r/EPA
United States
Environmental Protection
Agency
Office of Acid Deposition
Environmental Monitoring
and Quality Assurance
Washington DC 20460
EPA/600/3-89/037
March 1989
Research and Development
An Evaluation of
Trend Detection
Techniques for Use in
Water Quality Monitoring
Programs
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EPA/600/3-89/037
March 1989
AN EVALUATION OF TREND DETECTION TECHNIQUES
FOR USE IN WATER QUALITY MONITORING PROGRAMS
By
Jim C. Loftis, Robert C. Ward, Ronald D. Phillips
Department of Agricultural and Chemical Engineering
Charles H. Taylor
Department of Statistics
Colorado State University
Fort Collins, Colorado 80523
U.S. Environmental Prrsto^t
Region 5. T, •--,• ,.- ,,
230 ' D • . .-.- .. _t.c, :.:.
1L 6'..'ciG4
Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Corvallis, OR 97333
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Notice
The research described in this report has been funded by the
U.S. Environmental Protection Agency through a cooperative
agreement with Colorado State University (#CR813997). The
document has been subjected to the Agency's peer and adminis-
trative review, and it has been approved for publication as an
EPA document. Mention of trade names or commercial products
does not constitute endorsement or recommendation for use.
ii
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TABLE OF CONTENTS
Section Page
List of Illustrations v
List of Tables ix
Abstract xi
1. INTRODUCTION 1
1.1 Goals of Monitoring 1
1.2 Scope of Report 2
1.3 Overview of Study 2
2. RECOMMENDATIONS 5
3. JUSTIFICATION OF RECOMMENDATIONS 7
3.1 Monitoring Objective in Statistical Terms 7
3.2 Characteristics of Background or Historical Data 9
3.2.1 Description of Historical Data Sets 9
3.2.2 Results of Characterization 9
3.2.2.1 Seasonality 9
3.2.2.2 Normality 15
3.2.2.3 Serial Correlation 15
3.3 Alternate Methods for Trend Analysis 34
3.4 Monte Carlo Evaluation of Candidate Tests 38
3.5 Results of Monte Carlo Evaluation 39
4. EXPECTED PERFORMANCE OF MONITORING—POWER OF TREND
DETECTION 43
4.1 Individual Lakes 43
4.2 Multiple Lakes 48
4.2.1 Statistical Characteristics of the Regional Mean 49
4.2.2 Detectable Changes in Regional Means 51
4.3 Case Studies, Individual Lakes 58
4.3.1 Clearwater Lake, Ontario 58
4.3.
4.3.
4.3.
4.3.
4.3.
4.3.
.1 Sulfate Concentration 58
.2 Acid Neutralizing Capacity 64
.3 Sulfate/ANC 64
.4 Sulfate/(Calcium + Magnesium) 64
.5 Summary of Trend Testing Results 64
.6 Effect of Time of Sampling on Trend Detection,
Quarterly Sampling 80
4.3.1.7 Trend Detection, Annual Sampling 80
4.3.2 Twin Lakes, Colorado 89
111
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5. SPECIALIZED PROCEDURES FOR EXPLANATION OF TRENDS 103
5.1 Adjustment for Hydrologic Factors--Streamflow and Precipitation 103
5.2 Water Quality Indices 103
5.3 Multivariate Tests for Trend 104
5.4 Water Quality/Watershed Models 104
6. DETAILS OF TREND TESTING AND MONTE CARLO METHODS 105
6.1 Mann Kendall Tests 105
6.2 Linear Model Tests 106
6.3 Modified T-Test 107
6.4 Rank Transformations 108
6.5 Monte Carlo Simulations 109
6.5.1 Normal Uncorrelated Errors 114
6.5.2 Lognormal Errors 114
6.5.3 Normal Errors with Positive Correlation 115
6.5.4 Summary 116
REFERENCES 117
ABBREVIATIONS AND ACRONYMS 119
APPENDIX A - TIME GOALS AND OBJECTIVES 120
APPENDIX B - RESULTS OF SIMULATIONS 121
IV
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LIST OF ILLUSTRATIONS
Figure Page
3-1 Seasonal variation in quarterly means for SO4= and pH for
selected lakes in NLS subregion 1A 16
3-2 Seasonal variation in quarterly means for SO4= and pH for
selected lakes in NLS subregion 1C 17
3-3 Seasonal variation in quarterly standard deviations for
alkalinity, SO4=, and pH for selected lakes in NLS subregion 1A 18
3-4 Seasonal variation in quarterly standard deviations for
alkalinity, SO4=, and pH for selected lakes in NLS subregion 1C 19
3-5 Correlogram for conductivity at Upper Twin Lake, Colorado—
quarterly sampling, raw data 26
3-6 Correlogram for conductivity at Upper Twin Lake, Colorado—
quarterly sampling, seasonal means removed 27
3-7 Correlogram for conductivity at Lower Twin Lake, Colorado—
quarterly sampling, raw data 28
3-8 Correlogram for conductivity at Lower Twin Lake, Colorado—
quarterly sampling, seasonal means removed 29
3-9 Correlogram for alkalinity at Lake 1A1-105, quarterly
sampling, raw data 30
3-10 Correlogram for alkalinity at Lake 1A1-105, quarterly
sampling, seasonal means removed 31
3-11 Correlogram for alkalinity, Lake 1A1-102, quarterly
sampling, raw data 32
3-12 Correlogram for alkalinity, Lake 1A1-102, quarterly sampling,
seasonal means removed 33
4-1 Power of trend detection for trend = 0.005 standard deviations
per quarter and 0.02 standard deviations per year 44
4-2 Power of trend detection for trend = 0.02 standard deviations
per quarter and 0.08 standard deviations per year 45
4-3 Power of trend detection for trend = 0.05 standard deviations
per quarter and 0.20 standard deviations per year 46
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4-4 Power of trend detection for trend = 0.20 standard deviations
per quarter and 0.8 standard deviations per year. 47
4-5 Level of detectable trend for a=0.10 and 0=0.10 for five config-
urations of number of lakes and spatial correlation = 0.0 and 0.2 52
4-6 Level of detectable trend for a=0.10 and 0=0.20 for five config-
urations of number of lakes and spatial correlation = 0.0 and 0.2 53
4-7 Level of detectable trend for a=0.10 and 0=0.20 for five config-
urations of number of lakes and spatial correlation = 0.0 and 0.4 54
4-8 Level of detectable trend for a=0.10 and 0=0.40 for five config-
urations of number of lakes and spatial correlation = 0.0 and 0.2 55
4-9 Quarterly sulfate observations beginning in summer of 1973 59
4-10 Correlogram of quarterly sulfate data shown in Figure 4-9 60
4-11 Correlogram of quarterly sulfate data shown in Figure 4-9 after
detrending with ordinary least squares 61
4-12 Results of ANOCOV on raw sulfate data and on ranks of sulfate data 62
4-13 Results of seasonal Kendall (square symbols) and seasonal Kendall
corrected for serial correlation (plus symbols) on raw sulfate data 63
4-14 Quarterly ANC observations, mg L"1 as calcium carbonate, for
Clearwater Lake, Ontario, beginning fall of 1980 65
4-15 Correlogram of raw ANC data in Figure 4-14 66
4-16 Correlogram of ANC data shown in Figure 4-14 after
detrending by ordinary least squares 67
4-17 Results of ANOCOV on raw ANC data and on ranks of ANC data 68
4-18 Results of seasonal Kendall (square symbols) and seasonal Kendall
corrected for serial correlation (plus symbols) on raw ANC data 69
4-19 Quarterly sulfate/ANC ratios for Clearwater Lake, Ontario,
beginning fall of 1980 70
4-20 Correlogram of sulfate/ANC ratios shown in Figure 4-19 71
4-21 Correlogram of sulfate/ANC ratios shown in Figure 4-19 after
detrending by ordinary least squares 72
VI
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4-22 Results of ANOCOV on raw sulfate/ANC ratios and on ranks of
sulfate/ANC ratios 73
4-23 Results of seasonal Kendall (square symbols) and seasonal Kendall
corrected for serial correlation (plus symbols) on raw sulfate/ANC
ratios 74
4-24 Quarterly sulfate/(calcium + magnesium) ratios for Clearwater
Lake, Ontario, beginning fall of 1980 75
4-25 Correlogram of sulfate/(calcium + magnesium) ratios shown in
Figure 4-24 76
4-26 Correlogram of sulfate/(calcium + magnesium) ratios shown in
Figure 4-24 after detrending by ordinary least squares 77
4-27 Results of ANOCOV on raw sulfate/(calcium + magnesium) ratios
and on ranks of sulfate/(calcium + magnesium) ratios 78
4-28 Results of seasonal Kendall (square symbols) and seasonal Kendall
corrected for serial correlation (plus symbols) on raw
sulfate/(calcium + magnesium) ratios 79
4-29 Results of ANOCOV on raw sulfate data with quarters redefined as
Jan.-Mar., Apr.-June, etc., rather than Dec.-Feb., etc 81
4-30 Results of seasonal Kendall (square symbols) and seasonal Kendall
corrected for serial correlation (plus symbols) on raw sulfate data
with quarters redefined as in Figure 4-29 82
4-31 Least squares regression slope of the entire sulfate series as a
function of length of record 83
4-32 Results of ANOCOV on raw ANC data with quarters redefined as
Jan.-Mar., Apr.-June, etc., rather than Dec.-Feb., etc 84
4-33 Results of seasonal Kendall (square symbols) and seasonal Kendall
corrected for serial correlation (plus symbols) on raw ANC data
with quarters redefined as in Figure 4-32 85
4-34 Least squares regression slope of the entire ANC series as a
function of length of record 86
4-35 Time series plot of annual spring ANC at Clearwater Lake, Ontario 90
4-36 Time series plot of annual spring sulfate at Clearwater Lake,
Ontario 91
vn
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4-37 Time series plot of annual fall ANC at Clearwater Lake, Ontario 92
4-38 Time series plot of annual fall sulfate at Clearwater Lake,
Ontario 93
4-39 Quarterly time series of ANC data, in mg L"1 as bicarbonate,
from Twin Lakes, Colorado, Station 2, beginning in January 1977 94
4-40 Results of ANOCOV on raw ANC and on ranks of ANC time series shown
in Figure 4-39 95
4-41 Results of Seasonal Kendall (square symbols) and Seasonal Kendall
with correction for serial correlation (plus symbols) for raw
ANC data shown in Figure 4-39 96
4-42 Least squares regression slope of the entire ANC series at
Station 2 in Twin Lakes, Colorado, as a function of number of
observations 97
4-43 Quarterly time series of ANC data, in mg L"1 as bicarbonate,
from Twin Lakes, Colorado, Station 4, beginning in January 1977 98
4-44 Results of ANOCOV on raw ANC and on ranks of ANC time series
shown in Figure 4-43 99
4-45 Results of seasonal Kendall (square symbols) and seasonal Kendall
with correction for serial correlation (plus symbols) for raw ANC
data shown in Figure 4-43 100
4-46 Least squares regression slope of the entire ANC series at
Station 4 in Twin Lakes, Colorado, as a function of number of
observations 101
Vlll
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LIST OF TABLES
Table Page
3-1 The Best Data Records for Each NLS Region 10
3-2 The 19 Best Overall Data Records, ANC 11
3-3 Regional Means and Standard Deviations for (a) Alkalinity,
(b) sulfate, and (c) pH 14
3-4 Regional Maximum to Minimum Ratios of Quarterly Means and
Standard Deviations for Alkalinity 20
3-5 Regional Maximum to Minimum Ratios of Quarterly Means and
Standard Deviations for Sulfate 21
3-6 Regional Maximum to Minimum Ratios of Quarterly Means and
Standard Deviations for pH 22
3-7 Significant Skew Values of the Best Data Sets for Alkalinity 23
3-8 Significant Skew Values of the Best Data Sets for Sulfate 23
3-9 Significant Skew Values of the Best Data Sets for pH 24
3-10 Significant Kurtosis Values of the Best Data Sets for
Alkalinity 24
3-11 Significant Kurtosis Values of the Best Data Sets for Sulfate 25
3-12 Significant Kurtosis Values for the Best Data Sets for pH 25
3-13 Significant Correlations of the Best Data Sets for Alkalinity 35
3-14 Significant Correlations of the Best Data Sets for Sulfate 35
3-15 Significant Correlations of the Best Data Sets for pH 36
3-16 Significant Correlations of the Best Data Sets for All
Variables 36
3-17 Description of Simulations for Monte Carlo Testing Program 40
4-1 Between-lake Correlations from Monthly Data for ANC (« 40
Months of Data) for 11 Lakes in NLS Subregion 1A 56
IX
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4-2 Between-lake Correlations from Monthly Data for Sulfate (« 40
Months of Data) for 11 Lakes in NLS Subregion 1A 57
4-3 Results of Trend Detection with Annual Spring Subsampling of
ANC and Sulfate at Clearwater Lake, Ontario 87
4-4 Results of Trend Detection with Annual Fall Subsampling of
ANC and Sulfate at Clearwater Lake, Ontario 88
6-1 Simulated Powers for Normal Errors after Averaging Over
Ratio of Standard Deviations, Ratio of Means, and Patterns
of Seasonality 110
6-2 Simulated Powers for Lognormal Errors after Averaging Over
Ratio of Standard Deviations, Ratio of Means, and Patterns
of Seasonality Ill
6-3 Simulated Powers for Normal Errors with p=0.2 after Averaging
Over Ratio of Standard Deviations, Ratio of Means, and Patterns
of Seasonality 112
6-4 Simulated Powers for Normal Errors with p=0.4 after Averaging
Over Ratio of Standard Deviations, Ratio of Means, and Patterns
of Seasonality 113
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ABSTRACT
Information goals for a long-term water quality monitoring program to measure the
impacts due to acid precipitation were developed using the Acid Precipitation Act of 1980 (PL
96-294, Title VII) as a basis. These broad information goals were refined to obtain statistical
hypotheses for which statistical tests could be employed as part of a data analysis plan.
Seven statistical tests were identified as capable of providing the desired information
regarding trends in individual systems. The tests were evaluated under various conditions (i.e.,
distribution shape, seasonality and serial correlation) in order to determine how well they might
perform as part of a data analysis plan. A Monte Carlo simulation approach was used to
evaluate the tests.
For annual sampling, the Kendall-tau (also known as the Mann-Kendall) test is recom-
mended. For seasonal sampling, the Seasonal Kendall or analysis of covariance (ANOCOV) on
ranks test is recommended.
XI
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SECTION 1
INTRODUCTION
A major purpose of the Acid Precipitation Act of 1980 (PL 96-294, Title VII) is to evaluate
the environmental effects of acid precipitation. To accomplish this task, it is necessary to
detect and understand the nature of trends in water quality associated with acid precipitation.
The purpose of this final report is to examine the statistical characteristics of the water quality
variables most pertinent to acidification, for example, acid neutralizing capacity (ANC), pH, and
SO4, and to use these characteristics, along with estimates of our anticipated ability to detect
temporal trends of varying magnitudes, to develop a data analysis plan. The report focuses on
the detection of trends over time and does not deal directly with causes or effects.
The long-term trend monitoring of sensitive surface waters, in addition to examining the
water quality variables ANC, pH, and SO4, focuses strictly on water populations associated with
lakes and streams sensitive to acidification. These populations of concern have been defined by
other components of the TIME project. The statistical characteristics of existing data from
similar populations serve as the basis for selecting trend analysis approaches.
1.1 GOALS OF MONITORING
Section 702b of the Acid Precipitation Act of 1980 (PL 96-294) declares that one of its
purposes is to "...evaluate the environmental effects of acid precipitation..." This broad, legal
objective has been translated into more specific and detailed information requirements as part of
the implementation of PL 96-294. The TIME goals, as published in "The Concept of Time" and
repeated in Appendix A, make up part of this translation.
The TIME goal most relevant to the detection of long-term trends can be stated as follows:
"Estimate the regional trends of surface water quality—acidification or recovery." This form of
the goal statement avoids the concept of estimating the proportion of lakes exhibiting trends
and refers strictly to the collective trend in a region. This information goal must be further
refined into a statistically meaningful statement around which a statistically sound monitoring
system can be designed. This refinement means stating the goal as a hypothesis that can be
tested, using the data as it is collected.
The null hypothesis, as noted in the documents used to develop "The Concept of Time"
report (TIME goals distributed at the Corvallis Workshop on April 28, 1987), is: "There are no
long-term regional trends in acidification or recovery of surface waters." The alternate
hypothesis is that a trend exists.
More specific monitoring objectives may be stated as follows:
a. To detect monotonic trends (generally increasing or generally decreasing over time) in
data series, both seasonal and annual, for selected water quality variables in individual
lakes at the 90% confidence level.
b. To detect monotonic trends in time series consisting of weighted averages of water
quality observations over specified groups of lakes at the 90% confidence level.
This further refinement of the monitoring goals specifies the type of trends (monotonic)
and confidence level (90%) that the data analysis plan must account for in the statistical
procedures it employs. The water quality variables utilized in the data analysis plan, as noted
earlier, are ANC, pH, and SO4. It is assumed that the refinement of the legal goal to the TIME
goal and then to the statistical hypothesis is correct and that it has been approved by those
1
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responsible for enacting and implementing PL 96-294, Title VII. The data analysis plan is based
on this refinement. This design effort does not directly encompass the other goals listed in
"The Concept of Time" report. Other individuals working on TIME are addressing these goals.
1.2 SCOPE OF REPORT
This report provides only the information described in the discussion in Section 1.1 on
monitoring goals. The recommended data analysis plan is not designed to provide information
other than that on water quality trends in individual and groups of lakes and streams impacted
by acidic deposition. Physical, practical, hydrological, and statistical concerns developed
elsewhere in the TIME studies dictate that sampling must be performed on an annual basis for a
relatively large number of sites (lakes and streams) and on a seasonal basis—two to four samples
per year—for a smaller number of sites (TIME Conceptual Plan, 1987). In this report, therefore,
sampling frequencies are assumed to be limited to a range of quarterly to annually.
A univariate time series approach is used throughout. The single variable can be the
concentration of a water quality constituent, the ratio of concentrations of two constituents, or
a weighted average of concentrations over a group of lakes or streams. There is no explicit
attempt to detect trends on a population-wide basis. Changes in populations are covered
elsewhere within the TIME project.
This study concentrates on selecting trend analysis methods that are well matched to the
statistical characteristics of TIME data series. The primary characteristics of concern are
distribution shape (normality versus non-normality), seasonal variation, and serial correlation.
At this stage there has been no attempt to incorporate the effects of hydrologic variables such
as rainfall or acidic deposition into the recommended trend analysis procedures. The single
exception to this is that flow correction is recommended as part of trend analysis in streams.
Neither has there been an effort to explicitly account for inter-station correlation. The effect
of correlation between stations is, however, briefly discusssed in section 4.2, Multiple Lakes.
1.3 OVERVIEW OF STUDY
The Acid Precipitation Act of 1980 has been reviewed and its goals, as stated earlier, have
been identified. Ultimate users of the information to be derived from the monitoring program
have agreed with these goals. Their approval is critical, as alternative formulations of the
problem could lead to different data analysis plans. The National Research Council (1977, p. 32)
notes the need to clearly formulate monitoring purposes and criteria that are mutually under-
stood by the information users and the monitoring system designers and operators.
With the monitoring goals defined, the next step was to select data analysis procedures
that would provide the required information in a statistically sound manner. In order to select
statistical tests for the data analysis plan that would be well matched to both the goals and the
anticipated data attributes, background data from several sources were studied, including the
Long-term Monitoring (LTM) data set described in Newell et al. (1987), data from Environment
Canada for Clearwater Lake, Ontario, and data from the U.S. Bureau of Reclamation for Twin
Lakes, Colorado. From these data, we were able to infer the level of seasonal behavior, serial
correlation, and non-normality that could be anticipated from TIME data.
The conclusions of this study were that TIME data can be expected to be seasonal in both
mean and standard deviation, to be normally distributed in some cases and non-normally in
others, and to occasionally exhibit low level serial correlation for quarterly observations. No
conclusions regarding serial correlation of annual values were drawn.
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In view of these data characteristics, we selected and evaluated several candidate tests for
detecting trend. These tests included both parametric and nonparametric approaches. Several
options for dealing with seasonality were included, and one test included a correction for serial
correlation. The candidate trend tests were:
a. Analysis of covariance (ANOCOV)
b. Modified t-test
c. Kendall-tau following removal of seasonal means
d. Seasonal Kendall (SK)
e. Seasonal Kendall with serial correlation correction
f. ANOCOV on ranks
g. Modified "t" on ranks
We evaluated the candidate tests by comparing their performance under a Monte Carlo
simulation study designed to reproduce the data characteristics anticipated from TIME data. The
performance indices were actual significance level and power of trend detection. Based on
Monte Carlo results, we recommend a single trend test for annual data and two tests for
seasonal data.
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SECTION 2
RECOMMENDATIONS
For annual sampling, the recommended test is the Kendall-tau, sometimes called the
Mann-Kendall test for trend. The Mann-Kendall test is nonparametric and is a member of the
class of tests called rank correlation methods, meaning that the test checks for a correlation
between the ranks of data and time. The test is well known and is frequently recommended for
use in water quality trend analysis (Gilbert, 1987). The test does not account for seasonal
variation. Since, however, it is recommended for use with annual data only, no prior removal of
seasonal means is necessary.
For seasonal (generally quarterly) sampling, two alternative tests are recommended:
analysis of covariance (ANOCOV) on ranks or the seasonal Kendall test. Both tests are
nonparametric. and both tests performed very well under most of the conditions studied in the
Monte Carlo analysis, i.e., seasonal variation and both normal and lognormal noise. Neither test
performed well when observations were serially correlated. The only test that accounted for
serial correlation, the "corrected" version of seasonal Kendall, exhibited low power compared to
the other tests. It seemed to us that the reduction in power was too high a price to pay for
insensitivity to serial correlation. Therefore, we do not recommend the corrected test, except
for very long data records. This logic is discussed further in sections 3.1, 3.4, and 3.5, which
provide more information on statistical methods and Monte Carlo evaluations.
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SECTION 3
JUSTIFICATION OF RECOMMENDATIONS
3.1 MONITORING OBJECTIVE IN STATISTICAL TERMS
The goal of monitoring relevant to the trend detection portion of TIME is to determine
whether general increases or decreases in observed values of water quality variables are
statistically significant—as opposed to being the coincidental result of random or natural
variability. The term "trend detection" might therefore be somewhat misleading. It is not
generally possible for statistics to detect trends that are not apparent by inspection, especially
for data records of short to moderate length—say 20 years or less. It is preferable to think of
trend tests as a quantitative basis for deciding whether apparent trends are real.
Therefore in statistical terms, the objective of monitoring (and subsequent trend analysis)
is to accept the null hypothesis of no trend with specified (high) probability when no real trend
exists and to reject the null hypothesis with high probability when real trends do exist.
The definition of a "real" trend is somewhat subjective. For the purposes of this study,
trend is defined as a general increase or decrease in observed values of some random variable
over time. A real trend is one that results from physical or chemical changes, not from natural
hydrologic variability. In the present case, a "real" change would be one caused by acidic depo-
sition resulting from air pollution or by changes in acidic deposition rates resulting from
increasing or decreasing air pollution. Changes in water quality resulting from natural varia-
bility in precipitation patterns would not be considered "real" for the purposes of this study.
In view of this definition of a real change or trend, it is necessary to acknowledge a
significant limitation of the recommended statistical tests of trend. At this stage of the TIME
project, there is no attempt to directly relate hydrologic factors such as precipitation or acidic
deposition to water quality within the framework of routine data analysis, i.e., annual reports.
The recommended methods consider water quality variables individually and address the
following question: "Given the observed variability in a set of observations, what is the
probability that an observed pattern (of increases or decreases) resulted from a no-trend
situation?" If this probability is higher than some prespecified level, called the significance
level, the null hypothesis of no trend is accepted. If the probability is lower than the
significance level, the null hypothesis is rejected.
This limited univariate approach should be sufficient for routine (annual) reporting.
However, for selected subregions or watersheds, a more thorough level of trend analysis will be
undertaken by TIME. This will include modeling of hydrologic-chemical relationships in an
attempt to (1) explain trends that routine analyses have shown to be significant and (2) reduce
background variability of water quality in order to improve the ability of trend detection tests
to reject the null hypothesis when trends exist.
For clarity of future discussion, let us formally define three terms as follows:
a. "Trend" is a general increase or decrease in the value of observations of a particular
variable over time. For the purpose of comparing statistical tests, trends are later
assumed to be monotonic (one directional) and gradual (linear) for simulation purposes.
For the overall TIME project, however, the concept of trends is more general and
need not be limited to monotonic, linear, or even gradual trends. The procedures
recommended for data analysis are appropriate under the more general concept.
However, test performance in terms of ability to reject the null hypothesis depends on
the nature of real trends being examined.
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b. The "significance level" of a test, denoted by a, is the probability of rejecting the
null hypothesis of no trend when it is true. The significance level is also referred to
as the type I error. The "nominal significance level" of a test is the rejection proba-
bility when all of the assumptions underlying the test are satisfied and the test sta-
tistic follows its theoretical distribution. The nominal significance level is usually
specified before a test is run. In practice, the assumptions associated with a given
test are not satisfied exactly, and the true or "actual significance level" will be
different from the nominal level. In water quality monitoring, the assumptions under-
lying tests may be seriously violated, and the actual significance level may be quite
different from the nominal level. Since our knowledge of the variables being moni-
tored is quite limited, however, the actual significance level is never known. We can
only minimize the difference between nominal and actual levels through the use of
tests based on assumptions that closely match the data characteristics and/or tests
insensitive ("robust") to violations of underlying assumptions. In future discussion,
the terms "significance level" and "nominal significance level" will be used inter-
changably, whereas the modifier "actual" will be explicitly included when it is needed.
The term "confidence level" is defined as 1-a, where a is the nominal significance
level. Recommended pre-test confidence levels for the data analysis plan are 90% and
95%, using two-sided tests.
c. The "power" of a test is defined as the probability of rejecting the null hypothesis
when a real trend exists. Power may also be defined as l-£, where ft is the Type II
error or probability of accepting the null hypothesis when it is false (when a real
trend exists).
It would be ideal to be able maximize the power of a test and minimize the signifi-
cance level. Unfortunately, there is a direct relationship between the two. For given
population characteristics and given sample size, the power of a given test decreases
with decreasing significance level. Thus, we are forced to make do with some sort of
trade-off between the two (at least in the usual situation where both the resources
and direction of a monitoring program limit the effective independent sample size
available for testing).
The power of a test also depends on the nature of the trend that really occurs. By
"nature" of a trend, we mean the functional form (gradual, sudden, linear, polynomial,
monotonic sinusoidal, etc.), the magnitude and duration, and the population changes,
such as changing variance, that accompany or are considered part of the trend.
In ecosystem monitoring, the possibilities for the nature of trend are endless. There is,
therefore, no way to specify the power of a given test in the real world. The best we can do
is to consider a few types of trend (rigidly specifying functional form, magnitude, duration, and
accompanying population changes for each) that roughly represent the real world possibilities.
Using these hypothetical trend models, we can then identify the power of a given test under
those trend models and use the results to objectively compare the performance of alternative
statistical tests. We can also use models to represent the behavior of water quality under
no-trend conditions and use these to compare the empirical (actual) significance levels of
candidate statistical tests.
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This approach, limited simulation of water quality random variables under varying trend
magnitudes and assumed behavioral characteristics, was used to compare alternative trend tests.
Recommendations were formulated based on a comparison of empirical significance levels and the
power of candidate tests.
3.2 CHARACTERISTICS OF BACKGROUND OR HISTORICAL DATA
3.2.1 Description of Historical Data Sets
Historical data were used to establish a range of statistical properties that might be
expected from TIME data and to provide case studies demonstrating the application of recom-
mended statistical tests. Data were obtained from three sources: (1) LTM data (Newell et al.,
1987; Newell, 1987), (2) Twin Lakes data (Sartoris, 1987), and (3) Clearwater Lake data (Nicholls,
1987).
The LTM data were collected in a variety of studies that used differing sampling and lab-
oratory analysis techniques (Newell et al., 1987). Data from these different studies are therefore
not comparable, but since we are interested only in general descriptions of statistical behavior
at this point, the disparities are not of concern.
Table 3-1 lists the "best" LTM records in each National Lake Survey (NLS) region. Selec-
tion is based on length and completeness of record. Table 3-2 presents the 16 overall best LTM
lakes and adds Twin Lakes and Clearwater Lakes for a total study data set of 19 lakes. The
variables of primary interest are alkalinity or acid neutralizing capacity (ANC), sulfate
concentration, and pH. For Twin Lakes, sulfate data were not available, and conductivity data
were used instead.
3.2.2 Results of Characterization
3.2.2.1 Seasonality--
Several researchers have observed significant seasonal variation in lake water quality.
There are also hydrologic reasons to assume that lake water quality often varies seasonally.
The purpose of background data analysis was to establish the magnitude of seasonal changes that
we might expect in both the mean and standard deviation of the water quality in TIME series of
interest.
Before focusing on the 19 selected lakes in the study data set, researchers considered all
of the LTM data from regions 1A, 1C, 3A, IE, and 2 and streams. Regional mean values for
ANC, SO4=, and pH were computed for each season along with regional standard deviations by
season. Standard deviations were computed using deviations from the individual lake means.
Seasons were defined as quarters, starting with winter, consisting of December, January,
February.
The results are presented in Table 3-3 (a,b,c). Observe that variation in both means and
standard deviations across the four seasons ranges from minimal to very large. All three
variables show obvious seasonality in at least one region. Alkalinity shows more seasonality
than sulfate. The ratio of maximum to minimum ranges from just over 1.0 up to 5.0 or more for
both quarterly means and quarterly standard deviations in both ANC and SO4=. Negative ratios
are possible for ANC. The idea that one season might be identified as having consistently lower
variance does not seem to be supported.
-------
TABLE 3-1. THE BEST DATA RECORDS FOR EACH NLS REGION
The number of records for each lake was found using the ANC data for a series of quarterly
data and is given by: (Number of nonmissing points - number of missing points).
NLS-IE (Main)
Best 3:
NSL-3A (Southern Blue Ridee:
NC. TN. GA)
Best 3:
1E1-132
1E1-133
1E1-135
(9-2)
(9-2)
(9-2)
3A1-010
3A2-066
3A3-104
(8-3)
(9-4)
(8-3)
NLS-1A (Adirondacks: NYV
Best 11:
NLS-1C (NH. VT)
Best 5:
1A1-071
1A1-087
1A1-102
1A1-105
1A1-106
1A1-107
1A1-109
1A1-110
1A1-113
1A2-077
1A2-078
NLS-2 (Upper
MN. WI. MD
Best 3:
2A2-065
2A3-005
2C 1-029
(15-0)
(15-0)
(15-0)
(15-0)
(15-0)
(15-0)
(15-0)
(15-0)
(15-0)
(15-0)
(15-0)
Midwest:
(16-13)
(16-13)
(11-4)
1C1-091
1C1-093
1C1-097
1C1-064
1C3-075
LTM Streams
Best 7:
01434010
0210108450
0210289715
0213299630
03039420
03079520
03079700
(15-0)
(19-0)
(18-1)
(17-0)
(19-0)
(Southeastern U.S.)
(10-0)
(10-0)
(10-0)
(10-0)
(12-0)
(10-0)
(10-0)
Region NLS-1A also has a record of (40-3) with monthly data for each of the best data sets.
10
-------
TABLE 3-2. THE 19 BEST OVERALL DATA RECORDS, ANC
NLS - 1C
1C1-091 (15-0)
1C1-093 (19-0)
1C1-097 (18-1)
1C3-064 (17-0)
1C3-075 (19-9)
NLS 1A1
1A1-071 (15-0)
1A1-087 (15-0)
1A1-102 (15-0)
1A1-105 (15-0)
1A1-106 (15-0)
1A1-107 (15-0)
1A1-109 (15-0)
1A1-110 (15-0)
1A1-113 (15-0)
1A2-077 (15-0)
1A2-078 (15-0)
TWIN2
Site 2 (35-0)
Site 4 (35-0)
CLEARWATER2
(26-0)
1 Region NLS-1A also has a record of (40-3) with monthly data for each of the best data sets.
2 Twin and Clearwater Lakes were sampled more freqently than quarterly. Quarterly values
were obtained by choosing the observation closest to the middle of the quarter.
11
-------
TABLE 3-3. REGIONAL MEANS AND STANDARD DEVIATIONS
(a) Regional Means and Standard Deviations for Alkalinity (/zeq L"1)
Region
NLS-1A
NLS-1C
NLS-3A
NLS-1E
NLS-2
Streams
Season1
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
Mean
32.9
23.0
13.7
48.7
38.9
49.3
59.7
27.4
41.0
52.6
92.7
81.4
87.9
106.3
15.3
13.8
13.8
22.0
51.5
47.9
54.6
52.4
56.0
71.9
33.1
57.4
71.0
Stand. Dev.
29.8
11.4
16.9
27.8
28.5
17.8
19.3
7.2
9.3
11.2
21.4
14.9
21.4
10.2
7.0
3.5
6.0
5.0
20.3
25.5
9.7
11.9
44.0
52.4
13.0
23.7
39.2
Total Obs.2
242
57
48
75
62
448
113
86
123
117
73
0
24
14
26
44
0
15
15
10
292
0
116
84
89
162
23
21
21
61
1 The seasons are defined as follows: (1) December, January, and February, (2) March, April,
and May, (3) June, July, and August, (4) September, October, and November.
2 The number of observations for each of the seasons may not add up to the total number of
observations because of the short data records. Records with only one observation for a
season were not used for computing the seasonal means and standard deviations.
12
-------
TABLE 3-3. REGIONAL MEANS AND STANDARD DEVIATIONS (Continued)
(b) Regional Means and Standard Deviations for Sulfate
Region
NLS-1A
NLS-1C
NLS-3A
NLS-1E
NLS-2
Streams
Season1
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
Mean
121.6
131.7
119.2
115.1
121.8
112.4
123.6
104.3
108.4
111.4
38.8
39.9
40.5
38.7
70.2
69.7
69.0
70.6
84.5
85.5
83.8
86.5
95.5
159.9
155.4
158.1
89.8
Stand. Dev.
12.4
7.1
5.2
8.8
10.6
14.5
10.6
12.0
10.7
11.9
7.0
4.4
5.1
6.4
3.4
3.6
3.2
1.8
10.7
8.9
11.6
9.0
28.4
9.3
9.8
6.0
34.3
Total Obs.2
257
60
51
80
66
449
114
87
121
117
82
0
26
16
30
44
0
14
15
10
332
0
138
88
101
153
17
17
17
55
1 The seasons are defined as follows: (1) December, January, and February, (2) March, April,
and May, (3) June, July, and August, (4) September, October, and November.
2 The number of observations for each of the seasons may not add up to the total number of
observations because of the short data records. Records with only one observation for a
season were not used for computing the seasonal means and standard deviations.
13
-------
TABLE 3-3. REGIONAL MEANS AND STANDARD DEVIATIONS (Continued)
(c) Regional Means and Standard Deviations for pH
Region
NLS-1A
NLS-1C
NLS-3A
NLS-1E
NLS-2
Streams
Season1
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
All
1
2
3
4
Mean
5.77
5.52
5.51
6.03
5.88
5.92
5.78
5.70
6.02
6.05
6.51
6.40
6.78
6.42
5.82
5.75
5.88
5.89
6.08
6.07
6.16
5.99
5.38
5.75
5.69
6.09
5.45
Stand. Dev.
0.395
0.200
0.286
0.315
0.303
0.306
0.191
0.260
0.212
0.278
0.421
0.369
0.243
0.340
0.140
0.127
0.113
0.069
0.230
0.197
0.180
0.152
0.334
0.184
0.277
0.159
0.347
Total Obs.2
257
60
51
80
66
418
85
90
119
114
80
0
25
14
30
45
0
15
15
10
344
0
141
95
106
168
25
21
21
52
1 The seasons are defined as follows: (1) December, January, and February, (2) March, April,
and May, (3) June, July, and August, (4) September, October, and November.
2 The number of observations for each of the seasons may not add up to the total number of
observations because of the short data records. Records with only one observation for a
season were not used for computing the seasonal means and standard deviations.
14
-------
Figures 3-1 through 3-4 present ratios of maximum to minimum seasonal means and
standard deviations for selected lakes in NLS subregions 1A and 1C. Tables 3-4, 3-5, and 3-6
present composite results for the same two regions and for Twin Lakes. These results are
exemplary of the entire background data set, showing ANC as the most seasonal variable
followed by sulfate. Maximum to minimum ratios range generally from about 1.0 to 2.0 for
quarterly means and from 1.0 to 5.0 for standard deviations. Thus it appears that seasonal
variation ranges from none to the case where the maximum quarterly mean and/or standard
deviation is two to five times the minimum.
Due to small record length, no attempt was made to show that seasonally is statistically
significant. However, a reasonable range of seasonality has been established for use in
modeling.
There does not appear to be a consistent pattern or ordering of low or high values for any
variable in either the mean or standard deviation.
3.2.2.2 Normality-
Data records from the "best" lakes were checked for normality using tests for both
skewness and kurtosis. The skewness test determines whether the distribution is symmetrical
about the mean. Data sets that have sample skewness coefficients differing significantly from
zero (either positive or negative) are judged to be non-normal.
The sample kurtosis determines whether the distribution is too flat or too peaked compared
to the normal distribution. Sample kurtosis values that differ significantly from 3.0 are judged
to come from non-normal populations. From these results, the hypothesis of normality can be
rejected on only about 20% of the two-tailed tests when they are applied at significance levels
of 10% and 2% (5% and 1% for each tail).
Results are presented in Tables 3-7, 3-8, and 3-9 for skewness tests, and in Tables 3-10,
3-11, and 3-12 for kurtosis tests on raw data, log transformed data, and data with quarterly
means removed. In general, the log transformation and removal of quarterly means did not
increase or decrease the number of data records that appeared to be normal.
Although most of the records studied appeared to come from a normal distribution, a
blanket assumption of normality for TIME monitoring would be unwise. Several variables do not
appear to be normal, and many other studies of water quality random variables have shown that
non-normality is frequently encountered. Thus, in general studies that, like TIME, cover diverse
hydrologic conditions, we should be reluctant to place confidence in a normality assumption.
3.2.2.3 Serial Correlation--
Autocorrelation of a time series represents a carry-over of information from one obser-
vation to the next. Positive autocorrelation is the tendency of high values to follow high values
or low values to follow low values. Negative autocorrelation is the tendency of high values to
follow low values and vice versa. Autocorrelation in the absence of seasonality or trend is
referred to as serial correlation. Both seasonality and serial correlation make trend detection
more difficult.
The autocorrelation coefficient p(k) and its sample estimate r(k) range from +1 to -1.
Correlograms or plots of the sample autocorrelation function, r(k), are useful as tools to check
for seasonality and serial correlation. Figures 3-5 through 3-12 present correlograms for
selected variables and lakes.
15
-------
E
a
E
'c.
2
•x
o
n
o
V
1.7 •
1.6
1.5 •
1.4
1.3
1.2
1.1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
O.2
O.1
0
NLS 1A
Selected Best Lakes
^
/,$
I
^N
/S3
^
/,\
i
&
i
1
i
g
XN
l/,\
§
n
/,\
IZ23
PH
Figure 3- 1 . Seasonal variation in quarterly means for SO4~ and pH for selected lakes in NLS
subregion 1A.
16
-------
NLS 1C
Selected Best Lakes
E
3
E
x
o
n
c
o
W
IZZ) S04
PH
Figure 3-2.
Seasonal variation in quarterly means for SO4= and pH for selected lakes in NLS
subregion 1C.
17
-------
E
'c
is
x
o
2
n
c
o
V
Q
TJ
a
TJ
c
o
15 -
14 -
13 -
12 -
11 -
10 -
9 -
8 -
7 -
6 -
5 -
4 -
3 -
2
i ^
O
/
2
\
NLS 1A
Selected Best Lakes
y
ALK
S04 V77X PH
Figure 3-3. Seasonal variation in quarterly standard deviations for alkalinity, SO4~, and pH
for selected lakes in NLS subregion 1A.
18
-------
E
D
E
E
3
E
'x
o
m
c
o
Q
T)
L.
O
T3
C
O
•*-»
V)
NLS 1C
Selected Best Lakes
ALK
S04
PH
Figure 3-4. Seasonal variation in quarterly standard deviations for alkalinity, SO4=, and pH
for selected lakes in NLS subregion 1C.
19
-------
TABLE 3-4. REGIONAL MAXIMUM TO MINIMUM RATIOS OF QUARTERLY MEANS
AND STANDARD DEVIATIONS FOR ALKALINITY
Region
Lake
Mean
Stand. Dev.
NLS-1A
NLS-1C
Twin Lakes
Overall
1A1-071
1A1-087
1A1-102
1A1-105
1A1-106
1A1-107
1AI-109
1A1-110
1A2-077
1A2-078
Overall
1C1-091
1C1-093
1C1-097
1C3-064
1C3-075
Lower
Upper
3.70
1.21
4.91
1.74
2.74
-1.02
0.75
1.89
5.33
1.37
-7.96
2.18
1.51
-1.20
2.00
1.59
1.64
1.08
1.16
2.49
3.36
2.82
3.94
5.55
2.84
2.94
3.19
2.90
15.89
2.88
2.69
2.22
1.80
4.04
4.21
1.83
1.57
1.71
Note: The ratios of the means for alkalinity may not be representative of the system because
the value of alkalinity can be negative. Therefore, negative ratios are also possible.
20
-------
TABLE 3-5. REGIONAL MAXIMUM TO MINIMUM RATIOS OF QUARTERLY MEANS
AND STANDARD DEVIATIONS FOR SULFATE
Region
NLS-1A
NLS-1C
Lake
Overall
1A1-071
1A1-087
1A1-102
1A1-105
1A1-106
1A1-107
1A1-109
1A1-110
1A2-077
1A2-078
Overall
1C1-091
1C1-093
1C1-097
1C3-064
1C3-075
Mean
1.14
1.01
1.66
1.16
1.13
1.09
1.16
1.08
1.11
1.14
1.20
1.19
1.20
1.20
1.06
1.21
1.30
Stand. Dev.
2.03
2.85
3.05
2.23
1.89
1.80
3.20
3.07
5.18
3.27
2.81
1.13
2.23
1.98
2.28
1.47
4.00
21
-------
TABLE 3-6. REGIONAL MAXIMUM TO MINIMUM RATIOS OF QUARTERLY MEANS
AND STANDARD DEVIATIONS FOR pH
Region
NLS-1A
NLS-1C
Twin Lakes
Lake
Overall
1A1-071
1A1-087
1A1-102
1A1-105
1A1-106
1A1-107
1A1-109
1A1-110
1A2-077
1A2-078
Overall
1C1-091
1C1-093
1C1-097
1C3-064
1C3-075
Lower
Upper
Mean
1.09
.07
.15
.07
.11
.09
.04
.13
1.16
1.07
1.18
1.06
1.13
1.06
.08
.08
.07
.03
.05
Stand. Dev.
1.58
4.02
1.73
4.37
2.70
2.21
2.55
9.91
2.39
2.93
2.81
1.45
6.71
2.34
4.43
1.80
4.00
3.62
1.70
22
-------
TABLE 3-7. SIGNIFICANT SKEW VALUES OF THE BEST DATA SETS FOR
ALKALINITY
Region
NLS-1A
NLS-1C
Twin Lakes
Overall
Level
10%
2%
10%
2%
10%
2%
10%
2%
Raw
Data
2/10
1 / 10
1 /5
0/5
0/2
0/2
3/ 17
1 / 17
Logarithmic
Transform*
4/5
3/5
0/4
0/4
0/2
0/2
4/11
3/ 11
Quarterly
Means
Removed
2/10
0/10
1 /5
0/5
0/2
0/2
4/17
3/17
Note: The format of the table is (# significant values / # total observations), therefore 2/10
means that 2 of the 10 data sets tested were significantly skewed at that particular level.
* Because of negative alkalinity values, logarithmic transformations could not be performed on
some of the data sets.
TABLE 3-8. SIGNIFICANT SKEW VALUES OF THE BEST DATA SETS FOR SULFATE
Region
NLS-1A
NLS-1C
Overall
Level
10%
2%
10%
2%
10%
2%
Raw
Data
4/10
1 / 10
1 15
1 /5
5f 15
2/ 15
Logarithmic
Transform
2/10
1 / 10
1 / 5
1 / 5
3 / 15
2/15
Quarterly
Means
Removed
2/10
1 / 10
1 /5
1 15
3/ 15
2/ 15
Note: The format of the table is (# significant values / # total observations), therefore 2/10
means that 2 of the 10 data sets tested were significantly skewed at that particular level.
23
-------
TABLE 3-9. SIGNIFICANT SKEW VALUES OF THE BEST DATA SETS FOR pH
Region
NLS-1A
NLS-1C
Twin Lakes
Overall
Level
10%
2%
10%
2%
10%
2%
10%
2%
Raw
Data
2/10
1 / 10
1 /5
1 /5
0/2
0/2
3/ 17
2/17
Logarithmic
Transform
2/10
1 / 10
1 /5
1 /5
0/2
0/2
3/ 17
2/ 17
Quarterly
Means
Removed
0/10
0/10
1 /5
1 /5
0/2
0/2
1 / 17
1 / 17
Note: The format of the table is (# significant values / # total observations), therefore 2/10
means that 2 of the 10 data sets tested were significantly skewed at that particular level.
TABLE 3-10
Region
NLS-1A
NLS-1C
Twin Lakes
Overall
. SIGNIFICANT
Level
10%
2%
10%
2%
10%
2%
10%
2%
KURTOSIS VALUES OF THE BEST DATA SETS FOR ALKALINITY
Raw
Data
2/10
1 / 10
1 /5
0/5
0/2
0/2
3/ 17
1 / 17
Logarithmic
Transform*
4/5
3/5
1 /4
0/4
0/2
0/2
5/ 11
3/ 11
Quarterly
Means
Removed
3/10
1 / 10
1 /5
0/5
0/2
0/2
4/17
1 / 17
Note: The format of the table is (# significant values / # total observations), therefore 2/10
means that 2 of the 10 data sets tested were significantly skewed at that particular level.
* Because of negative alkalinity values, logarithmic transformations could not be performed on
some of the data sets.
24
-------
TABLE 3-11. SIGNIFICANT KURTOSIS VALUES OF THE BEST DATA SETS FOR SULFATE
Region
NLS-1A
NLS-1C
Overall
Level
10%
2%
10%
2%
10%
2%
Raw
Data
3/10
0/10
1 /5
1 /5
4/15
1 / 15
Logarithmic
Transform
2/10
0/10
1 /5
1 /5
3/15
1 / 15
Quarterly
Means
Removed
2/10
0/10
1 /5
1 /5
3/ 15
1 / 15
Note: The format of the table is (# significant values / # total observations), therefore 2/10
means that 2 of the 10 data sets tested were significantly skewed at that particular level.
TABLE 3-12. SIGNIFICANT KURTOSIS VALUES OF THE BEST DATA SETS FOR pH
Region
NLS-1A
NLS-1C
Twin Lakes
Overall
Level
10%
2%
10%
2%
10%
2%
10%
2%
Raw
Data
2/10
1 / 10
1 /5
1 /5
0/2
0/2
3/ 17
2/ 17
Logarithmic
Transform*
2/10
2/10
1 /5
0/5
0/2
0/2
3/ 17
2/17
Quarterly
Means
Removed
3/10
1 / 10
1 /5
0/5
0/2
0/2
4/17
1 / 17
Note: The format of the table is (# significant values / # total observations), therefore 2/10
means that 2 of the 10 data sets tested were significantly skewed at that particular level.
25
-------
-1
12
li
18
a
Figure 3-5. Correlogram for conductivity at Upper Twin Lake, Colorado--quarterly sampling,
raw data.
26
-------
I I
-1
3' 6'
4 i'z
1^5 l'fl ?'l ?i
1J 10 bl b'
Figure 3-6. Correlogram for conductivity at Upper Twin Lake, Colorado- -quarterly sampling,
seasonal means removed.
27
-------
-i
l'2
ii
18
2i
A
Figure 3-7. Correlogram for conductivity at Lower Twin Lake, Colorado—quarterly sampling,
raw data.
28
-------
I I . I
-I
12
ii
2T1
Figure 3-8. Correlogram for conductivity at Lower Twin Lake, Colorado--quarterly sampling,
seasonal means removed.
29
-------
tt
14
Figure 3-9. Correlogram for alkalinity at Lake 1A1-105, quarterly sampling, raw data.
30
-------
-1
? 2 ? 15 S
Figure 3-10. Correlogram for alkalinity at Lake 1A1-105, quarterly sampling, seasonal means
removed.
31
-------
-i-
41
12
14 ii
Figure 3-11. Correlogram for alkalinity, Lake 1A1-102, quarterly sampling, raw data.
32
-------
14
it
Figure 3-12. Correlogram for alkalinity, Lake 1A1-102, quarterly sampling, seasonal means
removed.
33
-------
The vertical lines represent the amount of correlation r(k) for each lag, k. If any of the
vertical lines cross either of the two outer horizontal lines, then correlation at that lag is
statistically significant at the 95% level. Significant values of the correlogram can result from
seasonality, serial correlation, or trend. The probability that a significant value results from
random chance is less than 5%. The correlation between one observation and the next is repre-
sented by r(l). For a trend free series, a significant lag 1 correlation and a decay for lags
following represents serial correlation.
The value of r(2) is the correlation between every second observation. For quarterly data,
this would be the correlation between adjacent springs and falls, and summers and winters. Lag
4 would then be the correlation between every fourth observation, or annual correlation for
quarterly data. A periodic cycle of a negative lag 2, a positive lag 4, a negative lag 6, and so
on indicates an annual seasonal cycle within the data set.
If the correlogram of the raw data shows an annual cycle and the correlogram of data with
quarterly means removed shows steady decay, we would conclude that both seasonality and serial
correlation are present.
Figure 3-5, depicting conductivity at Upper Twin Lake, Site 4, shows a dramatic annual
cycle. In Figure 3-6, the quarterly means have been removed, and we are left with an uncor-
related series. Figure 3-7 is a correlogram for conductivity at Lower Twin Lakes, Site 2.
Seasonality is not apparent, but serial correlation is significant. As we would expect, removing
the quarterly means does not greatly affect the correlogram (Figure 3-8). Figure 3-9, depicting
alkalinity at NLS-1A Lake 1 Al-105 is based on a smaller data set and shows significant but less
dramatic seasonality. Removing quarterly means (Figure 3-10) produces a series with no signifi-
cant serial correlation. For Lake 1A1-102, the alkalinity correlogram (Figure 3-11) shows no
significant serial correlation, but apparent seasonality is present. Removing quarterly means
reveals significant serial correlation of the residual series (Figure 3-12).
Tables 3-13 through 3-16 list the number of lakes (by region and variable) that showed
significant serial correlation of various types.
3.3 ALTERNATE METHODS FOR TREND ANALYSIS
Seven statistical tests for trend were selected as candidates for evaluation and possible use
within TIME. The selection was based on the results of background data characterization and
on a review of the statistical and hydrological literature. The candidate procedures were as
follows:
a. Analysis of covariance (ANOCOV) on raw data
b. Modified "t" on raw data
c. Kendall tau on deseasonalized data (also called the Kendall Rank Correlation test or
Mann-Kendall test for trend)
d. Seasonal Kendall with correction for serial correlation
e. Seasonal Kendall
f. Analysis of covariance (ANOCOV) on ranks of data
g. Modified "t" on ranks of data
A brief discussion of these procedures follows. A detailed description of the tests is
presented in Section 6.
34
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TABLE 3-13. SIGNIFICANT CORRELATIONS OF THE BEST DATA SETS FOR ALKALINITY
Region
NLS-1A
NLS-1C
Twin Lakes
Overall
Number
Lakes
10
5
2
17
Raw Data
Significant
Lag 1
0
0
2
2
Detectable
Seasonality
9
4
0
13
Significant
Seasonality
4
2
0
6
Deseason.
Significant
Lag 1
3
1
2
6
TABLE 3-14. SIGNIFICANT CORRELATIONS OF THE BEST DATA SETS FOR SULFATE
Region
Raw Data
Number Significant
Lakes Lag 1
Detectable
Seasonality
Significant
Seasonality
Deseason.
Significant
Lag 1
NLS-1A
NLS-1C
Overall
10
5
15
0
0
0
6
3
9
2
1
3
1
1
2
35
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TABLE 3-15. SIGNIFICANT CORRELATIONS OF THE BEST DATA SETS FOR pH
Region
Number
Lakes
Raw Data
Significant
Lag 1
Detectable
Seasonality
Significant
Seasonality
Deseason.
Significant
Lag 1
NLS-1A
NLS-IC
Twin Lakes
Overall
10
5
2
17
0
2
0
2
9
2
1
12
4
1
1
6
0
1
1
2
TABLE 3-16. SIGNIFICANT CORRELATIONS OF THE BEST DATA SETS FOR ALL
VARIABLES
Region
Number
Lakes
Raw Data
Significant
Lag 1
Detectable
Seasonality
Significant
Seasonality
Deseason.
Significant
Lag 1
NLS-1A
NLS-IC
Twin Lakes
Overall
30
15
4
49
0
2
2
4
24
9
1
34
10
4
1
15
4
3
3
10
36
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Analysis of covariance (ANOCOV) is based on a linear model and normal theory (Neter and
Wasserman, 1974). The trend test is simply multiple linear regression of a water quality
(dependent) variable against two or more predictive (independent) variables. One of the
dependent variables is time, and the rest are seasonal indicator variables. For quarterly
observations, three indicator variables are used, corresponding to any three of the four
seasons—for example, winter, spring, and summer. To indicate a winter observation, the first
indicator variable would be set equal to one and the rest equal to zero. For spring observa-
tions, the second indicator variable would equal one, the rest zero. If all three indicator
variables are zero, a fall observation is indicated.
The regression calculates three seasonal or coefficent terms, an overall intercept, and a
slope against time. The slope against time is tested for significance using a null hypothesis that
the slope is zero. If the null hypothesis is rejected, we conclude that there is a significant
(linear) trend in the data.
The ANOCOV procedure assumes homogeneous (constant) variance across all seasons. Since
TIME data are expected to exhibit seasonal changes in variance, however, another test was
developed that does not assume homogeneous variance. This test, called the modified "t",
involves a separate linear regression of the water quality variable against time in each season.
The regressions are followed by a test of the null hypothesis that the sum of the regression
slopes (four slopes for quarterly data) is equal to zero. If all slopes are assumed to be in the
same direction, this condition is satisfied only if there is not overall (linear) trend. If the null
hypothesis is rejected, we conclude that there is a significant overall linear trend.
The Kendall-tau procedure is described in Snedecor and Cochran (1980). The method is
nonparametric, meaning that it does not depend on an assumption of a particular underlying
distribution. The procedure tests for correlation between the ranks of data and time, and as
noted by Gilbert (1987), "can be viewed as a nonparametric test for zero slope of the linear
regression of time-ordered data versus time, as illustrated by Hollander and Wolfe (1973, p.
201)." Since the test depends only on relative magnitudes of data rather than actual values, it
may also be viewed as a test for general monotonic trends rather than specifically linear trends.
Seasonal variation should be removed from a data series prior to the application of this
test. This is accomplished by computing seasonal means (for example, the sample mean of all
fall values) and subtracting the appropriate seasonal mean from each observation. The procedure
utilized herein is to subtract seasonal means without any prior test for seasonality. In other
words, all data series are assumed to be seasonal and are "deseasonalized" prior to applying
Kendall-tau. Of course, for annual data in which all observations are from the same time of
year, the "deseasonalizing" step is not necessary.
The seasonal Kendall test as described in Hirsch et al. (1982) is an extension of the
Kendall-tau test. The seasonal Kendall test statistic is the sum of Kendall-tau statistics
computed for each season (month or quarter, for example) of the year. This test accounts for
seasonal variation directly and does not require prior removal of seasonal means.
The sixth and seventh procedures are identical to the first and second with one exception.
The data are first ranked, and the ranks are substituted for the original values of the observa-
tions. The rank transformation is suggested by Conover (1980). The fourth and fifth procedures
are identical with one exception. In the fourth procedure, the variance of the seasonal Kendall
test statistic is corrected by including a covariance term which reflects serial correlation of
observations. The modification is described in Hirsch and Slack (1984).
Two key assumptions of the various tests should be emphasized at this point. First is the
form of trend assumed. The analysis of covariance model assumes a linear trend component, and
the modified "t" assumes a linear trend within each season. Although the tests are certainly
37
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appropriate for more general trends (gradually increasing or decreasing but not necessarily in a
straight line), keep in mind that our comparisons were based on linear trend. Thus they slightly
favored these two tests. The ANOCOV and modified "t" on ranks make the same assumptions,
but on the ranks of data rather than the actual observations. Thus linear trends in the
observations are not assumed, but we do assume that the form of the trend is gradually
increasing or decreasing. The Kendall tau and seasonal Kendall tests are designed for general
monotonic trend. Thus they would be regarded as more general than the other tests.
In fact any of the tests could be applied to a wide variety of trend shapes, including
quadratic or step trends. If trends are not monotonic, meaning that the general tendency is in
one direction for a while and then the reverse, the tests would not be very sensitive, unless
there was a clear overall tendency in one direction or the other. Thus, we would generally
want to inspect time series plots before performing the tests, to identify segments with differing
trend directions or other characteristics indicating that more in-depth treatment is necessary.
The second key assumption is independence of observations. All of the tests account for
seasonal dependence in some way or other; however, only the corrected seasonal Kendall test
accounts for serial correlation--temporal dependence after seasonality is removed. All of the
other tests assume that samples are independent in the absence of seasonality, meaning that
there is not serial correlation.
Although we do not believe that quarterly observations will exhibit strong levels of serial
correlation, it should be understood that any serial correlation will affect the performance of
the tests. The tests will tend to reject the null hypothesis more often than they should. The
correlated series will tend to drift above or below the long-term mean and stay there for a
while, and this drift will sometimes be indistinguishable from trend, depending on the time
horizon over which the process is viewed. (The term "drift" has a particular connotation in
stochastic modeling, which is not intended here.)
In practice, the classification of observed patterns as either trend or serial correlation is
rather subjective. There is no way to overcome this difficulty without very long records or
physical explanations for observed patterns. Thus we have not spent a great deal of time
working on methods to account for serial correlation in trend analysis. More discussion of the
issue appears in the section 3.4 on the results of Monte Carlo evaluation of candidate tests.
3.4 MONTE CARLO EVALUATION OF CANDIDATE TESTS
Since analytical evaluations of power and significance levels are possible only in a limited
number of situations, a good comparison of trend testing methods is best achieved through
Monte Carlo testing. In a Monte Carlo evaluation, the significance level of a test is determined
by generating a large number (e.g., 500) of sequences of data with known characteristics and no
trend. The test is applied to each sequence, and the significance level is the fraction of trials
in which a trend is falsely detected. The power of a given test is determined in the same way,
except that a trend of known magnitude is added to each synthetic data sequence. The power
is the fraction of sequences in which the trend is correctly detected.
To compare alternative tests, we need to perform a very large number of simulation experi-
ments. Time series of several types must be considered. To adequately represent the range of
characteristics anticipated from TIME data, the parameters that should be varied are magnitude
of seasonality in mean, pattern of seasonality in mean, magnitude and pattern of seasonality in
variance, normal versus non-normal distribution, degree of serial correlation, trend magnitude,
and length of record. The number of possible combinations of these parameters is very large,
and rigorous testing of all possibilities would require an enormous amount of computer time.
38
-------
Therefore, we developed a limited Monte Carlo testing program that examined a few key
values of the above parameters, based on our historical data analysis, and tried all possible
combinations thereof. The simulations are described in Table 3-17. Only normal and lognormal
distributions were considered, and only simple AR(1) type autocorrelation was considered. Auto-
correlation was not considered for the lognormal case. Even with these simplifications, 3,024
combinations of parameters were evaluated. For each combination, at least 500 sequences were
generated to empirically determine the power or significance level of the candidate tests.
Details of the synthetic data generation procedure are presented in Section 6. All candidate
tests were applied to each synthetic data sequence.
3.5 RESULTS OF MONTE CARLO EVALUATION
Summary results of the Monte Carlo evaluation are presented in Tables 6-1 through 6-4 in
Section 6, along with a more complete discussion. Complete results are presented in Appendix
B, Tables B-l through B-6. Briefly, the most powerful tests over the range of conditions
studied appear to be the seasonal Kendall test and ANOCOV on ranks, although as expected, no
single test performs best under all conditions. Both of these tests performed as well as the
parametric tests when the data were normal and both out-performed (were more powerful than)
the parametric tests when the underlying distribution was lognormal. In a few cases, the
Kendall-tau on deseasonalized data was more powerful, but it did not generally preserve the
nominal significance level as well as the other tests. The modified "t" test on ranks performed
well, but was in most cases slightly less powerful than ANOCOV on ranks.
All tests except the corrected seasonal Kendall (d) suffered from inflated significance levels
under serial correlation. The corrected test, however, is much less powerful than the other
tests, except for very large trend magnitudes and/or long data records. As expected, it is very
difficult to distinguish between linear trend and serial correlation.
The corrected seasonal Kendall test is not recommended for routine application by TIME
until very long records (say 15-20 years of quarterly data) have been obtained. The correction
for serial correlation will cause the test to ignore trends of moderate magnitude and duration
that may be important from a management standpoint. The question of whether a change in
water quality is of interest is one of physical causality. Persistence in the series, as described
by a correlated process model, could be caused by some factors that are not of interest (e.g.,
long lake retention time) and others that are of interest (e.g., cycles of industrial activity). We
argue that it is wiser to detect more trends, some of which are false positives, and then to
screen according to probable cause, than to overlook changes that are really of interest.
In fact, we believe that it will probably not be possible to deal satisfactorily with the issue
of serial correlation with any routine time series approach until very long records are available.
Although several methods are available, such as ARIMA modeling of the time series and exten-
sions of linear regression, all require some type of estimation of the correlation structure of the
series of interest. The trend test is then modified in some way to account for the correlation.
In particular, the distribution of a test statistic will be obtained under a null hypothesis of
no deterministic trend plus some (estimated) serial correlation structure. The distribution of the
test statistic, under the null hypothesis, will depend on the true correlation structure. However,
the rejection value must depend on the estimated structure. Even if the true correlation struc-
ture were known exactly, the variance of the test statistic under the null hypothesis would be
increased (compared to the uncorrelated case), resulting in lower power. Also, for small to
moderate sample sizes, the estimated parameters of the correlation structure will have a high
variance, making matters ever worse. For example, assuming an AR(1) structure, the variance of
39
-------
TABLE 3-17. DESCRIPTION OF SIMULATIONS FOR MONTE CARLO TESTING PROGRAM
A. Seasonal patterns in mean
Pattern (1) Quarter 1 - Low
Quarter 2 - High
Quarter 3 - Low
Quarter 4 - Low
Pattern (2) Quarter 1 - Low
Quarter 2 - High
Quarter 3 - Low
Quarter 4 - High
B. Seasonal patterns in standard deviation
(same two patterns as in mean)
C. Ratios of largest to smallest quarterly standard deviation
1.0, 1.5, 3.0, 5.0
D. Ratio of largest to smallest quarterly mean
1.0, 1.5, 2.0
E. (change in mean per sampling interval)
Trend magnitude = ,
(average standard deviation over all quarters)
0.0, 0.002, 0.005, 0.02, 0.05, 0.2, 0.5
F. Length of record (years)
5, 15, 25
G. Underlying distribution
Normal, log normal
H. Lag-one autocorrelation coefficient p(l) »
0.2, 0.4
(Correlated sequences generated for normal data only.)
I. Nominal significance level = 0.05 for all tests.
40
-------
r(l), the sample estimate of p(l), is approximately equal to (l-p(l)2)/n. If p(l) = 0.2, and n =
24 (6 years of quarterly data), then the standard deviation of r(l) would be about 0.2--the same
magnitude as the true value of p(l).
In simpler terms, there is no way to uniquely characterize a correlated time series with
small sample sizes. For large sample sizes, more complex time series methods may be justified.
In the case of TIME, we have about 20 years to work on the problem.
In the meantime, we feel that other avenues, involving the development of closer links
between statistical and physical models of the system, offer greater promise of an effective
solution to the serial correlation problem. Statistical correlation should be due to physical
factors, such as multi-year weather patterns. Consequently, it should be possible to replace a
stochastic model of serial correlation structure with more physically based models, ranging from
multivariate linear models with additional predictive variables to detailed and complex watershed
models. We hope that the more physically based models can be formulated and calibrated with
intensive sampling over a shorter time frame, as opposed to sampling over a long period (> 20
years). These models also offer greater potential for transferability between watersheds than
univariate time series models.
As stated earlier, the two recommended tests out of the seven candidates are ANOCOV on
ranks and seasonal Kendall. ANOCOV on ranks offers the advantage of being insensitive to the
pattern and magnitude of seasonal change in variance. It is also easily applied by anyone who
has a microcomputer stat package with a multiple linear regression capability. ANOCOV is also
a very flexible method. The general ANOCOV model and procedure can be expanded by adding
covariates to achieve additional power or increased ability to explain trends. Additional
discussion of this topic follows in Section 5.
On the other hand, the seasonal Kendall test has a proven track record in water quality
data analysis (Smith et al., 1987) and offers slightly better performance under certain conditions,
notably the presence of serial correlation. For these reasons, the authors have a slight prefer-
ence for the seasonal Kendall test. We recommend including ANOCOV on ranks as an alterna-
tive because of its ease of application with statistical packages and its potential for extension
to multivariate tests. A further comparison of the two tests under alternate models of season-
ality is presented in Section 6.
41
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SECTION 4
EXPECTED PERFORMANCE OF MONITORING— POWER OF TREND DETECTION
The actual ability of TIME monitoring and data analysis to detect trends in water quality
will depend upon data characteristics, especially temporal variance, and upon the shape or
functional form and magnitude of the trend that actually occurs. Thus trend detection powers
cannot really be predicted in advance. It is informative, however, to assume a reasonable set of
data characteristics and trend characteristics and then to calculate detectable trend magnitudes
over various time horizons. Thus, the adequacy of a proposed monitoring network design can be
evaluated in objective terms.
An assumption of linear trend is used for this entire report. Actual trends may not be
linear; however, the type of trend anticipated from changing acidic deposition is one of fairly
gradual change over several years, as opposed to an abrupt shift in a year or two. Likewise,
only normal and lognormal underlying distributions are considered. Both logic and analysis of
historical data dictate that other distributions will be encountered by TIME. However, these
distributions have been proven to have fairly broad applicability for major ions, and can
therefore serve as a basis for evaluating network performance in the design stage. Furthermore,
the recommended trend testing methods are based on ranks of data and are thus insensitive to
the form of underlying distribution.
4.1 INDIVIDUAL LAKES
Under idealized conditions of independent, identically distributed normal samples, linear
trends in a time series may be detected using linear regression. Under these conditions, the
power of the test for significance of the regression slope may be estimated as follows
(Lettenmeier, 1976).
First, compute a dimensionless trend number Nt using
Nt - [n (n+1) (n-
where n = number of observations used in the regresssion
T « trend magnitude in units per sampling interval (for example Gieq L"1) per quarter)
a - standard deviation of the water quality variable in the absence of a trend.
Then, the power of the test is approximated by I- ft = F(Nt-t1.a/2,'»)» where F is the cumulative
distribution function of the student's "t" distribution with 7 = n-2 degrees of freedom.
Since the trend is linear, the total change in mean concentration occurring over n samples
is nT. Under real-world conditions of seasonality, non-normality, and serial correlation, the
power of the test will be different from that shown above. The simulation results reported in
the previous section provide powers for certain sets of more realistic conditions.
Figures 4-1 through 4-4 depict the power of trend detection for each region for 4 different
trend magnitudes (T/<7 - 0.005, 0.02, 0.05, and 0.20 for quarterly observations, or 4 times these
values for annual observations) over a 25-year time horizon. Each graph contains curves for
quarterly sampling and annual sampling. The curves were developed using the equation discussed
above for the power of the "t" test and linear regression.
43
-------
POWER OF TREND DETECTION
For Trend = 0.005 Std. Dev. per Quarter
Q)
O
OL
0.75
0.5
0.25
10 15
Time (years)
20
25
Trend = 0.005
Std. Dev./qtr.
Trend = 0.02
Std. Dev./yr.
O Nl
O p=0.2
O p=0.4
c £-"
§>
O 3
O ^
«
Ol
O
O
Figure 4-1.
20
15
10
High Var
SE streams
Med Var
NLS 1A
Low Var
NLS 1E
10 15
Time (years)
20
25
Power of trend detection for trend = 0.005 standard deviations per quarter and
0.02 standard deviations per year. The curves in the upper graph are the power
calculated with the trend number. The points are the results from the simula-
tion for ANOCOV on ranks for independent, rho = 0.2 and rho = 0.4
data, sampled quarterly. The lower graph shows the magnitude of the simulated
trend at three levels of variability.
44
-------
0)
O
a.
POWER OF TREND DETECTION
For Trend = 0.02 Sid. Dev. per Quarter
0.75
0.5
0.25
Trend = 0.02
Std. Dev./qtr.
Trend = 0.05
Std. Dev./yr.
Nl
p=0.2
5 10 15 20
Time (years)
25
100
en
I
O
High Var
SE streams
Med Var
HIS 1A
Low Var
NLS 1E
25 -
0 *£*-=.
10 15
Time (years)
Figure 4-2. Power of trend detection for trend = 0.02 standard deviations per quarter and
0.08 standard deviations per year. The curves in the upper graph are the power
calculated with the trend number. The points are the results from the simula-
tion for ANOCOV on ranks for independent, rho = 0.2 and rho = 0.4 data,
sampled quarterly. The lower graph shows the magnitude of the simulated trend
at three levels of variability.
45
-------
POWER OF TREND DETECTION
For Trend = 0.05 Std. Dev. per Quarter
0>
»
o
a.
10 15
Time (years)
20
Trend = 0.05
Std. Dev./qtr.
Trend = 0.2
Std. Dev./yr.
200
150
100
50
High Var
SE streams
Med Var
NLS 1A
Low Var
NLS 1E
10 15
Time (years)
20
25
Figure 4-3. Power of trend detection for trend = 0.05 standard deviations per quarter and
0.20 standard deviations per year. The curves in the upper graph are the power
calculated with the trend number. The points are the results from the
simulation for ANOCOV ranks for independent, rho = 0.2 and rho = 0.4 data,
sampled quarterly. The lower graph shows the magnitude of the simulated trend
at three levels of variability.
46
-------
0)
g
a.
POWER OF TREND DETECTION
For Trend = 0.2 Std. Dev. per Quarter
1
0.75
0.5
0.25
,
'• f (
- \
\ i
1 •
i i
•
_ i
i
•
i
i
•
i
i
.1 i
i
i
i
*
•
10 15
Time (years)
20
25
Trend = 0.2
Std. Dev./qtr.
Trend = 0.5
Std. Dev./yr.
o Nl
> p=0.2
0 p=0.4
g
I
8
<§
en
o
6
BOO
BOO -
400 -
200 -
High Var
SE streams
Med Var
NLS 1A
Low Var
NLS 1E
0 U^-.-.
10 15
Time (years)
Figure 4-4. Power of trend detection for trend = 0.20 standard deviations per quarter and 0.8
standard deviations per year. The curves in the upper graph are the power
calculated with the trend number. The points are the results from the
simulation for ANOCOV on ranks for independent, rho = 0.2 and rho = 0.4 data,
sampled quarterly. The lower graph shows the magnitude of the simulated trend
at three levels of variability.
47
-------
Anticipated powers under real-world conditions are shown as individual plotted points taken
from the simulation results reported earlier for ANOCOV on ranks over the whole range of
seasonality studied. Plotted points correspond to normal-independent, normal-p(l) = 0.2 and
normal-p(l) = 0.4 — all under quarterly sampling. Under annual sampling, seasonal variation and
serial correlation are not important, and the theoretical curve should give an adequate repre-
sentation of anticipated power. Powers for lognormal data are higher than those for normal
data, are not directly comparable, and are therefore not included.
In both the idealized and real-world cases, the trend magnitude is expressed in terms of a,
the temporal standard deviation of the water quality variable in the absence of trend. To
estimate power of trend detection as a function of total change for a specific region, we can
utilize the regional standard deviations shown earlier in Table 3-3.
For this analysis, let us choose one variable, ANC, and three regions to represent the
anticipated range of temporal variability. NLS-1E represents low temporal variability, whereas
NLS- 1 A and Eastern Streams represent medium and high variability, respectively. A cautionary
note is in order. In each case, the tables represent average values for the "better" locations in
the region in terms of data availability. There is no guarantee that these values are truly
representative of regional behavior, and certainly individual lake variances would differ from
these average values.
To reduce the number of cases to be considered, let us choose the arithmetic mean of the
four seasonal (quarterly) temporal standard deviations of ANC for each region. These average
standard deviations (in /ieq L'1) are 4.8 for NLS-1E (low variability), 27.2 for NLS-1A (medium
variability) and 32.1 for Streams (high variability).
Under the assumption of linear trend, we can plot the total change occurring over time for
a stated trend magnitude, T/a, and temporal standard deviation, a. We have included such plots
below each of the power curves (Figures 4- 1 through 4-4) using the three standard deviations
described in Section 3. These plots are specific to the assumed temporal standard deviations
and are included simply to illustrate how the power curves would be used in a given
situation- -not necessarily to suggest that these would be the actual powers anticipated for ANC
in the given regions.
An example might help to interpret the figures. Suppose that we are interested in a trend
magnitude of 0.2 standard deviations per year, equivalently 0.05 standard deviations per quarter
occurring over a 10-year period. From Figure 4-3 on page 46, we see that the total change
occurring (lower graph) in a medium variability system (NLS- 1 A, ANC standard deviation of 27.2
L"1) would be just over 50 /ieq L"1. A simple calculation verifies this.
(10 years) (0.2 a/year) (27.2 ^eq L'^/a = 54.2 /zeq L'1
From the upper graph we see that annual sampling (10 samples) would provide a power of
about 0.35 under the hypothesized trend, whereas quarterly sampling (40 samples) would provide
a power of about 0.95.
4.2 MULTIPLE LAKES
The primary monitoring objective discussed to this point has been detecting trends in
individual lakes. It will often be desirable, however, to also consider trends in average
conditions over several lakes.
Although there are many possible ways to define "average," it is reasonable at the network
design stage to limit discussion to simple arithmetic averages (regional sample mean) over a
48
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group of lakes with one observation per time period per lake. The statistical objective of
monitoring is then the detection of changes in the regional mean over time for specified water
quality variables.
4.2.1 Statistical Characteristics of the Regional Mean
Actually this objective amounts to simply replacing the individual lake water quality
variables considered thus far with new variables, which are regional means. Trends in the new
variables would be detected in exactly the same way as trends in the individual lake variables.
In order to determine the anticipated power of trend detection for regional means, however, we
need to determine the temporal variance and correlation structure of the regional means.
Although these characteristics could be determined from analysis of historical time series of
regional means, general results may be obtained more easily by relating regional mean character-
istics to those of individual lakes. Several simplifying assumptions are used in the following
development.
First, the temporal variance of the regional mean, var (x) is given by
var (X) = var [ I E xiit]
(^ [E var (xit) + 2 E cov (Xi,Xj)t]
lr ' J
where n = number of lakes sampled
xi|t - deseasonalized or residual observation at lake i, time t (after removal of
seasonal variation)
We assume that there is some regional temporal variance, a2, that may be used to replace the
terms var (xi|t) at this early, network design stage. We further assume that this variance is
stationary. The spatial covariance terms may be written as
cov (Xi.xpt = P(XJ,XJ)
where p-^ = the spatial correlation coefficient between stations i and j and
a2 = the regional temporal variance.
We can now write a simplified expression for the variance of the regional mean as follows.
var(x)= (£ )» [no2 + 2 E O»PU]
var(x)= jf[l + p-0(n-l)]
where p0 - the average interstation correlation at lag zero
var (x) = temporal variance of the regional mean
49
-------
Now we have the temporal variance of the "new" variables, and all that remains is to find the
temporal correlation structure.
We can write from the definition of temporal correlation,
... ._ _ . E[(xt - % )(x-t+k - % )]
(k) = p(xt,xt+k) = t t+k
t+k
After removal of season means, all ju's = 0. From stationarity assumptions, both variances
in the denominator are equal to var(x), given above. Therefore,
cov[xt,xt+k]
x var (x)
E[(xt )(xt+k)]
var (x)
Expanding in terms of individual lake observations we obtain the following:
-pu(k) + p12(k) + p13(k) + ... + pln(k) +
PS 00- *a
Pnl(k) + . . . + .pnn(k)
The entire term in brackets is a sum. The diagonal terms are lag-k autocorrelations at
each station (lake) l,...,n. The off-diagonal terms are lag-k cross correlations between stations.
Each station pair appears twice. Assuming that the cross correlation terms are small (negligible)
compared to the autocorrelation terms, we obtain
P,00- n(p-k)
where pk = the average of the n lag-k autocorrelations over all stations.
In particular for lag-one,
P-X0)
The lag-one autocorrelation coefficient of the regional mean is 1/n times the regional
average lag-one value. Since the term pi would typically be small for quarterly or less frequent
sampling, the value of p^U) would generally be negligible. Thus temporal correlation will be
ignored in the remainder of this section. Spatial correlation will be considered through its
effect on var (x) as developed above.
50
-------
4.2.2 Detectable Changes in Regional Means
The same approach used earlier for determining the power of detecting a linear trend may
now be applied to regional means. The individual lake temporal variance a2 is now replaced by
var (X). Everything else remains the same. Figures 4-5 through 4-8 present the detectable
change as a function of the number of observations for given numbers of stations (lakes). The
figures show two levels of average inter-lake (spatial) correlation, p0 = 0.2 and 0.4 and powers
(l-ft) of 0.90, 0.80, and 0.60. The case of p0 - 0.4 is considered only for 1-ft - 0.20. In all
cases, the significance level is constant at 0.10, and a linear trend (constant slope) occurs over
the entire period of monitoring.
In general, as the number of stations, n, increases, the var (x) decreases and the power
increases. However, as the spatial correlation increases, the var (X) also increases and power is
reduced.
The curves are generalized by presentation of the detectable change (vertical axis) in
standard deviations. The curves are independent of any time scale; thus they may be used for
quarterly, semi-annual, or annual sampling. However, the observations are assumed to be
temporally independent and equally spaced in time. The change indicated on the vertical axis is
assumed to occur as a linear trend over the number of observations indicated on the horizontal
axis. Application to a specific situation is best explained by example.
Suppose that in a given region, the average temporal standard deviation of sulfate con-
centration in individual lakes is 10 /*eq L"1. We can determine the detectable change with
stated power for various time periods and numbers of lakes as follows. Consider the case of a
- 0.10 and p = 0.20 (Figures 4-6 and 4-7). Also consider annual sampling. Over a period of 10
years (10 observations in each lake), the detectable change in sulfate concentration for one lake
would be three standard deviations (curve a, Figure 4-6), or 30 /xeq L"1. The trend magnitude
or slope over that time would be 3.0 /zeq L'Vyear. For four lakes with no spatial correlation,
the detectable change in 10 years would be 1.5 standard deviations (as in curve b, Figure 4-6),
or 15 /ieq L'1- The trend slope would be 1.5 /zeq L'Vyear. For 16 lakes, the detectable change
would be 7.5 /zeq L"1 (as in curve c, Figure 4-6), with a trend slope of 0.75 /jeq L'Vyear.
If spatial correlation exists, however, the detectable change is larger. For example, with
16 lakes and p0 = 0.4, curve e of Figure 4-7 applies, and the detectable change over 10 years of
annual sampling is 2.0 standard deviations or 20 /*eq L'1, a trend slope of 2.0 /xeq L'Vyear.
Figure 4-7 indicates that when the average inter-lake correlation is 0.4, the advantage of
increasing the number of lakes from four (curve d) to 16 (curve e) is slight. In the real world,
this effect would be even more pronounced, since the average inter-lake correlation would
increase as the number of lakes sampled within a region increased. Thus, the reduction in
detectable change would actually be less than that shown in the figures. Obviously, inter-lake
correlation has a significant impact on the power of detecting trends in regional means.
An important concern, therefore, is how large we might expect inter-lake correlations to
be. Given the paucity of data records for multiple lakes in a given region over long times, this
is a difficult question to address. For a single example, though, we computed inter-lake cor-
relations for 11 lakes (those with most complete data records) in NLS subregion 1 A. The values
for all possible pairings of lakes are shown in Tables 4-1 and 4-2 for ANC and sulfate,
respectively.
Each value is based on 40 monthly observations with a few missing values. For ANC, esti-
mated inter-lake correlations range from -0.090 to +0.841 with an average of +0.358. For sulfate
the range is -0.232 to +0.724 with an average of +0.252. As indicated by the map of sampling
locations in Newell (1987), however, many of the lakes are very close together. We therefore
51
-------
oJpha«-0,10 beta—0.10
J>
"o
s
c
5
W
c
9
CT
C
o
X.
o
— o1-0
0.5
10
b4-0
Number of Observations
— c16-0 —
d4-.2
Figure 4-5. Level of detectable trend for a=0.10 and /?=0.10 for five configurations of number
of lakes and spatial correlation = 0.0 and 0.2.
52
-------
olpha-0.10 beta-0.20
c
J3
53
jj
o
Q
•s
O
c
o
5)
c
o
£
o
— a1-0
b4-0
Number of Observations
— C16-0 —
44-J2
Figure 4-6. Level of detectable trend for a=0.10 and /9=0.20 for five configurations of number
of lakes and spatial correlation = 0.0 and 0.2.
53
-------
alpha-0.10 beto-0.20
o
+5
o
1
o
o
•o
c
o
5)
01
c
o
£
o
— a1-0
t>4-0
Number of Observations
— C16-0 —
d4-.4
Figure 4-7. Level of detectable trend for a=0.10 and £=0.20 for five configurations of number
of lakes and spatial correlation = 0.0 and 0.4.
54
-------
olpho-0.10 beta-0.40
c
0
"2
o
TJ
O
c
o
JC
o
3.5
2.5
10
i
20
i
30
— a1-0
b4-0
Number of Observations
— C16-0 —
50
e16-.2
Figure 4-8. Level of detectable trend for a=0.10 and 0=0.40 for five configurations of number
of lakes and spatial correlation = 0.0 and 0.2.
55
-------
TABLE 4-1. BETWEEN-LAKE CORRELATIONS FROM MONTHLY DATA FOR ANC
(w 40 MONTHS OF DATA) FOR 11 LAKES IN NLS SUBREGION 1A
Between
Lake
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
Average
and
Lake
2
3
4
5
6
7
8
9
10
11
3
4
5
6
7
8
9
10
11
4
5
6
7
8
9
10
11
Correlation
0.190
0.311
0.121
0.485
0.446
0.094
0.331
0.236
0.254
0.205
0.381
0.841
0.383
0.519
0.038
0.451
0.738
0.372
0.505
0.276
0.507
0.195
-0.090
0.376
0.251
0.114
0.393
correlation = 0.358
Lake designation, from Newell et al. (1987)
1A1-071
1A1-105
1A2-077
1A1-087
1A1-102
1A1-106
1A1-107
1A1-109
1A1-110
1A1-113
1A1-078
Between
Lake
4
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
7
7
7
7
8
8
8
9
9
10
Lake number
and
Lake
5
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
8
9
10
11
9
10
11
10
11
11
used in
1
2
3
4
5
6
7
8
9
10
11
Correlation
0.319
0.488
0.037
0.325
0.696
0.367
0.546
0.468
0.249
0.588
0.481
0.334
0.503
0.419
0.559
0.577
0.355
0.528
0.024
-0.018
0.228
0.238
0.611
0.284
0.477
0.525
0.467
0.118
Tables 4-1 and 4-2
56
-------
TABLE 4-2. BETWEEN-LAKE CORRELATIONS FROM MONTHLY DATA FOR SULFATE
(« 40 MONTHS OF DATA) FOR 11 LAKES IN NLS SUBREGION 1A
Between and
Lake Lake
2
3
4
5
6
7
8
9
10
11
2 3
2 4
2 5
2 6
2 7
2 8
2 9
2 10
2 11
3 4
3 5
3 6
3 7
3 8
3 9
3 10
3 11
Correlation
0.011
0.256
0.059
0.466
0.470
-0.156
0.224
0.161
0.366
0.288
0.542
0.609
0.463
0.256
0.085
0.415
0.246
0.059
0.145
0.474
0.156
0.438
0.065
0.325
0.410
0.512
0.004
Between
Lake
4
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
7
7
7
7
8
8
8
9
9
10
and
Lake
5
6
7
8
9
10
11
6
7
8
9
10
11
7
8
9
10
11
8
9
10
11
9
10
11
10
11
11
Correlation
0.561
0.344
-0.144
0.461
0.275
0.166
0.143
0.538
-0.226
0.364
0.036
-0.008
0.467
-0.114
0.464
0.442
0.249
0.466
-0.103
0.100
0.011
-0.232
0.724
0.250
0.459
0.334
0.311
0.192
Average correlation = 0.252
57
-------
regard these results as atypical and "on the high side," both in terms of maximum inter-lake
correlations and average inter-lake correlations for a region. In the example, no attempt was
made to relate inter-lake correlations to distance between lakes, although this would be a logical
next step in the analysis of a particular region (Hirsch and Gilroy, 1985). We present this
example, however, only to suggest that our choices of p0 = 0.0, 0.20, and 0.40 (in Figures 4-5,
4-6, and 4-7) have some relevance to a real world situation.
4.3 CASE STUDIES, INDIVIDUAL LAKES
4.3.1 Clearwater Lake. Ontario
In order to illustrate the application of statistical tests for trend, and to compare the
performance of alternative methods, we applied the tests to historical data from Clearwater
Lake, Ontario (Nicholls, 1987). The variables studied were ANC, sulfate, sulfate/ANC, and
sulfate/(calcium+magnesium). All samples were depth-integrated composites. The statistical
trend tests studied were Seasonal Kendall (SK), with and without correction for serial cor-
relation, and analysis of covariance (ANOCOV) on both raw data and ranks of data.
The Clearwater Lake data were used for the case study for two reasons. A long record is
available, and significant trends of increasing ANC and decreasing sulfate are known to exist.
Unfortunately, perhaps, the trend magnitudes are so large that all statistical tests indicate
significant trends in approximately the minimum time required to obtain stable values of the test
statistics. Thus little difference in performance between tests is apparent.
The case study assumes quarterly sampling. Since Clearwater Lake was sampled more fre-
quently, quarterly values were obtained by subsampling, that is, choosing the value that appears
closest to the center of the quarter. Quarters were defined as (1) December, January, February,
(2) March, April, May, etc.
4.3.1.1 Sulfate Concentration—
Figure 4-9 is a plot of quarterly sulfate observations beginning in summer of 1973. Figure
4-10 is the corresponding correlogram. No seasonal pattern is apparent, and the strong auto-
correlation is removed by detrending, using ordinary least squares (correlogram of Figure 4-11).
The detrended data have a skewness coefficient of -0.882, which is significant at the 1% level.
The trend magnitude, from Ordinary Least Squares (OLS), is 0.221 mg L"1/quarter or 0.161 stan-
dard deviations per quarter. Figure 4-12 portrays the results of applying ANOCOV to both raw
sulfate data and ranks in successive trials at a nominal significance level of 5%. Each trial
begins at the start of the historical record and ends with the quarter shown on the horizontal
axis. Missing values are not included in the number of quarters. The number of quarters used
ranges from 9 to 44. The value of the test statistic is shown on the vertical axis, and the
critical values are indicated with diamonds. For both the original data and ranks, ANOCOV
indicates a significant trend after 16 observations.
Figure 4-13 presents the results of applying the SK test, both with and without the serial
correlation correction, to the same data. The rejection value of the test statistic is 1.96 for a
nominal significance level of 5%. The SK tests indicate significant trend after 14 (with correc-
tion) and 13 (without correction) observations. For longer records, the value of the SK test
statistic is much larger (more negative) when the serial correlation correction is not used.
Since the detrended data are not correlated, the uncorrected test is preferred.
58
-------
30T
26--
16
14-
10
IB 27 36
OBSERVATION NUMBER
Figure 4-9. Quarterly sulfate observations beginning in summer of 1973.
59
-------
I I
-i
18
LAG
Figure 4-10. Correlogram of quarterly sulfate data shown in Figure 4-9.
60
-------
-1
ii
18
2i
LAG
Figure 4-11. Correlogram of quarterly sulfate data shown in Figure 4-9 after detrending with
ordinary least squares.
61
-------
ANOCOV of Sulfate
I
I
-19
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
910111213141516171819202122232425262728293031323334353637383940414243444546
Original
Number of Observations
+ Ranked O Crftkal Value
Figure 4-12. Results of ANOCOV on raw sulfate data and on ranks of sulfate data. Critical
value of the test statistic is shown for each number of observations. The test is
significant when the calculated statistic is more negative than the critical value.
62
-------
Seasonal Kendall on Sulfate
-i
-2
-3
-4
-5
-6
-7
-8
15
i
25
35
r
45
Number of Observations
D Original + Corrected
Figure 4-13. Results of seasonal Kendall (square symbols) and seasonal Kendall corrected for
serial correlation (plus symbols) on raw sulfate data.
63
-------
4.3.1.2 Acid Neutralizing Capacity--
Raw ANC data, beginning with fall of 1980, are plotted in Figure 4-14 and the correlogram
is presented in Figure 4-15. Lags one, three, and four show significant autocorrelation. This
correlation is removed upon detrending by OLS (Figure 4-16). No significant seasonality is
apparent. The detrended data do not show significant skewness at the 5% level. The OLS trend
magnitude over the period of record is 0.0348 mg L'1 as calcium carbonate per quarter or 0.170
standard deviations per quarter.
Results of ANOCOV are shown in Figure 4-17. The original data show a significant trend
after 11 observations, while the ranks show a significant trend after 16 observations. The test
statistic is extremely variable over the first four years of data for both tests. The SK test
statistics (Figure 4-18) are also quite variable over the first four years. The test without
correlation correction consistently shows a significant trend after 15 quarters. The corrected
test shows trend consistently after 16 quarters.
4.3.1.3 Sulfate/ANC--
The sulfate/ANC ratios beginning with fall of 1973 are plotted in Figure 4-19, and the
correlogram is presented in Figure 4-20. Only lag-one correlation appears significant, and
detrending results in significant lag-two correlation (Figure 4-21). Seasonality does not appear
to be significant. The detrended data are not significantly skewed at the 5% level. The OLS
trend magnitude over the period of record is 0.190 mg L'Vquarter or 0.102 standard deviations
per quarter.
The ANOCOV test statistics (Figure 4-22) are again highly variable for the first 14 obser-
vations. ANOCOV on ranks shows trend consistently after 14 observations, whereas ANOCOV on
raw data shows trend consistently after 17 observations. The SK test shows significant trend
consistently after 15 observations, without correction, and after 16 observations with correction
for serial correlation (Figure 4-23).
4.3.1.4 Sulfate/(Calcium + Magnesium)--
The sulfate/(calcium + magnesium) ratio (Figure 4-24, beginning with summer of 1973)
shows significant autocorrelation (Figure 4-25), which is removed by detrending (Figure 4-26).
The data are not significantly skewed. Trend magnitude from OLS is 0.0239 or 0.102 mg L"1 per
quarter.
ANOCOV finds significant trend after 21 observations for both raw data and ranks (Figure
4-27). The SK test (Figure 4-28), finds significant trend after 22 or 23 observations for the
uncorrected and corrected versions, respectively.
4.3.1.5 Summary of Trend Testing Results--
The example data are generally characterized by symmetric distributions, little apparent
seasonality, and little or no significant serial correlation. Trend magnitudes in the Clearwater
Lake data are quite large, ranging from 0.10 to 0.17 standard deviations per quarter. Conse-
quently, all four tests are able to detect trends quickly. Trends in sulfate, ANC, and
sulfate/ANC are detected by all tests in 13 to 16 observations. The sulfate/(calcium +
magnesium) ratio requires 21 to 23 observations.
64
-------
-0.50T
-1.0-
-1.8-
§ -2.0
-2.5-
-3.0
12
OBSERVATION NUMBER
IB
20
24
Figure 4-14. Quarterly ANC observations, mg L"1 as calcium carbonate, for Clearwater Lake,
Ontario, beginning fall of 1980.
65
-------
e
T~T~TT
-i
7 J5 3 ftaT
LAG
Figure 4-15. Correlogram of raw ANC data in Figure 4-14.
66
-------
e
"IF
LAG
Figure 4-16. Correlogram of ANC data shown in Figure 4-14 after detrending by ordinary
least squares.
67
-------
ANOCOV of ANC
0
+j
V)
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Original
Number of Observations
Ranked O
Critical Value
Figure 4-17. Results of ANOCOV on raw ANC data and on ranks of ANC data. Critical value
of the test statistic is shown for each number of observations. The test is
significant when the calculated statistic is larger than the critical value.
68
-------
Seasonal Kendall on ANC
.**
&
<3
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Number of Observations
D Original + Corrected
Figure 4-18. Results of seasonal Kendall (square symbols) and seasonal Kendall corrected for
serial correlation (plus symbols) on raw ANC data.
69
-------
-A-r
-7
-10-
-13
-IB
-19
12
OBSERVATION NUMBER
16
20
24
Figure 4-19. Quarterly sulfate/ANC ratios for Clearwater Lake, Ontario, beginning fall of
1980.
70
-------
e
21
24
LAG
Figure 4-20. Correlogram of sulfate/ANC ratios shown in Figure 4-19.
71
-------
0
-i
3' 6' i
10 is tfl 9
lb 1J 10 6
i 24
LAG
Figure 4-21. Correlogram of sulfate/ANC ratios shown in Figure 4-19 after detrending by
ordinary least squares.
72
-------
ANOCOV of S04/ANC
-5
Origlnol
Number of Observations
+ Ranked O Critical Value
Figure 4-22. Results of ANCOV on raw sulfate/ANC ratios and on ranks of sulfate/ANC ratios.
Critical value of the test statistic is shown for each number of observations.
The test is significant when the calculated statistic is more negative than the
critical value.
73
-------
Seasonal Kendall on SO4/ANC
M
*^
o
12
14
22
Number of Observations
D Original 4- Corrected
Figure 4-23. Results of seasonal Kendall (square symbols) and seasonal Kendall corrected for
serial correlation (plus symbols) on raw sulfate/ANC ratios.
74
-------
4.5
4.0
3.6
3.0
2.B
8.0
IB 27 36
OBSERVATION NUMBER
49 64
Figure 4-24. Quarterly sulfate/(calcium + magnesium) ratios for Clearwater Lake, Ontario,
beginning fall of 1980.
75
-------
e
I I
-i
12
18
Figure 4-25. Correlogram of sulfate/(calcium + magnesium) ratios shown in Figure 4-24.
76
-------
0
I I
-1
3'
6' 4
1'2 li
18 2
i 24
LAG
Figure 4-26. Correlogram of sulfate/(calcium + magnesium) ratios shown in Figure 4-24 after
detrending by ordinary least squares.
77
-------
ANOCOV of S04/(Ca+Mg)
o
+j
vt
000000000000
i i i i I i i i i i r i I i i i i i i i r i^i
91011121314151617181920212223242526272829303132333435363738394041424344
Original
Number of Observations
+ Ranked O Critical Value
Figure 4-27. Results of ANOCOV on raw sulfate/(calcium + magnesium) ratios and on ranks of
sulfate/(calcium + magnesium) ratios. Critical value of the test statistic is shown
for each number of observations. The test is significant when the calculated
statistic is more negative than the critical value.
78
-------
Seasonal Kendall on S04/(Ca+Mg)
IA
!£/
o
(A
O
-1
-2
-3
-4
-5
-6
-7
Number of Observations
D Original •f Corrected
Figure 4-28. Results of seasonal Kendall (square symbols) and seasonal Kendall corrected for
serial correlation (plus symbols) on raw sulfate/(calcium + magnesium) ratios.
79
-------
For the example data, there does not appear to be an advantage to using ratios for trend
detection as opposed to individual variables. In other situations, ratios may be advantageous for
detection and/or explanation of trends.
For the large trends seen here, ANOCOV on rank appears to work as well as ANOCOV on
raw data. The correlation-corrected version of the SK test detects trends one or two observa-
tions later than the uncorrected version.
As stated earlier, the case study illustrates that proposed tests for trend will work well for
large trends and time periods of four or more years. All of the test statistics were subject to
large variability in shorter data records. For this example, there was little difference in
performance among the tests studied.
4.3.1.6 Effect of Time of Sampling on Trend Detection, Quarterly Sampling—
For the Clearwater Lake data, the timing of sample collection has a notable effect on the
length of time required to detect trends in sulfate. This may be illustrated by redefining
quarters as (1) January, February, March, (2) April, May, June, etc., and choosing new data
points closest to the center of these redefined quarters. Figures 4-29 and 4-30 show the results
of applying ANOCOV and SK tests on the new series. In both cases, trends become significant
two or more quarters later than with the original data. Figure 4-31 presents the least squares
regression slope of the entire sulfate series as a function of length of record. Results are
highly variable for the first five years or so, but later stabilize around a value of 1.0 to 1.1 mg
L'Vyear.
For ANC, the redefined quarters do not affect the point at which trend becomes significant
using either ANOCOV or SK tests (Figures 4-32 and 4-33). However the SK test statistic
remains significant after 13 quarters in the new data series, whereas it temporarily dips and
becomes not significant in the original series. Figure 4-34 indicates that after year two, the
regression slope varies between 0.13 and 0.18 mg L'Vyear over the historical record.
We emphasize that seasons should be defined hydrologically or limnologically and could be
much shorter or longer than three months. A season could be defined in terms of streamflow
(spring freshet for example) or lake temperature profile rather than calendar date. The
recommended trend testing procedures would not be greatly affected by using sampling intervals
that were shorter or longer than three months or that varied somewhat from year to year, as
long as consideration was restricted to a fixed number of seasons per year and one observation
per season.
For the ANOCOV procedure, a day number could be used rather than an observation num-
ber for the time variable, t. This substitution would better reflect the exact time at which a
given sample was collected but would probably not have much impact on trend testing results.
4.3.1.7 Trend Detection, Annual Sampling--
In some, perhaps many, cases it may be desirable to collect a single spring or fall sample
rather than four quarterly samples. In order to determine whether annual sampling could detect
trends quickly, a simple study was performed using the Clearwater data set to obtain both spring
and fall annual series. Seasons were defined using limnological characteristics of lakes as
opposed to quarters. Table 4-3 presents the results of annual subsampling of the Clearwater
Lake data set, taking the ANC and sulfate observations closest to May 15 (spring). Table 4-4
repeats the exercise, in each case taking the observation closest to November 7.
80
-------
ANOCOV on S04
Ul
^j
JO
**
o
DOOOOOOOOOOOOOOOOOO
Orglnol
Number of Observations
+ Ranked O Critical Value
Figure 4-29. Results of ANOCOV on raw sulfate data with quarters redefined as Jan.-Mar.,
Apr.-June, etc., rather than Dec.-Feb., etc.
81
-------
Seasonal Kendall on S04
-B
Number of Observations
Orglnal •*• Corrected
Figure 4-30. Results of seasonal Kendall (square symbols) and seasonal Kendall corrected for
serial correlation (plus symbols) on raw sulfate data with quarters redefined as
in Figure 4-29.
82
-------
Slope Magnitude of S04
o
•O
*«
"E.
o>
o
a
£
(ft
10
Number of Observations
Figure 4-31. Least squares regression slope of the entire sulfate series as a function of length
of record.
83
-------
ANOCOV on ANC
5
V)
•8
V
23
Number of Observations
+ Ranked O
Critical Value
Figure 4-32. Results of ANOCOV on raw ANC data with quarters redefined as Jan.-Mar., Apr.
June, etc., rather than Dec.-Feb., etc.
84
-------
Seosonof Kendall on ANC
£
*5
2
w
e
fi
Number of Observations
Orglnal -f Corrected
Figure 4-33. Results of seasonal Kendall (square symbols) and seasonal Kendall corrected for
serial correlation (plus symbols) on raw ANC data with quarters redefined as in
Figure 4-32.
85
-------
l_
o
o
§
O>
E
9
TJ
*J
"c
o>
o
2
o
a
_o
in
0.18
0.17
0.16
0.15
0.14
0.13
0.12
0.11
0.1
0.09
0.08
0.07
Slope Magnitude of ANC
0.06 -U
11 13 15 17
Number of Observations
\
\
19
21
i
23
Figure 4-34. Least squares regression slope of the entire ANC series as a function of length
of record.
86
-------
TABLE 4 - 3. RESULTS OF TREND DETECTION WITH ANNUAL SPRING SUBSAMPLING
OF ANC AND SULFATE AT CLEARWATER LAKE, ONTARIO
SPRING ANC
DATE
5/13/81
5/13/82
5/02/83
5/08/84
5/22/85
5/07/86
SPRING
DATE
5/19/75
5/11/76
5/10/77
5/11/78
5/02/79
5/12/80
5/13/81
5/13/82
5/02/83
5/08/84
5/22/85
5/07/86
OBSERVATION
VALUE
mg L"1 as CaCos
-1.72
-1.56
-1.21
-1.21
-0.87
-1.00
SULFATE
OBSERVATION
VALUE
mg L"1
24.00
23.50
26.00
23.50
21.50
21.00
19.80
19.80
17.63
18.93
15.05
16.60
KENDALL TAU
Test
Statistic
--
--
6
10
13
Critical
Value
—
—
6
8
11
Significant Trend at
5% Two-sided
Yes
Yes
Yes
Test?
KENDALL TAU
Test
Statistic
—
—
—
-4
-9
-15
-20
-28
-35
-45
-54
Critical
Value
--
--
--
-8
-11
-13
-16
-18
-21
-25
-28
Significant Trend at
5% Two-sided
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Test?
87
-------
TABLE 4-4. RESULTS OF TREND DETECTION WITH ANNUAL FALL SUBSAMPLING OF
ANC AND SULFATE AT CLEARWATER LAKE, ONTARIO
FALL ANC
DATE
11/20/80
10/20/81
10/18/82
10/18/83
11/07/84
11/19/85
11/11/86
OBSERVATION
VALUE
mg L'1 as CaCo3
-2.20
-2.14
-1.83
-1.36
-1.03
-1.10
-1.10
KENDALL TAU
Test
Statistic
—
—
6
10
13
17
Critical
Value
—
—
6
8
11
13
Significant Trend at
5% Two-sided
Yes
Yes
Yes
Yes
Test?
FALL SULFATE
DATE
11/11/75
10/26/76
10/17/77
11/09/78
11/01/79
10/30/80
10/20/81
10/18/82
10/18/83
11/07/84
11/19/85
11/11/86
OBSERVATION
VALUE
mgL'1
21.00
27.00
26.00
24.00
22.50
21.00
21.00
19.40
18.86
17.00
16.75
16.59
KENDALL TAU
Test
Statistic
__
—
—
—
-2
-5
-7
-14
-22
-31
-41
-52
Critical
Value
— _
—
—
—
-8
-11
-13
-16
-18
-21
-25
-28
Significant Trend at
5% Two-sided
No
No
No
No
No
Yes
Yes
Yes
Yes
Test?
88
-------
Dates of observations are shown in the tables, with results of the Kendall-tau test for
trend on the annual values. For sulfate, trend is detected after seven years in the spring data
and nine years in the fall data. For ANC, trend is significant after four years, using the spring
data, and is also significant in the fall data after four years. Figures 4-35 through 3-38 are
time series plots of the annual (later winter and fall) ANC and sulfate data.
4.3.2 Twin Lakes. Colorado
A similar but less detailed study was performed on ANC data from Twin lakes, Colorado
(Sartoris, 1987). Quarterly data were used with quarters defined as (1) December, January,
February, (2) March, April, May, etc. Quarterly observations were obtained as subsamples from
a series of monthly means for the period January 1977 to September 1985. All samples were
depth integrated, and units are mg L"1 as bicarbonate. Station 2 was the middle of Lower Twin
Lakes and Station 4 was the middle of Upper Twin Lakes.
The quarterly time series of Figure 4-39 for Station 2 shows a generally decreasing trend
over the period; ANOCOV results (Figure 4-40) and SK results (Figure 4-41) confirm the
statistical signficance of the trend. Using ANOCOV on ranks, the trend is significant for
quarters 8 through 13 and 19 through 35. Using SK, the trend is significant for quarter 10 and
quarters 20 through 35. The SK test with correction for serial correlation does not show a
significant trend until quarter 24. A plot of regression slopes over time is presented in Figure
4-42.
Station 4 ANC data do not show a clear overall trend in the time series of Figure 4-43.
Some short-term "trends" are, however, apparent. The ANOCOV test indicates a significant
decreasing trend for quarters 7 through 12 and after quarter 22 (Figure 4-44). The SK test
indicates significant trend at quarter 10 and for quarters 22 through 32 (Figure 4-45). Both SK
and ANOCOV on ranks tests show that the overall trend through quarter 35 is not significant.
Both tests agree with the results of visual inspection of data and with the plot of regression
slopes over time (Figure 4-46). Since both tests identify significant short-term or temporary
trends in the series, a logical extension of the analysis would be to attempt to explain the
trends through analysis of other factors. Such extensions are discussed in the next section.
89
-------
Annual Spring ANC
O
O
o
O
B
o
0
ip
O
o
a
X)
O
Number of Years
Figure 4-35. Time series plot of annual spring ANC at Clearwater Lake, Ontario.
90
-------
Annual Spring Sulfate
o
53
I
9
D
JO
O
Number of Years
Figure 4-36, Time series plot of annual spring sulfate at Clearwater lake, Ontario.
91
-------
Annual Fall ANC
10
O
O
o
O
f>
o
01
o
o
a
.a
O
Number of Y«ara
Figure 4-37. Time series plot of annual fall ANC at Clearwater Lake, Ontario.
92
-------
Annual Fall Sulfate
o*
E
O
O
Number of Years
Figure 4-38. Time series plot of annual fall sulfate at Clearwater Lake, Ontario.
93
-------
•40.0000-r
30.0000+
QJ
4->
ra
i-
ro
o
20.0000-f
10.0000
IS IB
OBSERVATION NUMBER
24
30
38
Figure 4-39. Quarterly time series of ANC data, in mg L"1 as bicarbonate, from Twin Lakes,
Colorado, Station 2, beginning in January 1977.
94
-------
Analysts of Covariance at Site 2 on ANC
w
-6 -
-7
i i iiiiiiiiiiiiiii
678 91011121314151617181920212223242526272829303132333435
Origin
Number of Observations
+ Ranks O Critical Value
Figure 4-40. Results of ANOCOV on raw ANC and on ranks of ANC time series shown in
Figure 4-39. Critical value of the test statistic in shown for each number of
observations. The test is significant when the calculated statistic is more
negative than the critical value.
95
-------
Seasonal Kendall at Site 2 on ANC
O
+*
V)
.5
-3.5
-4
-4.5
I I I I T IITIIIIIIIIIIIIIIIITTT^
678 91011121314151617181920212223242526272829303132333435
Number of Observations
O Origin + Ranks
Figure 4-41. Results of seasonal Kendall (square symbols) and seasonal Kendall with correction
for serial correlation (plus symbols) for raw ANC data shown in Figure 4-39.
96
-------
Slope Magnitude at Site 2 on ANC
a
o
w
i i i I i i i i i i i i i i i i i i i i i i i i i I
7 8 91011121314151617181920212223242526272829303132333435
Number of Observations
Figure 4-42. Least squares regression slope of the entire ANC series at Station 2 in Twin
Lakes, Colorado, as a function of number of observations.
97
-------
40.0000T
30.0000-
<
-------
Analysis of Covortance at Site 4 on ANC
O
*»
I/)
*•
n
o
-5.5
i T i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—r
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Origin
Number of Observations
Ranks
Critical Value
Figure 4-44. Results of ANOCOV on raw ANC and on ranks of ANC time series shown in
Figure 4-43. Critical value of the test statistic is shown for each number of
observations. The test is significant when the calculated statistic is more
negative than the critical value.
99
-------
Seasonal Kendall at Sfte 4 on ANC
JD
53
D
I I I I I I I I I I I I I I I I I 1 I I
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Number of Observations
O Origin + Correct
Figure 4-45. Results of seasonal Kendall (square symbols) and seasonal Kendall with correction
for serial correlation (plus symbols) for raw ANC data shown in Figure 4-43.
100
-------
Slope Magnitude at Site 4 on ANC
a
o
w
-1
-2
-4
-5
-6
-7
-8
-9
II
L
I I I I I I I I I 1 I I I I I I I I ! I | I I I T I I
7 8 91011121314151617181920212223242526272829303132333435
Number of Observations
Figure 4-46. Least squares regression slope of the entire ANC series at Station 4 in Twin
Lakes, Colorado, as a function of number of observations.
101
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SECTION 5
SPECIALIZED PROCEDURES FOR EXPLANATION OF TRENDS
The trend testing procedures discussed thus far are designed for "quick-and-easy" analysis
of TIME series for individual variables at single locations (or average values over multiple
locations). They are broadly applicable and do not require local adjustment. Thus, they can be
applied over the entire country for the entire suite of TIME water quality variables. The pro-
cedures are ideal screening or exploratory tools for routine data anlysis in national or regional
routine monitoring programs--providing useful, comparable results quickly and efficiently.
However, because of their flexibility, the methods are not well suited for explaining the
causes of trends. They can tell only whether apparent trends are likely to be the result of
chance or of "real" change. Specialized techniques that consider interrelationships among
multiple water quality variables and/or local watershed conditions are sure to be more powerful
for detecting trends of a certain type (caused by acid rain, for example) and for explaining
possible causes of trends that are indicated by exploratory analysis. Of course, strictly
speaking, an observational study like TIME cannot establish "cause." It can only formulate
explanations of causal mechanisms that are consistent with observations.
5.1 ADJUSTMENT FOR HYDROLOGIC FACTORS—STREAM FLOW AND PRECIPITATION
For streams, the effect of flow/concentration relationships should always be considered in
trend monitoring. The simplest way to account for such relationships is to use an appropriate
data transformation to obtain flow-adjusted concentrations. Trend tests may then be applied to
both adjusted and nonadjusted data. Significant trends in nonadjusted data that do not appear
in flow adjusted data are deemed to be the result of changes in flow.
Since there are many causes of flow/quality relationships, the functional form of flow
adjustments is highly dependent on local factors (Hirsch et al., 1982). Local calibration, either
from long-term records or short-term intensive studies, is necessary. Thus flow adjustments
may not be possible, initially, at all TIME stream stations unless they correspond to USGS or
other longer term monitoring locations. Once calibrated, flow adjustments become a part of
routine data analysis, although periodic reevaluation and recalibration of flow/quality
relationships are required.
Lakes, of course, do not exhibit the same sort of flow/quality relationship as streams.
However, the same processes—dilution, washoff, etc.—affect lake quality. Thus, correction for
precipitation (or inflow or lake level) is appropriate and analogous to flow correction in streams.
The same requirements for local calibration over significant time periods apply.
5.2 WATER QUALITY INDICES
Observed values of multiple water quality variables (more than one constituent, time, or
location) may be combined to provide a single number that effectively represents water quality
relative to some particular standard, intended use, or external impact (Dinius, 1987). The
arithmetic average of ANC over several lakes for a single quarter is such an index, as is the
ratio of ANC to sulfate. Although any combination of water quality variables into a single one
may be thought of as an index, the more common forms are linear combinations or ratios (or
both) of two or more water quality constituents for one location at one time.
103
-------
As single variables, water quality indices can be tested for trend using the SK and
ANOCOV procedures. The nature of the index can provide considerable insight into the cause of
any significant trend, since the ideal index would be highly sensitive to the changes in water
quality in which we are interested—for example acid deposition effects—and very insensitive to
other changes or variability.
The construction of effective water quality indices is difficult, requiring considerable water
quality data and good understanding of the biological, chemical, physical, and hydrological
factors determining water quality at a particular location. The more detailed and discriminatory
an index, the more dependent it is on local factors; thus, the less transportable and broadly
applicable it becomes.
5.3 MULTIVARIATE TESTS FOR TREND
Both the ANOCOV and SK tests can be viewed as multivariate procedures in that they
consider water quality observations from different seasons simultaneously. For both tests,
extensions are possible such that more than one variable or more than one location may be
considered. An example of such a test is proposed by Dietz and Killeen (1981). Their test,
based on Kendall-tau, considers the possibility that some variables might have upward trends
while others trend downward. Hirsch and Slack (1984) suggest, though, that the Dietz and
Killeen test would be applicable only for very long records (at least when used on a single
variable as an alternative to SK).
ANOCOV is ideally suited for multivariate extensions. A vector of response variables can
be considered, as in [Ca"1"1", ANC, SO4=] instead of a single variable. Other predictive variables
can be added to the list of four that are used to indicate year and season. Possibilities for
additional covariates include stream flow or lake level, quarterly precipitation, acid deposition,
and additional water quality variables. In the general case, the covariates can be log linear
transformations (perhaps including rank transformations) of observed values of predictive
variables. However, comparatively little work has been done on multivariate trend detection and
additional research is needed on both general and site specific levels.
5.4 WATER QUALITY/WATERSHED MODELS
At a more intensive level of study, it will be possible to obtain process descriptions and
hydrologic/chemical/biological models of system behavior. Using such models, carefully cali-
brated for local geochemistry and other factors, it should be possible to identify specific and
quantitative input-output relationships between such factors as acid deposition over a watershed
and chemical response of receiving waters. Such relationships will be helpful in studying both
long-term trends and episodic responses. (The latter short-term effects may not result in
statistically significant trend.)
Only at this level is it possible to forecast water quality conditions and to examine "what
if scenarios that might be useful in developing management strategies. This level of data anal-
ysis, process modeling, is far from routine and is thus distinct from the earlier discussion.
However, routinely collected data are useful for calibration and verification of watershed type
models over a range of hydrologic conditions. Furthermore, background data sets are needed
when models are moved from one watershed to another. Therefore, routine monitoring plays an
important role, even at the level of intensive studies or research, in a monitoring program such
as TIME that is designed to support water quality management on a broad geographical and
long-term basis.
104
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SECTION 6
DETAILS OF TREND TESTING AND MONTE CARLO METHODS
In an attempt to find a good method of trend detection in seasonal time series data, seven
different tests were analyzed by means of simulation. Three of the tests are based on the
Mann-Kendall test (Mann, 1945; Kendall, 1975), which utilizes the sign of the pairwise differ-
ences of the data. Two of the tests, the analysis of covariance and a modified t-test, use a
linear model and normal theory. The final two tests are the application of the analysis of
covariance and the modified t-test to the ranks of the data.
6.1 MANN-KENDALL TESTS
The first method of trend detection discussed in this section consists of applying the
original Mann-Kendall test to deseasonalized data. That is, if an observation is made during
season i, i = l(l)p, where p is the number of seasons, then the mean of all observations made
during season i is subtracted from this observation. The Mann-Kendall test is then applied to
these differences.
The Mann-Kendall procedure tests the null hypothesis that the observations are randomly
permuted against the alternative hypothesis of a monotone trend. Define
1 if x > 0
sign(x) = 0 if x = 0
-1 if x < 0
If we apply the Mann-Kendall test to the sequence {Xj: i = 1(1 )n), then under the null
hypothesis, the test statistic
S = E sign (Xj - Xi)
is normally distributed with a mean of zero and a variance,
var (S) = n(n-l)(2n + 5)/18.
The seasonal Kendall test automatically compensates for seasonality in the mean of the
data. If our seasonal data is the sequence {Y^: i = l(l)n, j = l(l)p}, where n is the number of
years and p is the number of seasons, then for each season, j, compute
Sj = E sign (Ykj - Y,J)
(Sum is for k < 1)
The test statistic is the sum S = E Sj, which has a zero mean and a variance of
var (S) = E var(S=) + E cov(S:, Sk).
Note that each Sj is the original Mann-Kendall test statistic computed from the jth season's
data.
105
-------
We considered two versions of this seasonal Kendall test. The first (Hirsch et al., 1982)
assumes that the covariance terms in the var(S) are negligible. Thus
var (S) = E var(Sj) = E [n(n - 1) (2n + 5)/18]
- p[n(n - l)(2n
In the other version (Hirsch and Slack, 1984), the covariances are estimated as
cov(Sj,Sk) = Kjk/3 + (n3 - n)rjk/9
where
Kjk = E signRXy - Xmj)(Xlk - Xmk)]
rjk = 7 S sign[(Xy - Xmj)(Xlk - Xqk)]
(summed over 1, m, and q).
Adjustments to this variance are given in Hirsch and Slack (1984) to compensate for missing
values.
6.2 LINEAR MODEL TESTS
Two of our tests, the analysis of covariance and the modified t, are based on the
underlying linear model
(where i = 1(1 )n and j = 1(1 )p)
and normal theory. The seasonality of the quarterly means is modeled by the /*j, and the e^ are
assumed to be independently distributed normal with a zero mean and variance a?.
The analysis of covariance consists of a multiple regression of a dependent variable Yy on
p independent variables, t = [(i-l)p+j] and {ukij | k = l(l)p-l) where
1 if j = k
Uy = -1 if j = p
0 otherwise
In the regression equation, the coefficient of t, 0it represents a linear trend in Y, and the
coefficients of uki:, say Mk, compensate for any seasonality in the quarterly means of Y. (Only
three indicator variables are needed for four means, since M1 + ... + Mp = 0 by assumption.)
Hence, by usual linear methods, we may estimate £j and test for a linear trend while auto-
matically adjusting for seasonality in the means. This type of linear regression of a dependent
variable on a continuous independent variable and on a set of indicator variables is typically
called analysis of covariance and is discussed fully in Neter and Wasserman (1974).
106
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Note that, as in all standard linear regression, it is assumed the variances of all of the
observations are equal. Clearly, this is not the case. However, it is of interest to see how this
procedure performs when this assumption is not valid. We remove this assumption with the
modified t-test, which is discussed in the next section.
6.3 MODIFIED T-TEST
As mentioned in subsection 6.2, our water quality data generally do not satisfy the homo-
genity of variance assumption of the analysis of covariance. In this section, we remove this
assumption with the modified t-test.
The analysis of covariance procedure implicitly assumes that the linear trend, if it exists,
remains constant over the p seasons. Suppose this assumption is maintained, and a separate
simple regression of Yy versus t = [p(i-l) + j] is computed for each j = 1(1 )p. The sum of the
p resulting estimates of /3X (denoted by {bj | j = l(l)p}) may now be utilized for a test of /9X =
0, which no longer requires the assumption of equal seasonal variances.
This test incorporates Satterthwaite's approximation of the distribution of a linear com-
bination of independent x2 (chi-squared) random variables (Satterthwaite, 1946). Essentially, this
approximation assumes that if djZj/Cp-x2 (dj and Cj are constants and Z; is a random variable),
with degrees of freedom dt for i = l,...,m, then the sum, W = £ c^/d; is distributed as a scaled
X2 (i.e., for constants c and d, dW/c is distributed x2 with degrees of freedom d). The con-
stants c and d may be found by merely equating the first and second moments.
In order to derive this test, some definitions are needed. Define
E(j + p(i-l)) (n-l)p + 2j
= - (summing over i)
n
and
-.3 _ n\r>2
TT
(summing over i)
St. = £(j + P(i-l)--t)2 = ("3 - ">P2
i 11
Let MSE(j) be the mean squared error arising from regressing the observations belonging to
season j on t - [p(i-l)+j]. Since MSE(j) can be used to adequately estimate the variance o|, the
variance of bj,
V(bp = of/st ,
can be estimated as
s? = MSE(j)/st .
j
Since (n - 2) MSEQ/o? ~ x2 (degrees of freedom = n - 2), it is known that
107
-------
and
V(E
n - 2
Now, if we claim that W = E sj has a scaled x2 distribution with degrees of freedom d and a
scale parameter of c, then we can solve for c and d by equating the first two moments. That
is, let
E(E s?) = c and
V(E sf ) - 2c2/d
then
(Ea2/st)2
d = J
2(Ea?/s2 )
J j
n - 2
Although the cr? are not known, they can then be estimated by MSE(j). Therefore, we can
estimate the degrees of freedom as
(E s?)/(n-2)
Since the sum E ty is distributed normal with a mean E/Sj and variance c,
^ (E bs)
[(E s?)/c]*
Sb;
is approximately distributed t with df = d*. Thus, we can use T as our test statistic, and test
it against the Student's t distribution with d* degrees of freedom.
6.4 RANK TRANSFORMATIONS
Conover (1980) suggests using ranks instead of the raw data as a nonparametric extension
of multiple regression techniques. Hence, two additional tests involve using the analysis of
covariance and the modified "t" on the ranks of data, rather than on the raw data.
108
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6.5 MONTE CARLO SIMULATIONS
In order to compare the seven different tests, we performed an extensive simulation study.
Simulated data records consisting of 5, 15, and 25 years (n = 5, 15, 25) of quarterly (p = 4) data
were generated with a variety of different trend magnitudes and seasonality in the means and
the standard deviations.
The data were generated with models of the form
yy = mj + b [4(i-l)+j] + ey
where, as before, i indexes the year and j indexes the season. The parameters nij, j = 1,...,4,
introduce seasonality. The number b gives a trend to the data (for b £ 0), and the ey is a
simulated random error.
For each model, the seasonality parameters, nij, were constrained to sum to four, and each
nij took one of two possible values denoted by "high" or "low." Different ratios of "high" to
"low", denoted by rm, as well as two different patterns of seasonality, (low, high, low, low) and
(high, low, high, low), were considered. Therefore, the seasonality in the means is uniquely
specified by the pattern of seasonality and rm.
The errors, e^, were generated for the normal and the lognormal distributions. In the case
of the normal distribution, both uncorrelated and correlated errors were produced. Furthermore,
seasonality in the standard deviations of these errors was introduced. This seasonality followed
the same high/low pattern as for the means, with the ratio of high to low set to rs. However,
unlike the means, a- was set to 1.0 for seasons with "low" standard deviations and to rg for
seasons with "high" standard deviations.
For each set of parameters (the term "parameters" includes the number of years and the
pattern of seasonality, as well as rg, rm, and b), 500 data records were created. All seven tests
were applied to each record with a theoretical significance level of 0.05, and the total number
of rejections for each test was recorded. A stepwise logistic regression procedure was used to
determine which of the parameters significantly (at the 0.05 level) affected the simulated power.
In all cases examined, all parameters had a significant effect, except for rm, the ratio of season
means.
The results of the simulations are presented in terms of simulated power. In Appendix B,
Tables B-l through B-6, results are presented according to the parameters that were found to be
significant in the stepwise regression. For each combination of parameters, the number of
rejections are summed over the values of rm. The tabulated powers are this sum divided by the
product of the number of values that rm assumes and 500. Tables 6-1, 6-2, 6-3, and 6-4
summarize the combined results, according to length of record and trend magnitude only.
We attempted to use logistic regression techniques to model these powers as functions of
the applied test procedure, as well as of the significant parameters. We hoped that a model
could be established that would adequately explain the effects of the various parameters on
powers of the different tests. Unfortunately, our available computer resources restricted our
fitted model to second order or less, which was too crude for this application. Consequently,
we are left to compare the tabulated results directly.
109
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TABLE 6-1. SIMULATED POWERS FOR NORMAL ERRORS AFTER AVERAGING OVER
RATIO OF STANDARD DEVIATIONS, RATIO OF MEANS, AND
PATTERNS OF SEASONALLY
Years Slope
.000
.002
.005
5 .020
.050
.200
.500
.000
.002
.005
15 .020
.050
.200
.500
.000
.002
.005
25 .020
.050
.200
.500
Anal, of
covar.
.0440
.0482
.0456
.0612
.1418
.7642
.9815
.0518
.0520
.0705
.4338
.8725
1.0000
1.0000
.0495
.0709
.1793
.8224
.9978
1.0000
1.0000
Mann-Ken.
on deseas.
Mod. t data
.0463
.0488
.0452
.0562
.1216
.6956
.9629
.0506
.0528
.0697
.4222
.8607
1.0000
1.0000
.0497
.0703
.1773
.8158
.9977
1.0000
1.0000
.0598
.0651
.0598
.0774
.1723
.8486
.9930
.0578
.0623
.0807
.5118
.9608
1.0000
1.0000
.0512
.0788
.2056
.9307
1.0000
1.0000
1.0000
Seas. Ken.
w/correc.
for cov.
.0067
.0065
.0068
.0097
.0175
.3191
.9276
.0432
.0482
.0681
.4664
.9795
1.0000
1.0000
.0445
.0702
.1938
.9691
1.0000
1.0000
1.0000
Seas. Ken.
w/o correc.
for cov.
.0338
.0370
.0336
.0447
.1108
.8276
.9949
.0475
.0513
.0722
.5079
.9864
1.0000
1.0000
.0452
.0749
.2077
.9741
1.0000
1.0000
1.0000
Anal, of
cov. on
ranks
.0483
.0548
.0517
.0675
.1527
.8259
.9881
.0534
.0554
.0758
.4977
.9550
1.0000
1.0000
.0486
.0744
.2006
.9271
.9999
1.0000
1.0000
Mod. t
on ranks
.0433
.0465
.0440
.0560
.1281
.7816
.9838
.0523
.0545
.0753
.4915
.9573
1.0000
1.0000
.0495
.0739
.1990
.9278
1.0000
1.0000
1.0000
110
-------
TABLE 6-2. SIMULATED POWERS FOR LOGNORMAL ERRORS AFTER AVERAGING
OVER RATIO OF STANDARD DEVIATIONS, RATIO OF MEANS, AND
PATTERNS OF SEASONALITY
Years Slope
.000
.002
.005
5 .020
.050
.200
.500
.000
.002
.005
15 .020
.050
.200
.500
.000
.002
.005
25 .020
.050
.200
.500
Anal, of
covar.
.0388
.0426
.0405
.0738
.2523
.8278
.9764
.0448
.0508
.0894
.5368
.8957
.9988
.9999
.0460
.0724
.2331
.8531
.9908
1.0000
1.0000
Mod. t
.0369
.0398
.0383
.0616
.2004
.7613
.9560
.0442
.0512
.0851
.5212
.8879
.9987
.9999
.0445
.0712
.2275
.8459
.9899
1.0000
1.0000
Mann-Ken.
on deseas.
data
.0504
.0561
.0581
.1248
.3985
.9440
.9973
.0473
.0710
.1961
.8708
.9972
1.0000
1.0000
.0471
.1582
.6072
.9960
1.0000
1.0000
1.0000
Seas. Ken.
w/correc.
for cov.
.0058
.0096
.0103
.0214
.1033
.7902
.9833
.0448
.0738
.2312
.9498
1.0000
1.0000
1.0000
.0492
.2090
.7505
1.0000
1.0000
1.0000
1.0000
Seas. Ken.
w/o correc.
for cov.
.0362
.0383
.0425
.1148
.4234
.9688
.9993
.0490
.0842
.2591
.9615
1.0000
1.0000
1.0000
.0537
.2218
.7772
1.0000
1.0000
1.0000
1.0000
Anal, of
cov. on
ranks
.0509
.0540
.0615
.1383
.4403
.9474
.9988
.0502
.0829
.2313
.8842
.9975
1.0000
1.0000
.0518
.1870
.6541
.9968
1.0000
1.0000
1.0000
Mod. t
ranks
.0438
.0468
.0533.
.1174
.3878
.9270
.9984
.0499
.0813
.2294
.8843
.9983
1.0000
1.0000
.0516
.1852
.6537
.9973
1.0000
1.0000
1.0000
111
-------
TABLE 6-3. SIMULATED POWERS FOR NORMAL ERRORS WITH p = 0.2 AFTER
AVERAGING OVER RATIO OF STANDARD DEVIATIONS, RATIO OF
MEANS, AND PATTERNS OF SEASONALITY
Years Slope
.000
.002
.005
5 .020
.050
.200
.500
.000
.002
.005
15 .020
.050
.200
.500
.000
.002
.005
25 .020
.050
.200
.500
Anal, of
covar.
.1393
.1402
.1375
.1507
.2317
.7125
.9598
.1590
.1663
.1813
.4478
.7850
1.0000
1.0000
.1538
.1760
.2652
.7412
.9833
1.0000
1.0000
Mod. t
.1155
.1140
.1177
.1240
.1915
.6402
.9175
.1508
.1592
.1735
.4383
.7682
1.0000
1.0000
.1495
.1735
.2597
.7312
.9828
1.0000
1.0000
Mann-Ken.
on deseas.
data
.1722
.1700
.1733
.1877
.2735
.8008
.9837
.1813
.1922
.2040
.5288
.9118
1.0000
1.0000
.1765
.2020
.3048
.8778
.9987
1.0000
1.0000
Seas. Ken.
w/correc.
for cov.
.0130
.0107
.0108
.0128
.0248
.3178
.8803
.0743
.0792
.0840
.3480
.9043
1.0000
1.0000
.0730
.0903
.1683
.8630
1.0000
1.0000
1.0000
Seas. Ken.
w/o correc.
for cov.
.1230
.1212
.1202
.1317
.2073
.8023
.9882
.1657
.1755
.1900
.5385
.9553
1.0000
1.0000
.1760
.2025
.3068
.9352
1.0000
1.0000
1.0000
Anal, of
cov. on
ranks
.1617
.1587
.1582
.1662
.2560
.7858
.9757
.1732
.1842
.1978
.5215
.9050
1.0000
1.0000
.1732
.1972
.2995
.8767
.9982
1.0000
1.0000
Mod. t
ranks
.1302
.1358
.1308
.1385
.2187
.7325
.9603
.1720
.1808
.1920
.5143
.9052
1.0000
1.0000
.1692
.1932
.2955
.8750
.9988
1.0000
1.0000
112
-------
TABLE 6-4. SIMULATED POWERS FOR NORMAL ERRORS WITH p = 0.4 AFTER
AVERAGING OVER RATIO OF STANDARD DEVIATIONS, RATIO OF
MEANS, AND PATTERNS OF SEASONALITY
Years Slope
.000
.002
.005
5 .020
.050
.200
.500
.000
.002
.005
15 .020
.050
.200
.500
.000
.002
.005
25 .020
.050
.200
.500
Anal, of
covar.
.1393
.1402
.1375
.1507
.2317
.7125
.9598
.1590
.1663
.1813
.4478
.7850
1.0000
1.0000
.1538
.1760
.2652
.7412
.9833
1.0000
1.0000
Mod. t
.1155
.1140
.1177
.1240
.1915
.6402
.9175
.1508
.1592
.1735
.4383
.7682
1.0000
1.0000
.1495
.1735
.2597
.7312
.9828
1.0000
1 .0000
Mann-Ken.
on deseas.
data
.1722
.1700
.1733
.1877
.2735
.8008
.9837
.1813
.1922
.2040
.5288
.9118
1.0000
1.0000
.1765
.2020
.3048
.8778
.9987
1.0000
1.0000
Seas. Ken.
w/correc.
for cov.
.0130
.0107
.0108
.0128
.0248
.3178
.8803
.0743
.0792
.0840
.3480
.9043
1.0000
1.0000
.0730
.0903
.1683
.8630
1.0000
1.0000
1 .0000
Seas. Ken.
w/o correc.
for cov.
.1230
.1212
.1202
.1317
.2073
.8023
.9882
.1657
.1755
.1900
.5385
.9553
1.0000
1.0000
.1760
.2025
.3068
.9352
1.0000
1.0000
1.0000
Anal, of
cov. on
ranks
.1617
.1587
.1582
.1662
.2560
.7858
.9757
.1732
.1842
.1978
.5215
.9050
1.0000
1.0000
.1732
.1972
.2995
.8767
.9982
1.0000
1.0000
Mod. t
ranks
.1302
.1358
.1308
.1385
.2187
.7325
.9603
.1710
.1808
.1920
.5143
.9052
1.0000
1.0000
.1692
.1932
.2955
.8750
.9988
1.0000
1.0000
113
-------
6.5.1 Normal Uncorrelated Errors
In the case of normal errors with no correlation, we considered
b = 0.0, 0.002, 0.005, 0.02, 0.05, 0.2, 0.5
rm = 1.0, 1.5,2.0
rg= 1.0, 1.5, 3.0, 5.0
Observations from independently distributed standard normal random variables were
generated by the IMSL subroutine, GGNML. These observations are denoted by
where, as with the y^, i indexes the year and j indexes the season. The errors, e^ were
created as
As we can see from Tables 6-1, B-l, and B-2, the best tests appear to be the analysis of
covariance on ranks, the modified "t", and the seasonal Kendall test without correction for
correlation. The other procedures are considered to perform less well than these three tests
because either their powers were less or their simulated critical value (power when ftt = 0) was
greater than 0.05.
In general, these three tests gave similar performances. However, for short records (5
years), the analysis of covariance on ranks was the best test, except when rg = 5.0. With
longer records, the only discrimination between the performances of these tests occurs for rg =
3.0 and 5.0. In these cases, the seasonal Kendall test without correction for correlation has the
highest power.
As expected, for all of the tests we see a general decrease in power with an increase in
seasonality. Not only does the power drop with an increasing rg, but the powers are less for
the pattern of seasonality
{high, low, high, low)
than for the pattern
{low, high, low, low}
This also agrees with intuition, since the second pattern has higher "overall" variance.
6.5.2 Lognormal Errors
The seven tests were also compared when the errors were lognormally distributed and the
parameters had the same specifications as for the normal errors. The lognormal errors were
created by first generating the standard normal observations z^ and then letting
djj = [exp (Zjj) - exp (l/2)]/[exp (2)-exp (1)],
114
-------
which are distributed lognormal with a zero mean and a variance of one. Then let
which will be lognormally distributed with a zero expected value and variances corresponding to
the definition of rg and the patterns of standard deviations.
From Tables 6-2, B-3, and B-4, we can see that the best tests for lognormal errors were
the analysis of covariance on ranks, the modified "t" on ranks, and the seasonal Kendall test
without correction for correlation. The analysis of covariance on ranks has the highest powers
for short records, except for rg = 5.0. In this case, the seasonal Kendall test without correction
for correlation performed better than the other methods. For longer records, the seasonal
Kendall test without correction for correlation performed better than the other two methods.
As with normal errors, we see an overall decrease in power with an increase in seasonality.
Again, this increase in seasonality is given by both an increase in rg and a change in the
pattern of standard deviations from
{low, high, low, low}
to
{high, low, high, low)
6.5.3 Normal Errors with Positive Correlation
In this case, we introduce a positive correlation between e^ and the previously "observed"
error, e^ (ey rl = e^*, where
(i* j*) = 0, j - 1) if j > 1).
(i-l.p)ifj-l
First, a sequence of standard normal observations, zt (t = 1(1 )np), was generated with the IMSL
subroutine, GGNML. From the zt, an AR(1) process, dt, is generated with a lag-one correlation,
p = 0.2, 0.4. This is accomplished by letting
Thus, the sequence of dt's has a lag-one correlation, p, and a stationary variance of unity.
Since dt depends on dt.l5 da is set equal to a standard normal observation. Then the process is
"warmed up" by generating 500 preliminary observations.
After establishing the AR(1) process, e^ is generated as
This gives a positively correlated sequence with seasonal variances. Note, however, that the
lag-one correlation is no longer p, and it is now dependent on j. Specifically,
if j > 2
115
-------
if
Simulations for this error structure were run with
rm= 1.0, 1.5,2.0
r. » 1, 5
b = 0, 0.002, 0.005, 0.02, 0.05, 0.2, 0.5
Tables 6-3, B-5, and B-6 show that none of the seven tests performed very well. With the
exception of the seasonal Kendall test with correction for correlation, all of the test procedures
had excessive significance levels for both levels of rg and p, for all three record lengths. On
the other hand, the seasonal Kendall test with correction for correlation had very small
significance levels for five years of data. (For five years of data and p = 0.2, the significance
level for the seasonal Kendall test without correction for correlation is not much larger than
0.05. Hence, we might consider using this test for short records and low correlation.) For 15
and 25 years of data, the significance levels of the corrected SK test were acceptable for p =
0.2. However, for p = 0.4, the significance levels were large, but not nearly as large as those
for the other tests. Therefore, it appears that the best procedure for large data records and
small correlations is the Seasonal Kendall test with correction for correlations.
6.5.4 Summary
For uncorrelated errors, the best methods of detecting trends in time series are the
analysis of covariance on ranks and the seasonal Kendall without correction for correlation.
From the tables of simulated powers, we can see that for the most part, the difference in per-
formance between these two methods is not large. Hence, both of these two tests appear to be
appropriate for detection of trends in seasonal time series with uncorrelated errors. However, if
a choice must be made between these two methods, the seasonal Kendall is recommended, especi-
ally for long data records.
For correlated data, none of the tests appears to be very good. However, the seasonal
Kendall test with correction for correlation appears to be sufficient for large data records with
small correlations.
116
-------
REFERENCES
Conover, W.J. 1980. Nonparametric Statistics. Second Edition. John Wiley and Sons, New
York.
Dietz, G.J. and T.J. Killeen. 1981. A nonparametric multivariate test for monotone trend with
pharmaceutical applications. J. Am. Slat. Assn. 76(373): 169-174.
Dinius, S.H. 1987. Design of an index of water quality. Water Resources Bull. 23(5):833-843.
Gilbert, R.O. 1987. Statistical Methods for Environmental Pollution Monitoring. Van Nostrand
Reinhold, New York.
Hirsch, R.M. and E.J. Gilroy. 1985. Detectability of step trends in the rate of atmospheric
deposition of sulfate. Water Resources Bull. 21(5):733-784.
Hirsch, R.M. and J.R. Slack. 1984. A nonparametric trend test for seasonal data with serial
dependence. Water Resources Reseach. 20(6):727-732.
Hirsch, R.M., J.R. Slack and P.A. Smith. 1982. Techniques of trend analysis for monthly water
quality data. Water Resources Research. 18(1):107-121.
Hollander, M. and D.A. Wolfe. 1973. Nonparametric Statistical Methods. John Wiley, New York.
Kendall, M.G. 1975. Rank Correlation Methods. Fourth Edition. Charles Griffin, London.
Lettenmaier, D.P. 1976. Detection of trends in water quality data for records with dependent
observations. Water Resources Research. 12:1037-1046.
Mann, H.B. 1945. Nonparametric tests against trend. Econometrica. 13:245-274.
National Research Council. 1977. Analytical studies for the Environmental Protection Agency:
Vol. IV, Environmental Monitoring. Natl. Acad. Sci., Washington, D.C. (Library of Congress
Catalog Card No. 77-86463).
Neter, W.E. and J.A. Wasserman. 1974. Applied Linear Statistical Models. Irwin, Inc., New
York.
Newell, A.D., C.F. Powers and S.J. Christie. 1987. Analysis of data from long-term monitoring
of lakes. EPA600/4-87/014, U.S. EPA Environmental Research Laboratory, Corvallis,
Oregon,
Newell, A.D. 1987. Personal communication. U.S. EPA Environmental Research Laboratory,
Corvallis, Oregon. May.
Nicholls, A. 1987. Personal communication. Ministry of the Environment. Dorset, Ontario,
Canada. July 31.
117
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Sartoris, J. 1987. Personal communication. U.S. Department of Interior, Bureau of Reclamation,
Denver, Colorado. June 26.
Satterthwaite, F.G. 1946. An approximate distribution of estimates of variance components.
Biometrics. 2(6): 110-114.
Smith, R.A., R.B. Alexander and M.G. Wolman. 1987. Water quality trends in the Nation's
rivers. Science. 235:1607-1615.
Snedecor, G.W. and W.G. Cochran. 1980. Statistical Methods. The University of Iowa Press,
Ames, Iowa.
118
-------
ABBREVIATIONS AND ACRONYMS
ANC -- Acid neutralizing capacity
ANOCOV — Analysis of covariance
Ca -- Calcium
LTM -- Long-term Monitoring
NLS -- National Lake Survey
NLS-1A — National Lake Survey subregion 1A - Adirondacks
NLS-IB -- National Lake Survey subregion IB - Poconos/Catskills
NLS-1C -- National Lake Survey subregion 1C - Central New England
NLS-ID -- National Lake Survey subregion ID - Southern new England
NLS-IE -- National Lake Survey subregion IE - Maine
NRC — National Research Council
OLS — Ordinary least squares
SK -- Seasonal Kendall
SO4 — sulfate
TIME -- Temporally Integrated Monitoring of Ecosystems
119
-------
APPENDIX A
TIME GOALS AND OBJECTIVES
(Reprinted from "The Concept of Time" Report)
To provide a regional-scale assessment of the effects of acidic deposition on aquatic
ecosystems, a long-term monitoring program needs to incorporate representative site selection,
measurement of biologically relevant chemical variables, standardized analytical methods and
quality assurance protocols, and a sampling scheme that permits long-term changes in chemical
response to be differentiated from episodic changes and short-term daily, monthly, or annual
periodicities. The monitoring program must be predicted on a clear set of goals and objectives.
GOALS
The TIME Project has as its goals to:
• Estimate the regional proportion and subpopulation physiochemical characteristics of
lakes and streams that exhibit early and ongoing trends of surface water acidification
or recovery.
• Compare patterns and trends in observed surface water chemistry to forecasts made
using empirical or process-oriented procedures.
• Determine the relationships between patterns and trends in atmospheric deposition and
trends in surface water chemistry for defined subpopulations of aquatic resources in
areas particularly susceptible to acidification or recovery.
OBJECTIVES
In order to achieve these goals, the TIME project has the following objectives:
• Provide an early and ongoing indication of regional trends in surface water
acidification or recovery, using the most appropriate techniques to detect such trends.
• Quantify, with known certainty, for defined subpopulations of lakes and streams:
The rate at which changes in relevant chemistry are occurring.
The subpopulation characteristics of the affected lakes and/or streams.
The regional or subregional extent of these systems.
• Compare trends in local and regional atmospheric deposition with regional trends in
surface water chemistry.
120
-------
APPENDIX B
RESULTS OF SIMULATIONS
This appendix presents the results of simulations in terms of simulated power. Tables B-l
through B-6 show the results according to the parameters that were found to be significant in
the stepwise regression. For each combination of parameters, the number of rejections are
summed over the values of rm. The tabulated powers are this sum divided by the product of
the number of values that rm assumes and 500.
121
-------
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