DRAFT for Discussion - GW Statistical Methodology - October 20 2016

Technical Memorandum
STATISTICAL METHODS FOR
DEVELOPING REFERENCE
BACKGROUND CONCENTRATIONS
FOR GROUNDWATER AT COAL ASH
FACILITIES
October 2016 – DRAFT FOR DISCUSSION
PURPOSES ONLY

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Contents
Background ............................................................................................................................ 1
PART I – PRELIMINARY DATA ANALYSIS ........................................................................... 4
1. Descriptive Statistics ...................................................................................................... 4
2. Graphical Analysis ......................................................................................................... 6
3. Identify Outliers .............................................................................................................. 7
4. Identify Distributions ....................................................................................................... 8
5. Test for Serial Correlation .............................................................................................. 8
6. Test for Seasonality ....................................................................................................... 9
7. Test for Trend ...............................................................................................................10
8. Confirm Trend Using Piecewise Polynomial Regression ...............................................12
9. Determination of Baseline Period for Background Wells ...............................................13
PART II – TEST FOR SUB-GROUPS....................................................................................15
1. Graphical Analysis ........................................................................................................17
2. Tests for Sub-Groups ....................................................................................................17
3. Tests to Identify which Sub-Groups are Different ..........................................................20
PART III – DEVELOP BACKGROUND THRESHOLD VALUES (UPPER PREDICTION
LIMITS) .................................................................................................................................21

FIGURES
Figure 1. Decision Flow Chart for Part I – Preliminary Data Analysis ......................................... 5
Figure 2. Plot of Iron (Total) as a Function of Time .................................................................... 6
Figure 3. Scatter Plot of Boron (Dissolved) as Function of Time ................................................ 7
Figure 4. Box Plot of Monthly Observations of Dissolved Oxygen over Years 1998 through 2008
..................................................................................................................................10
Figure 5. Maximum Likelihood Estimation Regression using De-seasonalized Beryllium
(Dissolved) Concentrations (ug/L) .............................................................................11
Figure 6. Polynomial Piecewise Regression using Beryllium (Dissolved) Concentrations (ug/L)
..................................................................................................................................13
Figure 7. Decision Flow Chart for Part II – Determining Differences across Sub-Groups ..........16
Figure 8. Empirical Distribution Function Plot of De-seasonalized Observations (Specific
Conductance (µS/cm)) for Season=0 and Season=1 ................................................17
Figure 9. Decision Flow Chart for Part III - Developing Reference Background Concentration
Values .......................................................................................................................24

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

TABLES
Table 1. Monitored Constituents For Groundwater Assessments Required by the Coal Ash
Management Act of 2014 ........................................................................................................... 1
Table 2. Test for Normality Assumptions Required for ANOVA Tests .......................................18
Table 3. Kaplan-Meier Class of Test Results for Differences across Monthly Means ................20

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Background
Regulations providing North Carolina groundwater quality standards are provided in T15A
NCAC 02L .0202. Section (b)(3) of the regulation provides that:
Where naturally occurring substances exceed the established standard, the standard
shall be the naturally occurring concentration as determined by the Director.
This document provides the approach developed to establish the naturally occurring
concentrations of constituents included in the sampling required for the Duke Energy ash
basins. The naturally occurring concentration for a constituent is that value which represents the
upper threshold values from the upper tail of the data distribution. HDR will refer to this upper
threshold value as the reference background concentration value, that is, the concentration
of the constituent that represents ambient (i.e. naturally occurring) background conditions 1. The
monitored constituents for the groundwater assessments required by the Coal Ash Management
Act of 2014 are listed in Table 1 below.
Note that HDR will only use the non-filtered results. HDR will review and evaluate the
corresponding filtered results; however, they will not be used for compliance purposes. In
addition, samples will not be used in the development of the reference background
concentrations whenever turbidity conditions during the sampling events are greater or equal to
10 nephelometric turbidity units (NTUs)2.
Table 1. Monitored Constituents For Groundwater Assessments Required by the Coal
Ash Management Act of 2014
Aluminum Cobalt pH Uranium, Natural
Antimony Copper Selenium Uranium, 233
Arsenic Iron Strontium Uranium, 234
Barium Lead Sulfate Uranium, 236
Beryllium Manganese Total Dissolved Solids Radium 226
Boron Mercury Total Suspended Solids Radium 228
Cadmium Molybdenum Thallium
Chloride Nickel Vanadium
Chromium (Total) Nitrite Zinc
Chromium (VI) Nitrate

The steps described in this document serve as a guideline to develop upper prediction limits
(UPLs) for the constituents for the groundwater detection and assessment and monitoring
programs required by the Coal Ash Management Act at the Duke Energy ash basins in North
Carolina. They closely follow HDR’s proposed method to establish reference background
concentrations for constituents according to the Environmental Protection Agency’s Hazardous
and Solid Waste Management System; Disposal of Coal Combustion Residuals from Electric
1 Evaluating Metals in Groundwater at DWQ Permitted Facilities (NC DENR DWQ 2012). 2 The units of turbidity from a calibrated nephelometer are called Nephelometric Turbidity Units (NTU).
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Utilities; Final Rule (EPA CCR)3. The reference background concentrations determined by the
process presented in this document will be submitted to the North Carolina Department of
Environmental Quality (DEQ) Division of Water Resources for determination as to the naturally
occurring site reference background concentration for the specific constituent. A site-specific
report documenting the procedures, evaluations, and calculation will be prepared and submitted
to DEQ.
The steps in this document are based on the US Environmental Protection Agency (USEPA)
“Unified Guidance” (USEPA 2009), and the ProUCL (USEPA 2013). In addition, the North
Carolina Division of Water Quality (NCDWQ) technical assistance document for Evaluating
Metals in Groundwater at DWQ Permitted Facilities (NCDWQ 2012) was also referenced.
As recommended by the EPA Unified Guidance, HDR has selected the 95 percent upper
prediction limits (UPL95) to establish the reference background concentration value for each of
the constituents at each site 4. The UPL95 for each constituent represents a concentration value
with a low probability (i.e., 5 percent chance) of being exceeded when a future sample(s) for the
constituent is taken. If the concentration in the future sample is higher than the estimated
UPL95, then steps are taken to validate the observation by means of quality assurance checks
and/or by re-sampling and/or to investigate for possible natural or anthropogenic reasons for
observing a higher concentration than expected.
HDR will produce these limits for each of the constituents using their respective concentrations
observed in the samples taken from the set of background wells located at each Duke Energy
ash basin site. The data across these background wells per constituent and per site will be
pooled prior to estimating the reference background concentration using the UPL95.
Concentrations of the constituents measured in samples taken downgradient to the background
wells can then be compared to the background concentrations as part of any planned inter-well
testing regimen.
3 HDR modified its earlier methods to establish reference background concentration so that both state and federal
regulations are comparable. Having similar processes to address the two sets of regulations will minimize confusion.
4 There are different methods which can be used to estimate the reference background concentrations such as the
UPL and the upper tolerance limit (UTL). HDR chose the UPL as it is the statistic recommended by the Unified
Guidance (page 2-15). The Unified Guidance recommends the UPL over the UTL for the following reasons: 1. the
ability to estimate an UTL which can control for Type I error rates when simultaneously testing an exact number of
multiple future or independent observations is not precise as it is when estimating the appropriate UPL; 2. the
mathematical underpinnings of the UPLs under re-testing strategies are well established while those for re-testing
with tolerance limits are not. Re-testing strategies are now encouraged and sometimes required under assessment
monitoring situations; and finally, 3. statistically, the two limits are similar, especially under normal assumptions; and
to avoid confusion as to which statistic to use, the UPL is chosen.

If it is not known how many future samples will be tested for compliance at the next sampling event, one could use an
upper simultaneous limit (USL); however, the quality of this statistic rests heavily on a well-specified distribution and it
is not robust to outliers. ProUCL regularly cautions about the use of the USL in the presence of outliers and if there is
any question as to possible contamination in the assumed pristine background test sites. ProUCL has indicated that
the USL is preferred over the UTL (page 92) since the UTL does not address the increased chance of an exceedance
when multiple samples are being simultaneously tested. Then in a separate section, ProUCL states that the UTL is
preferred over the UPL since it is not dependent on the number of future observations being simultaneously tested
(page 100). The research team feels these recommendations are not based on sufficient evidence to discount the
use of the UPL to set reference background concentrations.
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Each site has between one and ten background wells; the wells have been drilled at different
points in time beginning as early as the late 1990s or early 2000s. In addition to the difference in
time when each well started being sampled, the wells were also sampled at different intervals,
and for different purposes, and were drilled into bedrock or in shallow or deep unconsolidated
materials. HDR reviewed and vetted these well locations from the perspective of the location
relative to possible sources and in adherence to the Comprehensive Site Assessment (CSA)
and/or Corrective Action Plan (CAP) specifications.
While there are differences in the locations and the sampling periods of the wells, the
fundamental assumption is that the constituent concentrations sampled at these background
wells, when pooled, serve as an estimate of overall well field conditions for a given constituent.
HDR will test this assumption using statistical tests. If distinct sub-groups exist as which may be
the case across different hydrostratigraphic units, HDR will develop separate background
concentrations for each distinct sub-group of wells.
The process described by this technical memo consists of three parts.
• Part I – Preliminary Data Analysis
Part I includes the analyses used to assess and transform the data where necessary such
that the data can be used to produce appropriate UPLs. HDR refers to this stage as the
preliminary data analysis (PDA). Consideration is given to issues related to outliers, serial
correlation, seasonality, spatial variability, trends, and appropriateness of the period of
record (sampling period).
• Part II – Test for Sub-Groups
Part II of this report describes the approach to test for sub-group differences. During the
statistical analyses in Part 1, questions may arise as to the appropriateness of assuming all
data from all wells at a given site for a given constituent truly represents the overall well field
condition at any point in time. There may be statistically significant differences in mean
concentrations among sub-groups of observations for a constituent.
Types of sub-groups to test include seasonal sub-groups (winter, spring, summer, and fall)
and well class sub-groups (bedrock, shallow, or deep). If after testing the groups are
statistically different, the same steps described in Part I can be applied to the partitioned
data to better understand the distribution of the samples within a sub-group for each
constituent.
The reference background concentration values using the UPL95 for a constituent will be
produced for each sub-group of samples, provided the sub-groups represent distinct
populations.
• Part III – Develop Background Threshold Values (Upper Prediction Limits)
Part III describes the statistical analyses used to produce UPLs for each constituent.

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

PART I – PRELIMINARY DATA ANALYSIS
The PDA phase contains nine distinct steps and is summarized in the decision flow chart in
Figure 1. The numbers alongside the shapes in Figure 1 represent where each of the PDA
steps occurs in the flow.
1. Descriptive Statistics
Descriptive statistics will be developed per background well, per constituent, and repeated
for data pooled across all background wells within a site. Purpose of descriptive analysis is
to characterize data and assess quality of information. The following statistics will be
produced per constituent, per well, and then pooled over all background wells.
• Sample size
• Number of detects
• Number of non-detects 5
• Percentage of non-detects
• Number of distinct observations
• Number of distinct method detection limits6 (MDLs)
• Range of collection period in months: Difference between last sampling date and first
sampling date
• Mean
• Median
• Minimum
• Maximum
• Standard deviation
• Coefficient of variation
• Median Absolute Deviation/0.6757
• Skewness
• Kurtosis

5 Trace concentrations of constituents may be present in groundwater samples but cannot be detected and are
reported as less than the detection limit of the analytical instrument or laboratory method used. Such an observation
is deemed to be a ‘non-detect’. Values which can be measured are called detects. 6 A method detection limit (MDL) is the single value deemed to be the minimum value that a specific test can detect
at the laboratory conducting the tests. Over time, a laboratory may have different MDLs for a constituent as the
laboratory’s equipment and protocols change. 7 The Median Absolute Deviation /0.675 is a robust estimate of variability of the population standard deviation. A
robust estimate of variability, which is less likely impacted by outliers in the sample, is derived by dividing the median
absolute deviation (MAD) with the constant of 0.675. The MAD is the sum of the differences between each of the
observations and the median. Any differences that are less then zero are converted to positive values before a total
sum is computed.
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Run Descriptive
Statistics Graphical Analysis
Identify
Distribution
(detects)
Deseasonalize
Seasonality
Yes No
Identify Outliers
Period of
Record
represents
background?
NoYes
Update period of
record (repeat
steps 1 – 7)
Run piecewise
regressions
Trend Analysis
1 2 3 4
6
7
8
9
Test for Serial
Correlation
5
Develop
background
(Go to Part III)
Sub-groups are
the same
Repeat above
sequence for each
sub-group
Yes No
Test for sub-
groups
(Go to Part II)
6 7

Figure 1. Decision Flow Chart for Part I – Preliminary Data Analysis
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

2. Graphical Analysis
Scatter plots of observations as a function of time will be developed. Different colors can be
used to differentiate detects from non-detects.

Figure 2. Plot of Iron (Total) as a Function of Time

The graphs visually provide clues as to whether the period of record is reflective of a steady
state baseline period. Multiple method detection limits over time can affect quality of the
data. One needs so decide at this point if all data can be incorporated into analysis or if
older historical data may need to be dropped. For example, Figure 3 below demonstrates a
time series of boron (dissolved). Although samples are observed starting in 1976 for boron
(dissolved), only the data from 2006 to 2012 were used to study trends and set UPLs,
because there was a change in the data reporting protocols for samples commencing in
2006 for boron.
Iron Total (ug/l)
Date
Iron Total (ug/l) - Raw Data
31/01/1993 05/02/1996 09/02/1999 13/02/2002 17/02/2005 22/02/2008
0
20000
40000
60000
80000
100000
120000
140000
Data Type
Detected
Not Detected
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Figure 3. Scatter Plot of Boron (Dissolved) as Function of Time
Outliers and seasonality can also be visually detected. Further statistical tests described in
sections 3 to 8 below will need to be conducted to confirm assumptions from visual
inspections.
3. Identify Outliers
A statistical outlier is defined as a value originating from a different statistical population than
the rest of the sample. Outliers or observations not derived from the same population as the
rest of the sample violate the basic statistical assumption of identically distributed
measurements. If an outlier is suspected, an initial helpful step is to construct a probability
plot of the ordered sample data versus the standardized normal distribution.
Two tests are available to test for possible outliers. Dixon’s Outlier Test is appropriate for
data series with sample sizes less than 25, and Rosner’s Outlier Test is applicable to those
with a sample size larger than 25. These outlier tests assume that the rest of the data
except for the suspect observation(s) are normally distributed.
In conjunction with Dixon’s and Rosner’s Tests, observations which are greater than three
standard deviations from the mean will be flagged and evaluated. Values greater than 3
standard deviations from the mean may be outliers.
Boron Dissolved vs Date
Slave River
Date
Boron Dissolved (ug/L) - Raw Data
Aug-1976 Nov-1980 Mar-1985 Jun-1989 Sep-1993 Dec-1997 Mar-2002 Jun-2006 Sep-2010 Dec-20140
6
13
19
25
32
38
44
51
57
63
69
76
82
88
95
101
107
114
120
Data Type
Detected
Not Detected
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

If outliers are found from the tests, HDR recommends that the anomalous numbers be
investigated. If they are correct values and collected under standard, consistent protocols,
they will remain in the data series. Otherwise, they can be dropped before proceeding.
Some distributions naturally have anomalously low or high values. The subsequent tests for
distribution types should find the best fitting distribution that can explain the anomalous
values.
Other literature suggests repeating the statistical procedures with and without the outliers.
After a comparison of the estimates is completed, a decision needs to be made as to which
data set is representative. HDR believes that a decision needs to be made, but at the data
collection and assessment stage. The risk of this method is that the estimated distributions
and statistics tend to be chosen to suit a goal. The decision is not necessarily objective. An
example would be where a sample was qualified as “J+” (biased high), due to equipment
blank contamination. If a sample such as this was seen as an outlier, it may be possible to
eliminate it from further analysis for this reason. If there is a doubt as to the authenticity and
reliability of the measured value, it should not be used. Otherwise, it is a value that was
generated by the system regulating the water quality conditions of the tested groundwater
well.
4. Identify Distributions
Since many tests make an explicit assumption concerning the distribution represented by
the sample data, the form and exact type of distribution often has to be checked using a
GOF test. A GOF test assesses how closely the observed sample data resemble a
proposed distributional model. The best GOF tests attempt to assess whether the sample
data closely resemble the tails of the candidate distributional model. The models under
consideration for water quality samples are normal, lognormal, or gamma distributions.
The Shapiro-Wilk (sample sizes <= 50) and Lilliefors (sample sizes >50) tests can be used
to test for normal distribution. Note that these two tests can be used to test for lognormal
distributions after the data is transformed using the natural log function. The two types of
empirical distribution function (EDF) based methods, the Kolmogorov-Smirnov (K-S) and
Anderson-Darling (A-D) test, are used to test for a gamma distribution.
The software package from the EPA, ProUCL8 has incorporated these methods to
automatically test for either normal, lognormal, or gamma distribution types. If all GOF tests
fail, non-parametric estimation methods will be used.
5. Test for Serial Correlation
Sources for serial correlation in water samples can be due to seasonal effects or temporal
effects related to the timing of the sample collections. Trend analysis using regression
techniques of a water quality constituent sampled over time is confounded if the time series
exhibits serial correlation. The regression errors from adjacent observations may be
8 Singh, Anita and Ashok K. Singh. 2013. ProUCL Version 5.0.00 User Guide: Statistical Software for Environmental
Applications for Data Sets with and without Non-detect Observations. U.S. Environmental Protection Agency,
EPA/600/R-07/041
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

correlated. For example, if the residual from a given month’s observation is high, then it is
likely that the residual from the next month’s observation will also be high. The same logic
follows for low residuals giving rise to other low residuals. This type of correlation is referred
to as serial correlation or autocorrelation. Run the autocorrelation function test for
successive lags (1-n). (HDR uses the statistical software application NCSS 9 to run this test.)
6. Test for Seasonality
As explained in step 5, there are different reasons why a series of water quality constituent
samples exhibit serial correlation. A common reason arises from changes in season as
evidenced from varying temperatures and precipitation. These changes impact water quality
constituents in a predictable and cyclical manner over the months. The study of water
quality changes over time is focused on the ability to discern true trend through regression
analysis amidst the cyclical nature of the data or its “seasonality”. The correct use of these
regression analyses rests on the crucial assumption that regression errors or residuals
arising from the model fitting are independent of each other. This is often not the case with
data that is seasonal in nature. If seasonality exists, then the autocorrelation function test
described in step 5 will pick up the behavior. To better understand the type of seasonality
(monthly, quarterly, bi-annually) which factors into the observed variability of a time series, a
visual inspection of the data as a function of time is recommended.
Box plots of observations on a monthly or quarterly basis (provided one has at least 8 to 10
observations per month or per quarter) will be plotted. Note that the side-by-side box plots
can also be used to visually inspect the level of variability per sub-group. See Figure 4 for an
example of box plots.
9 Hintze, J. (2013). NCSS 9. NCSS, LLC. Kaysville, Utah, USA. www.ncss.com.
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COAL ASH FACILITIES

Figure 4. Box Plot of Monthly Observations of Dissolved Oxygen over Years 1998
through 2008
7. Test for Trend
The samples from background wells represent water quality conditions exhibiting natural
variability and unaffected by anthropogenic activities. As such, one expects that the
measurements taken at regular intervals over time (three or more years) demonstrate a
steady or stationary time series. The data from the background wells will be tested to
determine whether a trend exists (values steadily increasing or steadily decreasing).
PAR_NAME=Oxygen Dissolved (mg/l)
Before Deseasonalization
Month
Raw Data
1 2 3 4 5 6 7 8 9 10 11 12
6
8
10
12
14
16
18
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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COAL ASH FACILITIES

Depending on the presence of non-detects and seasonality, one of the following regression
tests will be selected:
• Kendall Seasonal Regression if using full data series with seasonality present
• Maximum Likelihood Estimation (MLE) Regression 10 with best fitting distribution
on de-seasonalized data
• Mann-Kendall (non-parametric, no seasonality, only 1 MDL)
Figure 5 below demonstrates the estimated trend line for the beryllium (dissolved) over time.
In this particular regression, the slope is positive and statistically significant at the 0.05 level
of significance. The question arises is if this entire series due to the upwards trend should
be used to estimate the UPLs for the background wells.

Figure 5. Maximum Likelihood Estimation Regression using De-seasonalized
Beryllium (Dissolved) Concentrations (ug/L)
10 The MLE regression approach solves the “likelihood equation” to find values for mean and standard deviation that
are most likely to have produced both the non-detect and detect data.
Annual Trend Analysis - Deseasonalized Data vs. Date
Lognormal Distribution
Date
Deseasonalized Data
Mar-2006 Dec-2006 Sep-2007 Jun-2008 Feb-2009 Nov-2009 Aug-2010 May-2011
Jan-2012 Oct-20121.0000
1.0011
1.0021
1.0032
1.0042
1.0053
1.0063
1.0074
1.0084
1.0095
1.0105
1.0116
1.0126
1.0137
1.0147
1.0158
1.0168
1.0179
1.0189
1.0200
Data Type
Detected
Not Detected
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COAL ASH FACILITIES

8. Confirm Trend Using Piecewise Polynomial Regression
In the circumstances when changes in trend may occur within a parameter’s time series, the
piece-wise polynomial model has proven useful. This approach attempts to find an
appropriate mathematical model that expresses the relationship between the parameter
values and the sampling dates by using piece-wise regressions. Two types of piece-wise
models are used to study the trends: linear-linear model and linear-linear-linear model.
The linear-linear regression model assumes and identifies one structural break in a
parameter series, in which the two portions of the data separated by the break point follow
two different trends as modeled by two different linear equations. Similarly, the linear-linear-
linear model attempts to identify two structural breaks to separate three different linear
trends.
This approach is informative, but it has the disadvantage of not being able to account for
non-detects in a sample. Hence, HDR recommends implementing the piece-wise polynomial
models and MLE non-detects regression approach complementary to one another. HDR has
applied the piece-wise models mainly as a visual guide when selecting the baseline periods
for the parameters.
For example, while the MLE Regression approach suggested that the overall trend for
beryllium (dissolved) over time is steadily increasing (Figure 5), the polynomial piece-wise
regression with two structural breaks in Figure 6 suggests that recently the concentrations
are moving downwards to a level exhibited in the early part of the sampling period. HDR
concluded that the entire sample from the period of record can be used to produce UPLs for
this constituent.

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Figure 6. Polynomial Piecewise Regression using Beryllium (Dissolved)
Concentrations (ug/L)
9. Determination of Baseline Period for Background Wells
At this point, a decision is made if the entire period of record from which the background
samples were collected demonstrates a steady state and represents a baseline against
which future values can be tested. If both the linear regression models and the polynomial
piece-wise regression models suggest that over time the observations are steadily
increasing or decreasing, then a review of the data will be done to determine if a sub-
segment of the time series better represents the background period. This of course means
the final sample numbers are reduced.
Annual Trend Analysis: Deseasonalized Data vs Date
Piece-Wise (Linear-Linear-Linear)
Date
Deseasonalized Data
Mar-2006 Dec-2006 Sep-2007 Jun-2008 Feb-2009 Nov-2009 Aug-2010 May-2011
Jan-2012 Oct-2-0.0100
-0.0082
-0.0063
-0.0045
-0.0026
-0.0008
0.0011
0.0029
0.0047
0.0066
0.0084
0.0103
0.0121
0.0139
0.0158
0.0176
0.0195
0.0213
0.0232
0.0250
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

A consideration in the decision process is that if the number of samples in the period of
record is low (<10 samples) and over a short duration (< 2 years), then an observed trend
in the time series may not necessarily indicate changes to the natural variability in the water
quality of the background wells. In fact, the trend could be part of the natural variation; and if
the time series were longer, a steady state particular to the concentrations of the
constituents of interest could be observed.
As more sampling events occur, the background well data can be combined and tested
either for significant trends or for differences in means (see Part II below) between two time
periods. HDR recommends that at least two additional sampling events be completed before
amalgamating the new background data with the previous background data and testing
whether the new data from the background wells is from the same population as the older
data and hence can be combined to produce revised reference background concentration
values. As more background data is collected to produce the reference background
concentration values, the test statistics which are designed to detect statistically significant
exceedances become more powerful.

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
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PART II – TEST FOR SUB-GROUPS
After completion of the PDA and graphical analysis, the data findings may suggest that the
range of concentrations is not consistent over time. As discussed earlier, seasonal differences
often explain changes in concentrations over time. Other differences could be related to the
class of wells to which a background well belong. Examples of well classes observed across the
seven Duke Energy monitoring sites are bedrock wells and deep and shallow wells.
This part describes the steps required to validate if differences in mean concentrations across
the sub-groups are statistically significant or not. In other words, it is used to test if two or more
sample means are drawn from the same source population, i.e., same distribution function.
HDR uses the significance level of 0.05 to decide whether to accept or reject the null hypothesis
that there are no differences across the sub-population means.
Before proceeding to test for differences across the sub-group means, one needs a sufficient
sample size of at least 8 to 10 samples per sub-group according to the EPA’s 2009 Unified
Guidance Report 11 and ProUCL’s Technical Guide 12. (The 8 to 10 minimum is with reference to
producing background threshold values. If a sub-group is distinct enough to form its own time
series and background threshold values need to be estimated, then such a minimum must be
obtained. Tests of sub-group differences can be done on groups as low as 10; however, their
outcomes may not be reliable. HDR recommends that sample sizes per sub-group should be at
least 30.)
Testing for sub-groups can be done in three steps:
1. Graphical analysis,
2. Tests for sub-group differences, and
3. Tests to identify which sub-groups are different.
The testing steps are outlined in Figure 7.

11 2009. Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities: Unified Guidance. Office of
Resource Conservation and Recovery, Program Implementation and Information Division, U.S. Environmental
Protection Agency. EPA 530/R-09-007, page 5-3. 12 Singh, Anita, et al. 2010. ProUCL Version 5.00.0 User Guide. U.S. Environmental Protection Agency, EPA/600/R-
07/038, page 17
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

Censored
Observations
No Yes
Distribution Test
for ROS
Imputation
Test ANOVA
assumptions
Kruskal-Wallis
Test
ANOVA/Log
ANOVA Test
Test ANOVA
assumptions
KM Test of Sub-
Group Differences
ANOVA/Log
ANOVA
BDL >=50%
KM Test of Sub-
Group Differences
Dunn’s Test to
Identify Which
Sub-Groups are
Different
Sub-Groups
are Different
Dunn’s Test to
Identify Which
Sub-Groups are
Different
Sub-Groups
are Different
Tukey-Kramer
Test to Identify
Which Sub-Groups
are Different
Sub-Groups
are Different
Dunn’s Test to
Identify Which
Sub-Groups are
Different
Sub-Groups
are Different
No Yes
Yes
YesNo
Yes Yes
Tukey-Kramer’s
Test to Identify
Which Sub-Groups
are Different
Sub-Groups
are Different
Graphical Analysis
by Sub-Groups
Test for Sub-
Groups
1
2
3
YesNo
Yes Yes

Figure 7. Decision Flow Chart for Part II – Determining Differences across Sub-Groups
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
DRAFT FOR DISCUSSION PURPOSES ONLY –October 2016 - STATISTICAL METHODS FOR
DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
COAL ASH FACILITIES

1. Graphical Analysis
A useful visual test for sub-group differences is the EDF (see NCSS - Non-detects-Data
Group Comparison). This procedure computes summary statistics, generates EDF plots,
and computes hypothesis tests appropriate for two or more groups for data with non-detects
(left-censored) values (provided fewer than 50% of the values are non-detects (left-
censored)).
Figure 8 demonstrates that the two sub-groups representing samples taken during two
different seasons show similar distributions, i.e., no differences in concentrations between
the two seasons.

Figure 8. Empirical Distribution Function Plot of De-seasonalized Observations
(Specific Conductance (µS/cm)) for Season=0 and Season=1
2. Tests for Sub-Groups
The following methods can be used to detect for population differences across the sub-
groups:
• Analysis of variance (ANOVA) (under normal distribution assumptions)
• Log-ANOVA (under log-normal distribution assumption)
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• Kruskal-Wallis One-Way Analysis on Ranks (distribution free assumptions/non-
parametric, corrected for ties)
• Kaplan-Meir (Non-parametric, useful with heavy censoring. See NCSS for this test.)
HDR runs all four types of tests and compares the test outcomes across all distribution
assumptions as further lines of evidence; however, the test results for the appropriate
distribution assumption and level of censoring takes precedence.
The decision as to which test is to be used is predicated on the presence of censorship and
whether the distribution follows a parametric distribution of either normal, log-normal, or
gamma or does not have a discernable distribution and hence is non-parametric. Note that
the Log-ANOVA is simply the ANOVA approach applied to the natural-logarithm of the time
series.
The ANOVA tests do require that normality assumptions are valid for each sub-group. In
addition, the variances across the groups should be approximately equal. NCSS
automatically outputs tests which confirm if the normality assumptions hold. Below is an
example of such output. In this example, normality assumptions hold and an ANOVA can be
used to test for differences across sub-groups.
Table 2. Test for Normality Assumptions Required for ANOVA Tests
Assumption Test Value Prob Level Decision (5%)
Skewness Normality of Residuals -0.0424 0.966172 Accept
Kurtosis Normality of Residuals 1.1625 0.245037 Accept
Omnibus Normality of Residuals 1.3532 0.508348 Accept
Modified-Levene Equal-Variance Test 1.7228 0.085088 Accept

If the level of censoring is less than 50 percent, then the Regression on Order Statistics 13
(ROS) imputation technique can be used to impute values for the non-detects. Imputation
methods offer the advantage of a substitution method for non-detects that does not give all
samples the same value, hence reduces the impact of bias. ROS is a simple imputation
method that fills in non-detect data on the basis of a probability plot of detects.
ProUCL can test which distribution assumption can be used to impute using the ROS
approach. Once the non-detects have been imputed, the ANOVA assumptions can be
tested once more. If they are valid, then either ANOVA can be used to test for differences
across sub-groups. If not, HDR recommends using Kruskal-Wallis and Kaplan-Meier Tests
to test for sub-group differences.
Table 2 demonstrates the results of testing if monthly means are statistically different from
each other using the Kaplan-Meier approach (NCSS has modules which implement this
13 Regression on Order Statistics (ROS): A regression line is fit to the normal scores of the order statistics for the
uncensored observations and then to fill in values imputed from the straight line for the observations below the
detection limit.
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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test). In this demonstration, the means across the months are statistically different at the
significance level of 0.05 or 5 percent.

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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DEVELOPING REFERENCE BACKGROUND CONCENTRATIONS FOR GROUNDWATER AT
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Table 3. Kaplan-Meier Class of Test Results for Differences across Monthly Means
Hypotheses
H0: Distribution Functions are Equal Among Groups
HA: At Least One Group Distribution Functions Differs
Test Name Chi-Square DF
Prob
Level
Reject H0
(Alpha = 0.05)
Logrank 42.777 11 0 Yes
Gehan-Wilcoxon 42.772 11 0 Yes
Tarone-Ware 42.814 11 0 Yes
Peto-Peto 43.202 11 0 Yes
Mod. Peto-Peto 43.194 11 0 Yes

3. Tests to Identify which Sub-Groups are Different
Provided any of the four tests described above (ANOVA, Log-ANOVA, Kruskil-Wallis, and
Kaplan-Meier) show sub-group differences, HDR recommends the following tests to identify
which sub-group(s) is/are different from the others.
• Post-Hoc Test for Multiple Comparisons,
o Tukey-Kramer Test (parametric), and
o Dunn’s Test (non-parametric).

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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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COAL ASH FACILITIES

PART III – DEVELOP BACKGROUND THRESHOLD VALUES (UPPER
PREDICTION LIMITS)
The primary goal of water quality sampling is to determine whether or not a given sample or
collection of samples have exceeded some baseline set of samples or a particular previously
defined standard. In general, this is done by computing the UPL. The upper limit is computed
because the concern is generally for exceedances greater than the baseline or standard. A few
parameters, such as pH or dissolved oxygen may require both upper and lower prediction limits.
The formulation of the prediction limit may vary slightly with the particulars of the test to be
made and the characteristics of the data involved (see chapters 3 and 5 of ProUCL’s Version
5.0.00 Technical Guide 14), but in general, the formula for the prediction limit for k future or
independent observations is:
𝑈𝑈𝑈𝑈𝑈𝑈=𝑥𝑥̅+𝑡𝑡1−∝/𝑘𝑘,𝑛𝑛−1𝑆𝑆�1 +
1𝑛𝑛
Where
𝑥𝑥̅ = baseline (historical data) sample mean
S = baseline (historical data) standard deviation
t = Student’s t with 1−∝ degrees of freedom
∝ = Type I (false positive) error rate
n = number of observations in baseline dataset
k = number of future or independent obtained samples to be predicted 15.

HDR uses ProUCL to estimate the appropriate UPL given the type of parametric or non-
parametric distribution a constituent’s concentration follows, the level of censorship and the
number of future samples expected to be tested at a given point in time. Figure 9 describes the
steps and decision points required to select the appropriate UPL for each constituent at a given
site. The six key steps described below are flagged alongside their appropriate stage within the
decision flow chart in Figure 9.
1. The first step in applying ProUCL to produce the UPLs is to separate constituents into those
with at least one non-detect and those with no non-detects. ProUCL calculates UPLs
differently depending on whether there are non-detects in the data set. The algorithms in
ProUCL use imputation and modelling techniques to address the issue of non-detects. It
does not use the common practice of substituting a non-detect for a given constituent with
one-half of its MDL as this method introduces bias into the estimates for the UPL95, among
other estimates. Some constituents may have 50 percent or more of their observations as
14 Singh, Anita and Ashok K. Singh. 2013. ProUCL Version 5.0.00 Technical Guide: Statistical Software for
Environmental Applications for Data Sets with and without Non-detect Observations. U.S. Environmental Protection
Agency, EPA/600/R-07/041 15 Definition of Upper Prediction Limit (UPL): The upper boundary of a prediction interval for an independently
obtained observation (or an independent future observation), ProUCL Version 5.0.00, Technical Guide, page xx.
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non-detects. With so many observations below detection limits 16 (BDLs), it is difficult to
discern what distribution best fits the data. Provided the non-detects have been validated
through the data review process, if such a high percentage of non-detect measurements are
observed, HDR recommends applying non-parametric methods to estimate the background
concentration levels. If a sub-set of non-detects are suspect, they will be removed from the
analysis prior to estimating the percentage of non-detects (see graphical analysis section of
Part I). Once this step is completed, follow the steps below.
2. Produce 95 percent UPLs (UPL95) for one future (or independently obtained) observation
per constituent. Produce the UPL95 for m future observations where m represents the
number of different down gradient wells at each site where the constituent is sampled at
each sampling event. (Select ‘all’ option in ProUCL under Upper Limits/BTVs17 module.
ProUCL outputs UPLs, UTLs, and USLs). Save all output.
3. Record all UPL95s under all parametric distribution models and non-parametric distribution
models.
4. The following logic provides guidance in the situations where the parametric GOF tests
indicate that multiple distributions fit the observed distribution from the sample. Ideally, only
one or no distribution tests passes. If a distribution is strongly indicated by the empirical
distribution generated from the sample (i.e., follows the textbook shape for the distribution),
it is more likely that only that distribution will pass the GOF tests. However, when one is
presented with smaller datasets (n<30) or datasets with gaps between observations, or
shapes that almost look like the distribution being tested, more than one distribution GOF
test passes. In fact all three or any two of the GOF tests for the Gamma, Lognormal, or
Normal distribution can pass.
In these circumstances, preference is given to the Gamma distribution provided it has
passed its GOF test over the remaining two distributions (see point a below). If the Gamma
GOF test fails, then preference is given to the Lognormal over the Normal provided the
Lognormal has passed and is not heavily skewed (i.e., its standard deviation is <= 1) (see
point b). An interesting point is that in the situation where the Lognormal has passed and the
Gamma has failed, if the Lognormal is highly skewed (i.e., its standard deviation is > 1), then
the team uses the Gamma distribution to model the data and does not automatically default
to the Normal distribution even if the data sample also passed the Normal GOF test. The
Gamma and Lognormal distributions are closely related in shape; however, the Gamma has
the added advantage of not having extremely long tails to the right. This mitigates the
chance that the UPL95 will be extremely high relative to the median of the distribution as
could be the case if one used either the Lognormal or Normal distribution assumptions to
estimate the UPL95. At this point, the team is encouraged to study the empirical
16 Note that MDL and BDL mean different things. MDL is the “Method Detection Limit” and is the single value
specified as the minimum value that a specific test can detect. BDL is “Below Detection Limit” and indicates any result
less than or equal to the MDL. 17 ProUCL uses the term Background Threshold Value (BTV) to represent the reference background concentration
value.
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distributions produced during the PDA in light of the GOF test results and use professional
judgment as to the appropriateness of the GOF test results.
If at least one parametric GOF passes, then do the following for k=1 and k=m;
a. If Gamma passes, use the Gamma UPL95(k) as the background concentration level.
b. If Lognormal passes, then do the following;
i. If the standard deviation of the fitted lognormal distribution is less than or
equal to 1, then use lognormal UPL95(k) for the background concentration
level;
ii. If the standard deviation is > 1, use Gamma UPL95(k)18 for the background
concentration level.
c. Otherwise, use Normal UPL95(k) for the background concentration level.
If no parametric test passes, use the non-parametric UPL95 for the background
concentration level. Please note that the non-parametric UPL95 is constant for any number
of future or independent observations.
5. Given that measurements follow no discernable parametric distribution, if the sample size
<30 and distribution is skewed (i.e., skewness ne 0), use Chebyshev’s non-parametric
estimate (at a confidence coefficient=85 percent or 90 percent to be conservative, i.e., have
some chance an exceedance will occur); otherwise, use the non-parametric UPL95.
6. To develop UPLs for samples with insufficient data (i.e. < 8 observations). (Note that
ProUCL will provide a warning if you have insufficient samples for parametric estimates).
a. Rank order the observations from smallest to largest including a 120 percent of the
maximum detect observation to better understand the distribution.
b. Until sufficient samples are collected, assign the maximum detected value as the
background prediction limit. The maximum observation is used by some practitioners
to estimate an upper limit, especially for small samples (see U.S. Environmental
Protection Agency (EPA). 1992a. Supplemental Guidance to RAGS: Calculating the
Concentration Term. Publication EPA 9285.7-081, May 1992.).

18 The authors of ProUCL (2013) recommend for skewed data sets, parametric (e.g., gamma distribution on KM
estimates) or nonparametric methods (e.g., Chebyshev inequality) which account for data skewness should be used
to compute reference background concentration estimates. (See page 136). “The use of a parametric lognormal
distribution on a lognormally distributed data set tends to yield unstable impractically large upper confidence limit
values, especially when the standard deviation (sd) of the log-transformed data is greater than 1.0 and the data set is
of small size such as less than 30 inches to 50 inches (see page 257).
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Duke Energy Carolinas, LLC | TECHNICAL MEMORANDUM
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Censored
Observations
At least one
parametric GOF
passes
No Yes
Warning: Large
Loss of Power
Use Non-Parametric
Tests
Run UPLs(k) in
ProUCL for
constituents with no
non-detects
Record UPLs(k)
under different
distribution
assumptions and
distribution free
assumptions
No Yes
Run UPLs(k) in
ProUCL for
constituents with non-
detects
Record UPLs(k)
under different
distribution
assumptions and
distribution free
assumptions
At least one
parametric GOF
passes
Warning: Large
Loss of Power
Use Non-Parametric
Tests
No Yes
If sample size < 30,
provide warning that
more samples are
required
If sample size < 8,
estimate is unreliable
and recommend the
maximum value as the
upper background limit
until more samples can
be collected
sample size <=
30
sample size <=
30Use Non-Parametric Use Chebyshev
No Yes
No Yes
Use Non-Parametric
(ignores NDs)
Use Chebyshev
(imputes using KM)
Develop BTVs
If sample size < 30,
provide warning that
more samples are
required
If sample size < 8
estimate is unreliable
and recommend the
maximum value as the
upper background limit
until more samples can
be collected
1
2
3
5
6
Gamma
passes
Use Gamma
UPL95(k)
Lognormal
passes
Use Normal
UPL95(k)
Lognormal
sigma <=1
Use Lognormal
UPL95(k)Use Gamma
UPL95(k)
Gamma
passes
Use Gamma
UPL95(k)
Lognormal
passes
Lognormal
sigma <=1
Use Lognormal
UPL95(k)Use Gamma
UPL95(k)
Use Normal
UPL95(k)
4
>=50% BDLs
Use Non-
parametric
methods
No Yes
No Yes
No Yes No Yes
No Yes
No Yes
No Yes
No. Detects <4
YesNo
If 100% BDLs, team may
need to decide other site
specific values or
sources
Estimate is unreliable
regardless of the level of
censorship and recommend
the maximum detect value
as the upper background
limit until more samples can
be collected

Figure 9. Decision Flow Chart for Part III - Developing Reference Background Concentration Values

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