When is Menzerath-Altmann law mathematically trivial? A new approach.

Menzerath's law, the tendency of Z (the mean size of the parts) to decrease as X (the number of parts) increases, is found in language, music and genomes. Recently, it has been argued that the presence of the law in genomes is an inevitable consequence of the fact that Z=Y/X, which would imply that Z scales with X as Z ∼ 1/X. That scaling is a very particular case of Menzerath-Altmann law that has been rejected by means of a correlation test between X and Y in genomes, being X the number of chromosomes of a species, Y its genome size in bases and Z the mean chromosome size. Here we review the statistical foundations of that test and consider three non-parametric tests based upon different correlation metrics and one parametric test to evaluate if Z ∼ 1/X in genomes. The most powerful test is a new non-parametric one based upon the correlation ratio, which is able to reject Z ∼ 1/X in nine out of 11 taxonomic groups and detect a borderline group. Rather than a fact, Z ∼ 1/X is a baseline that real genomes do not meet. The view of Menzerath-Altmann law as inevitable is seriously flawed.