  Dependency distance minimization (DDm) is a well-established principle of
word order. It has been predicted theoretically that DDm implies compression,
namely the minimization of word lengths. This is a second order prediction
because it links a principle with another principle, rather than a principle
and a manifestation as in a first order prediction. Here we test that second
order prediction with a parallel collection of treebanks controlling for
annotation style with Universal Dependencies and Surface-Syntactic Universal
Dependencies. To test it, we use a recently introduced score that has many
mathematical and statistical advantages with respect to the widely used sum of
dependency distances. We find that the prediction is confirmed by the new score
when word lengths are measured in phonemes, independently of the annotation
style, but not when word lengths are measured in syllables. In contrast, one of
the most widely used scores, i.e. the sum of dependency distances, fails to
confirm that prediction, showing the weakness of raw dependency distances for
research on word order. Finally, our findings expand the theory of natural
communication by linking two distinct levels of organization, namely syntax
(word order) and word internal structure.
