STAND (version 2.0)

lnorm.ml: 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 x (exposure) variable in column 1, and det= 0 for non-detect or 1 for detect in Column 2

Value

• A list with components:
• mu
  ML estimate of $\mu$
• sigma
  ML estimate of $\sigma$
• logEX
  ML estimate of log of E(X)
• SigmaSq
  ML estimate of $\sigma^2$
• se.mu
  ML estimate of standard error of $\mu$
• se.sigma
  ML estimate of standard error of $\sigma$
• se.logEX
  ML estimate of standard error of log of E(X)
• se.Sigmasq
  ML estimate of standard error of $\sigma^2$
• cov.musig
  ML estimate of cov($\mu$,$\sigma)$
• m
  number of detects
• n
  number of observations in the data set
• m2log(L)
  -2 times the log-likelihood function
• convergence
  convergence indicator from optim

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.

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