MULTIVARIATE NONPARAMETRIC TESTS FOR TREND IN WATER QUALITY1

ABSTRACT: National and state fixed station stream quality monitoring networks have now been in existence for over ten years. The resulting data bases provide opportunities and challenges for statistical trend assessment. Although nonparametric tests have been developed that are well suited to such problems, the interpretation of variations in trend significance between seasons and variables remains a problem. One recently developed test is based on the sum of Mann-Kendall statistics over seasons or variables, with the test statistic variance computed as the sum of the covariances of the individual Mann-Kendall statistics. In this method, up- and downtrends can cancel, giving an overall indication of no trend. A related test which is sensitive to trend regardless of direction has been shown to behave poorly for typical stream quality record lengths. An alternative formulation which is sensitive to up- and downtrends and has power approaching that of the covariance sum method, is described. In addition, a variation of a contrast test for discriminating trend directions and magnitudes among variables or seasons where correlation between seasons or variables is present is described, and tests of its performance reported.     

DETECTING BASE FLOW IMPACTS IN COASTAL PLAIN STREAMS1

ABSTRACT: An established trend analysis methodology was applied to the problem of identifying and quantifying stream base flow impacts from water withdrawals and water loss through interbasin transfers. Impacts were simulated using base flow values selected from two U.S. Geological Survey (USGS) continuous record streamflow sites located within the Pinelands of southern New Jersey. Study site base flows were regressed against index site base flows with monotonic and step trend tests applied to the residuals from the regression model. The smallest, significantly detectable (α= 0.10) percentage reduction within a given simulation was used as an estimate of the sensitivity of a trend test. Evaluation of the trend analysis methodology led to the following practical considerations regarding trend test sensitivity. The proportion of study site base flow variability explained by index site base flows should be maximized, while at the same time minimizing positive, first-order autocorrelation in the regression residuals. Given the importance of detecting autocorrelation, missing values should be avoided or minimized. The quarterly (three-month) interval reduced the magnitude of autocorrelation relative to a shorter two-month sampling interval. Sensitivity appeared to improve when equalizing the number of values before and after a base flow impact(s) while seasonally biased sampling appeared to reduce sensitivity. Based primarily on past trend detection studies, nonparametric tests were deemed a better choice over their parametric counterparts, due to the lack of stringent data distributional requirements coupled with little or no loss of power even when applied to normally distributed data.     

NONPARAMETRIC APPROACHES TO ENVIRONMENTAL IMPACT ASSESSMENT1

ABSTRACT: The interesting developments in non-parainetric testing and estimation methods presented in the upcoming sequence of nine papers are evaluated, compared, and put into proper perspective. Because a deterioration in water quality constitutes a direct threat to human health, it is of utmost importance to have flexible non-parametric methods available for detecting and describing trends in water quality time series. A distinct advantage of nonparametric tests is that they are usually very effective when applied to “messy” environmental data which may, for example, contain many missing observations and not be normally distributed. By applying their enhanced approaches for nonparametric methods to water quality time series, as well as employing well designed simulation experiments, the authors of the papers clearly demonstrate the efficacy of utilizing nonparametric tests in environmental impact assessment.     

PARAMETRIC AND NONPARAMETRIC TESTS FOR DEPENDENT DATA1

ABSTRACT: Simulation and analytical results show that ignoring serial dependence can have serious effects on the performance of the t, sign, and Wilcoxen tests. In particular, the true significance levels of these tests are altered significantly from the intended nominal levels. Modifications for these tests are given and shown to have the correct significance levels. Furthermore, an estimate of serial correlation is suggested for binary data and evaluated by simulation. An application to the toxic contaminants data from the Niagara River concludes the paper.     

DETECTABILITY OF STEP TRENDS IN THE RATE OF ATMOSPHERIC DEPOSITION OF SULFATE1

A method is presented to assist policy makers in determining the combination of number of sampling stations and number of years of sampling necessary to state with a given probability that a step reduction in atmospheric deposition rates of a given magnitude has occurred at a pre-specified time. This pre-specified time would typically be the time at which a sulfate emission control program took effect, and the given magnitude of reduction is some percentage change in deposition rate one might expect to occur as a result of the emission control. In order to determine this probability of detection, a stochastic model of sulfate deposition rates is developed, based on New York State bulk collection network data. The model considers the effect of variation in precipitation, seasonal variations, serial correlation, and site-to-site (cross) correlation. A nonparametric statistical test which is well suited to detection of step changes in such multi-site data sets is developed. It is related to the Mann-Whitney Rank-Sum test. The test is used in Monte Carlo simulations along with the stochastic model to derive statistical power functions. These power functions describe the probability of detecting (α=0.05) a step trend in deposition rate as a function of the size of the step-trend, record length before and after the step-trend, and the number of stations sampled. The results show that, for an area the size of New York State, very little power is gained by increasing the number of stations beyond about eight. The results allow policy makers to determine the tradeoff between the cost of monitoring and time required to detect a step-trend of a given magnitude with a given probability.     

NONPARAMETRIC TEST PERFORMANCE FOR TRENDS IN WATER QUALITY WITH SAMPLING DESIGN APPLICATIONS1

ABSTRACT: In this paper four nonparametric tests for monotonic trend detection are compared with respect to their power and accuracy. The importance of comparing powers at equal empirical significance levels rather than nominal levels is stressed. Therefore, an appropriate graphical method is presented. The effect of the sampling frequency is also assessed using Monte Carlo simulations and a trajectory representation that visualizes the dynamics of the trade-off between the type I and type II errors. These methods are applied to compare four nonparametrical tests (seasonal Mann. Kendall, modified seasonal Mann-Kendall, covariance eigenvalue and covariance inversion) under several conditions. It is concluded with respect to the power that it is not worthwhile for the modified seasonal Mann-Kendall test applied to the AR(1) process considered in this paper to increase the sampling frequency from monthly to biweekly for detecting a monotonic trend of 5 percent, 10 percent, or 15 percent of the process variance. Under these conditions the seasonal Mann-Kendall test is highly liberal, while the covariance inversion and the covariance eigenvalue test are conservative. This research is situated in the development of an efficient sampling design for the Flemish water quality monitoring network.     

NONPARAMETRIC TESTS FOR TREND DETECTION IN WATER QUALITY TIME SERIES1

ABSTRACT: A review of nonparametric tests for trend leads to the conclusion that Mann-Whitney, Spearman, and Kendall tests are the best choice for trend detection in water quality time series. Recently these tests have been adapted to account for dependence and seasonality in such series (Lettenmaier, 1976; Hirsch, et al., 1972; Hirsch and Slack, 1984). For monotonic trends, a procedure allowing to select the pertinent tests considering the characteristics of time series is proposed and the practical limitations of the tests are also brought out. This procedure has been applied to identify the appropriate trend detection test for the time series of nine water quality parameters at Lake Laflamme (Québec). When a time series can be tested with the Mann-Whitney, Kendall, Spearman, or Lettenmaier (1976) test, the number of observations required to detect trends of a given magnitude, for selected significance and power levels can be calculated with the power function of the t test. When the test proposed by Hirsch, et al. (1984), Hirsch and Slack (1984), or Farrell (1980) need to be used, the number of observations can only be estimated approximately from the results of empirical power studies.     

MULTIVARIATE TREND TESTING OF LAKE WATER QUALITY1

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-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 monitoring study of acid precipitation effects.     

USING UNCENSORED TRACE-LEVEL MEASUREMENTS TO DETECT TRENDS IN GROUND WATER CONTAMINATION1

ABSTRACT: Environmental decision making involving trace-levels of contaminants can be complicated by censoring, the practice of reporting concentrations either as less than the limit of detection (LOD) or as not detected (ND) when a test result is less than the LOD. Censoring can result in data series that are difficult to meaningfully summarize, graph, and analyze through traditional statistical methods. In spite of the relatively large measurement errors associated with test results below the LOD, simple and meaningful analyses can be carried out that provide valuable information not available if data are censored. For example, an indication of increasing levels of contamination at the fringe of a plume can act as an early warning signal to trigger further study, an increased sampling frequency, or a higher level of remediation at the source. This paper involves the application of nonparametric trend analyses to uncensored trace-level groundwater monitoring data collected between March 1991 and August 1994 on dissolved arsenic and chromium for seven wells at an industrial site in New York.     

SUPPLEMENTATION OF MISSING VALUES IN WATER QUALITY DATA1

ABSTRACT The problem of estimating missing values in water quality data using linear interpolation and harmonic analysis is studied to see which one of these two methods yields better estimates for the missing values. The data used in this study consisted of midnight values of dissolved oxygen from the Ohio River collected over a period of one year at Stratton station. Various hypothetical cases of missing data are considered and the two methods of supplementing missing values are evaluated using statistical tests. The results indicate that when the percentage of missed data points exceeded ten percent of the total number in the original sample, harmonic analysis usually yielded better estimates for both the regularly and irregularly missed cases. For data that exhibit cyclic variation, examples of which are dissolved oxygen concentration and water temperature, harmonic analysis as a data generation technique appears to be superior to linear interpolation.     

EFFECT OF SERIAL CORRECTION ON GROUND WATER QUALITY SAMPLING FREQUENCY1

ABSTRACT: Methods of calculating uncertainty in estimates of serial correlation coefficients, and correcting for bias in short and medium length (less than 50 data point) records, are presented. Uncertainty and bias in the estimation of serial correlation coefficients for ground water quality data is shown to be considerable and to result in inaccurate calculation of the sampling frequencies for monitoring purposes. The methods are applied to a ground water data set consisting of 87 monthly measurements of nitrate concentrations. The variation in serial correlation coefficients with variation of record length is examined. The optimum sampling frequencies for detection of changes in ground water nitrate concentrations are estimated.     

CONSIDERATIONS OF SCALE IN WATER QUALITY MONITORING AND DATA ANALYSIS1

ABSTRACT: An assumption of scale is inherent in any environmental monitoring exercise. The temporal or spatial scale of interest defines the statistical model which would be most appropriate for a given system and thus affects both sampling design and data analysis. Two monitoring objectives which are strongly tied to scale are the estimation of average conditions and the evaluation of trends. For both of these objectives, the time or spatial scale of interest strongly influences whether a given set of observations should be regarded as independent or serially correlated and affects the importance of serial correlation in choosing statistical methods. In particular serial correlation has a much different effect on the estimation of long-term means than it does on the estimation of specific-period means. For estimating trends, a distinction between serial correlation and trend is scale dependent. An explicit consideration of scale in monitoring system design and data analysis is, therefore, most important for producing meaningful statistical information.     

ANTIDEGRADATION WATER QUALITY CRITERIA FOR THE DELAWARE RIVER: A DISTRIBUTION-FREE STATISTICAL APPROACH1

ABSTRACT: Existing water quality for the Middle Delaware Scenic and Recreational River is significantly better than is required by current standards, leaving a potential for degradation. A method is presented for deriving candidate antidegradation water quality criteria for this segment of the Delaware River using statistical analysis of historic (ambient) water quality data. Data for 34 water quality parameters are first evaluated for data density, serial correlation, trend, seasonality, and other factors. These preliminary analyses are based on observation of data plots and application of distribution-free statistical techniques that are insensitive to outliers and are robust to relatively mild violations of basic assumptions. Data for 12 of the parameters have sufficient density for further analysis and can reasonably be modeled as independent and identically distributed over time (either seasonally or for the entire data sets). For these 12 parameters, distribution-free statistical methods are developed and used to derive intervals within which there is high confidence (usually greater than 95 percent) that the quantiles with potential use as anti-degradation criteria (the 0.85^th, 0.90^th, and 0.95^th quantiles in this study) for a particular parameter lie.     

ANTIDEGRADATION WATER QUALITY CRITERIA FOR THE DELAWARE RIVER: A DISTRIBUTION-FREE STATISTICAL APPROACH1

ABSTRACT: Existing water quality for the Middle Delaware Scenic and Recreational River is significantly better than is required by current standards, leaving a potential for degradation. A method is presented for deriving candidate antidegradation water quality criteria for this segment of the Delaware River using statistical analysis of historic (ambient) water quality data. Data for 34 water quality parameters are first evaluated for data density, serial correlation, trend, seasonality, and other factors. These preliminary analyses are based on observation of data plots and application of distribution-free statistical techniques that are insensitive to outliers and are robust to relatively mild violations of basic assumptions. Data for 12 of the parameters have sufficient density for further analysis and can reasonably be modeled as independent and identically distributed over time (either seasonally or for the entire data sets). For these 12 parameters, distribution-free statistical methods are developed and used to derive intervals within which there is high confidence (usually greater than 95 percent) that the quantiles with potential use as anti-degradation criteria (the 0.85^th, 0.90^th, and 0.95^th quantiles in this study) for a particular parameter lie.     

EFFECTS OF IRRIGATION ON WATER QUALITY OF A SHALLOW UNCONFINED AQUIFER1

ABSTRACT: The ground water quality of a shallow unconfined aquifer was monitored before and after implementation of a border strip irrigation scheme, by taking monthly samples from an array of 13 shallow wells. Two 30 m deep wells were sampled to obtain vertical concentration profiles. Marked vertical, temporal, and spatial variabilities were recorded. The monthly data were analyzed for step and linear trends using nonparametric tests that were adjusted for the effects of serial correlation. Average nitrate concentrations increased in the preirrigation period and decreased after irrigation began. This was attributed to wetter years in 1978–1979 than in 1976–1977 which increased leaching, and to disturbance of the topsoil during land contouring before irrigation, followed by excessive drainage after irrigation. Few significant trends were recorded for other determinants, possibly because of shorter data records. Nitrate, sulphate, and potassium concentrations decreased with depth, whereas sodium, calcium, bicarbonate, and chloride concentrations increased. These trends allowed an estimation to be made of the depth of ground water affected by percolating drainage. This depth increased during the irrigation season and after periods of winter recharge. Furthermore, an overall increase in the depth of drainage-affected ground water occurred with time, which paralleled the development of the irrigation scheme.     

DATA ANALYSIS OF WATER QUALITY TIME SERIES IN LAKE ERIE1

ABSTRACT: A comprehensive data analysis study is carried out for detecting trends and other statistical characteristics in water quality time series measured in Long Point Bay, Lake Erie. In order to glean an optimal amount of useful information from the available data, the exploratory and confirmatory data anslysis stages are adhered to. To test a range of hypotheses regarding the statistical properties of the time series, a wide variety of both parametric and nonparametric techniques are employed. A particularly useful nonparametric method for discovering trends is the seasonal Mann-Kendall test.     

Hydrologic Effects of NDVI Time Series in a Context of Climatic Variability in an Upstream Catchment of the Minjiang River1

Abstract: Increasing regional vegetation activity has been frequently found especially in middle and high latitude and alpine areas, but the effects of which on regional hydrology is still highly uncertain. The Upstream Catchment of Minjiang River is a large mountainous catchment covering 22,919 km² with a diverse vegetation distribution pattern, including alpine group (A), subalpine group (SA), and temperate and subtropical group (T/ST). The Seasonal Mann‐Kendall test, a nonparametric trend test method, detected consistent upward trends in all groups in monthly accumulated growing degree days (AGDD_M) time series from 1982 to 2003, but no significant trend in mean monthly precipitation (MMP) time series in any group. The alpine group had a significant (p = 0.024) upward trend in monthly Normalized Difference of Vegetation Index (NDVI) time series from 1982 to 2003, in contrast, the SA and T/ST groups had decreasing (although not significant) trends. AGDD_M plays more important role than MMP in affecting NDVI change in alpine areas, indicating temperature was the main climatic driver. In contrast, water was the main driver for the T/ST group, as indicated by the significant correlation between NDVI and MMP and a weak correlation with AGDD_M. Correlation coefficients of NDVI and river flow varied with seasons, mostly negative, especially during the growing season (April to October). A significant (p = 0.025) correlation was found only in August, indicating that an increase in peak‐NDVI decreased high flow significantly. TI‐NDVI_c, which was developed in an attempt to track the vegetation change at the catchment scale, accounted for more than 40% of the evapotranspiration increase (r² = 0.43).     

CORRELATION OF ANNUAL PEAK FLOWS FOR PENNSYLVANIA STREAMS1

A common assumption in flood frequency analysis is that annual peak flows are independent events. This study was undertaken to investigate the validity of this assumption with regard to Pennsylvania streams by statistically analyzing the dependence between annual peak flows and to determine if basin carryover effects relate to the degree of dependence. Five tests of dependence, the autocorrelation test, the median crossing test, the turning points test, the rank difference test, and the Spearman rank order serial correlation coefficient test were applied to the series of annual peak flows for 57 streams. Of the 57 streams analyzed, only two exhibited signs of dependence by at least two of the tests performed, and the baseflow component of annual peak flows was found to be unrelated to the degree of dependence exhibited between annual peak flows. It was concluded that the assumption of independence of annual peak flows is valid in flood frequency analysis for Pennsylvania streams.     

A NEW LOOK AT FLOOD RISK DETERMINATION1

ABSTRACT: The probability distributions of annual peak flows used in flood risk analysis quantify the risk that a design flood will be exceeded. But the parameters of these distributions are themselves to a degree uncertain and this uncertainty increases the risk that the flood protection provided will in fact prove to be inadequate. The increase in flood risk due to parameter uncertainty is small when a fairly long record of data is available and the annual flood peaks are serially independent, which is the standard assumption in flood frequency analysis. But standard tests for serial independence are insensitive to the type of grouping of high and low values in a time series, which is measured by the Hurst coefficient. This grouping increases the parameter uncertainty considerably. A study of 49 annual peak flow series for Canadian rivers shows that many have a high Hurst coefficient. The corresponding increase in flood risk due to parameter uncertainty is shown to be substantial even for rivers with a long record, and therefore should not be neglected. The paper presents a method of rationally combining parameter uncertainty due to serial correlation, and the stochastic variability of peak flows in a single risk assessment. In addition, a relatively simple time series model that is capable of reproducing the observed serial correlation of flood peaks is presented.     

RELATING TRENDS OF PRINCIPAL COMPONENTS TO TRENDS OF WATER-QUALITY CONSTITUENTS1

ABSTRACT: The seasonal Kendall test is used for detecting water-quality trend or lack of trend for monthly data of 15 water-quality constituents at 15 sampling stations in the Arkansas River, the Neosho River, and the Verdigris River basins. Trends of individual constituents and the trends of the first four principal components for the correlation matrix of water-quality data at each station are determined, and the relationships between the trends of constituents and the trends of principal components are established. Using the principal components not only reduces the high dimensionality of the original data to a few principal components, but also presents an overall picture of water-quality trend of these river basins.