D E T E C T I O N OF WATER Q U A L I T Y CHANGES ALONG A R I V E R SYSTEM S.R. E S T E R B Y , 1 A N D A.H. E L - S H A A R A W I 2
Lakes Research Branch, 1 and Rivers Research Branch, 2 National Water Research Institute, Burlington, Ontario, Canada L7R 4A6
and
H.O. BLOCK Water Quality Branch, Western and Northern Region, Calgary, Alberta, Canada T2P 2M7 (Received: January 1992)
Abstract. Water quality monitoring networks are generally multi-purpose, and thus the data generated are expected to provide information on a set of objectives. Two questions that are fundamental to these objectives are the detection of long term t~ends and of differences between locations. The extent to which these questions can be answered and the types of statistical methods which can be used are considered in a case study of conventional parameters sampled monthly for nine years. Regression and nonparametric methods, which explicitly account for seasonality, are compared for both the determination of change over time and of differences between locations. Changes over time in the form of step changes and differences between locations which depend upon season were found.
1. Introduction Water quality monitoring networks are generally multi-purpose, and thus the data generated are expected to provide information on a set of objectives. Two questions that are fundamental to these objectives are the detection of long term trends and the determination of differences between locations. The extent to which these questions can be answered and the types of statistical methods which can be used depend upon the characteristics of the data set. Typically, fixed locations have been sampled at approximately equal intervals for ten to twenty years, with some variation in the sampling program from location to location. Other features include missing data, censored data, large seasonal variation and changes in laboratory techniques. Nonparametric techniques for the detection of trends in such water quality data sets have received considerable attention because fewer assumptions must be satisfied than with parametric methods. Hirsch et al. (1982) illustrated the use of a blocked Kendall's T, the seasonal Kendall test for trend, which involves testing the hypothesis of randonmess using a statistic powerful for the altemative of a monotonic change over time. These authors also extended the nonparametric point estimator of the slope given by Thiel (1958) and Sen (1968) to account for seasonality and called it the seasonal Kendall slope estimator. Gilbert (1987) gave a simple procedure for a confidence limit for the slope. Spearman's rank correlation coefficient is an alternative to 7- and it has been applied to water quality data (e.g. Environmental Monitoring and Assessment 23: 219-242, 1992. 1992 Kluwer Academic Publishers. Printed in the Netherlands.
220
S.R. ESTERBY ET AL.
Lettenmaier, 1976; EI-Shaarawi et al., 1983). Implicit to the test of Hirsch et al. is the assumption of homogeneity of trend over all seasons. Van Belle and Hughes (1984) showed how to test for homogeneity of trend over seasons and analogously, homogeneity of trend over a number of locations. Under homogeneity, a global test of trend is possible. Otherwise, trend tests are conducted separately by station or by season. All of the above mentioned nonparametric tests are briefly described by Wu and Zidek (1989). When the difference in mean or median concentration between locations is of interest, nonparametric procedures are available which are the analogues of the t test and, in some cases, of the least squares analysis of variance. For example, EI-Shaarawi et al. (1985) used the sign test to determine differences between contaminant concentrations at a pair of locations on the same river by matching dates of sampling. Gilbert (1987) illustrated the use of nonparametric procedures for comparing two locations when matching is or is not possible and for comparing k locations with or without blocking, which are analogues to two-way and one-way analysis of variance, respectively. Regression analysis is widely used for assessing trends and has been applied to water quality data, for example, by EI-Shaarawi et al. (1983). It also permits estimation of the point of change in a water quality variable, through the procedure of Esterby and E1-Shaarawi (1981). Further, location effects can easily be included in the regression model as can seasonal components and a variety of forms of change over time. Bodo (1989) found smoothing of an irregular water quality time series by robust locally weighted regression provides a useful method for graphical assessment of trend. All of the above procedures involve the assumption of serial independence, which may be tenable for water quality data collected over time if the sampling interval is long enough (Lettenmaier, 1976) or if aggregation is used (van Belle and Hughes, 1984). Hirsch and Slack (1984) proposed a modified seasonal Kendall test when independence between seasons is not tenable and EI-Shaarawi and Damsleth (1988) provide modifications for the t, sign and Wilcoxon tests in the presence of serial correlation. In regression analysis, given an estimate of the correlational structure of the errors, generalized least squares estimation can be performed. McLeod et al. (1983) developed a procedure based on seasonal adjustment to estimate the entries of an evenly spaced time series from data collected at irregular intervals and then used exploratory methods and intervention analysis to investigate changes in water quality variables. The present paper summarizes the results of a case study in which the objectives were to identify suitable statistical methods for the determination of water quality changes over time and between locations and to show what information on water quality changes could be obtained from the data set under study. The major ion concentrations measured by the Westem and Northern Region of the Water Quality Branch, Environment Canada, at five locations on the Bow and South Saskatchewan Rivers between 1978 and 1986 provide an example of data from a monitoring
WATERQUALITYCHANGESALONGA RIVERSYSTEM
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network with features as described above. Results from the analysis of nitrate plus nitrite, sodium and specific conductance are given here. The emphasis will be on the process of analyzing the data under the objectives stated above, and on the relationship between the characteristics of the data set and the statistical methods used. 1.1. SAMPLING LOCATIONS AND WATER QUALITY VARIABLES
The sampling locations will be referred to as BA0011, BE0013, BH0017, BN0001 and AK0001, where these span, in the above order, the headwater site on the Bow River at Lake Louise, BA0011, to the site, AK0001, on the South Saskatchewan River below major urban centres and the irrigation district of southem Alberta. Locations BE0013 and BH0017 are on the Bow River above the maj or urban centres and the irrigation district, while BN0001 is on the Bow River below Calgary and in the irrigation district. The flow conditions vary throughout the year and between locations, with the upper locations being characterized by low flow in the winer and a mid-year peak due to mountain snow melting. The lower two locations have an additional spring peak due to snow melting in the prairie regions and receive irrigation return flows. The locations were sampled at approximately monthly frequency. Only the record between 1978 and 1986 is considered here since 1978 was the first year of sampling at BH0017. The analytical methods are the standard methods of the Water Quality Branch (1979) and flow is the mean daily flow for the day of sampling at the sampling location, except for BE0013 and BH0017, which are values calculated from nearby gauging stations. Analytical method and laboratory changes occurred over the period considered and, from a preliminary examination of the data, water quality variables were grouped according to the type of variable and the number of methodological changes. The variables reported here, which are representative of these groups, are the nutrient, nitrate plus nitrite (NO3 + NO2), with numerous methodological changes, sodium (Na), thought to reflect urban and agricultural inputs, with only one laboratory change, and specific conductance, a general indicator of ion content, also with only one laboratory change. A regression analysis (Esterby et al., 1990a), based on a model which included year, seasonality and methodological and laboratory parameters, showed no indication of a change in the variable value due to laboratory change for Na and specific conductance, but did show an effect due to methodological and laboratory changes for NO3 + NO2. Thus the latter variable is evaluated within periods of constant analytical methodology at the same laboratory.
2. Statistical Methods The statistical methods are discussed here in the context of the case study data set. Only brief descriptions of the methods are given since details are available in the references cited. The most important features of the data set are the sampling
222
S.R. ESTERBY ET AL.
interval of approximately one month, the record length of nine years, seasonality, missing data and different sampling times at some of the locations. Seasonality accounts for the highest proportion of variability in the data. In the regression models constructed to test for an effect of method and laboratory changes in sodium and nitrate plus nitrite (Esterby et al., 1990a), seasonality terms accounted for 42 to 79 percent of the total variability, except for sodium at BN0001, where seasonality accounted for only 28 percent. If changes over time or differences between locations are to be detected and precise and unbiased estimates are to be obtained, the seasonality must be accounted for in the analysis. The two methods of modelling seasonality which will be considered here are blocking and the fitting of a smooth or piecewise smooth curve to represent seasonality. Under additivity, blocking to remove seasonality involves the assumption that the effect of the season is the same over all years. The seasonal Kendall test for trend involves this assumption. Let Y/j denote the value of the water quality variable at a station in season j of year i. The set of values Ylj, Y2j, ..., Yn D, where nj is the number of years with observations in the season j, can thus be represented as
Yij = ttj + ~if(i) + e i j . The effect for season j is given by #j and the change over time, in a very general form, b y / 3 i f ( i ) where f ( i ) denotes some function of time. The statistic calculated within each season in the seasonal Kendall test is obtained from the differences Yij - Ykj, the difference between the observations in years i and k of season j, and then a linear combination of these statistics is taken. If the assumption of constant seasonal effect over years is met, the seasonal effect will be removed from the differences. A practical consideration is how to define the seasons so that the above assumption is met. Hirsch et aL (1982) demonstrated the seasonal Kendall test using the month as the season since their data were monthly water quality time series. This involves the assumption that the seasonal component for a month (the component for January, for example) is constant over all years. Since the present set of data consists of samples collected at approximately monthly intervals, months are also used as seasons and it is assumed that an observation at any time within the month contains the same seasonal effect for that month. The seasonal Kendall test for trend, test for homogeneity of trend and nonparametric slope estimator are then applied. In the alternative method of accounting for seasonality, the date of sampling, which should always be available, is used. Thus, there is the pair (Yij, tij) consisting of the value of the water quality variable and, for example, the day in the year for the sample collected in month j of year i. A regression model is fitted which contains sinusoidal functions of fixed periods to describe the seasonal cycle within a year (Esterby et al., 1991). By allowing the phase and amplitude to vary from year to year, the differences in timing and amplitude of the peaks due to weather
WATER QUALITY CHANGES ALONG A RIVER SYSTEM
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differences from year to year can be accomodated. For the present data, day-withinyear (1 to 365 or 366) is used and regression models are fitted with a one or two component sinusoid for the seasonal component and either yearly mean levels or a linear change over years. Residual plots and diagnostics are used together with tests on parameters to arrive at the most satisfactory models. Similarly, seasonality has been taken into account by blocking or fitting curves in the comparison of locations. Let Ytij and Ykij be the values of a water quality variable at locations l and k in month j of year i. If the corresponding times of observation, ttij and tkij, are sufficiently close so that the locations are at approximately the same point on the hydrological cycle, methods based on the differences Yt~j - Ykij Can be used to determine differences between locations. Under the assumptions that the components due to season and to change over time are equal at both locations at matched sampling times and that the location difference is constant, the t test, sign test and Wilcoxon signed-rank test may be applied, with the further assumption for the t test that differences are normally distributed. These methods are applied to subsets of the data for the three upper locations. When sampling times do not match, some method of estimating the value of the water quality variable at the same time at both locations is required. It is possible to do this within the framework of regression by dividing the year into periods such that the water quality variable is approximately constant or changes linearly (Esterby et al., 1990b). A test that one fitted function is uniformly greater than the other over the interval (Tsutakawa and Hewett, 1978) can then be applied. The methods for the analysis of changes over time at one location and the comparison of locations, by fitting regression relationships within periods of the year, can accomodate missing data. Methods involving matching require observations for both members of a pair. The practical limitation of missing data, regardless of the method of analysis, is a reduced ability to detect or estimate the feature of interest. The adequacy of nine years of data to detect changes over time will be considered in the discussion of the results.
3. TemporalChanges The seasonal test for trend is the sum of statistics calculated for each month. To illustrate the data used in these individual statistics, the specific conductance over years is plotted separately by month for two locations (Figure 1). A visual assessment comparable to the test consists of examining the individual plots for changes over time and drawing some consensus from the plots for all months. The results of application of the Kendall test for trend for individual months, the seasonal Kendall test and the test for homogeneity of trend (Table I) show that there is evidence of a monotonic change over the period 1978 to 1986 for the three upper locations (BA0011, BE0013 and BH0017) and that there is no evidence of heterogeneity among months.
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TABLE I The Kendall test applied to individual months, the seasonal Kendall test and the test of homogeneity of trend for specific conductance data, 1978 to 1986. Period
BA0011
Sampling location BE0013 BH0017 BN0001
AK0001
Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
-0.31 0.00 -1.11 0.63 -0.10 0.00 -1.36 -0.21 0.32 - 1.15 -1.77 -1.56
-1.50 -1.25 -1.36 -0.10 -0.37 -0.63 0.25 0.00 -0.50 -0.94 -2.10" -1.47
-0.94 -0.25 0.31 -1.56 -0.62 -1.77 -0.52 -1.36 -0.94 -0.21 -0.62 -1.26
-0.94 -0.52 -1.56 -0.21 -0.52 -1.61 1.36 -0.84 0.00 0.00 0.12 -1.15
-0.52 -0.52 -1.15 -2.19" 0.73 1.15 -0.52 0.31 0.84 -0.31 0.52 -0.94
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Year
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Year 4.32" 8.33
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* Significant at the 0.05 level. a The entries in the table for the Kendall and seasonal Kendall tests are the standard normal deviate form, with continuity correction, as described by Hirsch et al. (1982). b The statistics for tests of trend and homogeneity, without continuity corrections, as described by van Belle and Hughes (1984), are distributed approximately as chi squares with 1 and 11 degrees of freedom, respectively.
The regression models which were fitted consist of a seasonal cycle for each year and either a mean for each year or a linear trend over years. The resulting models are superimposed on the specific conductance data for the same two locations in Figure 2. A visual assessment of the adequacy of the form of the model can be obtained from the proximity of the sampling points to the fitted curves and from the residual plots (Figure 2). The significance levels for sets of terms in the models and R 2, expressed as a percentage, are given for the two forms of regression models fitted to the data from all locations (Table II). There are significant differences between yearly means for all locations and, for all locations except AK0001, the linear term was significantly different from zero. Thus, the consensus from hypothesis testing under the two methods is that there is a monotonic change over years for the upper three locations. The regression analysis indicates that a linear term explains more variability for locations BE0013
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T A B L E II C o m p a r i s o n o f m o d e l s for specific c o n d u c t a n c e with yearly m e a n s and with trend in year.
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and BH0017 than for the others. If one accepts the form of change in the mean level over years as a linear change, then comparison of confidence intervals for the regression and nonparametric slope estimators allows a comparison which takes the variability of the estimates into account (Table III). The regression point estimator is always slightly lower as is the lower limit of the regression confidence interval. The agreement is remarkable in view of the differences in the method of accounting for seasonality and the expected influences of large observations on the regression methods. The former might be expected to impair the nonparametric method by not taking into account differences in the seasonal cycle from year to year and the latter to produce distorted estimates due to the influence of larger observations. However, the residual plots from the regression models with linear terms in year (Figure 2) show that the linear term does not account for some of the yearly changes in mean level, since residuals are predominantly positive in some years and negative in others, particularly for BH0017. The lower percentage of variation explained and the presence of fewer runs in the residuals (100R 2 and p runs, Table II) for regression models with the linear term in years, also indicate that the linear term is inadequate. Multiple comparison procedures, guided by a plot of the estimates of yearly means levels (Figure 3), leads to the conclusion that there are sets of high years and of low years, with some variation from location to location, but with 1982 having low specific conductance relative to the other years at all locations (Table IV). These results help to explain why the three upper locations were found to exhibit a monotonic change which could be modelled to varying
230
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TABLE III Point estimate and confidence limits for the slope from the model in Table II,/3, and the nonparametric slope estimator, B. Nonparametric Slope Estimation
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95% Confidence Interval -1.8, -5.2, -4.8, -4.6, -5.1,
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TABLE IV Tests for significant differences between the estimated mean specific conductance for the indicated groupings of years. Mean for Indicated Years with Significant Differences at 0.05 Level Station
1
2
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82,82 163.5
84,85,86 170.9
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78,79,80,81
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78,81,82,83,84,85,86 394.3
3 78,79,80,81 178.0
79,80 436.3
The means are arithmetic means of the year effects shown in Table V. Underlined years are not significantly different based upc- ~ method.
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degrees as a linear trend. The drop in specific conductance in 1982 is essentially a step change, with the step size larger for BE0013 and BH0017 than BA0011. Such a change can be fitted as a linear decline, provided the step size is large enough.
4. Location Differences If it is reasonable to formulate the test for location differences as a test of the hypothesis of the same monotonic trend at all locations, then the method described by van Belle and Hughes (1984) can be used. From the above results, it is clear that this is not the best approach here. A further consideration for these five locations is the complex hydrological cycle with different sources of water at different times of the year. To show the nature of the variability of two other variables, sodium and nitrate plus nitrite were analyzed over the period 1984 to 1986, a period of constant methodology and laboratory for nitrate plus nitrite and a period in which mean levels of specific conductance were constant over time at all locations. To examine the changes in concentration of the variables within intervals in which different flow events are occurring, the sodium and nitrate plus nitrite concentrations have been plotted with the year divided into three intervals (Figures 4 and 5). These intervals correspond approximately to: (1) constant flow at the three upper locations and increased flow at the two lower locations due to the snow melting on the prairies; (2) a peak in flow at all locations due to the snow melting in the mountains; and (3) approximately constant flow at the three upper locations and a small increase at the lower locations (Figure 6). Clearly, the sodium concentration is always higher at the two lower locations than at the three upper locations and similarly for nitrate plus nitrite, except in mid-year when all five locations have similar concentrations. Because of this and the fact that the three upper locations can be matched by sampling date but the two lower cannot, the analyses will be done separately for these sets of three and two locations. 4.1. MATCHED SAMPLING DATES
Differences in concentrations between pairs of locations over the entire period, 1984 to 1986, were used to test the hypothesis of no difference between locations for the three possible pairs (Table V). From this it can be concluded that there are location differences between the sodium and nitrate plus nitrite concentrations which are highly significant on the basis of all tests except for nitrate plus nitrite at BA0011 and BH0017. This conclusion holds whether a comparison-wise or family error rate is adopted (see Table 5 footnote for an explanation). An assumption underlying all these tests is that the set of differences {dimj}, where dimj is the differefice between the concentration at locations i and m at time j, is a random sample from one population. Plots of the differences show seasonal effects (Figure 7). To make the above assumption tenable, the data were separated into the three intervals within the year and the results, based on the Wilcoxon signed-rank test
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TABLE V Tests for a difference in concentration between locations and estimation of the average difference, for years 1984 to 1986.
Result Number of pairs Significance probability ~ t test Wilcoxon signed ranks Sign test Confidence limits b t test
k sample sign test
Sodium BA0011 BE0013 BA0011 -BE0013 - B H 0 0 1 7 - B H 0 0 1 7 34 33 33
Lakes Research Branch, 1 and Rivers Research Branch, 2 National Water Research Institute, Burlington, Ontario, Canada L7R 4A6
and
H.O. BLOCK Water Quality Branch, Western and Northern Region, Calgary, Alberta, Canada T2P 2M7 (Received: January 1992)
Abstract. Water quality monitoring networks are generally multi-purpose, and thus the data generated are expected to provide information on a set of objectives. Two questions that are fundamental to these objectives are the detection of long term t~ends and of differences between locations. The extent to which these questions can be answered and the types of statistical methods which can be used are considered in a case study of conventional parameters sampled monthly for nine years. Regression and nonparametric methods, which explicitly account for seasonality, are compared for both the determination of change over time and of differences between locations. Changes over time in the form of step changes and differences between locations which depend upon season were found.
1. Introduction Water quality monitoring networks are generally multi-purpose, and thus the data generated are expected to provide information on a set of objectives. Two questions that are fundamental to these objectives are the detection of long term trends and the determination of differences between locations. The extent to which these questions can be answered and the types of statistical methods which can be used depend upon the characteristics of the data set. Typically, fixed locations have been sampled at approximately equal intervals for ten to twenty years, with some variation in the sampling program from location to location. Other features include missing data, censored data, large seasonal variation and changes in laboratory techniques. Nonparametric techniques for the detection of trends in such water quality data sets have received considerable attention because fewer assumptions must be satisfied than with parametric methods. Hirsch et al. (1982) illustrated the use of a blocked Kendall's T, the seasonal Kendall test for trend, which involves testing the hypothesis of randonmess using a statistic powerful for the altemative of a monotonic change over time. These authors also extended the nonparametric point estimator of the slope given by Thiel (1958) and Sen (1968) to account for seasonality and called it the seasonal Kendall slope estimator. Gilbert (1987) gave a simple procedure for a confidence limit for the slope. Spearman's rank correlation coefficient is an alternative to 7- and it has been applied to water quality data (e.g. Environmental Monitoring and Assessment 23: 219-242, 1992. 1992 Kluwer Academic Publishers. Printed in the Netherlands.
220
S.R. ESTERBY ET AL.
Lettenmaier, 1976; EI-Shaarawi et al., 1983). Implicit to the test of Hirsch et al. is the assumption of homogeneity of trend over all seasons. Van Belle and Hughes (1984) showed how to test for homogeneity of trend over seasons and analogously, homogeneity of trend over a number of locations. Under homogeneity, a global test of trend is possible. Otherwise, trend tests are conducted separately by station or by season. All of the above mentioned nonparametric tests are briefly described by Wu and Zidek (1989). When the difference in mean or median concentration between locations is of interest, nonparametric procedures are available which are the analogues of the t test and, in some cases, of the least squares analysis of variance. For example, EI-Shaarawi et al. (1985) used the sign test to determine differences between contaminant concentrations at a pair of locations on the same river by matching dates of sampling. Gilbert (1987) illustrated the use of nonparametric procedures for comparing two locations when matching is or is not possible and for comparing k locations with or without blocking, which are analogues to two-way and one-way analysis of variance, respectively. Regression analysis is widely used for assessing trends and has been applied to water quality data, for example, by EI-Shaarawi et al. (1983). It also permits estimation of the point of change in a water quality variable, through the procedure of Esterby and E1-Shaarawi (1981). Further, location effects can easily be included in the regression model as can seasonal components and a variety of forms of change over time. Bodo (1989) found smoothing of an irregular water quality time series by robust locally weighted regression provides a useful method for graphical assessment of trend. All of the above procedures involve the assumption of serial independence, which may be tenable for water quality data collected over time if the sampling interval is long enough (Lettenmaier, 1976) or if aggregation is used (van Belle and Hughes, 1984). Hirsch and Slack (1984) proposed a modified seasonal Kendall test when independence between seasons is not tenable and EI-Shaarawi and Damsleth (1988) provide modifications for the t, sign and Wilcoxon tests in the presence of serial correlation. In regression analysis, given an estimate of the correlational structure of the errors, generalized least squares estimation can be performed. McLeod et al. (1983) developed a procedure based on seasonal adjustment to estimate the entries of an evenly spaced time series from data collected at irregular intervals and then used exploratory methods and intervention analysis to investigate changes in water quality variables. The present paper summarizes the results of a case study in which the objectives were to identify suitable statistical methods for the determination of water quality changes over time and between locations and to show what information on water quality changes could be obtained from the data set under study. The major ion concentrations measured by the Westem and Northern Region of the Water Quality Branch, Environment Canada, at five locations on the Bow and South Saskatchewan Rivers between 1978 and 1986 provide an example of data from a monitoring
WATERQUALITYCHANGESALONGA RIVERSYSTEM
221
network with features as described above. Results from the analysis of nitrate plus nitrite, sodium and specific conductance are given here. The emphasis will be on the process of analyzing the data under the objectives stated above, and on the relationship between the characteristics of the data set and the statistical methods used. 1.1. SAMPLING LOCATIONS AND WATER QUALITY VARIABLES
The sampling locations will be referred to as BA0011, BE0013, BH0017, BN0001 and AK0001, where these span, in the above order, the headwater site on the Bow River at Lake Louise, BA0011, to the site, AK0001, on the South Saskatchewan River below major urban centres and the irrigation district of southem Alberta. Locations BE0013 and BH0017 are on the Bow River above the maj or urban centres and the irrigation district, while BN0001 is on the Bow River below Calgary and in the irrigation district. The flow conditions vary throughout the year and between locations, with the upper locations being characterized by low flow in the winer and a mid-year peak due to mountain snow melting. The lower two locations have an additional spring peak due to snow melting in the prairie regions and receive irrigation return flows. The locations were sampled at approximately monthly frequency. Only the record between 1978 and 1986 is considered here since 1978 was the first year of sampling at BH0017. The analytical methods are the standard methods of the Water Quality Branch (1979) and flow is the mean daily flow for the day of sampling at the sampling location, except for BE0013 and BH0017, which are values calculated from nearby gauging stations. Analytical method and laboratory changes occurred over the period considered and, from a preliminary examination of the data, water quality variables were grouped according to the type of variable and the number of methodological changes. The variables reported here, which are representative of these groups, are the nutrient, nitrate plus nitrite (NO3 + NO2), with numerous methodological changes, sodium (Na), thought to reflect urban and agricultural inputs, with only one laboratory change, and specific conductance, a general indicator of ion content, also with only one laboratory change. A regression analysis (Esterby et al., 1990a), based on a model which included year, seasonality and methodological and laboratory parameters, showed no indication of a change in the variable value due to laboratory change for Na and specific conductance, but did show an effect due to methodological and laboratory changes for NO3 + NO2. Thus the latter variable is evaluated within periods of constant analytical methodology at the same laboratory.
2. Statistical Methods The statistical methods are discussed here in the context of the case study data set. Only brief descriptions of the methods are given since details are available in the references cited. The most important features of the data set are the sampling
222
S.R. ESTERBY ET AL.
interval of approximately one month, the record length of nine years, seasonality, missing data and different sampling times at some of the locations. Seasonality accounts for the highest proportion of variability in the data. In the regression models constructed to test for an effect of method and laboratory changes in sodium and nitrate plus nitrite (Esterby et al., 1990a), seasonality terms accounted for 42 to 79 percent of the total variability, except for sodium at BN0001, where seasonality accounted for only 28 percent. If changes over time or differences between locations are to be detected and precise and unbiased estimates are to be obtained, the seasonality must be accounted for in the analysis. The two methods of modelling seasonality which will be considered here are blocking and the fitting of a smooth or piecewise smooth curve to represent seasonality. Under additivity, blocking to remove seasonality involves the assumption that the effect of the season is the same over all years. The seasonal Kendall test for trend involves this assumption. Let Y/j denote the value of the water quality variable at a station in season j of year i. The set of values Ylj, Y2j, ..., Yn D, where nj is the number of years with observations in the season j, can thus be represented as
Yij = ttj + ~if(i) + e i j . The effect for season j is given by #j and the change over time, in a very general form, b y / 3 i f ( i ) where f ( i ) denotes some function of time. The statistic calculated within each season in the seasonal Kendall test is obtained from the differences Yij - Ykj, the difference between the observations in years i and k of season j, and then a linear combination of these statistics is taken. If the assumption of constant seasonal effect over years is met, the seasonal effect will be removed from the differences. A practical consideration is how to define the seasons so that the above assumption is met. Hirsch et aL (1982) demonstrated the seasonal Kendall test using the month as the season since their data were monthly water quality time series. This involves the assumption that the seasonal component for a month (the component for January, for example) is constant over all years. Since the present set of data consists of samples collected at approximately monthly intervals, months are also used as seasons and it is assumed that an observation at any time within the month contains the same seasonal effect for that month. The seasonal Kendall test for trend, test for homogeneity of trend and nonparametric slope estimator are then applied. In the alternative method of accounting for seasonality, the date of sampling, which should always be available, is used. Thus, there is the pair (Yij, tij) consisting of the value of the water quality variable and, for example, the day in the year for the sample collected in month j of year i. A regression model is fitted which contains sinusoidal functions of fixed periods to describe the seasonal cycle within a year (Esterby et al., 1991). By allowing the phase and amplitude to vary from year to year, the differences in timing and amplitude of the peaks due to weather
WATER QUALITY CHANGES ALONG A RIVER SYSTEM
223
differences from year to year can be accomodated. For the present data, day-withinyear (1 to 365 or 366) is used and regression models are fitted with a one or two component sinusoid for the seasonal component and either yearly mean levels or a linear change over years. Residual plots and diagnostics are used together with tests on parameters to arrive at the most satisfactory models. Similarly, seasonality has been taken into account by blocking or fitting curves in the comparison of locations. Let Ytij and Ykij be the values of a water quality variable at locations l and k in month j of year i. If the corresponding times of observation, ttij and tkij, are sufficiently close so that the locations are at approximately the same point on the hydrological cycle, methods based on the differences Yt~j - Ykij Can be used to determine differences between locations. Under the assumptions that the components due to season and to change over time are equal at both locations at matched sampling times and that the location difference is constant, the t test, sign test and Wilcoxon signed-rank test may be applied, with the further assumption for the t test that differences are normally distributed. These methods are applied to subsets of the data for the three upper locations. When sampling times do not match, some method of estimating the value of the water quality variable at the same time at both locations is required. It is possible to do this within the framework of regression by dividing the year into periods such that the water quality variable is approximately constant or changes linearly (Esterby et al., 1990b). A test that one fitted function is uniformly greater than the other over the interval (Tsutakawa and Hewett, 1978) can then be applied. The methods for the analysis of changes over time at one location and the comparison of locations, by fitting regression relationships within periods of the year, can accomodate missing data. Methods involving matching require observations for both members of a pair. The practical limitation of missing data, regardless of the method of analysis, is a reduced ability to detect or estimate the feature of interest. The adequacy of nine years of data to detect changes over time will be considered in the discussion of the results.
3. TemporalChanges The seasonal test for trend is the sum of statistics calculated for each month. To illustrate the data used in these individual statistics, the specific conductance over years is plotted separately by month for two locations (Figure 1). A visual assessment comparable to the test consists of examining the individual plots for changes over time and drawing some consensus from the plots for all months. The results of application of the Kendall test for trend for individual months, the seasonal Kendall test and the test for homogeneity of trend (Table I) show that there is evidence of a monotonic change over the period 1978 to 1986 for the three upper locations (BA0011, BE0013 and BH0017) and that there is no evidence of heterogeneity among months.
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TABLE I The Kendall test applied to individual months, the seasonal Kendall test and the test of homogeneity of trend for specific conductance data, 1978 to 1986. Period
BA0011
Sampling location BE0013 BH0017 BN0001
AK0001
Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
-0.31 0.00 -1.11 0.63 -0.10 0.00 -1.36 -0.21 0.32 - 1.15 -1.77 -1.56
-1.50 -1.25 -1.36 -0.10 -0.37 -0.63 0.25 0.00 -0.50 -0.94 -2.10" -1.47
-0.94 -0.25 0.31 -1.56 -0.62 -1.77 -0.52 -1.36 -0.94 -0.21 -0.62 -1.26
-0.94 -0.52 -1.56 -0.21 -0.52 -1.61 1.36 -0.84 0.00 0.00 0.12 -1.15
-0.52 -0.52 -1.15 -2.19" 0.73 1.15 -0.52 0.31 0.84 -0.31 0.52 -0.94
Seasonal Kendall a
Year
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van Belle and Hughes b Trend Homogeneity
Year 4.32" 8.33
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0.66 12.32
Test Kendall a
* Significant at the 0.05 level. a The entries in the table for the Kendall and seasonal Kendall tests are the standard normal deviate form, with continuity correction, as described by Hirsch et al. (1982). b The statistics for tests of trend and homogeneity, without continuity corrections, as described by van Belle and Hughes (1984), are distributed approximately as chi squares with 1 and 11 degrees of freedom, respectively.
The regression models which were fitted consist of a seasonal cycle for each year and either a mean for each year or a linear trend over years. The resulting models are superimposed on the specific conductance data for the same two locations in Figure 2. A visual assessment of the adequacy of the form of the model can be obtained from the proximity of the sampling points to the fitted curves and from the residual plots (Figure 2). The significance levels for sets of terms in the models and R 2, expressed as a percentage, are given for the two forms of regression models fitted to the data from all locations (Table II). There are significant differences between yearly means for all locations and, for all locations except AK0001, the linear term was significantly different from zero. Thus, the consensus from hypothesis testing under the two methods is that there is a monotonic change over years for the upper three locations. The regression analysis indicates that a linear term explains more variability for locations BE0013
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T A B L E II C o m p a r i s o n o f m o d e l s for specific c o n d u c t a n c e with yearly m e a n s and with trend in year.
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and BH0017 than for the others. If one accepts the form of change in the mean level over years as a linear change, then comparison of confidence intervals for the regression and nonparametric slope estimators allows a comparison which takes the variability of the estimates into account (Table III). The regression point estimator is always slightly lower as is the lower limit of the regression confidence interval. The agreement is remarkable in view of the differences in the method of accounting for seasonality and the expected influences of large observations on the regression methods. The former might be expected to impair the nonparametric method by not taking into account differences in the seasonal cycle from year to year and the latter to produce distorted estimates due to the influence of larger observations. However, the residual plots from the regression models with linear terms in year (Figure 2) show that the linear term does not account for some of the yearly changes in mean level, since residuals are predominantly positive in some years and negative in others, particularly for BH0017. The lower percentage of variation explained and the presence of fewer runs in the residuals (100R 2 and p runs, Table II) for regression models with the linear term in years, also indicate that the linear term is inadequate. Multiple comparison procedures, guided by a plot of the estimates of yearly means levels (Figure 3), leads to the conclusion that there are sets of high years and of low years, with some variation from location to location, but with 1982 having low specific conductance relative to the other years at all locations (Table IV). These results help to explain why the three upper locations were found to exhibit a monotonic change which could be modelled to varying
230
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TABLE III Point estimate and confidence limits for the slope from the model in Table II,/3, and the nonparametric slope estimator, B. Nonparametric Slope Estimation
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95% Confidence Interval
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B -1.0 -3.0 -3.0 -2.0 -1.9
95% Confidence Interval -1.8, -5.2, -4.8, -4.6, -5.1,
0.0 -1.0 -1.3 0.0 1.7
TABLE IV Tests for significant differences between the estimated mean specific conductance for the indicated groupings of years. Mean for Indicated Years with Significant Differences at 0.05 Level Station
1
2
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82,82 163.5
84,85,86 170.9
BE0013
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78,79,80,81
285.1
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79,80,81
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324.3
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79,80 406.8
370.7
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78,81,82,83,84,85,86 394.3
3 78,79,80,81 178.0
79,80 436.3
The means are arithmetic means of the year effects shown in Table V. Underlined years are not significantly different based upc- ~ method.
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degrees as a linear trend. The drop in specific conductance in 1982 is essentially a step change, with the step size larger for BE0013 and BH0017 than BA0011. Such a change can be fitted as a linear decline, provided the step size is large enough.
4. Location Differences If it is reasonable to formulate the test for location differences as a test of the hypothesis of the same monotonic trend at all locations, then the method described by van Belle and Hughes (1984) can be used. From the above results, it is clear that this is not the best approach here. A further consideration for these five locations is the complex hydrological cycle with different sources of water at different times of the year. To show the nature of the variability of two other variables, sodium and nitrate plus nitrite were analyzed over the period 1984 to 1986, a period of constant methodology and laboratory for nitrate plus nitrite and a period in which mean levels of specific conductance were constant over time at all locations. To examine the changes in concentration of the variables within intervals in which different flow events are occurring, the sodium and nitrate plus nitrite concentrations have been plotted with the year divided into three intervals (Figures 4 and 5). These intervals correspond approximately to: (1) constant flow at the three upper locations and increased flow at the two lower locations due to the snow melting on the prairies; (2) a peak in flow at all locations due to the snow melting in the mountains; and (3) approximately constant flow at the three upper locations and a small increase at the lower locations (Figure 6). Clearly, the sodium concentration is always higher at the two lower locations than at the three upper locations and similarly for nitrate plus nitrite, except in mid-year when all five locations have similar concentrations. Because of this and the fact that the three upper locations can be matched by sampling date but the two lower cannot, the analyses will be done separately for these sets of three and two locations. 4.1. MATCHED SAMPLING DATES
Differences in concentrations between pairs of locations over the entire period, 1984 to 1986, were used to test the hypothesis of no difference between locations for the three possible pairs (Table V). From this it can be concluded that there are location differences between the sodium and nitrate plus nitrite concentrations which are highly significant on the basis of all tests except for nitrate plus nitrite at BA0011 and BH0017. This conclusion holds whether a comparison-wise or family error rate is adopted (see Table 5 footnote for an explanation). An assumption underlying all these tests is that the set of differences {dimj}, where dimj is the differefice between the concentration at locations i and m at time j, is a random sample from one population. Plots of the differences show seasonal effects (Figure 7). To make the above assumption tenable, the data were separated into the three intervals within the year and the results, based on the Wilcoxon signed-rank test
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TABLE V Tests for a difference in concentration between locations and estimation of the average difference, for years 1984 to 1986.
Result Number of pairs Significance probability ~ t test Wilcoxon signed ranks Sign test Confidence limits b t test
k sample sign test
Sodium BA0011 BE0013 BA0011 -BE0013 - B H 0 0 1 7 - B H 0 0 1 7 34 33 33
