EnvStats pp 113-148 | Cite as

Prediction and Tolerance Intervals

  • Steven P. Millard


Any activity that requires constant monitoring over time and the comparison of new values to “background” or “standard” values creates a decision problem: if the new values greatly exceed the background values, has a change really occurred, or have the true underlying concentrations stayed the same and this is just a “chance” event? Statistical tests are used as objective tools to decide whether a change has occurred (although the choice of Type I error level and acceptable power are subjective decisions). For a monitoring program that involves numerous tests over time, figuring out how to balance the overall Type I error with the power of detecting a change is not a trivial problem, but it is also a problem that has been dealt with for a long time in the statistical literature under the heading of “multiple comparisons.” Prediction intervals and tolerance intervals are two tools that you can use to attempt to solve the multiple comparisons problem. This chapter discusses the functions available in EnvStats for constructing prediction and tolerance intervals. See Millard et al. (2014) for a more in-depth discussion of this topic.


Confidence Level Lognormal Distribution Gamma Distribution Prediction Interval Sampling Occasion 
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  1. Chou, Y.M., and D.B. Owen. (1986). One-sided Distribution-Free Simultaneous Prediction Limits for p Future Samples. Journal of Quality Technology 18, 96−98.Google Scholar
  2. Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley & Sons, New York, 493 pp.Google Scholar
  3. Davis, C.B., and R.J. McNichols. (1987). One-sided Intervals for at Least p of m Observations from a Normal Population on Each of r Future Occasions. Technometrics 29, 359−370.MathSciNetGoogle Scholar
  4. Davis, C.B., and R.J. McNichols. (1994b). Ground Water Monitoring Statistics Update: Part II: Nonparametric Prediction Limits. Ground Water Monitoring and Remediation 14(4), 159−175.CrossRefGoogle Scholar
  5. Davis, C.B., and R.J. McNichols. (1999). Simultaneous Nonparametric Prediction Limits (with Discusson). Technometrics 41(2), 89−112.CrossRefGoogle Scholar
  6. Guttman, I. (1970). Statistical Tolerance Regions: Classical and Bayesian. Hafner Publishing Co., Darien, CT, 150 pp.Google Scholar
  7. Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley & Sons, New York, 392 pp.Google Scholar
  8. Hawkins, D. M., and R.A.J. Wixley. (1986). A Note on the Transformation of Chi-Squared Variables to Normality. The American Statistician 40, 296–298.Google Scholar
  9. Hollander, M., and D.A. Wolfe. (1999). Nonparametric Statistical Methods. Second Edition. John Wiley & Sons, New York, 787 pp.Google Scholar
  10. Krishnamoorthy K., T. Mathew, and S. Mukherjee. (2008). Normal-Based Methods for a Gamma Distribution: Prediction and Tolerance Intervals and Stress-Strength Reliability. Technometrics 50(1), 69–78.MathSciNetCrossRefGoogle Scholar
  11. Kulkarni, H.V., and S.K. Powar. (2010). A New Method for Interval Estimation of the Mean of the Gamma Distribution. Lifetime Data Analysis 16, 431−447.MathSciNetCrossRefGoogle Scholar
  12. Millard, S.P., P. Dixon, and N.K. Neerchal. (2014). Environmental Statistics with R. CRC Press, Boca Raton, Florida.Google Scholar
  13. Singh, A., A.K. Singh, and R.J. Iaci. (2002). Estimation of the Exposure Point Concentration Term Using a Gamma Distribution. EPA/600/R-02/084. October 2002. Technology Support Center for Monitoring and Site Characterization, Office of Research and Development, Office of Solid Waste and Emergency Response, U.S. Environmental Protection Agency, Washington, D.C.Google Scholar
  14. Singh, A., R. Maichle, and N. Armbya. (2010a). ProUCL Version 4.1.00 User Guide (Draft). EPA/600/R-07/041, May 2010. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.Google Scholar
  15. USEPA. (1994b). Statistical Methods for Evaluating the Attainment of Cleanup Standards, Volume 3: Reference-Based Standards for Soils and Solid Media. EPA/230-R-94-004. Office of Policy, Planning, and Evaluation, U.S. Environmental Protection Agency, Washington, D.C.Google Scholar
  16. USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities: Unified Guidance. EPA 530-R-09-007, March 2009. Office of Resource Conservation and Recovery, Program Implementation and Information Division, U.S. Environmental Protection Agency, Washington, D.C.Google Scholar
  17. Wilson, E.B., and M.M. Hilferty. (1931). The Distribution of Chi-Squares. Proceedings of the National Academy of Sciences 17, 684–688.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Steven P. Millard
    • 1
  1. 1.Probability, Statistics and InformationSeattleUSA

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