Figure 1

Symbolic illustration of the advantage with the new herein presented 2D approach to combining test statistics. The values of the two test statistics, A and B, are plotted on the positive x- and y-axes, respectively. The points correspond to bootstrap samples of pairs of these values, and the color of the cloud represent the probability density at the point: red means high density, i.e. a high probability to find a point there, and blue low. The 1D projections of the cloud are plotted on the negative x- and y-axes. The difference between a 1D analysis of these tests, considered independently, and the herein considered 2D approach, is found by comparing the two points p ₁ and p ₂. These two points correspond to two different hypothetical pairs of (A,B)-values, as calculated from the original data. If such a data point lies sufficiently outside the empirical distribution, the null hypothesis used to generate the empirical distribution is rejected. As can be seen, both p ₁ and p ₂ lies within the 1D distributions, and have essentially the same p-values, if the tests are two-sided. This stands in stark contrast to the situation in 2D: there p ₂ lies within the cloud, but p ₁ lies clearly outside. For this reason, the observation p ₁ would only be rejected in a 2D analysis, and not in a 1D analysis. Note that the main reason for this 2D advantage to be exploited is both that the 2D cloud does not lie parallel to either of the axes, and that the considered point just like p ₁ lies in a place that exploits the thinly populated areas that only are revealed in 2D.