Statistical Analysis of Non-detect Data


Environmental exposure measurements are, in general, positive and may be subject to left censoring; i.e., the measured value is less than a "detection limit", and is referred to as a non-detect or "less than" value. This package calculates the censored data equivalent of a number of statistics that are used in the analysis of environmental data that do not contain non-detects, i.e. the usual complete data case.


Package: stand
Type: Package
Version: 2.0
Date: 2015-09-10
License: GPL version 2 or newer

In occupational monitoring, strategies for assessing workplace exposures typically focus on the mean exposure level or the probability that any measurement exceeds a limit. Parametric methods used to determine acceptable levels of exposure are based on a two parameter lognormal distribution. The mean exposure level, an upper percentile, the exceedance fraction, and confidence limits for each of these statistics are calculated. Statistical methods for random samples (without non-detects) from the lognormal distribution are well known for each of these situations–see e.g., Lyles and Kupper (1996). In this package the maximum likelihood method for randomly left censored lognormal data is used to calculate these statistics, and graphical methods are provided to evaluate the lognormal assumption. Nonparametric methods based on the product limit estimate for left censored data are used to estimate the mean exposure level, and the upper confidence limit on an upper percentile (i.e., the upper tolerance limit) is obtained using a nonparametric approach.

The American Industrial Hygiene Association (AIHA) has published a consensus standard with two basic strategies for evaluating an exposure profile—see Mulhausen and Damiano(1998), Ignacio and Bullock (2006). Most of the AIHA methods are based on the assumptions that the exposure data does not contain non-detects, and that a lognormal distribution can be used to describe the data. Exposure monitoring results from a compliant workplace tend to contain a high percentage of non-detected results when the detection limit is close to the exposure limit, and in some situations, the lognormal assumption may not be reasonable. The function IH.summary calculates most of the statistics proposed by AIHA for exposure data with non-detects. All of the methods are described in the report Frome and Wambach (2005).


This work was supported in part by the Office of Health, Safety, and Security of the U. S. Department of Energy and was performed in the Computer Science and Mathematics Division (CSMD) at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725. Additional funding and oversight have been provided through the Occupational Exposure and Worker Studies Group, Oak Ridge Institute for Science and Education, which is managed by Oak Ridge Associated Universities under Contract No. DE-AC05-060R23100.

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

The work has been authored by a contractor of the U.S. Government. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this work, or to allow others to do so for U. S. Government purposes.


Throughout this document and the online help files the greek letter γ is used to represent the confidence level for a one-sided confidence limit (default value 0.95). This is represented by gam or gamma in the argument list and value of functions that compute confidence limits.


Aitchison, J. and J. A. C. Brown (1969), The Lognormal Distribution, Cambridge, U.K., Cambridge University Press.

Akritas, M. G., T. F. Ruscitti, and G. S. Patil (1994), "Statistical Analysis of Censored Environmental Data," Handbook of Statistics, Vol. 12, G. P. Patil and C. R. Rao (eds), 221-242, Elsevier Science, New York.

American Conference of Governmental Industrial Hygienists (ACGIH) (2004), "Notice of Intended Change In: 2004 TLVs and BEIs," ACGIH, p. 60, Cincinnati, OH.

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

Armstrong, B. G. (1992), "Confidence Intervals for Arithmetic Means of Lognormally Distributed Exposures," American Industrial Hygiene Association Journal, 53(8), 481-485.

Chambers, J. M., W. S. Cleveland, B. Kleiner, and P. A. Tukey (1983), Graphical Methods for Data Analysis, Duxbury Press, Boston.

Clopper, C. J. and Pearson, E. S. (1934), "The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial," Biometrika, 26, 404-413.

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

Crow, E. L. and K. Shimizu (1988), Lognormal Distribution, Marcel Decker, New York.

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

Cox, D. R. and D. Oakes (1984), Analysis of Survival Data, Chapman and Hall, New York.

Department of Energy (December, 1999), "Chronic Beryllium Disease Prevention Program, Federal Register," 10 CFR Part 850, volume 64, number 235, 68854-68914.

Fowlkes, E. B. (1979), "Some Methods for Studying the Mixture of Two Normal (Lognormal) Distributions," Journal of the American Statistical Association, 74, 561-575.

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:

Hahn, G. J. and W. Q. Meeker (1991), Statistical Intervals, John Wiley and Sons, New York.

Hewett, P. and G. H. Ganser, (1997), "Simple Procedures for Calculating Confidence Intervals Around the Sample Mean and Exceedance Fraction Derived from Lognormally Distributed Data," Applied Occupational and Environmental Hygiene, 12(2), 132-147.

Helsel, D. (1990), "Less Than Obvious: Statistical Treatment of Date Below the Detection Limit," Environmental Science and Technology, 24(12), 1767-1774.

Hesel, D. R. and T. A. Cohn (1988), "Estimation of Descriptive Statistics for Multiply Censored Water Quality Data," Water Resources Research, 24, 1997-2004.

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

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

Kalbfleisch, J. D. and R. L. Prentice (1980), The Statistical Analysis of Failure Time Data, John Wiley and Sons, New York.

Kaplan, E. L. and Meir, P. (1958), "Nonparametric Estimation from Incomplete Observations," Journal of the American Statistical Association, 457-481.

Land, C. E. (1972), "An Evaluation of Approximate Confidence Interval Estimation Methods for Lognormal Means," Technometrics, 14(1), 145-158.

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.

Meeker, W. Q. and L. A. Escobar (1998), Statistical Methods for Reliability Data, John Wiley and Sons, New York.

Moulton, L. H. and N. A. Halsey (1995), "A Mixture Model with Detection Limits for Regression Analysis of Antibody Response on Vaccine," Biometrics, 51, 1570-1578.

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

Neuman, M. C., P. M. Dixon, B. B. Looney, and J. E. Pinder (1989), "Estimating Mean and Variance for Environmental Samples with Below Detection Limit Observations," Water Resources Bulletin, 25, 905-916.

Ng, M. P. (2002), "A Modification of Peto's Nonparametric Estimation of Survival Curves for Interval-Censored Data," Biometrics, volume 58, number 2, pp. 439-442.

Odeh, R. E. and D. B. Owen (1980), Tables for Normal Tolerance Limits, Sampling Plans, and Screening, Marcel Deker, New York.

Peto, R. (1973), "Experimental Survival Curves for Interval-censored Data," Applied Statistics, volume 22, number 1, pp. 86-91.

Schmee, J., D. Gladstein, and W. Nelson (1985), "Confidence Limits for Parameters of a Normal Distribution From Singly Censored Samples, Using Maximum Likelihood," Technometrics, 27, 119-128.

Schmoyer, R. L., J. J. Beauchamp, C. C. Brandt and F. O. Hoffman, Jr. (1996), "Difficulties with the Lognormal Model in Mean Estimation and Testing," Environmental and Ecological Statistics, 3, 81-97.

Sommerville, P. N. (1958), "Tables for Obtaining Non-Parametric Confidence Limits," Annals of Mathematical Statistics, 29, 599-601.

Taylor, D. J., L. L. Kupper, S. M. Rappaport, and R. H. Lyles (2001), "A Mixture Model for Occupational Exposure Mean Testing with a Limit of Detection," Biometrics, 57, 681-688.

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

Turnbull, B. W. (1976), "The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data," Journal of the Royal Statistical Society, Series B (Methodological), 38(3), 290-295.

Venables, W. N. and B. D. Ripley (2002), Modern Applied Statistics with S, 4th edition. Springer-Verlag, New York.

Verrill, S. and R. A. Johnson (1998), "Tables and Large-Sample Distribution Theory for Censored-Data Correlation Statistics for Testing Normality," Journal of the American Statistical Association, 83(404), 1192-1197.

Waller, L. A., and B. W. Turnbull, (1992), "Probability Plotting with Censored Data," The American Statistician, 46(1), 5-12.


# Example 1 from Frome and Wambach (2005) ORNL/TM-2005/52
# the results are the same. For example replaces mlndln().
unlist(mle[1:4])             # ML estimates mu sigma E(X) and sigma^2
unlist(mle[5:8])            # ML Estimates of standard errors
unlist(mle[9:13])            # additional  output from ORNL/TM-2005/52
IH.summary(SESdata,L=0.2)    #  All sumarry statistics for SESdata
#  lognormal q-q plot for SESdata Figure in ORNL/TM-2005/52
title("SESdata: Smelter-Elevated Surfaces")

Questions? Problems? Suggestions? or email at

Please suggest features or report bugs with the GitHub issue tracker.

All documentation is copyright its authors; we didn't write any of that.