The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to multivariate data. This is not straightforward because the maximum difference between two joint cumulative distribution functions is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of formula_54 or formula_55 or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.