STAND (version 2.0)

percentile.ml: 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

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

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).

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))