percentile.ml {STAND}

Calculate ML Estimate of Xp and Confidence Limits

Description

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

Usage

percentile.ml(dd, p = 0.95, gam = 0.95, dat = TRUE)

Arguments

dd

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

p

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

gam

one-sided confidence level \gamma. Default is 0.95

dat

if dat is FALSE then dd is a list from lnorm.ml. Default is TRUE

Details

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

var(\mu + z\sigma ) = var(\mu ) + Z^2p var (\sigma )+ 2z cov(\mu ,\sigma)

The 100\gamma {\%} LCL and UCL for Xp are

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

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

The ML estimates of var(\mu), var(\sigma), and cov(\mu ,\sigma) are obtained from the ML variance-covariance matrix using lnorm.ml. The null hypothesis Ho: Xp \ge Lp is rejected at the \alpha = (1- \gamma ) significance level if the 100\gamma\% UCL for Xp < Lp (indicating the exposure profile is acceptable).

Value

A LIST with components:

Xp

ML estimate of the pth percentile of lognormal distribution

Xp.LCL

100*\gamma% lower confidence limit for Xp

Xp.UCL

100*\gamma% upper confidence limit for Xp

p

probability for Xp the 100pth percentile. Default 0.95

gam

one-sided confidence level \gamma. Default is 0.95

Note

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%.

Author(s)

E. L. Frome

References

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: http://www.csm.ornl.gov/esh/aoed/ORNLTM2005-52.pdf

See Also

Help files for lnorm.ml,efraction.ml

Examples

data(beTWA)
# calculate ML estimate of 95th percentile and CLs for Example 2 in ORNLTM2005-52 
unlist(percentile.ml(beTWA,0.95,0.95))