In Reynold J. Stone’s (RJS) comments (Stone 2011) on a trend analysis of daily temperature extreme indices in South America by Vincent et al. (2005), he suggested that the authors may have used a statistical method incorrectly, which could have led to misleading results. Specifically, RJS indicated that Vincent et al. (2005) did not assess the presence of changepoints before applying Sen’s method for estimating linear trends (Sen 1968). As a result, findings of a significant decreasing trend of 5.5% in the percentage of cold nights and increasing trend of 5.6% in the percentage of warm nights from 1960–2000 could have been an artifact of inappropriate statistical analysis. RJS suggested, instead, a decreasing step of 3.5% in 1976 for the percentage of cold nights and an increasing step of 3.8% for 1979 in the percentage of warm nights, without any significant trends on either side of the changepoint. RJS speculated that these abrupt changes could have been a response to the well-known large-scale climate shift around 1976/1977, corresponding to a phase shift in the Pacific decadal oscillation (PDO) (Miller et al. 1994; Agosta and Compagnucci 2008).
As a matter of fact, Vincent et al. (2005) carefully considered possible step changes in the station data by examining the annual mean of daily maximum and minimum temperatures using a two-phase regression-based approach (Vincent 1998). Stations with major problems were excluded in the computation of regionally averaged temperature indices; therefore, there was no need to consider step changes when trend analysis of the regional series was conducted. In the following sections, we show that the steps identified by RJS could be artifacts of using the Pettitt test, which assumes that there is not a trend in the series, and thus a trend could be falsely identified as a step. We also investigate a possible influence of large-scale circulation as represented by the PDO. We show that the PDO indeed has a significant influence on regional temperature indices and that a statistically significant trend is still present after the PDO effect is considered.
2. The Pettitt test
The Pettitt test (Pettitt 1979) has been used for the detection of changepoints in climatological time series along with other methods (Wijngaard et al. 2003). Under the null hypothesis, all values are considered to have a common distribution function, whereas under the alternative hypothesis, there is a break in the time series for which the distribution function of the two subseries, prior and after the break, are different. Therefore, the difference in the distribution may arise either from a change in the mean, a significant temporal trend, or both. As climate varies at different time scales, temporal trend cannot simply be assumed nonexistent and a climate application of the Pettitt test should involve use of a reference series to account for possible trends. Without using a reference series, acceptance of the alternative of the Pettitt test could also indicate that the time series has a significant trend and does not necessarily indicate a change in the mean.
A monotonic trend alters the sample mean of the time series, making it dependent on the period for which it is estimated. In this case, if the time series is divided into two subperiods, the two subperiod averages would be different. Intuitively, if the Pettitt test is applied to such series, a step could be identified from one subperiod average to the next, even if there is not a real step. In the following, we apply the Pettitt test to simulated series that contains a trend and no step to demonstrate that this is really the case.
We simulated three sets of 10 000 time series with the same sample size and statistical properties (i.e., standard deviation, trend, and first-order autocorrelation) as those of the regional temperature indices used in Vincent et al. (2005). Specifically, we first generated a series of 41 data points using an autoregressive lag-1 [AR(1)] process with a standard deviation of 2.0 and a lag-1 autocorrelation of 0.1. A slope of 0.0 (no trend), 0.05, and 0.10 was introduced in the simulated series of the three sets, respectively. For each of the three slope values, we applied the Pettitt test to each of the 10 000 series and obtained the percentages of the series that have a statistically significant break (a step as interpreted by RJS) at the 5% significance level. The result percentages are shown in Fig. 1 (gray bars).
When there is no trend in the series, detection rate of a significant step within the position of 21–25 is the highest but much less than the expected 5%. Detection rate becomes greater than 25% within the positions of 16–20 and 21–25 when the slope is 0.10. This means that there is more than 50% chance that a trend of 0.10 will be mistakenly identified as a significant step using the Pettitt test if the series is assumed to not contain any trends. Results also clearly show that the location of a significant step detected by the Pettitt test is often in the middle of the series. As a comparison, we also applied a method that does not assume stationarity of the series (Wang 2003) but considers both step and trend in a two subseries divided by a step. Results from this analysis, shown in Fig. 1 (black bars), indicate that false detection rate of a step is less than 5% as what would be expected and much less than the false detection rates obtained with the Pettitt test. Clearly, when used without a proper reference series, the Pettitt test easily identifies a genuine trend as a significant step around the middle of the time series.
3. Using the PDO to explain the variability in cold and warm nights
As speculated by RJS, the well-known phase shift in the PDO around 1976/77 (Agosta and Compagnucci 2008) may have resulted in a step in the regionally averaged temperature indices. To investigate if the trend identified in Vincent et al. (2005) could be due to this circulation change, we conducted a trend analysis with the PDO index explicitly considered in a regression model:
where yi is the annual percentage of cold and warm nights, i is the year, xi is the annual mean of the PDO index, and ei is the error. The standardized monthly values of the PDO were retrieved from the Web site http://jisao.washington.edu/pdo/ (Zhang et al. 1997; Mantua et al. 1997). The annual means of the standardized monthly PDO index clearly indicate a change from negative to positive phase in 1976 (Fig. 2), and the annual percentage of cold nights and warm nights (Figs. 1a and 1b in RJS) are correlated with the annual means of the PDO index from 1960 to 2000 with a coefficient of −0.8 and 0.6, respectively.
The results of fitting the regression model Eq. (1) to the data show that the estimated parameter b is significantly different from zero (at the 5% significance level) for the annual percentage of cold and warm nights with p values of 0.0002 and 0.0019, respectively (Table 1). The estimated parameter c is also significantly different from zero for both cold and warm nights. This preliminary analysis suggests that variation in the PDO has a significant influence on regional temperature indices in South America, as speculated by RJS. It also shows that the PDO alone does not explain long-term changes in the temperature indices as a significant decreasing trend in the cold nights and a significant increasing trend in the warm nights are also identified. There was only a small reduction in the trends when the PDO influence was considered; the trend was reduced from 5.5% to 4.1% and from 5.6% to 4.9% for cold nights and warm nights, respectively.
4. Summary and discussion
The proper use of any statistical test requires careful consideration of its underlying assumptions. A step change in a series, if not properly considered, can be falsely identified as a trend, as pointed out by RJS. A trend can also be falsely identified as a step by an improper application of the Pettitt test, as shown in section 2. As climate varies at different space and time scales with potential trends and steplike changes, the assumptions of no trend or no changepoint have to be verified prior an analysis. Vincent et al. (2005) considered the possibility of changepoints in the series by examining the annual mean of the daily maximum and minimum temperatures at individual stations, and applied trend analysis only after they were convinced that data homogeneity was not an issue. RJS assumed that there was not a trend in the temperature indices, without any due justification, and indicated step changes in the indices series. He also argued that no significant trends were identified from two subperiods, but this can simply be due to a sample size that is too small since power of detecting a significant trend in a series is drastically reduced with decrease in sample size (Zhang and Zwiers 2004). Furthermore, it is not possible to discriminate whether the step identified by RJS is a real step or a reflection of a long-term trend since the Pettitt test as used by RJS cannot make such a distinction. Here, we expanded Vincent et al.’s (2005) analysis by explicitly considering the influence from the PDO and found that the shift in the PDO does not fully account for the trends in regional temperature indices.
By considering possible step changes in the station data as was done in Vincent et al. (2005), and explicitly modeling the influence of the PDO on the regional temperature indices series, we have shown that the presence of changepoints was considered during the analysis. We therefore conclude that findings of a significant decreasing trend in the percentage of cold nights and a significant increasing trend in the percentage of warm nights in South America for 1960–2000 are not artifacts of inappropriate statistical analysis. In fact, care was taken in Vincent et al. (2005) to ensure that statistical analyses were conducted appropriately.
The authors greatly appreciate the comments and suggestions provided by Ewa Milewska of the Climate Research Division of Environment Canada, and those of an anonymous reviewer, which led to an improved version of this paper.
Corresponding author address: Lucie A. Vincent, 4905 Dufferin St., Toronto ON M3H 5T4, Canada. E-mail: firstname.lastname@example.org
The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JCLI3589.1.