Bounds and approximations of p -value distribution. (A) Strict bounds and approximations (geometric mean of upper and lower bound, and gamma) for n = 10000 molecules and k = 4 experiments, on the left-hand side over the whole range of rank products, on the right-hand side for small rank products only (gamma approximation is outside the figure). It can be seen that, on the log scale, the bounds are very tight. Zooming in on small rank products, the bounds are on average about a factor 2.5 off (i.e., higher/lower than the exact p-value), yet the geometric mean approximation is still very close to the exact p-value. (B) Same as (A), but for n = 10000 and k = 20. The curve on the left looks more or less the same but, as is best seen on the right, the bounds are much further off (almost a factor 1000). (C) Same as (A), but for n = 10 and k = 4. The curve on the left may look worse, but that is mainly because of the scaling of the y-axis. Relatively speaking, the bounds are still on average about a factor 2.5 off. (D) Same as (A), but for n = 10 and k = 20. With very small n and relatively large k, we get the worst of both worlds.