Calculate ML Estimate of Xp and Confidence Limits

Description Usage Arguments Details Value Note Author(s) References See Also Examples

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Calculate the ML estimate of Xp the 100pth percentile of the lognormal distribution, and the lower and upper 100*γ% confidence limits LX(p,γ) and UX(p,γ). The upper confidence limit is used to test the null hypothesis that the exposure profile is "unacceptable". If UX(p,γ) < L the null hypothesis is rejected and workplace is considered "safe" or the object/area is not contaminated. The Type I error is ≤ α = 1 - γ. The resulting interval (LX,UX) is an approximate 100*(2γ - 1) percent confidence interval for Xp.


1, p = 0.95, gam = 0.95, dat = TRUE)



An n by 2 matrix or data frame with
x (exposure) variable in column 1, and
det= 0 for non-detect or 1 for detect in column 2


is probability for Xp the 100pth percentile. Default is 0.95


one-sided confidence level γ. Default is 0.95


if dat is FALSE then dd is a list from Default is TRUE


The point estimate of Yp = log(Xp) is μ + z σ where μ and σ are ML estimates and z is qnorm(p). The variance of the estimate is

var(μ + zσ ) = var(μ ) + Z^2p var (σ )+ 2z cov(μ ,σ)

The 100γ {\%} LCL and UCL for Xp are

LX(p,γ ) = exp[Yp- t(γ ,(m-1))var(Yp)^{1/2}],

UX(p,γ ) = exp[Yp + t(γ ,(m-1))var(Yp)^{1/2}].

The ML estimates of var(μ), var(σ), and cov(μ ,σ) are obtained from the ML variance-covariance matrix using The null hypothesis Ho: Xp ≥ Lp is rejected at the α = (1- γ ) significance level if the 100γ\% UCL for Xp < Lp (indicating the exposure profile is acceptable).


A LIST with components:


ML estimate of the pth percentile of lognormal distribution


100*γ% lower confidence limit for Xp


100*γ% upper confidence limit for Xp


probability for Xp the 100pth percentile. Default 0.95


one-sided confidence level γ. Default is 0.95


The UCL is also referred to as an upper tolerance limit(UTL), i.e., if p = 0.95 and gam = 0.99 then Xp.UCL is the UTL-95%-99%.


E. L. Frome


Cohen, A. C. (1991), Truncated and Censored Samples, Marcel Decker, New York

Cox, D. R. and D. V. Hinkley (1979), Theoretical Statistics, Chapman and Hall, New York.

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:

See Also

Help files for,


# calculate ML estimate of 95th percentile and CLs for Example 2 in ORNLTM2005-52 

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