Quantification of relationship between annual daily maximum temperature and annual daily maximum rainfall in South Australia

YE Qianping and AHAMMED Faisal

School of Natural and Built Environments,University of South Australia,Adelaide,Australia

ABSTRACT The empirical relationship between annual daily maximum temperature(ADMT)and annual daily maximum rainfall (ADMR) was investigated. The data were collected from four weather stations located in Adelaide, South Australia, from 1988 to 2017. Due to the influence of sea surface temperature on rainfall and temperature, the distance from the weather station to the sea was considered in the selection of weather stations.Two weather stations near the sea and two inland weather stations were selected. Three non-parametric statistical tests (Kruskal-Wallis, Mann-Whitney, and correlation) were applied to perform statistical analysis on the ADMT and ADMR data. It was revealed that the temperature and rainfall in South Australia varies according to weather station location. The distance from the sea to the weather station was found to have limited influence on temperature and rainfall. Meanwhile, with the 0.05 level of significance, the association between ADMT and ADMR near sea stations is not as significant as the association between the two inland weather stations.It is relatively unrealistic to use ADMR to predict ADMT,or vice versa, since their correlation is not statistically significant (Spearman’s rank correlation coefficient:−0.106).

KEYWORDS Annual daily maximum rainfall;annual daily maximum temperature;kruskal-Wallis;Mann-Whitney;correlation

1. Introduction

Studies predict that global warming in the 21st century will be far beyond the natural variability over the past 100 years(Crowley 2000;Pfleiderer et al.2019).Also,the influence of the greenhouse effect on temperature is beyond the natural variability in the weather system.Held and Soden (2006) and Shen et al. (2018) additionally found that climate change has a major impact on the global hydrological cycle. Despite international actions to reduce global greenhouse gas emissions, it is predicted that the global climate will undergo significant changes (Ibne Amir et al. 2012; Njijsse et al. 2019).Existing literature suggests that the increase in temperature will cause highly intensive rainfall in many parts of the world (Westra,Alexander, and Francis 2013;Herath,Sarukkalige, and Nguyen 2018). Pfleiderer et al. (2019)reported that summer temperatures across the globe are more persistently 2°C above pre-industrial levels, based on daily observed temperature data from 1950 to 2014.In Australia, climate change is expected to have an important influence on communities.For instance,Innes et al. (2015) found that high temperatures had a major harmful impact on wheat yields from 1922 to 2008 in New South Wales,based on historical data,and that the most detrimental combination was low rainfall with high temperatures.Nicholls and Collins (2006)observed that temperatures in Australia increased over the 20th century, as did the sea surface temperatures in nearby oceans. It was also found that the rainfall trend has decreased in southwestern Australia, while it has increased in northeastern Australia.Murphy and Timbal(2008) analyzed the climate change in southeastern Australia from 1997 to 2006 and found a downward trend of rainfall along with rising daily maximum temperature. On the other hand, some studies have suggested that analysis of climate variables should be based on a catchment scale (Sharma and Shakya 2006). For instance, Ibne Amir et al. (2012) studied climatic variables for the Fitzroy basin, which is the second largest coastal river system in Australia.Their study analyzed the rainfall,maximum and minimum temperatures and evaporation trend from 1954 to 2010,revealing an insignificant decreasing trend of rainfall in summer and increasing trends of maximum and minimum temperature and evaporation. Herath, Sarukkalige, and Nguyen(2018)studied the relationship between daily maximum temperature and different rainfall durations across seven major cities in Australia for a 10-year period.Their results showed that the rainfall-temperature scaling relationship is restricted to a certain range of temperature.Besides,other factors influence the relationship,such as the percentile and duration of the rainfall, the study period and the season. According to Guerreiro et al.(2018), changes of hourly rainfall extremes in Australia are due to changes in atmospheric circulation patterns,atmospheric stability, latent heat, moisture coverage,and upward motions. Meanwhile, Cong and Brady(2012)found that temperature and rainfall correlate significantly over sea and land. For instance, Aldrian and Susanto(2003)analyzed three major Indonesian rainfall regions and their relationship to sea surface temperature,revealing that the rainfall variability correlated with sea surface temperature in the nearby Indian and Pacific Ocean waters. Hence, extreme rainfall and temperature events usually do not occur at the same time due to their correlation relationship(Adler et al.2008).

Worku et al.(2018) investigated the extremes of daily rainfall and temperature for Jemma Sub-Basin,Upper Mile Basin,Ethiopia,for the period 1981-2014.They estimated annual trends of annual daily maximum rainfall and temperature using non-parametric tests such as Mann-Kendall, seasonal Mann-Kendall and Sen’s slope. Their results showed an increasing trend of annual and summer rainfall at more than 78%of the stations and decreasing trends of spring extreme rainfall at most stations of Jemma Sub-Basin, Upper Blue Nile Basin. Almeida et al.(2017) studied the spatiotemporal rainfall and temperature trends(minimum,maximum,and average)throughout the Brazilian Legal Amazon using Mann-Kendall and Sen’s slope tests.Their results showed increasing trends of annual maximum temperature of 0.04°C yr−1,and annual maximum rainfall showed an insignificant trend. Rastogi et al.(2017)investigated the effects of climate change on probable maximum precipitation (PMP) over the Alabama-Coosa-Tallapoosa River Basin, and the results showed that the PMP driven by projected future climate forcing was higher than the 1981-2010 baseline values by 20%in the near future(2021-50)and 44%in the far future(2071-2100).Kamruzzaman et al.(2019)studied the maximum rainfall at seven sites in South Australia over a period of 59 years and observed that rainfall anomalies were not associated with past values of temperature or the Southern Oscillation Index.

For analyzing climate data,statistical hypothesis tests are generally used. Thet-test, ANOVA, Mann-Whitney,and Kruskal-Wallis tests work for the same purpose,which is to identify the relationship between the sample and the population, and to see whether the sample is from the population (Guo, Zhong, and Zhang 2013).Both thet-test and ANOVA are parametric tests and assume the data to have a normal distribution.The nonparametric versions of these tests are the Mann-Whitney and Kruskal-Wallis tests. These versions do not assume the data to be normally distributed.Thet-test specifically deals with comparing two samples, while ANOVA can deal with multiple samples. Likewise, Kruskal-Wallis is a simplification of the Mann-Whitney test from two samples to multiple samples (Guo, Zhong, and Zhang 2013).Zhao and Huang(2007)studied different hypothesis tests in terms of identifying the trends of climate variables and found that parametric tests are restricted by the assumption of a normal distribution, while nonparametric tests are easier to use for analyzing climate variables. The Mann-Whitney and Kruskal-Wallis tests are well-known non-parametric statistical tests.The former,introduced by Mann and Whitney(1947),compares whether the distributions of two samples of continuous observations are equivalent, as in their medians are equal. It relies on a comparison between the sums of the ranks related to these two samples. Besides, when analyzing the trends of temperature and rainfall over time, the Mann-Whitney test can be used for step changes (Abolverdi et al. 2014). The Kruskal-Wallis test,meanwhile, introduced by Kruskal and Wallis (1952), is a nonparametric test, and a substitute for one-way ANOVA.It is designed to identify any statistically significant difference between the distributions of three or more sample groups. The test is similar to the Mann-Whitney test,apart from the latter can only compare two groups of data(Al-Ahmadi and Al-Ahmadi 2013).Zhang et al. (2006) used a simple linear regression method to identify trends in hydrological datasets. Generally, it is believed that rainfall,temperature, and evaporation are the main factors of influence on regional hydrological cycles. However, Murphy and Timbal (2008) found evaporation makes a minor contribution to a changing climate system, since in reality it is controlled by the existing water amount.

This paper is a case study of the association between extreme rainfall and temperature events in South Australia based on different hypothesis tests. To the best of our knowledge, it is the first work of its kind in a South Australian context. The Kruskal-Wallis and Mann-Whitney tests are used to perform the hypothesis testing. Though these tests are appropriate for nonparametric data, applications of these tests are very rare in rainfall and temperature data. Statistical misuse in research could be minimised by applying these tests to understand the variability of extreme rainfall and temperature. Also, policymakers could use these tests for climate projection and drought forecasting.

2. Study area

Four weather stations, shown in Table 1 and Figure S1,were chosen for this study. Two of them (Adelaide Airport and Cape Willoughby) are located within 5 km from the sea, while the other two (Mt Barker and Rosedale) are located more than 40 km from the sea.The annual daily maximum temperature (ADMT) and annual daily maximum rainfall(ADMR)data for the period 1988-2017 were collected from the Australian Bureau of Meteorology.

Table 1.Details of the selected weather stations.

3. Data and methods

3.1 Selection of statistical tests

The Statistical Package for the Social Sciences (SPSS),version 25.0(IBM Corporation 2017)was used to perform the data analysis.Table 2 shows the hypothesis tests and their data requirements.The Chi-Square,Cramer’s V,and Phi tests require the independent variable and dependent variable to be categorical data.Other tests require the dependent variable to be scale data.

The collected ADMT and ADMR datasets are scale data.Apart from correlations and regressions,other tests require the independent variable to be categorical.Regarding the weather station locations, data were categorized as 1.00 = Adelaide Airport, 2.00 = Cape Willoughby,3.00 =Mt Barker, and 4.00 =Rosedale. Meanwhile, based on the shortest distance between the weather stations and the sea, Adelaide Airport and Cape Willoughby were recoded as 1,while Mt Barker and Rosedale were recoded as 2.

In contrast,thet-test and ANOVA require the scale data to be normally distributed. The Mann-Whitney and Kruskal-Wallis tests are used when the scale data are nonnormal or uncertain. Table 3 shows the requirements of the dataset for the various hypothesis tests. Frequency analysis using SPSS was performed on the collected data to understand the distribution pattern.

It was observed from the frequency analysis that the ADMT data were normally distributed, while the ADMR data were non-normally distributed. Hence, thet-test and ANOVA were rejected,since both these tests require the data to be normally distributed. Therefore, we decided to apply the Kruskal-Wallis and Mann-Whitney tests; and the null and alternative hypotheses of these tests were determined using SPSS. Meanwhile,the ADMT and ADMR data were used to perform correlation and regression analysis.There are two main types of correlation test; namely, Pearson’s correlation and Spearman’s correlation. Pearson’s assumption requires a linear relationship between two variables. However,Spearman’s assumption requires a monotonic relationship between two variables.Therefore,it was important to identify the relationship between the ADMT and ADMR data before performing the correlation test.

Figure 1 shows that the ADMT and ADMR data do not have a linear relationship (R2= 0.011), but that there is a relationship between them to a certain degree.A monotonic relationship between the ADMT and ADMR datasets was assumed, and Spearman’s correlation was performed.

Table 2.Hypothesis tests for different data types.

Table 3.Dataset requirements for hypothesis tests.

3.2 Null and alternative hypotheses

The null and alternative hypotheses for the Kruskal-Wallis and Mann-Whitney tests were prepared by the tests themselves in SPSS. The decision against the null hypothesis was observed from the test results. The significance level was 0.05 (i.e., the null hypothesis was rejected if thep-value was less than 0.05).

For the Kruskal-Wallis test,the null hypothesis,H0,was that the distribution of ADMT/ADMR is the same across categories of weather stations.The alternative hypothesis,HA, was that the distribution of ADMT/ADMR is not the same across categories of weather stations.

For the Mann-Whitney test, the null hypothesis, H0,was that the distribution of ADMT/ADMR is the same across the categories of sea locations. The alternative hypothesis, HA, was that the distribution of ADMT/ADMR is not the same across the categories of sea locations.

Figure 1.Scatter plot with no linear relationship between ADMT and ADMR for the duration of 1988-2017.

Table 4.Hypothesis testing using Kruskal-Wallis.

Figure 2.Kruskal-Wallis test results for ADMT.

4. Results and discussion

4.1 Kruskal-Wallis test

The Kruskal-Wallis test results(Table 4,Figures 2-5)are statistically significant,as the significance levels for both cases are less than 0.05, (p=0.001 for ADMT;p=0.001 for ADMR).The outputs also show that the null hypotheses were rejected for both cases. Therefore, there are significant differences in the distribution patterns for ADMT and ADMR across weather stations in South Australia.These differences are also observed in the box plots shown in Figures 2 and 3 and pairwise comparisons shown in Figures 4 and 5. The highest median value of ADMT was observed for weather station 4.00(Rosedale),and almost similar median values of ADMR were observed for the other three weather stations.

Figure 3.Kruskal-Wallis test results for ADMR.

Figure 4.Pair-wise comparisons of ADMT for station category.

4.2 Mann-Whitney test results

Table 5 and Figures 6 and 7 show the SPSS outputs for the Mann-Whitney tests.It can be observed from Table 5 that the significance level is less than 0.05(p=0.001 for ADMT;p= 0.006 for ADMR), and therefore the test results are statistically significant. The decisions of the hypothesis testing were that the null hypotheses were rejected for both cases. Therefore, the distribution patterns for ADMT and ADMR are different across the categories,whether the weather stations are located near or far from the sea.These differences can also be observed from Figures 6 and 7.

Figure 5.Pair-wise comparisons of ADMR for station category.

Table 5.Hypothesis testing using Mann-Whitney test.

4.3 Correlation and regression

Using Spearman’s rank correlation coefficient,the correlation between ADMT and ADMR is −0.106, which is close to 0. Thus, there is a weak correlation between ADMT and ADMR. TheP-value is 0.251 > 0.05, which means it is statistically insignificant(Table 6).

5. Conclusions

Quantification of the association between ADMT and ADMR was performed using the Kruskal-Wallis, Mann-Whitney and correlation tests. It is revealed from the analysis that temperature and rainfall in Adelaide vary according to weather station locations. The distance from the sea to the weather station has limited influence on the temperature-rainfall link. The distributions of ADMT and ADMR across various categories of stations were observed as different using the Kruskal-Wallis and Mann-Whitney tests. Weak correlations between ADMT and ADMR for the selected weather stations were also observed.Therefore,it is not appropriate to use ADMR to predict ADMT,or vice versa,in South Australia.

Figure 6.Mann-Whitney test results of ADMT for distance from sea category.

Figure 7.Mann-Whitney test results of ADMR for distance from sea category.

Table 6.Correlation between ADMT and ADMR.

Application of the Kruskal-Wallis or Mann-Whitney test in rainfall and temperature data is very rare,though these are appropriate tests for nonparametric data. Application of these tests to understand the variability of ADMT and ADMR could help to minimize statistical misuse in research. The use of the Kruskal-Wallis and Mann-Whitney tests in this study could be applied to extreme temperature and rainfall data for any other location. Also, policymakers could use the approach to understand the availability of water and forecast drought in Australia and other countries.

Disclosure statement

No potential conflict of interest was reported by the authors.