ProUCL Version 3.0 Statistical Software to Compute Upper Confidence Limits of the Unknown Population Mean


v>EPA
U.S. Environmental
Protection
Agency
ProUCL	Version	3
Statistical	Software
Upper	Confidence
of the Unknown Population Mean
Introduction
The EPA Technical Support Center (TSC) in Las
Vegas has developed ProUCL Version 3.0
software to support risk assessment and cleanup
decisions at contaminated sites. Risk
assessments, exposure evaluations, and cleanup
decisions are often made based upon the mean
concentrations of the contaminants of potential
concern (COPCs). The true population mean
concentrations of the COPCs at a contaminated
site are often unknown, and are frequently
estimated by the respective sample means based
upon the data collected from the site under
investigation. In order to address the
uncertainties associated with the estimates of the
true unknown mean concentrations of the
COPCs, appropriate 95% upper confidence limits
(UCLs) of the respective unknown means are
frequently used in many environmental
applications. The computation of an appropriate
95% UCL of practical merit depends upon the
data distribution and the skewness associated
with the data set under study. ProUCL can be
used to compute an appropriate UCL of the
unknown population mean using a discernible
probability distribution (e.g., normal, lognormal,
gamma) and/or a suitable non-parametric
distribution-free method.
The Need for UCL Computational
Software
A 95% UCL of the unknown population
arithmetic mean, fiv of a COPC is often used in
environmental applications to:
•	Estimate the exposure point concentration
(EPC) term,
•	Support risk assessment applications,
•	Determine the attainment of cleanup
standards,
•	Estimate background level mean contaminant
concentrations, or
•	Compare the soil mean concentrations with
site-specific soil screening levels.
In December 2002, the EPA revised the Guidance
Document to Calculate the Upper Confidence
Limits for Exposure Point Concentrations at
Hazardous Waste Sites (OSWER 9285.670).
ProUCL, Version 3.0 consists of all parametric
and non-parametric UCL computation methods
as described in this revised EPA UCL Guidance
Document.
ProUCL also computes the UCLs of the unknown
population mean based upon the positively
skewed gamma distribution, which is often better
suited to model environ men Lai data sets than the
lognormal distribution. For positively skewed
data sets, the default use of a lognormal
distribution often results in unpractically large
UCLs, especially when the data sets are small.
This is also illustrated in the example included in
this fact sheet. In order to obtain accurate and
stable UCLs of practical merit, other distributions
such as a gamma distribution should be used to
model positively skewed data sets. ProUCL,
Version 3.0 has procedures to perform the gamma
goodness-of-fit test and to compute UCLs of the
population mean based upon gam ma distributed
data sets.
ProUCL Version 3.0 Capabilities
Performs Goodness-of-Fit Tests to Assess
Normality/Lognormality of a Data Set Using:
•	Informal graphical quantile-quantile (Q-Q) plot
(normal probability plot) and histogram.
•	Shapiro - Wilk test: to be used when the
sample size is less than or equal to 50.
179CMB04.FS 4/20/2004

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ProUCL Version 3.0 Statistical Software to
Compute Upper Confidence Limits of the
Unknown Population Mean
•	LUliefors test: to be used when the sample size
is large (e.g., greater than 50).
Performs Goodness-of-Fit Test for Gamma
Distribution Using:
•	Informal graphical quantile-quantile (Q-Q) plot
(gamma probability plot) and histogram.
•	Kolmogorov-Smirnov test: available for sample
sizes in the range 4-2500 (critical values
computed using Monte Carlo simulations) and
values of the estimated shape parameter, k in
the interval [0.01,100.0],
•	Anderson-Darling test: available for sample
sizes in the range 4-2500 (critical values
computed using Monte Carlo simulations) and
values of the estimated shape parameter, k in
the interval [0.01,100.0],
Computes the Estimates of Relevant Population
Parameters:
•	Computes all relevant descriptive summary
statistics for raw and log-transformed data.
•	Computes the maximum likelihood (ML) and
minimum variance unbiased (MVU) estimates
of the various population parameters such as
the mean, standard deviation, quantiles,
coefficient of variation, skewness, and also the
MLEs of the shape parameter k and scale
parameter 0 of a gam ma distribution.
Computes Five Parametric UCLs:
A (l-a) 100% (for all values of the confidence
coefficient, (l-a) in the interval [0.5,1.0] including
0.95	except for a couple of methods) UCL of the
unknown population mean, /i,, using five (5)
parametric methods for normal, lognormal, and
gamma distributions. The five parametric UCL
computation methods incorporated in ProUCL
are:
1.	Student's-t UCL: to be used for normally (or at
least approximately normally) distributed data
sets. Student's-t UCL is available for all
confidence coefficients, (l-a) in the interval
[0.5,1.0],
2.	Approximate Gamma UCL: to be used for
gamma distributed data and is typically used
when k hat (ML estimate of the shape
parameter, k) is greater than or equal to 0.5.
Approximate gamma UCL is available for all
confidence coefficients (l-a) in the interval [0.5,
1.0],
3.	Adjusted Gamma UCL: to be used for gamma
distributed data sets and should be used when
k hat is greater than 0.1 and less than 0.5.
Adjusted gamma UCL is available only for
three confidence coefficients: 0.90, 0.95, and
0.99.
4.	H-UCL based upon Land's H-statistic: to be
used for lognormally distributed data sets. In
ProUCL, H-UCL is available only for two
confidence coefficients: 0.90 and 0.95. ProUCL
can compute H-UCL for samples of size up to
2001.
Caution: For highly skewed data sets, H-UCL
should be avoided as the H-statistic often
results in unrealistically large, impractical
and unusable H-UCL values. ProUCL
provides warning messages and recommends
the use of alternative UCLs for such highly
skewed lognormally distributed data sets.
5.	Chebyshev (MVUE) UCL: to be used for
lognormally distributed data sets. This UCL
computation method uses the MVU estimates
of the standard deviation of the mean and of
other parameters of a lognormal distribution.
Chebyshev (MVUE) UCL is available for all
confidence coefficients, (l-a) in the interval
[0.5,1.0],
Computes Ten Non-parametric UCLs Based
Upon Bootstrap Procedures and Chebyshev
Inequality:
These UCLs can be computed for all confidence
coefficients, (l-a) in the interval [0.5,1.0].
1.	Central Limit Theorem (CLT) based UCL: to
be used when the sample size is large.
2.	Adjusted-CLT (Adjusted for skewness) UCL:
to be used for skewed data sets of large sizes.
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ProUCL Version 3.0 Statistical Software to
Compute Upper Confidence Limits of the
Unknown Population Mean
3.	Modified-t statistic (Adjusted for skewness)
based UCL: may be used for mildly skewed
data.
4.	Chebyshev (Mean, Sd) UCL: based upon the
sample mean and standard deviation, Sd.
5.	Jackknife UCL for mean (same as Student's-t
UCL).
6.	Standard Bootstrap UCL.
7.	Bootstrap-t UCL.
8.	Hall's Bootstrap UCL.
9.	Percentile Bootstrap UCL.
10. Bias-corrected accelerated (BCA) Bootstrap
UCL.
As mentioned before, for most of the UCL
computation methods, ProUCL can compute the
UCLs for all confidence coefficients, (l-a) in the
interval [0.5,1.0]. However, since in most
environmental applications (e.g., estimation of
EPC), a 95% UCL of mean is used, therefore,
ProUCL makes recommendations for the most
appropriate 95% UCL(s) which may be used to
estimate the unknown population mean
concentration. The basis and theoretical
justification for these recommendations can be
found in the references listed in this fact sheet.
Example: An example illustrating the importance
of the use of gamma distribution based UCL
computation methods is discussed next. Consider
a simulated positively skewed data set of size 20:
0.0086284103,16.18078972, 7.334853523,
6.12856E-005,1.756500498,1.394359005,
23.41857632, 7.516539628, 0.8594623274,
39.06134332,17.97357103,114.885481,
9.251610362, 39.44123801, 71.64271025,
6.271065467, 0.9742964478, 0.1558884758,
0.4817911951, 0.0065875373. This data set was
generated from a gam ma distribution with the
shape parameter, k = 0.25 and the scale
parameter, 0 = 50. This can be seen from the two
enclosed figures, a histogram and the associated
gamma probability plot generated by ProUCL.
This data set does not follow a normal or a
lognormal distribution. However, using the
default lognormal (as some Limes done in practice
for positively skewed environ men Lai data seLs)
distribution, one will get an unrealistically large
95% H-UCL (=498390.58) as given in the enclosed
ouLpuL generaLed by ProUCL. This in turn will
lead to Lhe use of Lhe maximum value (=114.885)
as an estimate of Lhe EPC Lerm - a pracLice often
used in environmental applications. However,
since Lhe daLa seL does follow a gamma
disLribution, Lherefore, an appropriaLe and more
accuraLe estimate of Lhe EPC term (representing
Lhe average exposure over a long period of Lime
over an exposure area) will be obLained by using
Lhe 95% UCL (=45.067) based upon Lhe gamma
disLribution as recommended in Lhe enclosed
ouLpuL Lable generaLed by ProUCL, Version 3.0.
Summary
ProUCL compuLes parameLric UCLs based upon a
normal, lognormal, and a gamma disLribution.
ProUCL also compuLes UCLs using several non-
parameLric methods. The compuLaLion of an
appropriaLe UCL of Lhe unknown population
mean depends upon Lhe daLa disLribuLion,
Lherefore goodness-of-fiL LesLs need Lo be
performed Lo assess Lhe daLa disLribution before
using one of Lhe UCL compuLaLion meLhods
available in ProUCL. Based upon an appropriaLe
daLa disLribuLion and Lhe associaLed skewness,
ProUCL provides recommendations abouL one or
more 95% UCL compuLaLion meLhods LhaL may
be used Lo estimate Lhe unknown mean
concenLraLion of a COPC.
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ProUCL Version 3.0 Statistical Software to
Compute Upper Confidence Limits of the
Unknown Population Mean
~l File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests UCLs Window Help
~ & I x uHlf
Histogram of G(.25,50)
Close window
For Helpj press F1

Gamma Q-Q Plot for G[. 25,50]
40	GO	80
Theoretical Quantiles of Gamma Distribution
N = 20, Mean = 17.931, k hat = 0.301
Slope = 1.029, Intercept = 0.414, Correlation, R = 0.996
K-S Test Statistic = 0.103, Critical Value(0.05) = 0.210, Data are Gamma Distributed
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ProUCL Version 3.0 Statistical Software to
Compute Upper Confidence Limits of the
Unknown Population Mean
ProUCL Version 3.0

llii
File Edit View Options Summary Statistics Histogram Goodness-of-Fit Tests
UCLs Window Help
Da? jt ^ H fl f V?
P UCL Statistics for G(.25,50)

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B | C D
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F
G
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Data File
D:\PData\Gamma-Test-File2.txt

Variable:
G(.25,50)



2


3
Raw Statistics

Normal Distribution Test

4
Number of Valid Samples
20
Shapiro-Wilk Test Statisitic
0.6627653

5
Number of Unique Samples
20
Shapiro-Wilk 5% Critical Value
0.905

6
Minimum
6.13E-005
Data not normal at 5% significance level


7
Maximum
114.88548


8
Mean
17.930768
95% UCL (As
suming Normal Distribution)

9
Median
6.8029595
Student's-t UCL
29.275688

10
Standard Deviation
29.341896


11
Variance
860.94688
Gamma Distribution Test
12
Coefficient of Variation
1.6363993
A-D Test Statistic
0.1492209

13
Skewness 2.3948728
A-D 5% Critical Value
0.8445354

14

K-S Test Statistic
0.1026697

15
Gamma Statistics
K-S 5% Critical Value
0.2101034

16
k hat
0.3008281
Data follow gamma distribution

17
k star (bias corrected)
0.2890372
at 5% significance level
18
Theta hat
59.604702

19
Theta star
62.036194
95% UCLs (Assuming Gamma Distribution)
20
nu hat
12.033123
Approximate Gamma UCL
41.976248

21
nu star
11.561488
Adjusted Gamma UCL
45.066538

22
Approx.Chi Square Value (.05)
4.9386585


23
Adjusted Level of Significance
0.038
Lognormal Distribution Test
24
Adjusted Chi Square Value
4.6000062
Shapiro-Wilk Test Statisitic
0.8641123

25

Shapiro-Wilk 5% Critical Value
0.905
26
Log-transformed Statistics
Data not lognormal at 5% significance level
27
Minimum of log data
-9.699966

28
Maximum of log data
4.7439358
0.5953246
95% UCLs (Assuming Lognormal Distribution)

29
Mean of log data
95% H-UCL
498390.58

30
Standard Deviation of log data
3.6226212 95% Chebyshev (MVUE) UCL
1438.355

31
Variance o
flog data
13.123384
97.5% Chebyshev (MVUE) UCL
1932.73

32

99% Chebyshev (MVUE) UCL
2903.834

33



34




95% Non-parametric UCLs

35




CLT UCL
28.72273

36




Adj-CLT UCL (Adjusted for skewness)
32.476962
37



Mod-t UCL (Adjusted for skewness)
29.861273

38




Jackknife UCL
29.275688

39




Standard Bootstrap UCL
28.161388

40



Bootstrap-t UCL
39.768558

41
RECOMMENDATION
Hall's Bootstrap UCL
71.935053

42
Data follow gamma distribution (0.05)
Percentile Bootstrap UCL
29.194241
43

BCA Bootstrap UCL
34.604005

44
Use Adjusted Gamma UCL
95% Chebyshev (Mean, Sd) UCL
46.529711

45




97.5% Chebyshev (Mean, Sd) UCL
58.904496

46



99% Chebyshev (Mean, Sd) UCL
83.212366

17






|< 1 ~ l\ General Statistics /
For Help, press F1	^
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ProUCL Version 3.0 Statistical Software to
Compute Upper Confidence Limits of the
Unknown Population Mean
Computer Requirements to Operate
ProUCL
Installation of ProUCL Version 3.0 requires a
microprocessor speed of at least 200MHz, 12 MB
of hard drive space, 48 MB of memory (RAM),
and Windows 98 (or newer) operating system.
ProUCL is compatible with Windows NT-4,
Windows 2000, Windows XP, and Windows ME.
Installation
ProUCL can be downloaded from the TSC
website at www.epa.gov/nerlesdl /tsc/tsc.htm.
The website contains download and usage
instructions.
Find More Information About ProUCL
The TSC website at
www.epa.gov/nerlesdl / tsc/tsc.htm provides
additional information. EPA technical issue
papers used in the development of ProUCL are
also available at the TSC website. For additional
information, contact:
Gareth Pearson, TSC Director
E-mail: pearson.gareth@epa.gov
Phone: 702-798-2270
The website containing information on the 2002
EPA guidance for calculating the 95% UCLs is:
www.epa.gov/superfund/programs/risk/
ragsa/ucl.pdf.
References
EPA (2002), Calculating Upper Confidence
Limits for Exposure Point Concentrations at
Hazardous Waste Sites, OSWER 9285.6-10,
December 2002.
Schulz, T. W., and Griffin, S. (1999), Estimating
Risk Assessment Exposure Point
Concentrations When Data are Not Normal
or Lognormal. Risk Analysis, Vol. 19, No. 4,
1999.
Singh, A. K., Singh, A., and Engelhardt, M.
(1997), "The Lognormal Distribution in
Environmental Applications," EPA/600/R-
97/ 006, December 1997.
Singh, A., Singh, A. K., and lad, R. J. (2002).
"Estimation of the Exposure Point
Concentration Term Using a Gam ma
Distribution." EPA/600/R-02/084.
Singh, A., Singh, A. K., Engelhardt, M., and
Nocerino, J.M. (2003), "On the Computation
of the Upper Confidence Limit of the Mean
of Contaminant Data Distributions." Under
EPA Review.
Singh, A. and Singh, A.K. (2003). Estimation of
the Exposure Point Concentration Term
(95% UCL) Using Bias-Corrected
Accelerated (BCA) Bootstrap Method and
Several Other Methods for Normal,
Lognormal, and Gamma Distributions. Draft
EPA Internal Report.
Notice
The U. S. Environmental Protection Agency through its Office of Research and Development funded and
managed the research described here under Interagency Agreement DW47939416 to the Government
Services Administration (GSA). It has been subject to the Agency's peer and administrative review and
has been approved for publication as an EPA document.
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