efraction.ml {STAND}

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

dd

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

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

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.

Value

A LIST with components:

f

is the ML estimate of exceedance fraction for lognormal distribution

f.LCL

is the 100*\gamma% lower confidence limit for f

f.UCL

is the 100*\gamma% upper confidence limit for f

L

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

gam

one-sided confidence level \gamma. Default is 0.95

Note

(f.LCL, f.UCL) is an 100(2\gamma -1) percent confidence interval for F

Author(s)

E. L. Frome

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

See Also

lnorm.ml,percentile.ml

Examples

# 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))