efraction.ml: Calculate ML Estimate of Exceedance Fraction and Confidence Limits

Description

Calculate the ML estimate of the exceedance fraction $F = Pr [X > L]$ and "large sample" confidence limits for lognormal data with non-detects.

Usage

efraction.ml(dd, gam = 0.95, L = 5, dat = TRUE)

Arguments

if dat is TRUE dd is an n by 2 matrix or data frame with x in column 1 det in column 2

gam

one-sided confidence level $\gamma$. Default is 0.95

L is specified limit for the exceedance fraction; e.g., the occupational exposure limit

dat

if dat is FALSE, then dd is a list from lnorm.ml. Default is TRUE

Value

A LIST with components:
fis the ML estimate of exceedance fraction for lognormal distribution
f.LCLis the 100*$\gamma$% lower confidence limit for f
f.UCLis the 100*$\gamma$% upper confidence limit for f
LL is specified limit for the exceedance fraction; e.g., the occupational exposure limit
gamone-sided confidence level $\gamma$. Default is 0.95

Details

The exceedance fraction FL represent the proportion of the X's that exceed a given limit Lp. The null hypothesis of interest is $Ho: FL \ge Fo= 1-p$; i.e., Fo is the maximum proportion of the population that can exceed the limit Lp. The ML point estimate of FL is $f = 1 - N(v)$ where $v = [log(L)-\mu ] /\sigma$ , and N(v) is the standard normal distribution function. The large sample $100\gamma%$ LCL for $V = [log(L) - \mu ]/\sigma$ is LCLv $= v - t(\gamma , m-1) var(v)^{1/2}$, where $$var(v)= p1^2 var(\mu )+ p2^2 var(\sigma)+ 2p1p2 cov( \mu, \sigma)$$, and p1 and p2 are partial derivatives of $v$ with respect to $\mu$ and $\sigma$. The $100\gamma%$ UCL for FL is $UF( L, \gamma) = 1 - N(LCLv)$. The $100\gamma%$ LCL for FL is $LF( L, \gamma) = 1 - N(UCLv)$, where $UCLv = u + t(\gamma, m-1) var(v)^{1/2}$. The null hypothesis $Ho: FL = 1 - p$ is rejected if the $100\gamma%$ UCL for FL is less than Fo, indicating that the exposure profile is acceptable. The large sample ML estimates of the exceedance fraction and $100\gamma%$ confidence limits for lognormal data are calculated using the output from lnorm.ml.

References

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

Examples

Run this code

# calculate ML estimate of exceedance fraction and CLs for Example 2 in ORNLTM2005-52 
data(beTWA)
unlist(efraction.ml(beTWA,L=0.2))
#  calculate nonparametric CLs 
unlist(efclnp(beTWA,L=0.2))

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