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

TL;DR

This paper critically examines the mathematical triviality of Menzerath-Altmann law in genomes, introducing new statistical tests that challenge the assumption that the law's presence is inevitable, and demonstrating its non-trivial nature across various taxonomic groups.

Contribution

It introduces a new non-parametric correlation ratio test to evaluate the relationship in genomes, showing that Menzerath-Altmann law is not a trivial consequence and varies across groups.

Findings

Most tests reject the Z ~ 1/X hypothesis in genomes.

The correlation ratio test is the most powerful among those used.

Menzerath-Altmann law is not an inevitable consequence in genomes.

Abstract

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 based upon the correlation ratio, which is able to reject Z ~ 1/X in nine out of eleven 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.