United States
Environmental Protection
Agency
Office of Environmental
Information
Washington, DC 20460
EPA/240/B-06/002
February 2006
a EPA
Data Quality Assessment:
A Reviewer's Guide
EPA QA/G-9R
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FOREWORD
This document is the 2006 version of the Data Quality Assessment: A Reviewer's Guide
which provides general guidance to organizations on assessing data quality criteria and
performance specifications for decision making. The Environmental Protection Agency (EPA)
has developed a process for performing the Data Quality Assessment (DQA) Process for project
managers and planners to determine whether the type, quantity, and quality of data needed to
support Agency decisions have been achieved. This guidance is the culmination of experiences
in the design and statistical analyses of environmental data in different Program Offices at the
EPA. Many elements of prior guidance, statistics, and scientific planning have been incorporated
into this document.
This document is one of a series of quality management guidance documents that the
EPA Quality Staff has prepared to assist users in implementing the Agency-wide Quality
System. Other related documents include:
EPA QA/G-4 Guidance on Systematic Planning using the Data Quality Objectives Process
EPA QA/G-4D DEFT Software for the Data Quality Objectives Process
EPA Q A/G-5 S Guidance on Choosing a Sampling Design for Environmental Data
Collection
EPA QA/G-9S Data Quality Assessment: Statistical Methods for Practitioners
This document is intended to be a "living document" that will be updated periodically to
incorporate new topics and revisions or refinements to existing procedures. Comments received
on this 2006 version will be considered for inclusion in subsequent versions. Please send your
written comments on the Data Quality Assessment: A Reviewer's Guide to:
Quality Staff (2811R)
Office of Environmental Information
U.S. Environmental Protection Agency
1200 Pennsylvania Avenue N.W.
Washington, DC 20460
Phone: (202) 564-6830
Fax: (202)565-2441
E-mail: quality@epa.gov
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TABLE OF CONTENTS
Page
INTRODUCTION 1
0.1 Purpose of this Guidance 1
0.2 DQA and the Data Life Cycle 2
0.3 The Five Steps of Statistical DQA 3
0.4 Intended Audience 4
0.5 Organization of this Guidance 5
STEP 1: REVIEW THE PROJECT'S OBJECTIVES AND SAMPLING DESIGN 7
1.1 Review Study Objectives 8
1.2 Translate Study Objectives into Statistical Terms 8
1.3 Developing Limits on Uncertainty 9
1.4 Review Sampling Design 10
1.5 What Outputs Should a DQA Reviewer Have at the Conclusion of Step 1? 12
STEP 2: CONDUCT A PRELIMINARY DATA REVIEW 13
2.1 Review Quality Assurance Reports 13
2.2 Calculate Basic Statistical Quantities 14
2.3 Graph the Data 14
2.4 What Outputs Should a DQA Reviewer Have at the Conclusion of Step 2? 14
STEP 3: SELECT THE STATISTICAL METHOD 15
3.1 Choosing Between Alternatives: Hypothesis Testing 15
3.2 Estimating a Parameter: Confidence Intervals and Tolerance Intervals 17
3.3 What Output Should a DQA Reviewer Have at the Conclusion of Step 3? 17
STEP 4: VERIFY THE ASSUMPTIONS OF THE STATISTICAL METHOD 19
4.1 Perform Tests of Assumptions 19
4.2 Develop an Alternate Plan 20
4.3 Corrective Actions 20
4.4 What Outputs Should a DQA Reviewer Have at the End of Step 4? 20
STEPS: DRAW CONCLUSIONS FROM THE DATA 21
5.1 Perform the Statistical Method 21
5.2 Draw Study Conclusions 21
5.3 Hypothesis Tests 21
5.4 Confidence Intervals 23
5.5 Tolerance Intervals 23
5.6 Evaluate Performance of the Sampling Design 24
5.7 What Output Should the DQA Reviewer Have at the End of Step 5? 24
INTERPRETING AND COMMUNICATING THE TEST RESULTS 25
6.1 Data Interpretation: The Meaning ofp-values 25
6.2 Data Interpretation: "Accepting" vs. "Failing to Reject" the Null Hypothesis 26
6.3 Data Sufficiency: "Proof of Safety" vs. "Proof of Hazard" 26
6.4 Data Sufficiency: Quantity vs. Quality of Data 28
6.5 Data Sufficiency: Statistical Significance vs. Practical Significance 28
6.6 Conclusions 29
REFERENCES 25
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Page
Appendix A: Commonly Used Statistical Quantities 33
Appendix B: Graphical Representation of Data 37
Appendix C: Common Hypothesis Tests 43
Appendix D: Commonly Used Statements of Hypotheses 47
Appendix E: Common Assumptions and Transformations 49
Appendix F: Checklist of Outputs for Data Quality Assessment 55
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CHAPTER 0
INTRODUCTION
0.1 Purpose of this Guidance
Data Quality Assessment (DQA) is the scientific and statistical evaluation of
environmental data to determine if they meet the planning objectives of the project, and thus are
of the right type, quality, and quantity to support their intended use. This guidance describes
broadly the statistical aspects of DQA in evaluating environmental data sets. A more detailed
discussion about DQA graphical and statistical tools may be found in the companion guidance
document, Data Quality Assessment: Statistical Methods for Practitioners (Final Draft) (EPA
QA/G-9S) (U.S. EPA 2004). This guidance applies to using DQA to support environmental
decision-making (e.g., compliance determinations), and to using DQA in estimation problems in
which environmental data are used (e.g., monitoring programs).
DQA is built on a fundamental premise: data quality is meaningful only when it relates to
the intended use of the data. Data quality does not exist in a vacuum, a reviewer needs to know
in what context a data set is to be used in order to establish a relevant yardstick for judging
whether or not the data is acceptable. By using DQA, a reviewer can answer four important
questions:
1. Can a decision (or estimate) be made with the desired level of certainty, given the
quality of the data?
2. How well did the sampling design perform?
3. If the same sampling design strategy is used again for a similar study, would the
data be expected to support the same intended use with the desired level of
certainty?
4. Is it likely that sufficient samples were taken to enable the reviewer to see an
effect if it was really present?
The first question addresses the reviewer's immediate needs. For example, if the data are
being used for decision-making and provide evidence strongly in favor of one course of action
over another, then the decision maker can proceed knowing that the decision will be supported
by unambiguous data. However, if the data do not show sufficiently strong evidence to favor
one alternative, then the data analysis alerts the decision maker to this uncertainty. The decision
maker now is in a position to make an informed choice about how to proceed (such as collect
more or different data before making the decision, or proceed with the decision despite the
relatively high, but tolerable, chance of drawing an erroneous conclusion).
The second question addresses how robust this sampling design is with respect to
changing conditions. If the design is very sensitive to potentially disturbing influences, then
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interpretation of the results may be difficult. By addressing the second question the reviewer
guards against the possibility of a spurious result arising from a unique set of circumstances.
The third question addresses the problem of whether this could be considered a unique
situation where the results of this DQA only applies to this situation only and the conclusions
cannot be extrapolated to similar situations. It also addresses the suitability of using this data
collection design cannot reviewer's potential future needs. For example, if reviewers intend to
use a certain sampling design at a different location from where the design was first used, they
should determine how well the design can be expected to perform given that the outcomes and
environmental conditions of this sampling event will be different from those of the original
event. As environmental conditions will vary from one location or one time to another, the
adequacy of the sampling design should be evaluated over a broad range of possible outcomes
and conditions.
The final question addresses the issue of whether sufficient resources were used in the
study. For example, in an epidemiological investigation, was it likely the effect of interest could
be reliably observed given the limited number of samples actually obtained.
0.2 DQA and the Data Life Cycle
The data life cycle (depicted in Figure 0-1) comprises three steps: planning,
implementation, and assessment. During the planning phase, a systematic planning procedure
(such as the Data Quality Objectives (DQO) Process) is used to define criteria for determining
the number, location, and timing of samples (measurements) to be collected in order to produce a
result with a desired level of certainty.
This information, along with the sampling methods, analytical procedures, and
appropriate quality assurance (QA) and quality control procedures, is documented in the QA
Project Plan. Data are then collected following the QA Project Plan specifications in the
implementation phase.
At the outset of the assessment phase, the data are verified and validated to ensure that
the sampling and analysis protocols specified in the QA Project Plan were followed, and that the
measurement systems were performed in accordance with the criteria specified in the QA Project
Plan. Then the statistical component of DQA completes the data life cycle by providing the
evaluation needed to determine if the performance and acceptance criteria developed by the
DQO planning process were achieved.
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QUALITY ASSURANCE ASSESSMENT
PLANNING
Data Quality Objectives Process
Quality Assurance Project Plan Developmen
I
IMPLEMENTATION
Field Data Collection and Associated
Quality Assurance / Quality Control Activities
I
ASSESSMENT
Data Verification/Validation
Data Quality Assessment
Routine Data
/QC/Performance
Evaluation Data /
INPUTS
DATA VERIFICATION /VALIDATION
• Verify measurement performance
• Verify measurement procedures and
reporting specifications
L
VERIFIED /VALIDATED DATA
j
DATA QUALIT
• Review project c
sampling desig
• Conduct prelimi
• Select statistica
• Verify assumptic
• Draw conclusion
i
Y ASSESSMENT
bjectives and
nary data review
method
ns of the method
s from the data
OUTPUT
PROJECT CONCLUSIONS
Figure 0-1. Data Life Cycle
0.3 The Five Steps of Statistical DQA
The statistical part of DQA involves five steps that begin with a review of the planning
documentation and end with an answer to the problem or question posed during the planning
phase of the study. These steps roughly parallel the actions of an environmental statistician
when analyzing a set of data. The five steps, which are described in more detail in the following
chapters of this guidance, are briefly summarized as follows:
1 Review the project's objectives and sampling design: Review the objectives
defined during systematic planning to assure that they are still applicable. If
objectives have not been developed (e.g., when using existing data independently
collected), specify them before evaluating the data for the projects objectives.
Review the sampling design and data collection documentation for consistency
with the project objectives observing any potential discrepancies.
2. Conduct apreliminary data review: Review QA reports (when possible) for the
validation of data, calculate basic statistics, and generate graphs of the data. Use
this information to learn about the structure of the data and identify patterns,
relationships, or potential anomalies.
3. Select the statistical method: Select the appropriate procedures for summarizing
and analyzing the data, based on the review of the performance and acceptance
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criteria associated with the projects objectives, the sampling design, and the
preliminary data review. Identify the key underlying assumptions associated with
the statistical test.
4. Verify the assumptions of the statistical method: Evaluate whether the
underlying assumptions hold, or whether departures are acceptable, given the
actual data and other information about the study.
5. Draw conclusions from the data: Perform the calculations pertinent to the
statistical test, and document the conclusions to be drawn as a result of these
calculations. If the design is to be used again, evaluate the performance of the
sampling design.
Although these five steps are presented in a linear sequence, DQA is by its very nature
iterative. For example, if the preliminary data review reveals patterns or anomalies in the data
set that are inconsistent with the project objectives, then some aspects of the study analysis may
have to be reconsidered. Likewise, if the underlying assumptions of the statistical test are not
supported by the data, then previous steps of the DQA may have to be revisited. The strength of
DQA Process is that it is designed to promote an understanding of how well the data satisfy their
intended use by progressing in a logical and efficient manner.
Nevertheless, it should be realized that DQA cannot absolutely prove that the objectives
set forth in the planning phase of a study have been achieved. This is because the reviewer can
never know the true value of the item of interest, only information from a sample. Sample data
collection provides the reviewer only with an estimate, not the true value. As an reviewer makes
a determination based on the estimated value, there is always the risk of drawing an incorrect
conclusion. Use of a well-documented planning process helps reduce this risk to an acceptable
level.
0.4 Intended Audience
This guidance is written as a general overview of statistical DQA for a broad audience of
potential data users, reviewers, data generators and data investigators. Reviewers (such as
project managers, risk assessors, or principal investigators who are responsible for making
decisions or producing estimates regarding environmental characteristics based on environmental
data) should find this guidance useful for understanding and directing the technical work of
others who produce and analyze data. Data generators (such as analytical chemists, field
sampling specialists, or technical support staff responsible for collecting and analyzing
environmental samples and reporting the resulting data values) should find this guidance helpful
for understanding how their work will be used. Data investigators (such as technical investigators
responsible for evaluating the quality of environmental data) should find this guidance to be a
handy summary of DQA-related concepts. Specific information about applying DQA-related
graphical and statistical techniques is contained in the companion guidance, Data Quality
Assessment: Statistical Methods for Practitioners (Final Draft) (EPA QA/G-9S).
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0.5 Organization of this Guidance
Chapters 1 through 5 of this guidance address the five steps of DQA in turn. Each chapter
discusses the activities expected and includes a list of the outputs that should be achieved in that
step. Chapter 6 provides additional perspectives on how to interpret data and understand/
communicate the conclusions drawn from data. Finally, Appendices A through E contain non-
technical explanatory material describing some of the statistical concepts used. Appendix F is a
checklist that can be used to ensure all steps of the DQA process have been addressed.
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CHAPTER 1
STEP 1: REVIEW THE PROJECT'S OBJECTIVES AND SAMPLING DESIGN
DQA begins by reviewing the key outputs from the planning phase of the data life cycle
such as the Data Quality Objectives, the QA Project Plan, and any related documents. The study
objective provides the context for understanding the purpose of the data collection effort and
establishes the qualitative and quantitative basis for assessing the quality of the data set for the
intended use. The sampling design (documented in the QA Project Plan) provides important
information about how to interpret the data. By studying the sampling design, the reviewer can
gain an understanding of the assumptions under which the design was developed, as well as the
relationship between these assumptions and the study objective. By reviewing the methods by
which the samples were collected, measured, and
reported, the reviewer prepares for the preliminary
data review and subsequent steps of DQA.
Step 1. State the Problem
Define the problem that motivates the study;
Identify the planning team; examine budget, schedule.
T
Step 2. Identify the Goal of the Study
State how environmental data will be used in solving the
problem; identify study questions; define alternative outcomes
Step 3. Identify Information Inputs
Identify data and information needed to answer study questions.
Step 4. Define the Boundaries of the Study
Specify the target population and characteristics of interest;
define spatial and temporal limits, scale of inference.
Step 5. Develop the Analytic Approach
Define the parameter of interest; specify the type of inference
and develop logic for drawing conclusions from the findings.
Statistical
Hypothesis Testing
Estimation and other
analytical approaches
Systematic planning improves the
representativeness and overall quality of a sampling
design, the effectiveness and efficiency with which
the sampling and analysis plan is implemented, and
the usefulness of subsequent DQA efforts. For
systematic planning, the Agency recommends the
DQO Process, a logical, systematic planning process
based on the scientific method. The DQO Process
emphasizes the planning and development of a
sampling design to collect the right type, quality, and
quantity of data for the intended use. Employing both
the DQO Process and DQA will help to ensure that
projects are supported by data of adequate quality; the
DQO Process does soprospectively and DQA does so
retrospectively. Systematic planning, whether the
DQO Process or other, can help assure that data are
not collected spuriously. The DQO Process is
discussed in Guidance on the Data Quality Objectives
Process (QA/G-4) (U.S. EPA 2000a).
In instances where project objectives have not
been developed and documented during the planning
phase of the study, it is necessary to recreate some of
the project objectives prior to conducting the DQA.
These are used to make appropriate criteria for
evaluating the quality of the data with respect to their intended use. The most important
recreations are: hypotheses chosen, level of significance selected (tolerable levels of potential
decision errors), statistical method selected, and number of samples collected. The seven steps
of the DQO Process are illustrated in Figure 1-1.
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Step 6. Specify Performance or Acceptance Criteria
Develop performance criteria for new data being collected,
acceptance criteria for data already collected.
Step 7. Develop the Detailed Plan for Obtaining Data
Select the most resource effective sampling and analysis plan
that satisfies the performance or acceptance criteria.
Figure 1-1. The Data Quality
Objectives Process
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1.1 Review Study Objectives
First, the objectives of the study should be reviewed in order to provide a context for
analyzing the data. If a systematic planning process has been implemented before the data are
collected, then this step reduces to reviewing the documentation on the study objectives. If no
clear planning process was used, the reviewer should:
• Develop a concise definition of the problem (e.g. DQO Process Step 1) and of the
methodology of how the data were collected (e.g. DQO Process Step 2). This should
provide the fundamental reason for collecting the environmental data and identify all
potential actions that could result from the data analysis.
• Identify the target population (universe of interest) and determine if any essential
information is missing (e.g. DQO Process Step 3). If so, either collect the missing
information before proceeding, or select a different approach to resolving the problem.
• Specify the scale of determination (any subpopulations of interest) and any boundaries on
the study (e.g. DQO Process Step 4) based on the sampling design. The scale of
determination is the smallest area or time period to which the conclusions of the study
will apply. The apparent sampling design and implementation may restrict how small or
how large this scale of determination can be.
1.2 Translate Study Objectives into Statistical Terms
In this activity, the reviewer's objectives are used to develop a precise statement of how
environmental data will be evaluated to generate the study's conclusions. If DQOs were
generated during planning, this statement will be found as an output of DQO Process Step 5.
In many cases, this activity is best accomplished by the formulation of statistical
hypotheses, including a null hypothesis, which is a "baseline condition" that is presumed to be
true in the absence of strong evidence to the contrary, as well as an alternative hypothesis, which
bears the burden of proof. In other words, the baseline condition will be retained unless the
alternative condition (the alternative hypothesis) is thought to be true due to the preponderance
of evidence. In general, such hypotheses often consist of the following elements:
• a population parameter of interest (such as a mean or a median), which describes the
feature of the environment that the reviewer is investigating;
• a numerical value to which the parameter will be compared, such as a regulatory or risk-
based threshold or a similar parameter from another place (e.g., comparison to a reference
site) or time (e.g., comparison to a prior time); and
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• a relationship (such as "is equal to" or "is greater than") that specifies precisely how the
parameter will be compared to the numerical value.
Section 3.1 provides additional information on how to develop the statement of hypotheses, and
includes a list of commonly encountered hypotheses for environmental projects.
Some environmental data collection efforts do not involve the direct comparison of
measured values to a fixed value. For instance, for monitoring programs or exploratory studies,
the goal may be to develop estimates of values or ranges applicable to given parameters. This is
best accomplished by the formulation of confidence intervals or tolerance intervals, which
estimate the probability that the true value of a parameter is within a given range. In general,
confidence intervals consist of the following elements:
• a range of values with in which the unknown population parameter of interest (such as
the mean or median) is thought to lie; and
• a probabilistic expression denoting the chance that this range captures the parameter of
interest.
An example of a confidence interval would be 'We are 95% confident that the interval 47.3 to
51.8 contains the population mean.'
Tolerance intervals are used with proportions. Here, we wish to have a certain level of
confidence that a certain proportion of the population falls in a certain region. An example of a
tolerance interval would be 'We are 95% confident that at least 80% of the population is above
the threshold value.' Section 3.2 provides additional information on confidence intervals and
tolerance intervals.
For discussion of technical issues related to statistical testing using hypotheses or
confidence/tolerance intervals, refer to Chapter 3 of Data Quality Assessment: Statistical
Methods for Practitioners (Final Draft) (EPA QA/G-9S).
1.3 Developing Limits on Uncertainty
The goal of this activity is to develop quantitative statements of the reviewer's tolerance
for uncertainty in conclusions drawn from the data and in actions based on those conclusions.
These statements are generated during DQO Process Step 6, but they can also be generated
retrospectively as part of DQA.
If the project has been framed as a hypothesis test, then the uncertainty limits can be
expressed as the reviewer's tolerance for committing false rejection (Type I, sometimes called a
false positive) or false acceptance (Type II, sometimes called a false negative) decision errors1.
1 Decision errors occur when the data collected do not adequately represent the population of interest. For example,
the limited amount of information collected may have a preponderance of high values that were sampled by pure
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A false rejection error occurs when the null hypothesis is rejected when it is, in fact, true. A
false acceptance error occurs when the null hypothesis is not rejected (often called "accepted")
when it is, in fact, false. Other related phrases in common use include "level of significance"
which is equal to the Type I error (false rejection) rate, and "power" which is equal to 1 - Type II
error (false acceptance) rate. When a hypothesis is being tested, it is convenient to summarize
the applicable uncertainty limits by means of a "decision performance goal diagram". For
detailed information on how to develop false rejection and false acceptance decision error rates,
see Chapter 6 of Guidance on the Data Quality Objectives Process (QA/G-4) (U.S. EPA 2000a).
If the project has been framed in terms of confidence intervals, then uncertainty is
expressed as a combination of two interrelated terms:
• the width of the interval (smaller intervals correspond to a smaller degree of
uncertainty); and
• the confidence level (typically stated as a percentage) indicating the chance this
interval captures the unknown parameter of interest (a 95% confidence level
represents a smaller degree of uncertainty than, say, a 90% confidence level).
If the project has been framed in terms of tolerance intervals, then uncertainty is
expressed as a combination of confidence level and:
• the proportion of the population that lies in the interval.
Note that there is nothing inherently preferable about obtaining a particular probability,
such as 95% for the confidence interval. For the same data set, there can be a 95% probability
that the parameter lies within a given interval, as well as a 90% probability that it lies within
another (smaller) interval, and an 80% probability of being in even a smaller interval. All the
intervals are centered on the best estimate of that parameter usually calculated directly from the
data (see also Chapter 3.2).
1.4 Review Sampling Design
The goal of this activity is to familiarize the reviewer with the main features of the
sampling design that was used to generate the environmental data. If DQOs were developed
during planning, the sampling design will have been summarized as part of DQO Process Step 7.
The design and sampling strategy should be discussed in clear detail in the QA Project Plan or
Sampling and Analysis Plan. The overall type of sampling design and the manner in which
samples were collected or measurements were taken will place conditions and constraints on
how the data can be used and interpreted.
chance. A decision maker could possibly draw the conclusion (decision) that the target population was high when,
in fact, it was much lower. A similar situation occurs when the data are collected according to a plan that is too
limited to reflect the true underlying variability.
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The key distinction in sampling design is between judgmental (also called authoritative)
sampling (in which sample numbers and locations are selected based on expert knowledge of the
problem) and probability sampling (in which sample numbers and locations are selected based
on randomization, and each member of the target population has a known probability of being
included in the sample).
Judgmental sampling has some advantages and is appropriate in some cases, but the
reviewer should be aware of its limitations and drawbacks. This type of sampling should be
considered only when the objectives of the investigation are not of a statistical nature (for
example, when the objective of a study is to identify specific locations of leaks, or when the
study is focused solely on the sampling locations themselves). Generally, conclusions drawn
from judgmental samples apply only to those individual samples; aggregation may result in
severe bias due to lack of representativeness and lead to highly erroneous conclusions.
Judgmental sampling, although often rapid to implement, precludes the use of the sample for any
purpose other than the original one.
If the reviewer elects to proceed with judgmental data, then great care should be taken in
interpreting any statistical statements concerning the conclusions to be drawn. Using a
probabilistic statement with a judgmental sample is incorrect and should be avoided as it gives
an illusion of correctness where there is none. The further the judgmental sample is from a truly
random sample, the more questionable the conclusions.
Probabilistic sampling is often more difficult to implement than judgmental sampling due
to the difficulty of locating the random locations of the samples. It does have the advantage of
allowing probability statements to be made about the quality of estimates or hypothesis tests that
are derived from the resultant data. One common misconception of probability sampling
procedures is that these procedures preclude the use of expert knowledge or important prior
information about the problem. Indeed, just the opposite is true; an efficient sampling design is
one that uses all available prior information to stratify the region (in order to improve the
representativeness of the resulting samples) and set appropriate probabilities of selection.
Common types of probabilistic sampling designs include the following:
• Simple random sampling - the method of sampling where samples are collected at
random times or locations throughout the sampling period or study area.
• Stratified sampling - a sampling method where a population is divided into non-
overlapping sub-populations called strata and sampling locations are selected randomly
within each stratum using some sampling design.
• Systematic sampling - a randomly selected unit (in space or time) establishes the starting
place of a systematic pattern that is repeated throughout the population. With some
important assumptions, can be shown to be equivalent to simple random sampling.
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• Ranked set sampling- a field sampling design where expert judgment or an auxiliary
measurement method is used in combination with simple random sampling to determine
which locations should be sampled.
• Adaptive cluster sampling - a sampling method in which some samples are taken using
simple random sampling, and additional samples are taken at locations where
measurements exceed some threshold value.
• Composite sampling - a sampling method in which multiple samples are physically
mixed into a larger sample and samples for analysis drawn from this larger sample. This
technique can be highly cost-effective (but at the expense of variability estimation) and
had the advantage it can be used in conjunction with any other sampling design.
The document Guidance on Choosing a Sampling Design for Environmental Data Collection
(EPA QA/G-5S) (U.S. EPA 2002) provides extensive information on sampling design issues and
their implications for data interpretation.
Regardless of the type of sampling scheme, the reviewer should review the sampling
design documentation and look for design features that support the project's objectives. For
example, if the reviewer is interested in making a decision about the mean level of contamination
in an effluent stream over time, then composite samples may be an appropriate sampling
approach. On the other hand, if the reviewer is looking for hot spots of contamination at a
hazardous waste site, compositing should be used with caution, to avoid "averaging away" hot
spots. Also, look for potential problems in the implementation of the sampling design. For
example, if simple random sampling has been used, can the reviewer be confident this was
actually achieved in the actual selection of data point? Small deviations from a sampling plan
probably have minimal effect on the conclusions drawn from the data set, but significant or
substantial deviations should be flagged and their potential effect carefully considered. The most
important point is to verify that the collected data are consistent with how the QA Project Plan,
Sampling and Analysis Plan, or overall objectives of the study stated them to be.
1.5 What Outputs Should a DQA Reviewer Have at the Conclusion of Step 1?
There are three outputs a DQA reviewer should have documented at the conclusion of
Step 1:
1. Well-defined proj ect obj ectives and criteria,
2. Verification that the hypothesis or estimate chosen is consistent with the project's
objective and meets the project's performance and acceptance criteria, and
3. A list of any deviations from the planned sampling design and the potential
effects of these deviations.
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CHAPTER 2
STEP 2: CONDUCT A PRELIMINARY DATA REVIEW
The principal goal of this step of the process is to review the calculation of some basic
statistical quantities, and review any graphical representations of the data. By reviewing the data
both numerically and graphically, one can learn the "structure" of the data and thereby identify
appropriate approaches and limitations for using the data.
There are two main elements of preliminary data review: (1) basic statistical quantities
(summary statistics) and (2) graphical representations of the data. Statistical quantities are
functions of the data that numerically describe the data and include the sample mean, sample
median, sample percentiles, sample range, and sample standard deviation. These quantities,
known as estimates, condense the data and are useful for making inferences concerning the
population from which the data were drawn. Graphical representations are used to identify
patterns and relationships within the data, confirm or disprove assumptions, and identify
potential problems.
The preliminary data review step is designed to make the reviewer familiar with the data.
The review should identify anomalies that could indicate unexpected events that may influence
the analysis of the data.
2.1 Review Quality Assurance Reports
When sufficient documentation is present, the first activity is to review any relevant QA
reports that describe the data collection and reporting process as it was actually implemented.
These QA reports provide valuable information about potential problems or anomalies in the
data set. Specific items that may be helpful include:
• Data verification and validation reports that document the sample collection, handling,
analysis, data reduction, and reporting procedures used;
• Quality control reports from laboratories or field stations that document measurement
system performance.
These QA reports are useful when investigating data anomalies that may affect critical
assumptions made to ensure the validity of the statistical tests.
In many cases, such as the evaluation of data cited in a publication, these reports may be
unobtainable. Auxiliary questions such as "Has this project or data set been peer reviewed?",
"Were the peer reviewers chosen independently of the data generators?", and "Is there evidence
to persuade me that the appropriate QA protocols have been observed?", should be asked to
assess the integrity of the data. Without some form of positive response to these questions, it is
difficult to assess the validity of the data and the resulting conclusions. The purpose of this
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validity inspection of the data is to assure a firm foundation exists to support the conclusions
drawn from the data.
2.2 Calculate Basic Statistical Quantities
The basic quantitative characteristics of the data using common statistical quantities ato
be expected of almost any quantitative study. It is often useful to prepare a table of descriptive
statistics for each population when more than one is being studied (e.g., background compared to
a potentially contaminated site) so that obvious differences between the populations can be
identified. Commonly used statistical quantities and the differences between them are discussed
in Appendix A
2.3 Graph the Data
The visual display of data is used to identify patterns and trends in the data that might go
unnoticed using purely numerical methods. Graphs can be used to identify these patterns and
trends, to quickly confirm or disprove hypotheses, to discover new phenomena, to identify
potential problems, and to suggest corrective measures. In addition, some graphical
representations can be used to record and store data compactly or to convey information to
others. Plots and graphs of the data are very valuable tools for stakeholder interactions and often
provide an immediate understanding of the important characteristics of the data.
Graphical representations include displays of individual data points, statistical quantities,
temporal data, or spatial data. Since no single graphical representation will provide a complete
picture of the data set, the reviewer should choose different graphical techniques to illuminate
different features of the data. At a minimum, there should be a graphical representation of the
individual data points and a graphical representation of the statistical quantities. If the data set
consists of more than one variable, each variable should be treated individually before
developing graphical representations for the multiple variables. If the sampling plan or
suggested analysis methods rely on any critical assumptions, consider whether a particular type
of graph might shed light on the validity of that assumption. Usually, graphs should be applied
to each group of data separately or each data set should be represented by a different symbol.
There are many types of graphical displays that can be applied to environmental data; a variety
of data plots are shown in Appendix B.
2.4 What Outputs Should a DQA Reviewer Have at the Conclusion of Step 2?
At the conclusion, two main outputs should be present:
1. Basic statistical quantities should have been calculated, and
2. Graphs showing different aspects of the data should have been developed.
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CHAPTER 3
STEP 3: SELECT THE STATISTICAL METHOD
This step concerns the selection of an appropriate statistical method that will be used to
draw conclusions from the data. Detailed technical information that reviewers can use to select
appropriate procedures may be found in Chapter 3 of Data Quality Assessment: Statistical
Methods for Practitioners (Final Draft) (EPA QA/G-9S). The statistical method will be selected
based on the sampling plan used to collect the data, the type of data distribution, assumptions
made in setting the DQOs, and any deviations from assumptions noted from Chapter 2.
If a particular statistical procedure has been specified in the planning process, the
reviewer should use the results of the preliminary data review to determine if it is appropriate for
the data collected. If not, then the reviewer should document what the anomaly appears to be,
and then select a different method. Chapter 3 of Data Quality Assessment: Statistical Methods
for Practitioners (Final Draft) (EPA QA/G-9S) provides alternatives for several statistical
procedures. If a particular procedure has not been specified, then the reviewer should select one
based upon the reviewer's objectives, the preliminary data review, and the key assumptions
necessary for analyzing the data.
All statistical tests make assumptions about the data. For instance, so-called parametric
tests assume some distributional form, e.g., a one-sample t-test assumes the sample mean has an
approximate normal distribution. The alternative, nonparametric tests, make much weaker
assumptions about the distributional form of the data. However, both parametric and
nonparametric tests assume that the data are statistically independent or that there are no trends
in the data. While examining the data, the reviewer should always list the underlying
assumptions of the statistical test. Common assumptions include distributional form of the data,
independence, dispersion characteristics, approximate homogeneity, and the basis for
randomization in the data collection design. For example, the one-sample Mest needs a random
sample, independence of the data, that the sample mean is approximately normally distributed,
that there are no outliers, and that there are few "non-detects".
Statistical methods that are insensitive to small or moderate departures from the
assumptions and are called robust, but some tests rely on certain key assumptions. The reviewer
should note any sensitive assumptions where relatively small deviations could jeopardize the
validity of the test results.
Appendix C shows many standard statistical tests and lists the assumptions needed for
each. The remainder of this chapter focuses on the two major categories of procedures that were
presented in Section 1.2: hypothesis tests and confidence interval/tolerance interval estimation.
3.1 Choosing Between Alternatives: Hypothesis Testing
The full statement of a statistical hypothesis has two major parts: the null hypothesis and
the alternative hypothesis. For both, a population parameter (such as a mean, median, or upper
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proportion) is compared to either a fixed value or to the same population parameter. Although
the language of hypothesis testing is somewhat arcane, it does describe precisely what is being
done in choosing between alternatives.
It is important to take care in defining the null and alternative hypotheses because the null
hypothesis will be considered true unless the data demonstratively shows proof for the
alternative. In layman's terms, this is equivalent of an accused person appearing in civil court;
the accused is presumed to be innocent unless shown by the evidence to be guilty by a
preponderance of evidence. Note the parallel: "presumed innocent" & "null hypothesis
considered true", "evidence" & "data", "preponderance of evidence" & "demonstratively
shows". It is often useful to choose the null and alternative hypotheses in light of the
consequences of making an incorrect determination between them. The true condition that
occurs with the more severe decision error is often defined as the null hypothesis thus making it
hard to make this kind of decision error. The statistical hypothesis framework would rather allow
a false acceptance than a false rejection. As with the accused and the assumption of innocence,
the judicial system makes it difficult to convict an innocent person (the evidence must be very
strong in favor of conviction) and therefore allows some truly guilty to go free (the evidence was
not strong enough). The judicial system would rather allow a guilty person to go free than have
an innocent person found guilty.
If the reviewer is interested in drawing inferences about only one population, then the
null and alternative hypotheses will be stated in terms that relate the true value of the parameter
to some fixed threshold value (this is known as a one-sample test). An example of this type of
problem is the comparison of pollutant levels in an effluent stream to a regulatory limit. If the
reviewer is interested in comparing two populations, then the null and alternative hypotheses will
be stated in terms that compare the true value of one population parameter to the corresponding
true parameter value of the other population (this is called a two-sample test). An example of a
two-sample problem is the comparison of a potentially contaminated waste site to a reference
area using samples collected from the respective areas
It is worth noting that all hypothesis tests have a similar structure and follow five general
steps:
1. Set up the null hypothesis
2. Set up the alternative hypothesis
3. Choose a test statistic
4. Select the critical value or/>-value
5. Draw a conclusion from the test
Appendix D gives examples of commonly used statements of statistical hypotheses and
the technical aspects are discussed in Chapter 3 of Data Quality Assessment: Statistical Methods
for Practitioners (Final Draft) (EPA QA/G-9S).
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3.2 Estimating a Parameter: Confidence Intervals and Tolerance Intervals
Estimation is used when the purpose of a project is to estimate a parameter together with
an indication of the uncertainty of that estimate. For example, the project's objective may be to
estimate the average level of pollution for a particular contaminant. A reviewer can describe the
desired (or achieved) degree of uncertainty in the estimate by establishing confidence limits
within which one can be reasonably certain that the true value will lie.
The most common type of interval estimate for the value of interest is a confidence
interval. A confidence interval may be regarded as combining a numerical "error" around an
estimate with a probabilistic statement about the unknown parameter. When interpreting a
confidence interval statement such as "The 95% confidence interval for the mean is 19.1 to
26.3", the implication is that the best estimate for the unknown population mean is 22.7 (halfway
between 19.1 and 26.3), and that we are 95% certain that the interval 19.1 to 26.3 captures the
unknown population mean. In this case, the "error" (width of the confidence interval) is a
function of the natural variability in data, the sample size, and the percentage degree of certainty
chosen.
Another type of interval estimate is the tolerance interval. A tolerance interval specifies a
region that contains a certain proportion of the population with a certain confidence. For
example, the statement 'A 99% tolerance interval for 90% of the population is 5.7 to 9.3 ppm',
means that we are 99% confident that 90% of the population lies between 5.7 and 9.3 ppm.
In general, confidence/tolerance intervals may be applied to any project whose goal is to
estimate the value of a given parameter (such as mean, median, or upper percentile). Chapter 3
of Data Quality Assessment: Statistical Methods for Practitioners (Final Draft) (EPA QA/G-9S)
has advice on the statistical formulation of confidence/tolerance intervals.
3.3 What Output Should a DQA Reviewer Have at the Conclusion of Step 3?
There are two important outputs that the reviewer should have documented from this step:
1. the chosen statistical method, and
2. a list of the assumptions underlying the statistical method.
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CHAPTER 4
STEP 4: VERIFY THE ASSUMPTIONS OF THE STATISTICAL METHOD
In this step, the reviewer should assess the validity of the statistical test chosen in Step 3
by examining its underlying assumptions. This step is necessary because the validity of the
selected method depends upon the validity of key assumptions underlying the test. The data
generated will be examined by graphical techniques and statistical methods to determine if there
have been serious deviations from the assumptions. Minor deviations from assumptions are
usually not critical as the robustness of the statistical technique used is sufficient to compensate
for such deviations.
If the data do not show serious deviations from the key assumptions of the statistical
method have occurred, then the DQA process continues to Step 5, 'Drawing Conclusions from
the Data.' However, it is possible that one or more of the assumptions may be called into
question, and this could result in a reevaluation of one of the previous steps. It is important to
note that statistically significant deviations are not always serious deviations that invalidate a
statistical test. For example, a statistical determination of a deviation from normality may not be
seriously important for a very large sample size, but critically so for a small sample size. This
iteration in the DQA process is an important check on the validity and reliability of the
conclusions to be drawn.
4.1 Perform Tests of Assumptions
Most of the commonly used hypothesis test procedures require a random sample together
with the independence of data. There are two commonly encountered departures from
independence: serial patterns in data collection (autocorrelation), and clustering (clumping
together) of data. Some need further assumptions to make them valid; Appendix C contains
most of the commonly encountered tests together with their needed assumptions. Before
implementing the statistical method selected, it is important to attain assurance that the
assumptions needed for that method has been met. For example, a one-sample t-tesi uses the
sample mean and variance and requires the data be independent and come from an approximately
normal distribution. Independence may be checked qualitatively by reviewing the sampling plan
and quantitatively by applying a test of 'independence'. If only a small amount of data is
available, then the normality assumption may be checked qualitatively by inspecting the shape of
a histogram of the data and quantitatively by applying an appropriate test for distributional
assumptions.
For any statistical test selected it is necessary for the reviewer to assess the
appropriateness of the level of significance (Type I error rate) with respect to the risk to human
health or resource expenditure if such a decision error were to be made. The level of
significance is the chance that the null hypothesis is rejected when it is actually true. The choice
of specific level of significance is up to the principal investigator and is a matter of experience or
personal choice. It does not have to be the same as that chosen in Step 3 of the DQA Process and
it is common for a value of 5% to be chosen, although there is no compelling reason to do so.
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4.2 Develop an Alternate Plan
If it is determined that one or more of the assumptions is not met, then an alternate plan is
needed. This means the selection of a different statistical method or the collection of additional
data to verify the assumptions; Chapter 3 of Data Quality Assessment: Statistical Methods for
Practitioners (Final Draft) (EPA QA/G-9S) provides a detailed list of alternative methods.
4.3 Corrective Actions
A common distributional assumption is normality of the underlying populations. If this
assumption is not valid, then the general corrective course of action is to use a corresponding
nonparametric procedure or investigate the use of some form of transformation of data. There
are many parametric tests that have nonparametric counterparts. For example, suppose a one-
sample t-tesi was selected and it was found that the data didn't follow an approximate normal
distribution. An alternative plan would be to use the Wilcoxon Signed Rank test if the data
follow an approximate symmetric distribution (which can be checked by inspecting a histogram
of the data). Parametric tests generally have more statistical power than the nonparametric tests
when the key assumptions hold but have difficulty dealing with outliers and non-detects. Should
these be found in the data, then a possible alternative would be to use the corresponding
nonparametric method as such tests handle outliers and non-detects better than parametric
methods. It is recommended that if anomalous data are included in the data set, analyses be
conducted both with and without those results to understand the implications they have on
meeting the project objectives.
One of the most important assumptions underlying statistical procedures is that there is no
inherent bias (systematic deviation from the true value) in the data. If bias is present, substantial
distortion of the false rejection and false acceptance decision error rates can occur and so the
level of significance may be different than that assumed, and the statistical power weakened. In
general, bias cannot be discerned by examination of routine data and special studies are needed to
estimate the magnitude of the bias. Bias is of great concern when comparing data to a fixed or
regulatory standard. It is of lesser concern when comparing two or more populations as the bias
tends to be in the same direction and so the effects usually cancel out.
If a trend in the data is detected or the data are found not to be independent, then basic
statistical methods should not be applied. Time series analysis or geostatistical method
investigations may be needed and a statistician should be consulted. Common assumptions and
the use of transformations are presented in Appendix E.
4.4 What Outputs Should a DQA Reviewer Have at the End of Step 4?
There are two important outputs:
1. documentation of the method used to verify each assumption together with the
results from these investigations, and
2. a description of any corrective actions that were taken.
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CHAPTER 5
STEP 5: DRAW CONCLUSIONS FROM THE DATA
In this, the final step of the DQA, the reviewer now performs the statistical hypothesis
test or computes the confidence/tolerance interval, and draws conclusions that address the
projects objectives. This step represents the culmination of the planning, implementation, and
assessment phases of the project operations. The reviewer's planning objectives will have been
reviewed (or developed retrospectively) and the sampling design examined in Step 1. Reports on
the implementation of the sampling scheme will have been reviewed and a preliminary picture of
the sampling results developed in Step 2. In light of the information gained in Step 2, the
statistical test will have been selected in Step 3. To ensure that the chosen statistical methods are
valid, the underlying assumptions of the statistical test will have been verified in Step 4.
Consequently, all of the activities conducted up to this point should ensure that the calculations
performed on the data set and the conclusions drawn here in Step 5 address the reviewer's needs
in a scientifically defensible manner.
5.1 Perform the Statistical Method
Here the statistical method selected in Step 3 is actually performed and the hypothesis
test completed or confidence/tolerance interval calculated. The calculations for the procedure
should be clearly documented and easily verifiable. In addition, documentation of the results
should be understandable so they can be communicated effectively to those who may hold a
stake in the resulting decision. If computer software is used to perform the calculations, ensure
that the procedures are adequately documented, particularly if algorithms have been developed
and coded specifically for the project.
5.2 Draw Study Conclusions
Whether hypothesis testing is performed or confidence/tolerance intervals are calculated,
the results should lead to a conclusion about the study questions. The conclusion should be
expressed in plain English and not just as a statistical statement, e.g., "it is statistically
significant".
5.3 Hypothesis Tests
The goal of this activity is to translate the results of the statistical hypothesis test so that
the reviewer may draw a conclusion from the data. Hypothesis tests can only be used to show
there is evidence for or against the alternative. Failing to reject the null hypothesis does not
prove or demonstrate there is evidence that the null is true, only that there is not sufficient
evidence that the alternative is true.
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The results of the statistical hypothesis test will be either:
(a) reject the null hypothesis, in which case there is sufficient evidence in favor of the
alternative hypothesis. The reviewer should be concerned about a possible false
rejection error.
(b) accept (fail to reject) the null hypothesis, in which case there is not sufficient
evidence in favor of the alternative hypothesis. The reviewer should be
concerned about a possible false acceptance error.
In case (a), the data have provided the evidence for the alternative hypothesis, so the
decision can be made with sufficient confidence and without further analysis. This is because
the statistical tests described in this document inherently control the false rejection error rate
within the reviewer's tolerable limits when the underlying assumptions are valid.
In case (b), the data do not provide sufficient evidence for the alternative hypothesis. An
initial step is to reexamine the false rejection rate and ascertain how strictly this value is to be
interpreted. If it has been somewhat arbitrarily selected (by custom or precedent) then the data
should be statistically analyzed further. If the false rejection rate has a stricter interpretation then
data are said not to support rejecting the null hypothesis and two outcomes considered:
(1) The false acceptance decision error limits were satisfied. In this case, the
conclusion is drawn in favor of the null hypothesis, since the probability of
committing a false acceptance error is believed to be sufficiently small in the
context of the current study (see Section 5.2).
(2) The false acceptance decision error limits were not satisfied. In this case, the
statistical test was probably not powerful enough to satisfy the reviewer's
performance criteria. The reviewer may choose to tolerate a higher false
acceptance decision error rate than previously specified and draw the conclusion
in favor of the null hypothesis, or instead implement an alternate approach such as
obtaining additional data before drawing a conclusion and making a decision.
When the test fails to reject the null hypothesis, the most thorough procedure for
verifying whether the false acceptance error limits have been satisfied is to compute the
estimated power of the statistical test. The power of a statistical test is the probability of
rejecting the null hypothesis when the null hypothesis is false and is also equal to one minus the
false acceptance error rate. Computing the power of the statistical test across the full range of
possible parameter values can be complicated and usually needs statistical software.
An approximate method that can be used for checking the performance of the statistical
test utilizes the actual data generated. Using an estimate of the variance obtained from the actual
data or an upper confidence limit on variance, the sample size needed that satisfies the reviewer's
objectives can be calculated retrospectively. If this theoretical sample size is less than or equal
to the number of samples actually taken, then the test is probably sufficiently powerful. If the
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theoretical number of samples is greater than the number actually collected, then additional
samples should be collected to satisfy the reviewer's performance criteria for the statistical test.
The method gives only approximate power as actual sample estimates are used in a retroactive
manner as if they were known parameter values.
5.4 Confidence Intervals
A confidence interval is simply an interval estimate for the population parameter of
interest. The interval's width is dependent upon the variance of the point estimate, the sample
size, and the confidence level. More specifically, the width is relatively large, if the variance is
large, the sample size is small, or the confidence level is large.
The interpretation of a confidence interval makes use of probability in an intuitive sense.
When a confidence interval has been constructed using the data, there is still a chance that the
interval does not include the true value of the parameter estimated. For example, consider this
confidence interval statement: "the 95% confidence interval for the unknown population mean is
43.5 to 48.9". It is interpreted as, "I can be 95% certain that the interval 43.5 to 48.9 captures the
unknown mean." Notice how there is a 5% chance that the interval does not capture the mean.
The confidence level is the 'confidence' we have that the population parameter lies
within the interval. This concept is analogous to the false rejection error rate. The width of the
interval is related to statistical power, or the false acceptance error rate. Rather than specifying a
desired false acceptance error rate, the desired targeted interval width can be specified with an
expectation that the final interval will approximately have this desired width.
A confidence interval can be used to make to decisions and in some situations a test of
hypothesis is set up as a confidence interval. Confidence intervals are analogous to two-sided
hypothesis tests. If the threshold value lies outside of the interval, then there is evidence that the
population parameter differs from the threshold value. In a similar manner, confidence limits
can also be related to one-sided hypothesis tests. If the threshold value lies above (below) an
upper (lower) confidence bound, then there is evidence that the population parameter is less
(greater) than the threshold.
5.5 Tolerance Intervals
A tolerance interval is an interval estimate for a certain proportion of the population. The
interval's width is dependent upon the variance of the population, the sample size, the desired
proportion of the population, and the confidence level. More specifically, the width is large if the
variance is large, the sample size is small, the proportion is large, or the confidence level is large.
When a tolerance interval has been constructed using the data, there is still a chance that
the interval does not include the desired proportion of the population. For example, consider this
tolerance interval statement: "the 99% tolerance interval for 90% of the population is 7.5 to 9.9".
It is interpreted as, "I can be 99% certain that the interval 7.5 to 9.9 captures 90% of the
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population." Notice how there is a 1% chance that the interval does not capture the desired
proportion.
The confidence level is the 'confidence' we have that the desired proportion of the
population lies within the interval. This concept is analogous to the false rejection error rate.
The width of the interval is partially related to statistical power (false acceptance error rate).
Rather than specifying a desired false acceptance error rate, the desired interval width can be
specified.
A tolerance interval can be used to make to decisions and in some situations a test of
hypothesis is set up as a tolerance interval. Tolerance intervals are analogous to a hypothesis
test. If the threshold value lies outside of the interval, then there is evidence that the desired
proportion of the population differs from the threshold value. In a similar manner, tolerance
limits can also be related to one-sided hypothesis tests. If the threshold value lies above (below)
an upper (lower) tolerance limit, then there is evidence that the desired proportion of the
population is less (greater) than the threshold.
5.6 Evaluate Performance of the Sampling Design
If the sampling design is to be used again, either in a later phase of the current study or in
a similar study, the reviewer will be interested in evaluating the overall performance of the
design. To evaluate the sampling design, the reviewer performs a statistical power analysis that
describes the estimated power of the statistical test over the range of possible parameter values.
The estimated power is computed for all parameter values under the alternative hypothesis to
create a power curve. A power analysis helps the reviewer evaluate the adequacy of the
sampling design when the true parameter value lies in the vicinity of the action level (which may
not have been the outcome of the current study). In this manner, the reviewer may determine
how well a statistical test performed and compare this performance with that of other tests.
The calculations needed to perform a power analysis can be relatively complicated,
depending on the complexity of the sampling design and statistical test selected. A further
discussion of power curves (performance curves) is contained in the Guidance on the Data
Quality Objectives Process (QA/G-4) (U.S. EPA 2000a), and Visual Sample Plan (VSP). VSP is
free software (http://dqo.pnl.gov/vsp/) that can be used to determine theoretical sample sizes for
determination of whether enough data is available to meet the specified decision error tolerances.
5.7 What Output Should the DQA Reviewer Have at the End of Step 5?
At the end of Step 5, there should be several outputs regarding conclusions based on the
data:
1. Statistical results with a specified significance level,
2. study conclusion in plain English, and
3. an assessment of performance of the sampling design.
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CHAPTER 6
INTERPRETING AND COMMUNICATING THE TEST RESULTS
At the conclusion of DQA Step 5, the reviewer has performed the applicable statistical
test, and has drawn conclusions from this test. In many cases, the conclusions are so
straightforward and convincing that they readily lead to an unambiguous path forward for the
project. There are occasions where difficulties may arise in interpreting or explaining the results
of a statistical test, or issues arise related to the scope and nature of the data set. This chapter
looks at some issues relating to data interpretation and data sufficiency.
6.1 Data Interpretation: The Meaning of/>-values
The classical approach for hypothesis tests is to pre-specify the significance level of the
test, i.e., the false rejection error rate (Type I error rate). This rate is used to define the decision
rule associated with the hypothesis test. For instance, in testing whether the population mean
exceeds a threshold level (e.g., 100 ppm), the test statistic usually involves the average of the
results obtained. Now due to random variability, it is quite possible to have a sample average
slightly greater than lOOppm even though the true (but unknown) mean concentration is less than
or equal to lOOppm. However, if the sample mean is "much larger" than 100 ppm, then there is
only a small chance that the true site mean concentration is below the threshold. Hence the
decision rule might take the form "reject the null hypothesis if the sample average exceeds 100 +
C", where C is a positive quantity that depends on the specified acceptable false rejection rate
and on the variability of the data. If this does happen, then the result of the statistical test is
reported as "reject the null hypothesis"; otherwise, the result is reported as "do not reject the null
hypothesis."
The conclusions of the hypothesis test have to be presented in plain English to avoid
misinterpretation. The phrase "reject the null hypothesis" can be explained in plain English as
"it is highly unlikely the base line assumption (null hypothesis) is true". The phrase "fail to
reject the null hypothesis" or equivalently, "do not reject the null hypothesis" can be explained in
plain English as "there is insufficient evidence to disprove the base line assumption (null
hypothesis)".
An alternative way of reporting the result of a statistical test is to report its/?-value, which
is defined as the probability, assuming the null hypothesis to be true, of observing a test result at
least as extreme as that found in the data. Many statistical software packages reports-values,
rather than adopting the classical approach of using a pre-specified false rejection error rate. In
the above example, for instance, the/?-value would be the probability of observing a sample
mean as large or larger than as the sample mean obtained if in fact the true mean was equal to
100 ppm. Obviously, in making a decision based on the/?-value, one should reject the null
hypothesis when/? is small and not reject it if/? is large. Thus the relationship between/^-values
and the classical hypothesis testing approach is that one rejects the null hypothesis if the/?-value
associated with the test result is less than the agreed upon false rejection rate. If an analyst had
chosen the false rejection error rate as 0.05 before the data were collected and reported a/?-value
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of 0.12, then the conclusion would be "do not reject the null hypothesis"; if the/>-value had been
reported as 0.03, then the conclusion would be "reject the null hypothesis." An advantage of
reporting ^-values is that they provide a measure of the strength of evidence for or against the
null hypothesis, which allows reviewers to establish their own false rejection error rates. The
significance level can be interpreted as that/?-value that divides "do not reject the null
hypothesis" from "reject the null hypothesis."
6.2 Data Interpretation: "Accepting" vs. "Failing to Reject" the Null Hypothesis
The classical approach to hypothesis testing results in one of two conclusions: "reject the
null hypothesis" (called a significant result) or "do not reject the null hypothesis" (a
nonsignificant result). In the latter case one might be tempted to equate "do not reject" with
"accept." Strictly speaking this not correct because of the philosophy underlying the statistical
testing procedure. This philosophy places the burden of proof on the alternative hypothesis; that
is, the null hypothesis is rejected only if the sample result convinces us that the alternative
hypothesis is the more likely state of nature. If a nonsignificant result is obtained, it provides
evidence that the null hypothesis could sufficiently account for the observed data, but it does not
imply that the hypothesis is the only hypothesis that could be supported by the data. In other
words, a highly nonsignificant result (e.g., a p-value of 0.80) may indicate that the null
hypothesis provides a reasonable model for explaining the data, but it does not necessarily imply
that it is the only reasonable model, and therefore does not imply that the null hypothesis is true.
It may, for example, simply indicate that the sample size was not large enough to establish
convincingly that the alternative hypothesis was more likely. When the phrase "accept the null
hypothesis" is encountered, it should be considered as "accepted with the preceding caveats."
6.3 Data Sufficiency: "Proof of Safety" vs. "Proof of Hazard"
The establishment of null and alternative hypotheses is not simply an arbitrary exercise;
the manner in which hypotheses are framed can have consequences for the expense of data
collection, for the adequacy of the collected data, and ultimately for the outcome of the project.
This is because the null hypothesis will be allowed to stand unless the data convincingly
demonstrate that it should be rejected in favor of the alternative (in other words, the "burden of
proof is on the alternative hypothesis). During DQA, the reviewer should consider this issue
and its impact on the conclusions of the study, if it was not resolved through the DQO Process.
In general, this question can be considered as a tradeoff between "proof of safety" (i.e.,
the null hypothesis assumes the existence of an environmental problem, and the alternative
position will be accepted only if we can reject the null), versus "proof of hazard" (i.e., the null
hypothesis assumes that there is no environmental problem). Formulating a set of hypotheses
unavoidably builds into them an implicit preference about what outcome we can "live with" in
the absence of compelling evidence to the contrary. This can lead to consequences such as:
• Environmental contamination may remain undetected, or a mitigation effort may be
launched unnecessarily.
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• The degree to which a cleanup level has been achieved may be greater or lesser.
• Depending on the range of measured values compared to threshold values, there may be a
need for additional data collection to resolve the hypothesis.
As there are potential "real-world" consequences of hypothesis formulation, some
environmental programs determine in advance (by either regulation or guidance) how hypotheses
will be defined, rather than leave it to a case-by-case determination. In effect, this can be viewed
as a programmatic policy on the "proof of safety" vs. "proof of hazard" tradeoff. See Table 6-1
for some examples.
Table 6-1. Selected Guidelines for Establishment of Hypotheses
Program
Sample Provision
Reference
Radiation Protection
"The objective of final status (decommissioning)
surveys is typically to demonstrate that residual
radioactivity levels meet the release criterion. In
demonstrating that this objective is met, the null
hypothesis...tested is that residual contamination
exceeds the release criterion; the alternative
hypothesis... is that residual contamination meets
the release criterion."
Multi-Agency
Radiation Survey
and Site
Investigation
Manual
(MARSSIM) (U.S.
EPA 2000b)
Whole Effluent
Toxicity Testing
"The concept of hypothesis testing relies on the
ability to distinguish statistically significant
differences between a control treatment and other
test treatments... .hypothesis testing techniques...
test the null hypothesis.. .that there is no difference
between the control treatment and other test
treatments (the effluent is not toxic). This null
hypothesis is rejected (the effluent is determined to
be toxic) if the difference between the control
treatment and any other test treatment is statistically
significant."
Method Guidance
and
Recommendations
for Whole Effluent
Toxicity (WET)
Testing (40 CFR
Part 136) (U.S.
EPA 2000c)
Brownfields
"Generally, the more severe consequences of
making the wrong decision at a Brownfields site
occur when the site is actually contaminated above
established health limits, but the decision-maker
acts on data that erroneously indicate that the site is
clean. In this situation, human health could be
endangered if reuse occurs without cleanup.
Therefore, the null hypothesis is likely to be 'the site
is too dirty for the reuse scenario', and the site
assessment is then designed to show that the site is
clean, which is the alternative hypothesis "
Quality Assurance
Guidance for
Conducting
Brownfields Site
Assessments
(U.S. EPA 1998)
EPA QA/G-9R
27
February 2006
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In cases where a planner (or the reviewer, when hypotheses are being generated
retroactively) does have flexibility in formulating hypotheses, one difficulty may be obtaining a
consensus on which error should be of most concern. The ideal approach is not only to set up the
direction of the hypothesis in such a way that controlling the false rejection error protects the
health and environment, but also to set it up in a way that minimizes uncertainty as well as
expenditure of resources in situations where decisions are relatively "easy" (e.g., all observations
are far from the threshold level of interest).
6.4 Data Sufficiency: Quantity vs. Quality of Data
With environmental data collection, there are often a variety of methods available for
determining the results. For example, with chemical measurements of environmental media,
several different analytical methods for determining the concentrations of chemicals in the
sample are available. Project teams encounter difficult decisions in the planning phase when
they have to decide whether to gather more samples using inexpensive analytical methods or
fewer samples using expensive methods. The trade-off between quantity and quality of data is
complex.
It is intuitive that the more data that are available, the stronger certainty there can be in
the decision that is reached. However, it is also possible that much less data, but of higher
quality, could improve the certainty in the decision. This is especially true if the precision differs
greatly between the available sample analysis methods. When debating selection of analytical
method and the choice between quality and quantity of data, the statistical methods that will be
used to determine the answer to the study questions should be considered and the analytical
method that maximizes the expected certainty in decision-making should be selected. There are
techniques, such as Collaborative Sampling, that can assist in addressing this dilemma but it
rarely can be applied after the sample has been collected (see Guidance on Choosing a Sampling
Design for Environmental Data Collection, U.S. EPA QA/G-5S).
6.5 Data Sufficiency: Statistical Significance vs. Practical Significance
Statistical significance is a concept based on the weight of evidence that a hypothesis is
valid. It is never possible to have perfect knowledge about a population being studied, but it is
possible to learn enough about it to be able to say with confidence that a particular hypothesis
concerning that population cannot be true. However, one should be very careful not to allow the
statistics to dictate decisions without recourse to common sense. In particular, as more and more
data are collected, it becomes easier and easier to achieve statistical significance. The concern is
that at some point it may be possible to determine statistical significance at levels that are not of
practical significance. This can be illustrated through the following example:
Based on operations at an industrial plant, and their waste release permit, it is expected that
the pH of water leaving the plant will be 5.9. The releases are monitored by weekly
collections and each week these data are combined with all previous data and the average
pH is compared to 5.9. After the first few months, the average release pH is 5.88, which is
not statistically significantly different from 5.9 and the conclusion of no real difference
EPA QA/G-9R 28 February 2006
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justified. After several years have elapsed the average release pH is 5.8996 and this is
statistically significantly different from the permitted value of 5.9, but yet a conclusion of a
real difference be justified? This is a case where having so much data allows the reviewer
to identify very small differences from the expected level, but the statistically significant
result may very well not have any practical significance (in this case a difference in pH of
0.0004, which is barely measurable).
While statistics provide a strong and essential tool for environmental decision-making, the
science of statistics is not a substitute for common sense and can lead to bad decisions if not
tempered with practicality.
6.6 Conclusions
This document may be used to either assist in conducting a DQA, or in reviewing an
existing DQA. Steps 1-5 should be followed roughly in the order presented. However, it may
occasionally be productive to revisit earlier steps based on information gleaned during the DQA
process. For that reason it is often beneficial to view this as an iterative process rather than one
for which all inputs should be gathered sequentially. Data quality assessment should be
conducted on all data intended for use in decision-making, regardless of the level of planning
defined prior to data collection.
The information contained in this document is meant to provide an overview of the DQA
process. There are several levels of assistance available from EPA for those conducting DQAs:
1. The checklist in Appendix F provides a de minimus list of outputs necessary for a
complete DQA. This can be used on its own to check that the DQA is complete.
2. This document provides further explanation for each of the outputs on the checklist.
The user can either begin with the checklist and refer back to this document as
necessary, or follow the steps as laid out in chapters 1-5 of this document directly.
3. Data Quality Assessment: Statistical Methods for Practitioners (Final Draft) (EPA
QA/G-9S) provides much more detail for implementation of a DQA. Again the DQA
reviewer can either refer to that document as necessary for details of implementation
of selected methods, or can perform a DQA by following chapters 1-5 of that
document directly.
4. EPA Quality Staff offers an introductory course in Data Quality Assessment. If the
course is not being offered at a time and location convenient for you, it may be
downloaded from http://www.epa.gov/quality/trcourse.htmltfintro dqa.
EPA QA/G-9R 29 February 2006
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EPA QA/G-9R 30 February 2006
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U.S. Environmental Protection Agency, 1998. Quality Assurance Guidance for Conducting
Brownfields Site Assessments EPA 540-R-98-038. Office of Solid Waste and Emergency
Response
U.S. Environmental Protection Agency, 2000a. Guidance on the Data Quality Objectives Process
(QA/G-4)
U.S. Environmental Protection Agency, 2000b. Multi-Agency Radiation Survey and Site
Investigation Manual (MARSSIM)
.S. Environmental Protection Agency, 2000c. Method Guidance and Recommendations for
Whole Effluent Toxicity (WET) Testing (40 CFR Part
56)
U.S. Environmental Protection Agency, 2002. Guidance on Choosing a Sampling Design for
Environmental Data Collection (EPA QA/G-5S)
U.S. Environmental Protection Agency, 2004. Data Quality Assessment: Statistical Methods for
Practitioners (EPA QA/G-9S) (Final Draft)
U
136)
EPAQA/G-9R 31 February 2006
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EPA QA/G-9R 32 February 2006
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APPENDIX A: COMMONLY USED STATISTICAL QUANTITIES
Measures of Central Tendency:
Measures of the center of a sample of data points
Mean: The most commonly used measure of the center of a sample is the sample mean, denoted
by X. This estimate of the center of a sample can be thought of as the "center of gravity" of the
sample. The sample mean is an arithmetic average for simple sampling designs; however, for
complex sampling designs, such as stratification, the sample mean is a weighted arithmetic
average. The sample mean is influenced by extreme values (large or small) and non-detects.
Median: The sample median is the second most popular measure of the center of the data. This
value falls directly in the middle of the data when the measurements are ranked in order from
smallest to largest. This means that !/2 of the data are smaller than or equal to the sample median
and !/2 of the data are larger than or equal to the sample median. The median is another name for
the 50th percentile. The median is not influenced by extreme values and can easily be used in the
case of censored data (non-detects).
Mode: The third method of measuring the center of the data is the mode. The sample mode is
the value of the sample that occurs with the greatest frequency. Since this value may not always
exist, or if it does it may not be unique, this value is the least commonly used. However, the
mode is useful for describing qualitative data.
Measures of Relative Standing:
Relative position of one observation in relation to all observations
Percentiles: A percentile is the data value that is greater than or equal to a given percentage of
the data values. Stated in mathematical terms, the/>th percentile is a data value that is greater
than or equal to p% of the data values and is less than or equal to (!-/>)% of the data values.
Therefore, if V is the/>th percentile, then/>% of the values in the data set are less than or equal to
x, and (100-^)% of the values are greater than or equal to x. A sample percentile may fall
between a pair of observations. For example, the 75th percentile of a data set of 10 observations
is not uniquely defined as it falls between the 7th and 8th largest values. Important percentiles
usually reviewed are the quartiles of the data, the 25th, 50th, and 75th percentiles. Also important
for environmental data are the 90th, 95th, and 99th percentile where a decision maker would like to
be sure that 90%, 95%, or 99% of the contamination levels are below a fixed risk level. There
are several methods for computing sample percentiles.
Quantiles: A quantile is very similar in concept to a percentile; however, a percentile represents
a percentage whereas a quantile represents a fraction. If x is the/>/100 quantile of the data, then
the fraction/7/100 of the data values lie at or below x and the fraction (l-p)/100 of the data values
lie at or above x, whereas if V is the//1 percentile, then at least p% of the values in the data set
lie at or below x, and at least (100-/>)% of the values lie at or above x. For example, the 0.95
EPAQA/G-9R 33 February 2006
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quantile has the property that 0.95 of the observations lie at or below x and 0.05 of the data lie at
or above x.
Measures of Dispersion:
Measures of how the data spread out from the center
Range: The easiest measure of dispersion to compute is the sample range, maximum -
minimum. For small samples, the range is easy to interpret and may adequately represent the
dispersion of the data. For large samples, the range is not very informative because it only
considers (and therefore is greatly influenced) by extreme values which could be outliers.
Variance and Standard Deviation: The variance measures the dispersion of the data from the
mean and is denoted by s2. A large variance implies that there is a large spread among the data
so that the data are not clustered around the mean. A small variance implies that there is little
spread among the data so that most of the data are near the mean. The variance is affected by
extreme values and by a large number of non-detects. The standard deviation (s) is the square
root of the sample variance and has the same unit of measure as the data.
Coefficient of Variation: The coefficient of variation (CV) is a unitless measure that allows the
comparison of dispersion across several sets of data. The CV is the standard deviation divided
by the mean. It is also called the relative standard deviation (RSD).
Interquartile Range: When extreme values are present, the interquartile range may be more
representative of the dispersion of the data than the standard deviation. It is the difference
between the first and third quartiles (25th and 75th percentiles) of the data. This statistical
quantity does not depend on extreme values and is therefore useful when the data include a large
number of non-detects.
Measures of Association:
The relationship between two or more variables
Pearson's Correlation Coefficient: The Pearson (often "Pearson" is omitted) correlation
coefficient measures a linear relationship between two variables. Values of the correlation
coefficient close to +1 (positive correlation) imply that as one variable increases so does the
other, the reverse holds for values close to -1 (negative correlation). Values close to 0 imply
little correlation between the variables. The correlation coefficient does not detect nonlinear
relationships so it should be used only in conjunction with a scatterplot graph of the data. A
scatterplot can be used to determine if the correlation coefficient is meaningful or if some
measure of nonlinear relationships should be used. The correlation coefficient can be
significantly changed by extreme values so a scatter plot should be used first to identify such
values. Note that correlation does not imply cause and effect.
Spearman's Rank Correlation Coefficient: An alternative to the Pearson correlation is
Spearman's rank correlation coefficient. It is calculated by first replacing each lvalue by its
rank (i.e., 1 for the smallest lvalue, 2 for the second smallest, etc.) and each 7 value by its rank.
These pairs of ranks are then treated as the (X,Y) data and Spearman's rank correlation is
EPA QA/G-9R 34 February 2006
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calculated using the same formulae as for Pearson's correlation. Spearman's correlation will not
be altered by nonlinear increasing transformations of the Xs or the 7s.
EPAQA/G-9R 35 February 2006
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EPA QA/G-9R 36 February 2006
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APPENDIX B: GRAPHICAL REPRESENTATION OF DATA
This appendix contains examples of several types of
common graphical representations of data.
Histogram/Frequency Plots
Description: Divide the data range into units, count the
number of points within the units, and display the data as the
height (frequency plot) or area (histogram) within a bar graph.
Drawbacks: Demands the reviewer make arbitrary choices to
partition the data.
Uses: Distribution - A normal distribution will be bell-shaped.
Symmetry - Symmetric data has the same amount of
data either side of the center point.
Variability - Both plots indicate the spread of the data
(standard deviation, variance).
Skewness - Right (positive) skewed data have a large
number of low values, relatively few high values.
NOTE: The j/-axis of a histogram can also represent relative
frequencies which are frequencies divided by the sample size.
Box- and-Whiskers Plot
Description: Composed of a central box divided by a horizontal line
representing the median and two lines extending out from the box
called whiskers. The length of the central box indicates the spread of
the bulk of the data (the central 50%) while the length of the whiskers
show how stretched the tails of the distribution are. The sample mean
is displayed using a '+' sign and any unusually small or large data
points are displayed by a '*' on the plot.
Drawbacks: Schematic diagram instead of numerical.
Uses: Statistical Quantities - Visualize the relationships.
Symmetry - If the distribution is symmetrical, the box is divided
in two equal halves by the median, the whiskers will be the
same length and the number of extreme data points will be
distributed equally on either end of the plot for symmetric data.
Outliers - Values that are unusually large or small are easily identified.
10 15 20 25 30 35 «
Example Histogram
1 Percentage of Observations (per ppm)
0 tO *. 0> CO |
5 10 15 20 25 30 35 40
Concentration (ppm)
Example Frequency Plot
Example Box-
and-Whiskers Plot
EPA QA/G-9R
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February 2006
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Stem-and-LeafPlot
Description: Each observation in the stem-and-leaf
plot consists of two parts: the stem of the observation
and the leaf. The stem is usually made up of the
leading digit of the numerical values while the leaf is
made up of trailing digits in the order that
corresponds to the order of magnitude from left to
right. The stem is displayed on the vertical axis and
the data points displayed from left to right.
Advantages: Stores data in a compact form while, at
the same time, sorts the data from smallest to largest.
Non-detects can be placed in a single stem.
10.00 I 14.00
20.00 I 21.00 24.00 24.00 26.00 28.00 29.00 29.00
30.00 | 33.41 35.00 38.00 38.00 38.19 39.00
40.00 I 40.00 40.00 41.00 41.00 42.00 43.0£ 43.10 43.49
50.00 I 50.00 50.00 50.39 51.00 51.00 SS.OO 53.00 55.00
60.00 | 61.00 61.00 63.00 63.00 65.46 68.00 68.00
70.00 I "74.00 75.00
80.00 I 34.00
Example Stem-and-Leaf Plot
Drawbacks: Demands the reviewer make arbitrary choices to partition the data.
Uses: Distribution - Normally distributed data is approximately bell shaped.
Symmetry - The top half of the stem-and-leaf plot will be a mirror image of the bottom
half of the stem-and-leaf plot for symmetric data.
Skewness - Left skewed (negative) data have many high values, relatively few low
values. Left skewed data are relatively rare with environmental measurements.
Ranked Data Plot
Description: A plot of the data from smallest to
largest at evenly spaced intervals.
Advantages: Easy to construct, easy to interpret,
makes no assumptions about a model for the data, and
shows every data point.
Uses: Density - A large amount of data values have a
flat slope, i.e., the graph rises slowly. A small
amount of data values have a large slope, i.e.,
the graph rises quickly.
Skewness - A plot of data that are skewed to the
Smallest
Largest
Example Ranked Data Plot
right (many low values, but few high) extends mores sharply at the top giving the graph
a ' J' shape. A plot of data that are skewed to the left (few low values, but many high)
increases sharply near the bottom giving the graph an inverted ' J' shape.
Symmetry - The top portion of the graph will stretch to upper right corner in the same
way the bottom portion of the graph stretches to lower left, creating an S-shape, for
symmetric data.
EPA QA/G-9R
38
February 2006
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Quantile Plot
Description: A graph of the data against the quantiles.
Advantages: Easy to construct, easy to interpret,
makes no assumptions about a model for the data,
and displays every data point.
Uses: Density - A large amount of data values
has a flat slope, i.e., graph rises slowly. A
small amount of data values has a large
slope, i.e., the graph rises quickly.
Skewness - A plot of data that are skewed
to the right (many low values, but few
high) is steeper at the top right than the
bottom left. A quantile plot of data that
are skewed to the left (few low values, but
many high) increases sharply near the
bottom left of the graph.
Symmetry - The top portion of the graph will stretch to the upper right corner in the same
way the bottom portion of the graph stretches to the lower left, creating an S-shape for
symmetric data.
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Example Quantile Plot
Normal Probability Plot (Two Variables)
Description: The graph of the quantiles of a data set
against the quantiles of the normal distribution plotted
on normal probability graph paper.
Drawbacks: Tedious to generate by hand, but can
be created with most statistical software.
Uses: Normality - The graph of normally distributed
data should be linear.
Symmetry - The degree of symmetry can be
determined by comparing the right and left
sides of the plot.
Outliers - Data values that are much larger or
much smaller than rest will cause the other
data values to be compressed into the middle
of the graph, ruining the resolution.
Example Normal Probability Plot
EPA QA/G-9R
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February 2006
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Scatterplot (Two Paired Variables)
Description: Paired values are plotted on separate axes.
Advantages: Clearly shows the relationship between
two variables, easy to construct.
Uses: Correlation/Trends - Linearly correlated
variables cluster around straight line. Nonlinear
patterns may be obvious.
Outliers - Potential outliers from a single variable
or from paired variables may be identified.
Clustering - Points clustered together can be
easily identified.
Time Plot (Temporal Data)
Description: A plot of the data over time.
Advantages: Easy to generate and interpret.
Uses: Trends - Including large-scale and small-
scale, seasonal (patterns that repeat over
time), and directional (downward/upward
trends) including subtle shifts.
Serial Correlation - Shows relationship
between successive observations.
Variability - Look for increasing or
decreasing variability over time.
Outliers - Values that are unusually large
or small are easily identified.
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EPA QA/G-9R
40
February 2006
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Posting Plots (Spatial Data)
Description: Map of data locations together
with corresponding data values.
Drawback: May not be feasible for a large
amount of data
Uses: Errors - Identify obvious errors in data
location and values.
Sampling Design - Easy way to review
design.
Trends - Obvious trends can be
identifified.
4.0 11.6 14.9 17.4 17.7 12.4
7.7 15.2 35.5 14.7 16.5
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Example Posting Plot
EPA QA/G-9R
41
February 2006
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EPA QA/G-9R 42 February 2006
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APPENDIX C: COMMON HYPOTHESIS TESTS
Compare a mean to a fixed number
- for example, to determine whether
the mean contaminant level is greater
than 10 ppm
Compare a median to a fixed
number - for example, to determine
whether the median is greater than
8 ppm
Compare a proportion or
percentile to a fixed number - for
example, to determine if 95% of all
companies emitting sulfur dioxide
into the air are below a fixed
discharge level.
Compare a variance to a fixed
number - for example, to determine
if the variability of an analytical
method exceeds a fixed number.
One-Sample f-Test
Wilcoxon Signed
Rank Test
Chen Test
Wilcoxon Signed
Rank Test
Sign Test
One-Sample
Proportion Test
Chi-squared test
Random Sample
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Other Assumptions
• Not many data values are identical
• Symmetric
• Data come from a right-skewed
distribution (like a lognormal
distribution)
Not many data values are identical
• Symmetric
Not many sample values equal to the
fixed level (reduces efficiency)
Not many sample values equal to the
fixed level (reduces efficiency)
EPA QA/G-9R
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February 2006
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Compare a correlation coefficient
to a fixed number - for example,
determine if the correlation between
two contaminants exceeds 0.5.
Compare two means - for example,
to compare the mean contaminant
level at a remediated Superfund site
to a background site or to compare
the mean of two different drinking
water wells.
Compare several means against a
control population - for example, to
compare different analytical methods
to the standard method.
Compare two proportions or
percentiles - for example, to compare
the proportion of children with
elevated blood lead in one area to the
proportion of children with elevated
blood lead in another area.
Compare two correlations - for
example, to determine which of two
contaminants is a better predictor of a
third
z-Test of a
Correlation
Coefficient
Student's
Two-Sample f-Test
Satterthwaite's
Two-Sample f-Test
Dunnett's Test
Two-Sample Test for
Proportions
Kendall's Test
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• Linear relationship
• Approximately the same underlying
variance for both populations
• All group sizes are approximately
equal
• Linear relationships
EPA QA/G-9R
44
February 2006
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Compare the variances of 2 or
more populations - for example, to
compare the variances of several
analytical methods.
Determine if one population
distribution differs from another
distribution - for example, to
compare the contaminant levels at
a remediated Superfund site those
of a background area.
F-Test
Bartlett's Test
Levene's Test
Wilcoxon Signed
Rank Test
Quantile Test
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• 2 populations only
2 or more populations
2 or more populations
The two distributions have the same
shape and dispersion (approximately)
• Only a few identical values
• The difference is assumed to be some
fixed amount
• Equal variances
Data generated using systematic or
simple random sampling design
• The difference is assumed to be only
in the upper part of the distributions
EPA QA/G-9R
45
February 2006
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EPA QA/G-9R 46 February 2006
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APPENDIX D: COMMONLY USED STATEMENTS OF HYPOTHESES
Type of Decision
Compare environmental conditions to a fixed
threshold value, such as a regulatory standard or
acceptable risk level; presume that the true
condition is at most the threshold value.
Compare environmental conditions to a fixed
threshold value; presume that the true condition is
at least the threshold value.
Compare environmental conditions to a fixed
threshold value; presume that the true condition is
equal to the threshold value and the reviewer is
concerned whenever conditions vary significantly
from this value.
Compare environmental conditions associated with
two different populations to a fixed threshold value
such as a regulatory standard or acceptable risk
level; presume that the true condition is at most the
threshold value. If it is presumed that conditions
associated with the two populations are the same,
the threshold value is 0.
Compare environmental conditions associated with
two different populations to a fixed threshold value
such as a regulatory standard or acceptable risk
level; presume that the true condition is at least the
threshold value. If it is presumed that conditions
associated with the two populations are the same,
the threshold value is 0.
Compare environmental conditions associated with
two different populations to a fixed threshold value
such as a regulatory standard or acceptable risk
level; presume that the true condition is equal to
the threshold value. If it is presumed that
conditions associated with the two populations are
the same, the threshold value is 0.
Null Hypothesis
The value of the
measured parameter
is at most the
threshold value.
The value of the
measured parameter
is at least the
threshold value.
The value of the
measured parameter
is equal to the
threshold value.
The difference
between the two
measured parameters
is at most the
threshold value.
The difference
between the two
measured parameters
is at least the
threshold value.
The difference
between the two
measured parameters
is equal to the
threshold value.
Alternative
Hypothesis
The value of the
measured parameter
is greater than the
threshold value.
The value of the
measured parameter
is less than the
threshold value.
The value of the
measured parameter
is not equal to the
threshold value.
The difference
between the two
measured parameters
is greater than the
threshold value.
The difference
between the two
measured parameters
is less than the
threshold value.
The difference
between the two
measured parameters
is not equal to the
threshold value.
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APPENDIX E: COMMON ASSUMPTIONS AND TRANSFORMATIONS
Independence
The assumption of independence of data is the key to the validity of the false rejection
and false acceptance error rates associated with a selected statistical test. Environmental data are
particularly susceptible to correlation problems due to the fact that data are collected under a
spatial pattern (for example a grid) or sequentially over time (for example, daily readings from a
monitoring station). Readings taken close together can be correlated (known as autocorrelation)
and the effectiveness of statistical tests is diminished. When data are truly independent between
themselves, the correlation between data points is by definition zero and statistical tests have the
desired chosen decision error rates (given appropriate other assumptions have been satisfied).
The reason non-independence is an issue for statistical testing situations is that if
observations are positively correlated over time or space, then the effective sample size for a test
tends to be smaller than the actual sample size - i.e., each additional observation does not
provide as much "new" information because its value is partially determined by (or a function of)
the value of adjacent observations. This smaller effective sample size means that the degrees of
freedom for the test statistic decreases, or equivalently, the test is not as powerful as originally
thought. In addition to affecting the false acceptance error rate, applying the usual tests to
correlated data tends to result in a test whose actual significance level (false rejection error rate)
is larger than the nominal error rate.
Non-independence also occurs when data are actually collected in groups and not
randomly over the entire sample. For example, suppose there are 5 locations each with 4
measurements on a contaminant. This is different from 20 locations each with 1 measurement on
a contaminant. If the 5-location data are analyzed as if it was 20-location data incorrect
conclusions may occur. In the 5-location data set, each group of 4 measurements could be
expected to be approximately the same and, overall, not as diverse as the 20-location data set. In
statistical terms, this is because the 5-location data set variability can be partitioned into two
sources of variability: between locations, and within locations. The 20-location data set cannot
be partitioned into the two sources as only 1 measurement was taken at each location. Between
locations variability is usually much larger than within location variability and so the 5-location
data can be considered "more lumpy" than the 20-location data and this will affect the analysis.
One of the most effective ways to determine statistical independence for data collected
over time is through use of the Rank von Neumann Test. Compared to other tests of statistical
independence, the rank von Neumann test has been shown to be more powerful over a wide
variety of cases. This means that very little effectiveness is lost by always using the ranks in
place of the original concentrations; the rank von Neumann ration should still correctly detect
non-independent data. Details on how to use the rank von Neumann test are to be found in Data
Quality Assessment: Statistical Methods for Practitioners (Final Draft) (EPA QA/G-9S).
EPA QA/G-9R 49 February 2006
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NmdE5slribulian
Irgrmal Efetributicn
Distributional Assumptions
Many statistical tests and models are only
appropriate for data that follow a particular distribution.
Two of the most important distributions for tests
involving environmental data are the normal
distribution and the lognormal distribution.
The assumption of normality is very important,
as it is the basis for the majority of statistical tests. A
normal distribution is a reasonable model of the
behavior of certain random phenomena and can often
be used to approximate other probability distributions.
In addition, the Central Limit Theorem shows that as
the sample size gets large, some of the sample summary statistics (e.g., the sample mean) behave
as if they are a normally distributed variable. As a result, a common assumption associated with
parametric tests or statistical models is that the errors associated with data, or a proposed model,
approximate a normal distribution.
Environmental data commonly exhibit distributions that are non-negative and skewed
with heavy or long right tails. Several standard probability models have these properties,
including the Weibull, gamma, and lognormal distributions. The lognormal distribution is a
commonly used distribution for modeling environmental contaminant data. The advantage to
this distribution is that a simple (logarithmic) transformation will transform a lognormal
distribution into a normal distribution. So, methods for testing for normality can be used to test
for lognormality if a logarithmic transformation has been used.
Tests for Normality
Test
Shapiro-Wilk Test
Filliben's Statistic
Geary's Test
Studentized Range
Test
Chi-Square Test
Sample
Size
#50
#100
>50
#1000
Large
Recommended Use
Highly recommended but difficult to compute by
hand.
Highly recommended but difficult to compute.
Useful when tables for other tests are not available.
Highly recommended if the data are symmetric, the
tails of the data are not heavier than the normal
distribution, and there are no extreme values.
Useful for grouped data and when the comparison
distribution is known. May be used for other
distributions besides the normal distribution
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Outliers
Outliers are measurements that are extremely large or small relative to the rest of the data
and, therefore, are suspected of misrepresenting the population from which they were collected.
Outliers may result from transcription errors, data-coding errors, or measurement system
problems such as instrument breakdown. However, outliers may also represent true extreme
values of a distribution (for instance, hot spots) and indicate more variability in the population
than was expected. Not removing true outliers and removing false outliers both lead to a
distortion of estimates of population parameters.
Statistical outlier tests give the reviewer probabilistic evidence that an extreme value
(potential outlier) does not "fit" with the distribution of the remainder of the data and is therefore
a statistical outlier. These tests should only be used to identify data points that need further
investigation. The tests alone cannot determine whether a statistical outlier should be discarded
or corrected within a data set; this decision should be based on expert or scientific grounds.
Potential outliers may be identified through a graphical representation of the data. If a
data point is found to be an outlier, the reviewer may either: 1) correct the data point; 2) discard
the data point from analysis; or 3) use the data point in all analyses. This decision should be
based on scientific reasoning in addition to the results of the statistical test. One should never
discard an outlier based solely on a statistical test. Instead, the decision to discard an outlier
should be based on some scientific or quality assurance basis. Discarding an outlier from a data
set should be done with extreme caution, particularly for environmental data sets, which often
contain legitimate extreme values. If an outlier is discarded from the data set, all statistical
analysis of the data should be applied to both the full and truncated data set so that the effect of
discarding observations may be assessed. If scientific reasoning does not explain the outlier, it
should not be discarded from the data set.
If no data points are discarded, document the identification of any "statistical" outliers by
documenting the statistical test performed and the possible scientific reasons investigated. If any
data points are discarded, document each data point, the statistical test performed, the scientific
reason for discarding each data point, and the effect on the analysis of deleting the data points.
Statistical Tests for Outliers
Sample Size
n#25
n#50
n325
n350
Test
Extreme Value Test
Discordance Test
Rosner's Test
Walsh's Test
Assumes
Normality
Yes
Yes
Yes
No
Multiple
Outliers
Yes
No
Yes
Yes
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Values below Detection Limits
Data generated from chemical analysis may fall below the detection limit (DL) of the
analytical procedure. These measurement data are generally described as not detected, or non-
detects, (rather than as zero or not present) and the appropriate limit of detection is usually
reported. In cases where measurement data are described as not detected, the concentration of
the chemical is unknown although it lies somewhere between zero and the detection limit. Data
that includes both detected and non-detected results are called censored data in the statistical
literature.
There are a variety of ways to evaluate data that include values below the detection limit.
However, there are no general procedures that are applicable in all cases. All of the suggested
procedures for analyzing data with non-detects depend on the amount of data below the detection
limit. For relatively small amounts below detection limit values, replacing the non-detects with a
small number and proceeding with the usual analysis may be satisfactory. For moderate amounts
of data below the detection limit, a more detailed adjustment is appropriate. In situations where
relatively large amounts of data below the detection limit exist, one may need only to consider
whether the chemical was detected as above some level or not. The interpretation of small,
moderate, and large amounts of data below the DL is subjective.
In addition to the percentage of samples below the detection limit, sample size influences
which procedures should be used to evaluate the data. For example, the case where 1 sample out
of 4 is not detected should be treated differently from the case where 25 samples out of 100 are
not detected. In some cases, the data investigator should consult a statistician for the most
appropriate way to evaluate data containing values below the detection level. In no case should
the non-detects be discarded and the resulting data set analyzed as if the non-detects had never
been recorded. Serious bias will result, leading to questionable conclusions.
Guidelines for Analyzing Data with Non-detects
Approximate
Percentage of Non-
detects
< 15%
15% -50%
> 50% - 90%
Statistical Analysis Method
Replace non-detects with 0,
DL/2, DL; Cohen's Method
Trimmed mean, Cohen's
adjustment, Winsorized mean
and standard deviation.
Use tests for proportions
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Transformations
Data that do not satisfy statistical assumptions can sometimes be converted or
transformed mathematically into a form that allows standard statistical tests to perform
adequately. Any mathematical function that is applied to every point in a data set is called a
transformation and the most commonly used transformation is:
Logarithmic (LogXor Ln X): This transformation may be used when the original
measurement data follow a lognormal distribution or when the standard deviation of
measurements is proportional to the mean of the data points at that level.
By transforming the data, assumptions that are not satisfied in the original data can be
satisfied by the transformed data. For instance, a right-skewed distribution can be transformed to
be approximately Gaussian (normal) by using a logarithmic or square-root transformation. Then
the normal-theory procedures can be applied to the transformed data. If data are lognormally
distributed, then apply procedures to logarithms of the data. However, selecting the correct
transformation may be difficult and the reviewer should consult a statistician.
Once the data have been transformed, all statistical analysis should be performed on the
transformed data. Very rarely should data be transformed back to the original units after analysis
and conclusions drawn with the transformed data set this can lead to biased estimates. For
example, estimating quantities such as means, variances, confidence limits, and regression
coefficients in the transformed scale typically leads to biased estimates when transformed back
into original scale. However, it may be difficult to understand or apply results of statistical
analysis expressed in the transformed scale. Therefore, if the transformed data do not give
noticeable benefits to the analysis, it is usually better to use the original data with a different
technique of analysis.
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APPENDIX F: CHECKLIST OF OUTPUTS FOR DATA QUALITY ASSESSMENT
Step
1
1
1
2
2
3
3
4
4
5
5
5,6
6
Input
Well-defined project objectives and
criteria
Verification that the hypothesis chosen
is consistent with the objective and
criteria
A list of any deviations from the
planned sampling design and the effects
of these deviations
Statistics of interest have been
calculated
Graphs and plots of the data are
available
The statistical method for data analysis
has been selected
The assumptions underlying the method
have been identified
Documentation of the method used to
verify each assumption and the results
from these investigations
A description and rationale for any
corrective actions that were taken, if
any were necessary
Statistical results with a specified
significance level
An assessment of the performance of
the sampling design
Interpretation of the statistical result and
study conclusions
A final product or decision
G-9R
Section
1.1
1.2
1.4
2.2
2.3
3.0
3.0
4.1
4.2 & 4.3
5.1
5.5
5.3-5.4,
6.1 & 6.2
6.3-6.5
G-9S
Section
1.1
1.1
1.1
2.2
2.3
3.1
3.2-3.4
4.1
4.1
5.2
5.4
5.5
5.5
G-9R: Data Quality Assessment: A Reviewer's Guide
G-9S: Data Quality Assessment: Statistical Methods for Practitioners (Final Draft)
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