## 5.5 Trend Tests A trend refers to an association or correlationAn estimate of the degree to which two sets of variables vary together, with no distinction between dependent and independent variables (USEPA 2013b). between concentration and time or spatial location, but can also refer to any population characteristic changing in some predictable manner with another variable. Trends take various forms, such as increasing, decreasing, or periodic (cyclic).

Detecting and assessing temporal and spatial trends is important for many environmental studies and monitoring programs. Trend tests are generally recommended as an intrawellComparison of measurements over time at one monitoring well (Unified Guidance). alternative to prediction limits or control charts for use in detection monitoring. Trend evaluations are frequently used to determine whether it is reasonable to assume concentrations are temporally stationaryA distribution whose population characteristics do not change over time or space (Unified Guidance). (for example, to perform statistical evaluations that require stationary means) and to detect or model decreasing trends to support natural attenuation studies. Table F-1 includes information about checking assumptions for Trend Tests.

### 5.5.1 Linear Regression (Parametric Methods to Test and Model Trends)

Linear regression is used to test for linear temporal trends. Ordinary least squares regression is used to fit the “best” straight line. A linear trend is reported when the slope of the regression line is demonstrated to be statistically different from zero (using a t-testA t-test, or two-sample test, is a statistical comparison between two sets of data to determine if they are statistically different at a specified level of significance (Unified Guidance).); a positive slope indicates an increasing trend and a negative slope a decreasing trend. The linear correlation coefficient Pearson’s r (Equation 3.5 in Chapter 3.3, Unified Guidance), which is the correlation coefficient between the observed and calculated concentrations, provides information about the direction and “strength” of the linear trend. A positive value of r indicates an increasing linear trend and a negative value a decreasing linear trend. The trend is “strong” if the absolute value of r (which ranges from – 1 to 1) is near one.

A regression line models concentrations for the period of time over which the concentrations were measured. However, this method is often used to predict concentrations at future times as well, under the assumption that the same linear relationship will be observed—an assumption that may not be valid for monitoring events in the distant future). For a decreasing trend, the regression line is often extrapolated to estimate the time at which 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. will be met. However, even when it is assumed that the regression line is valid for future monitoring, this approach will not necessarily result in conservative estimates (underestimate the actual time required to achieve a criterion), because it does not take into account the uncertainty of the regression fit that arises from the variability of the data around the calculated regression line. A set of concentrations that fall nearly on the calculated regression line will result in estimates that are more reliable than concentrations that exhibit much larger scatter about the same regression line. A confidence intervalStatistical interval designed to bound the true value of a population parameter such as the mean or an upper percentile (Unified Guidance). is often calculated for the regression line (Chapter 5.2) to account for the uncertainty of the meanThe arithmetic average of a sample set that estimates the middle of a statistical distribution (Unified Guidance). concentration as it varies linearly with time (for instance, to provide upper bound estimates of cleanup times or contaminant concentrations).

Many commercial statistical software packages offer multiple options for nonlinear regression fits. For example, many provide quadratic and cubic polynomials that can be used to model nonlinear trends. Many software packages also calculate confidence limits for nonlinear regression fits.

### 5.5.2 Mann-Kendall Test (Nonparametric Method to Test and Model Trends)

The Mann-Kendall test is a nonparametricStatistical test that does not depend on knowledge of the distribution of the sampled population (Unified Guidance). test for monotonic trends, such as concentrations that are either consistently increasing or decreasing over time. Therefore, the test is not appropriate when there are cyclic trends (where concentrations are alternatively increasing and then decreasing). The Mann-Kendall statistic provides an indication of whether a trend exists and whether the trend is positive or negative. Subsequent calculation of Kendall’s Tau permits a comparison of the strength of correlation between two data series.

The Mann-Kendall test can be used to evaluate the following:

• Are contaminant concentrations increasing or decreasing in upgradient or downgradient wells?
• Does contaminant flux, as measured across a plume cross section, indicate an increasing or decreasing trend?
• Are concentrations within a well stable?

### 5.5.3 Theil-Sen Trend Lines (Nonparametric Method to Test and Model Trends)

When a 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) is demonstrated using the Mann-Kendall test and the trend appears to be linear, you can use a Theil-Sen line to estimate the slope of the trend. The Theil-Sen line is a nonparametric alternative to the parametric ordinary least squares regression line. An ordinary least squares regression line models how the mean concentration changes linearly with time; a Theil-Sen line models how the medianThe 50th percentile of an ordered set of samples (Unified Guidance). (50th percentile) concentration changes linearly with time. Therefore, the approach may not be appropriate when more than 50% of the concentration measurements are nondetects. The slope of the Theil-Sen line will be significantly different from zero when Kendall’s tau is significantly different from zero (and vice versa). Like the parametric linear regression line, confidence intervals can be calculated for the nonparametric Theil-Sen line.

### 5.5.4 Spearman’s Rank Correlation Test

Spearman’s rank correlation coefficient rho (ρ) is a nonparametric correlation coefficient that can be used to test for monotonic trends. The Spearman rank correlation test is discussed further in Section 5.12.2 of this document.

Publication Date: December 2013

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