# 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

1 | ```
efraction.ml(dd, gam = 0.95, L = 5, dat = TRUE)
``` |

### Arguments

`dd` |
if |

`gam` |
one-sided confidence level |

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

`dat` |
if |

### 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* |

`f.UCL` |
is the 100* |

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

`gam` |
one-sided confidence level |

### Note

(f.LCL, f.UCL) is an 100*(2γ -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

1 2 3 4 5 |