# efraction.exact: Exceedance Fraction and Exact Confidence Limits In STAND: Statistical Analysis of Non-Detects

## Description

Calculate estimate of the exceedance fraction FL = Pr [X > L] and exact confidence limits for random sample from normal/lognormal distribution. This function should only be used for complete samples.

## Usage

 `1` ```efraction.exact(x, gam = 0.95, L=NA ,logx=TRUE,wpnt=FALSE) ```

## Arguments

 `x` vector of data values `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 `logx` If `TRUE`, sample is from lognormal, else normal. Default is `TRUE` `wpnt` if `TRUE`, show warning from pnt. Default is FALSE

## Details

The exceedance fraction represent the proportion of the X's that exceed a given limit Lp. The null hypothesis of interest is Ho: F ≥ Fo = 1-p; i.e., Fo is the maximum proportion of the population that can exceed the limit Lp. The null hypothesis is rejected if the 100 γ\% UCL for FL is less than Fo , indicating that the exposure profile is acceptable. The type I error rate for this test is less than or equal to α = 1 - γ.

## Value

A LIST with components:

 `f` estimate of exceedance fraction for lognormal distribution as % `fe.LCL` 100*γ% exact lower confidence limit % units `fe.UCL` 100*γ% exact upper confidence limit % units `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 `Logx` If `TRUE`, sample is from lognormal, else normal. Default is `TRUE`

## Note

(fe.LCL, fe.UCL) is an approximate 100(2γ -1) percent confidence interval for F. The R function `uniroot` is used to find the noncentrality parameter of noncentral t distribution to calculate CL's for U = (L - μ) / σ where F = pnorm(U). In some versions of R this may cause a warning message. See R bug report RP 9171 full precision was not achieved in 'pnt'. This warning message may occur in `uniroot` calls to `pt` and does not effect the precision of the final result

E. L. Frome

## References

Johnson, N. L. and B. L. Welch (1940), "Application of the Non-Central t-Distribution," Biometrika, 31(3/4), 362-389.

Lyles, R. H. and L. L. Kupper (1996), "On strategies for comparing occupational exposure data to limits," American Industrial Hygiene Association Journal. 57:6-15.

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

Ignacio, J. S. and W. H. Bullock (2006), A Strategy for Assesing and Managing Occupational Exposures, Third Edition, AIHA Press, Fairfax, VA.

Mulhausen, J. R. and J. Damiano (1998), A Strategy for Assessing and Managing Occupational Exposures, Second Edition, AIHA Press, Fairfax, VA.

Help files for `efraction.ml`,`efclnp`, `percentile.exact`, `efraction.exact`, `aihand`
 ```1 2 3 4 5 6 7 8``` ```# calculate exceedance fraction and exact CLs for Example data # Appendix Mulhausen and Damiano(1998) Ignacion and Bullock (2006 data(aihand) x <- aihand\$x ; det <- rep(1,length(x)) aiha<-data.frame(x,det) # complete data unlist(efraction.exact(x,gam=0.95,L=5) ) # exact CLs unlist(efraction.ml(aiha,gam=0.95,L=5)) # ML CLs unlist(efclnp(aiha,L=5)) # nonparametric CLs ```