When is Menzerath-Altmann law mathematically trivial?
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 ~ 1/X. Z ~ 1/X 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 provide rigorous statistical arguments to support the correctness of rejecting Z ~ 1/X when X and Y are significantly correlated. Furthermore, we correct the recent claim that Z ~ 1/X holds if and only if X and Y are independent (Baixeries et al. 2012, Biosystems 107 (3), 167-173). Indeed, Z ~ 1/X if and only if Y is mean independent of X, a statistical property of intermediate strength between independence and uncorrelation. However, it is still true that the random breakage model proposed to explain Z ~ 1/X with independence between X and Y does not fit real genome data. We reject Z ~ 1/X in ten out of eleven taxonomic groups by means of a new correlation ratio test.