Top Twelve Tip #10
Know Your Target for Trend Analysis
(As with our other Top Twelve Tips, you'll get much more detail about these 12 in our Applied Environmental Statistics course, soon to be available at our Online Training Center.)
Trend analysis is any test where one of the explanatory (x) variables is time. Continuous trends are often measured using linear regression or the Theil-Sen line (the line that goes with the “Mann-Kendall test for trend”). Step trends are tested using two group tests (the t-test or Wilcoxon rank-sum tests) comparing conditions before vs. after a specific date or event.
What is a trend? In statistics, a trend is monotonic -- it goes in predominantly one direction, an overall increase or decrease. Patterns that go up and then down again are technically not trends, even though they can be modeled with polynomial regressions or other techniques. One of the important cyclical (non-trend) patterns in environmental systems is seasonality, the tendency at one time of year for values to be higher than average, and at another time of the year to be lower.
Parametric trend analysis incorporates seasonality by building a more complex regression model. Sine and cosine terms are added to the regression equation where time is the explanatory (x) variable. Sine and cosine model a wave pattern that is fit to the pattern of the data, peaking at the time of year observed at the site. This provides better predictions, as well as removing noise in order to see the trend more clearly (see Tip #7). Nonparametric trend analysis (usually methods based on Kendall’s tau correlation) split up the data into seasonal “blocks”, performing the test separately on each block and combining the results into one overall test. No comparisons across seasons are made -- no comparison is made of June data in later years to January data in earlier years, etc.
The most prevalent trend test in environmental studies is a nonparametric test with blocking, the Seasonal Kendall test, developed at the US Geological Survey by Hirsch and others in the 1980s. Computed on the original data, it tests whether values are increasing, decreasing, or staying the same. Just as commonly, the trend test is computed on data adjusted for a covariate such as depth to water or another possible cause of change. By adjusting for or ‘factoring out’ that other variable, the test focuses on the targeted cause of the trend. This again utilizes Top Twelve Tip #7, factoring out the ‘noise’ due to changing (often natural) influences to focus on trends due to variables (human effects) that might be able to be managed. For more information on trend analysis, see the chapter in the free textbook Statistical Methods in Water Resources by Helsel and Hirsch, available as book #2 at http://practicalstats.com/books/ .