Table 4 Models testing whether changes in cetacean cranial asymmetry are associated with other discrete traits

Ancestral state reconstruction (‘ancestral’)

Species belong to one of three ancestral categories: ‘archaeocete’, ‘odontocete’, and ‘mysticete’

The placing of species into ‘archaeocete’, ‘odontocete’, and ‘mysticete’ was based on the literature and published fossil descriptions [26, 37]

‘Regime’ model (‘regime’)

Assumes selective evolutionary regimes. Archaeocetes are assigned to ‘regime1’, mysticetes to ‘regime2’, and most odontocetes to ‘regime3’. The highly asymmetric monodontids, platanistids, and superfamily physeteroids are classified as a separate ‘regime4’

Regimes are based on a preliminary trait plot (Fig. 3) which shows that the monodontids, platanistids, and superfamily physeteroids have a much higher trait value (sum radii for the individual specimen (Σρ_spec)) (≥ 0.42, Fig. 3) than other odontocetes and therefore may be evolving asymmetry under one different selective regime

‘Regime-split’ model (‘regime-split’)

As in the regime model, archaeocetes are assigned to ‘regime1’, mysticetes to ‘regime2’, odontocetes in general to ‘regime3’, and the highly asymmetric odontocetes (monodontids, platanistids, and physeteroids) are placed in their own separate selective regimes

Each highly asymmetric group is evolving under its own separate selective regime: (1) monodontids, (2) platanistids, and (3) physeteroids

Echolocation model (‘echo’)

Species assigned to one of four groups depending on whether the species could echolocate

Band0: Cannot echolocate

Band1: Not capable of echolocation, although reception of ultrasonic signals cannot be ruled out

Band2: Early echolocation, e.g. Cotylocara macei [4] and Echovenator [5, 38]

Band3: Fully echolocating

i. Although rudimentary, echolocation evolved very early in whale evolution, likely soon after odontocetes diverged from the ancestors of baleen whales [4]

ii. The ability to produce ultrasonic sounds, and therefore echolocate, has been inferred for almost all fossil odontocetes [9] although Odobenocetops likely had greatly reduced echolocation abilities [26]

iii. Mysticetes do not echolocate

iv. All extant odontocetes echolocate [39]

Echolocation-frequency model (‘echo-freq’)

Categorising by echolocation in the extant odontocetes and sound production in the extant mysticetes

i. Data on frequency specifics is not available for fossils

ii. Narrowband high-frequency (NBHF) cetaceans designated according to Kastelein et al. [40] and Khyn et al. [41, 42]

iii. The non-NBHF delphinids were assigned to broadband low frequency (BBLF) according to Jensen et al. [43] and Turl et al. [44]

iv. The sperm whale sits in its own category. The hypertrophied nasal structures and deep-diving behaviour produce a low-frequency multi-pulsed sound [45]

v. Ziphiids sit in their own category. They produce frequency-modulated buzz clicks (FM-buzz) [46,47,48,49,50]

vi. Mysticetes do not echolocate and produce low-frequency sound [24, 51]

vii. The Monodontidae sit in their own category. They produce narrowband structured (NBS) pulses [52,53,54]

See Additional file 1: Table S8 for further details