When rainfall or the water balance is converted into a drought index such as SPI or SPEI, the first decision that determines the accuracy of the results is the choice of the probability distribution used to transform the values into the standard normal distribution. If the distribution is unsuitable, drought events will appear more severe or milder than they really are. For this reason, the distribution is not chosen by intuition but is decided by statistical tests known as Goodness of Fit Tests. The idea behind these tests rests on a null hypothesis stating that the observed data are indeed drawn from the assumed distribution. A statistic is calculated that measures the distance between the data and the theoretical distribution, and it is compared with a critical value at a specified significance level, which in the study under discussion is 0.05. If the statistic exceeds the critical value, the hypothesis is rejected, and we conclude that the distribution does not represent the data. If it remains below it, there is not enough evidence to reject it. The study adopted two common tests. The first is the Kolmogorov–Smirnov (KS) test, which calculates the maximum absolute difference between the empirical cumulative distribution function of the data and the assumed theoretical distribution function; the smaller this difference, the better the fit. The second is the Anderson–Darling (AD) test, which is more sensitive because it gives greater weight to the tails of the distribution. This feature is particularly important in drought studies, because severe drought lies in the lower tail of the distribution, which is the part where a distribution that is acceptable in the middle may fail. The two tests were applied in three stages in the thesis. In the first stage, the distribution of each station was selected for rainfall accumulated over twelve months, with the result that Lognormal was selected for Karbala and Samawah, Logistic for Hilla, Gamma for Najaf, and Weibull for Diwaniyah. In the second stage, the distribution of the water balance (precipitation minus evapotranspiration) was selected, with Genlog chosen for four stations, namely Hilla, Karbala, Diwaniyah, and Samawah, and GEV for Najaf. In the third stage, the selection was repeated for the climate series projected under the moderate and high scenarios, and the most suitable distributions changed from one scenario to another. This confirms that the choice of distribution must not be carried over from one period to another without testing. The results of the June tests, presented as an example, show how these tests are read. For accumulated rainfall, the maximum difference in the KS test ranged between 0.076 and 0.123, with p-values of approximately 0.71 to 0.99. The AD statistic ranged between 0.17 and 0.50, with p-values between 0.19 and 0.93. For the water balance, the results were stronger: the difference ranged between 0.070 and 0.121, with p-values between 0.72 and 0.997, and the AD statistic between 0.16 and 0.41, with p-values between 0.84 and 0.998. The null hypothesis was not rejected at any station, meaning that the selected distributions are statistically acceptable. One noteworthy detail is that the AD test gave the lowest p-values for rainfall at Hilla, about 0.21, and at Samawah, about 0.19, which are much lower than the KS values for the same two stations (0.94 and 0.85). The reason is that AD detects deviation in the tails that KS overlooks. Therefore, using both tests together gives a more accurate picture than relying on either one alone. In the same chapter, the KS test was also used to evaluate the LARS-WG weather generator, as the distributions of generated wet and dry spells and daily rainfall were compared with their observed counterparts, and the results were classified from “excellent” to “poor”. Performance was good to excellent in winter, autumn, and spring, and poor in summer because rainfall is almost nonexistent then. The same test thus serves two functions: selecting the most suitable distribution for the index, and judging the efficiency of the model that generates the data. Despite the value of these tests, they must be interpreted with caution. Failing to reject the hypothesis means that the distribution is acceptable, not necessarily that it is the correct one, as the test may accept more than one distribution for the same data. Moreover, the statistical power of the tests decreases when the number of values is small, and a 31-year series for each month is not large. It is therefore preferable to support the test with a visual inspection of the distributions and to pair it with other criteria for comparing distributions. Thus, goodness of fit tests are the link that ensures drought indices are built on a sound statistical foundation. They do not change the general concept of the index, but they protect its results from bias and make comparisons between stations, periods, and scenarios more reliable. Presented by the lecturer (Assist. Lect. Zahraa Hussein Mohammed), Department of Civil Engineering. Source: Zahraa Hussein Mohammed Aziz (2024). Projections of Climate Changes in Hydrological Regime of the middle Mesopotamian plain