# Exceedance Fraction and Exact Confidence Limits

### 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 |

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

`logx` |
If |

`wpnt` |
if |

### 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* |

`fe.UCL` |
100* |

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

`gam` |
one-sided confidence level |

`Logx` |
If |

### 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

### Author(s)

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.

### See Also

Help files for `efraction.ml`

,`efclnp`

,
`percentile.exact`

, `efraction.exact`

,
`aihand`

### Examples

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
``` |