Water Resources Research

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] Simulation experiments of Yue and Wang [2002] show that the existence of positive serial correlation significantly increases the probability of rejecting the null hypothesis of no trend while it is actually true, no matter what the sample size is. Positive serial correlation also increases the probability of rejecting the null hypothesis when some trend is present for time series with short record length. Their results indicate that removal of positive AR(1) by prewhitening greatly reduces the rejection rate or power of the test of rejecting the null hypothesis while it might be false.

[2] The authors provide no information about the effect of prewhitening on the rejection rate of the null hypothesis when there is no trend. This is an important question because positive serial correlation without prewhitening could lead to a type 1 error (rejecting the null hypothesis when it is true) much larger than the significance level, leading to rather frequent detection of a trend when it is not actually present.

[3] We have repeated the authors' simulation study to estimate the rejection rate of the null hypothesis for prewhitened series in the absence of trend. Parameters of the AR(1) process, sample sizes, the significance level, and the number of generated series are the same as in the authors' study. Figure 1 shows the simulation results, rejection rate versus serial correlation for two sample sizes, n = 20 and 50, before and after prewhitening.

Figure 1.

Rejection rate for the series with no trend, without and with prewhitening (α = 0.05).

[4] It is seen that prewhitening reduces the probability of rejecting the null hypothesis of no trend when it is true for series with positive serial correlation. Even for small samples (n = 20), the rejection rate is almost the same as the significance level (α = 0.05) up to a value of the serial correlation coefficient equal to 0.5. For a very high correlation of 0.8 the rejection rate after prewhitening is 0.12 compared with 0.40 before prewhitening. For larger samples (n = 50) the effect of prewhitening is stronger; the rejection rate reduced to 0.09 from 0.48 for the serial correlation equal to 0.8.

[5] It can be concluded that prewhitening of series with positive serial correlation brings down the rejection rate of the null hypothesis almost to the selected level of significance, thus preventing the detection of a false trend more frequently than the selected level.

[6] The Mann-Kendall (MK) test is applied to the annual streamflow series of 106 sites in Turkey. Record lengths vary between 25 and 66 years. The null hypothesis of no trend is rejected at 31 sites at the significance level α = 0.05. Trend slopes are in the range β = 0.0034–0.075. Lag 1 serial correlation coefficient is significantly different from zero at 21 sites. Prewhitening is applied to those series. With prewhitening the null hypothesis of no trend is not rejected only at four of these sites. The result of the MK test is not changed with prewhitening for the remaining 17 series. The series for which the result is affected by prewhitening have trend with slopes ρ = 0.008–0.027 before prewhitening.

[7] The authors argue that prewhitening is not suitable for eliminating the effect of positive serial correlation on the MK test when trend exists because it will remove a portion of trend and hence reduces the possibility of rejecting the null hypothesis while it might be false. On the other hand, however, prewhitening has the advantage of significantly reducing the magnitude of type 1 error when the time series is free of trend. In practice, we do not know if there is a trend. It could be preferred to apply prewhitening when a significant positive serial correlation exists so that a false detection of trend is prevented. This would make it more difficult to find out a weak trend by applying the MK test. However, the prevention of an error of false detection of trend when it does not exist may be preferred to an error of not being able to detect a weak trend.