Comment on “Applicability of prewhitening to eliminate the influence of serial correlation on the Mann‐Kendall test” by Sheng Yue and Chun Yuan Wang

[1] Statistics is a powerful tool for hydroclimatology. However, misuse of a statistical technique, often caused by ignoring unavoidable underlying assumptions that must be made to use the technique, can render the analysis meaningless and could also result in wrong conclusions. The Mann-Kendall (MK) test [Mann, 1945; Kendall, 1975] for trend has frequently been used to detect trend in hydroclimatological time series because it requires fewer assumptions than some of its parametric counterparts. However, this test does require the assumption that the observations that are to be analyzed are the realizations of a collection of independent random variables. If the test is applied to serially correlated data (which is often the case for hydroclimatological time series), the trend detection results may not be reliable because the test may then reject the null hypothesis of no trend (H0) more often than specified by the significance level [von Storch, 1995]. [2] One way to avoid this problem is to apply the test only on data that have been made serially independent [Lettenmaier et al., 1994; Gan, 1998; von Storch, 1995]. A simple remedy exists if there is a sufficient physical reason to assume that the time series to be tested is a sum of a deterministic trend and serially correlated noise generated by an AR(1) (or higher order) process. In this case, the test may be conducted after prewhitening the time series [von Storch, 1995]. The prewhitening procedure has been used in many applications for detecting trends [e.g., Douglas et al., 2000; Zhang et al., 2001; Wang and Swail, 2001]. [3] Yue and Wang [2002] (referred to as YW hereinafter) question the validity of the prewhitening serially correlated data before applying the MK test. They argue that prewhitening reduces the magnitude of any trend that may be present and that it therefore reduces the power, or sensitivity, of the test. They suggest that ‘‘when sample size and magnitude of trend are large enough, serial correlation does not significantly influence the MK test,’’ and in such case, ‘‘it is better to use the MK test on the original data rather than after prewhitening’’. Unfortunately, this advice could lead unsuspecting users to make serious errors in the interpretation of their data because it requires the user to first judge visually whether trend is present in the data that is to be analyzed. Such subjective assessments cannot be performed reliably when data are serially correlated. For example, the upper panel of Figure 1 displays a sample of length 80 from a pure red noise process with lag 1 correlation coefficient r1 = 0.4. Trend is visually apparent, even though the red noise process has no trend. Once a trend of 0.02 per time step is added to the same time series, no trend is visually apparent (bottom panel of Figure 1), even though the red noise process now has a trend. Moreover, the overall performance of the combined procedure, consisting of subjective trend assessment followed by objective estimation of the trend magnitude, cannot be assessed. Thus this advice leaves the user with a trend analysis procedure with unknown characteristics. Good science should always use analytical tools that have known performance characteristics. Also, we note that this advice contracts advice given by Yue et al. [2002]: they recommend the removal of autocorrelation before assessing the statistical significance of a trend in that article. [4] In this note, we correct some misconceptions of YW, and provide methods for determining the magnitude and the statistical significance of linear trend in serially correlated data. The performance of the different methods of trend detection that we consider are assessed using Monte-Carlo simulation. We describe the methods for trend detection, and the approach used to assess these methods in the following section. Results are presented in section 3, and we conclude with comments on YW in section 4.