EPA 600/R-11/204 I December 2011 I www.epa.gov/ada
United States
Environmental Protection
Agency
An Approach for Evaluating the
Progress of Natural Attenuation in
Groundwater
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May-90 Oct-95 Apr-01 Oct-06 Apr-12 Sep-17 Mar-23 Sep-28
Date Sampled
May-90 Oct-95 Apr-01 Oct-06 Apr-12 Sep-17 Mar-23 Sep-2i
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Office of Research and Development
.da, Oklahoma 74820
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An Approach for Evaluating the
Progress of Natural Attenuation in
Groundwater
John T. Wilson
U.S. EPA/NRMRUORD/Ground Water and
Ecosystems Restoration Division
Office of Research and Development
National Risk Management Research Laboratory, Ada, Oklahoma 74820
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Notice
The U.S. Environmental Protection Agency through its Office of Research and
Development conducted the research described here as an in-house effort in
collaboration with the Ground Water Forum. The Forum is a group of ground-
water scientists that support the Superfund and RCRA programs in each of the ten
EPA Regional Offices. This Report has been subjected to the Agency's peer and
administrative review and has been approved for publication as an EPA document.
Nothing in this Report changes Agency policy regarding remedial selection criteria,
remedial expectations, or the selection and implementation of MNA. This document
does not supersede any previous guidance and is intended for use in conjunction
with the OSWER Directive 9200.4-17P, Use of Monitored Natural Attenuation at
Superfund, RCRA Corrective Action, and Underground Storage Tank Sites (U.S. EPA,
1999) and Performance Monitoring of MNA Remedies for VOCs in Ground Water.
EPA/600/R-04/027 (Pope et al., 2004).
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Foreword
The U.S. Environmental Protection Agency (EPA) is charged by Congress with protecting the Nation's land, air, and water
resources. Under a mandate of national environmental laws, the Agency strives to formulate and implement actions leading
to a compatible balance between human activities and the ability of natural systems to support and nurture life. To meet
this mandate, EPA's research program is providing data and technical support for solving environmental problems today
and building a science knowledge base necessary to manage our ecological resources wisely, understand how pollutants
affect our health, and prevent or reduce environmental risks in the future.
The National Risk Management Research Laboratory (NRMRL) is the Agency's center for investigation of technologi-
cal and management approaches for preventing and reducing risks from pollution that threatens human health and the
environment. The focus of the Laboratory's research program is on methods and their cost-effectiveness for prevention
and control of pollution to air, land, water, and subsurface resources; protection of water quality in public water systems;
remediation of contaminated sites, sediments and ground water; prevention and control of indoor air pollution; and resto-
ration of ecosystems. NRMRL collaborates with both public and private sector partners to foster technologies that reduce
the cost of compliance and to anticipate emerging problems. NRMRL's research provides solutions to environmental
problems by: developing and promoting technologies that protect and improve the environment; advancing scientific and
engineering information to support regulatory and policy decisions; and providing the technical support and information
transfer to ensure implementation of environmental regulations and strategies at the national, state, and community levels.
Monitored Natural Attenuation (MNA) is widely applied to ground water contamination at hazardous waste sites. Under
the Comprehensive Environmental Response, Compensation, and Liability Act (CERCLA), MNA is considered to be
a remedy like any other remedy. When MNA has been selected as a remedy, concentrations of contaminants in the
groundwater are expected to achieve a clean-up goal at the site within a reasonable time frame. At many CERCLA
sites, the time by which the goals are to be obtained is specified in the Record of Decision (ROD). At CERCLA sites, the
performance of the remedy or combination of remedies that were selected under the ROD is reviewed every five years.
At present, there is no generally accepted approach to evaluate long term monitoring data and establish a time by which
clean up goals should be attained. At present, there is no generally accepted approach that can be used to determine
whether the extent of attenuation within a particular five year review period is adequate to allow the site to attain the
clean-up goal at the site within a specified time frame.
This report presents a simple, statistically based approach for evaluating the progress of natural attenuation from the
data collected during site characterization and long term monitoring. The report provides an approach to establish the
time that should be required to attain the clean up goals, and an approach to evaluate attenuation within a review cycle
to determine whether attenuation within that period of time is adequate to allow the site to attain the ultimate clean-up
goal by a specified time.
David G. Jewett,
Ground Water an
s
.cting Director
Ecosystems Restoration Division
National Risk Management Research Laboratory
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Contents
Notice ii
Foreword iii
Acknowledgements xi
Abstract xiii
1.0 Introduction 1
1.1. Data analysis best performed before selection of MNA as remedy 4
1.2. Data analysis performed after selection of MNA as remedy 5
2.0 Illustration of the First Phase of Analysis 7
2.1. Identifying the rate law for natural attenuation 9
2.2. Estimating time required to reach clean up goals 10
2.3. Estimating uncertainty in the rate of natural attenuation 11
2.4. Shortcuts to estimate the rate of attenuation 12
3.0 Illustration of the Second Phase of Analysis 13
3.1. Testing whether the reduction in concentration is statistically significant 14
3.2. Testing whether the reduction in concentration is adequate to meet goals 14
3.3. Estimating the probability that the reduction in concentration is not adequate to meet goals . . 15
3.3.1. Establishing a decision criterion 15
3.3.2. Balancing the confidence in the test and the power of the test 16
3.3.3. Applying the decision criterion 17
4.0 Regression as an Alternative Phase Two Analysis 19
4.1. Establishing a decision criterion for regression 19
4.2. Regression with only one sample each year 21
5.0 Putting the Statistical Analysis into a Geohydrological Framework 23
5.1. Monitoring to document trends in attenuation in three dimensional space 23
5.2. Evaluation of whether attenuation is adequate at a site 24
5.2.1. Evaluation of the site as a whole 24
5.2.2. Evaluation of individual wells 27
5.2.3. Uncertainty in estimates of time to attain cleanup goal 28
5.2.4. Interpretation of projections 28
5.2.5. Transformation products from natural attenuation may be a special case 29
5.2.6. Rates of attenuation reported in the literature 30
5.2.7. Dealing with data quality issues 31
6.0 Suggestions and Recommendations 33
7.0 References 35
Appendix AFirst Phase Analysis, Using Linear Regression to Extract Rate Constants 37
A.I.Use of the spreadsheet to calculate a linear regression 37
A. 1.1. Express the sample dates as decimal years 37
A. 1.2.Calculate natural logarithms 38
A.I.3. Run the regression 39
A. 1.4.Examine the results 40
A.2. Statistical background on linear regression 41
A.2.1. Linear relationship between variables 42
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A.2.2. Uniform variance in data 42
A.2.3. Normal distribution of residuals 42
A.3. Using Excel to generate a Q-Q plot and using the One-Sample Kolmogorov-Smirnov
Goodness-of-Fit Test 44
A.4. Using ProUCL for a Q-Q plot and Goodness-of-Fit testing 48
References 49
Appendix B Second Phase Analysis 53
B.I.Statistical Approach 53
B. 1.1. Statistical background: Use of the Student's ^-statistic 53
B.I.2.Theoretical basis of the statistical comparison of means 54
B.I.3.Use of a spreadsheet to calculate the difference ofmeans 55
B.I.4.Modifying the spreadsheet to accommodate samples 61
B.2.Independence of samples and seasonal effects 62
B.3. Selecting the appropriate value of a 63
Reference 65
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Figures
Figure 1. Distribution of TCE, czs-DCE and Vinyl Chloride in the most contaminated well in a
transect of monitoring wells arranged between a source of contamination of TCE in
ground water and Lake Michigan 7
Figure 2. Monitoring record for concentrations of TCE in well MW-3B, a Point of Compliance
Well in a transect of monitoring wells 9
Figure 3. Comparison of the means for the first year of the review cycle (2001) and the final
year of a five year review cycle (2006) to the clean up goals 13
Figure 4. A comparison of the concentrations in samples in the initial and final year of the
review cycle to the interim goals in the final year of the cycle (2006) to be adequate
for the long term clean up goal (5 (J-g/L in 2017) 16
Figure 5. A comparison of the regression line through the natural logarithm of the
concentrations of TCE, and the 80% confidence interval on the line to interim goal at
the end of the review cycle (2007) and the long term clean up goal (5 (J-g/L in 2017) 20
Figure 6. Effect of number of samples on the confidence belts on regression lines fit through the data. 21
Figure 7. Projected 60% confidence belts on the regression of the natural logarithm of
concentration of vinyl chloride in MW-3A on date 28
Figure A. 1. A spreadsheet to calculate the decimal date of sampling and the natural logarithm of
the concentration of TCE 38
Figure A.2. The Regression dropdown menu under the Data Analysis tab in the Tools menu 39
Figure A.3. The input menu for linear regression 40
Figure A.4. The summary output of the linear regression 41
Figure A.5. Fit of a linear trend and an exponential trend to concentration of TCE 43
Figure A.6. Fit of a linear trend to the natural logarithm of concentration of TCE 44
Figure A.7. Report in Excel 2003 of the Residuals and Standard Residuals from the Linear
Regression of the example data set 45
Figure A.8. Calculation of the values of the standard normal distribution (z-distribution)
corresponding to the rank and quantile of the standard residuals 46
Figure A.9. Normal Probability Plot (also called a Q-Q plot) comparing the distribution of the
residuals from the regression to the normal probability distribution 47
Figure A. 10. Data entry form in an application to perform the Kolmogorov-Smirnov Test 48
Figure A.I 1. Results returned from the Kolmogorov-Smirnov test 49
Figure A.12. Using ProUCL to generate a Q-Q plot 50
Figure A. 13. Using ProUCL to evaluate goodness-of-fit to a normal distribution 51
Figure B.I. Populating the spreadsheet Evaluation ofMNA to calculate the mean concentrations of
contaminant in the first and in the final year of the review cycle 55
Figure B.2. Calculating the standard deviation of the means, the difference between the means,
and the standard deviation of the difference between the means 56
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Figure B.3. Calculating the difference between the means necessary to be statistically significant at
a predetermined probability of error. 57
Figure B.4. Setting interim clean-up goals for the final year of the review cycle 59
Figure B.5. Calculating the mean of the interim goals, the standard deviation of the interim goals,
the difference between the mean of samples and the mean of the interim goals, and the
standard deviation of the difference between the mean of samples in the final year and
the mean of the interim goals 60
Figure B.6. Evaluation whether attenuation is adequate to attain goals 61
Figure B.7. Data input screen and subscreen for G*Power 3.1.2 64
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Tables
Table 1. Maximum concentrations of TCE, cis-DCE and Vinyl Chloride in a transect of
monitoring wells (12/11/2000 to 10/03/2003) 8
Table 2. Second phase of analysis 26
Table 3. Using the first phase of analysis to project when concentrations of TCE, cis-DCE and
Vinyl Chloride in monitoring wells will attain the clean up goal 27
Table 4. Typical rates of attenuation of concentrations over time in monitoring wells 31
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Acknowledgments
The Groundwater Forum is a group of ground-water scientists that support the Superfund and
RCRA programs in each of the ten EPA Regional Offices. This report was developed for and in
close consultation with the Groundwater Forum. Luanne Vanderpool with U.S. EPA Region 5
managed the review of the report for the Groundwater Forum, and provided the EPA internal
peer review. External peer reviews were provided by Thomas McHugh and Charles Newell of
Groundwater Services Inc, Houston, Texas; Yue Rong of the California Regional Water Quality
Control Board, Los Angeles, California; and Aristeo (Resty) Pelayo of the Wisconsin Department
of Natural Resources, Madison, Wisconsin.
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Abstract
Monitored Natural Attenuation (MNA) remedies monitor the results of natural processes and
utilize these data to predict future conditions in the aquifer. Performance monitoring to evaluate
MNA effectiveness is a critical element for MNA remedies.
Monitoring over time insures that the future behavior of the plume is consistent with past behav-
ior and that the risk of exposure to the contaminants is managed. The trend of contaminant
concentrations over time in a particular well can be used to forecast the future concentrations in
that well and predict when concentrations will attain a selected concentration level. The purpose
of this document is to present a simple, statistically based approach for evaluating the progress
of natural attenuation from the data collected during site characterization and long term monitor-
ing. The intended audience is technical professionals that actually perform the data analyses (i.e.,
hydrogeologists, engineers) as well as project managers who review those analyses and/or make
decisions based on those analyses.
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1.0
Introduction
Monitored Natural Attenuation (MNA) was a
component of more than 20 percent of rem-
edies implemented between 1982 and 2005
at National Priority List (NPL) sites where
groundwater is contaminated (U.S. EPA, 2007).
In the interval 2005 through 2008, MNA was
a component of more than 18% of NPL sites
where groundwater is contaminated (U.S. EPA,
2010). The percentage of sites using MNA is
likely to increase as source areas are depleted
by more aggressive remedies and MNA is used
to remediate remaining contamination.
MNA remedies monitor the results of natural
processes and utilize these data to predict future
conditions in the aquifer. MNA remedies for
contaminated groundwater are expected to
demonstrate that MNA will "... attain cleanup
levels (or other remedial action objectives) in a
time frame that is reasonable when compared
to the cleanup time frames of the other alterna-
tives and when compared to the time frame of
the anticipated resource use" (U.S. EPA, 1999,
p. 9-7). This statement defines key expectations
which EPA has for MNA remedies and is a
measure of success for all MNA remedies. This
paper describes methods which can be used to
define "... a timeframe that is reasonable ..."
and methods which quantify demonstrations
that MNA will achieve the remedial action
objectives.
Performance monitoring to evaluate MNA
effectiveness is a critical element for MNA
remedies. One objective for the performance
monitoring program of an MNA remedy is
to "Demonstrate that natural attenuation is
occurring according to expectations" (U.S.
EPA, 1999, p. 22). The expectations generally
include being able to achieve the concentra-
tion based clean-up goals for the contaminants
in the groundwater at the site. A Record of
Decision (ROD) for a site regulated under
the Comprehensive Environmental Response,
Compensation, and Liability Act (CERCLA)
may specify a particular date by which specific
clean-up goals must be reached. Whether a
specific date has been specified or whether the
cleanup time must simply be reasonable, the
approach presented in this document describes
a methodology using simple statistics which
objectively demonstrates whether a MNA
remedy is proceeding at a rate which will
achieve the goals specified in the ROD and is
meeting expectations. The method is specifi-
cally intended to support the five-year remedy
review process under CERCLA; however, it is
applicable to any site that has concentration-
based clean up goals for the contaminants, and
a date (whether specified or simply "reason-
able") by which the goals must be attained.
Monitored Natural Attenuation (MNA) is
generally considered in enforcement actions
regulated under CERCLA to be a remedy like
any other remedy. The Record of Decision
(ROD) will specify clean up goals for each
contaminant at the site. These goals are usually
expressed in terms of the maximum concen-
tration of the contaminants in soil or ground
water that will be tolerated at the site when the
remedy is complete.
Monitoring over time insures that the future
behavior of the plume is consistent with past
behavior and that the risk of exposure to the
contaminants is managed. The trend of con-
taminant concentrations over time in a par-
ticular well can be used to forecast the future
concentrations in that well and predict when
concentrations will attain a selected concentra-
tion level. The approach presented in this paper
can be used to estimate cleanup times by MNA
even where clean up goals are not specified by
time or concentration.
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The approach is intended to be used in the con-
text of existing U.S. EPA guidance and recom-
mendations for MNA. These U.S. EPA docu-
ments can be found at http://www.epa.gov/oust/
cat/mna.html and an associated link at http://
www. epa.gov/ada/gw/mna.html. Two of the
U.S. EPA documents are widely applicable: Use
of Monitored Natural Attenuation at Superfund,
RCRA Corrective Action, and Underground
Storage Tank Sites (U.S. EPA, 1999) and
Performance Monitoring of MNA Remedies
for VOCs in Ground Water (Pope et al, 2004).
U.S. EPA (1999) lays out a general framework
for the appropriate implementation of MNA at
EPA enforcement actions. Pope et al. (2004)
summarize that comprehensive framework in
the following steps:
1) demonstrate that natural attenuation is
occurring according to expectations,
2) detect changes in environmental condi-
tions (e.g., hydrogeologic, geochemical,
microbiological, or other changes) that
may reduce the efficacy of any of the
natural attenuation processes,
3) identify any potentially toxic and/or
mobile transformation products,
4) verify that the plume(s) is not expanding
downgradient, laterally or vertically,
5) verify no unacceptable impact to down-
gradient receptors,
6) detect new releases of contaminants to
the environment that could impact the
effectiveness of the natural attenuation
remedy,
7) demonstrate the efficacy of institutional
controls that were put in place to protect
potential receptors, and
8) verify attainment of remediation
objectives.
Performance Monitoring of MNA Remedies
(Pope et al., 2004) briefly discusses the
use of statistics to evaluate trends. In their
Section 3.3.34 they state:
Statistical methods are also available to
facilitate analysis and comparison of trends
by considering data variability through
time. For instance, changes in contaminant
concentrations over space or time can be
used to calculate attenuation rates, and the
variability associated with those rates can be
quantified with confidence intervals about the
rates. These confidence intervals can be used
to determine the likelihood of attaining site-
related remedial goals. If all values of the
attenuation rate falling within the confidence
intervals lead to predictions that site reme-
dial goals will be attained in the desired time
frame, then confidence that MNA can attain
remedial goals is increased.
At many sites, the evaluation of trends in con-
centrations stops with an evaluation of whether
natural attenuation is occurring, and the evalua-
tion does not proceed to the next step of evalu-
ating whether MNA meets expectations. This
Report provides an approach to carry out the
first step in the comprehensive evaluation of the
performance of MNA, which is to demonstrate
that natural attenuation is occurring according
to expectations.
In CERCLA, the clean-up goals must be
met throughout the plume of contamination.
Progress should not be evaluated at only a few
selected wells. Care must be taken to avoid
making generalizations about clean-up prog-
ress without careful consideration of the entire
monitoring network. While the example used
to demonstrate the methodologies outlined in
this paper focuses on a single well, the method-
ologies should be applied to the entire monitor-
ing network. Many sites will have problematic
wells where the trends in concentration within
the five year review cycle are not adequate.
Depending on circumstances, the trends in the
problematic wells may or may not indicate that
MNA is inadequate for the plume as whole.
Interpretation of data from problematic wells is
discussed in Section 5.2.
The approach presented in this document is
divided into two phases. The first phase is
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best undertaken before MNA is selected as part
of the remedy at a site. The second phase is
undertaken as a part of performance monitoring
after the selection of MNA as a remedy.
In the first phase, the trend in concentrations
at individual wells is evaluated to estimate
the date when concentrations can be expected
to meet the clean up goal at each well. The
approach in the first phase can be used to evalu-
ate in a general manner whether natural attenu-
ation processes have a reasonable likelihood of
reaching concentration based goals in a reason-
able time period. If a date to attain the clean
up goals has not been established at a site, the
first phase also provides one approach that can
be used to select a date (see Sections 1.1 and
5.2.3).
In the second phase of analysis, data are
evaluated over the interval of time in a review
cycle, such as the five-year review cycle that is
common in CERCLA. The attenuation in con-
centration between the initial and final years of
the interval is compared to the attenuation that
would be required in order to attain the clean up
goal by the specified date. If the concentration
in the final year of the review cycle is higher
than the concentration that would be required to
attain the goal by the specified date, the second
phase estimates the probability that natural
attenuation will not meet the clean up goal.
Before this approach (or any other approach
to evaluate MNA) is applied to data gener-
ated from monitoring wells, the user should
consider the representativeness of the existing
wells in portraying the groundwater condi-
tions, and whether the existing monitoring wells
adequately characterize the plume vertically
and horizontally. This type of data analysis and
forecasting is only as reliable as the representa-
tiveness of data that is subjected to the analy-
sis. Subtle deterioration in the condition of the
well, incorrect interpretations of groundwater
flow directions or changes in the groundwater
direction over time can be falsely interpreted as
"attenuation". The approach presented in this
document is only appropriate when it has been
determined that the monitoring well network
accurately represents the ground water flow
system. Consult Performance Monitoring of
MNA Remedies (Pope et al., 2004) for recom-
mendations on the design of a monitoring
system.
The purpose of this document is to present a
simple, statistically based approach for evaluat-
ing the progress of natural attenuation from the
data collected during site characterization and
long term monitoring. The intended audience is
technical professionals that actually perform the
data analyses (i.e., hydrogeologists, engineers)
as well as project managers who review those
analyses and/or make decisions based on those
analyses. In this document the two phases of
the approach are described and illustrated with
an example. Since the approach outlined in
this document uses statistical methods, even the
simplified description in the body of the report
is somewhat technical. Non-technical readers
may be best advised to read Sections 1 and 5
first to understand the kind of decisions to be
made using the methods described in this paper.
The two appendices are more technical and
provide additional information regarding the
statistical methods used in this approach as well
as detailed step by step instructions to apply the
methods to a data set.
Section 1 introduces the approach for evaluat-
ing the progress of Natural Attenuation.
Section 2 introduces the statistics which are
used in Phase 1 to perform the evaluation of the
progress of MNA and describes the application
of these methods to characterize the behavior of
contaminants at the site before MNA is selected
as a remedy for the site. Methods are presented
to evaluate trends in concentrations over time
and to estimate the time required to reach clean-
up goals. The statistical methods are described
in much more detail in the appendixes which
accompany this report, but Section 2 is a statis-
tical discussion and is somewhat technical.
Section 3 applies the statistical methods used
in Phase 2 to evaluate data from a long-term
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monitoring program in order to determine
whether the progress of natural attenuation is
sufficient to achieve the clean-up goals set for
the site. The methods are built on the /-test for
the difference of means. The statistical methods
are described in much more detail in the appen-
dixes which accompany this report, but Section
3 is a statistical discussion and like Section 2 is
somewhat technical.
Section 4 describes the use of linear regression
techniques to evaluate the progress of Natural
Attenuation and for documenting the level of
confidence which can be placed in decisions
made from the data. This is an alternative
approach to the method using the /-statistic pre-
sented in Section 3.
Section 5 describes the application of the
decision process for evaluating the progress
of Natural Attenuation using the results from
methods described in Sections 2 through 4.
Section 5 uses technical terms from earlier
sections, but presents more general discussion
regarding the performance of MNA at the site.
These are issues which decision makers must
recognize and consider even if they do not per-
form the statistical analyses themselves.
1.1. Data analysis best performed
before selection of MNA as remedy
The first phase of data analysis determines,
based on the available monitoring record,
whether natural attenuation processes appear
to be capable of reducing the concentrations of
contaminants to the clean up goals within a rea-
sonable time period. This evaluation requires
sufficient monitoring data to indentify temporal
trends in contaminant concentration. This first
phase of evaluation should be conducted before
MNA is selected as part of the remedy at a site.
The first phase of data analysis can be used to
meet the particular requirements set out in the
OSWER Directive (U.S. EPA 1999) regard-
ing the use of MNA. Text quoted from the
Directive is in bold font. The Directive outlined
a three-tiered approach to site characteriza-
tion. The first line of evidence presented in
the Directive was: (1) Historical groundwater
and/or soil chemistry data that demonstrate
a clear and meaningful trend of decreasing
contaminant mass and/or concentration over
time at appropriate monitoring or sampling
points. In addition, the Directive states that...
EPA expects that documenting the level of
confidence on attenuation rates will provide
more technically defensible predictions of
remedial timeframes and form the basis for
more effective performance monitoring pro-
grams. The Directive also states: Statistical
confidence intervals should be estimated for
calculated attenuation rate constants (includ-
ing those based on methods such as historical
trend data analysis ...).
The first phase of the approach presented in this
document uses regression analysis to evaluate
natural attenuation. Regression analysis pro-
vides rate constants for attenuation that provide
a precise definition of a "clear and meaningful
trend of decreasing contaminant concentration
over time." Because higher rate constants mean
faster remediation time frames, simple calcula-
tions in a spread sheet are used to determine
whether the rate constants are different from
zero at some predetermined level of statistical
confidence and to extract the statistical confi-
dence interval on the rate of attenuation.
At many sites where MNA has been selected as
remedy, a concentration-based goal for cleanup
has been identified, but the date to attain that
goal is not identified. Ideally, the target date to
attain the goal should be established based on
factors such as the timeframe for anticipated
future use of the resource, a comparison to
the time that would be required for alternative
remedial techniques to attain the goal, and the
reliability of existing institutional controls and
other controls on exposure to contamination.
One option to establish a target date is to
use regression analysis to extrapolate natural
attenuation in the historical record and predict
the time when the goal can be expected to be
attained. This date can then be compared to the
requirements imposed by site specific factors
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to determine if it is appropriate. This option
is straightforward, but it has a serious implica-
tion. If the rate of attenuation continues into
the future at the same rate as in the past, on
average, half the time the true date to attain the
goal will be further in the future than the target
date, and half the time continued long term
monitoring will reveal that MNA is not ade-
quate to attain the final goal by the target date.
If regression analysis is used to select a target
date, that date should acknowledge and make
provision for the uncertainty in the regression
(see Section 5.2.3).
1.2. Data analysis performed after
selection of MNA as remedy
The second phase would apply to evaluations
after approval of MNA as a remedial measure.
Fitting one trend to all the data assumes that the
rate of natural attenuation does not change over
the history of the site. However, the rate of
attenuation can change for a variety of reasons.
The monitoring record is subject to temporal
variations in concentrations that create tempo-
rary or "apparent" trends. High water tables
can inundate residual contamination that was
previously in the unsaturated zone, producing
slugs of contaminants at high concentration.
At some sites, critical substrates are required
by the bacteria that degrade the contaminants.
Examples would include molecular hydrogen
and acetate used by Dehalococcoides bacteria
to degrade trichloroethylene, or sulfate used by
bacteria to degrade benzene. These substrates
can be depleted over time, which may change
the rate of biodegradation. These interac-
tions and more are discussed in greater detail
in Performance Monitoring of MNA Remedies
(Pope et al., 2004). Statistical analysis is one
approach to recognize a change in the trend in
concentrations over time.
Active source control or source remediation will
reduce concentrations of contaminants, and the
benefit of this active treatment will be included
in the overall trend in contaminant concentra-
tions. Biostimulation with growth substrates
and nutrients or bioaugmentation with active
microorganisms will increase the rate of bio-
degradation. Some vendors even talk about
enhanced natural attenuation. These active
measures will interfere with a statistical evalua-
tion of MNA. It is best to restrict an evaluation
of natural attenuation to the time periods before
active remediation has been implemented, or
to time periods after the benefits of the active
remediation have been realized.
The purpose of the second phase of analysis
is to determine whether MNA continues to
perform as expected, and whether the recent
performance of MNA at the site is adequate to
meet the clean up goals. Depending on the out-
come, the analyses in the second phase might
support a decision to allow a MNA remedy to
proceed as defined in the ROD. Depending on
the outcome, the analyses may also identify a
need for additional sampling and data evalua-
tion, or identify the need for additional active
remediation at a site.
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2.0
Illustration of the First Phase of Analysis
Figure 1 shows the location of clusters of
monitoring wells at a CERCLA enforcement
action where MNA has been selected as part of
the remedy. Figure 1 also presents the high-
est concentrations of contaminants observed in
the well clusters. The highest concentrations
of contaminants are in the MW-3 well cluster.
Table 1 presents the highest concentrations of
contaminants observed in the individual wells.
The well with the highest concentrations of
vinyl chloride (the compound of greatest con-
cern) is well MW-3B, the middle well of the
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MW5
A \A
200 400 600 800 1000
Position along POCW transect (feet SW to NE)
Lake / Q
Michigan ,**' Q ^_
~ MW5
MW6
/ ° ^
/
,' ''O " MW 1
- MW4
MW3
/ Shore Line
Current "Hot Spot"
O
200 Feet
Source O
Figure 1. Distribution of TCE, cis-DCE and Vinyl Chloride in the most contaminated well in a transect of
monitoring wells arranged between a source of contamination of TCE in ground water and Lake
Michigan.
-------
Table 1. Maximum concentrations of TCE, cis-DCE and Vinyl Chloride in a transect of monitoring wells
(12/11/2000 to 10/03/2003).
Monitoring Well
MW-1B
MW-1A
MW-2B
MW-2A
MW-3C
MW-3B
MW-3A
MW-4C
MW-4B
MW-4A
MW-5B
MW-5A
MW-6
Depth
Shallow
Deep
Shallow
Deep
Shallow
Intermediate
Deep
Shallow
Intermediate
Deep
Shallow
Deep
TCE
ug/L
18
17
10
53
8600
2300
20
18
11
2
1
1
1
cis-DCE
Hg/L
4
37
3
290
3300
3200
1900
12
19
1
1
1
4
Vinyl Chloride
ug/L
2
2
2
200
1200
2200
1500
2
4
4
2
2
2
MW-3 well cluster. Figure 2 presents long term
monitoring data of TCE concentrations in well
MW-3B. This data set will be used to illus-
trate the first phase of the analysis, determining
whether natural attenuation processes appear
to be capable of reducing the concentrations
of contaminants to the clean up goals within a
reasonable time period.
Changes in the direction of ground water flow
can cause the centerline of a plume to shift
away from a monitoring well, and give the false
impression that concentrations in the plume are
declining. The procedures in this Section are
only appropriate when it has been determined
that ground water flow to the monitoring well is
stable. For this example, assume that the flow
of ground water at the site has been evaluated
and the direction of flow has not changed to an
appreciable extent.
Performance Monitoring of MNA Remedies
(Pope et al., 2004) emphasizes the need for
evaluating the monitoring well network to
determine if it is sufficient for MNA. At any
site where this statistical approach might be
applied, the monitoring network should first
be evaluated to determine that it is sufficient
to that purpose. For this example, assume that
the monitoring network has been evaluated and
determined to be sufficient to document the
performance of MNA and it is appropriate to
move to the next step of evaluating trends in
concentrations.
-------
3000
g
ro
0)
o
o
O
ro
en
2000.0 2002.0 2004.0 2006.0
Date of Sampling
2008.0
2010.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
Y=-0.3260X+659.7
B
10000
- 1000
100 .9
ro
10
0)
o
2000.0 2005.0 2010.0
Date of Sampling
2015.0
1
2020.0
Figure 2. Monitoring record for concentrations of TCE in well MW-3B, a Point of Compliance Well in a tran-
sect of monitoring wells.
2.1. Identifying the rate law for natural
attenuation
Panel A of Figure 2 presents the monitoring
data on an arithmetic scale, where concentra-
tion is plotted as a function of time. Over
seven years of monitoring, the concentration
of TCE (the parent compound) declined from
2,300 (J-g/L to 163 (J-g/L. The solid curved line
is an exponential trend line that is fit to the
data. In this example the fit of the trend line is
in reasonable agreement with the data. Notice
that the dates are plotted as decimal years. The
slope of the line is the instantaneous rate of
attenuation (dC/df) in (J,g/L per year. Note that
as the concentrations decline, so does the rate of
attenuation.
These data follow what is known as first order
rate law. The rate of attenuation is described
in Equation 1, where r is the rate of change in
concentration with time ((J-g/L per year), C is
the concentration of contaminant (|j,g/L), and
-------
k is the rate constant. The unit for a first order
rate constant is reciprocal time. In this case, the
unit is (per year).
r = dC/dt = klC]1 = k\C] 1
A familiar example of attenuation under a
first order rate law is the decay of radioactive
nuclides.
The rate of change in concentration at any
instant in time under a first order rate law is
proportional to the concentration at that instant
(r = k[C] ). This rate law is called "first" order
because the exponent on the concentration C of
contaminant is one. The change in the con-
centration of contaminant in Equation 1 can
be integrated over time to produce Equation 2,
where t is an interval of time, C is the final
concentration at the end of the interval, and C0
is the initial concentration at the beginning of
the time interval.
If the natural logarithms of the concentrations
are plotted as a function of time, the slope of
the line is k.
If the exponent on the concentration of con-
taminant in the rate law were zero, the rate law
would be "zero" order and attenuation would
follow Equation 3.
r= dC/dt= k'[C]° = k' 3
Under a zero order rate law, the rate of degrada-
tion ((J-g/L per year) is a fixed number, the zero
order rate constant k'.
2.2. Estimating time required to reach
clean up goals
Panel B of Figure 2 compares the example data
on a logarithmic scale. The y-axis on the left
side of the figure plots the data as the natural
logarithm of the concentrations in (J,g/L. For
convenience of comparison, the y-axis on
the right side presents the actual concentra-
tions plotted on a logarithmic scale. The data
in Panel B appear to fall along a straight line.
The general form for an equation that follows a
straight line is presented as Equation 4, where
(a) is the slope of the line and (b) is the Y inter-
cept, the value of 7 when X=Q.
Y=a*X + b 4
If attenuation follows a first order rate law,
a plot of the natural logarithm of concentra-
tions on elapsed time will lie along a straight
line. The log-transformed data in Panel B of
Figure 2 appear to lie along a straight line. A
linear regression was used to fit an equation that
described the relationship between the natural
logarithm of the concentrations of TCE and
the date of sampling. The slope of the regres-
sion line is the first order rate constant for the
change in concentration over time. Details of
fitting the line are discussed in Appendix A.
The equation for the line as presented in Panel
B of Figure 2 is Equation 5 below, where Y is
the natural logarithm of the concentration of
TCE in (J,g/L and X is the date in decimal years.
Y =-0.3260*^ + 659.7 5
Solving Equation 4 for X provides Equation 6.
Solving Equation 5 for X provides Equation 7.
X = (Y -659.7)7 (-0.3260) 7
For purposes of illustration, we will assume that
the desired clean up goal for TCE is 5 jag/L,
and we will assume that it is desired that this
goal will be attained by 2017. The natural
logarithm of 5 is 1.609. Substituting 1.609 for
Y in Equation 7 predicts that natural attenuation
will attain the goal by the year 2018.7, which is
only slightly later than the desired cleanup date
of 2017. The same relationships are presented
graphically in Panel B of Figure 2.
As described in Equation 7, the date by which
the concentration based cleanup goal must
be attained is "key" to the evaluation of the
progress of MNA. The statistical evaluation
of the progress of MNA in a review cycle is
-------
determined by only four parameters: the date
when cleanup is to be attained, the cleanup
goal, the historical concentrations of the con-
taminant and the rate of attenuation.
Note that the value of Y in Equation 7 when X
is zero is 659.7 (see the equation in Panel B of
Figure 2) which corresponds to a TCE concen-
tration at year zero of 3 x 10286 ug/L (Panel A
of Figure 2). This impossibly high value for
the initial concentration of TCE was calculated
from the regression because the zero value for
the Date of Sampling was the beginning of
the Common Era. If the regression had been
performed on the date expressed in years since
the date of the first sample, the value of Y in
Equation 7 would have been near to the con-
centration in the first samples taken.
Trends in contaminant concentrations over time
have been reviewed at hundreds of hazardous
waste sites. A first order rate law is almost
always a better fit than a zero order rate law.
The calculations and procedures in this docu-
ment will assume that attenuation at the site
of interest follows a first order law. Equations
4 and 6 can be used to compare the long term
performance of natural attenuation to the
established clean up goals, or they can be used
to predict a time by which the goal may be
attained.
No real data set is ever a perfect match to the
theoretical assumptions for a particular statisti-
cal test, and a certain amount of judgment is
required to select the most appropriate data
and the most appropriate statistical procedure.
In the example data in Panel B of Figure 2,
the apparent good agreement with a straight
line indicates that (1) the rate of attenuation in
concentration follows a first order rate law and
(2) the first order rate constant did not change
over the time period in the monitoring record.
Both these conditions must be true if the calcu-
lations in this Section are to be used to forecast
a time in the future at which the concentration
will attain the clean up goal.
In the illustration, the entire record for the
monitoring wells was evaluated. The entire
record should be evaluated unless there is a
significant change in conditions at the site that
justifies using some smaller portion of the
record. If the time period in the record includes
efforts at source removal or source remediation,
the argument could be made that the best data
to evaluate the progress of natural attenuation is
the portion of the record that occurs after active
source control was completed. If the source
was remediated by a pump and treat system, the
best data to evaluate natural attenuation would
be the portion after pumping ceased and the
water table returned to its natural condition. In
the illustration, there are seven years of quarter-
ly monitoring data. This amount of monitoring
data is not always available when the feasibil-
ity of MNA as a remedial action needs to be
evaluated. However sufficient data should be
available to demonstrate a clear and meaningful
trend of decreasing contaminant mass and/or
concentration over time as specified in the
OSWER Directive (U.S. EPA 1999).
If visual examination indicates that a plot of the
natural logarithm of concentrations on time is
best fit by a curved line, that observation would
indicate that either (1) the rate of attenuation
does not follow a first order rate law or (2) the
first order rate constant is changing over time.
If this is the situation, the calculations in this
approach should not be used to forecast a time
in the future at which the concentration will
attain the clean up goal.
2.3. Estimating uncertainty in the rate
of natural attenuation
Appendix A of Performance Monitoring of
MNA Remedies (Pope et al., 2004) discusses
a variety of factors that add uncertainty to an
estimate of the rate of natural attenuation. The
OSWER Directive (U.S. EPA 1999) specifies
that this uncertainty in the rate constant be
described as a confidence interval on the rate
constant. Many evaluations of trends in MNA
will express the uncertainty in the fitted trend
line as the Coefficient of Determination (r 2).
-------
This is probably done because r 2 is a familiar
statistic, and because it is readily obtained from
the computer application used to create the
figure showing the trend line. The r 2 statistic
is not related in a simple way to the confidence
interval on the rate of attenuation, and should
not be used to evaluate the uncertainty in the
trend line. The spreadsheet used to fit the line
through the data also has the capacity to put
statistical confidence intervals on the slope of
the line. Details for this process are illustrated
in Appendix A. For the data set in Panel B of
Figure 2, the first order rate constant for attenu-
ation of concentrations of TCE was 0.326 per
year. As described in Appendix A (Figure A4
and section A. 1.4), the rate constant was no
slower than 0.281 per year at 90% confidence.
2.4. Shortcuts to estimate the rate of
attenuation
The Panels in Figure 2 were generated as charts
in Microsoft Excel 2003. The spreadsheet,
like many other spreadsheets, allows the user
to conveniently fit trend lines to a data set in
a chart, without going through the procedures
using a spreadsheet to fit a regression line to
data as is illustrated in Appendix A. Note, this
shortcut approach does not produce a confi-
dence interval for the rate constant as is done
using a spreadsheet to calculate the regression.
Options in the chart menu were used to fit an
exponential trend line to the data in Panel A of
Figure 2. The resulting equation for the line is
Y = 3E + 286e'0326X . The date (Xin. the equa-
tion) is an exponent, and the coefficient of X
is -0.326. Similarly, options in the chart menu
were used to fit a linear trend line to the natural
logarithm of the concentration data as shown in
Panel B of Figure 2. As discussed earlier, the
equation for the line is Y = -0.3260X+ 659.7 .
The date is a factor instead of an exponent, but
the coefficient of Xis the same. In both equa-
tions, the coefficient of the time elapsed is the
first order rate constant for change in concen-
tration over time. Because time was plotted in
decimal years, the unit for the rate constant is
reciprocal years. The negative of the coefficient
is the first order rate constant for attenuation
over time. The rate constant is negative because
concentrations are decreasing.
-------
3.0
Illustration of the Second Phase of Analysis
The same data will be used to illustrate the
second phase of the evaluation. Panel A of
Figure 3 identifies the data that are involved in
the comparison. The mean of concentrations
of TCE determined in water samples acquired
in each of the four quarters of 2001 will be
compared to the mean of concentrations in each
of the four quarters of 2006. The data and the
means are enclosed in dashed lines. The mean
concentration of TCE in 2001 was 1404 (J,g/L
10000
and the mean concentration in 2006 was
308
Data in Figure 3 are expressed on a log
scale, and all statistical calculations in the
Second Phase were done on the natural loga-
rithms of the concentrations. As a result, the
means in Figure 3 are geometric means instead
of the more familiar arithmetic means. To cal-
culate an arithmetic mean, one adds the values
and divides the sum by the number of values.
'H 1000
100
o Samples
* Means
Goals
Initial Year of Review Cycle
Final Year of Review Cycle
I I O O
-i-
2000.0 2001.0 2002.0 2003.0 2004.0 2005.0 2006.0 2007.0 2008.0
Date of Sampling
1000
'o n /
j f ' oo Final Year of Review Cycle
e» i & i -X^
o Samples
Means
^o!>
» o
Clean up Goal^
2000.0 2005.0 2010.0 2015.0
Date of Sampling
2020.0
Figure 3. Comparison of the means for the first year of the review cycle (2001) and the final year of a five
year review cycle (2006) to the clean up goals.
-------
An arithmetic mean was calculated for the natu-
ral logarithms of the concentrations, and then
the anti-natural logarithm was taken to express
the geometric mean as a concentration. This
is equivalent to multiplying the original values
and taking the nthroot of the product, where n
is the number of values. Note that for a set of
positive values, the geometric mean is less than
the arithmetic mean. The larger numbers have
less influence on a geometric mean.
3.1. Testing whether the reduction
in concentration is statistically
significant
First, the data were evaluated to determine
whether there was a statistically significant
decline in the average concentration of contami-
nants over the time interval in the review cycle.
Over the time period in the review cycle, was
there any significant attenuation? The compari-
son is made using the t statistic for the differ-
ence of means. The theoretical background
on the t statistic and details on using a spread-
sheet to make the calculations are presented in
Appendix B.
The t statistic allows the user to determine
whether means are different from each other
if the user is willing to accept some previ-
ously selected probability that the results of the
statistical test may be in error. If the statisti-
cal test indicates that attenuation is significant
at the selected probability of error, then it is
appropriate to go forward with the remainder of
the evaluation. If the test fails to indicate that
attenuation is significant over the time interval
in the review cycle, then it is not appropriate
to interpret any apparent decline in concentra-
tions over the time interval in the review cycle.
Attenuation may have happened, but the attenu-
ation is not documented by the monitoring data.
Imagine that the acceptable probability of error
for the attenuation of concentrations of TCE
as presented in Figure 3 was 10%. Appendix
B presents detailed calculations that deter-
mine that the extent of attenuation between the
mean of samples taken in 2001 and the mean
of samples taken in 2006 is in fact statistically
significant with a probably of error of 10%, and
it is appropriate to go to the next step in the
evaluation.
3.2. Testing whether the reduction in
concentration is adequate to meet
goals
The next step is to determine whether the
observed mean of the concentration data in the
final year of the review cycle is consistent with
meeting the clean up goal in the predetermined
time interval. To do this, it is necessary to
calculate a goal in the final year of the review
cycle that is consistent with the predetermined
ultimate long term goal for clean up. Because
natural attenuation can be expected to follow
a first order rate law, we will assume the same
extent in attenuation (Ci/Co) in each review
cycle. Equation 2 can be rearranged to produce
Equation 8.
ln(C,/C0) =
8
When the review cycles are the same length, the
rate of attenuation ((J-g/L per year) decreases as
the concentrations go down, but the extent of
removal (Ci/Co) in each review cycle stays the
same.
The consequence of this assumption is illustrat-
ed in Equation 9, where for purposes of illus-
tration there are six review cycles between the
first year of the first review cycle and the date
by which the clean up goals are to be attained.
The product of the attenuation in each cycle
will be the overall attenuation, where Co is the
concentration at the start of the first review
cycle, and Cr C2, C3, C4 and C5 are the concen-
trations at the end of the current, second, third,
fourth and fifth review cycles respectively, and
C is the ultimate long term goal at the end of
the sixth review cycle.
-------
If
In general, if there are n review cycles between
the first year of the first cycle and the goal, then
C,
'c.
C
and
C
'C.
This final term can be rearranged to produce
Equation 10, which calculates the interim con-
centration goal C. from the initial concentration
Cg, the final goal C , and the number of review
cycles n.
C, = C
C,
10
For purposes of illustration, we will assume
that the monitoring wells must reach the MCL
for TCE by 2017, and that there will be five
years in each review cycle. The initial year
in the first review cycle started in 2001: the
cleanup goal must be attained within 16 years
of the initial year of the review cycle. There is
time for 16/5 = 3.2 review cycles (n) including
the current cycle.
The concentration of TCE in the first sample
taken in 2001 was 1200 (J-g/L. For purposes of
illustration, we will assume that the clean up
goal is the MCL for TCE, which is 5 (J-g/L. In
the example, l/n = 1/3.2 . Using these assump-
tions, Equation 10 becomes:
C, = 1200
1200
1/3.2
Panel A of Figure 4 compares the samples in
the initial year of the first 5-year review cycle
to the interim goals set using Equation 10 for
the final year of the review cycle and to the
final clean up goal. Panel B of Figure 4 com-
pares the interim goals to the actual samples
taken in 2006. The concentration of TCE in
the samples was higher than the concentra-
tion in the interim goals, raising the possibility
that natural attenuation might not be adequate
to attain the ultimate clean up goals. If the
concentration of TCE were lower than interim
goals, this fact would indicate that natural
attenuation over the review cycle was adequate
to meet the goals and no further statistical
evaluation of the data would be necessary.
3.3. Estimating the probability that the
reduction in concentration is not
adequate to meet goals
The comparison in Panel B of Figure 4 does
not take into account the natural variation in
the data. If MNA were exactly adequate to
meet the goal, but subject to random variation
from one review cycle to the next, approxi-
mately one-half the time the samples would be
greater than the concentration required to meet
the goal, and one-half the time they would be
less. In Panel B of Figure 4, the interim goal
for each sample taken in 2006 was lower than
the actual concentration, and the mean of the
interim goals was lower than the mean of the
actual samples, but the concentration in some
of the samples taken in 2006 is less than the
interim goals for other samples. It is not obvi-
ous from direct inspection whether the mean of
the samples is so much greater than the mean of
the goals as to be statistically significant. How
much concern should we have that the concen-
trations in 2006 were not adequate to meet the
clean up goal within the specified timeframe?
3.3.1. Establishing a decision criterion
The following decision criterion will be used to
determine if attenuation is not adequate to meet
the long term goal. If the mean of the interim
goals in the final year of this review cycle is
less than the mean of the samples in the final
year of this review cycle at some predeter-
mined level of confidence, then attenuation
in this review cycle is not adequate to attain
the goal. The null hypothesis in the compari-
son is "The mean of the interim goals is not
less than the mean of the samples in the final
year of the review cycle." The interpretation
-------
10000
1000 -
'is 100
i=
-------
in fact the means are not different. According
to the decision criterion, the statistical test
would indicate that attenuation is not adequate
to meet the goal, when in truth it is adequate.
The probability of a Type I error is represented
by a. The confidence in the test is described by
(1 - a), usually expressed as a percentage.
Type II error is a false negative, in this case
failing to reject the null hypothesis when in
truth the null hypothesis is false. In this appli-
cation the error would be to fail to indicate that
natural attenuation was not adequate to meet
the goal, when in truth it is not adequate. The
probability of a Type II error is represented by
P. The statistical power of the test to recog-
nize when natural attenuation is not adequate is
described by (1 - P).
The null hypothesis (HQ ) is-
The rate of attenuation is adequate to attain the
cleanup goal by the time specified.
Accept Hfl
Reject H0
HQ is true
Correct
Decision
Type I error
(probability a)
HQis false
Type II error
(probability P)
Correct Decision
For a given data set and statistical test, a and
P stand in an inverse relationship to each other.
As the value of a becomes lower (the test is
more stringent, and confidence is higher), the
value of P becomes higher and the test has less
power to recognize when apparent attenuation
is not adequate. The smaller the value of a, the
less it is likely that the statistical analysis will
warn that attenuation is not adequate to meet
the long term goal.
Both types of error are important to an evalua-
tion of natural attenuation; a value for a should
not be selected based on some previous rule-
of-thumb or default value. One who is trying
to support the use of MNA might want to be
able to recognize MNA when it is happening.
This person would want to minimize Type I
error, and would select small values of a and
corresponding large values of p. Alternatively,
one who is concerned that attenuation is not
adequate to meet the long term goal might
want to be sure that the statistical test warns
that attenuation is not adequate, when in fact,
attenuation is not adequate. This person would
want to minimize Type II error, and would
select large values of a and corresponding
small values of p.
One objective approach is to balance Type I
and Type II error and to perform the test for
values of a and P where a = p. Appendix B
describes in detail one process to determine
these values. There are many site-specific
issues and other factors that need to be con-
sidered while selecting the appropriate values
for a and p. This document does not discuss
the factors that might go into the decision. If
site specific concerns make either Type I error
or Type II more important, the ratios of a to
P can be adjusted accordingly as described in
Appendix B. With small data sets, balancing
Type I and Type II error will result in relatively
large values of both a and p.
3.3.3. Applying the decision criterion
The /-test for the difference of means will be
used to determine if the mean of goals is less
than the mean of the samples. Appendix B
includes a short discussion of the assumptions
behind the /-statistic, and provides instructions
to set up a spreadsheet (EvalMNA) to make
the calculations. As illustrated in Appendix B,
EvalMNA first determines geometric means of
the samples in the initial year ( Co) and final
year (Cz) of the review cycle. In order to make
the data more closely follow the underlying
assumptions of the /-statistic, all the calcula-
tions are done using the natural logarithms of
the concentrations and not the concentrations
themselves. A mean calculated on the natural
logarithms is equivalent to a geometric mean of
the original data set, not an arithmetic mean of
the original data set.
For the data presented in Panel B of Figure 3,
these geometric means are 1404 (J,g/L and
-------
308 (J-g/L respectively. EvalMNA then calcu-
lates the attenuation factor ( Ci/Co ) that would
be significant at various levels of the probability
of error. At a value of ( Ci/Co ) = 1.0 , there is
no attenuation. The attenuation of the means
over the review cycle (Ci/Co) = 0.219 was
significantly different from one at a probability
of error that was less than 0.0025.
Then EvalMNA compares the geometric mean
of the interim goals to the geometric mean
of the samples in the final year of the review
cycle. The mean of the interim goals that
would be necessary to be adequate for MNA
was 241 (J,g/L. Panel B of Figure 3 compares
the means of the samples to the mean of goals.
For the data set in Panel B of Figure 3, when
a = P, then a = P = 0.26 (See Section B.3 in
Appendix B for details on how the values of
a and P were determined for this data set). At
this value of a, there is 26% chance of accept-
ing that attenuation is not adequate to meet the
goal, when in truth it is adequate. At this level
of P, there is only a 74% chance of recognizing
that attenuation is not adequate, when it is not
adequate. Based on the decision criterion, the
/-test for the difference of means indicated that
attenuation of concentrations of TCE in the well
during the review cycle was not adequate to
meet the long term goal. See Section B.I.3. for
details of the calculations.
-------
4.0
Regression as an Alternative Phase Two Analysis
One potential criticism of the approach suggest-
ed for the second phase of analysis is that the
comparison of the initial and final year of the
review cycle does not take advantage of all the
data that have been collected. The concern does
merit consideration. Regression is designed
to evaluate the entire range of the data, and is
not specifically designed for the evaluation of
the changes from the beginning to the end of
a review cycle. However, a comparison of an
interim goal to a statistical confidence belt on
the regression line can be used to determine if
the rate of attenuation is not adequate at some
level of statistical confidence. If the regres-
sion line falls below the interim goal, the rate
of attenuation is faster than the rate required to
meet the goal, and there is no indication from
the trend in concentrations that the progress
of MNA is anything other than satisfactory. If
the regression line falls above the interim goal,
the rate of attenuation is too slow to attain the
goal. However, this simple criterion ignores
the uncertainty in the regression line. A better
approach is to compare the interim goal to a
statistical confidence belt on the regression line.
If regression is to be used to determine whether
attenuation is adequate for the long term goal,
the interim goal is compared to a line instead
of a mean. The interim goal should represent
the properties of the line, instead of the mean
condition in the first year of the review cycle.
The average properties of all the data in the
regression are described by the midpoint of the
regression line. The interim goal should be
calculated by comparing the midpoint of the
regression line to_the final clean up goal, using
Equation 10. If X is the mean of the sampling
dates in the regression, X. . .is the last
° ' interim goal
date in the regression, and Xf , , is the date
° ' final goal
by which the concentration based goal is to be
attained, the value of n in Equation 10 would
be determined following Equation 11:
n ~ (^"interim goal ~ X ) / (^f
inal goal
11
The value of Co in Equation 10 is 7 the mean
of the values of Y in the regression, where Y is
the natural logarithm of the concentrations of
contaminants.
There may be negative consequences to a MNA
remedy that is working too slowly, but no
negative consequences if the remedy is work-
ing faster than necessary to attain the goals.
For this reason, the statistical confidence belts
should be one-tailed estimates on the regression
line; all the probability for error in the regres-
sion should be assigned to the lower belts.
These belts describe the rates of attenuation
that are faster than the regression line. If the
confidence belt is higher than the value of the
interim goal, then at that level of confidence the
rate of natural attenuation over the time period
represented in the regression is too slow to
attain the clean up goal by the time specified.
4.1. Establishing a decision criterion
for regression
The decision criterion would be as follows. If
the interim goal lies below the confidence
belt, then natural attenuation over the time
interval included in the regression is not
adequate to meet the goal. Figure 5 compares
the final clean up goal and the interim goal for
the example data set to a linear regression on
data collected in the years 2001 through 2006.
The solid line in Figure 5 is the best estimate
of the "true" line through the data. The regres-
sion line is slightly above the interim goal for
the end of the monitoring period. In Figure 5,
it is difficult to see the relationship between the
regression line and the interim goal. Panel A
of Figure 6 plots the same data on a time scale
-------
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Regression Line
80% confidence interval
O Interim Goal
Clean Up Goal
o
2000.0 2005.0 2010.0 2015.0 2020.0
Date Sampled
Figure 5. A comparison of the regression line through the natural logarithm of the concentrations of TCE,
and the 80% confidence interval on the line to interim goal at the end of the review cycle (2007)
and the long term clean up goal (5 fjg/L in 2017).
often years. Note that the interim goal falls
between the 80% and 95% confidence belts. At
80% confidence we can say that attenuation is
not adequate to meet the long term goal, but we
cannot say the same thing at 95% confidence.
If the value of a for the one-tailed confidence
interval on the regression line is adjusted until
the confidence belt runs through the interim
goal, that value of a is 0.127 corresponding to
a probability of error of 87.3%. In comparison,
the Mest for difference of means determined
that the confidence level that could support a
determination that natural attenuation was not
adequate fell somewhere between 85% and
90% (In Appendix B, compare cell X15 to cells
AA16 and AA17 in Figure B6 and cells AB16
and AB 17 in Figure B6.). If values of a are
adjusted in EvalMNA to determine the value
where the prediction changes from No evidence
that attenuation is not adequate to Attenuation
is not adequate to attain goal, that value of
a is near 0.125 corresponding to a probabil-
ity of error of 87.5%. The results of the two
approaches were similar for the example data
set.
Figures 5 and 6 were created in EXCEL.
Equations 10 and 11 were used to calculate a
value of the interim goal. The regression line
and the confidence belts on the regression line
were calculated following the standard formulas
for linear regression. The EXCEL file used to
create Figure 5 and 6 can be used as a template
to calculate an interim goal and compare the
interim goal to confidence belts from another
monitoring record. To use the file as a tem-
plate, download the file from http://www.epa.gov/
nrmrl/gwerd/csmos/models/RegressionMNA.html Open
the tab Data and Calculations and follow the
instructions to copy the data from a new moni-
toring record over the data in the template.
-------
4.2. Regression with only one sample
each year
McHugh et al. (2011) evaluated three large
monitoring records and found that there was a
characteristic time-independent variability asso-
ciated with subsequent samples from the same
well. For most wells the long-term trend in
attenuation could not be distinguished from the
10000 n
time-independent variability unless the samples
were separated in time by 320 to 400 days.
Although this relationship was first documented
in a formal manner in 2011, many site manag-
ers and regulatory staff have had an intuitive
understanding of this interaction for years. As
a result, many data sets only have one sample
'{3 1000
I
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O
100
A
o Samples
Regression Line
80% confidence interval
95% confidence interval
O Interim Goal
All Data
"::''
2000.0 2001.0 2002.0 2003.0 2004.0 2005.0
Date Sampled
2006.0 2007.0 2008.0
10000
2 1000
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c
o
O
100
B
«- Samples
Regression Line
80% confidence interval
95% confidence interval
O Interim Goal
Fall Data Only
2000.0 2001.0 2002.0 2003.0 2004.0 2005.0
Date Sampled
2006.0 2007.0 2008.0
Figure 6. Effect of number of samples on the confidence belts on regression lines fit through the data.
-------
in each year and the alternative approach using
regression is the only approach available.
This situation is illustrated in Panel B of
Figure 6. The data set in Panel A was collapsed
by selecting only the fall sample for each year
of the review cycle. Because there were fewer
data in the regression, the confidence belts in
Panel B are wider than in Panel A. The interim
goal is above the 80% confidence belt in the
regression in Panel B. Contrary to the case for
the regression with twenty-four samples, the
regression with six samples failed to indicate
that attenuation was not adequate for the goal at
80% confidence. This illustrates the potential
problem of performing statistical data analysis
on smaller data sets. Smaller data sets have less
capacity to recognize when attenuation is not
adequate to attain the goal.
-------
5.0
Putting the Statistical Analysis into a
Geohydrological Framework
Any statistical analysis can be no better than the
monitoring data. Statistics cannot substitute for
an inadequate monitoring record. The statisti-
cal approach described here can only serve its
purpose when it is applied to a system of moni-
toring wells that is adequate to describe the
plume of contamination. Consult Performance
Monitoring ofMNA Remedies (Pope et al.,
2004) for U.S. EPA recommendations con-
cerning the design of a network of monitoring
wells, and a process for developing site-specific
monitoring objectives and performance criteria
for MNA. Statistics can only provide a reli-
able interpretation when they are applied to
data from wells that are representative of the
groundwater conditions and adequately char-
acterize the plume vertically and horizontally.
Pope et al. (2004) put emphasis on the need for
a monitoring network that can track the plume
of contamination in three dimensions.
The example used to demonstrate the meth-
odologies outlined in this paper focuses on
a single well, but the clean-up goals must be
met throughout the plume. The methodologies
described in this approach should be applied to
the entire monitoring network and not at only a
few selected wells. Similarly, degradation rates
may vary in different portions of the plume and
over time, so the monitoring well providing the
least favorable evaluation of MNA progress
may not be the same well from one evaluation
cycle to the next. Further, those applying the
methods described in this paper should recog-
nize that large scale environmental conditions
may affect the results of any MNA evaluation.
For example Section 3 uses data from the first
and last year of a CERCLA 5-Year Review
cycle to evaluate the progress of MNA and
predict whether cleanup goals will be attained
on time. This approach might overestimate
or underestimate the effectiveness of MNA if
the site is impacted by either severe drought
or flooding during the first or final year of a
5-Year Review cycle. This concern diminishes
as the number of observations (samples) and the
number of review cycles becomes larger over
time, but the impact of factors external from the
site should not be ignored especially when the
database is small. Statistical methods can not
compensate for bad data, inadequate data, an
inaccurate concept model, or insufficient experi-
ence in the interpretation of groundwater flow
systems and contaminant migration.
5.1. Monitoring to document trends in
attenuation in three dimensional
space
As described in Section 2.5 of Performance
Monitoring of MNA Remedies (Pope et al.,
2004), the direction of ground water flow can
change from season to season and from year to
year. A subtle shift in the direction of ground
water flow can move a plume away from a
monitoring well, giving the false impression
that the concentrations of contaminants are
attenuating. Pope et al. (2004) offer sugges-
tions for dealing with these changes. If moni-
toring wells are arranged into transects that are
aligned perpendicular to ground water flow, it
is often possible to recognize when a plume has
shifted, as opposed to being attenuated. As the
plume moves away from one well it will move
toward an adjacent well in the transect. Over
time, plumes can shift up and down in aquifers
as well as side to side. To deal with this possi-
bility, well clusters can be installed at different
elevations in the monitoring well transect.
At the site illustrated in Figure 1, each monitor-
ing location is a cluster of wells which sample
water from different portions of the aquifer.
-------
Figure 1 presents the data from the most con-
taminated well in the cluster at each monitoring
location. Table 1 presents data on the vertical
distribution of contamination in the monitoring
well transect. At the most contaminated loca-
tion, there was significant variation in concen-
trations with depth; the highest concentrations
of contaminants were at the shallowest depth
interval.
The monitoring wells in the transect capture
flow lines of contaminated water in the plume
as well as clean or much cleaner ground water
that surrounds the plume in three-dimensional
space. To the extent that the monitoring wells
provide a comprehensive sample of flow in the
plume, the trend analysis of concentrations in
the monitoring wells will provide a comprehen-
sive description of attenuation of contaminants.
Because concentration trends in monitoring
wells may vary independently of initial contam-
inant concentrations, it is important to evaluate
concentration trends across the entire contami-
nant plume.
5.2. Evaluation of whether attenuation
is adequate at a site
A remedy is put in place to protect a valued
resource from contamination or to restore a
valued resource, an aquifer, to beneficial use. If
necessary, natural attenuation can be replaced
with some form of active remediation that is
designed to treat the entire plume, or active
remediation can be applied to those portions
of the contaminant plume that are not perform-
ing as required in the ROD. Performance
Monitoring ofMNA Remedies (Pope et al.,
2004) identifies several decisions that can be
made as part of a performance review: continue
monitoring without change, modify the moni-
toring program, modify institutional controls,
implement a contingency or alternative remedy,
and terminate performance monitoring. Pope et
al. (2004) identify six criteria that may trigger a
decision. Three of the criteria address trends in
concentrations. These criteria are:
1) contaminant concentrations in soil or
ground water at specified locations
exhibit an increasing trend not originally
predicted during remedy selection,
2) near-source wells exhibit large concen-
tration increases indicative of a new or
renewed release, and
3) contaminant concentrations are not
decreasing at a sufficiently rapid rate
to meet the remedial objectives. The
approach presented in this Report is
designed to evaluate the third criterion.
As discussed in Section 2.6.1.2 of Pope et al.
(2004), most reviews of the progress on natural
attenuation attempt a comprehensive assessment
of the performance of the plume as a whole.
The assessment may use Thiessen polygons
to associate concentrations in individual wells
with a volume of water, multiply concentration
by volume to estimate the total mass of con-
taminant in the polygon sampled by the well,
and then add the masses of all the polygons to
estimate the overall mass of contaminant in the
plume at any given time (Dupont et al., 1998a,
1998b). The overall rate of natural attenuation
is calculated by comparing the rate of reduc-
tion of total mass over time. Other assessments
simply tally the number of wells where concen-
trations are declining, stable, or increasing.
Under most regulatory programs, the site will
not be considered clean until the entire site is
clean. Consequently, MNA can be demonstrated
to be an acceptable remedy when the methods
described in this paper show that all of the
monitoring wells can be expected to achieve the
site-specific clean up goals for each contami-
nant within an acceptable timeframe.
5.2.1. Evaluation of the site as a whole
Any large data set will likely have some prob-
lematic wells that:
l)are not meeting expectations, or
2) show an increase in concentrations over
the review cycle, or
3) have so much variation in concentration
over time that it is impossible to discern
a trend.
-------
A comparison of any problematic well to the
most contaminated wells at the end of the
review cycle will put the problematic well in
context. If the final concentration in a prob-
lematic well at the end of the review cycle is
low with respect to the most contaminated
wells at the site, then that particular prob-
lematic well, alone, does not put the site at
risk for not attaining the final goal, at least
not in the review cycle under consideration.
If the problematic well is one of the most con-
taminated wells at the end of the review cycle,
then the problematic well indicates a risk that
the remedy for the site will not attain the final
goal. A decision is required as to whether an
additional monitoring cycle is appropriate or
whether a more active remedial measure should
be implemented in all or a portion of the plume.
The second phase of analysis facilitates these
comparisons. The interim goal that is set for
the well that has the highest initial concentra-
tion of a particular contaminant can be consid-
ered the site-wide interim goal for that con-
taminant for the review cycle. If no well has
a concentration of that particular contaminant
at the end of the review cycle that is above the
site-wide interim goal, then attenuation across
the entire site can be considered adequate to
attain the final clean up goal. If a different well
exceeds the site-wide interim goal at the end
of the review cycle, a re-evaluation of MNA as
a potential remedy should be conducted. The
Conceptual Site Model (CSM) should be reeval-
uated in light of the new information regarding
the distribution of contamination. The possibil-
ity of additional source areas or residual source
material should be evaluated. The possibility
that groundwater flow directions have changed,
due to pumping or simply a re-interpretation
of the water level data, should be evaluated. If
the exceedance of the interim goal is small, if
revised estimates of the final clean-up time do
not conflict with the conditions in the ROD or
with other expectation for use of the resource,
and if the explanations (CSM, sources, flow
direction changes due to pumping) are reason-
able and manageable, then monitoring for an
additional cycle may be appropriate.
If the contaminant concentration in the well
newly observed to exceed the interim goal
would jeopardize the cleanup time expectations
of the ROD or the reuse expectations for the
site, and if additional source areas or residual
source material are not found and removed,
then additional monitoring for another cycle
may not be justified and more active reme-
diation should be implemented as described
in Section 5.2. This situation is described in
the next few paragraphs using data from the
example site. A method for quantifying the
uncertainty related to the decision to continue
MNA for another review cycle is described in
Section 5.2.2.
Table 2 illustrates these well-to-well compari-
sons for concentrations of TCE, cis-DCE, and
vinyl chloride in the three cluster wells in loca-
tion MW 3 in Figure 1. For each compound
in each monitoring well, the EvalMNA spread-
sheet was used to estimate the geometric means
of concentrations at the beginning and end of
the review cycle, as well as the interim goal.
The mean concentration in the final year was
compared to the interim goal to determine if
attenuation was adequate to meet the long term
goal. If the calculations in EvalMNA indicated
that attenuation was not adequate, then Table 2
presents the probability that the calculations
were in error, and attenuation truly was ade-
quate. If the probability of error is small, that
is a strong indication that attenuation was not
adequate.
The highest initial concentration of TCE was
found in well MW-3C, and the interim goal
for MW-3C (582 (J-g/L) can be considered
the interim goal for TCE for the entire site.
Fortunately, attenuation of TCE in MW-3C was
adequate to attain the long term goal. The final
mean for TCE in MW-3C was 259 jag/L. In
contrast, attenuation of TCE in well MW-3B
was not adequate to meet the long term goal;
and there is only a 15% chance that attenuation
would truly reach the goal. However, the final
-------
Table 2. Second phase of analysis. Evaluation of whether the reductions in concentrations of TCE, c/s-DCE
and Vinyl Chloride in monitoring over the review cycle will not meet the goals by the specified date.
Well
MW-3C
MW-3B
MW-3A
MW-3C
MW-3B
MW-3A
MW-3C
MW-3B
MW-3A
Compound
TCE
TCE
TCE
cis-DCE
cis-DCE
cis-DCE
Vinyl
Chloride
Vinyl
Chloride
Vinyl
Chloride
Mean Cone.
in 2001
C
o
"£/L
5055
1404
Mean Cone.
in 2006
C.
"£/L
259
308
Below MCL
1648
2319
1282
384
1652
1096
35
781
493
10
559
604
Interim Goal
for 2006
C
g
"£/L
582
241
614
777
517
74
203
153
Is attenuation
adequate to
meet long
term goals?*
Yes
No
Yes
No
Yes
Yes
No
No
ft*
a
O.15
>0.4
O.0025
O.0025
* If G! (the geometric mean for 2006)
-------
Along with Monitoring well MW-3A,
Monitoring well MW-3B also had the highest
concentrations of vinyl chloride at the begin-
ning and end of the review cycle, and the
second phase of analysis indicates that attenua-
tion is not adequate to meet the long term goal.
In contrast to the situation with cis-DCE, the
probability of error is very low. There is little
chance that the attenuation of concentrations of
vinyl chloride over the current review cycle is
adequate to meet the goals, and the behavior of
vinyl chloride deserves attention. The behavior
of vinyl chloride will be discussed further in the
next section.
5.2.2. Evaluation of individual wells
If the analysis in the second phase indicates
that attenuation over the review cycle is not
adequate to attain the clean up goal, the next
step is to evaluate the potential consequences.
The simplest and most straightforward way to
make that evaluation is to compare the date
when concentrations might achieve the clean up
goal to the date specified in the ROD. Again,
the process will be illustrated with the example
data set. Equation 4 was used to estimate the
date to attain the MCL for each contaminant in
each well.
Table 3 compares the maximum concentra-
tions and rate constants for attenuation to the
date when the goal might be attained. All three
contaminants might fail to meet the goal by
2017. Projected concentrations of TCE miss
the goal by less than two years while projected
concentrations of cis-DCE miss the goal by a
little over six years and projected concentra-
tions of vinyl chloride miss the goal by a little
over thirty one years. Concentrations of TCE
are projected to be below the MCL by the end
of the final review cycle, while concentrations
of cis-DCE and vinyl chloride above the MCL
are expected to persist into subsequent review
cycles.
Table 3. Using the first phase of analysis to project when concentrations of TCE, cis-DCE and Vinyl Chloride
in monitoring wells will attain the clean up goal.
Well
Compound
Maximum
Cone.
"S/L
Rate
Constant for
Attenuation
per year
Rate Constant at
90%
Confidence
per year
Target Date to Attain the Clean up Goal
End of Fourth Review Cycle
MW-3C
MW-3B
MW-3C
MW-3B
MW-3A
MW-3C
MW-3B
MW-3A
TCE
TCE
cis-DCE
cis-DCE
cis-DCE
Vinyl Chloride
Vinyl Chloride
Vinyl Chloride
8600
2300
3300
3200
1900
1200
2200
1500
0.588
0.326
0.563
0.153
0.154
0.715
0.206
0.130
>0.523
>0.281
>0.380
>0.106
>0.105
>0.601
>0.152
>0.0832
Date to
Attain MCL
years
2017
2021
2013.5
2018.9
2006.2
2023.3
2018.6
2009.3
2032.7
2048.5
-------
5.2.3. Uncertainty in estimates of time to
attain cleanup goal.
When Equation 4 is used to estimate the date to
attain the clean up goal, there is uncertainty in
the estimate, and the uncertainty should be con-
sidered in the evaluation. Figure 7 compares
the trend in concentrations of vinyl chloride
in monitoring well MW-3 A to the MCL for
vinyl chloride. In Figure 7, the left hand 7-axis
scales the data as the natural logarithm of vinyl
chloride concentration, and the 7-axis on the
right hand side scales the actual concentrations
of vinyl chloride. A regression was used to
predict the date when the concentration should
equal the MCL. The uncertainty is evaluated
by comparing the date when the confidence belt
on the regression line reached the MCL to the
date when the regression line reached the MCL.
The MCL for vinyl chloride is 2 (J,g/L, and the
natural logarithm of 2 is 0.69. The regression
line crosses the MCL at the projected date of
2048. The confidence belts were calculated for
a = 0.40. There is a 20% chance that the true
line runs above the upper confidence belt and
a 20% chance that the true line runs below the
lower confidence belt. The lower confidence
belt crosses the MCL in 2040. There is only a
20% chance that concentrations will attain the
MCL before 2040.
Similarly, the upper confidence belt in Figure 7
crosses the MCL in 2063. There is a 20%
chance of attaining the goal sometime after
2063, which means there is an 80% chance of
attaining the goal before 2063. The value of a
can be adjusted based on site specific concerns
to identify an appropriate target date to attain
the cleanup goal.
5.2.4. Interpretation of projections
The interpretation of contaminant concentra-
tion trends in an evaluation of the progress of
natural attenuation depends on two separate
evaluations. The first is a simple comparison
of the concentrations at the end of the review
cycle (such as a 5-Year Review) to the inter-
im concentration goal to determine whether
attenuation during the review cycle is adequate
to attain the final clean up goal by the date
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regression line
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Figure 7. Projected 60% confidence belts on the regression of the natural logarithm of concentration of
vinyl chloride in MW-3A on date.
-------
specified in the ROD. If the concentrations can
be expected to meet the clean up goal on or
before the specified date, there is no need for
further action until the next review.
If the comparison shows that attenuation cannot
be expected to meet the clean up goal, the next
step is to determine the statistical validity of
that result. To make this determination, it is
necessary to select an acceptable probability of
error a to use with the statistical test (the con-
fidence level of the test will be equal to (1-a)).
There are two approaches for obtaining a. In
one approach, the value of a is selected (or
negotiated) based on a willingness to modify
the ROD and implement an active remedy. The
other approach recognizes that a (the prob-
ability of Type I error) and P (the probability
of Type II error) are inversely linked. [See
Section 3.3.2 for a discussion of Type I and
Type II error.] In the second approach, the ratio
of P to a is selected. This may also involve
negotiations between the responsible parties and
regulatory agencies. Appendix B describes one
process to determine these values.
To complete the evaluation, the selected level
of a is used in the statistical tests built into
EvalMNA. There can be two results: either
the monitoring data indicate that attenuation
is not adequate to meet the goal, or the data
cannot provide evidence that attenuation is not
adequate at the selected level of confidence (the
statistical null hypothesis).
If the evaluation of the data indicates that
attenuation is not adequate to meet the goal at
the selected level of statistical confidence, this
provides documentation that there is a problem
with monitored natural attenuation as a remedy,
and some sort of active corrective action may
be indicated. If the data does not provide
evidence that attenuation is not adequate, there
is no basis to require additional active correc-
tion action at this time. When the data does
not provide evidence that attenuation is not
adequate, one cannot necessarily presume that
adequate attenuation is in fact occurring. There
is little choice except to continue to collect and
evaluate monitoring data. Even though natural
attenuation appears to be adequate to achieve
the remedial goals, the site is not yet clean and
subsequent data may demonstrate inadequate
attenuation.
Once confidence in the contaminant con-
centration trends and rates of attenuation are
established, collecting data over a longer time
interval may be appropriate and sufficient to
demonstrate achievement of the objectives of
the following review cycle(s). If the true rate
of attenuation is slower than the rate needed to
meet the clean up goal, the discrepancy between
the samples and the interim goals will grow
larger over time and be easier to discern. When
the data are highly variable and there are too
few samples, the statistical power of the test
described in this paper will be low; so increas-
ing the interval between samples (decreasing
sample frequency) may not be appropriate. The
monitoring strategy should be evaluated. If an
acceptable probability of error (a) for use with
the statistical tests cannot be achieved with the
current sample frequency, perhaps samples for
performance monitoring should be collected
more frequently than was originally anticipated.
5.2.5. Transformation products from natural
attenuation may be a special case
The concentration of vinyl chloride is con-
trolled by the rate of production of vinyl
chloride from cis-DCE as well as the rate of
degradation of vinyl chloride to ethylene. The
rate of attenuation is a composite rate which
includes the rate of production of the degrada-
tion product from the parent compound. In the
example used in this document the actual rate
of removal is higher than the compound specific
rate of attenuation because the input of vinyl
chloride from degradation of cis-DCE offsets
some of the degradation of vinyl chloride to
ethylene, resulting in an apparent degradation
rate which is less than would be attained if
there was no cis-DCE. As the concentrations of
TCE and cis-DCE decline over time, the rate of
attenuation of cis-DCE and vinyl chloride can
be expected to increase as the mass of parent
-------
compound is exhausted. But with the data
available in early review cycles, a projection of
the regression line may tend to overestimate the
time required for transformation products such
as vinyl chloride to attain the clean up goal. As
a consequence, concentrations of cis-DCE and
vinyl chloride may attain their MCL at some
time before the date projected by Equation 4.
This pattern of a transient accumulation of
the degradation product depends on an active
and ongoing mechanism for further degrada-
tion of the product. If there is no mechanism
for further degradation of the product, the
product (such as vinyl chloride from cis-DCE)
will simply accumulate. To understand if this
pattern of transient accumulation applies to a
particular site, it is necessary to know if the
degradation product is being degraded at the
current time.
The most direct line-of-evidence for degrada-
tion of a compound is the production of the
degradation product such as vinyl chloride from
cis-DCE and ethylene (or ethane) from vinyl
chloride. There are other lines of evidence that
can substantiate degradation of organic com-
pounds. A variety of molecular biological tools
are available to document the density of micro-
organisms in ground water that can degrade
contaminants (Azadpour-Keeley et al., 2009).
These tools are particularly useful to under-
stand the natural attenuation of PCE and TCE
to ethylene or ethane. The only organisms that
are known to degrade cis-DCE to vinyl chloride
or to degrade vinyl chloride to ethylene belong
to the Dehalococcoides group of bacteria. This
group of bacteria can easily be recognized and
enumerated in ground water with commercially
available molecular biological tools (Lu et al.,
2006). Often, the ratio of stable isotopes of
carbon, hydrogen, and chlorine will change in a
predictable fashion as compounds are degraded
in groundwater, and changes in the ratio of iso-
topes in the organic compound as water moves
along a flowpath can be used to document deg-
radation (Hunkeler et al. 2008).
There may be justification for making a distinc-
tion between a primary contaminant and its
transformation products. In the example data,
TCE was the material that was released, and
there is a possibility that TCE remains in the
aquifer as a non aqueous phase liquid (NAPL)
which could act as a continuing source of TCE
contamination in ground water. If attenuation
of the primary contaminant is not adequate to
meet the goal, some sort of active corrective
action may be indicated.
Often, the only plausible source of the transfor-
mation product in ground water is degradation
of the primary contaminant. If the attenuation
of the transformation products is not adequate
to meet the goal, there may be justification to
continue monitoring to see if the rate of attenua-
tion of the transformation products will increase
as the concentration of the primary contaminant
decreases.
5.2.6. Rates of attenuation reported in the
literature
One approach to evaluate and validate rate con-
stants that are extracted from data at a particular
site is to compare those constants to constants
extracted by others at other sites. Farhat et al.
(2004) analyzed temporal records and extracted
rate constants for petroleum hydrocarbons
from 366 sites and for chlorinated solvents at
40 sites. Newell et al. (2006) reported tempo-
ral records for concentrations of PCE, TCE,
DCE or TCA in monitoring wells: 52 records
were available from 23 sites. Kampbell et al.
(2000) reported temporal records for concen-
trations of benzene, toluene, ethylbenzene and
total xylenes at five large fuel spills. Peargin
(2001) reported temporal records for MTBE in
22 wells, benzene in 39 wells, and xylene in
34 wells that were screened across a smear zone
at fuel spill sites. Wilson et al. (2005) reported
rate constants from six MTBE spill sites. These
rates are compiled in Table 4. In general, the
rate constants for attenuation in the absence of
active remediation were in the order of 0.1 to
1.0 per year. The rates extracted in this docu-
ment from the example data for attenuation of
-------
Table 4. Typical rates of attenuation of concentrations over time in monitoring wells.
Chemical
Benzene
Benzene
Benzene*
Toluene
Toluene*
Ethylbenzene
Ethylbenzene*
Xylenes
Xylenes
Xylenes
PCE
PCE*
TCE
TCE*
c/s-DCE
cis-DCE*
Vinyl Chloride*
1,1,1-TCA
1,1,1-TCA*
1,2-DCA*
MTBE
MTBE
MTBE*
Median Rate
Maximum Rate
per year
0.22
0.12
0.22
0.31
0.41
0.087
0.18
0.093
0.07
0.25
0.23
0.11
0.11
0.15
0.16
0.62
0.07
0.34
0.15
0.14
0.04
0.23
0.08
0.41
4.49
0.44
6.42
0.41
2.96
0.43
7.10
0.97
1.84
0.60
2.42
0.17
1.75
1.44
0.61
1.50
1.25
0.70
6.9
Number
of Sites
5
39
359
5
89
5
90
5
34
89
9
32
13
37
2
11
18
6
23
13
22
6
78
Reference
Kampbell et al. (2000)
Peargin (2001)
Farhat et al. (2004)
Kampbell et al. (2000)
Farhat et al. (2004)
Kampbell et al. (2000)
Farhat et al. (2004)
Kampbell et al. (2000)
Peargin (2001)
Farhat et al. (2004)
Newell et al. (2006)
Farhat et al. (2004)
Newell et al. (2006)
Farhat et al. (2004)
Newell et al. (2006)
Farhat et al. (2004)
Farhat et al. (2004)
Newell et al. (2006)
Farhat et al. (2004)
Farhat et al. (2004)
Peargin (2001)
Wilson et al. (2005)
Farhat et al. (2004)
* Rate constants for well at each site with the highest concentration.
TCE, czs-DCE and vinyl chloride ranged from
0.1 to 0.7 per year (Table 3). These rates are
consistent with rates extracted by Newell et al.
(2006) and Farhat et al. (2004) at other sites
(Table 4).
The range of rate constants reported in Table 4
is narrow. The lower bond on the reported
rate constants may be caused by an inability to
resolve rate constants that are much below 0.1
per year with current approaches to monitoring.
Of the 52 temporal records reported by Newell
et al. (2006) for chlorinated solvents, attenu-
ation in 28 of the records was not statistically
significant at 95% confidence. As described by
Newell et al. (2002), the rate of natural attenu-
ation in a monitoring well more closely tracks
the rate of weathering of the source of contami-
nation as opposed to the rate of natural degra-
dation of the contaminant once it enters ground
water. The upper bound on the rate constants
may be limited by the rate of mass transfer of
contaminant from the residual source of con-
tamination to ground water or soil gas.
5.2.7. Dealing with data quality issues
When the progress of natural attenuation is
substantial and concentrations approach the
Maximum Contaminant Level (MCL) that is
allowed in drinking water, the data set will
-------
often include samples with concentrations that analyze it using a protocol that is less likely to
are below the method detection limit (MDL) or produce a flag.
the reporting limit (RL). If the data are evaluat-
ed by regression analysis and the samples with
concentrations below the MDL or RL occur
after the mean date in the regression, include
the sample and use the MDL or RL as value
for the sample. This will respect the number
of degrees of freedom in the data set, but will
bias the regression line and confidence bands to
project concentrations that are higher than the
true concentration. This biased analysis is con-
servative in that it will indicate that there is less
attenuation than the true extent of attenuation.
If this biased analysis indicates that attenuation
is still adequate to attain the goal, there is no
concern with the biased analysis. If the biased
analysis indicates that attenuation is not ade-
quate to attain the goal, disregard the analysis
and refer the statistical analysis to a profession-
al statistician who will apply methods that are
appropriate for data sets that contain censored
data. If the samples with concentrations below
the MDL or RL occur before the mean date in
the regression, refer the statistical analysis to a
professional statistician.
If the data are evaluated by comparing the mean
of samples in the final year of the monitoring
period to the mean of interim goals, and the
samples with concentrations below the MDL
or RL occur in the final year, conduct a biased
comparison using the MDL or RL as the value
for the sample. If this biased analysis indicates
that attenuation is adequate to attain the goal,
there is no concern with the biased analysis. If
samples with concentrations below the MDL or
RL occur in the initial year used to establish the
interim goals, abandon the approach of compar-
ing means in the final year of the monitoring
period and analyze the data set using regression
analysis.
If the value is flagged with an EPA Data
Qualifier "J" indicating that it is an estimated
value, use the value in the statistical analysis.
If possible, obtain another water sample and
-------
6.0
Suggestions and Recommendations
Follow the process outlined in Performance
Monitoring ofMNA Remedies (Pope et al.,
2004) to develop site-specific monitoring
objectives and performance criteria for MNA.
Conduct the performance review with those
objectives and criteria in mind.
Prior to a review of the performance of MNA,
analyze the monitoring record to reveal:
l)the attenuation in concentration of each
contaminant in each monitoring well
over the review cycle (Cz'/Co); and
2) determine whether the attenuation is
adequate to attain the long term goal by
the date specified in the decision docu-
ments; or alternatively,
3) estimate the probability that attenuation
is not adequate to attain the goal.
If the concentrations in a well are not expected
to attain the clean up goal by the specified time,
associate a level of statistical confidence and
statistical power with that determination. When
the confidence in a statistical test and the power
of the test are high, the results of the test are
more compelling, and provide more justifica-
tion to initiate active (more aggressive) clean up
actions rather than MNA.
Identify those wells that are more likely to pre-
vent a site from attaining the goals for MNA.
Then identify and assign priority to those areas
of the aquifer that are the best candidates for
focused active remediation. Re-evaluate prog-
ress toward clean-up and site closure at the end
of the next review cycle to determine the impact
of the focused active remediation efforts.
-------
-------
Azadpour-Keeley, A., M.J. Barcelona, K.
Duncan and J.M. Suflita. 2009. The Use
of Molecular and Genomic Techniques
Applied to Microbial Diversity,
Community Structure, and Activities at
DNAPL and Metal Contaminated Sites.
EPA/600/R-09/103. Available: http://www.
epa.gov/nrmrl/pubs/600r09103/600r09103.
Dupont, R.R., D.L. Sorensen, M. Kemblowski,
M. Bertleson, D. McGinnis, I. Kamil and
Y. Ma. 1998a. Monitoring and Assessment
of In-Situ Biocontainment of Petroleum
Contaminated Ground- Water Plumes.
EPA/600/R-98/020. Available: http://uwrl.
usu.edu/people/faculty/dupont/epamonitor-
ing.pdf.
Dupont, R.R., D.L. Sorensen, M. Kemblowski,
M. Bertleson, D. McGinnis, I. Kamil
and Y. Ma. 1998b. Project Summary:
Monitoring and Assessment of In-Situ
Biocontainment of Petroleum Contaminated
Ground- Water Plumes. EPA/600/
SR-98/020. Available: http://www.p2pays.
org/ref/07/06450.pdf.
Farhat, S.K., PC. de Blanc, C.J. Newell, J.R.
Gonzales and J. Perez. 2004. SourceDK
Remediation Timeframe Decision Support
System: User's Manual. Available: http://
www. gsi-net. com/en/software/free-software/
sourcedk.html.
Paul, F., E.Erdfelder, A. Buchner and A.-G.
Lang. 2009. Statistical power analyses
using G*Power 3.1: Tests for correlation
and regression analyses, Behavior Research
Methods, 41 (4), 1149-1160, doi:10.3758/
BRM.41.4.1149. Both the journal article
and the computer application are available:
http://www.psycho.uni-duesseldorf.de/aap/
vroiects/svower/.
7.0
References
Hunkeler, D., R.U. Meckenstock, B. Sherwood
Lollar, T.C. Schmidt and J.T. Wilson. 2008.
A Guide for Assessing Biodegradation and
Source Identification of Organic Ground
Water Contaminants Using Compound
Specific Isotope Analysis (CSIA). EPA
600/R-08/148 December. Available: http://
www. epa. gov/ada/gw/mna.html.
Kampbell, D.H., C.B. Snyder, D.C. Downey
and J.E. Hansen. 2001. Light nonaqueous-
phase weathering at some JP-4 fuel release
sites. Journal of Hazardous Substance
Research 3(4), 1-7 . Available: http://www.
engg. ksu. edu/hsrc/JHSR/v3 _no4.pdf.
Lu, X., D.H. Kampbell and J.T. Wilson. 2006.
Evaluation of the Role of Dehalococcoides
Organisms in the Natural Attenuation of
Chlorinated Ethylenes in Ground Water.
EPA/600/R-06/029. Available: http://www.
epa. gov/ada/gw/mna. html.
McHugh, I.E., L.M. Beckley, C.Y. Liu and C.J.
Newell. 2011. Factors influencing variabil-
ity in groundwater monitoring data sets.
Ground Water Monitoring & Remediation
31(2): 92-101.
Newell, C.J., H.S. Rifai, J.T. Wilson, J.A.
Connor, J.A. Aziz and M.P. Suarez.
2002. Calculation and Use of First-Order
Rate Constants for Monitored Natural
Attenuation Studies. EPA/540/S-02/500.
Available: http://www.epa.gov/ada/gw/mna.
html.
Newell, C.J., I. Cowie, T.M. McGuire, and W.
McNab. 2006. Multi-year temporal changes
in chlorinated solvent concentrations at
23 MNA sites. Journal of Environmental
Engineering 132(6): 653-663.
Peargin, T.R. 2001. Relative depletion
rates of MTBE, benzene, and xylenes
-------
from smear zone non-aqueous phase
liquid. In: Bioremediation of MTBE,
Alcohols, and Ethers. Proceedings of the
Sixth International In Situ and On-Site
Bioremediation Symposium. San Diego,
California, June 4-7. Battelle Press
6(l):67-74.
Pope, D., S. Acree, H. Levine, S. Mangion,
J. van Ee, K. Hurt and B. Wilson.
2004. Performance Monitoring of MNA
Remedies for VOCs in Ground Water.
EPA/600/R-04/027. Available: http://www.
epa.gov/ada/gw/mna.html
U.S. EPA. 1999. Use of Monitored Natural
Attenuation at Superfund, RCRA Corrective
Action, and Underground Storage Tank
Sites. Office of Solid Waste and Emergency
Response, Directive 9200.4-17P. Available:
http://www. cluin. org/download/reg/
d9200417.pdf.
U.S. EPA. 2007. Treatment Technologies for
Site Cleanup: Annual Status Report (Twelfth
Edition) EPA-542-R-07-012. Available:
http://www.clu-in.org/download/remed/
asr/12/asrl 2_main_body.pdf.
U.S. EPA. 2010. Superfund Remedy Report
(Thirteenth Edition) EPA-542-R-10-004.
Available: http://www.epa.gov/tio/download/
remed/asr/13/srr_13th_maindocument.pdf.
Wilson, J.T., P.M. Kaiser and C. Adair. 2005.
Monitored Natural Attenuation of MTBE
as a Risk Management Option at Leaking
Underground Storage Tank Sites EPA-
600/R-04/179. Available: http://www.epa.
eov/ada/sw/mna. html.
-------
Appendix A
First Phase Analysis,
Usins Linear Regression to Extract Rate Constants
This appendix provides further explanation
about the statistical methods being used and
detailed instructions on the use of a spreadsheet
to extract the rate of attenuation from monitor-
ing data.
The time required to reach a clean-up goal for
a contaminant in ground water sampled by a
monitoring well is directly related to the exist-
ing concentration of the contaminant in water,
the value of the clean-up goal, and the rate
constant for attenuation of that contaminant in
water sampled by that well. Rate constants that
are extracted from trends in concentration over
time can be used to forecast the time that will
be required for MNA to attain a particular clean
up goal for a contaminant. The uncertainty in
the estimate of the time required to reach the
goal is directly related to the uncertainty in
the estimate of the rate constant for attenua-
tion in concentration over time. The OSWER
Directive (U.S. EPA 1999) specifically requires:
Statistical confidence intervals should be
estimated for calculated attenuation rate
constants (including those based on meth-
ods such as historical trend data analysis,
analysis of attenuation along a flow path in
groundwater, and microcosm studies). This
appendix provides one approach to extract
information from historical trend analysis in
order to satisfy this requirement of the OSWER
Directive.
The monitoring data are analyzed using linear
regression. While specialized statistical soft-
ware is not always available, spreadsheets are
common computer applications. The simple
statistical tests outlined here will be calcu-
lated using a spreadsheet rather than statistical
software. The illustrations will be taken from
calculations done with Microsoft Excel 2003;
however, any spreadsheet can be used.
Section A. 1 provides instructions to perform
the linear regression on a particular data set.
Section A.2 provides instructions on the use of
Excel 2003 and links to computer applications
available on the internet to determine whether
a particular data set is appropriate for analysis
using linear regression.
A. 1. Use of the spreadsheet to
calculate a linear regression
Open a spreadsheet to a blank sheet.
A.1.1. Express the sample dates as decimal
years
Enter the dates when the samples were collect-
ed in a column, as was done in Column D of
Figure A.I. Be sure that the cells are formatted
as a date. To format as a date, left click with
the mouse on the D column header to select
the column, open Format from the menu bar,
open the Number tab and select Date from the
drop down menu. We intend to perform the
regression on elapsed years in the data set, not
elapsed days. To do that, we need to express
the data as a decimal year. Copy the data from
cells D5 to D31 and paste the data into cells
C5 through C31. Following the process above,
format the cells in column C as a Number
instead of a Date. Excel expresses the Date
as the number of days since January 1, 1900.
Insert a formula that converts the number
of days in the C column into decimal years.
This is done in cell B5 in Figure A. 1. Click
on the cell to select it, then type the formula
=1900+C5/365.25, which will appear in thefx
box above the cells. Click on the check mark
next to thefx box to accept the formula. Then
left click on the square at the lower right of the
cell with the formula [cell B5 in Figure A.I],
and drag it with the mouse to copy the formula
into the other cells in the column. Column B
-------
now contains the dates that the samples were
collected expressed as decimal years.
(EXCEL USER NOTE: Microsoft Excel
can calculate dates using two different date
systems. One system (the one used here) sets
January 1, 1900 as day 1. The other system
sets January 1904 as day 1. Make sure you
are not using the 1904 date system. Search
on "date system " in Microsoft Office Help
for instructions for your version of EXCEL.)
A. 1.2. Calculate natural logarithms
Next, enter the reported values for the con-
centrations of TCE, as was done in cells E5
through E31 of Figure A.I. Make sure the cells
are formatted as a Number. In cell F5 insert
a formula =LN(E5) to calculate the natural
logarithm of the value in the adjacent cell. As
above, copy the formula to the remaining cells
in column F. Column F now contains the natu-
ral logarithm of the concentrations of TCE in
column E in the spreadsheet.
=1900+05/365.25
=LN(E5)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A\
\
B
Date
\Decimal Year
1
1 2000.9
2001.3
2001.5
2001.7
2001.9
2002.3
2002.5
2002.7
2002.9
2003.2
2003.4
2003.7
2003.9
2004.3
2004.4
2004.7
2004.9
2005.3
2005.5
2005.7
2005.9
2006.3
2006.5
2006.7
2006.9
2007.3
2007.5
C
Date
Number
36870
36985
37061
37157
37223
37350
37431
37523
37594
37712
37777
37894
37960
38084
38146
38252
38322
38454
38516
38604
38698
38812
38889
38967
39057
39180
39252
D
Date
12/10/2000
4/4/2001
6/19/2001
9/23/2001
11/28/2001
4/4/2002
6/24/2002
9/24/2002
12/4/2002
4/1/2003
6/5/2003
9/30/2003
12/5/2003
4/7/2004
6/8/2004
9/22/2004
12/1/2004
4/12/2005
6/13/2005
9/9/2005
12/12/2005
4/5/2006
6/21/2006
9/7/2006
12/6/2006
4/8/2007
6/19/2007
E\
\
TCE\
M9/L \
1800
1200
2300
1600
880
1100
580
870
1400
1400
980
520
530
700
730
400
400
672
724
306
169
276
388
357
234
232
163
F
LNTCE
^ 7.496
7.090
7.741
7.378
6.780
7.003
6.363
6.768
7.244
7.244
6.888
6.254
6.273
6.551
6.593
5.991
5.991
6.510
6.585
5.724
5.130
5.620
5.961
5.878
5.455
5.447
5.094
G
Figure A.1. A spreadsheet to calculate the decimal date of sampling and the natural logarithm of the concen-
tration of TCE.
-------
A. 1.3. Run the regression
Next, click on Tools in the menu bar, and
select the tab Data Analysis, then scroll down
and select Regression from the menu. See
Figure A.2. This will open an input screen as
illustrated in Figure A.3.
(EXCEL USER NOTE: The instructions are
for EXCEL 2003. If you are using another
version of EXCEL, search on "data analy-
sis " in Microsoft Office Help for instructions.
The Data Analysis ToolPak is not installed
by default when EXCEL is installed. If the
Data Analysis tab is not shown under Tools,
select Tools Add-Ins and check the two
Analysis Tools options).
Input the data range of cell in column F (LN
TCE) as the Input Y Range and the data range
of cells in column B (Date Decimal Year)
as the Input X Range. Input the desired con-
fidence interval, and then click OK. The
spreadsheet calculates the regression and
presents output in a new tab in the spreadsheet
(Figure A.4).
D
Date
Date
Date
TCE
LNTCE
Decimal Year
Number
2000.9
36870
12/10/2000
1800
7.496
2001.3
36985
4/4/2001
1200
7.090
2001.5
37061
6/19/2001
2300
7.741
2001.7
37157
9/23/2001
1600
7.378
2001.9
37223
11/28/2001
880
6.780
10
2002.3
37350
4/4/2002
1100
7.003
11
2002.5
37431
6/24/2002
580
6.363
12
2002.7
37523
9/24/2002
870
6.768
13
16
17
18
19
20
21
22
23
24
I Covariance
Descriptive Statistics
Exponential Smoothing
F-Test Two-Sample for Variances
Fourier Analysis
Histogram
Moving Average
Random Number Generation
Rank and Percentile
Cancel
Help
25
2005.9
38698
12/12/2005
169
5.130
26
2006.3
38812
4/5/2006
276
5.620
27
2006.5
38889
6/21/2006
388
5.961
23
2006.7
38967
9/7/2006
357
5.878
29
2006.9
39057
12/6/2006
234
5.455
30
2007.3
39180
4/8/2007
232
5.447
31
2007.5
39252
6/19/2007
163
5.094
Figure A.2. The Regression dropdown menu under the Data Analysis tab in the Tools menu.
-------
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Date
Decimal Year
2000.9
2001.3
2001.5
Date
Number
36870
36985
37061
Date
12/10/2000
4/4/2001
6/19/2001
TCE
1800
1200
2300
LNTCE
7.496
7.090
7.741
Labels
0 Confidence Level:
Output options
O Output Range:
0 New Worksheet Ply:
O New Workbook
Residuals
0 Residuals
0 Standardized Residuals
0 Residual Plots
0 Line Fit Plots
Normal Probability
0 Normal Probability Plots
2007.3
2007.5
39180
39252
4/8/2007
6/19/2007
232
163
5.447
5.094
Figure A.3. The input menu for linear regression.
A. 1.4. Examine the results
Examine Figure A.4. The slope of the regres-
sion is in cell B18, identified as the Coefficient
for X Variable 1. The coefficient in the regres-
sion analysis is the first order rate constant for
the rate of change. The rate of attenuation is
the negative of the rate of change. For this par-
ticular data set, the first order rate of attenuation
is +0.326 per year.
Examine Figure A.4. The values in cells F18
and G18 are the upper and lower 95% confi-
dence intervals on the rate of change. The 95%
confidence intervals are provided as a default
in Excel 2003. Because we asked for an 80%
confidence interval, the values in cells HIS and
118 are the 80% confidence intervals. Because
of the sign change, the "upper" confidence
interval in cells is the slower confidence inter-
val and the "lower" confidence interval in cells
is the faster confidence interval.
The confidence intervals on the rates are cal-
culated by Excel using the Student's t distribu-
tion. The calculations use a two-tailed value
for the t statistic. That means that one half the
error is in rates that are faster than the "faster"
confidence interval, and one half is in rates that
are slower than the "slower" confidence inter-
val. A rate faster than is expected is a desirable
-------
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
X\
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
AN OVA
Regression
Residual
Total
Intercept
X Variable 1
/
B
0.886
0.784
0.776
0.346
27.000
df
1.000
25.000
26.000
Coefficients
659.700
4 -0.326
/
F
Significance F
0.00000000083
Lower 95%
518.563
-0.396
G
Upper 95%
800.837
-0.256
H
Lower 80.0%
569.493
-0.371
y
/
Upper 80.0%
749.907
4 -0.281
/
7
7
Rate of attenuation
= 0.326 per year
Slower 90%
confidence internal
on the rate of
attenuation
= 0.281 per year
Figure A.4. The summary output of the linear regression.
outcome, and there is no need to use statistics to
"protect" ourselves from the possibility of faster
rates of attenuation. The best course is to put
all the uncertainty on the side with the slower
rate. Statisticians refer to this as a one-tailed
test because all the uncertainty is in one tail of
the frequency distribution. The term 80.0%
confidence in the Excel spreadsheet means we
are willing to accept a 20% chance of error,
10% of the chance in the faster tail and 10%
in the slower tail. As a result, the two-tailed
Upper 80.0% confidence interval in cell 118 of
Figure A.4 is also the slower one tailed 90%
confidence interval on the rate. As revealed by
the regression analysis in Figure A.4, the rate
constant for attenuation of TCE concentrations
over time is 0.326 per year (cell B18), and that
rate constant is at least 0.281 per year at 90%
confidence (cell 118). Remember that the rate
of attenuation is the negative of the rate of
change as calculated in EXCEL.
A.2. Statistical background on linear
regression
Data analysis using linear regression makes
three major assumptions:
1) there is a linear relationship between the
dependent variable (concentration) and
independent variable (time or date of
sampling),
2) the variance in the data is constant over
time, and
3) the residual errors follow a normal prob-
ability distribution.
The regression in Section A. 1 was conducted on
a natural logarithm transformation of the origi-
nal concentration data. This section explains
why the transformation was appropriate for the
example data set, and provides an approach to
determine if the natural logarithm transforma-
tion is appropriate for other data sets.
-------
A.2.1. Linear relationship between variables
If the relationship is not linear, this is usually
most evident in a plot of the observed versus
predicted values. The data points should be
symmetrically distributed around the regression
line. Look carefully for evidence that the data
follow a "bowed" pattern, which would indicate
that the data do not follow a linear model. This
approach is illustrated in Figure A. 5.
Panels A and B of the figure were created as
charts in Excel 2003. The spreadsheet provides
a convenient option of fitting linear, exponen-
tial, logarithmic and polynomial trend lines to
the data in the chart, and extracting an equa-
tion and values of r2 for the trend line. The
Coefficient of Determination (r2) is an estimate
of the goodness of fit of the data to the equa-
tion. The value of the coefficient is the frac-
tion of the total variation in the data that is
explained by the equation. If r2 were 0.800,
then 80% of the variation in the dependant vari-
able is explained by the equation that is fit to
the data, and 20% remains unexplained by the
equation.
Panel A of Figure A. 5 plots the data on an
arithmetic scale, and fits a linear regression
line. Notice that the distribution of the example
data seems to be bowed. Most of the data from
the very earliest dates are above the line, and
in one case the datum is far above the line,
most of the data from the intermediate dates
are below the line, and most of the data from
the latest dates are again above the line. The
data are better fit with an exponential trend
line than a linear trend line (r2 =0.7843 for the
exponential trend line in Panel B of Figure A.5
compared to r2 =0.6848 for the linear trend line
in Panel A of Figure A.5). The value of r2 for
a logarithmic trend line was 0.6849, and the
value for a polynomial trend line varied from
r2 =0.7199 for a second degree polynomial to
r2 =0.7266 for a six degree polynomial (data
not presented). Of the available trend lines
in Excel 2003, the exponential trend line has
the highest r2 and provides the best fit to the
example data set.
Figure A. 6 is a linear trend line on the natu-
ral logarithm of the concentrations of TCE.
Notice that the data now follow the linear trend
line without any apparent curvature. Taking a
logarithm reduces an exponent in the dependant
variable to a factor, making the relationship
between the variables a linear function of the
independent variable. A logarithmic transfor-
mation of the concentration data will provide a
better fit to the assumptions of linear regression.
The value of r2 for the linear trend of the natu-
ral logarithm of TCE on date of sampling is the
same as the exponential trend of concentration
on date. The natural logarithm transformation
of the data provides a better fit of the assump-
tion of a linear relationship.
A.2.2. Uniform variance in data
The vertical distance between each data point
and the regression line is called the residual
variation or residual for that data point. If the
variance in the data is constant over time, the
residuals will be uniformly distributed along the
regression line. Notice in Panel A of Figure A.5
that the distances are greater at earlier dates and
smaller at later dates, while in Panel B the dis-
tances are variable but uniformly spread along
the line. Obviously, the natural logarithms of
the concentrations provide a better fit to the
assumption of uniform variance in the data with
date of sampling.
A.2.3. Normal distribution of residuals
The third criterion, that the residuals from the
regression follow a normal probability distri-
bution, is not readily apparent from a simple
inspection of a plot of the data. Excel 2003
gives the user an option to view a table with
the residuals from the regression. The menu
depicted in Figure A.3 provides an option to
see a table of the Residuals, and the Standard
Residuals from the regression. Figure A.7
presents the RESIDUAL OUTPUT report
provided in Excel 2003 when these options are
selected. As discussed above, the numbers in
the column labeled Residuals are simply the
difference between the value of the depen-
dant variable and the predicted value of the
-------
3000
0
2000.0
2002.0 2004.0 2006.0
Date of Sampling
2008.0
3000
2500 -
15> 2°°° "
^.
c
= 1500 H
(U
o
o 1000 -
o
500 -
0
2000.0
TCE
Expon. (TCE)
r2=0.7843
-i-
-i-
2002.0 2004.0
Date of Sampling
2006.0
2008.0
Figure A.5. Fit of a linear trend and an exponential trend to concentration of TCE.
-------
C
o
"ro
*-»
CD
O
c
o
O
m
D5
o
ro
9.0
8.5 -
8.0
7.5 -
7.0 -
6.5 -
6.0 -
5.5 -
5.0 -
4.5 -
4.0
» TCE
Linear (TCE)
r2=0.7843
2000.00 2002.00 2004.00 2006.00
Date of Sampling
Figure A.6. Fit of a linear trend to the natural logarithm of concentration of TCE.
2008.00
dependant variable (Predicted Y) in the report
as depicted in the figure.
The assumption of a normal distribution of
residuals is evaluated by comparing the dis-
tribution of the residuals to the normal prob-
ability distribution. The first step is to examine
a normal probability plot or Q-Q plot to see
if there are obvious differences between the
distribution of the residuals of regression and
the normal distribution. The second step is
a goodness-of-fit test between the residuals
and the normal distribution. Two approaches
to make these comparisons are discussed in
Sections A.3. and A.4. below. Examine both
sections to identify the approach that is most
appropriate for you.
A.3. Using Excel to generate a Q-Q
plot and using the One-Sample
Kolmogorov-Smirnov Goodness-
of-Fit Test
The standard probability distribution has a mean
of zero and a standard deviation of one. The
regression calculates an equation such that the
mean of the Residuals is zero. In the column
labeled Standard Residuals, the Residuals
have been scaled to make their standard devia-
tion also equal to one. The numbers labeled
PROBABILITY OUTPUT are not needed for
this analysis. Delete this information to create
space in the spreadsheet for a calculation of
the values of the standard normal probability
distribution that correspond to the Standard
Residuals.
Select the values under Standard Residuals,
then open the Data drop down menu from
the menu bar, select Sort, select Expand the
Selection, and then the Sort button. Indicate
that you intend to sort with a header, and
sort ascending. The Observation number,
Predicted Y, Residuals, and Standard Residuals
should all sort together.
Label the column immediately to the right of
Standard Residuals as Rank, and insert numbers
to order the Standard Residuals from least to
greatest, as illustrated in Figure A. 8. Label the
next column to the right as Quantile. In the
first cell under Qauntile insert a formula for the
-------
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
A
B
RESIDUAL OUTPUT
Observation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Predicted Y
7.476268292
7.373639551
7.305815339
7.220142651
7.161242677
7.04790485
6.975618519
6.893515526
6.830153433
6.72484742
6.666839871
6.562426282
6.503526308
6.392865752
6.337535474
6.242938547
6.180468879
6.062668932
6.007338654
5.928805356
5.844917515
5.743181198
5.674464562
5.604855503
5.524537357
5.414769225
5.350514709
C
Residuals
0.01927365
-0.28356272
0.43484906
0.15761626
-0.38132077
-0.04483939
-0.61259042
-0.12502231
0.41407408
0.5193801
0.2207127
-0.30859747
-0.2306493
0.15821458
0.25550906
-0.251474
-0.18900433
0.44758941
0.57745274
-0.20522025
-0.7150188
-0.12278033
0.28654078
0.27288028
-0.06921624
0.03196815
-0.25676451
D
Standard Residuals
0.056823429
-0.836012096
1.282041175
0.464691199
-1.124226702
-0.132197469
-1.806066066
-0.368596297
1.220791463
1.531259292
0.650714914
-0.909820664
-0.680010438
0.466455207
0.753303074
-0.741406732
-0.557230901
1.319602825
1.702471618
-0.605039399
-2.108049945
-0.361986387
0.844792151
0.804517668
-0.204066375
0.094249899
-0.75700444
E
F
G
PROBABILITY OUTPUT
Percentile
1.851851852
5.555555556
9.259259259
12.96296296
16.66666667
20.37037037
24.07407407
27.77777778
31.48148148
35.18518519
38.88888889
42.59259259
46.2962963
50
53.7037037
57.40740741
61.11111111
64.81481481
68.51851852
72.22222222
75.92592593
79.62962963
83.33333333
87.03703704
90.74074074
94.44444444
98.14814815
Y
5.093750201
5.129898715
5.446737372
5.455321115
5.620400866
5.723585102
5.877735782
5.96100534
5.991464547
5.991464547
6.253828812
6.272877007
6.363028104
6.510258341
6.551080335
6.584791392
6.593044534
6.768493212
6.779921907
6.887552572
7.003065459
7.090076836
7.244227516
7.244227516
7.377758908
7.495541944
7.740664402
Figure A.7. Report in Excel 2003 of the Residuals and Standard Residuals from the Linear Regression of the
example data set.
center of the interval represented by rank. The
formula is the rank minus 0.5, then that differ-
ence divided by the number of observations.
Drag the formula to paste into all the cells with
an observation.
Then label the column immediately to the right
of Quantile as z-Score. Insert a formula that
returns the value of the z-distribution that cor-
responds to the appropriate quantile. For Excel
2003, the formula is =NORMSINV(column
row). Drag the formula to paste into all the
cells with an observation.
The distribution of the Standard Residuals will
be compared to the standard normal distribu-
tion (z-Score), to determine if the distributions
are different. The first comparison is graphical.
Create a chart with two series of data. The first
series plots values under z-Score as the x-axis
and the Standard Residuals as the y-axis. The
second series plots z-Score as both the x-axis
and y-axis. The resulting chart compares the
distribution of the Standard Residuals to the
normal probability distribution. The normal
probability distribution will lie along a straight
line. If the Standard Residuals are normally
distributed, they will lie along the same line.
Jumps or breaks between adjacent data points
may indicate that the data are sampled from two
different populations with different properties.
A sample that is well separated from the line
may be an "outlier" (Singh et al. 2010). The
-------
=(E25-0.5)/27
=NORMSINV(F25)
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
A
B
RESIDUAL OUTPUT
Observation
21
7
5
12
2
27
16
13
20
17
8
22
25
6
1
26
4
14
11
15
24
23
9
3
18
10
19
Predicted Y
5.845
6.976
7.161
6.562
7.374
5.351
6.243
6.504
5.929
6.180
6.894
5.743
5.525
7.048
7.476
5.415
7.220
6.393
6.667
6.338
5.605
5.674
6.830
7.306
6.063
6.725
6.007
C
Residuals
-0.715
-0.613
-0.381
-0.309
-0.284
-0.257
-0.251
-0.231
-0.205
-0.189
-0.125
-0.123
-0.069
-0.045
0.019
0.032
0.158
0.158
0.221
0.256
0.273
0.287
0.414
0.435
0.448
0.519
0.577
D >
Standard Residuals
-2.108
-1.806
-1.124
-0.910
-0.836
-0.757
-0.741
-0.680
-0.605
-0.557
-0.369
-0.362
-0.204
-0.132
0.057
0.094
0.465
0.466
0.651
0.753
0.805
0.845
1.221
1.282
1.320
1.531
1.702
E
\
\
\
Ranl\
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
F \
\
\
\
Quantile
* 0.019
0.056
0.093
0.130
0.167
0.204
0.241
0.278
0.315
0.352
0.389
0.426
0.463
0.500
0.537
0.574
0.611
0.648
0.685
0.722
0.759
0.796
0.833
0.870
0.907
0.944
0.981
G
( z-Score
* -2.085
-1.593
-1.325
-1.128
-0.967
-0.828
-0.704
-0.589
-0.482
-0.380
-0.282
-0.187
-0.093
0.000
0.093
0.187
0.282
0.380
0.482
0.589
0.704
0.828
0.967
1.128
1.325
1.593
2.085
Figure A.8. Calculation of the values of the standard normal distribution (z-distribution) corresponding to the
rank and quantile of the standard residuals.
data in the outlier could be real, or it could be
an error in labeling the sample in the field, an
error in reading the label in the laboratory, an
error in dilution of the sample in the laboratory,
or an error in entering the data in the data file.
The plot for the example data is presented in
Figure A.9. There is little evidence that the
Standard Residuals are not normally distributed.
Excel 2003 has a menu option for Normal
Probability Plots. Excel does not provide a
normal probability plot or Q-Q plot as the terms
are usually understood. Instead Excel provides
a plot of the original data against the quantile
expressed as percent of the number of samples.
Construct your own normal probability plot as
described above, and ignore the "normal prob-
ability" plot provided by Excel.
Interpreting a normal probability plot requires
a certain amount of judgment, which in turn
is based on experience. If you want a more
objective means to compare the distributions,
there are a number of statistical tests that can
be used to compare the distribution of Standard
-------
C/3
03
1
V)
CD
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D
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03
D
f~
03
CO
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1.5 -
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-1.5 -
-2 -
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0^»*
X
^s^
^
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0 o $* <" Standard Residuals
o » » Normal Distribution
*
o
*
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
Rank Based z-Score
Figure A.9. Normal Probability Plot (also called a Q-Q plot) comparing the distribution of the residuals from
the regression to the normal probability distribution.
Residuals to the normal probability distribu-
tion. One useful statistic is the One-Sample
Kolmogorov-Smirnov Goodness-of-Fit Test.
The test is also called the Kolmogorov-Smirnov
D test or Kolmogorov-Smirnov Z test. The
Kolmogorov-Smirnov Test provides an estimate
of the probability that a given distribution is not
significantly different from a second distribu-
tion. In this case, the first distribution will be
the Standard Residuals from the regression and
the second will be the normal distribution as
described by the z-Scores.
Applications to perform the Kolmogorov-
Smirnov Test are readily available on the inter-
net. One that is particularly useful can be found
at http://www.physics.csbsju.edu/stats/KS-test.
html. To use the application, open the KS-test
Data Entry form provided as a link from the
web page and copy the values for Standard
Residuals from the Excel spreadsheet and paste
into Dataset 1 and copy values from z-Score
from Excel and paste into Dataset 2, then
calculate (see Figure A. 10). The application
returns with the following statement regarding
distribution 1, which is the Standard Residual
after regression: KSfinds the data is consistent
with a normal distribution: P = 0.93 where the
normal distribution has mean= -2.6199E-02
and sdev= 1.106. See Figure A. 11.
For any value of P above 0.50, the
Kolmogorov-Smirnov Test finds that it is
more likely that the distribution of Standard
Residuals is not significantly different from a
normal distribution. This leads to the following
suggested decision criterion. For any value
of P above 0.50, it is most likely that the
Standard Residuals follow a normal distribu-
tion, and the approach and procedures in
Appendix A are appropriate. If the value of
P is less than 0.50, it is most likely that the
-------
KS-test Data Entry
Use the below form to enter your data for a Kolmogorov-Smirnov test.
The KS-test seeks differences between your two datasets; it is non-
parametric and distribution free. Reject the null hypothesis of no
difference between your datasets if P is "small". In addition this page
reports if your datasets seem to have normal or lognormal distribution.
This may allow you to use other tests like the t-test.
For each dataset, enter your data into the given box separating each
datum from its neighbor with tabs, commas, or spaces. Very commonly
you will already have the data in your computer in some format. You
should be able to just copy and paste that data into the appropriate
area. This KS-test form is designed to handle datasets with between 10
and 1024 items in each dataset.
Dataset 1:
-2.108049355
-1.806066066
-1.124226702
-0.909820564
Standard Residuals
-0.75700444
Dataset 2: «
-1.593218318
-1.324957589
-1.12S143S45
-0.967421566
-U.H2B464049
Calculate Please
Z-Scores
|
| ClearAIICata
Figure A.10. Data entry form in an application to perform the Kolmogorov-Smirnov Test.
Standard Residuals do not follow a normal
distribution, and the approach and proce-
dures in Appendix A should not be used with
that data set to extract a first order rate
constant for natural attenuation.
A.4. Using ProUCL for a Q-Q plot and
Goodness-of-Fit testing
U.S. EPA provides free software that can easily
construct a Q-Q plot and test for goodness-
of-fit. Download and install ProUCL Version
4.00.05 (Singh et al, 2010). Copy the column
heading and data for residuals from regression
from column C in the Excel spreadsheet as
depicted in Figure A. 8 and paste the data into
a new Excel workbook. Format the data as
a number and the column label as text. Save
the workbook as an Excel 97-2003 workbook
(*.xls) in the data file in the directory contain-
ing ProUCL. Open the File menu in ProUCL,
select the Load Excel Data option, select the
workbook and open the data into ProUCL.
Open the menu Graphs and select Multi Q-Q
to open the menu Select Variables. Select
Residuals and click OK. ProUCL generates a
Q-Q plot (Figure A. 12, compare to Figure A.9).
ProUCL also will do goodness-of-fit testing.
Open the menu Goodness-of-Fit, to open the
-------
KS Test: Results
Kolmogorov-Smirnov Comparison of Two Data Sets
The results of a Kolmogorov-Smirnov test performed at 17:08 on 6-JUL-2011
The maximum difference between the cumulative distributions, D, is: 0.0741 with a corresponding P of: 1.000
Data Set 1:
27 data points were entered
Mean = 1.1111E-10
95% confidence interval for actual Mean: -0.3956 thru 0.3956
Standard Deviation = 1.000
High =1.70 Low = -2.11
Third Quartile = 0.805 First Quartile = -0.741
Median = -0.1322
Average Absolute Deviation from Median = 0.824
KS finds the data is consistent with a normal distribution: P= 0.93 where the normal distribution has mean= -2.6199E-02 and sdev= 1.106
Figure A.11. Results returned from the Kolmogorov-Smirnov test.
menu Select Variables. Select Residuals.
Open Options and select the desired level of
confidence. Click OK to Select Confidence
Level and OK to Select Variables and ProUCL
produces values for the Shapiro Wilk (SW)
Test Statistic and the Lilliefors Test Statistic
(Figure A. 13). According to Singh et al. (2010)
ProUCL 4.0 provides S-W test only for samples
of sizes up to 50. Lilliefors test (along with
graphical Q-Qplot) seems to perform fairly
well for samples of size 50 and higher. Both
tests are restricted in ProUCL to confidence
levels of 90%, or 95% or 99%. If the data
appear to be normal at a = 0.10, there is no
reason to reject the Approach in this Report.
However, the limited menu for statistical confi-
dence provided by ProUCL 4.0 does not allow
ProUCL to evaluate the distribution of residuals
against the decision criterion of a probability of
50%.
References:
Singh, A., R. Maichle, N. Armbya, A.K.
Singh and S.E. Lee. 2010. ProUCL
Version 4.00.05 User Guide (Draft).
EPA/600/R-07/038. Available http://www.
epa.gov/esd/tsc/software. htm. updates avail-
able at http://www.epa.gov/osp/hstl/tsc/soft-
ware.htm.
-------
P? PrnlJCI 4.0
Fib Edit ConFigurc Number of Samples Summary Statistics 3 Outlier Tests Goodness of Fit
BOX Plot
HbLuur dm
Hesiduals
3^
0.0 1927-36515855012
0.13 4849062801062
0.157C1C257C22102
-0.381320763782425
-0.61 259041 "5274539
i ID 1 c«i 1
Theoretical Quantiles (Standard Normal)
--H Residuals
ID | Count |
Group hy
Figure A.12. L/s/ng ProUCL to generate a Q-Q plot.
-------
P? ProUCL 4.0 - [Sheetl .wst]
B File Edit Configure Number of Samples Summary Statistics Graphs
Outlier Tests
CLl'&l BJHlHl EJ
1
2
,-.
0
Residuals
0.01 9273651 585501 2
-0.283562715077704
n vn^Q^Qncooninco
1
234
5
6
i
//
n Hypothesis
Normal
Gamma
Lognorrnal
'^^^H
9
TZ
N
^gl^Tiyn^^j
Variables
^^^^H
Name | ID I Count
| Bi Select Confidence Level X
« 90%
r 95%
r 99 x
OK
Cancel
.-
^^^
»
«
^^^^^^^^^H
Selected
Name | ID | Count
Residuals
Group b
0 27
Correlation Coefficient R 0.93S
S hapiio Wilk T est S tatisdc 0.969
Shapiro Wilk Critical (0.9) Value 0.935
Approximate Shapiro Wilk P Value 0.592
Lilliefors Test Statistic 0.0863
Lilliefors Critical (0.9) Value 0.155
Data appear Normal at (0.1 ) Significance Level
y Variable
3
Options
OK Cancel
1
Figure A.13. Using ProUCL to evaluate goodness-of-fit to a normal distribution.
-------
-------
Appendix B
Second Phase Analysis
B.1. Statistical Approach
The purpose of the second phase analysis is to
compare conditions at the beginning of a review
cycle to conditions at the end of the review
cycle. Instead of a linear regression of all the
concentration data collected in the review cycle
over time, we will only compare data collected
in the first year of the cycle to data collected in
the final year of the cycle. To have statistical
control on the extent of attenuation from the
first to the last year, it is necessary to have mul-
tiple samples of the concentration of the con-
taminant in the water from the well in both the
first year and in the final year. The samples in
the first year will be used to calculate a popula-
tion of expected concentrations in the last year
of the review cycle, based on
l)a concentration-based long term goal for
clean up,
2) a date by which the goal is to be
obtained, and
3) the assumption that attenuation will
follow first order kinetics.
These expectations will be taken as interim
goals.
The approach assumes that each year of the
monitoring record can be treated as a single
entity, and the effects of attenuation within each
year can be ignored. McHugh et al. (2011)
evaluated three large monitoring records, and
found that there was a characteristic time-inde-
pendent variability associated with subsequent
samples from the same well, and that long term
trend in attenuation in concentration could not
be distinguished from the time-independent
variability unless the samples were separate
in time by more than 320 to 400 days. As an
acceptable approximation, the effects of attenu-
ation within a monitoring year will be ignored.
All calculations are performed on the natural
logarithm of the concentrations in the samples.
A geometric mean will be calculated for the
interim goals and for the samples in the final
year of the review cycle. A Student's t test for
the difference of means will be used to compare
the mean for the interim goals to the geometric
mean of samples in the final year of the review
cycle. This test provides a confidence inter-
val on the difference between the mean of the
interim goals and the mean of the final year of
data. If the mean of samples in the final year
is greater than the mean of the interim goals at
some predetermined level of confidence, that
fact will indicate that attenuation in the review
cycle is not adequate to meet the long term
goal. The Student's t test for the difference of
means performed on the natural logarithm of
the concentration data is equivalent to compar-
ing the ratio of the means to determine if the
attenuation (CB ,/C. ,.,) is adequate to attain the
^ final initial' ~
goals.
B.1.1. Statistical background: Use of the
Student's f-statistic
The Student's /-statistic is simply the ratio of
"signal" to "noise" in a data set. In this appli-
cation, the calculated value of the /-statistic is
the difference between the means divided by
the standard deviation of the difference. The
difference in the means is the signal and the
standard deviation is a measure of the noise. A
difference between means can be tested to see if
it is statistically different from zero by compar-
ing the calculated value of the /-statistic to the
critical value of t. The critical value of t is the
maximum value of t that can be expected from
a theoretical distribution of t where:
l)the variation in the measured values is
due only to normal random variation,
and
-------
2) the user is willing to accept a prede-
termined chance that the test will fail
and conclude that the difference in the
means is different from zero when in
fact it is not different.
If the predetermined chance that the test will
fail is a, then the level of confidence in the
test is (1 - a). As an example, a value of a of
0.05 would correspond to a confidence level of
95%; a value of a of 0.20 would correspond to
a confidence level of 80%. If the calculated t is
greater than the critical value of t, the difference
in the means is statistically different from zero
at that predetermined level of confidence.
Statistics text books provide tables of the
critical value of t calculated for predetermined
values of a such as 0.01, 0.05, 0.10, or 0.15.
Spreadsheets have macros that can calculate the
critical value of t for any arbitrary value of a.
B.1.2. Theoretical basis of the statistical
comparison of means
In this application, the signal is the difference
in the means of the initial year and final year
of the review cycle. Student's t is a calculated
statistic that will be compared to the critical
value of t which depends on the acceptable
probability of error and the number of degrees
of freedom in the comparison (ta df). For this
case, t is defined in Equation B.I, whereX} is
the mean of the samples collected in the initial
year, X2 is the mean of the samples collected in
the final year, \\.l is the true but unknown mean
concentration in the initial year, |u,2 is the true
but unknown mean in the final year, and S(^l-^2>
is the standard deviation of the difference of the
means.
t = -
B.I
S(Xl-X2)
If we postulate that the real difference in the
means in the initial and final year is zero, where
(uru2) = 0, then
(X,-X^-0
The difference of means is significantly greater
than zero whenever Equation B.2 pertains.
(Xl-X2)>S(I^xt^,, B.2
J«
In equation B.2, ta is the critical value of
the /-statistic for a particular value of a and
the appropriate number of degrees of freedom
in the comparison. When there are the same
number of samples in each mean and the vari-
ance of the two means are the same, the degrees
of freedom are simply 2(n-l) where n is the
number of samples in each mean. When there
are a different number of samples in each mean
or the variance of the means is different, the
calculations are more complex.
The demonstration spreadsheet (EvalMNA.xls)
which accompanies this appendix allows for
the possibility that there are a different number
of samples in each mean, and that the variance
of the means is different. When the number of
samples in the means are different, or the vari-
ances are different, the standard deviation of
the difference between the means is calculated
following equation B.3, where sx} is the sample
standard deviation of the first mean, sX2 is the
sample standard deviation of the second mean,
n1 is the number of measurements in the first
mean, and n2 is the number of measurements in
the second mean:
n.
ft,
B.3
The number of degrees of freedom in s(
can be approximated by the Welch-Satterthwaite
equation, Equation B.4 below.
d.f. = -
(sx2
x* /«2)2
B.4
S(Xl-X2)
-------
B.1.3. Use of a spreadsheet to calculate the
difference of means
The following provides a detailed step by step
explanation of the application of the ^-statistic
for the difference of means to determine wheth-
er the difference in concentrations between the
initial and final years of the review cycle are
consistent with meeting the cleanup goals. If
you have access to the internet, download the
spreadsheet from http://www.epa.gov/nrmrl/
gwerd/csmos/models/EvalMNA.html. Open the
tab MNA Evaluation and enter data from the
evaluation. If the spreadsheet is not available,
reconstruct the spreadsheet from a blank spread-
sheet, following instructions provided below.
To set up the spreadsheet from a blank
spreadsheet, enter equations in the appropri-
ate cells. Figure B.I shows the equations to
be entered in cells D13 through DIG, cells
D23 through V2S, ceJJE15, ceJJE25, cell
F15and ce//F25.
The equation in D13 should be copied into
cells D14 through DIG andD23 through
D26. Click on ce//D13 to select it, and
then type in the formula =LN(C13). The
1
2
3
4
5
G
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
28
29
30
A
B
Date
C
D
E
F
Comparison of Initial Year of Review Cycle t
Concentration
MP/L
=LN(C13) =AVEI
\
Initial Year"""
ni=?
4
4/4/2001
6/19/2001
9/23/2001
11/28/2001
Final Year
n2=?
4
4/5/2006
6/21/2006
9/7/2006
12/6/2006
sjnitial Year
\
\
1200 >
2300
1600
880
LN Cone.
RAGE(D13:
\
Mean
LN Cone.
D16) =E
\
Geometric
Mean of
Cone. (pg/L)
XP(E15)
'
Initial Year \ Initial Year \ Initial Year
I I
\\ \\
k. 7.090
7.741
7.378
6.780
=LN(C23)
\
Final Year ^
276
386
357
234
=AVERP
Final Year
\
\
V 5.620
5.961
5.878
5.455
kGE(D23:D
\
\
* 7.247
Final Year
5.729
t
/
\ = Co
I
^1404
Final Year
= C/
A 308
t
1
26) =EXP(L25)
Figure B.1. Populating the spreadsheet Evaluation of MNA to calculate the mean concentrations of contami-
nant in the first and in the final year of the review cycle. The data entry cells are formatted in red.
The formulas in the cells identified with arrows are provided in the blue text boxes.
-------
formula will appear in the fx box above the
cells. Click on the check mark to accept the
formula. Then left click on the square at
the lower right of the cell with the formula
[cell D13 in Figure B.I], and drag it with
the mouse to copy the formula into cells D14
through D16. Copy the formula in cell D13
and paste into cell D23. Then drag the for-
mula into cells D24 through D26.
Continue to create the spreadsheet by
entering formulas in cells as described in
Figures B.2 through B.5. Formulas will
have to be copied and dragged into cells as
described above in columns M, N, O, S, T,
Z, AB, AC and AD. The use of the $ in the
formulas fixes the row number as a formula
is copied into other cells. Site specific data
must be entered into cells P13, Q13, and
R13. The user must enter data in Column K
to specify the acceptable probability of error
in the comparison. The values in Figure B.3
are for illustration only. The font used in
Excel was Arial, and as a result it is impor-
tant to distinguish the letter I used in the for-
mulas from the number 1. If the spreadsheet
1
2
3
4
5
G
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
D
LN Cone.
Initial Year
7.090
7.741
7.378
6.780
Final Year
5.620
5.961
5.878
5.455
E
F
G
H
I
J
Comparison of Initial Year of Review Cycle to Final Year of Revie1
Mean
LN Cone.
Geometric
Mean of
Cone. (pg/L)
=STDEV(D13:D16)
Initial Year
7.247
Final Year
5.729
-S
:
Stan Dev
LN Cone.
=(G1b
Initial Year \ Initial Year
= Co
1404
Final Year
= Ci
308
TDEV(D2C
\
\
\
I
* 0.410
Final Year
0.233
/
/
:D26)
Difference
of Means
(LN Cone.)
A2/B12+G;
V
V
\
V
)
v -1.518
T
\
V
V
\
\
\
Stan Dev
Difference
of Means
25A2/B22)'
Attenuation
Factor
Ci'Co
'0.5
=F25/F15
I
I
\
\
I
1 0.236
=E25-E15
\
\
^ 0.219
Figure B.2. Calculating the standard deviation of the means, the difference between the means, and the stan-
dard deviation of the difference between the means. Calculations are performed on the natural
logarithms of the concentration data. Also calculated is an attenuation factor between the mean
of samples in the initial year and mean of samples in the final year of the review cycle.
-------
was constructed from a blank spreadsheet,
remember to label the column headings as
appropriate.
Figure B.I presents the first data entry cells in
the spreadsheet. The sampling dates in the first
year go in cells B13 through B16 and the dates
for the final year go into cells B23 through B26.
Enter the number of sampling dates in the first
year in cell B12 and the number of sampling
dates in the final year in cell B22. These num-
bers will be used later to calculate the degrees
of freedom in the /-statistic. Concentration data
for the first year go in cells C13 through C16,
and for the final year in cells C23 through C26.
Notice that concentrations are in units of (J-g/L.
Once this data is entered, the spreadsheet uses
the formulas already embedded in the spread-
sheet to make a number of calculations.
Enter a, the Acceptable Probability of Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
J \
Attenuation!
K
Probability
Factor II of Error
Ci/Co \| a one-tailed
^0.219
I
* 0.4
0.3
0.2
0.15
0.1
0.05
0.025
0.010
0.005
0.0025
L
Degrees
Freedom
in
Student's
t
A 4.755
M
Critical Value
Student's
t
[2a, d.f.)
/
0.271 *
0.569
0.941
1.190
1.533
2.132
2.776
3.747 I
4.604 /
5.598 /
/
~T
N
Difference
of Means
required to be
signficant at
various levels
of a one-tailed
=TINV(2*K12,LJ
A -0.064
7 -0.134
/ -0.222
/ -0.280
/ -0.361
/ -0.502
f -0.654
-0.883
-1.085 I
-1.319 /
/
±
O
Attenuation
Factor (Ci/Co)
required to be
signficant at
various levels
of a one-tailed
610)
A 0.938
7 0.875
/ 0.801
/ 0.756
/ 0.697
/ 0.605
/ 0.520
0.414
0.338
0.267
/ /
=F25/F15
=-l$17*M12
=EXP(N12)
=((G15A2/B12+G25A2/B22)A2)/((G15A2/B12)A2/(B12-1 )+(G25A2/B22)A2/(B22-1))
Figure B.3. Calculating the difference between the means necessary to be statistically significant at a prede-
termined probability of error. Calculations are performed on the natural logarithms of the concen-
tration data. The differences in means are then back transformed into the attenuations factors
that are significant at a predetermined probability of error.
-------
The spreadsheet calculates the natural logarithm
of the concentrations in column C, and presents
the arithmetic mean of the natural logarithms
of the concentrations in the first year and final
year of the review cycle in cells E15 and E25
respectively (see Figure B.I).
In column F, the spreadsheet calculates the
geometric mean of the samples in the initial
year (Co ) and final year (Ci) of the sampling
interval (Figure B.2). Notice that in the review
cycle, there was roughly a four-fold reduc-
tion in the concentrations of TCE between the
original concentration at the beginning of the
interval (Co) and the concentration at the end of
the interval (Ci ). The spreadsheet calculates an
attenuation factor (C./C ) in cell J17.
^ I 0/
The next step is to determine whether that
extent of attenuation is statistically significant.
The spreadsheet will perform calculations to
determine whether the difference between the
means of the transformed data is significant.
The variance of the difference is simply the
sum of the variances of the individual means.
The variance is estimated by the square of the
standard deviation. In cells G15 and G25 the
spreadsheet calculates the standard deviation
of the transformed data in the first year and in
the final year of the review cycle. The differ-
ence between the means is calculated in cell
HI7 (Figure B.2). Note that the first mean is
subtracted from the final mean. If concentra-
tions attenuate, the difference will be a negative
number.
In cell 117, the spreadsheet calculates the
standard deviation of the difference between
the means, using equation B.3. In cell L10
(Figure B.3), the spreadsheet estimates the
number of degrees of freedom in t using
Equation B.4. Excel 2003 can calculate the
critical value of the /-statistic for any prob-
ability of error and for any number of degrees
of freedom. In Figure B.3, Cells K12 through
K21 contain example values for the probability
of error (a). In Figure B.3, these range from
0.4 to 0.0025. The user can enter any desired
value for a in the spreadsheet. The correspond-
ing level of confidence is (1- a ) . If a is 0.05,
then the level of confidence is 95%.
In column M, the spreadsheet calculates the
critical value of the /-statistic. The function
TINV calculates the inverse of the /-statistic,
which is another way to say the critical value
oft.
The formula in Excel for TINV has the format
TINV(/zrs/ parameter, second parameter). The
first parameter is the probability of error and
the second parameter is the degrees of freedom
in the comparison. Excel assumes that the user
is making a two-tailed comparison. In other
words, half the distribution of error is in values
greater than the mean, and half is in values less
than the mean. A rate of attenuation that is
faster than expected is an acceptable outcome.
There is no reason to use statistics to protect
against a rate that is faster than expected. We
are only interested in rates that are slower than
expected. It is more appropriate to assign all of
the uncertainty to the slower rate. To use Excel
to calculate a one-tailed value of the critical
value of the /-statistic, the first parameter in the
formula for TINV must be twice the probability
of error that would be expected for a two tailed
test. This is why the formula multiplies the
value of a by 2.
Equation B.2 can be rearranged as follows:
(X, -X2} > s(Ii_l2) x/ad/ is equivalent to
\X2 -Xlj< -s^_x2) x *a,
,d.f.
lates a value for ~s(xl-x2) x
Column N calcu-
*M/ . If the value
ofX2- Xl is less (more negative) than the value_
of -e = . x / , , in column N, then X. and X.
(AJ A 2 ) OC,a./ . L 1
are significantly different at the corresponding
level of a in column K.
The actual difference of means X2- Xl was
-1.518 (cell H17 in Figure B.2). This differ-
ence is significant with a probability of error
-------
less than 0.0025 (compare cell N22 to cells HI7
in Figures B.2 and B.3).
The difference of means is the natural logarithm
of the attenuation factor (Ci/Co). The formula
in Column O takes the antilogarithm of the
differences to recover the attenuation factors
that are significant for various levels of a. The
attenuation factors in column O are offered to
illustrate the capacity of the monitoring data to
resolve the extent of natural attenuation over
the review cycle. The measured attenuation
factor was 0.219 (cell J17 in Figure B.2). As
an example, at a preselected probability of error
of 0.05, the variation in the samples makes it
possible to recognize any attenuation factor that
is less than 0.605 over the five year interval in
the review cycle (cell O18 in Figure B.3).
Figure B.4 shows data input cells for the final
concentration based goal, the time interval
involved in the review cycle, and the total
time interval from the initial year of review
cycle to the date when the clean up goal must
be reached. The spreadsheet uses Equation
8 to calculate an interim concentration in the
final year of the review cycle that corresponds
to each sample in the first year of the review
cycle. The interim goals in cells S13 through
S16 correspond to the concentrations in cells
C13 through C16. The equation in column T
calculates the natural logarithm of the interim
goals.
As depicted in Figure B.5, the equation in
cell U15 calculates an arithmetic mean of the
natural logarithms, and the equation in cell VI5
recovers the geometric mean of the interim
goals. The geometric mean of the interim goals
was 241 (J,g/L, compared to a geometric mean
of samples in the final year of 308 (J-g/L (see
cell F25 in Figure B.2). The sample mean is
larger than the goal. There is a possibility that
natural attenuation is not adequate to meet the
goal by the time specified.
We will use the same approach with the /-sta-
tistic for the difference of means to compare
the mean of the interim goals to the mean of
the samples in the final year. The equation in
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
P
Q
R
S
T
Setting Interim Goals (Cig) for Final Year of Review Cycle
Final Goal
or MCI
ftjg/L)
Time Interval*
between years
in review cycle
(years)
Time Interval
from initial
year to goal
(years)
* length of review cycle /
/
/
/
Interim
Goal (Cig)
required
to be on track
to meet
Final Goal
(M9/L)
216
/ 339
264
175
LN Cig required
to be adequate
to meet goal
-LN(S13)
/
/ 5.377
5.825
5.575
5.164
=C13*(P$13/C13)A(Q$13/R$13)
Figure B.4. Setting interim clean-up goals for the final year of the review cycle. A separate interim goal is
calculated for each sample in the initial year of the review cycle, assuming first order degradation
to attain the final clean up goal by the specified date.
-------
1
2
3
4
5
6
7
8
9
10
11
U
Interim Goals
Mean
LN Cig
12 _AX/CDA^
13
14
15
16
17
18
19
20
21
22
23
24
/
r 5.485
V
W
X
Y
Comparison of Samples in the Final Year of
to Interim Goals for Final Year of Revic
Interim Goals
Geometric Mean
Cone. (|jg/L)
=STDE
E(T13:T16)
f 241
/
-EXP(irio)
=(G?!V
Interim Goals
Stan Dev
LN Cig
:V(T13:T16)
/
/
/
* 0.282
Difference
of Means
(LN Cone.)
4 -0.243
j_
-U15-bzo i
1
V2/B22+W15A2/B12)A0
Stan Dev
Difference
of Means
4 0.183
/
.5
Figure B.5. Calculating the mean of the interim goals, the standard deviation of the interim goals, the differ-
ence between the mean of samples and the mean of the interim goals, and the standard deviation
of the difference between the mean of samples in the final year and the mean of the interim goals.
Calculations are performed on the natural logarithms of the concentration data or the goals.
cell W15 calculates the standard deviation of
the natural logarithm of the interim goals, the
equation in cell XI5 calculates the difference
between the mean of the goals and the mean
of the samples in the final year of the review
cycle, and the equation in cell Y15 calculates
the standard deviation of the difference in the
means.
As depicted in Figure B.6, the equation in
column AA calculated the degrees of freedom
in the comparison of the interim goals to the
samples in the final year, and the equation
in column AB calculates the critical value of
Student's t. Note that Excel does not calculate
the critical value for a fractional value of the
degrees of freedom. Instead it calculates the
value for the next lowest integer. The equa-
tion in column AC calculates the minimum
difference required for the sample mean to
be statistically different from the mean of
the interim goals. As the probability of error
decreases, the difference of the means become
more negative. The actual difference was
-0.243 (cell X15 in Figure B.5). The minimum
difference necessary to be statistically signifi-
cant at a probability of error of 0.15 was only
-0.235 (cell AC16 in Figure B.6). The mean
of the interim goals was less than the mean of
samples in the final year with a probability of
error of 0.15 (cell Z16 in Figure B.6).
Notice that the minimum difference necessary
to be statistically significant at a probability of
error of 0.10 was -0.270 (cell AC17), which is
more negative than the actual difference of the
means. There was no evidence that the mean
of the interim goals was less than the mean of
samples in the final year with a probability of
error of 0.10.
-------
=((G25A2/B22+W1 5A2/B1 2) A2)/((G25A2/B22) A2/(B22-1 )
2/B1 2) A2/(B1 2-1 ))
=IF(X$15-AC13<0, "No","No evidence not adequate")
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
_24_
Z \
I
Probability \
of Error \
a one-tailed\
=K12
/
0.4
0.3
0.2
0.15
0.1
0.05
0.025
0.01
0.005
0.0025
AA
Degrees
Freedom
in
Student's
, t
\
* 5.795
/
/
/
/
/
/
/
/
/
AB
Critical Value
Student's
t
(2a, d.f.]
4 °'267
T 0.559
1 0.920
1.288
1.476
2.015
2.571 ,
3.365 /
4.032 /
4.773 /
=-AB13
AC
Difference
of Means
required for
C; to be
statistically
different
from Cig
4 -0.049
/ -0.102
/ -0.168
/ -0.235
/ -0.270
/ -0.368
' -0.470
-0.615
-0.737
-0.872
T$15
\ AD
\ Attenuation
\ Adequate
\ to Attain Goal?
I
T No
No
No
No
No evidence not adequate
No evidence not adequate
No evidence not adequate
No evidence not adequate
No evidence not adequate
No evidence not adequate
=TINV(2*$Z13,$AA$11)
Figure B.6. Evaluation whether attenuation is adequate to attain goals. C. is the mean of the samples in the
final year and C. is the mean of the interim goals. If the difference of means required for C. to be
statistically different from C. is less than the difference between C. and C., then the extent of at-
tenuation in concentrations over the review cycle is not adequate to attain the clean up goal in the
specified time.
Based on this comparison, we can say with 85%
confidence that the trend in natural attenua-
tion over the review cycle from 2001 to 2006
was not adequate to attain the clean up goal.
However, we can not make the same statement
at 90% confidence.
B.1.4. Modifying the spreadsheet to
accommodate samples
The spreadsheet is set up with four samples in
the first year and four samples in the final year
as a default. If fewer samples or more samples
are available, the spreadsheet must be modified.
If there are fewer than four samples in the ini-
tial year and final year, the pre-existing example
values remaining in cells in Column B should
-------
be deleted. If there are more than four samples,
the extra samples can be inserted in the blank
cells below the existing example data. Insert
the correct number of samples in cells B12 and
B22. Modify the formulas in cells El5, E25,
G15, G25, T21, U15 and W15 to reflect to
correct range of the data. To modify a formula,
click on the cell containing the formula, and
then edit the formula in thefx box above the
cells of the spreadsheet. You will only edit the
second parameter, to reflect the correct range of
row numbers in the data set.
As an example, if you only had three samples
in the first year, then data would be entered in
cells B13 through B15 and cells C13 through
CIS. The equation in cell E15 would be modi-
fied to read =AVERAGE (D13:D15).
Three samples are probably the minimum
number of samples in the mean for a compari-
son of means using the /-test. With only two
samples, the ability to resolve differences is
very low. With more samples, the ability to
resolve differences increases. At many sites,
one sample each quarter is a reasonable sched-
ule to evaluate natural attenuation, particularly
if one sample is collected each quarter in the
initial year and final year of the review cycle.
However, there are certain hydrogeologic set-
tings such as bare karst terrain with conduit
flow where groundwater movement and con-
taminant concentrations can be extremely vari-
able over short time intervals. For these types
of very dynamic hydrogeologic settings, larger
sample sizes are typically needed in order to
have reasonable assurance that the true popula-
tion is captured by the sample data. Selection
of quarterly sampling for groundwater moni-
toring is typically done, but there are clearly
hydrogeologic settings where other sampling
frequencies are more technically justifiable.
B.2. Independence of samples and
seasonal effects
A further assumption of the /-statistic is that the
samples are independent of each other. If repli-
cate samples were taken and analyzed on a par-
ticular day, they only count as one sample. You
can average the samples, or randomly choose
one sample to enter into the spreadsheet. Wells
are usually purged before they are sampled,
and the sample represents a composite of the
volume of water surrounding the well screen
that was produced from the well during purging
and sampling. A sample taken on the following
day may not be an independent sample, because
it could have some component of the water in
the previous day's sample. For a sample to be
independent, you must allow the ambient flow
of the ground water to move all the water that
might have been produced in an earlier sample
away from the well screen.
One approach is to estimate the seepage veloc-
ity of the ground water in the aquifer, and com-
pare the distance ground water travels along a
flow path over a time interval to the distance
around a well that contributes ground water
during a purging and sampling event.
The distance around a well that contributes
ground water during a purging and sampling
event can be estimated from information on the
construction of the well, the volume of water
produced during purging and sampling, and an
estimate of effective porosity of the filter pack
and the aquifer material. The formula for the
volume of cylinder can be used to estimate the
volume of water contained within a particu-
lar radius of capture of the monitoring well
(Equation B.5). The equation calculates the
volume of water (V ) that would be contained
in aquifer material within a particular radius
(ra). Equation B.5 corrects that volume for
the additional water that would be contained
in the radius of the filter pack (r ), and for the
additional water that would be contained in the
radius of the well itself (rw ). In Equation B.5,
(d) is the length of the filter pack (not the
screen) or the length of the screen if there is no
filter pack, and 0a is the porosity of the aquifer
material and 0 is the porosity of the filter pack
if any. If there is no filter pack, assume 0 = 0a.
B.5
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Solving equation B.5 for (ra) produces
Equation B.6.
B.6
The diameter of capture of water in the aquifer
by the well (Z)a) is described by Equation B.7.
= 2r
B.7.
Divide the diameter of capture of water in the
aquifer (Da) by the seepage velocity of ground
water in the aquifer to estimate the time that
must elapse to be able to collect an independent
sample. It would be wise to add a safety factor
of a few-fold.
Some sites show strong seasonal effects on con-
centrations. This may be due to natural hetero-
geneity or even changes in the elevation of the
water table. If you are missing a sample, sea-
sonal effects could add an additional source of
uncertainty that is not accounted for in the /-test
for the difference of means. If possible, balance
the samples across the seasons. If a sample is
missing, one alternative to be considered is to
shift the entire analysis forward or backward
one season to provide a complete sample set for
one year of monitoring, even though the "one
year of monitoring" is not contained within one
calendar year.
B.3. Selecting the appropriate value
of a
Paul et al. (2009) provide G*Power 3.1.2, a
convenient application that can be used to deter-
mine the associated values of a (probability of
a Type I error) and p (probability of a Type II
error) in a /-test for the difference of means.
A screen shot of the input and results screen
of the application is provided in Figure B.7.
Download and open the application. Open
the drop down menu Test Family and select t
tests, open Statistical Test and select Means:
Difference between two independent means (two
groups), open Type of power analysis and select
Compromise: Compute implied a and power
- given p/a ratio, sample size, and effect size.
Open the Determine => Effect size d submenu
and populate the menu. In mean group 1 enter
the arithmetic mean of the natural logarithm
of samples in the last year of the review cycle
(cell E25 in the spreadsheet EvalMNA) and in
SD o group 1 enter the standard deviation of the
natural logarithm of the samples (cell G25 in
EvalMNA). Similarly, in mean group 2 enter
the arithmetic mean of the natural logarithm of
the interim goals (cell U15 in EvalMNA) and
in SD o group 2 enter the standard deviation
of the natural logarithm of the interim goals
(cell W15). Select Calculate and transfer to
main window. If for some reason, the appro-
priate ratio of p/a is something other than 1,
insert the appropriate value for p/a ratio. Then
enter the appropriate values for Sample size
group 1 (number of samples in the final year)
and Sample size group 2 (number of samples in
the initial year which is equal to the number of
interim goals). Select Calculate. The err prob
and p err prob are presented in the appropriate
windows. For the example data, a = p = 0.256.
This value can be inserted into any cell of
column K row 13 to 22 of EvalMNA to evalu-
ate whether the means are different for this
precise value of a.
The null hypothesis of the /-test for the differ-
ence in means is: The means are not different.
The interpretation of the null hypothesis is
that the attenuation in concentrations over the
time interval in the review cycle is adequate to
attain the clean up goal within the time period
expected. At this value of a, the /-test for the
difference of means rejects the null hypoth-
esis (Figure B.6), indicating that the extent of
attenuation is not adequate to meet the long
term goals.
Notice that the degrees of freedom in the
G*Power 3.1.2 is 6, (Figure B.7) while the
degrees of freedom in the comparison in
EvalMNA is 5.795, which Excel will truncate
to 5 in the calculation of the critical value of
Student's /. As a result, for a particular value of
a, the critical value of Student's / in EvalMNA
-------
File Edit View Tests Calculator Help
Central and noncentral distributions
Protocol of power analyses
critical t =0.695502
-4
-3
Test family
Statistical test
ttests
Means: Difference between wo independent means (two groups)
Type of power analysis
Compromise: Compute implied a & power - given pjct ratio, sample size, and effect size
Input Parameters
Determine =>
Tlil(s)
Effect size d
p/a ratio
Sample size group 1
Sample size group 2
© nl = nZ
Mean group 1
Mean group 2
SD crgroup 1
SO crgroup 2
Effect sized 0,9433127
Calculate and transfer to main window
Output Parameters
Noncentralitv parameter &
Critical t
Df
a err prob
f. err prob
Power (1-fs err prob)
X-Yplotfor a range of values
-
Calculate
Figure B.7. Data input screen and subscreen forG*Power 3.1.2.
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will be a larger number, making the difference
in means required for C. to be statistically dif-
ferent from C. a larger number. For a = 0.256,
ig & '
the corresponding critical value of t in G*Power
3.1.2 is 0.696 compared to the critical value
of t in EvalMNA of 0.706. The critical value
of t in EvalMNA is conservative in that more
attenuation will be required before the statistical
analysis will determine that attenuation is not
adequate to attain the goal.
Reference:
Paul, R, E. Erdfelder, A. Buchner and A.-G.
Lang. 2009. Statistical power analyses
using G*Power 3.1: Tests for correlation
and regression analyses, Behavior Research
Methods, 41 (4), 1149-1160, doi: 10.37587
BRM.41.4.1149. Both the journal article
and the computer application are available
at http://www.psycho.uni-duesseldorf.de/
aav/vroiects/svower/.
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