About-STAND: Statistical Analysis of Non-detect Data
In STAND: Statistical Analysis of Non-Detects
Description
Details
Acknowledgements
Note
References
Examples
Description
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.
Details
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).
Acknowledgements
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.
Note
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.
References
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: http://www.csm.ornl.gov/esh/aoed/ORNLTM2005-52.pdf
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.
Examples
1
2
3
4
5
6
7
8
9
10
11
12
# Example 1 from Frome and Wambach (2005) ORNL/TM-2005/52
# NOTE THAT FUNCTIONS NAMES AND DETAILS HAVE BEEN REVISED IN THIS PACKAGE
# the results are the same. For example lnorm.ml() replaces mlndln().
data(SESdata)
mle<-lnorm.ml(SESdata)
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
qq.lnorm(plend(SESdata),mle$mu,mle$sigma)
title("SESdata: Smelter-Elevated Surfaces")
Example output
Loading required package: survival
mu sigma logEX Sigmasq
-2.290764 1.276000 -1.476678 1.628180
se.mu se.sigma se.logEX se.Sigmasq
0.2311395 0.1754489 0.3137301 0.4477474
cov.musig m n m2log(L) convergence
-0.002005525 28.000000000 31.000000000 -12.852885390 0.000000000
SESdata
mu -2.2907643
se.mu 0.2311395
sigma 1.2760000
se.sigma 0.1754489
GM 0.1011891
GSD 3.5822818
EX 0.2283952
EX.LCL 0.1338480
EX.UCL 0.3897286
KM.mean 0.2030645
KM.LCL 0.1254281
KM.UCL 0.2807009
KM.se 0.0455803
Xp.obs 0.6502500
Xp 0.8253637
Xp.LCL 0.4464973
Xp.UCL 1.5257099
NpUTL NA
Maximum 1.1400000
NonDet% 9.7000000
n 31.0000000
Rsq 0.9830338
m 28.0000000
f 29.6686380
f.LCL 19.4593597
f.UCL 41.8076231
fnp 29.0322581
fnp.LCL 16.0611091
fnp.UCL 45.1904417
m2logL -12.8528854
L 0.2000000
p 0.9500000
gamma 0.9500000
STAND documentation built on May 2, 2019, 3:39 p.m.
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Description Details Acknowledgements Note References Examples
Description
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.
Details
| 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).
Acknowledgements
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.
Note
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.
References
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: http://www.csm.ornl.gov/esh/aoed/ORNLTM2005-52.pdf
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.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 | # Example 1 from Frome and Wambach (2005) ORNL/TM-2005/52
# NOTE THAT FUNCTIONS NAMES AND DETAILS HAVE BEEN REVISED IN THIS PACKAGE
# the results are the same. For example lnorm.ml() replaces mlndln().
data(SESdata)
mle<-lnorm.ml(SESdata)
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
qq.lnorm(plend(SESdata),mle$mu,mle$sigma)
title("SESdata: Smelter-Elevated Surfaces")
|
Example output
Loading required package: survival
mu sigma logEX Sigmasq
-2.290764 1.276000 -1.476678 1.628180
se.mu se.sigma se.logEX se.Sigmasq
0.2311395 0.1754489 0.3137301 0.4477474
cov.musig m n m2log(L) convergence
-0.002005525 28.000000000 31.000000000 -12.852885390 0.000000000
SESdata
mu -2.2907643
se.mu 0.2311395
sigma 1.2760000
se.sigma 0.1754489
GM 0.1011891
GSD 3.5822818
EX 0.2283952
EX.LCL 0.1338480
EX.UCL 0.3897286
KM.mean 0.2030645
KM.LCL 0.1254281
KM.UCL 0.2807009
KM.se 0.0455803
Xp.obs 0.6502500
Xp 0.8253637
Xp.LCL 0.4464973
Xp.UCL 1.5257099
NpUTL NA
Maximum 1.1400000
NonDet% 9.7000000
n 31.0000000
Rsq 0.9830338
m 28.0000000
f 29.6686380
f.LCL 19.4593597
f.UCL 41.8076231
fnp 29.0322581
fnp.LCL 16.0611091
fnp.UCL 45.1904417
m2logL -12.8528854
L 0.2000000
p 0.9500000
gamma 0.9500000
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