Calculate the ML estimate of the exceedance fraction F = Pr [X > L] and "large sample" confidence limits for lognormal data with nondetects.
1  efraction.ml(dd, gam = 0.95, L = 5, dat = TRUE)

dd 
if 
gam 
onesided confidence level γ. Default is 0.95 
L 
L is specified limit for the exceedance fraction; e.g., the occupational exposure limit 
dat 
if 
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= 1p; 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 , m1) 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, m1) 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
.
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.UCL 
is the 100*γ% upper confidence limit for 
L 
L is specified limit for the exceedance fraction; e.g., the occupational exposure limit 
gam 
onesided confidence level γ. Default is 0.95 
(f.LCL, f.UCL) is an 100(2γ 1) percent confidence interval for F
E. L. Frome
Frome, E. L. and Wambach, P. F. (2005), "Statistical Methods and Software for the Analysis of Occupational Exposure Data with NonDetectable Values," ORNL/TM2005/52,Oak Ridge National Laboratory, Oak Ridge, TN 37830. Available at: http://www.csm.ornl.gov/esh/aoed/ORNLTM200552.pdf
lnorm.ml
,percentile.ml
1 2 3 4 5 
Questions? Problems? Suggestions? Tweet to @rdrrHQ or email at ian@mutexlabs.com.
All documentation is copyright its authors; we didn't write any of that.