455 
 
International Conference on Sustainable Built Environment (ICSBE-2010) 
Kandy, 13-14 December 2010 
ON THE VARIABILITY OF TREND TEST RESULTS 
 
1A. Ramachandra Rao and M. Azli 
 
1Department of Civil Engineering 
Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia. 
1Tel: (603) 7967-5266, (6012) 314-5757 ; Fax: (603) 7967-5318 
1E-mail: rao@um.edu.my, rao@ecn.purdue.edu 
 
1Corresponding author: A. Ramachandra Rao 
 
Abstract 
 
Trend tests are used to investigate statistical significance of trends. The popular Mann–Kendall (MK) trend test 
was originally proposed for random data. It was later modified to handle correlated data. After the scaling 
hypothesis was introduced, the MK test was further modified to accommodate it. The results from these three 
versions of the MK test can be very different. The objective of the present paper is to illustrate these variations 
in the MK trend test results. Not considering these variations would lead to spurious conclusions about statistical 
significance of trends in data with associated erroneous deductions. Monthly temperature data from Malaysia 
are used for illustration. 
 
 
Keywords: Mann–Kendall trend test; Correlations; Scaling hypothesis; Monthly temperatures; Malaysia 
 
 
1.0 Introduction 
 
Trend tests have been used to investigate the impacts of climate change and variability in hydrologic 
time series in different parts of the world. Trends in various series have been investigated: in Japanese 
precipitation series (Xu et al., 2003); in Yangtze basin in China (Zhang et al., 2006); in precipitation 
in Seoul, Korea (Wang et al., 2006). Earlier studies include those by World Meteorological 
Organization (1988), Mitosek (1992), Chiew and McMahon (1993) and Burn (1994). In many of these 
studies tests based on assumption of randomness in data are used. With the exception of papers by 
Hamed (2008) and Kumar et al. (2009) the effect of scaling on trend detection is not considered. 
A widely used non-parametric test for detecting trends in time series is the Mann–Kendall 
(MK) test (Mann, 1945; Kendall, 1975).The null hypothesis in the MK test is that data are random 
and independent, i.e. there is no trend or serial correlation among observations. However, observed 
hydrologic and climatic time series, especially monthly data, are generally autocorrelated. The 
autocorrelations in observed data will lead to misinterpretation of results of trend tests. This situation 
was recognized early by Cox and Stuart (1955) who stated that “positive serial correlation among the 
observations would increase the chance of significant answer even in the absence of a trend”. 
Problems in interpreting confusing trend test results explain in part the variety and even contradictory 
results reported from them. 
Modifying the tests for trends to account for the effect of serial correlation in data and using 
the modified tests has been the approach used by several investigators. Lettenmaier (1976) and Hirsch 
and Slack (1984) were early investigators who considered the effect of serial correlation on the results 
from trend tests. Hamed and Rao (1998) introduced a modified MK trend test for autocorrelated data 
with arbitrary correlation structure. 
The effect of scaling on trend detection was investigated by Hamed (2008). By using 
simulated fractional Gaussian series, Hamed (2008) demonstrated that the null hypothesis of no trend 
was rejected by the MK trend test by as small a percentage as ten percent for random data to as high 
as sixty percent for data with the Hurst parameter H of 0.9. The number of rejections increases with 
increasing H and decreases with lower significance levels. Because of the symmetry of the test 
statistic, which is not affected by scaling, both the false positive and negative trends occur in equal 
proportions. These results point out the importance of testing for scaling effects in trend tests. The 
objective of the research reported herein is to present the variation in results from trend tests 
depending on the assumptions on which the tests are based. 
456 
 
International Conference on Sustainable Built Environment (ICSBE-2010) 
Kandy, 13-14 December 2010 
Monthly Malaysian temperature data from two stations are used in the study. Temperature 
data from the past three decades have been selected for study because global warming and its effects 
became prominent during this period (Fig. 1) Climate change and its effects started attracting attention 
and investigation during this period. Monthly temperature data from Alor Setar in Kedah and Senai in 
Johor are used in the study (Fig. 2). Alor Setar is located in the north of Peninsular Malaysia while 
Senai is located in the south. The duration of data is from 1979 to 2007. 
 
Fig. 1 
Fig. 2 
 
Three tests, the MK test (Mann, 1945; Kendall, 1975), the modified MK test (Hamed and 
Rao, 1998), and the MK test under the scaling hypothesis (Hamed, 2008), are used in the study. 
Because the details of the test are available in these references, they are briefly discussed next. 
 
2.0 Tests used in the study 
 
2.1  MK test 
 
Consider a time series . The test statistic S is computed by Eq. (1). 
 
 (1) 
 (2) 
 
where Ri and Rj in Eq. (1) are the ranks of observations xi and xj respectively of the time series. 
Assuming that the data are independent and identically distributed, Kendall (1975) showed that 
 
 (3) 
 
Kendall (1975) also showed that the significance of trends can be tested by comparing the 
standardised variable u1 in Eq. (4) with the standard normal variate at a significance level α. 
 
 (4) 
 
The basic assumption in this test is that the data are random. If the data are correlated then the 
correlation may be removed by pre-whitening the data. Alternatively, the variance V0(S) may be 
modified to account for the correlation. Such a modification to the MK test proposed by Hamed and 
Rao (1998) is discussed below. 
 
2.2 Modified MK test 
 
V0(S) in Eq. (4) is recalculated in this test as V*(S) by using Eq. (5). 
 
 (5) 
 
In Eq. (5), (n/ns*) represents a correction to V0(S) because of the autocorrelations in the data. The 
approximation used for (n/ns*) is the empirical expression in Eq. (6). 
 
457 
 
International Conference on Sustainable Built Environment (ICSBE-2010) 
Kandy, 13-14 December 2010 
 (6) 
 
In Eq. (6), ρs(i) are the autocorrelation coefficients of the ranks of the data. 
 
As the ranks of the observations of observations are used in Eq. (6), V*(S) is computed without using 
either the data or their autocorrelation function. In the present study, significant correlation 
coefficients up to N/10 of N ranks are used. The modified statistic u2 is computed and tested for 
significance. 
 
  (7) 
 
2.3 MK test under the scaling hypothesis 
 
In this test, the data are detrended by using Sen’s (1968) non-parametric trend estimator. The scaling 
coefficient H is obtained by maximising log likelihood function in McLeod and Hipel (1978). This 
estimate of H is approximately normally distributed for the uncorrelated case when true H is 0.5 with 
the mean and variance given by Eqs. (8). 
 
 (8) 
 
The significance of H is tested by using 
 
and  in Eqs. (8). If H is significant, the trend test under 
the scaling hypothesis is conducted. The modified variance of the test statistic is computed by using 
Eq. (9). 
 (9) 
where: 
 
 (10) 
The variance V(S) in Eq. (9) is corrected for bias by multiplying it with the factor B in Eq. (11). 
 
 (11) 
 
The coefficients a0, a1,... a4 in Eq. (11) are functions of the sample size n and are found in Hamed 
(2008). The modified test statistic u3 is computed by using the modified variance and Eq. (4). If u3 is 
significant, then the trend is significant; otherwise, it is not. The test under the scaling hypothesis is 
conducted only if the decisions from MK or modified MK tests are significant. 
 
 
3.0 Data analysis and results 
 
3.1  Results of the MK test 
 
The values of the statistic S and the variance V0(S) for the data from Alor Setar are 10,457 and 
4,702,775, respectively. The statistic u1 is 4.822, and is significant at 10%, 5% and 2.5% levels. 
Therefore the conclusion is that the Alor Setar temperatures have a strong positive trend. The values 
of S and V0(S) for data from Senai in Johor are, respectively, 11,650 and 4,702,775. The value of u1 
for Senai is 5.372 which is larger than u1 for Alor Setar. Therefore the conclusion from this test may 
be that the positive trend in Senai data is stronger than that in Alor Setar. Depending only on these 
458 
 
International Conference on Sustainable Built Environment (ICSBE-2010) 
Kandy, 13-14 December 2010 
results one may conclude that there is a north-south gradient in the Malaysian temperature trend. But 
we will have to consider the strong correlation in monthly temperature data and perform the modified 
MK test. 
 
3.2 Results of the modified MK test 
 
For the data from Alor Setar, the values of the modified variance V*(S), the variance inflation factor 
V*(S)/V0(S), and the statistic u2 are 10,452,906, 2.223, and 3.234, respectively. u2 is smaller than u1 
which is 4.822 due to the effect of correlation in the data. u2 is also significant at 10%, 5%, and 2.5% 
levels, and is positive which indicates an increasing trend in temperature. The values of V*(S), 
V*(S)/V0(S), and u2 for Senai are 28,410,496, 6.041, and 2.186, respectively. u2 for Senai has decreased 
from 5.372 to 2.186, a reduction of 59.3%. u2 is positive for both data sets indicating increasing trends 
in temperature. For both sets of data, u2 is significant at 10%, 5%, and 2.5% significance levels. u2 for 
Senai is smaller than that for Alor Setar which is opposite to the behaviour of u1. As u2 is statistically 
significant, the MK test under the scaling hypothesis is conducted to test the significance of the test 
statistic. 
 
3.3 Results from the MK test under the scaling hypothesis 
 
Before performing the MK test under the scaling hypothesis, Hurst’s parameter H, and mean and 
standard deviation of H are estimated. The statistical significance of H is tested and if H is found 
significant, the MK test is performed under the scaling hypothesis. Otherwise, inferences from the 
previous tests are accepted. Accordingly, the H value for Alor Setar, its mean and standard deviation 
are estimated to be 0.92, 0.486, and 0.035, respectively. The H value for Senai is 0.90, and its mean 
and standard deviation are the same as for Alor Setar data. The H estimates for both data sets are 
statistically significant. They are also close to unity which indicates that MK test should be run under 
the scaling hypothesis. 
 
The bias-corrected variance V(S), the variance inflation factor V(S)/V0(S), the bias correction 
factor B, and the statistic u3 for the test under the scaling hypothesis are computed. For the Alor Setar 
data, the values of V(S), V(S)/V0(S), B, and u3 are 36,290,000, 7.717, 3.196, and 0.971, respectively. u3 
is statistically insignificant, and has decreased to 0.971 from u2 of 3.234. Because of the high H value, 
the variance inflation factor V(S)/V0(S) is quite large and so is the bias correction factor B. 
Consequently the trend is statistically insignificant. The values of V(S), V(S)/V0(S), B, and u3 for the 
Senai data are 54,190,000, 11.523, 2.255, and 1.054 respectively. In this case, u3 is also insignificant 
indicating the statistical insignificance of the trend in the temperature data. 
 
 
4.0 Summary and conclusions 
 
As the example discussed above clearly illustrates, the MK test statistic is strongly affected by 
correlation in the data and by the scaling factor H. Conclusions drawn without considering these 
factors can be misleading or even wrong. Although the trend statistic u3 is insignificant, plots of the 
temperature data in Malaysia during these years show an overall, general, gradual warming trend. But 
this trend is statistically insignificant in the data from all the stations. The situation is “mixed” in the 
sense that there are increasing but statistically insignificant trends in Malaysian monthly temperature 
data. 
 
 
Acknowledgements 
 
We would like to extend our gratitude Dr. Khaled H. Hamed of Cairo University for clarifying several 
points in his paper, to the Vice-Chancellor Professor Datuk Dr. Ghauth Jasmon and the Dean of the 
Faculty of Engineering Professor Dr. Mohd Hamdi Abdul Shukor of the University of Malaya for 
their support. The Department of Irrigation and Drainage, Ministry of Natural Resources and 
459 
 
International Conference on Sustainable Built Environment (ICSBE-2010) 
Kandy, 13-14 December 2010 
Environment, Malaysia, provided the data used in this study. Their help is acknowledged. We would 
also like to thank the University of Malaya for partially supporting this work.  
 
 
References 
 
Burn, D.H. (1994). Hydrologic effects of climatic change in West Central Canada. J. Hydrol., 160, pp. 53–70. 
Chiew, F.H.S., McMahon, T.A. (1993). Detection of trend or change in annual flow of Australian rivers. Int. J. 
Climatol., 13, pp. 643–653. 
Cox, D.R., Stuart, A. (1955). Some quick sign tests for trend in location and dispersion. Biometrika, 42, pp. 80–
95. 
Hamed, K.H. (2008). Trend detection in hydrologic data: the Mann-Kendall trend test under the scaling 
hypothesis. J. Hydrol., 349, pp 350–363. 
Hamed, K.H., Rao, A.R. (1998). A modified Mann-Kendall trend test for autocorrelated data. J. Hydrol., 204, 
pp. 182–196. 
Hansen, J., Ruedy, R., Sato, M., Lo, K. (2010). Global surface temperature change. Draft only (dated 1 June 
2010). NASA Goddard Institute for Space Studies, New York, New York, USA. Available at: 
http://data.giss.nasa.gov/gistemp/ [Accessed 1 July 2010]. 
Hirsch, R.M., Slack, J.R. (1984). Non-parametric trend test for seasonal data with serial dependence. Water 
Resour. Res., 20(6), pp. 727–732. 
Kendall, M.G. (1975). Rank correlation methods. Griffin, London. 
Kumar, S., Merwade, V., Kam, J., Thurner, K. (2009). Streamflow trends in Indiana: effects of long term 
persistence, precipitation and subsurface drains. J. Hydrol., 374, pp. 171–183. 
Lettenmaier, D.P. (1976). Detecting trends in water quality data from records with dependent observations. 
Water Resour. Res., 12(5), pp. 1037–1046. 
Mann, H.B. (1945). Nonparametric tests against trend. Econometrica, 13, pp. 245–259. 
McLeod, A.I., Hipel, K.W. (1978). Preservation of the rescaled range: 1. a reassessment of the Hurst 
phenomenon. Water Resour. Res., 14(3), pp. 491–508. 
Mitosek, H.T. (1992). Occurrence of climate variability and change within the hydrologic time series: a 
statistical approach. CP-92-05, International Institute for Applied Systems Analysis, Laxenburg, 
Austria. 
Sen, P.K. (1968). Estimates of the regression coefficient based on Kendall’s tau, J. Amer. Statistical Assoc., 63, 
pp. 1379–1389. 
Wang, B., Ding, Q., Jhun, J.G. (2006). Trends in Seoul (1778–2004) summer precipitation. Geophys. Res. Lett., 
33, L15803 
World Meteorological Organization. (1988). Analyzing long time series of hydrological data with respect to 
climate variability. WCAP-3 (WMO/TD-No.224). 
Xu, Z.X., Takeuchi, K., Ishidaira, H. (2003). Monotonic trend and step changes in Japanese precipitation. J. 
Hydrol., 279, pp. 144–150. 
Zhang, Q., Liu, C., Xu, C.Y., Xu, Y., Jiang, T. (2006). Observed trends of annual maximum water level and 
stream flow during past 130 years in the Yangtze River basin, China. J. Hydrol., 324, pp. 255–265. 
 
460 
 
International Conference on Sustainable Built Environment (ICSBE-2010) 
Kandy, 13-14 December 2010 
Figures 
 
 
Fig. 1. Global land-ocean temperature index for 1977–2009 (Hansen et al., 2010) 
 
 
 
 
 
Fig. 2. Locations of two meteorological stations in Peninsular Malaysia: Alor Setar in Kedah 
(48603) and Senai in Johor (48679)