United States
Environmental Protection
Agency
Office of
Research and
Development
Office of Solid Waste
and Emergency Response
EPA/600/S-99/006
November 1999
&EPA Technology Support Center Issue
Some Practical Aspects of Sample Size
and Power Computations for Estimating
the Mean of Positively Skewed
Distributions in Environmental
Applications
Ashok K. Singh1, Anita Singh2, and Max Engelhardt3
The Technology Support Projects,
Technology Support Center (TSC) for
Monitoring and Site Characterization was
established in 1987 as a result of an agreement
between the Office of Research and
Development (ORD), the Office of Solid Waste
and Emergency Response (OSWER) and all ten
Regional Offices. The objectives of the
Technology Support Project and the TSC were
to make available and provide ORD's state-of-
the-science contaminant characterization
technologies and expertise to Regional staff,
facilitate the evaluation and application of site
characterization technologies at Superfund and
RCRA sites, and to improve communications
between Regions and ORD Laboratories. The
TSC identified a need to provide federal, state,
and private environmental scientists working on
hazardous waste sites with a technical issue
paper that identifies data assessment applications
that can be implemented to better define and
identify the distribution of hazardous waste site
contaminants. The examples given in this Issue
paper and the recommendations provided were
the result of numerous data assessment
approaches performed by the TSC at hazardous
waste sites.
This paper was prepared by Ashok K. Singh,
Anita Singh, and Max Engelhardt. Support for
this project was provided by the EPA National
Exposure Research Laboratory's Environmental
Sciences Division with the assistance of the
Superfund Technical Support Projects
Technology Support Center for Monitoring and
Site Characterization, OSWER's Technology
Innovation Office, the U.S. DOE Idaho National
Engineering and Environmental Laboratory, and
the Associated Western Universities Faculty
Fellowship Program. For further information,
contact Ken Brown, Technology Support Center
Director, at (702) 798-2270, Ashok K. Singh at
(702) 895-0364, Anita Singh at (702) 897-3234,
or Max Engelhardt at (208) 526-2100.
Purpose and Scope
Often in Superfund applications of the U.S.
EPA, exposure assessment and cleanup deci-
sions are made based upon the mean concentra-
tions of the contaminants of potential concern
(COPC) at a polluted site. The objective may be
to 1) compare the soil concentrations with site
specific or generic soil screening levels (SSLs),
*
cf
o
s^C
\
echnology
i upport
reject
1 Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154
2 Lockheed Martin Environmental Systems & Technologies, 980 Kelly Johnson Dr., Las Vegas, NV 89119
3 Lockheed Martin Idaho Technologies, P.O. Box 1625, Idaho Falls, ID 83415-3730
Technology Support Center for Technology Innovation Office
Monitoring and Site Characterization Office of Solid Waste and Emergency
National Exposure Research Laboratory Response
Environmental Sciences Division U.S. EPA, Washington, D.C.
Las Vegas, Nevada 89193-3478 Walter W. Kovalick, Jr., Ph.D., Director
Printed on Recycled Paper
Rev. 2/23/0Ǥ25CMB99.RPT
-------
2) compute the exposure point concentration
(EPC) term used as one of several parameters to
estimate the contaminant intake for an individual,
or 3) verify the attainment of cleanup goals
(CUGs) or cleanup standard as set forth in the
Record of Decision (ROD) agreed upon by all
concerned parties, such as the USEPA and the
party responsible for introducing contamination at
the site. The CUG of a COPC has been denoted
by Cs throughout this article. Suppose that a
COPC is believed to be present at a certain site
and its concentration varies according to a
probability distribution with an unknown mean, //.
The mean, //, is one of the commonly used
measures of the central tendency of a distribution
and is often used to represent the EPC term or
some cleanup standard at a site. The mean, //, is
typically estimated by the sample mean and some
upper confidence limit (UCL) of the mean, which
are obtained using the sampled data.
The decisions about the population mean are
made using testing of hypotheses about the
population mean. In general, there are two
hypotheses of interest, the null hypothesis,
denoted by H0, and the alternative hypothesis,
denoted by Ha. In Superfund applications, such as
the determination of exposure assessment or the
attainment of cleanup levels, it is of interest to test
one-sided hypotheses about the population mean;
therefore, all sample size and power computation
discussions in this paper have been done for one-
sided hypotheses. For example, suppose a
regulator suspects that the mean concentration of
the contaminant exceeds a specified level, say //0
(the CUG, Cs), but the party responsible for
introducing the contaminant claims that the mean
concentration is below //0. A mathematical
formulation of these hypotheses would be the null
hypothesis, H0: // >CS, which is the regulator's
claim, and the alternative hypothesis, Ha: //
-------
null hypothesis is designated, an environmental
chemist may find this definition of the false
positive error rate contradictory to his intuitive
definition of a false positive. As adopted in EPA
documents (1989b), the statistical convention has
been followed in this paper as well.
One of the inherent assumptions required to
determine the sample size is that one is dealing
only with a single statistical population (e.g., one
remediated part of the site). Violation of this
assumption can lead to invalid applications of a
statistical model (e.g., lognormal) and technique.
For example, a normally distributed data set with
a few outliers can be incorrectly modeled by the
lognormal distribution with the lognormal
assumption hiding the outliers. Also, the mixture
of two or more datasets with significantly
different mean concentrations, such as one coming
from a clean part and the other taken from a
contaminated part of the site, can also be
incorrectly modeled by a lognormal distribution.
These are frequent occurrences in environmental
applications as discussed by Singh, Singh, and
Engelhardt (1997). It appears that the use of the
lognormal distribution and the H-statistic based
UCL tend to hide contamination rather than find
it. Actually, under the assumption of lognormal
distribution, one can get away with very little or
no cleanup (Bowers, Neil, and Murphy, 1994) at a
polluted site. Moreover, there are practical
problems which can occur even when the
lognormal assumption is correct; especially when
the distribution is highly skewed and the number
of samples taken (available data) is small. In such
a case the H-UCL of the mean can be orders of
magnitude greater than the true mean
concentration, making the UCL of little practical
use for the intended purpose.
The main objective of this article is to discuss
some problems regarding the lognormal
assumption and how they relate to sample size
determination needed to draw reliable inference
about hypotheses testing for the population mean
with prespecified performance parameters.
Methods for computing the number of samples
from a normally distributed population are
available in the literature (Bain and Engelhardt,
1992). The use of the standard sample-size
formula when the population variance is unknown
has been discussed by Kupper and Hafher (1989),
who proposed a simple adjustment for sample size
determination when the population variance is
unknown. For a lognormal distribution, many
times, the practitioners like to use the standard
formula given by equation (5) below, as an
approximation, but it has been recommended
(Stewart, 1994) that caution should be exercised
while using the standard formulas for computing
the number of samples.
In general, when a statistical procedure is
based on correct assumptions, by taking a
sufficiently large number of samples it is possible
to make decisions about the parameters (e.g., the
mean) with whatever level of confidence is
prescribed. However, in real applications, taking a
large number of samples may not be practical as it
may be time consuming with an unacceptably
high cost. When approximate formulas are used
for determining the number of samples, or if the
lognormal assumption is wrong, it is possible to
end up with either too many or too few samples.
In the former case, the cleanup expense will be
too high, and in the latter case, the actual level of
confidence may fall short of what was prescribed
by the regulators. It is very well possible that it
may not be feasible to achieve the desired
performance parameters without taking an
enormous number of samples. This is especially
true when a lognormal model is assumed.
Therefore, it becomes necessary to find a balance
between the choice of performance parameters
(error rates, power) and the number of samples
needed for hypothesis testing. For example, when
more (or less) samples are taken, then the gain (or
loss) in levels of performance standards for the
various approximate formulas need to be
investigated. Keeping some of these practical
considerations in mind, the regulators may have to
settle for reduced values of performance
standards. A multi-phase approach may have to
be adopted. In this article, several UCL of the
mean computation methods have been compared
via Monte Carlo simulations.
A convenient way to perform a test of
hypotheses about an unknown parameter is to first
compute a confidence interval for the parameter,
-3-
-------
and then reject H0 if the hypothesized value, in
this case the cleanup standard, Cs, lies outside of
the interval. For the verification of the attainment
of a cleanup goal, the test based upon the one-
sided UCL of the mean is typically used. A one-
sided UCL is a statistic such that the true
population mean is less than the UCL with a
prescribed level of confidence, say (l-oc)100%.
The associated test rejects H0, suggesting that the
site is clean if UCL < Cs. The choice of an
appropriate statistical procedure depends on the
distributional assumptions and knowledge of the
variance, o2.
2.0 Normal and Lognormal Distribution
Let xl5 x2, ..., xn be a random sample from a
population with unknown mean, //, and variance,
a2. Denote the sample mean and sample variance,
respectively, as
x = -
=1
xp and
(1)
Sx =
n-\
=1
(2)
An important feature of the normal model is
that the mean and standard deviation (sd) are
location and scale parameters, respectively. In
particular, if a normally distributed random
variable is transformed by adding a constant, the
effect on the density function is a simple linear
translation without changing the shape of the
density function. For example, given two normal
densities with the same sd, if the means differ by 3
units, then the 90th percentiles also differ by 3
units, the 95th percentiles differ by 3 units, and so
forth. This makes it possible to derive a sample
size formula in terms of the difference, or the
error margin (limit), A = Cs- //;. Figure 1 exhibits
normally distributed densities for three different
sites, each with the same sd, o= 0.5 ppm, but with
different means (designated by dashed vertical
lines), // = 2 ppm at site A, // = 5 at site B, and // =
10 at site C.
Normal density functions
Figure 1. Normal density functions with
different means, // = 2, 5, 10.
However, the mean of a lognormal
distribution is not a location parameter. It is
possible for lognormal distributions which appear
to be located at roughly the same place to have
very different means and, conversely, there exist
lognormal distributions with the same mean, //;,
which appear to differ in location. This makes it
impossible to derive a sample size formula in
terms of the error margin, A. Thus, for a
lognormal model, the problem reduces to
distinguishing between the two populations with
mean, // (> CJ, and n1 (
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graphical paradigm, to recognize that all five
distributions have the same mean. It can be
shown mathematically that larger values of a
yield distributions not only with greater skewness,
but also with a thicker right tail. The distribution
mean tends toward the thicker tail of a skewed
distribution, but tail thickness is generally hard to
spot through visual inspection.
Lognormal probability density functions
01234567
9 10 11 12
Figure 2. Lognormal density functions with
means, //] = 2, 5, 10.
Lognormal density functions with
= 0.1. /i = 0.69
= 0.3, u, = 0.65
C: a = 0.5. ^ = 0.57
D: a = 1.0. ji = 0.1
E: a = 1.5. u = -0.43
Figure 3. Lognormal density functions, all with
A = 2.
The mean is an intuitive and a commonly used
measure of central tendency of a distribution. The
sample mean and the associated UCL are often
used to verify the attainment of cleanup goals and
SSLs, and to estimate the EPC terms in exposure
and risk assessment studies. USEPA guidance
documents recommend the use of H-statistics to
compute a UCL of the mean of a lognormal
distribution (EPA, 1989a, 1992, 1996). A detailed
discussion of H-statistics is given in Gilbert
(1987). Even though, for a lognormal
distribution, the test based on the H-statistic is
uniformly most powerful (UMP), it has little
practical merit for positively skewed data sets of
small sizes, such as 20-30 or less. Also, recent
work by Gilbert (1993) and Singh, Singh, and
Engelhardt (1997) indicates that statistical tests of
hypotheses based on H-statistics can yield
unusually high false negatives (not rejecting H0 as
defined above, when in fact it is false), which
would result in unnecessary cleanup. This is
especially true for samples of small sizes from
skewed populations with a exceeding 1 as can be
seen in Figures 3A-3C, 4A-4D, 11A-11C, and
12A-12D. These comments suggest that for large
values of a, mean is not a good measure of central
tendency for a lognormal distribution. Other
parameters which are sometimes used as measures
of central tendency are the median (50th
percentile) and the mode (maximum of the
density) of the distribution. For a unimodal
symmetric distribution, such as the normal
distribution, the mean, median, and mode are the
same. However, the mean, median, and mode can
be quite different for a highly skewed distribution.
For example, the mean, median, and mode of
population E in Figure 3 are 2.0, 0.65, and 0.05,
respectively.
Singh, Singh, and Engelhardt (1997) note that
ordinarily one would expect the mean and the
associated 95% UCL of the mean to be smaller
than the 95th percentile of the sampled population.
While this is a likely occurrence when the
population is normal, the situation can be
somewhat different with lognormal populations.
For example, it has been observed (see Example
1) that the mean and, consequently, the H-statistic
based UCL of the mean can exceed the 90% or
95% percentiles of the lognormal distribution by
orders of magnitude, especially for skewed
datasets of small sizes. This fact can be easily
seen by comparing percentiles with the mean of a
lognormal distribution. The population mean, //b
is greater than xp, the 100pth percentile of a
lognormal distribution if and only if a > 2zp ,
where z is the 100 pth percentile of the standard
-5-
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normal distribution. For example, when p = 0.80,
zp = 0.842, then ft exceeds x 0 80, the 80th
percentile, if and only if a > 1.68, and ft will
exceed the 95th percentile if and only if a > 3.29.
This observation and the simulation results
summarized in Section 4 suggest that for a
exceeding 2 and samples of a size as large as 50,
the sample mean and the associated H-statistic
based UCL for the lognormal mean becomes
unrealistically large and cannot be considered a
reliable estimate of a cleanup standard or of an
EPC term. Due to these reasons, the 1996 EPA
Soil Screening Guidance Document abandoned
the use of H-UCL to compare the soil
concentrations with the SSLs.
Using Monte Carlo simulation experiments, it
has been observed that the H-statistic based UCL
for the mean is greater than the true mean and the
CUG by orders of magnitude even when the
sample was drawn from a population with a mean
smaller than the cleanup standard, as can be seen
in Figures 7A, 7B, 8A-8D, and 15A, 15B, 16A-
16D. It is, therefore, desirable to have procedures
which work better (achieving the pre-specified
performance measures, approximately) than the
H-statistic based UCL.
Example 1. Consider a simulated dataset of size
n = 15 from a lognormal distribution, LN(5,
(1.71)2). The generated data range from 16.52 to
1498.61, and are given in Singh, Singh, and
Engelhardt (1997). The mean and sd of the
lognormal distribution are //; = 629.55, and ol =
2595.18, and the 80th, 90th, and the 95th
percentiles for this lognormal distribution are
626.29, 1329.05, and 2472.41, respectively. Note
that the sd, 1.71, exceeds the 2*z080 = 2* 0.842 =
1.68; therefore, the mean, ft,, already exceeds the
80th percentile of the lognormal distribution. The
95% UCL of the mean based on the t-distribution,
central limit theorem (CLT), the Chebychev
theorem (based on the minimum variance
unbiased estimates of mean and sd of the
lognormal distribution), and the H-statistic are:
749.31, 731.14, 2059.47, and 4613.32, respective-
ly. Notice that the 95% H-UCL is 4613.32 which
exceeds the 95th percentile value of 2472.41 for
the lognormal distribution. Thus, even though the
H-UCL is theoretically sound and possesses
optimal properties, the practical merit of the use of
H-UCL is questionable, as it becomes quite large
when the sd of the log-transformed variable starts
to exceed 1.0. This is especially true for samples
of small sizes (viz., <30).
In light of the above remarks, it is crucial that
great care should be exercised in choosing an
appropriate model and in understanding the
potential problems associated with the chosen
model when attempting to make decisions about a
population mean. Other measure of central
tendency, such as the median or some other
quantile (e.g., 75%, 85%) and other distributions
(e.g., Weibull, Gamma) need to be considered for
highly skewed data sets, which will be discussed
in a sequel article. In addition to describing the
complexity of interpreting statistical evidence
about the mean of a lognormal, this article also
discusses the difficulties involved in selecting an
adequate number of samples when drawing an
inference about the mean of a lognormal
distribution. In order to shed some light on these
issues, the coverage probability, statistical power,
and the UCLs for the various procedures have
been compared via Monte Carlo simulation
experiments. Section 2 has a brief description of
normal and lognormal distributions, and Section 3
discusses the methods for computing the UCL of
the mean of a lognormal distribution. In Section
4, the power and the UCL of the mean obtained
using these procedures have been compared via
Monte Carlo simulation experiments, and
conclusions have been summarized in Section 5.
-6-
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2.1 Normally Distributed Datasets
Variance, a2, is known
If a2 is known, then a (1 -a) 100% one-sided UCL
for the mean is given by the equation:
UCL = x +
(3)
where z^ is the (l-oc)th quantile of the standard
normal distribution (SND) and H0 (defined earlier)
is rejected if UCL < Cs with a false positive rate of
a. In order to study the false negative rate, P, it is
necessary to consider the power function of the
test. In general, the power function, denoted by
= P[H0 is rejected given // is true] = Pfreject
H0 //], is the probability of rejecting H0. Using
the properties of the SND, the power function is
given as follows:
DO*) =
(4)
False positive
rate, a = 0.05 ' 0.0
Cleanup standard
9.0 9.2 9.4 9.6 9.B 10.0 10.2
Population mean, ft (ppm)
Figure 4. Power function for test of hypothesis
H0: //> 10 ppm.
where O is the standard normal cumulative
distribution function. Thus, if it is critical to
detect when the difference between the true mean
and the hypothesized mean is at least A, then it
would be desirable to have a low false negative
rate, P at //l5 with HO";) = 1 ~ P and //j = Cs -
A. For example, suppose it is required to perform
a test of H0: // > 10 ppm, and the test is based on a
sample from a normal population with known o =
1, and it is required to have no more than 5%(a =
0.05) false positive decisions and no more than
10% (/?= 0.10) false negative decisions if the true
mean is // < 9 ppm, a scenario given in Figure 4.
For a given choice of the standardized
difference, d = (Cs - //)/<7, and error rate, a, there
is no guarantee that a prescribed false negative
rate, /3, will be achieved. Note that for normal
distributions, the power function in equation (4) is
an increasing function of the sample size, n, and
difference, d. Thus the power function can be
made arbitrarily close to 1 by choosing
sufficiently large n (this may not be practical).
Thus, equating the power in equation (4) to
1 - P and solving it for n results in the following
formula,
n
(5)
Note that in equation (5), one can also use the
critical values based upon the Chebychev bound,
which will only result in a higher sample size as
the critical values based on the Chebychev
inequality will be higher than those based upon
the normal distribution. For example, for a, = 0.05
and P = 0. 10, zj _ K = 1 .645, and zj _p = 1 .282,
whereas the corresponding conservative cut offs
based upon the Chebychev inequality are 4.47 and
31.6, respectively.
Example 2. Suppose it is required to have a test
in which the false positive rate is a, = 0.05 and the
false negative rate is P = 0.10 when the true mean,
//, is 0.5 sd below Cs with d = ( Cs - p)la= 0.5.
The sample size required to achieve these
performance parameters is = [(1.645 +
1.282)/0.5]2 = 34.3, which when rounded up yield
a value of 35. For a less stringent condition, with
a and P as 0.05 and 0.10, respectively, and d =
1.0, the sample size is given by n = [(1.645 +
1.282)/1.0]2 = 8.6, which can be rounded up to 9.
Variance, a2, is unknown
If o2 is unknown, then a (1 - #r)100% one-
sided UCL for the mean is provided by
UCL= JF+ f^.^/V^, (6)
where ^ _„_„_! is the (1 -
-------
freedom (df). The test rejects H0 and declares the
site clean if UCL < Cs, with a false positive error
rate, a. In this case, the power function can still
be expressed as the probability that the UCL
defined by equation (6) is less than Cs, but its
evaluation becomes complicated requiring the use
of the noncentral t distribution. Tables of sample
size based on the noncentral t distribution are
given in Bain and Engelhardt (1992).
Power (a known)
Power [u unknown)
Folse positive
rate, a = 0.05 \ Q_Q
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Standardized difference d
Figure 5. Power functions for tests of mean
withn= 11 and cc = 0.05.
Example 2 (continued). Consider Example 1,
withJ= 1.0, a = 0.05, (3 = 0.10, and ais
unknown. Using the non-central t distribution
based tables (Bain and Engelhardt), the required
sample size comes out to be n =11, which is only
two more than the sample size obtained using
equation (5). For n =11, the power functions for
the two cases, a known and a unknown, have
been plotted in Figure 5. Kupper and Hafner
(1989) note that the actual power attained using
sample sizes computed using equation (5) is
generally quite close to the desired power, as can
also be seen in Figure 5. The sample size required
for the test when o is unknown will always be
larger than when it is known. Specifically, when
a is unknown, Kupper and Hafner recommended
adjusting the sample size by adding 2 or 3 to the
result obtained by using equation (5).
2.2 Positively Skewed Datasets
For a lognormal population, the skewness is a
function of o, as can be seen in Figure 3. A
lognormal distribution is typically used for highly
positively skewed datasets. If there is evidence
that the population distribution is positively
skewed, especially when a is larger than 1, then a
lognormal distribution is often assumed. Let xb x2,
... , xn be a random sample from a lognormal
population, LN(//, <72). In other words, the natural
logarithm of data are normal with mean // and
variance o2, N(/u, o2). Lety and sy, denote the
sample mean and sample sd, respectively, of the
log-transformed data j, = Info); / = 1, 2, ... , n.
For a lognormal population, the mean is fa =
exp(// + 0.5 o2) and the median is given by M =
exp(//). Note that fa is the mean of the lognormal
distribution and // is the mean of the transformed
distribution, and for positive values of a, fa is
always greater than M
Variance, a2, is known
When <72 is known, the problem can be
converted to the sample size determination for a
normal population by means of a log-
transformation. The null hypothesis, H0: fa > Cs,
is equivalent to the hypothesis, H0: // > ln(CJ -
0.5<72, which can be tested by comparing the
normal distribution based UCL applied to the
transformed data with the hypothesized mean,
//0 = ln(Q-0.5a2.
Variance, 02, is unknown
There is no easy solution to compute the
sample size, power, and the UCL of the lognormal
mean when a2 is unknown. Some of the available
procedures are discussed in the following.
Test based upon the median, M
A test for the median, M = exp(//), is
sometimes used to test for the mean of a
lognormal distribution. The hypothesis for the
median is H0: M > Cs, which is equivalent to H0:
// > ln(CJ, and the results for the normal
population can be applied with //0 = ln(Q). If o2
is known, then a test (or the UCL) based on the
transformed data for a normal distribution may be
used; on the other hand, if o2 is unknown, then a
test based on the Student's -1 distribution can be
used. It should be stressed that this only provides
-------
a test for the median, M, and does not provide a
test of the lognormal mean, //]. If the amount of
skewness is small, then the distribution is
approximately symmetrical, as can be seen in
Figure 3, M closely approximates the mean, //j,
and a test of the median may reasonably be used
as an approximate test of the mean. To study how
well this approximation works, a small numerical
study was carried out to compare the nominal and
actual significance levels when the test of a
median is used as a test of the mean when a is
unknown. The technical details are given in the
Appendix. The numerical results are given below
in Table 1 for a sample of size n = 10.
From Table 1, it is clear that if the value of o
is small, say less than 0.10, then the nominal and
actual significance levels don't appear to differ a
great deal. However, for am the neighborhood of
0.25, the actual level is about double the nominal
level, and if o is as large as 1.0, they differ by
roughly an order of magnitude. This means that
even for small values of o, such as 0.2-0.25, a test
based on the median will have a higher false
positive rate than the pre-specified a for declaring
a site to be clean when, in fact it is polluted. This
discrepancy increases dramatically with an
increase in o. Assuming the resulting
approximate test is deemed usable (at least for
small values of o), the sample size formula as
given by equation (5) above (or using Kupper and
Hafner,1989 adjustment) could be applied to
determine an appropriate sample size.
As pointed out earlier, a lognormal
distribution is often used for large values of o
such as those exceeding 0.75-1.0; therefore, the
use of the median-based test for the mean of a
lognormal distribution does not seem to be a
feasible or desirable option. For values of o
smaller than 0.5, a normal distribution is often
used and one needs not consider a lognormal
distribution. Also, many exposure and risk
assessment applications do not recommend the
use of median based estimates of the exposure
point concentration term (EPA 1992, 1996).
Moreover, for skewed datasets with o exceeding
0.5, the median is much smaller than the mean of
the distribution. For example, when // = 3.594
and o=l, the mean, //;, of the lognormal
distribution is 60, and median, M, is 36.4. Thus a
test based on equation (6) for testing the median
does not provide an adequate approximate test of
the lognormal mean when a is larger than 0.5.
Therefore, the power and the UCLs for the test
based on median have not been plotted in the
figures given in Section 4.
3.0 Computation of the UCL of the Mean
Modified t - test for asymmetrical populations
If o2 is unknown, then Johnson (1978) and
Chen (1995) suggested the use of a modified t-
statistic for testing the mean of a positively
skewed distribution. Using Johnson's modified t -
statistic, a (1 -
-------
The test rejects the hypothesis, H0: //; > (Q)
if the UCL < Cs . The UCL of the mean given by
equation (7) has been used in the simulations
discussed in Section 4. It has been observed that,
for all practical purposes, the difference between
the values of the UCL and the error rates (and,
consequently the sample size) based upon the
Student's t-statistic given by equation (6) and the
modified t-statistic given by equation (7) is
negligible. Therefore, in order to avoid the
cluttering of graphs, the power and the UCL of the
mean based upon the modified t-statistic have not
been plotted in the figures given in Section 4.
H-statistics based UCL when the variance, a2,
is unknown
Land (1971, 1975) derived the uniformly most
powerful unbiased (UMPU) test for the mean, //1;
of a lognormal distribution, which is based on the
H-statistic. As mentioned earlier, EPA guidance
documents (1980a,1992) recommend the use of
the H-statistic based UCL of the mean for
positively skewed distributions. The (l-oc)100%
UCL for the mean, //b based on the H-statistic is
given as follows.
UCL =
(8)
The critical values, H1_a, used in equation (8)
are not readily available and are not available for
n larger than 100. A subset of tables of critical
values, H1_a,as computed by Land (1975) can be
found in Gilbert (1987). Although it is shown by
Land (1971, 1975) that the test based on the H-
statistic has optimal properties, Singh, Singh, and
Engelhardt (1997) point out that this procedure
can sometimes lead to UCLs which are too large
to be of any practical value. This is particularly
true for samples of small sizes and large values of
a. For example, for a in the interval 0.5-1.0, a
sample of size 15 or less is considered small; for o
in the interval, 1-1.5, a sample of size 30 or less
can be considered small, and so forth. This
sample size requirement increases with o. This is
further discussed in the simulation Section 4.0.
Power and sample size for H-statistic based test
Despite the problems which accompany the
lognormal distribution, if the user is certain that
the population is lognormally distributed, then
theoretically, a UCL based on the H-statistic is the
optimal (although, it may not be realistic or
practical) choice for achieving a pre-specified
false positive error rate, a. For a lognormal
distribution, no simple established sample size
formula, such as given by equation (5), is
available which can be used to determine the
number of samples needed to test a lognormal
mean with a pre-specified error limit, A = Cs- n2,
where //2 < Cs (the symbol //, is reserved as a
general term for a lognormal mean in this section).
Since a lognormal mean is not a location
parameter, the error limit can not be handled
simply in terms of the difference, Cs - //2. Thus,
the problem is to determine an adequate number
of samples to collect to be able to detect when the
lognormal mean approaches Cs - A with
prespecified error rates,
-------
H-test is needed, the details for which are given in
the Appendix. A search routine with this power
function can be used to derive the sample size
based on the H-statistic. Some examples
illustrating these points follow.
Example 3. This example provides a comparison
of power functions for H-tests based on three
different sample sizes obtained using the
approximation formula discussed above. The
example is based on the simple choices: Cs = 10, a
= 0.05, p = 0.10, o= 1, and for 6= ln(Q - ln(C,
-A) = 1, d = 1, the error limit in the real domain is
relatively large, A = 6.32. The population sd of
the lognormal distribution in real space is o1 =
13.11. Note that the problem is to obtain an
adequate number of samples to identify when the
mean of the lognormal distribution starts to
approach 3.68 given that the hypothesized mean is
lOppmormore. The power functions are
graphed in Figure 6. Using the incorrect error
limit as ln(J) = ln(6.32) = 1.84 in equation (5),
one gets the erroneous sample size given by n =
[(1.645 + 1.282)(1)/1.84]2 = 2.53, which is
rounded up to 3 (the power for this size is not
plotted in Figure 6). Using the error limit of 6= 1
in equation (5), the approximate sample size is n =
9, and the dashed curve in Figure 6 is the graph of
the power function for an H-test based on this
sample size. A search routine for the power
function of the H-test as given in Appendix A was
also used to determine the sample size, n = 23.
This is the smallest integer such that the power >
1 -/?= 0.90 when 6= 1. Another possible
approximation would be to apply equation (5)
directly mean and sd, fa and ol. That is, even
though equation (5) is derived for use with a
normal population, we plug the error limit and sd
in the real domain, A = 6.32 and o1 = 13.11 into
(5). The result would be n = [(1.645 +
1.282)(13.11)/6.32]2 = 36.9, which is rounded up
to 37. While the number of samples for the
approximation based on <5was too small, this
approximation, as expected yields a value which
False positive
rate, a = 0.05 ( Q Q
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Standardized difference d
Figure 6. Comparison of power functions of H-
test for three sample sizes: n = 9, 23
and 37.
is too large.
From this discussion, one can conclude that
using 6 in equation (5) does not provide the best
approximation because the sample size it yields is
less than half of the one actually required and the
power resulting from this approximation is 0.46,
or roughly half of the nominal value of 0.90.
However, it should be pointed out that the sample
size came out to be 23 because the alternative
mean selected is 3.68 ppm, which is much lower
than the hypothesized mean of 10 ppm. In
practice, many times, the alternative mean is
closer to the hypothesized mean, in which case the
number of samples needed to achieve a pre-
specified power will increase dramatically. For
example, if the alternative mean, //2, is 7, then the
sample size based on the H-statistic will be 122;
whereas, if //2 = 8, then the sample size based on
the H-statistic will become 286 to achieve a power
of 0.9, which may not be practical! Also using 6
in equation (5) for //2 = 7, the approximate sample
size is 67, and for //2 = 8, the sample size is 172!
Example 4. A more practical example with
overlapping densities and a higher value of a is
discussed. LetQ= 100, a= 0.05, /?= 0.10, o =
1.5, A =100-80 = 20 resulting in 6= ln(Q -
\n(Cs-A) = 0.2231, with d = 6la= 0.1488. The
problem is to obtain an appropriate number of
samples to distinguish between two overlapping
log-normal distributions. As shown below, a large
number of samples will be needed to meet these
performance standards. Using the incorrect error
limit as \n(A) = ln(20) = 2.9957 in equation (5),
the sample size comes out to be 2 (rounded from
-11-
-------
2.15). Using 6= 0.2231 as the error limit in
equation (5), the sample size comes out be 387,
which is fairly large. Next, using the search
routine with the H-statistic based power, the
sample size comes out to be 877! Taking about
877 samples to achieve a power of 0.9 is probably
not practical. Another example from a Superfund
site is considered next.
Example 5. This problem was encountered when
working on a dataset from a Superfund site with
benzo(a)pyrene equivalents (BaPE) being the
main COPC. The data came from three areas of
the site. Using the historical data, the sample
mean and sd are 17.872 ppm and 40.41 ppm. The
data do not follow a normal distribution but
follow a lognormal distribution where the mean
and sd of the log transformed data are 1.449 and
1.708, respectively. The CUG for the site is 60
ppm. The objective is to verify the attainment of
the CUG by the three areas of the site. Enough
samples are needed to be collected to be able to
detect when the mean becomes 50 ppm (so that
the areas can be considered clean) with a
confidence coefficient of 0.95 and a power of 0.9.
Use of the incorrect error limit, ln(Zl) = ln(60-
50) = ln(10) = 2.303 in equation (5) resulted in n
= 4.71; and the consultants for the site suggested
that 4.7-5 samples would be sufficient to verify
the attainment of the cleanup standard with the
pre-specified performance objectives, which is
obviously incorrect. Using 6= ln(CJ - \n.(Cs-A)
= In (60) - ln(50) = 0.182 as the error limit in
equation (5) with o= 1.7, the sample size comes
out to be 752. Next using the search routine with
the H-statistic based power, the sample size comes
out to be 1808! Apparently taking 752 or 1808
samples to achieve a power of 0.9 is neither
practical nor desirable. The influence of the
sample size on the power needs to be determined.
Is it really worth taking a large number of samples
such as 1808 in an effort to achieve the high
power of 0.9? From a practical point oj"view', one
needs to know how to compute the UCL of the
mean correctly with maximum practical power for
a given sample size and level of significance.
Obviously, there is no simple solution to the
problem. It is observed that serious
underestimation of the sample size can occur
when ln(Zl) is substituted for the error margin in
equation (5). However, the error limit, 6, based
sample size determination procedure can also lead
to a substantial underestimation of the sample size
for larger values of 6, especially when Qis much
larger than Cs - A, as is the case in Example 3.
For large values of sd, the sample size obtained
using the normal distribution based power or the
H-statistic based power can be unreasonably large
to be of practical merit. The question is, can we
use approximate tests based upon the t-test, the
CLT, or the Chebychev theorem to meet
(approximately) the pre-specified performance
standards? Does there exist a UCL computation
procedure (or model) which yields practical
values for the sample size and the UCL? Does
there exist a procedure which has higher power
(that may not be equal to 0.9, or some other pre-
specified level) than the other procedures for a
given value of n, the sample size? In order to
investigate the power behavior of the various
procedures, a Monte Carlo simulation study has
been performed which is discussed in Section 4.
Chebychev Inequality based UCL of the mean
The Chebychev theorem as discussed by
Singh, Singh, and Engelhardt (1997) to obtain a
conservative estimate of the UCL of the mean of a
lognormal distribution has also been included in
this study. The two-sided Chebychev theorem
states that, given a random variable, X, following
any distribution, continuous or discrete, with a
finite mean, //1; and a sd, <71; we have:
-------
data, the result is no longer guaranteed to be
conservative.
In general, if fa denotes an estimate of the
unknown mean, //b and cr(f^l) is an estimate of the
standard error of fa, then the quantity UCL = fa +
4A7a(fa) will give a 95% UCL for //b which
should tend to be conservative. The power of the
test based on this Chebychev bound has been
included in the simulation experiments as well.
The Chebychev UCL can be computed using the
estimates of mean and sd as given by equations
(1) and (2) above (denoted by Cheb_M) or can be
obtained using the MVUE given by equations (8)
and (10) of Singh, Singh and Engelhardt (1997)
based on the lognormal distribution theory
(denoted by Cheb_MV) to obtain the estimates //j
and cr(fa). The power of the test based on the
Chebychev UCL has been computed using both
sets of estimates, Cheb_M and Cheb_MV.
From the simulation experiments discussed in
Section 4.0 below, it is observed that for all
sample sizes, the Chebychev - UCL is more
conservative (higher) than all other UCLs except
for the UCL based on the H-statistic. The H-UCL
tends to be much larger than the Chebychev-UCL
for large values of sd (> 1.5) and samples of small
sizes (n<30), and as sd increases, this sample size
requirement becomes larger than 30. However, it
is also observed that the H-UCL based upon
samples of small sizes from populations with large
values of o, tend to be unrealistically large
(Singh, Singh, Engelhardt, 1997) to be of any
practical use. Inference based upon such large H-
UCL values would result in a large number of
false negatives as can be seen from Figures 3A-
3C, 4A-4D, 11A-11C, and 12A-12D.
4.0 Monte Carlo Simulation Experiments
From the discussion presented here, it is
obvious that there does not exist a single sample
size determination formula which can be preferred
over the other formulas while sampling from a
lognormal population. Therefore, extensive
Monte Carlo simulation experiments were carried
out and several procedures to compute the 95%
UCL of the mean with a test size of 0.05 have
been compared in terms of their power. In order
to study the power of the various UCL
computation methods, 10,000 samples of sizes 10,
15, 20, 30, 50, and 100 were generated from a
variety of lognormal populations for various
values of the mean above and below the CUG
value. The simulations have been performed for
two CUG values: 10 and 60. Several combinations
of mean and sd have been considered. For CUG =
10, the samples were generated from lognormal
populations with means of 15, 12, 10, 8.2, 6.7,
and 3.7, and for CUG = 60 ppm, samples were
generated from lognormal populations with means
of 100, 70, 60, 50, 40, and 30. The sd values were
0.5,1.0,1.5, and 2.0. Higher sample sizes
including, 150, 200, 300, 400, have been also tried
for larger values of o such as 1.5, and 2.0. The H-
UCL and power could not be computed for the
larger sample sizes as the critical values for the H-
statistic are not available for samples of sizes
larger than 100. Therefore, graphs for values of n
larger than 100 are not included here.
Procedures considered in the simulation
experiments include the Students's t-test,
modified t-test for asymmetric distributions, the
CLT based normal test, a test based on the
median, a test based on the H-statistic, and tests
based on the Chebychev theorem. A close look at
the simulation results suggests that the differences
in the values (power, UCL) obtained using the
Student's t-test and the modified t-test are not
significant. The UCLs based on the modified li-
test are slightly higher than those based on
Student's t-test. For example, for a sample of size
10 and the population mean of 100, the 95% UCL
of the mean using the Student's t-test are 128.77
(a= 0.5), 157.79 (a= 1.0), 190.01 (a= 1.5),
223.27 (a= 2.0), and 242.70 (a= 2.5), whereas
the corresponding 95% UCL using the modified t-
testare 129.39 (a= 0.5), 159.99 (a= 1.0), 194.45
(a= 1.5), 230.24 (a= 2.0), and 251.61 (a= 2.5),
respectively. Also, for small samples, the t-test
results are preferred over the CLT results, and the
t-test results approach the CLT results as the
sample size increases. As observed above, the test
for the mean based on the median does not work
well for values of a larger than 0.5. Therefore, in
order to avoid cluttering the graphs, the
computations based upon the CLT, modified t-
-13-
-------
test, and the median based test are not included in
the graphs presented in this paper. The power and
the UCLs are plotted only for four methods,
namely, the t-test (denoted by t), H-statistic
(denoted by H), the Chebychev UCL using the
simple sample arithmetic mean and sd (denoted by
Cheb_M), and the Chebychev UCL based on the
MVU estimates obtained using the lognormal
theory (denoted by Cheb_MV).
Graphs for the power and the UCL of the
mean for the four procedures have been plotted.
ForCUG= 10, and o= 0.5, 1.0, 1.5, and 2.0, the
power function is displayed in Figures 1A-1F,
2A-2F, 3A-3F, and 4A-4F, respectively, and for
CUG = 60 ppm, and o= 0.5, 1.0, 1.5, and 2.0, the
power function is given in Figures 9A-9F, 10A-
10F, 11A-11F, and 12A-12F, respectively. For
each sample size, the plotted UCLs are the
average values of the respective UCLs over
10,000 iterations for each combination of mean
andsd. For CUG = 10, and a= 0.5, 1.0, 1.5, and
2.0, the UCLs are displayed in Figures 5A-5F,
6A-6F, 7A-7F, and 8A-8F, respectively, and for
CUG = 60 ppm, and a= 0.5, 1.0, 1.5, and 2.0, the
UCLs are given in Figures 13A-13F, 14A-14F,
15A-15F, and 16A-16F, respectively. From these
graphs, the following observations can be been
made.
1. From these graphs, it is clear that the H-
statistic based test does possess the pre-
specified size of 0.05 (significance level) for
all values of n and o.
2. The size of the t-test is larger than the other
three procedures. As the sample size
increases, the differences between the H-
statistic and the t-test based results decrease.
3. From Figures 1A-1F, 2A-2F, 3A-3F, 9A-9F,
10A-10F, and 11A-11F, it is observed that,
for a < 1.5, the test based on the Chebychev
bound has the realized test size (level of
significance) smaller than the pre-specified
test size of 0.05 for samples of all sizes
considered (except for n = 10 and o= 1.5),
which is a desirable property for the
Chebychev UCL to possess. This is
especially true for the Chebychev results
based on the MVU estimates of the mean and
sd of the lognormal distribution.
4. For small valuesof<7(<0.5), the power and
the UCL values based upon the t-test and the
H-statistic are quite close, even for samples of
a size as small as 15 for both CUG values as
can be seen in Figures 1A-1F, 5A-5F, 9A-9F,
and 13A-13F. The discrepancy between the
UCLs and power based on the t-test and the
H-statistic decreases as the sample size
increases.
5. From Figures 2A-2C, 6B-6C, 10A-10C, and
14B-14C, it is observed that for a -1 and
samples of a size smaller than 20, there is not
much difference in the power and the 95%
UCL of the mean based on the H-test and the
Cheb-UCL tests. Actually, the power of H-
test is smaller than the Cheb-UCL for samples
of a size smaller than 10.
6. For o exceeding 1.5, the size of the Cheb-MV
test becomes larger than the pre-specified
level of significance, 0.05 for samples of size
30 or smaller, as can be seen in Figures 4A-
4C and 12A-12C. As the sample size
increases, the size of the test based on the
Cheb-MV comes close to the pre-specified
size of 0.05 and then becomes smaller than
prespecified size as can be seen in Figures
4A- 4F and 12A-12F. A similar pattern will
be observed for larger values of o and the
sample size requirement for the size of the
Cheb-MV test to reach 0.05 will also increase.
7. From Figures 3A-3C, 4A-4D, 11A-11C, and
12A-12D, it can be seen that for o > 1.5, the
test based on the H-statistic yields powers
smaller than the Cheb-MV test for samples of
sizes smaller than 30, which will result in a
large number of false negatives. A similar
pattern will be observed for larger values of o
and the sample size requirement for the power
of the H-test to reach the power of the Cheb-
MV test will also increase.
8. As the sample size increases, the power based
on the Chebychev UCL decreases. For small
sample sizes, the power comes quite close to
the power based on the H-statistics and then
-14-
-------
as the sample size increases, the power
becomes smaller than the power for the H-
test. However, these decrements are not so
dramatic as can be seen in Figures 4A-4F and
12A-12F. Also, as the sample size increases,
the Cheb-MV UCL becomes larger than the
H-UCL. However, those increases are
consistent without any dramatic changes. A
similar pattern will be observed for larger
values of a and the sample size requirement
for the power of the Cheb-MV test to reach
the power of the H-test will also increase.
9. For values of a larger than 1, the H-statistic
based UCL becomes unrealistically large, at
least for samples of sizes smaller than 30 for
both values of the CUG, as can be seen in
Figures 7A-7D, 8A-8D, 15A-15D, and 16A-
16D. Cleanup decisions based on such
unreasonably large H-UCL values cannot be
considered reliable. For example, when //; =
60 and o= 2.0, the 95% quantile of this
lognormal distribution is about 218, whereas
the 95% UCLs of the mean are about 1300010
and 1488 for samples of sizes 10 and 20,
respectively (Figures 16A and 16C). A
similar pattern will be observed for larger
sample sizes as (/increases. For example, for
o= 2.0, the H-UCL is much higher than the
rest of the UCLs even for samples of size 50,
as can be seen in Figures 8A-8D and 16A-
16D. For p, = 60 and a= 2.0, the 95% Cheb-
MV UCL of mean are 244, 228, and 222 for
samples of size 10, 15, and 20, respectively.
5.0 Summary and Recommendations
The discussion presented here leads to the
conclusion that there does not exist a single
sample size determination formula which can be
preferred over the other formulae while sampling
from a lognormal population. From the simulation
results, it appears that it is not feasible to achieve
the desired error rates and critical difference
without taking an enormous number of samples.
This is especially true when a lognormal model is
assumed. Keeping these practical considerations
in mind, the regulators may have to settle for
reduced values of performance standards.
From equation (9), it is concluded that, even
though the H-UCL based test is the test achieving
the pre-specified Type I error rate, for samples of
small size, the associated H-UCL is unreasonably
large, even when samples are obtained from a
lognormal distribution. For example, for o= 2.0,
n = 15, and mean = 60, the H-UCL of the mean is
6493 (Fig.l6B), which is an unlikely event to
happen at a Superfund site. For highly skewed
populations, a large number of samples is needed
to obtain a UCL of practical value.
It is concluded that for less skewed data with
o <0.5, a test based on normal distribution (t-test)
may be used to determine the sample size and
power, and consequently one can use the t-test
based UCL of the mean to verify the attainment of
cleanup standards. For uinthe interval (0.5-1.0),
and for samples of a size smaller than 30, the
Chebychev bound gives reasonable and reliable
results in terms of power and the UCL of the
mean; as the sample size becomes larger than 30,
one can use the test based on the t-statistic. For
samples of small sizes (less than 50) and am the
interval (1.0-1.5), the H-UCL of the mean
becomes large, and the Cheb-MV UCL can be
used to verify the cleanup standard; for samples of
large sizes (greater than 50), the central limit
theorem can be used to compute the UCL of the
mean. For samples of small sizes (less than 100)
and am the interval (1.5-2.0), the H-UCL of the
mean becomes too large, and the Cheb-MV UCL
can be used to verify the cleanup standard (with
higher Type I error rates); for samples of large
sizes (greater than 100) due to the central limit
theorem, one can compute the UCL of the mean
based on the normal theory. Similar patterns will
be observed as the sd increases.
Based on the Monte Carlo simulation results
and the authors' experience with Superfund site
work, the following recommendations are made.
1. A multi-phase approach may be used when
the sample size formulas discussed here result
in an unrealistically high number of samples
(e.g., exceeding 100-200) needed to achieve
the desired performance parameters. One can
start with taking a reasonable and
economically possible number of samples.
The power and size of the tests can then be
-15-
-------
computed. A procedure with maximum power
may be chosen to compute the UCL of the
mean. If deemed necessary for increased
power, more samples can be taken in the next
phases and the UCL of the mean can be re-
computed using the procedure yielding the
maximum power.
2. Low values of error rates, a, and P, and the
error margin, A, result in large sample sizes
which, in reality, may not be achievable. For
example, for a lognormal distribution, with
a= 0.05, ft = 0.1, sd = 1.7, and A = 10, the
sample size needed is 1808, as seen in
Example 5 above, which in not feasible to
collect. For a sample of size 100, and sd = 1.5
and 2.0, and A = 10, the size of the H-UCL
test is about 0.05 but the powers are only
about 0.20 and 0.17 (see Figures 1 IF, 12F),
respectively. If possible, one should consider
reducing the performance objectives.
3. It is recommended to avoid the use of the
lognormal distribution. The appropriate use of
the lognormal distribution is not clear to most
people and it can very easily be used
incorrectly, which may lead to incorrect
conclusions.
4. For highly skewed populations, the arithmetic
mean becomes larger than higher quantiles of
the distribution, such as 90% and 95% (e.g., o
exceeds 3.26), etc. Other measures of central
tendency, such as the median, or some other
quantile (e.g., 80%, 90%) need to be
considered for highly skewed datasets for the
verification of the achievement cleanup goals.
It is crucial that great care should be exercised
in choosing an appropriate model and in
understanding the potential problems associated
with the chosen model when attempting to make
decisions about a population mean. It is also
recommended that some additional Monte Carlo
simulations be done to assess the performance of
the various methods for a variety of skewed
population distributions, such as the Weibull and
the Gamma, and of the sort common with
contamination data (e.g., mixtures and with
outliers).
-16-
-------
Figure 1 A:CUG=10, n=10, sigma=O.S
Figure 1B: CUG=10, n=15, sigma=0.5
4 6 8 10 12 14 16
Arithmetic Mean
8 10 12 14 16
Arithmetic Mean
Figure 1C: CUG=10, n=20, sigma=0.5
Figure 1D: CUG=10, n=30, sigma=0.5
4 6 8 10 12 14 16
Arithmetic Mean
4 6 8 10 12 14
Arithmetic Mean
Figure 1E: CUG=10, n=50, sigma=0.5
Figure 1F: CUG=10, n=100, sigma=0.5
H 1 1-
4 6 8 10 12 14
Arithmetic Mean
8 10 12 14 16
Arithmetic Mean
Figures 1A - IF
-17-
-------
Figure 2A: CUG=10, n=10, sgma=1.0
Figure 2B: CUG=10, n=15, sigma=1.0
4 6 8 10 12 14
Arithmetic Mean
6 8 10 12 14
Arithmetic Mean
Figure 2C: CUG=10, n=20, sigma=1.0
Figure 2D: CUG=10, n=30, sigma=1.0
4 6 8 10 12 14 16
Arithmetic Mean
4 6 8 10 12 14 16
Arithmetic Mean
Figure 2E: CUG=10, n=50, sigma=1.0
Figure 2F: CUG=10, n=100, sigma=1.0
6 8 10 12 14
Arithmetic Mean
6 8 10 12 14 16
Arithmetic Mean
Figures 2A - 2F
-18-
-------
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Figure 3A: CUG=10, n=10, sigma=1.5
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Figure 3E: CUG=10, n=50, sigtna=1.S
2
3S
N
4
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k
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6 8 10 12
Arithmetic Mean
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Figure 3D: CUG=10, n=30, sigma=1.5
N
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Figure 3F: CUG=10, n=100, sigma=1.5
}
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4
s
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u ^
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6 8 10 12 14 16
Arithmetic Mean
Figures 3A - 3F
-19-
-------
Figure 4A: CUG= 10, n = 10, sigma=2.0
^
AO
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N,
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Figure 4B: CUG=10, n=15,
\
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2 4 6 8 10 12 14 16 2
Arithmetic Mean
Figure 4C: CUG=10, n=20, sjgma=2.0
_. 60 -
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1
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Figure 4E: CUG=10, n=50, sigma=2.0
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Figure 4D: CUG=10, n=30,
Si,
1.
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Figure 4F: CUG=10, n=100, sigma=2.0
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ti
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Figures 4A - 4F
-20-
-------
Figure 5A: CUG=10, n=10, sigma=0.5 Figure 5B: CUG=10, n=15, sigma=0.5
D •
ra
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2 4 6 8 10 12 14 16 2 4 6 8 10 12
Arithmetic Mean Arithmetic Mean
J
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16
Figures 5A - 5F
-21-
-------
Figure 6A: CUG=1 0, n=1 0, sigma=1 .0 Figure 6B: CUG=1 0, n=1 5, sigma=1 .0
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2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
Arithmetic Mean Arithmetic Mean
Figure 6C: CUG=10, n=20, sigma=1.0 Figure 6D: CUG=10, n=30, sigma=1.0
25 -
8 '
5 15 -
10 -
2
25 -
to
2
5 -
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4 6 8 10 12 14 «
Arithmetic Mean
6E: Figure CUG=10, n=SO, sigma=1.0
-x
x1
o^^
<=^
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X
x^
^x
*t&r
2 4 6 8 10 12
Arithmetic Mean
J
X
xl
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14
t
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t 25
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f c=15 -
Cheb M | •
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i^
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2 4 6 8 10 12 14 16
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Figure 6F: CUG=1 0, n=1 00, sigma=1 .0
-El-
t
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Cheb M 510
i^
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^^
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d33
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^
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16 2 4 6 8 10 12 14
Arithmetic Mean
s
-E3-
t
H
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16
Figures 6A - 6F
-22-
-------
3000
2500
2000
d
"^1500
8
1000
500
0
70 -i
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5
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2
Figure 7A: CUG=10, n=10, sigma=1.5
®—
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4 6 8 10 12 14 1
Arithmetic Mean
Figure 7C: CUG=10, n=20, sigma=1.5
J-^
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Figure 7B: CUG=1 0, n=1 5, sigma=1 .5
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Figure 7D: CUG=10, n=30, sigma=1.5
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4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
Arithmetic Mean Arithmetic Mean
Figure 7E: CUG=10, n=50, sigma=1.5 Figure 7F: CUG=10, n=100, sigma=1.5
jf'
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m
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4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
Arithmetic Mean Arithmetic Mean
Figures 7A - 7F
-23-
-------
„ Mean UCL
5-1. -i ro K)
en o cji O cji
300 -
_i
°200 -
S 150 -
5
Mean UCL c
000
000
000
000
000
0
Figure 8A: CUG=10, n=10, sigma=2.0 Figure 8B: CUG=10, n=15, sigma=2.0
J
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2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
Arithmetic Mean Arithmetic Mean
Figure 8C: CUG= 1 0, n = 20, sigma=2.0 Figure 8D: CUG=1 0, n=30, sigma=2.0
X
^
^
=ff
*&
=1
I—
X
=^
^
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. — -1
4 6 8 10 12 14 16
Arithmetic Mean
Figure 8E: CUG=10, n=50, sigma=2.0
"0
iU - -
a- -
0 -I-
2
>i
«*r
ET""
&
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^zr
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^
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I/
I I I I I I I I I I I
4 6 8 10 12
Arithmetic Mean
/
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III
14 16
--
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2 4 6 8 10 12 14 16
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Figure 8F: CUG=10, n=100, sigma=2.0
-El- 35 _
t
30 -
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u _l • '
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"T" C
Cheb M S 20
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10
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a
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Cheb MV
2 4 6 8 10 12 14 16
Arithmetic Mean
Figures 8A - 8F
-24-
-------
Figure 9A: CUG=60, n=10, sigma=0.5
Figure 9B: CUG=60, n=15, sigma=0.5
40 60 80
Arthmetic Mean
60
Arthmetic Mean
Figure 9C: CUG=60, n=20, sigma=0.5
Fgure 9D: CUG=60, n=30, sigma=0.5
60 80
Arthmetic Mean
40 60 80 100
Arthmetic Mean
100
Fgure 9E: CUG=60, n=50, sigma=0.5
Figure 9F: CUG=60, n=100, sigma=0.5
40 60 80 100 120
Arthmetic Mean
40
60 80
Arthmetic Mean
100 120
Figures 9 A - 9F
-25-
-------
Figure 10A: CUG=60, n=10, sgma=1.0
Figure 10B: CUG=60, n=1S, sigma=1.0
60 80
Arithmetic Mean
60 80
Arthmetic Mean
Fgure 10C: CUG=60, n=20, sigma=1.0
Figure 10D: CUG=60, n=30, sigma=1.0
40 60
Arthmetic Mean
40 60 80
Arthmetic Mean
100 120
80-
Figure 10E: CUG=60, n=50, sigma=1.0
_ 60-
40-
20-
20
Figure 10F: CUG=60, n=100, sigma=1.0
40 60 80 100 120
Arthmetic Mean
40 60 80 100 120
Arthmetic Mean
Figures 10A - 10F
-26-
-------
Figure 11 A: CUG=60, n=10, sigma=1.5
Figure 11B: CUG=60, n=15, sigma=1.5
t
-®-
H
-V-
Cheb_M
Cheb MV
60 80 100
Arithmetic Mean
60 80
Arthmetic Mean
Figure 11C: CUG=60, n=20, sigma=1.5
Figure 11D: CUG=60, n=30, sigma=1.5
o4
t
-®-
H
Cheb_M
Cheb MV
20 40 60 80 100 120
Arthmetic Mean
20 40 60 80 100
Arthmetic Mean
Figure 11E: CUG=60, n=50, sigma=1.5
80
._ 60
20
Figure 11F: CUG=60, n=100, sigma=1.5
20 40 60 80 100 120
Arthmetic Mean
20 40 60 80 100 120
Arthmetic Mean
Figures 11A - 11F
-27-
-------
Figure 12A: CUG= 60, n = 10, sigma=2.0
Figure 12B: CUG=60, n=15, sigma=2.0
20 40 60 80
Arithmetic Mean
40 60 80 100
Arthmetic Mean
120
Fgure 12C: CUG=60, n=20, sigma=2.0
60 80
Arthmetic Mean
120
Figure 12D: CUG=60, n=30, sigma=2.0
20 40 60 80 100 120
Arthmetic Mean
H
Cheb_M
Cheb MV
Fgure 12E: CUG=60, n=50, sigma=2.0
04-
20 40 60 80 100
Arthmetic Mean
Figure 12F: CUG=60, n=100, sigma=2.0
80
H
Cheb_M
Cheb MV
._ 60
40
120
80 100
Arthmetic Mean
t
-®-
H
Cheb_M
Cheb MV
Figures 12A - 12F
-28-
-------
Figure 13A: CUG=60, n=10, sigma=0.5 Figure 13B: CUG=60, n=1S, sigma=0.5
160 -
140-
o
D
«
60 -
40-
j
J SA
^
^
6$
%>
/
/ /
Y/
V
s
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H ° 100 -
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A 60-
Cheb MV 4Q ;
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/
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Cheb MV
20 40 60 80 100 120 20 40 60 80 100 120
Arthmetic Mean Arthmetic Mean
Figure 13C: CUG=60, n=20, sigma=0.5 Figure 13D: CUG=60, n=30, sigma=0.5
_i
^ 100 -
c
<5 80
5
\
20
o
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t
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pr
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x^x0"
t
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H ^ 100 -
_ c
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1
8
40 60 80 100 120 20
Arthmetic Mean
Figure 13E: CUG=60, n=50, sigma=0.5
/>
#
Y
/.
r\
T
X
•/
s
s
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t
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H 0 •
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Cheb M 1
-*-
Cheb MV 40 .
(
20 40 60 80 100 120 20
Arthmetic Mean
s.
<&
r
s,
-------
Figure 14A: CUG=60, n=10, sgma=1.0
Figure 14B: CUG=60, n=15, sigma=1.0
60 80
Arithmetic Mean
60 80
Arthmetic Mean
100
Figure 14C: CUG=60, n=20, sigma=1.0
Figure 14D: CUG=60, n=30, sigma=1.0
60 80
Arthmetic Mean
60 80 100
Arthmetic Mean
Figure 14E: CUG=60, n=50, sigma=1.0
Figure 14F: CUG=60, n=100, sigma=1.0
60 80
Arthmetic Mean
t
H
Cheb_M
Cheb MV
40 60 80 100 120
Arthmetic Mean
Figures 14A - 14F
-30-
-------
Figure 15A: CUG=60, n=10, sigma=1.5
Figure 15B: CUG=60, n=15, sigma=1.5
5000
Q3000
D
2000
1000
20
A
40 60 80 100
Arithmetic Mean
t
r o
•*• I'
Cheb_M I
Cheb MV
600
120
20
t
-®-
H
Cheb_M
Cheb_MV
40 60 80 100 120
Arthmetic Mean
Figure 15C: CUG=60, n=20, sigma=1.5
Figure 150: CUG=60, n=30, sigma=1.5
40 60 80 100
Arthmetic Mean
60 80
Arthmetic Mean
Figure 15E: CUG=60, n=50, sigma=1.5
Figure 15F: CUG=60, n=100, sigma=1.5
40
60 80
Arthmetic Mean
Cheb M 2
100
Cheb_MV
40 60 80 100
Arthmetic Mean
Cheb M
Cheb_MV
120
Figures ISA - 15F
-31-
-------
Figure 16A: CUG= 60, n = 10, sigma=2.0
20000000
15000000
O
5000000
Figure 16B: CUG=60, n=15, sigma=2.0
20 40
60 80 100
Arithmetic Mean
10000
8000
O 6000
D
4000
2000
40 60 80 100
Arthmetic Mean
t
-®-
H
Cheb_M
Cheb MV
120
Figure 16C: CUG=60, n=20, sigma=2.0
2000
O
:1000
500
60 80 100
Arthmetic Mean
t
-®-
H
Cheb_M
-*-
Cheb MV
Figure 16D: CUG=60, n=30, sigma=2.0
40 60 80 100
Arthmetic Mean
120
t
-»-
H
Cheb_M
Cheb_MV
Figure 1GE: CUG=60, n=50, sigma=2.0
Figure 16F: CUG=60, n=100, sigma=2.0
60 80
Arthmetic Mean
80 100
Arthmetic Mean
Figures 16A - 16F
-32-
-------
Notice
The U.S. Environmental Protection Agency (EPA), through its Office of Research and Development
(ORD), funded and prepared this Issue Paper. It has been peer reviewed by the EPA and approved for
publication. Mention of trade names or commercial products does not constitute endorsement or
recommendation by EPA for use.
-33-
-------
References
Bain, L. J., and Engelhardt, M. (1992), Introduction to Probability and Mathematical Statistics, Boston:
Duxbury Press.
Blackwood, L. G. (1991), "Assurance Levels of Standard Sample Size Formulas," Environmental Science
and Technology, Vol. 25, No. 8, pp. 1366-1367.
Bowers, T., Neil, S., and Murphy, B. (1994), "Applying Hazardous Waste Site Cleanup Levels: A
Statistical Approach to Meeting Site Cleanup Goals on Average." Environmental Science and
Technology.
Chen, L. (1995), "Testing the Mean of Skewed Distributions," Journal of the American Statistical
Association, 90, 767-772.
EPA (1989a), "Risk Assessment Guidance for Superfund: Volume 1. Human Health Evaluation Manual
(Part A)," Publication EPA 540/1-89/002.
EPA (1989b), "Methods for Evaluating the Attainment of Cleanup Standards. Volume 1. Soils and Solid
Media," Publication EPA 230/2-89/042.
EPA (1992), "Supplemental Guidance to RAGS: Calculating the Concentration Term," Publication EPA
9285.7-081.
EPA (1991), "A Guide: Methods for Evaluating the Attainment of Cleanup Standards for Soils and Solid
Media," Publication EPA/540/R95/128.
EPA (1996), "A Guide: Soil Screening Guidance: Technical Background Document," Second Edition,
Publication 9355.4-04FS.
Gilbert, R.O. (1987), Statistical Methods for Environmental Pollution Monitoring, New York: Van
Nostrand Reinhold.
Gilbert, R.O. (1993), "Comparing Statistical Tests for Detecting Soil Contamination Greater than
Background," Pacific Northwest Laboratory, Technical Report No. DE 94-005498.
Hogg, R.V., and Craig, A.T. (1978), Introduction to Mathematical Statistics, New York: Macmillan
Publishing Company.
Kupper, L. L. and Hafner, K. B. (1989), "How Appropriate Are Popular Sample Size Formulas?", The
American Statistician, Vol. 43, No. 2, pp. 101-105.
Johnson, N.J. (1978), "Modified t-Tests and Confidence Intervals for Asymmetrical Populations," The
American Statistician, Vol. 73, pp. 536-544.
Land, C. E. (1971), "Confidence Intervals for Linear Functions of the Normal Mean and Variance,"
Annals of Mathematical Statistics, 42, 1187-1205.
-34-
-------
Land, C. E. (1975), "Tables of Confidence Limits for Linear Functions of the Normal Mean and
Variance," in Selected Tables in Mathematical Statistics, Vol. Ill, American Mathematical Society,
Providence, R.I., 385-419.
Singh, A. K., Singh, Anita, and Engelhardt, M., "The Lognormal Distribution in Environmental
Applications," EPA/600/R-97/006, December 1997.
Stewart, S. (1994), "Use of Lognormal Transformations in Environmental Statistics," M.S. Thesis,
Department of Mathematics, University of Nevada, Las Vegas.
-35-
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Appendix
Approximate test of the mean of a lognormal distribution
As discussed above, for a lognormal distribution that is not highly skewed, such as o < 0. 5, it might
be reasonable to use a test of the median as an approximate test of the mean. Let the x/s be a random
sample from a lognormal population, with both // and <72 unknown, and with yi = ln(X;). Denote the
standardized difference by d= 61 o= [ln(Cs) - \a\.(jj,^\lo . The lognormal mean is ^ = exp(// + 0.5<72), and
the parameter // can be written in the equivalent form, // = ln(//j) - 0.5 o1. The power function is given by
) = P[UCL
= P[(y - P) + fi-
= P[\/w(y -
= P[Z + fj_B n_^WI(n-\) < v + O.Sy] (Al)
where Z = \fn(x -//)/<7,and JT= (« - 1)^2/!?2 are independent random variables; Zis standard normal and
chi-square distributed with « - 1 degrees of freedom. Equation (Al) can be written as follows:
, a) =P[Z < fiid - gl(Wj\ = *(fid-glM)fwMdw (A2)
where w =t__l(n-\ -O.Sa, and
is the probability density function of a chi-square distribution with v=n-\ degrees of freedom. Note
that equation (A2) depends not only on d, but also separately on o because the function g^(w) depends on
o. The power can be evaluated by numerical integration and it yields the significance level of the test of a
lognormal mean if d = 0. This approach was used to compute the significance levels shown in Table 1.
Results about the noncentral t distribution, with the appropriate choice of the noncentrality parameter, can
also be used to evaluate this power function.
The power function of the H-test
An approach similar to the derivation which yielded equation (A2) can be used to derive the power
function of the H-test as a function of uand the standardized difference, d = 61 o= [ln(Cs) - ln(//j)]/<7.
Note, this can also be written as ln(Cs) = ln(//]) + od = // + <72/2 + ad. Recall, that the H-factors depend
on sy and «; that is, Hl_K=Hl_ K(sy, ri). It follows that the power function, as a function of both // and o,
is given by
7) = P[UCL < C>,a] = P[exp(j7
-36-
-------
= v[(y ~
-1) + V»(sld)Hl_J^frl::\<^lnd\n,d\
= P[Z
(A3)
g2(w) =Jn(o/2)[w/(n - 1) - 1] +
(n - 1) , «)/(« - 1) ,
Z and fFare the random variables defined in equation (Al). The function g2(w) depends not only on
d, but also separately on o. Thus, in order to perform numerical evaluation with equation (A3), it is
necessary to specify both d and o. The integral on the right of equation (A3) can be evaluated by
numerical integration in a manner similar to the evaluation of equation (A2).
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