## Overview

The trends displayed in the national maps and on the station data plots represent 1) magnitudes of trends in data, and 2) whether or not these trends are statistically significant at different confidence levels.

## Mann-Kendall Trend Test

The Mann-Kendall Test was used to determine trend significance, or lack there-of, for the trends calculated in SPI, SPEI, PDSI, and precipitation data. To paraphrase the description of the Mann-Kendall test found in Hamed and Rao 1998: The Mann-Kendall test ranks the data points in a time series, compares the ranks of all points in the time series to each other, and calculates a test statistic from a summation of these comparisons. For a large sample size, the test statistic tends towards a normal distribution, so statistical significance is determined by comparing a standardized test statistic (dependent on the variance of the test statistic) to a standard normal distribution. The null hypothesis for the test is that data are randomly ordered, while the alternative hypothesis is that there is a monotonic trend in the data, either positive or negative. The different statistical significance levels that the user can pick are used to determine statistical significance for the Mann-Kendall test, which is run as a two-tailed test.

While interpreting the results of the trend tests from the time series and national maps, the following should be kept in mind:

- Shorter time periods (particularly the 35- and 30-year analysis periods) are likely sensitive to over-estimating the presence of a long-term trend (or lack thereof) due to single events.
- The presence of serial correlation in a time series (a time series correlating with itself at some time lag) can violate the assumption that all data points in a time series are independent samples, which can lead to mis-evaluation of trend significance. The SPI and SPEI trend significance calculations should be less sensitive to this problem because of the way those data are sub-sampled, but the serial correlation problem may be present with trend significance calculations for PDSI trends.