The signed Kolmogorov-Smirnov test: why it should not be used

Abstract

The two-sample Kolmogorov-Smirnov (KS) test is often used to decide whether two random samples have the same statistical distribution. A popular modification of the KS test is to use a signed version of the KS statistic to infer whether the values of one sample are statistically larger than the values of the other. The underlying hypotheses of the KS test are intrinsically incompatible with this approach and the test can produce false positives supported by extremely low p-values. This potentially makes the signed KS test a tool of p-hacking, which should be discouraged by replacing it with standard tests such as the t-test and by providing confidence intervals instead of p-values.