# efraction.ml: Calculate ML Estimate of Exceedance Fraction and Confidence... In STAND: Statistical Analysis of Non-Detects

## 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

 `1` ```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 γ. 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 ≥ 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)-μ ] /σ , and N(v) is the standard normal distribution function. The large sample 100γ\% LCL for V = [log(L) - μ ]/σ is LCLv = v - t(γ , m-1) var(v)^{1/2}, where

var(v)= p1^2 var(μ )+ p2^2 var(σ)+ 2p1p2 cov( μ, σ)

, and p1 and p2 are partial derivatives of v with respect to μ and σ. The 100γ\% UCL for FL is UF( L, γ) = 1 - N(LCLv). The 100γ\% LCL for FL is LF( L, γ) = 1 - N(UCLv), where UCLv = u + t(γ, m-1) var(v)^{1/2}. The null hypothesis Ho: FL = 1 - p is rejected if the 100γ\% UCL for FL is less than Fo, indicating that the exposure profile is acceptable. The large sample ML estimates of the exceedance fraction and 100γ\% 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*γ% lower confidence limit for `f` `f.UCL` is the 100*γ% 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 γ. Default is 0.95

## Note

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

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

`lnorm.ml`,`percentile.ml`
 ```1 2 3 4 5``` ```# 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)) ```