## C.5 Study Question 5. Is there a trend in contaminant concentrations?

Whether concentrations are increasing, decreasing, periodic, or stable over time is a question that generally requires analysis beyond simple graphical methods, especially when data fluctuate or exhibit high variability. The tests described below for general trend testing are closely related to the tests used for season or period trend analyses or for calculating attenuation rates (Study Question 6, Study Question 7). Statistical trend tests can be used as a diagnostic tool to determine if the meanThe arithmetic average of a sample set that estimates the middle of a statistical distribution (Unified Guidance). of the population is stationaryA distribution whose population characteristics do not change over time or space (Unified Guidance). to qualify the use of the distribution for many other statistical tests. Trend tests can also be used to demonstrate decreases in contaminant concentrations over time. Temporal trend analysis of groundwater monitoring results often reveals differences in results between wells. Even at sites with overall decreasing chemical concentrations, the trend analysis often identifies some wells with statistically significant decreasing concentrations, some wells with decreasing concentrations that are not statistically significant, and some wells with increasing concentrations.

This question is usually relevant in the remediation, monitoring, and closure stages of the project life cycle.

Selecting and Characterizing the Data Set

Verify that the data set can support trend analyses and modeling. Refer to Section 3.4: Common Statistical Assumptions for further discussion of how the following requirements may impact statistical analysis results.

- Check for outliersValues unusually discrepant from the rest of a series of observations (Unified Guidance). using box plots, probability plots, Dixon's test, or Rosner's test.
- Check for autocorrelationCorrelation of values of a single variable data set over successive time intervals (Unified Guidance). The degree of statistical correlation either (1) between observations when considered as a series collected over time from a fixed sampling point (temporal autocorrelation) or (2) within a collection of sampling points when considered as a function of distance between distinct locations (spatial autocorrelation). between successive sampling events.
- Ability to detect trends can be impacted by pooling data across wells.
- In general, longer records of data are better at detecting trends but in many cases trends will differ based on remedial methods.
- See also Section 4.1: Considerations for Statistical Analysis.

Statistical Methods and Tools

To determine if there is a temporal change or pattern to the data, first use simple graphical techniques to observe significant trends. However, if cyclical effects complicate the pattern of the data consider other statistical methods to answer this study question. The statistical methods described below focus on the monotonic trendThe long-term movement in an ordered series, which regarded together with the oscillation and random component, generates observed values that are entirely increasing or decreasing. (EPA 2006c)s, as well as systematic variation in a temporal setting.

- These plots show concentration on the y-axis versus time on the x-axis.
- You must assign values to nondetects for these plots.

Use this test to visualize temporal changes of concentrations of a single chemical. Data from multiple time periods can show existing patterns in the data. By combining multiple sample points, the temporal patterns usually show up as parallel traces. The spread of the lines would indicate the dispersion of the data over time. Qualitative comparison of the slopes can identify individual trends between wells.

- Data must follow a normal distributionSymmetric distribution of data (bell-shaped curve), the most common distribution assumption in statistical analysis (Unified Guidance). and have a constant varianceThe square of the standard deviation (EPA 1989); a measure of how far numbers are separated in a data set. A small variance indicates that numbers in the dataset are clustered close to the mean. or spread of sample results.
- You must assign values to nondetectsLaboratory analytical result known only to be below the method detection limit (MDL), or reporting limit (RL); see "censored data" (Unified Guidance)..

This test identifies temporal differences among sample periods or nonstationarity (change in means over time). Evaluate the temporal effects of individual sampling events or cyclical event (season) by grouping concentrations across monitoring wells for each sampling date, or season.The one-way ANOVAone-way analysis of variance for temporal effects can formally identify cyclical or nonmonotonic trends.

Significant cyclical variation usually tends to inflate the estimate of the current population variance. If the test identifies a significant temporal effect, the data set can be adjusted to account for seasonality or other cyclical patterns. Study Question 6 addresses how to identify and correct for seasonality or other periodic changes in concentrations.

- This nonparametricStatistical test that does not depend on knowledge of the distribution of the sampled population (Unified Guidance). test does not require that data are derived from a particular statistical distribution.
- Spearman's test is not influenced by outliers or extreme values.
- This test is not appropriate for data sets with a large number nondetects.

This test provides information on the direction of the trend (increasing or decreasing) and whether or not the trend is significant. This test is closely related to Pearson’s test discussed below. This test is not recommended if there are seasonal or periodic fluctuations in concentrations. Study Question 6 addresses how to identify and correct for seasonality or other periodic changes in concentrations. If there is autocorrelation among successive sampling events then the samples are not independent and the degrees of freedomThe number of ways which members of a data set or data sets can be independently varied (Unified Guidance). for evaluating statistical significanceStatistical difference exceeding a test limit large enough to account for data variability and chance (Unified Guidance). A fixed number equal to alpha (α), the false positive rate, indicating the probability of mistakenly rejecting the stated null hypothesis (H₀) in favor of the alternative hypothesis (Hᴀ). Or, the p-value sufficiently low such that the analyst will reject the null hypothesis (H₀). are overstated.

- This nonparametric test does not require that data are derived from a particular statistical distribution.
- This test is not influenced by outliers or extreme values.
- This test is not appropriate for data sets with a large number of nondetects.

This test identifies significant changes in the mean over time using at least eight sample points (not assuming a particular distribution). The slope of the concentration over time is monotonic. Trends are significant when the absolute value of the S statistic is greater than the critical value. This result indicates that the mean is not stationary over time at this sampling pointA specific spatial location from which groundwater is being sampled.. The total of the pair differences (S statistic) results in a large negative or positive value for decreasing or increasing trends, respectively. The size of the S statistic is not a measure of the magnitude of the slope. You can calculate the monotonic trends of concentrations over time at a single point to identify statistically significant concentration trends.

- This test identifies the slope of the trend line.
- Confidence limits can be calculated around the slope of the trend line to ascertain whether the trend is statistically significant.
- Like the Mann-Kendall, this test does not require a normal distribution and can handle extreme values.
- This test is not appropriate for data sets with a large number of nondetects.

This test estimates the slope of a trend line, which can be used to predict the mean concentration at some point in time. If the slope is positive and upper and lower confidence limits around the slope are also positive, the test indicates that there is a statistically significant increasing trend, and the opposite is true for a negative slope and confidence limits. If the upper confidence limit is positive and the lower is negative, there is insufficient evidence to indicate a statistically significant trend.

- This test requires a normal distribution.
- This test is sensitive to outliers or extreme values.
- This test requires a constant dispersion of the data over sampling events.

This test provides information on the direction of the trend (increasing or decreasing) and whether or not this trend is significant. This test is closely related to the Spearman’s test discussed above. This test is not recommended if there are seasonal or periodic fluctuations in concentrations. Study Question 6 addresses how to identify and correct for seasonality or other periodic changes in concentrations. If there is autocorrelation among successive sampling events then the samples are not independent and the degrees of freedom for evaluating statistical significance are overstated.

- This test requires a normal distribution.
- This test is sensitive to outliers.
- This test requires constant dispersion of the data.

Regression is an easy procedure to apply and shows the relationship of pairs of data (time and concentration) to obtain the slope and intercept of a line. To apply a linear regression appropriately, the relationship must be linear (monotonic increasing or decreasing). When the slope is not zero, either a positive or negative trend is present. The statistical significance of the trend is identified by the slope being significantly different from zero.

Interpretation of Results and Associated Uncertainty

Temporal trend analyses show whether and how chemical concentrations are changing over time. As discussed above, nonparametric tests, such as the Mann-Kendall test and Theil-Sen trend line, do not require assumptions regarding the data distribution. In contrast, linear regression analysisA parametric statistical method to measure the linear trend of a data set using data point regression residuals that are based on assumptions of normality, homoscedasticity, and independence (Unified Guidance). requires an assumption regarding the pattern of change over time (such as monotonically decreasing). Additionally, parametricA statistical test that depends upon or assumes observations from a particular probability distribution or distributions (Unified Guidance). regression analysisA statistical tool for evaluating the relationship of one of more independent variables to a single continuous dependent variable (Kleinbaum et al. 2007). assumes that the variability not associated with the temporal trend is normally-distributed. If the required assumptions are not satisfied, then the accuracy of the regression analysis is reduced. However, if the assumptions are satisfied, regression analysis will be more accurate than the Mann-Kendall test because the regression analysis uses the information concerning data distribution as part of the test.

If the p-valueIn hypothesis testing, the p-value gives an indication of the strength of the evidence against the null hypothesis, with smaller p-values indicating stronger evidence. If the p-value falls below the significance level of the test, the null hypothesis is rejected. is less than 0.05, then typically the change in concentration over time is statistically significant. A statistically-significant trend depends on a number of factors including the length of the monitoring record and the magnitude of variability not associated with the long-term trend relative to the magnitude of the long-term trend. In data sets with high variability, longer monitoring records are needed to identify statistically significant trends. See also Section 4.5.1: Monitoring for Concentration Changes and Section 4.6.2: Trends Toward Compliance Criteria.

Related Study Questions

Study Question 3: Are concentrations above or below a criterionGeneral term used in this document to identify a groundwater concentration that is relevant to a project; used instead of designations such as Groundwater Protection Standard, clean-up standard, or clean-up level.?

Study Question 6: Is there seasonality in the concentrations?

Study Question 7: What are the contaminant attenuation rates in wells?

Key Words: Temporal Trends, Remediation, Monitoring, Closure

Publication Date: December 2013