percentile.exact {STAND}

Estimate of Xp and Exact Confidence Limits for Normal/Lognormal

Description

Calculate estimate of Xp the 100*p percentile of the normal/lognormal distribution, and the lower and upper 100*\gamma% exact confidence limits. The resulting interval (Xp.LCL,Xp.UCL) is an approximate 100*(2\gamma - 1) percent confidence interval for Xp the 100*p percentile. This function should only be used for complete samples.

Usage

percentile.exact(x, p = 0.95, gam = 0.95,logx=TRUE,wpnt=FALSE)

Arguments

x

vector of positive data values

p

probability for Xp the 100pth percentile. Default is 0.95

gam

one-sided confidence level \gamma. Default 0.95

logx

If TRUE, sample is from lognormal, else normal. Default is TRUE

wpnt

if TRUE, show warning from pnt. Default is FALSE

Details

A point estimate of Xp, the 100pth percentile of a normal/lognormal distribution is calculated. Exact confidence limits for Xp are calculated using the quantile function of the non-central t distribution. The exact UCL is m + K*s, where m is the sample mean, s is the sample standard deviation, and the K factor depends on n, p, and \gamma. The exact LCL is m + K'*s. The local function kf calculates K and K' using the quantile function of the non-central t distribution qt.

The null hypothesis Ho: Xp \ge Lp is rejected at the \alpha = (1- \gamma ) significance level if the 100\gamma\% UCL for Xp is less than the specified limit Lp (indicating the exposure profile is acceptable).

Value

A LIST with components:

Xp

estimate of the pth percentile of the distribution

Xpe.LCL

100*\gamma% exact lower confidence limit for Xp

Xpe.UCL

100*\gamma% exact 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

Logx

If TRUE, sample is from lognormal, else normal. Default is TRUE

n

sample size

Ku

the K factor used to calculate the exact UCL

Kl

the K' factor used to calculate the exact LCL

Note

The UCL is also referred to as an upper tolerance limit, i.e., if p = 0.95 and \gamma = 0.99 then Xpe.UCL is the exact UTL 95% - 99%.

Author(s)

E. L. Frome

References

Burrows, G. L. (1963), "Statistical Tolerance Limits - What are They," Applied Statistics, 12, 133-144.

Johnson, N. L. and B. L. Welch (1940), "Application of the Non-Central t-Distribution," Biometrika, 31(3/4), 362-389.

Lyles R. H. and L. L. Kupper (1996), "On Strategies for Comparing Occupational Exposure Data to Limits," American Industrial Hygiene Association Journal, 57:6-15.

Tuggle, R. M. (1982), "Assessment of Occupational Exposure Using One-Sided Tolerance Limits," American Industrial Hygiene Association Journal, 43, 338-346.

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

Ignacio, J. S. and W. H. Bullock (2006), A Strategy for Assesing and Managing Occupational Exposures, Third Edition, AIHA Press, Fairfax, VA.

Mulhausen, J. R. and J. Damiano (1998), A Strategy for Assessing and Managing Occupational Exposures, Second Edition, AIHA Press, Fairfax, VA.

See Also

Help files for percentile.ml, efraction.exact, aihand

Examples

#                      EXAMPLE 1
# calculate 95th percentile and exact CLs for Example data
# Appendix  Mulhausen and Damiano (1998)
data(aihand)
x <- aihand$x ;  det <- rep(1,length(x))
aiha <- data.frame(x,det) #  complete data
unlist(percentile.exact(x,gam=0.95,p=0.95) )[1:5]  #  exact CLs
unlist(percentile.ml(aiha,gam=0.95,p=0.95))   #  ML CLs
#                      EXAMPLE 2
#  Ignacio and Bullock (2006) Mulhausen and Damiano (1998)
#  Calculate TABLE VII.3 (page 272) Factor for One-Sided Tolerance
#  Limits for Normal Distribution (Abridged Version)
#  Same as Table III Burrows(1963) Panel 3 Page 138
nn <- c(seq(3,25),seq(30,50,5))
pv <-c(0.75,0.9,0.95,0.99,0.999)
tab <- matrix(0,length(nn),length(pv))
  for( k in (1:length(nn) ) ){
  xx <- seq(1,nn[k])
  for(j in (1:length(pv))) {
  tab[k,j ]<- percentile.exact(xx,pv[j],gam=0.95,FALSE)$Ku
}}
dimnames(tab)<-(list(nn,pv)) ; rm(nn,pv,xx)
round(tab,3)
#
#                      EXAMPLE 3
#  Calculate TABLE I One Sided Tolerance Factor K'
#  Tuggle(1982) Page 339 (Abridged Version)
nn <- c(seq(3,20),50,50000000)
pv <-c(0.9,0.95,0.99)
tab <- matrix(0,length(nn),length(pv))
  for( k in (1:length(nn) ) ){
  xx <- seq(1,nn[k])
  for(j in (1:length(pv))) {
  tab[k,j ]<- percentile.exact(xx,pv[j],gam=0.95,FALSE)$Kl
}}
dimnames(tab)<-(list(nn,pv)) ; rm(nn,pv,xx)
round(tab,3)