frome, E. L. and Wambach, P. F. (2005), "Statistical Methods and Software for the Analysis of Occupational Exposure Data with Non-Detectable Values," ORNL/TM-2005/52,Oak Ridge National Laboratory, Oak Ridge, TN 37830. Available at: http://www.csm.ornl.gov/esh/aoed/ORNLTM2005-52.pdf
See Also
optim
, efraction.ml
, percentile.ml
Examples
# Calculate MLE for Example 2 in ORNLTM2005-52
data(beTWA)
mle.TWA<- unlist(lnorm.ml(beTWA)) # ML for Be monitoring data
mle.TWA[1:4] # ML estimates of parameters
mle.TWA[5:8] # Standard errors of ML estimates
mle.TWA[9:13] # additional results from lnorm.ml