VOL. 27, NO. 3
WATER RESOURCES BULLETIN
AMERICAN WATER RESOURCES ASSOCIATION
JUNE 1991
EPA/600/J-94/510
MULTIVARIATE TREND TESTING OF LAKE WATER QUALITY1
Jim C. Loftis, Charles H. Taylor, Avis D. Newell, and Phillip L. Chapman2
ABSTRACT: Multivariate methods of trend analysis offer the
potential for higher power in detecting gradual water quality
changes as compared to multiple applications of univariate tests.
Simulation experiments were used to investigate the power advan-
tages of multivariate methods for both linear model and Mann-
Kendall based approaches. The experiments focused on quarterly
observations of three water quality variables with no serial correla-
tion and with several different intervariable correlation structures.
The multivariate methods were generally more powerful than the
univariate methods, offering the greatest advantage in situations
where water quality variables were positively correlated with
trends in opposing directions. For illustration, both the univariate
and multivariate versions of the Mann-Kendall based tests were
applied to case study data from several lakes in Maine and New
York which have been sampled as part of EPA's long term monitor-
ing study of acid precipitation effects.
(KEY TERMS: trend analysis; multivariate analysis; statistics;
lake water quality; acid precipitation.)
INTRODUCTION
Trend analysis is a formal approach to deciding
whether an apparent change in water quality is likely
due to random noise or to an actual underlying
change in water quality. The use of formal statistics
does not generally imply that one will be able to
detect trends that are not apparent from inspection of
the data. However it does mean that different data
analysts will be able to reach the same conclusions,
given the same data and acceptance of a common set
of assumptions.
Trend analysis may be defined for present purposes
as a formal test of the null hypothesis that the long-
term mean(s) of a given water quality variable (or
vector of variables) are not changing over time. A "sig-
nificant" trend is said to exist when the null hypothe-
sis is rejected. Whether a trend is "significant,"
therefore, depends on its magnitude relative to the
variance of observations, how long it persists, how
many observations are collected, and on the signifi-
cance level of the test. Unfortunately perhaps, "signif-
icance" is not defined directly in terms of the
importance of a given change relative to its impact on
aquatic habitat or other beneficial uses of the water
resource. A change of any magnitude will be deemed
statistically significant if enough samples are collect-
ed.
Trend analysis can be applied in many water quali-
ty monitoring programs where data are collected at
fixed locations over a long period of time. Long-term
studies of lakes (or streams) to assess the impacts of
acid precipitation are a prime example. A major goal
of monitoring lakes might be to determine whether
apparent changes in water quality were the result of
acidic deposition. However, trend analysis, as we have
defined it, cannot establish cause and effect relation-
ships and cannot, therefore,, accomplish this objec-
tive.
Trend tests can, however, serve as screening tools,
indicating the likelihood of an observed change occur-
ring as a result of natural variation when there was,
in fact, no underlying change in the mean. Those
changes which cause rejection of the null hypothesis
would be targeted for further study from chemical,
physical, and biological standpoints to establish
causal relationships.
Trend analysis has become an accepted part of
many monitoring programs and has been the subject
of considerable research over the past dozen years.
Most applications of trend analysis deal with a single
water quality variable or constituent at a single loca-
tion. However, attention has recently been directed to
!Paper No. 90060 of the Water Resources Bulletin. Discussions are open until February 1, 1992.
Respectively, Professor, Dept. of Agricultural and Chemical Engineering, 100 Engineering S., Colorado State University, Ft. Collins,
Colorado 80523; Senior Research Biostatistician, Meirell Dow Research Institute, 2110 E. Galbraith Rd., Cincinnati, Ohio 45215; Scientist,
Mantech Environmental Technology, Inc., 200 S.W. 35th St., Corvallis, Oregon 97333; and Associate Professor, Dept. of Statistics, Colorado
State University, 101 Statistics Bldg., Ft. Collins, Colorado 80523.
461
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Loflis, Taylor, Newell, and Chapman
analysis of multivariate trends, defined here as con-
sidering more than one constituent and/or more than
one location at a time.
Multivariate approaches have two advantages over
the more traditional approach of performing univari-
ate tests on several variables for the same site and
period of record. If a small, yet statistically insignifi-
cant, trend is apparent in multiple variables, then,
depending on the intervariable correlation structure,
a multivariate approach might detect a significant
trend when the variables are considered together. In
addition, the multivariate test controls the overall sig-
nificance level or probability of a false trend detection.
Application of several univariate tests results in an
inflation of the overall significance level unless the
significance levels of the individual tests are adjusted,
as we shall later discuss.
Recent works (Lettenmaier, 1988; Loftis et al.,
1991) have suggested several methods for multivari-
ate trend analysis, comparing their performance to
formulate recommendations of which is best in a
given set of circumstances. Both multivariate and
univariate tests were applied to selected data sets in
a study of stream quality trends across the U.S. for
the period 1978-1987 (Lettenmaier et al., 1991).
However, these papers did not examine the funda-
mental question of whether multivariate methods
offer an important advantage over the more tradition-
al univariate approach. We now attempt to answer
this question for a particular type of monitoring pro-
gram, focusing on detection of trends in lakes charac-
terized by seasonal sampling.
We consider two classes of methods, those based on
linear models or regression and those based on rank
correlation or the Mann-Kendall test for trend. A sim-
ulation study is used to compare powers of trend
detection for both classes of multivariate methods
against the powers of corresponding univariate meth-
ods in which the nominal significance levels are
adjusted using a Bonferroni inequality. For practical
illustration, we then apply univariate and multivari-
ate methods to real data, using the Mann-Kendall
based tests. The case study data were collected in a
study of the long-term effects of acid deposition.
The simulation study was confined to gradual,
monotonic changes, simulated as linear trends occur-
ring over 10 or 20 years. This type of change might be
encountered, for example, when acid deposition rates
are increased as a result of industrialization or
decreased as a result of air quality control measures.
We also focus on the issue of hypothesis testing and
exclude the important related question of estimating
the magnitude of trend.
BACKGROUND
Linear Models and Normal Theory
The concept of multivariate trend analysis is easily
introduced using linear models. Let us assume that
we are interested in a set of three water quality vari-
ables, say sulfate concentration, calcium concentra-
tion, and acid neutralizing capacity (ANC). We can
write a set of linear equations describing seasonal
variation of the variables and linear trend over time
as follows:
y =u+Bi + L . + e, . .
l.i.j 1 1 I.J 1, i. J
y =u+Bi+L . + e„ . .
2, i.j 2 2 2 ,j 2, i,j
y = u + B i + L +e
J 3, i, j 3 3 3 ,j 3, i,j
(1)
(2)
(3)
where:
yk j j = concentration of variable k in year i, season
j; k=l for sulfate; k=2 for calcium; and k=3
for ANC; i=l,2,. . . ,N (N=number of years
considered); j=l, 2, 3, 4.
uj( = mean of constituent k for season 4 in ab-
sence of trend.
Lkj
seasonal adjustment to the mean for con-
stituent k and season j. Ljg = the mean of
constituent k for season j minus the mean of
constituent k for season 4. L^ 4 = 0 since 114
= the mean for season 4.
ek,ij = error term with mean zero and unspecified
distribution. Errors may be correlated be-
tween variables and over time (serial cor-
relation).
Bfc = time trend for constituent k in units per
year.
The multivariate formulation considers the variables
and model parameters as vectors. Thus we have y, u,
B, L, and e, each a vector of dimension three, and the
multivariate model is
yiJ = u + Bj + Lj + eg
(4)
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Multivariate Trend Testing of Lake Water Quality
Other model forms are possible. For example, one
* could let each constituent by season combination be
represented by a separate independent variable.
A univariate null hypothesis that any of the indi-
vidual B)< equal zero can be tested using multiple lin-
ear regression and the analysis of covariance
described in Taylor and Loftis (1989) (see Anderson,
1984, for a more general reference). The regression
will also provide estimates of the parameters B^, u^,
and L^j. Standard regression techniques are based on
normality of the error terms e^jj. Taylor and Loftis
(1989) found that a simple extension of analysis of
covariance using the ranks of data, as suggested by
Conover (1980), provides a powerful nonparametric
test. However, the classical parametric approach is
used here.
When testing for trend in more than one con-
stituent or location, the usual approach is to perform
multiple univariate tests, each at some significance
level, a, of say 5 percent. The probability of rejecting
the null hypothesis when true is therefore no greater
than 5 percent for any single test, but the probability
of at least one rejection in K tests when all K null,
hypotheses are true may be much greater than a. If
N=3 as in our example, the probability of at least one
rejection must be no greater than 3a-Sa^-a3, depend-
ing on the correlation between variables. With a =
0.050000, this probability is 0.142625.
To control the overall significance level (probability
of at least one false rejection), one could apply a
Bonferroni inequality and perform K univariate tests,
each at a significance level of a/K (Snedecor and
Cochran, 1980). When this approach is used with
multiple applications of univariate analysis of covari-
ance, we shall refer to the test as the
ANOCOV/Bonferroni approach.
Alternatively, one can use a single multivariate test
of the null hypothesis that all K of the trend slopes
are zero, i.e.,
Bi, . . . , Bk, . . . , Bk = 0. (5)
The multivariate test would reject the null hypothesis
if any of the slopes were found to be different from
zero.
A procedure for performing the multivariate test,
referred to as MANOVA for multivariate analysis of
variance, is described in Loftis et al. (1991), and more
generally in Anderson (1984). Conveniently, the mul-
tivariate estimates of the parameters B, u, and L, are
the same as the univariate estimates. A classical
parametric procedure, MANOVA assumes normality
of the error terms.
However, for the more general case of non-normal
errors, robust procedures for estimation and testing of
linear models have been developed. One of these, an
aligned rank order procedure suggested by Sen and
Puri (1977), was adapted to the water quality trend
detection problem (Loftis et al., 1991). Under the con-
ditions studied, the Sen and Puri (SP) test was very
nearly as powerful as MANOVA for normal data and
as powerful as rank correlation methods (based on the
Mann-Kendall test for trend) for lognormal data. The
SP procedure is not pursued here as it is much more
complicated and difficult to apply than either MANO-
VA or the Mann-Kendall based procedures described
below.
Mann-Kendall Based Procedures
One of the most popular tests for trend in a single
water quality variable at a single location is a rank
correlation method called the seasonal Kendall (SK)
test (Hirsch et al., 1982). The study cited and another
by Taylor and Loftis (1989), reported that the SK test
compared favorably with available alternatives in
terms of both power and actual significance level for
normal and lognormal data with no serial depen-
dence. A "corrected" form of the test (SKC), presented
in Hirsch and Slack (1984) accounts for correlation
between seasons and provides conservative signifi-
cance levels in the presence of moderate serial corre-
lation. The SKC test has low power compared to the
original SK test when the data are serially indepen-
dent, especially when record lengths are ten years or
less.
To test for trend in more than one constituent
and/or location at a time, one can use the test sug-
gested by Dietz and Killeen (1981) from which the
SKC test was developed. However, as observed by
Lettenmaier (1988), this test has low power for record
lengths shorter than about 20 years of monthly data.
Lettenmaier referred to the Dietz and Killeen test as
the covariance inversion (CI) method since computa-
tion of the test statistic involves inversion of the
covariance matrix associated with the vector of sea-
sonal Kendall test statistics. In order to improve the
power performance of the test, Lettenmaier (1988)
suggested a new approach, called the covariance
eigenvalue (CE) method which avoids the matrix
inversion step.
In a Monte Carlo study of the CE test using
constant between season correlations, Lettenmaier
observed significantly improved power compared to
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Loflis, Taylor, Newell, and Chapman
the CI test. Loflis et al. (1991), found similar improve-
ment in power using first-order, autoregressive -
AR(1) — serial correlation with AR parameter (lag-one
correlation coefficient) values of 0.34 and smaller.
The latter authors also suggested modifications of
the CI and CE tests to improve their power under
serially independent observations. These modified
tests, called MCI and MCE, are related to the original
CI and CE tests in the same way that the SK test is
related to the SKC test. More specifically the MCI,
MCE, and SK tests assume that between season cor-
relations for each constituent are zero. The CI, CE,
and SKC tests account for this correlation.
Correlations between constituents are still accounted
for in the modified tests.
Loflis et al. (1991), found, not surprisingly, that the
MCI and MCE tests were somewhat less powerful in
detecting linear trends than methods based on linear
models. However, the Mann-Kendall based proce-
dures, MCI and MCE, are more general in that they
do not assume a linear model and are simpler to com-
pute. Of the two tests, MCI and MCE, neither seemed
to have a strong advantage over the other in terms of
power. Therefore, the simpler MCI test is suggested
here as a useful multivariate alternative to the uni-
variate seasonal Kendall test. A description of the test
follows the paper as an appendix.
We should note here that the MCI test will produce
seasonal Kendall test statistics for each of the water
quality variables of concern, as well as an overall
multivariate test statistic. Therefore, all of the infor-
mation developed in the univariate approach is also
produced in the multivariate approach. The individu-
al SK statistics could be used, when the multivariate
null hypothesis is rejected, to indicate which of the
individual variables had significant trends.
CHARACTERISTICS OF LAKE
QUALITY DATA
Before moving on to a Monte Carlo comparison of
univariate and multivariate tests, let us first examine
the characteristics of lake quality data. This examina-
tion should focus attention on the range of character-
istics which would be most important to study in the
simulation experiments.
The characteristics most likely to bear on the rela-
tive performance of univariate v. multivariate trend
testing methods are the intervariable correlation
structure and trend directions. The shape of the
underlying distribution would also be important in
choosing between parametric and nonparametric
approaches. We considered both classes of methods in
our simulation study. However, our case study appli-
cations were limited to nonparametric methods since
non-normal random errors are common in water qual-
ity, and nonparametric methods are preferable for
routine applications.
The presence of serial correlation might also affect
our choice of methods. Under such conditions, the
choice should be based on the costs of failing to reject
the null hypothesis when it is not true versus those of
rejecting the null hypothesis when it is in fact true.
When data are serially correlated, but there is no
trend, methods which assume independence between
seasons tend to reject the null hypothesis more fre-
quently than suggested by their nominal significance
levels. On the other hand, the methods which accom-
modate serial correlation tend to have lower power to
detect real trends.
To obtain some idea of the range of between-vari-
able correlations and serial correlation which was
likely to be encountered in practice, we examined a
few data records of excellent quality from Maine and
New York. The data were obtained from EPA's Long
Term Monitoring (LTM) project (Newell et al., 1987).
This EPA project has funded the continuation of
water quality monitoring in various regions since
1983. Major ion chemistry, ANC, pH, conductance,
DOC and Al are measured in these lakes to monitor
the effects of acidic precipitation on surface water
chemistry. Data from two regions, 15 lakes in the
Adirondacks, and five lakes in the Tunk Mountain
Watershed of Maine are considered here for estimates
of between-variable correlation and serial correlation.
The Adirondack Lakes have been monitored by
Charles Driscoll at the University of Syracuse since
1982 with funding from the Electrical Power Research
Institute through 1985, and EPA funding since that
time (Driscoll, 1991). Adirondack lakes were sampled
monthly at the outlet. In order to make the
Adirondack sampling schedule resemble the seasonal
schedule followed in other regions, we subsampled the
monthly data, selecting the observations closest to
January 15, April 15, July 15, and October 15.
The Maine sites have been monitored by the U.S.
Fish and Wildlife Service in cooperation with the
University of Maine since 1982, with EPA funding
beginning in 1983 (Kahl et al., 1991). These lakes are
sampled from the epilimnion in the spring, summer,
and fall.
For both the Maine and New York data sets, we
studied the serial and intervariable correlation struc-
ture of the seasonal series. Our objective was to
roughly establish a range of correlations which would
be realistic for a Monte Carlo evaluation of trend
WATER RESOURCES BULLETIN
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Multivariate Trend Testing of Lake Water Quality
testing methods. The data were preprocessed by
s removing seasonal means and an estimated linear
trend component in every case prior to estimation of
correlations.
Serial Correlation
Lag-one autocorrelation coefficients were estimated
from the case study data after preprocessing. These
estimates were then compared with standard er-
ror limits to assess their significance. Of 65 series
examined, each approximately six years in length,
only eight has estimated values, rj, which were out-
side the range of one standard error from zero. Of
these eight, two were positive and six were negative.
None of the estimates fell outside plus or minus two
standard errors from zero.
Since the probability of r^ falling outside one stan-
dard error is 32 percent for a serially independent
series, we would not be able to infer that serial corre-
lation is significant in the example data. Of course,
with sufficiently long record lengths, we would be able
to reject a null hypothesis of zero serial correlation,
even if the level of correlation were very small. We
should also note that we checked for serial correlation
in the original monthly series from New York and
found a number of significant r^ values, typically in
the range of 0.2 to 0.5.
Intervariable Correlation
We calculated intervariable correlations for several
variables in the LTM data sets. The strongest rela-
tionship was observed between calcium plus magne-
sium and ANC, where the median correlation was
0.50. The weakest relationship was found between
calcium plus magnesium and nitrate, where the medi-
an correlation was 0.17. For ANC and sulfate, the
median correlation was -0.23.
Time Trends
Logic dictates that trends of interest would include
simultaneous increases in some variables and
decreases in others. For example, acidification of a
lake would often be accompanied by increasing sulfate
concentration and decreasing ANC. Examination of
the case study data sets revealed that many combina-
tions of increases and decreases are possible. We shall
see several of these later in our case study applica-
tions to real data.
Consequently, it appears that we should consider
as the general case unspecified trend directions for all
variables. We shall formulate tests such that the null
hypothesis should be rejected when any of the vari-
ables included have a trend in either direction.
MONTE CARLO STUDY
Simulation Approach, Linear Models
Our initial simulation study considered linear
model based approaches only. Quarterly data for three
variables at one location were generated using the
procedure described in Loftis et al. (1990). The follow-
ing five ratios of trend slopes among the three vari-
ables were considered:
1:1:1, 1:1/2:0; 1:0:0, 1:-1:0, 1:1:—1.
In the first combination, the slopes for all three
variables are equal and in the same direction; in the
second combination, the second variable has a slope
which is one half that of the first variable; and the
third variable has no trend; and so on.
Seven different intervariable correlation struc-
tures, shown in Table 1, were generated. To limit the
size of the study, only normal, serially independent
errors, were considered. Our two linear model based
approaches, MANOVA and ANOCOV/Bonferroni,
would not perform well for skewed distributions, as
shown by Loftis et al. (1991), for the case of lognormal
errors. The same study investigated the effect of seri-
al correlation on both linear model and Mann-Kendall
based tests. However, our earlier examination of the
LTM data suggests that an assumption of serial inde-
pendence is justified for the present study of quarter-
ly observations of lake quality.
Each experiment involved generating 500 tri-
variate sequences, each 10 or 20 years in length and
having a constant slope vector B. In each trial, the
entire sequence was tested for trend under the null
hypothesis of no trend in all three variables using
both MANOVA and the ANOCOV/Bonferroni meth-
ods. The empirical power at a given slope is the num-
ber of rejections of the null hypothesis divided by the
number of trials.
By performing the experiment over a range of
slopes, indexed by the slope of the first variable, a
power curve can be drawn. The empirical significance
level is equal to the power when the slope vector is
the zero vector. All tests were run at a nominal signif-
icance level of 5 percent.
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Loftis, Taylor, Newell, and Chapman
TABLE 1. Intervariable Correlation Structures Simulated in Monte Carlo Study.
Correlation
Structure Description
1. corr between all variables within a season = 0.2
2. corr between all variables within a season = 0.5
3. corr between variables 1 and 2 within a season = 0.2
corr between variables 1 and 3 within a season = -0.2
corr between variables 2 and 3 within a season = -0.2
4. corr between variables 1 and 2 within a season = 0.0
corr between variables 1 and 3 within a season = 0.2
corr between variables 2 and 3 within a season = 0.5
5. corr between variables 1 and 2 within a season = 0.8
corr between variables 1 and 3 within a season = -0.2
corr between variables 2 and 3 within a season = 0.2
6. corr between variables 1 and 2 within a season = 0.8
corr between variables 1 and 3 within a season = 0.2
corr between variables 2 and 3 within a season = 0.2
7. same as number 1 but with seasonal variance
Results of Simulations
Table 2 presents results for the ten-year simula-
tions. A plus or minus in the table indicates that
MANOVA recorded at least 25 more or fewer, respec-
tively, rejections than the ANOCOV/Bonferroni
method at the point where the curve for the more
powerful test crossed 80 percent power or 400 rejec-
tions. A difference of 25 rejections corresponds to a 5
percent difference in power.
TABLE 2. Comparison of Simulated Powers for MANOVA and
ANOCOV/Bonferroni Methods of Trend Detection. A plus (minus)
indicates that MANOVA produced at least 25 more (fewer)
rejections than the ANOCOV/Bonferroni method at 80 percent
power. All tests were run on 500 quarterly sequences, each 10
years in length with normal, serially independent data.
Intervariable correlation structures are shown in Table 1.
Slope
Combination
Intervariable Correlation Structure Number
1
2
3
4
5
6
7
1:1:1
0
_
+
0
+
0
0
1:0:0
0
+
+
0
+
+
0
1:-10
+
+
+
+
+
+
+
1:1:—1
+
+
-
~
+
+
~
1:1/2:0
0
+
+
+
+
0
0
This method of comparison is consistent with an
interpretation of the empirical powers as estimates of
binomial proportions, p. The 5 percent difference rep-
resents two standard errors of the difference between
two binomial proportions, each estimated from 500
independent trials when the true value of the binomi-
al parameter, p, is 0.80. We suggest that this method
of comparison is objective and consistent, although
heuristic.
MANOVA was found to have significantly greater
power than the ANOCOV/Bonferroni method in 24 of
the 35 cases studied and significantly lower power in
two cases. In all cases, the empirical significance lev-
els of both methods were less than or equal to nomi-
nal levels. Interestingly, the slope combination which
showed the smallest advantage for MANOVA was
1:1:1. The slope combination which showed the most
consistent advantage for MANOVA was 1:—1:0. The
intervariable correlation structure which showed the
smallest advantage for MANOVA was #1, while corre-
lation structure #5 showed the greatest advantage.
In 20-year simulations (not shown), MANOVA had
significantly higher power for the 1:1:1 slope combina-
tion with correlation structure #7. Otherwise, the
results are the same as for ten years.
In Figures 1, 2, and 3, we compare empirical power
curves for MANOVA and the ANOCOV/Bonferroni
method, using selected slope combinations for each
intervariable correlation structure. From the figures,
interesting results emerge.
For correlation structure #1, the two power curves
are very close together for slope combinations 1:1:1
and 1:0:0 (Figure la, b). However, for correlation
structure #2, the ANOCOV/Bonferroni method has
WATER RESOURCES BULLETIN
466
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Multivariate Trend TwUng of Lake Water Quality
higher power than MANOVA for 1:1:1 slopes and
jlower power than MANOVA for 1:0:0 slopes (Fig-
ure 2a, b). For correlation structure #4 (Figure 3a, b),
MANOVA is more powerful in both slope combina-
tions.
In Figure 3b the Bonferroni method shows much
lower power than MANOVA This situation occurred
in several other cases, but the reverse, MANOVA hav-
ing much lower power, was never observed.
Overall, MANOVA has a definite advantage over
the Bonferroni method. Although there are a very few
cases (notably those of homogeneous trend) where the
Bonferroni method has slightly higher power, these
represent the exception rather than the general rule.
Since it would be prudent to assume non-homoge-
neous trends for real situations, the multivariate
approach would be the method of choice.
Mann-Kendall Based lists
We later performed a similar simulation study of
the MCI test in comparison to application of multiple
seasonal Kendall (SK) tests, the latter having signifi-
cance levels adjusted via the Bonferroni inequality.
10 yrs, 3 variables, correlation # 1
slopes are 1:1:1
jco!
JOCI
IXI
/
/
I
10 yrs, 3 variables, correlation tj2
slopes are 1:1:1
aoo:
I*
/
yPOr
o oot 01 cm o:
10 yrs, 3 variables, correlation -1
slopes are 1:0:0
Gis ::•>
•SCO* vfO
Figure la, b. Power Curves from Monte Carlo Comparison of
MANOVA and ANOCOV/Bonferroni Methods. Each point repre-
sents 500 simulations of three variables over 10 years (quarterly)
with normal data. Correlation structure #1 is described in Table 1.
The slope plotted on the horizontal axis is the largest of the three
and is given in units per year.
10 yrs, 3 variables, correlation =2
slopes are 1:0:0
ictj zx zi:
SXK vejr
Figure 2a, b. Power Curves from Monte Carlo Comparison of
MANOVA and ANOCOV/Bonferroni Methods. Each point repre-
sents 500 simulations of three variables over 10 years (quarterly)
with normal data. Correlation structure #2 is described in Table 1.
The slope plotted on the horizontal axis is the largest of the three
and is given in units per year.
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Loftis, Taylor, Newell, and Chapman
The data generation procedure was the same as that
of the linear models study, again using normal errors.
However, since the Mann-Kendall procedures use only
the ranks or relative magnitudes of the data, exactly
the same results would have been obtained using log-
normal errors or any other distribution.
10 yrs, 3 variables, correlation
slopes are 1:—1:0
oct ::ce o-e
9CCC -T:ts/ywr
10 yrs, 3 variables, correlation #4
slopes are 1:11
VAf/;
Figure 3a, b. Power Curves from Monte Carlo Comparison of
MANOVA and ANOCOV/Bonferroni Methods. Each point repre-
sents 500 simulations of three variables over 10 years (quarterly)
with normal data. Correlation structure #4 is described in Table 1.
The slope plotted on the horizontal axis is the largest of the three
and is given in units per year.
For three water quality variables at a single loca-
tion, we evaluated six different slope patterns for
intervariable correlation structures #1 and #2 (from
Table 1) and eight slope patterns for correlation
structure #6. We then extended the study to five
water quality variables, considering only three slope
patterns and correlation structures #1 and #2.
Results of the study are summarized in Table 3
using the same format as Table 2 for the linear model
results. The results are quite similar for both the lin-
ear model and Mann-Kendall based approaches. Here
again, the multivariate method has a definite power
advantage overall. The MCI test had higher power by
our previous criterion in 15 of 26 cases studied, while
the SK/Bonferroni method had higher power in only
one case.
TABLE 3. Comparison of Simulated Powers for the Modified Cov
ariance Inversion (MCI) and SK/Bonferroni Methods of Trend
Detection Using Three and Five Variables. A plus (minus) indicates
that the MCI test produced at least 25 more (fewer) rejections than
the SK/Bonferroni method at 80 percent power. All tests were run
on 500 quarterly sequences, each 10 years in length with normal,
serially independent data. Intervariable correlation structures are
shown in Table 1.
Slope
Combination
Intervariable Correlation Structure Number
1
2
3
1:1:1
0
0
0
1:0:0
0
+
+
1:0:1
0
+
+
1:-1:0
+
+
+
1:1:—1
~
+
+
1:1:0
0
1:—1:1
+
1:0:—1
+
1:1:1:1:1
0
_
1:1:—1:—1:0
+
+
1:-1K):0K)
+
+
When all of the trends are in the same direction
(1:1:1 slope pattern), the two approaches give similar
performance, and the Bonferroni method sometimes
has a slight advantage. An example is presented in
Figure 4a for five variables with correlation structure
#1 and homogeneous trend. However, when some of
the trends are in opposing directions, the multivariate
method generally has a large power advantage.
Figure 4b, for the 1:1:-1:-1:0 slope combination, is an
example.
Analysis of the simulation results for correlation
structure #6 indicates a relationship between power
advantage and the strength of (positive) intervariable
correlation. In this structure, the first and second
variables have noise terms with a correlation coeffi-
cient of 0.8 while the other intervariable correlations
are 0.2. Comparing the results for the 1:0.-1 and
1:-1:0 patterns in Figure 5a, b, we find that the MCI
test shows a larger power advantage, compared to the
WATER RESOURCES BULLETIN
468
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Multivariate n,gting of uke Water «y
Figure 4a, b. Power Curves from Monte Carlo Comparison of MCI
Test and SK/Bonferroni Test. Each point represents 500 simula-
tions of five variables over 10 years (quarterly) with normal data.
Correlation structure #1 is described in Table 1. The slope plotted
on the horizontal axis is the largest of the three and is given in
units per year.
Figure 5a, b. Power Curves from Monte Carlo Comparison of MCI
Test and SK/Bonferroni Test. Each point represents 500 simula-
tions of three variables over 10 years (quarterly) with normal data.
Correlation structure 06 is described in Table 1. The slope plotted
on the horizontal axis is the largest of the three and is given in
units per year.
SK/Bonferroni method, when opposing trends occur
coincident with larger positive correlation.
APPLICATIONS TO LTM DATA
As illustrative case studies, we applied both the
multivariate MCI test and univariate SK/Bonferroni
tests to the LTM data from Maine and New York.
These applications were performed for the sole pur-
pose of comparing the performance of the two tests on
real data. Detailed trend results and interpretation of
the Adirondack and Maine data are presented in
Driscoll (1991) and Kahl et al. (1991), respectively.
The MCI test is a logical choice over MANOVA for
routine applications since it is less restrictive in its
assumptions regarding the distribution of the noise
terms and of the functional form of the trend. In keep-
ing with an overall monitoring objective of evaluating
the effects of acid precipitation on lake water quality,
we considered a set of three variables: ANC, sulfate,
and calcium.
Each test considered only one location (lake) at a
time and was performed over the entire length of re-
cord. The Maine data consisted of three observations
469
WATER RESOURCES BULLETIN
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Loflis, Taylor, Newell, and Chapman
per year (April, July, and October) while the New
York data were a quarterly subsample (January,
April, July, and October) of the original monthly
series.
Unlike the simulation study, this application to
real data is uncontrolled in the sense that we do not
know the true values of population parameters such
as intervariable correlations, serial correlation struc-
ture, or underlying distribution. Furthermore, since
we do not know the true trend magnitudes, we cannot
say whether our tests obtain the "correct" result in a
given case. However, based on the earlier simulations,
we would expect the MCI test to detect trends more
often than the SK/Bonferroni test.
Table 4 presents the trend testing results. The
directions of the trends in the individual variables are
indicated in the three columns following the lake
identifier. These were determined by seasonal
Kendall tests using a nominal significance level of
0.20 to obtain high power. The last two columns give
the rejection significance levels (sometimes called the
"p" values) for both the MCI and SK/Bonferroni tests.
The rejection significance level for each test should
be interpreted in the following way. If the test were
performed at a nominal (a priori) significance level
greater than the given rejection level, then the test
would have resulted in rejection of the null hypothesis
of no trend, and vice versa.
In 16 of 20 cases, the MCI test produced a rejection
significance level equal to or smaller than that of the
SK/Bonferroni test, suggesting that the MCI test
would tend to reject the null hypothesis more often. If
we choose a specific significance level, say 0.10, we
see that the two tests would have given similar
results. There are three cases in which the MCI test
would have rejected the null hypothesis when the
Bonferroni approach did not: lakes NY4, NY5, and
ME1. There is one case where the reverse is true, lake
NY8.
There was considerable variety in the patterns of
slope directions, supporting the general assumption of
non-homogeneous trend. In some cases, opposing
trends were noted in ANC and sulfate concentration,
and, in others, the two variables moved in the same
direction over time. Figure 6a, b presents examples of
the two situations. For the Maine Lake of Figure 6a,
ANC and sulfate have opposing trends. In the New
York Lake data shown in Figure 6b, ANC and sulfate
have trends in the same direction.
TABLE 4. Results of Applying Trend Tests to Case Study LTM Data. For individual variables *+" indicates an increasing trend,
indicates a decreasing trend, and "0" indicates no significant trend at 0.20 significance level using a univariate SK test.
Rejection significance levels are given for the SK test adjusted via the Bonferroni inequality and for the multivariate MCI
(modified covariance inversion) test. All Maine lake observations are from the epilimnion. The New York data sets
consisted of six years of quarterly data, and the Maine sets consisted of seven years of observations three times per year.
Lake ID Variable* Rejection Significance Level*
Number
ANC
Sulfate Calcium
SK/Bonferroni
MCI
NEW YORK LAKES
NY1
_
_
0.019
0.021
NY 2
0
-
~
0.418
0.116
NY 3
0
-
+
0.034
0.026
NY4
-
-
+
0.146
0.067
NY5
-
-
0
0.146
0.014
NY6
0
0
0
1.000
0.647
NY 7
-
-
0
0.223
0.114
NY8
-
-
0
0.092
0.113
NY9
-
0
0
0.223
0.164
NY10
0
-
+
0.011
0.009
NY11
-
-
~
0.011
0.013
NY12
0
-
~
0.223
0.110
NY13
-
-
~
0.011
0.010
NY 14
-
-
-
0.334
0.217
NY15
-
-
0
0.019
0.012
MAINE LAKES
MEl
+
0
0
0.136
0.071
ME2
+
-
+
0.005
0.005
ME3
+
0
0
0.009
0.005
ME4
+
0
0
0.026
0.056
ME 5
+
-
0
0.005
0.002
WATER RESOURCES BULLETIN
470
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Multivariate Trend Testing of Lake Water Quality
Ukc- MK:i
•QO4 i G
Lake NV1
i — •
; u»;
¦ *e:
Figure 6a, b. Time Scries Plots for ANC, Sulfate Concentration,
and Calcium Concentration for the LTM Lakes ME2 and NY1.
SUMMARY AND CONCLUSIONS
Simulation and case study applications were used
to compared the performance of univariate and multi-
variate methods of testing for trend in seasonal water
quality data. Monte Carlo studies suggested that mul-
tivariate analysis of variance (MANOVA) and the
modified covariance inversion (MCI) tests were gener-
ally more powerful than their univariate counterparts
applied using the Bonferroni inequality.
This conclusion applied over the studied range of
between-variable correlations, both positive and nega-
tive. Multivariate methods had the greatest power
advantage in the case of positive correlations between
variables with opposing slope directions. The Bonfer-
roni methods performed better in only a few cases
where trends were homogeneous across variables.
However, non-homogeneous trends should be
assumed for most real applications.
Application of the MCI and seasonal Kendall (SK)
tests to lake quality data from EPA's Long Term
Monitoring program in Maine and New York yielded
comparable results. The three variables selected to
evaluate acid deposition impacts exhibited a variety
of patterns in trend directions.
Based on these results we can make a very positive
recommendation for general application of multivari-
ate approaches. There are very few cases where the
univariate methods perform better, and, in those
cases, the power advantage is small. Therefore, it
should not be necessary to examine a data set for
between-variable correlation structure or trend direc-
tions before choosing the multivariate approach. Of
the two multivariate methods studied, we recommend
the MCI test for routine applications because of its
robust performance (as demonstrated elsewhere for
rank-based methods in general) and its simplicity.
ACKNOWLEDGMENTS
The work described in this article was funded in part by the
Colorado Experiment Station Project No. 349 and by the
Environmental Protection Agency Contract No. 68-03-3439. The
research has not been subject to review by the agency, and on offi-
cial endorsement should be inferred.
LITERATURE CITED
Anderson, T. W., 1984. An Introduction of Multivariate Statistical
Analysis. John Wiley and Sons, New York, New York.
Conover, W. J., 1980. Practical Nonparametric Statistics, Second
Edition. John Wiley and Sons, New York, New York.
Dietz, E. J. and T. J. Killeen, 1981. A Nonparametric Multivariate
Test for Monotone Trend with Pharmaceutical Applications. J.
American Statistical Assoc. 76:169-174.
Driscoll, C. T., 1991. Processes Regulating Seasonal Variations in
the Water Chemistry of Adirondack Lakes. Water, Air and Soil
Pollution (in preparation).
Hirsch, R. M., and J. R. Slack, 1984. A Nonparametric Test for
Seasonal Data with Serial Dependence. Water Resources
Research 29(6):727-732.
Hirsch, R. M., J. R. Slack, and P. A. Smith, 1982. Techniques of
Trend Analysis for Monthly Water Quality Data. Water Re-
sources Research 18(1):107-121.
Kahl, J. S., T. A. Haines, S. A. Norton, and R. B. Davis, 1991.
Recent Trends in the Acid-Base Status of Surface Waters in
Maine, U.S.A. Water, Air, and Soil Pollution (in press).
Lettenmaier, D. P., 1988. Multivariate Nonparametric Tests for
Trend in Water Quality. Water Resources Bulletin 24(3):505-512.
Lettenmaier, D. P., E. R. Hooper, C. Wagoner, and K. B. Faris, 1991.
Trends in Stream Quality in the Continental United States,
1978-1987. Water Resources Research 27(3):327-339.
Loflis, J. C., C. H. Taylor, and P. L. Chapman, 1991. Multivariate
Tests for Trend in Water Quality. Water Resources Research
27(7):1419-1429.
471
WATER RESOURCES BULLETIN
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Loflis, Taylor, Newell, and Chapman
Newell, A. D., C. F. Powers, and S. J. Christie, 1987. Analysis of
Data from Long Term Monitoring of Lakes. EPA/600/4-87/014,
U.S. Environmental Protection Agency, Washington, D.C.
Sen, P. K. and M. L. Puri, 1977. Asymptotically Distribution-Free
Aligned Rank Order Tests for Composite Hypotheses for
General Multivariate Linear Models. Zeischrifl fur Wahrschein-
lichkeitstheorie und Verwandte Gebiete 39:187-195.
Snedecor, G. W. and W. G. Cochran, 1980. Statistical Methods. The
Iowa State University Press, Ames, Iowa.
Taylor, C. H. and J. C. Loflis, 1989. Testing for Trend in Lake and
Ground Water Quality Times Series. Water Resources Bulletin
25(4 ):715-726.
where:
and
Ugh+fgh) 13
ifg=h
ifg*h
l* = 1
& i 0
0 if x = 0
—1 if x < 0
Under the null hypothesis of no trend
K= I signiy - y .)
i+1
-------
Multivariate Trend Testing of Lake Water Quality
The "covariance inversion" test (a term used by Let-
tenmaier, 1988) is based on the work of Dietz and
Killeen (1981) who suggested that
is asymptotically distributed chi-squared with to
degrees of freedom provided that T is of full rank.
Otherwise, if the rank of r is q<8, then
is asymptotically chi-squared distributed with q
degrees of freedom where r_1 is a generalized inverse
of T. Consequently, the null hypothesis may be reject-
ed if iff is large when compared to the appropriate chi-
squared statistic.
The covariance inversion test accounts for between-
season correlation via the variance-covariance matrix
X in Equation (A5). However our formulations of the
MANOVA and seasonal Kendall methods assume
independence between seasons. We can easily con-
struct a modified version of the covariance inversion
test which effectively assumes independence between
seasons and, therefore, more closely parallels the
MANOVA and seasonal Kendall methods.
The modified test is performed simply by setting
the appropriate off-diagonal elements of Z equal to
zero and proceeding as in the original test. The appro-
priate elements are those corresponding to between-
season covariances. The elements corresponding to
between-constituent covariance for the same season
are left as is.
V=STP*S
(A12)
sTr-is
(A13)
473
WATER RESOURCES BULLETIN
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing) iii |||i ii ii
llllllllllllllllllll -
535-148722
1. REPORT NO. 2.
EPA/600/J-94/510
'¦""I. HI Mil IIII
PF
4. TITLE AND SUBTITLE
Multivariate trend testing of lake water
quality
5. REPORT DATE
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
'j.C.Loftis, 2C.H.Taylor, 3A.D.Newell
'P.L.Chapman
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
'CSU, FT.Collins, CO, ^errel Dow Rsch
Inst.,Cincinnati, OH, 3ManTech,
ERL-Corval1is,OR
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
US EPA ENVIRONMENTAL RESEARCH LABORATORY
200 SW 35th Street
Corvajfc, OR 97333
13. TYPE OF REPORT ANDJ,ER|OD,COVERED
Journal Article
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
1991 Water Resources Bulletin, 27(3) 461-473
16. ABSTRACT
Multivariate methods of trend analysis offer the potential for higher power in
detecting gradual water quality changes as compared to multiple applications of
univariate tests. Simulation experiments were used to investigate the power
advantages of multivariate methods for both linear model and Mann-Kendall based
approaches. The experiments focused on quarterly observations of three water
quality variables with no serial correlation and with several different
intervariable correlation structures. The multivariate methods were generally
more powerful than the univariate methods, offering the greatest advantage in
situations where water quality variables were positively correlated with trends
in opposing directions. For illustration, both the univariate and multivariate
versions of the Mann-Kenadall based tests were applied to case study data from
several lakes in Maine and New York which had been sampled as part of EPA's long
term monitoring study of acid precipitation effects.
17. KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
b. IDENTIFIERS/OPEN ENDED TERMS
c. COSATl Field/Group
trend analysis, multivariate analysis,
statistics, lake water quality, acid
precipitation, probability, population sp
ice
18. DISTRIBUTION STATEMENT
19. SECURITY CLASS (This Report)
21. NO. OF PAGES
13
20. SECURITY CLASS (This page)
22. PRICE
EPA Form 2220-1 (R»». 4-77) previous edition is obsolete
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