Progress on the small cetacean body mass database, with special reference to the genus, Kogia

Over the past month, I’ve been devoting more of my effort towards developing a database for the body masses of small toothed whales that I’ve began about 15 months ago. I’ve briefly mentioned this project in last month’s post dedicated to the body mass of river dolphins. I will now provide a broader overview of the database, along with some focus on the Kogiids. This can be considered a sequel to a twitter post I did about a year ago.

The diminutive sperm whales

Before getting into the dataset, I would like to provide a summary on the species of small sperm whales of the Kogia genus. These whales are recognized as the closest living relatives of the sperm whale (Physeter macrocephalus), though are currently assigned to a separate family, Kogiidae. They share some key similarities to Physeter, such as a singular blowhole, the spermaceti organ, and the reduction / lack of functional teeth in the upper jaw (Plön, 2004; Ross, 1979). Nonetheless, they aren’t close relatives, and the morphological differences suggest a divergence of over 20 millon years (Alfsen et al., 2021). The most obvious difference is that Kogiids are much smaller than Physeter, being only about the size of a typical dolphin at 2-4 m. The existence of two species, the pygmy sperm whale (K. breviceps) and the dwarf sperm whale (K. sima), was only officially recognized less than 60 years ago (Handley & Norris, 1966).

The two species are very similar, with the key differences being that K. sima is much smaller, possesses a larger dorsal fin, and appears to occasionally exhibit vestigial teeth in its upper jaw (Handley, 1966; Ross, 1979). The morphometry and growth for both species has only been extensively examined for the populations along Southern Africa (Plön, 2004; Ross, 1979, 1984). The only known set of growth parameters were yielded from the stranding records from this region coupled with a smaller sample from Australia (Table 1).

Table 1: Size parameters for Kogia species from South Africa and Australia (Plön, 2004)

Parameters	K. breviceps (M/F) (cm)	K. sima (M/F) (cm)
Size at birth	120	103
Size at Sexual maturity	242 / 262	215 / 197
Asymptotic size	286 / 306	263 / 249
Maximum size	327.6 / 330.5a	260.4 / 274.3b

a. Maximum length cited in text, though appendix records an individual of unknown sex measuring 339 cm (SAM 80/03).
b. Maximum length cited in text, but appendix records female measuring 286 cm (PEM N2773).

No precise length measurements exceeding 350 cm were recorded for K. breviceps within South African records (Plön, 2004). This species may grow larger in the Southeastern U.S. given records of females up to 427 cm (MME 8843) and males reaching 411 cm  (Credle, 1988). The maximum length for K. sima in the USNM database (285 cm, MME 8069) more closely agrees with the South African data (Plön, 2004).

Building the database

In the current version of my compendium of body weights, I’ve divided my work into 14 somewhat arbitrary groups.

• Kogia
• River dolphins
• Narwhals
• Belugas
• Pilot whales
• False Killer whales
• Other Blackfish
• Risso’s
• Bottlenose
• Stenella
• Delphinus
• Porpoises
• Current and Former members of Lagenorhynchus ‘’Lags and Ex-Lags’’.
• Other dolphins (Cephalorhynchus, Steno, Lagenodelphis, Lissodephis, etc.)

Now you might notice two major groups not on here. I’m not going to name them, but many should who’s missing. Bear with me and trust me when I say I’ve excluded them for a good reason. It’ll make sense later.

Anyways, one of the major challenges in compiling body weight data for smaller cetaceans is that published data is far more abundant and widespread across sources. Because weighing larger species like humpback, blue and sperm whales is much more challenging, most of the existing data for a single species can be covered across several sources and nearly all the data was collected from whaling stations. Since weighing a dolphin is much easier, nearly every stranding record in the 20^th century becomes a potential source for weight data. I must therefore be more thorough than I am used to. Compilations of records provided by museum databases, like that of the Smithsonian, makes things easier. However, I wish to not solely rely on this as even these aren’t totally comprehensive.

Under the right circumstances, I will also digitize plots for weight data that that’s not directly provided in text (Figure 1). This was how I’ve obtained 118 of the 126 body weights analyzed in a review for Platanista (Braulik et al., 2021). My calibration and manual extraction appear to have been sufficiently precise, as my extracted data points perfectly matched the individuals that were listed in the review and other literature to the nearest cm and kg. I was able to identify about 13 of these replicates. I can only use this method for weight data plotted on the arithmetic scale below, as any imprecision for log-transformed values will introduce more error when back converting.

Figure 1: Data extraction for Platanista

My general process while collecting data is once I feel that I’ve comprehensively covered as much as I could from published literature for a certain group, I proceed to download the data from museum databases. For each group, I keep separate tallies for data I’ve compiled directly from literature versus data from databases. For Kogiids, I’ve compiled 127 records from literature and 118 records from the USNM database. I plot the data distinguishing both species in Figure 2. Some caution should be exercised when describing weight relationships primarily from stranding data, which is prone to emaciated outliers (Ross, 1984). My dataset of 245 kogiids excludes about 10 that I felt were obvious outliers.

Figure 2: Weight data for Kogia

The largest weights for K. sima were two males that were 260.4 cm / 272 kg and 256 cm / 303 kg. The largest female measured 264 cm and weighed 264 kg. Males were also the heaviest for K. breviceps (313 cm/ 700 kg, 315 cm / 680 kg, 351 cm / 680 kg) and the heaviest female was 285 cm / 540 kg. The heaviest females for both species were lactating.

Figure 3: Fields for data entries

Figure 3 shows the fields for each entry in the dataset. I record the species, sex, reproductive status for females (pregnant or lactating), measurements and their logarithmic transformations, sources, and lastly notes for specific details on each individual when warranted. The first few fields allow me to perform comparisons, such as the weight relationships between K. breviceps and K. sima. When comparing two models that did and did not include species ID as a variable, AIC model selection preferred the model that didn’t use species ID (delta for separate species model = 3.65, weight = 0.84).

As I’ve alluded to in my river dolphin post, AIC model criterion penalizes models that use too many variables. The difference (delta) in the AIC scores suggests that including species as a variable makes the model more complicated than it needs to be. This indicates that there’s not a significant difference in the inherent length-weight relationship between the two species. Pooling the data for both species yields the regression below in meters and kilograms. Figure 4 shows that the relationship for Kogiids is very similar for that of P. macrocephalus than it is for other groups (Brodie, 1971; Bryden, 1972; Cockcroft & Ross, 1989; Lockyer, 1976, 1993) .

Mass =18.1 (Length) ^2.777, R² = 0.918

Figure 4: Weight relationships for some odontocetes

Besides the data itself being abundant, the actual trickiest aspect of compiling this database is that a lot of data, especially museums specimens, are prone to being republished in reviews. Since the exact measurements for the same individual aren’t always consistent across sources, it’s very easy to have replicates of the same individuals if I’m not paying attention to dates, locations, and catalog IDs (Figure 5). This was especially the case for the South African samples of the Kogiid dataset (Plön, 2004; Ross, 1979, 1984).

Figure 5: Example of entries found in multiple sources.

I try to list all the notable references in the ‘’Source’’ field for each entry as it helps others using my dataset to be mindful of replicates. This also provides options for those who wish to check the literature directly, as some sources are less accessible than others. Since museum specimens are themselves often cited in literature, I primarily use the ‘’Notes’’ field for catalog IDs. This is especially important for filtering replicates when merging data from online museum records.

Current Progress

While I think my work is almost done for river dolphins and Kogiids, I still have some finalizing to do with some of the other groups I’ve mentioned earlier. I will say that in total, I have slightly over 2,600 individual records with extraction still underway for additional sources. Not sure what it would look like for the future, but I should be doing a new post each time I feel I’ve covered about 95% of what’s out there for a certain group. Once I’m done with every group, I’ll do a public release upon which I’ll continue to add incremental updates. Anyone with any questions, sources they wish to share with me, or requests for the database before it goes public can contact me at the email below.

 cetologyH@gmail.com

References

Alfsen, A., Bosselaers, M., & Lambert, O. (2021). New sperm whale remains from the late Miocene of the North Sea and a revised family attribution for the small crown physeteroid Thalassocetus Abel, 1905. Comptes Rendus Palevol, 39. https://doi.org/10.5852/cr-palevol2021v20a39

Braulik, G. T., I. Archer, F., Khan, U., Imran, M., Sinha, R. K., Jefferson, T. A., Donovan, C., & Graves, J. A. (2021). Taxonomic revision of the South Asian River dolphins (Platanista): Indus and Ganges River dolphins are separate species. Marine Mammal Science, 37(3), 1022–1059. https://doi.org/10.1111/mms.12801

Brodie, P. F. (1971). A Reconsideration of Aspects of Growth, Reproduction, and Behavior of the White Whale (Delphinapterus leucas), with Reference to the Cumberland Sound, Baffin Island, Population. Journal of the Fisheries Research Board of Canada, 28(9), 1309–1318. https://doi.org/10.1139/f71-198

Bryden, M. M. (1972). Growth and development of marine mammals. In R. J. Harrison (Ed.), Functional anatomy of marine mammals (Vol. 1, pp. 1–80). Academic Press.

Cockcroft, V. G., & Ross, G. J. B. (1989). Age, Growth, and Reproduction of Bottlenose Dolphins Tursiops truncatus from the East Coast of Southern Africa. Fishery Bulletin, U.S., 289–302.

Credle, V. (1988). Magnetite and Magnetoreception in Stranded Dwarf and Pygmy Sperm Whales, Kogia simus and Kogia breviceps [MSc thesis]. University of Miami.

Handley, C. O., Jr. (1966). A synopsis of the genus Kogia (pygmy sperm whales). In K. S. Norris (Ed.), Whales, dolphins and porpoises (pp. 62–69). University of California Press.

Lockyer, C. (1976). Body weights of some species of large whales. ICES Journal of Marine Science, 36(3), Article 3. https://doi.org/10.1093/icesjms/36.3.259

Lockyer, C. (1993). Seasonal Changes in Body Fat Condition of Northeast Atlantic Pilo Whales, and their Biological Signficance. Report of the International Whaling Commission (Special Issue), 14, 325–350.

Plön, S. (2004). The status and natural history of pygmy (Kogia breviceps) and dwarf (K. sima) sperm whales off Southern Africa / [Thesis (Ph.D. (Zoology & Entomology))]. Rhodes University.

Ross, G. J. B. (1979). Records of pygmy and dwarf sperm whales, genus Kogia, from southern Africa, with biological notes and some comparisons. Annals of the Cape Provincial Museum (Natural History), 11, 259–327.

Ross, G. J. B. (1984). The smaller cetaceans of the south east coast of southern Africa. Annals of the Cape Provincial Museum (Natural History), 15, 173–410.

Brief note: On the mass of the large fossil river dolphin, Pebanista yacuruna.

One of my ongoing side projects is developing a comprehensive database of weight data for small odontocetes ranging from belugas, dolphins, pygmy & dwarf sperm whales, etc. It is naturally very challenging as such data is more widely collected as opposed to that of large species. Every now and again, I may be encouraged to dedicate a post to a certain portion of this dataset. Now appears to be a good time with the recent discovery and description of a prehistoric species of river dolphin (Benites-Palomino et al., 2024).

Introduction

‘River dolphin’ is a term used to describe several species of small odontocetes that have adapted to freshwater and estuarine environments, and are found in various parts of Asia and South America. Despite their name, most of these species exist outside of the families of oceanic dolphins and porpoises. Four these genera were once united under one family, Platanistidae, however they were eventually placed in their own families (Zhou, 1982).

Modern phylogenetic trees suggest that the La Plata dolphin (Pontoporia blainvillei), boto (Inia spp.), and the Chinese baiji (Lipotes vexifiller) form their own clade, while the South Asian species (Platanista spp.) evolved from a very basal lineage of toothed whales (McGowen et al., 2020). The Tucuxi (Sotalia spp.) actually are ‘’true dolphins’’ of the Delphinidae family.

Below I provide a summary of the average size of physical maturity (L_∞) and largest reliable sizes for some of the major species of each genus (L_max).

Table 1: Size parameters for different species of River dolphins

Species	L∞ (cm) M/F	Lmax (cm) M/F	Source
Inia geoffrensisa	231.5 / 199.8	255 / 225	(Best & da Silva, 1984; Martin & Da Silva, 2006)
Platanista gangetica	205b / 250b	212 / 267	(Anderson, 1878; Braulik et al., 2021)
Pontoporia blainvilleic	130 / 155	152 / 177	(Barreto & Rosas, 2006; Beneditto & Ramos, 2001; Botta et al., 2010; Kasuya & Brownell, 1979)
Sotalia guianensis	185d	222 / 208	(De O. Santos et al., 2003; Meirelles et al., 2010; Ramos et al., 2010; Ramos & Lima, 2000)
Sotalia fluviatilis	—b / —b	149 / 152	(Silva, 1994)
Lipotes vexiffiler	—b / —b	229 / 253	(Brownell & Herald, 1972; Chen & Chen, 1975)

a. Corresponds specifically to those sampled from the Amazon River basin.
b. Data deficient.
c. Best reflects populations along coasts of southern Brazil, Uruguay, and Argentina. Populations from Rio de Janeiro, São Paulo, and Espírito Santo are quite smaller (Barreto & Rosas, 2006; Ramos et al., 2002)
d. There’s generally no consistent hints of sexual dimorphism as noted for other species (Lima et al., 2016; Rosas et al., 2003).

Caution should be placed in extending the values I report here to congeneric species or different morphs, as adult size and sexual dimorphism can vary considerably (Braulik et al., 2021; Silva, 1994; Silva et al., 2023). For Inia geoffrensis, older literature acknowledged a 274 cm male and 228 cm female respectively recorded in Peru (Layne, 1958) and the Orinoco river basin (Trebbau, 1975). However, it is suspected that these were not standardized measurements (Silva, 1994). So, for the time being, I will stick firmly with the largest individuals documented in more modern research.

Despite having been discovered in South America, the newly described fossil species Pebanista yacuruna is a close relative of the extant Ganges (P. gangetica) and Indus River (P. minor) dolphins (Benites-Palomino et al., 2024). On top of that very interesting detail, Pebanista is also quite large compared to any of the modern species, at an estimated range of 2.8-3.5 m based on the bizygomatic width (BZW) of these two specimens (Benites-Palomino et al., 2024). However, the authors consider these estimates to be conservative, as the BZW extrapolation is prone to underestimating the true size in similar taxa.

Mass data

I’ve currently assembled 414 individual weights for the above-mentioned genera that I’ve perused from existing literature and the USNM records. As can be seen in Figure 1, there’s a lot of variation at equal length. AIC model selection suggests that there’s a significant difference in the weight relationships between these species (ΔAICc= 72.09). This basically means that including the species as an additional variable justifies the cost of making the model more complicated, which is usually the case when the difference between two model’s AIC scores (ΔAICc) exceeds 2.

Figure 1: Mass data for River dolphin species.

As of now, I believe my dataset is comprehensive of the existing data for South Asian River dolphins, though it’s still a work in progress for other genera. I still lack a rather large sample (n=378) collected for the boto between 1994-2004 (Martin & Da Silva, 2006). I’m also aware of some larger datasets for Lipotes and Pontoporia, which I provide the existing regressions for in Table 2 (Brownell, 1984).

Update: 4:40 PM EST March 24th, 2024- I’ve since updated my dataset , which I can now proudly say is comprehensive for the existing weight data for the baiji (L. vexillifer) . I will leave Brownell’s regression, but will update the parameters for my baiji weight formula and the pooled sample. The corresponding mass estimates for Pebanista will also be updated.

Table 2: Mass-length formulae parameters for river dolphins

Genus	a*	b	N	r2	Source
Pooled sample	13.30	2.672	430	0.90	Current post
Inia	15.37	2.553	129	0.84	Current post
Platanista	12.45	2.519	141	0.90	Current post
Pontoporia	14.16	2.231	89	0.88	Current post
Pontporia	15.43 (M) 13.42 (F)	1.518 (M) 2.512 (F)	75 (M) 56 (F)	0.78 (M) 0.95 (F)	(Brownell, 1984)
Sotalia	15.47	2.491	42	0.90	Current post
Lipotes	20.23	2.370	28	0.76	Current post
Lipotes	17.47 (M) 3.739 (F)	2.445 (M) 4.218 (F)	12 (M) 8 (F)	0.91 (M) 0.90 (F)	(Brownell, 1984)

*All constants of proportionality converted to units of meters and kilograms.

The maximum weight recorded from a non-pregnant river dolphin was 207 kg for a male boto from the Amazon (Martin & Da Silva, 2006). The corresponding length was not cited, so it was not included in the regression. Two baiji weighing 224 kg and 237 kg appear to be the heaviest pregnant individuals recorded (Chen & Chen, 1975).

Table 3: Mass estimates for Pebanista

Formula type	Holotype (280 cm) Mean mass (kg) (95% Prediction interval)	MUSM 3953 (347 cm) Mean mass (kg) (95% Prediction interval)
Pooled sample	208.9 (173.8-251.1)	371.2 (308.1-447.2)
Inia	212.9 (172.0-263.6)	368.2 (295.2-459.3)
Platanista	166.7 (132.5-209.6)	286.1 (226.0-362.2)
Pontoporia	140.9 (114.4-173.5)	227.4 (182.1-284.0)
Sotalia	201.1 (158.8-254.6)	343.1 (265.9-442.7)
Lipotes	202.7 (162.9-252.0)	337.0 (266.0-426.9)

Table 3 clearly shows that the expected mass for Pebanista varies greatly depending on the sample. In my opinion, the mass of Pebanista is probably best represented by either its closest relative, Platanista, or its modern ecogeographical counterpart, Inia. This would likely mean the typical adult of this species ranged from 150-300 kg.

Concluding thoughts

Aside from its large size and unusual geographic location, Pebanista is a very foundational discovery for our knowledge on the independent marine-freshwater transitions within Cetacea. This was an interesting departure from my typical focus on larger species. I got this post out late due to me trying to compare all the growth studies for the La Plata and Guiana dolphins. I’m looking forward to future opportunities to share progress on what will probably remain an ongoing effort until I’m dead.

References

Anderson, J. (1878). Anatomical and zoological researches: Comprising an account of the zoological results of the two expeditions to western Yunnan in 1868 and 1875; and a monograph of the two cetacean genera, Platanista and Orcella. B. Quaritch. https://doi.org/10.5962/bhl.title.50434

Barreto, A. S., & Rosas, F. C. W. (2006). Comparative Growth Analysis of Two Populations of Pontoporia blainvillei on the Brazilian Coast. Marine Mammal Science, 22(3), 644–653. https://doi.org/10.1111/j.1748-7692.2006.00040.x

Beneditto, A. P. M. di, & Ramos, R. M. A. (2001). Biology and conservation of the franciscana (Pontoporia blainvillei) in the north of Rio de Janeiro State, Brazil. J. Cetacean Res. Manage., 3(2), 185–192. https://doi.org/10.47536/jcrm.v3i2.889

Benites-Palomino, A., Aguirre-Fernández, G., Baby, P., Ochoa, D., Altamirano, A., Flynn, J. J., Sánchez-Villagra, M. R., Tejada, J. V., de Muizon, C., & Salas-Gismondi, R. (2024). The largest freshwater odontocete: A South Asian river dolphin relative from the proto-Amazonia. Science Advances, 10(12), eadk6320. https://doi.org/10.1126/sciadv.adk6320

Best, R. C., & da Silva, V. M. F. (1984). Preliminary Analysis of Reproductive Parameters of the Boutu, Inia geoffrensis, and the Tucuxi, Sotalia fluviatilis, in the Amazon River System. Report of the International Whaling Commission (Special Issue), 6, 361–369.

Botta, S., Secchi, E. R., Muelbert, M. M. C., Danilewicz, D., Negri, M. F., Cappozzo, H. L., & Hohn, A. A. (2010). Age and growth of franciscana dolphins, Pontoporia blainvillei (Cetacea: Pontoporiidae) incidentally caught off southern Brazil and northern Argentina. Journal of the Marine Biological Association of the United Kingdom, 90(8), 1493–1500. https://doi.org/10.1017/S0025315410001141

Braulik, G. T., I. Archer, F., Khan, U., Imran, M., Sinha, R. K., Jefferson, T. A., Donovan, C., & Graves, J. A. (2021). Taxonomic revision of the South Asian River dolphins (Platanista): Indus and Ganges River dolphins are separate species. Marine Mammal Science, 37(3), 1022–1059. https://doi.org/10.1111/mms.12801

Brownell, R., & Herald, E. (1972). Lipotes vexillifer. Mammalian Species, 44. https://doi.org/10.2307/3503836

Brownell, R. Jr. (1984). Review of reproduction in Platanistid dolphins. Report of the International Whaling Commission (Special Issue), 6, 149–158.

Chen, W., & Chen, Y.-Y. (1975). Notes on some morphological and anatomical features of the white-flag dolphin, Lipotes vexillifer, Miller. Acta Hydrobiologica Sinica, 5, 360–370.

De O. Santos, M. C., Rosso, S., & Ramos, R. M. A. (2003). Age estimation of marine tucuxi dolphins ( Sotalia fluviatilis ) in south-eastern Brazil. Journal of the Marine Biological Association of the United Kingdom, 83(1), 233–236. https://doi.org/10.1017/S0025315403007021h

Kasuya, T., & Brownell, R. Jr. (1979). Age determination, reproduction, and growth of the Franciscana Dolphin, Pontoporia Blainvillei. Scientific Reports of The Whales Research Institute Tokyo, Japan, 31, 45–67.

Layne, J. N. (1958). Observations on Freshwater Dolphins in the Upper Amazon. Journal of Mammalogy, 39(1), 1–22. https://doi.org/10.2307/1376605

Lima, J., Carvalho, A., Azevedo, C., Barbosa, L., & Serafim, L. (2016). Variation of age and total length in Sotalia guianensis (Van Bénéden, 1864) (Cetacea, Delphinidae), on the coast of Espírito Santo state, Brazil. Brazilian Journal of Biology, 77. https://doi.org/10.1590/1519-6984.13215

Martin, A. R., & Da Silva, V. M. F. (2006). Sexual dimorphism and body scarring in the Boto (Amazon river dolphin) Inia geoffrensis. Marine Mammal Science, 22(1), 25–33. https://doi.org/10.1111/j.1748-7692.2006.00003.x

McGowen, M. R., Tsagkogeorga, G., Álvarez-Carretero, S., dos Reis, M., Struebig, M., Deaville, R., Jepson, P. D., Jarman, S., Polanowski, A., Morin, P. A., & Rossiter, S. J. (2020). Phylogenomic Resolution of the Cetacean Tree of Life Using Target Sequence Capture. Systematic Biology, 69(3), 479–501. https://doi.org/10.1093/sysbio/syz068

Meirelles, A. C. O., Ribeiro, A. C., Silva, C. P. N., & Soares-Filho, A. A. (2010). Records of Guiana dolphin, Sotalia guianensis, in the State of Ceará, Northeastern Brazil. Latin American Journal of Aquatic Mammals, 97–102. https://doi.org/10.5597/lajam00157

Ramos, R. M. A., Beneditto, A. P. M. D., Siciliano, S., Santos, M. C. O., Zerbini, A. N., Bertozzi, C., Vicente, A. F. C., Zampirolli, E., Alvarenga, F. S., & Lima, N. R. W. (2002). Morphology of the franciscana (Pontoporia blainvillei) off southeastern Brazil: Sexual dimorphism, growth and geographic variation. Latin American Journal of Aquatic Mammals, 129–144. https://doi.org/10.5597/lajam00017

Ramos, R. M. A., Beneditto, A. P. M. D., Siciliano, S., Santos, M. C. O., Zerbini, A. N., Vicente, A. F. C., Zampirolli, E., Alvarenga, F. S., Fragoso, A. B., J. Lailson-Brito, J., Azevedo, A. F., Barbosa, L., & Lima, N. R. W. (2010). Morphology of the Guiana dolphin (Sotalia guianensis) off southeastern Brazil: Growth and geographic variation. Latin American Journal of Aquatic Mammals, 137–149. https://doi.org/10.5597/lajam00163

Ramos, R. M. A., & Lima, N. R. W. (2000). Growth parameters of Pontoporia blainvillei and Sotalia fluviatilis (Cetacea) in northern Rio de Janeiro, Brazil. Aquatic Mammals, 26(1), 65–75.

Rosas, F., Barreto, A., & Monteiro‐Filho, E. (2003). Age and growth of the estuarine dolphin (Sotalia guianensis) (Cetacea, Delphinidae) on the Paraná coast, southern Brazil. Fishery Bulletin 101 (2): 377-383. Fishery Bulletin- National Oceanic and Atmospheric Administration, 101, 377–383.

Silva, V. M. F. da. (1994). Aspects of the biology of the amazonian dolphin genus Inia and Sotalia fluviatilis [Ph.D dissertation, University of Cambridge]. https://repositorio.inpa.gov.br/handle/1/38235

Silva, V. M. F. da, Brum, S. M., Mello, D. M. D. de, Amaral, R. de S., Gravena, W., Campbell, E., Gonçalves, R. da S., & Mintzer, V. (2023). The Amazon River dolphin, Inia geoffrensis: What have we learned in the last two decades of research? Latin American Journal of Aquatic Mammals, 18(1), Article 1. https://doi.org/10.5597/lajam00298

Trebbau, P. (1975). Measurements and some observations of the freshwater dolphin, Inia geoffrensis,  in the Apure River, Venezuela. Zoologische Garten Jena, 45, 153–167.

Zhou, K. (1982). Classification and phylogeny of the superfamily Plantanistoidea, with notes on evidence of the monophyly of the cetacea. Scientific Reports of The Whales Research Institute Tokyo, Japan, 34, 93–108.

Comparing methods for fitting the allometric formula, with special reference to the weight relationship of the sperm whale

Biological allometry is the study of the different growth rates of body parts, or in other words, the change in anatomical proportions as body size increases. Allometry contrasts with isometry, where an organism’s proportions remain the same with increasing size. This concept was largely popularized by the early 20th century (Huxley, 1932), and the following power function is used to describe most allometric relationships.

y= ax^b

The conventional means of obtaining the “a” and “b” parameters is by transforming the data to their logarithmic values and fitting them to the linear model below. The log-linear model is then back-transformed to the above power function.

log ⁡(y) = b ∙ log⁡(x) + log⁡(a)

This formula is often attributed to Huxley himself, however, it was actually discovered earlier by a few other authors (Froese, 2006; Gayon, 2000). To predict the weight of fish, researchers developed the allometric power function as a modification of another formula (Thompson, 1917), which adhered to Galileo’s description of volume/mass varying with the cube of a body’s linear dimensions.

W= k ∙ L³

The simple cube law was insufficient as most fish species slightly deviate from a strictly cubic relationship due to changes in external proportions and body condition as they grow (Froese, 2006). Given that an exponent of 3 describes an isometric change in mass, the exponent ‘’b’’ is interpreted to indicate whether an animal’s body became relatively leaner ( < 3, negative allometry) or stockier (> 3, positive allometry) with increasing length (Froese, 2006). This method was then eventually applied to cetaceans (Schultz, 1938), and became the canonical means for cetologists to describe weight-length relationships. The linear model enabled ease of use for fitting and analyzing parameters.

Flaws in the traditional approach?

Despite being a deeply entrenched practice in many biological fields, the method of back-transforming the log-linear parameters has been the center of some criticism and controversy (Cawley & Janacek, 2010; Packard, 2012, 2023; Xiao et al., 2011). The log-transformation method inherently introduces bias in two manners: the linear regression is made to fit the geometric mean of the data, rather than the arithmetic mean (J. P. Hayes & Scott Shonkwiler, 2006; Smith, 1993) and smaller values are emphasized in the regression (Jansson, 1985; Packard et al., 2009; Packard & Birchard, 2008, 2008; Packard & Boardman, 2009).

Table 1

	Arithmetic values	Logarithmic values
	10	1
	100	2
	1,000	3
	10,000	4
Mean	11,110/4= 2,777.5	1010/4= 102.5 = 316.23

Comparisons of arithmetic (left) and geometric (right) means of the same data

Table 1 illustrates the principle of the downward bias of geometric means and Figure 1 illustrates how the equal spacing of residuals (difference between observations and the predicted values) on the logarithmic scale is not maintained in the back transformation. In a regression, the general goal is to minimize the total sum of residuals between the fitted formula and the data. On the log scale (Figure 1A), fitting to the midpoint of the smaller values has the same impact as large values. On the arithmetic scale (Figure 1B), variance for the residuals increases greatly along the x-axis (known as heteroscedasticity). This means that directly fitting a nonlinear model on the arithmetic scale would place more emphasis on the larger values. By instead placing more weight on smaller values, the log-transformation method has been criticized for ‘’rotating’’ the model parameters from values that would provide a better fit on the arithmetic scale (Packard et al., 2009) .

Figure 1

 Remedies for these problems are mainly split between the inclusion of correction factors (Beauchamp & Olson, 1973; D. B. Hayes et al., 1995; J. P. Hayes & Scott Shonkwiler, 2006; Smith, 1993) or directly fitting the nonlinear model on the arithmetic scale without any data transformation (Packard, 2012, 2023; Packard et al., 2009). Both approaches are largely ignored in recently published allometric formulae for whales (Agbayani et al., 2020; Christiansen et al., 2022; Fortune et al., 2021). Across nearly all the cetacean weight-length models in older literature, none of them were non-linear and only one used a correction factor (Odell et al., 1980).

Comparing fitting methods

To illustrate how the traditional log-linear fitting method compares to nonlinear regression, I’ve ran some analyses of all the piecemeal weight data for sperm whales that I’ve managed to compile from literature (n=58). All data were adjusted upwards by assuming 12% of the intact mass was lost to body fluids (Sleet et al., 1980). The traditional approach was performed by converting all the length and weight data to logarithmic values (base 10) and fitting the least-squares regression for the linear relationship. The linear model was then back transformed to the allometric power function. The nonlinear regression was performed by directly fitting the power function to the non-transformed data. The Levenberg–Marquardt algorithm through the glsnls function in R was used to find the least-squares solution for the nonlinear model. Table 2 provides a summary of the parameters.

Table 2

Type	a	b	units
Nonlinear model	0.0961	2.209	meters/ tonnes
Back-transformed linear model	0.0350	2.582	meters/ tonnes

The first thing that sticks out is how surprisingly low the ‘’b’’ parameter is in the nonlinear regression. Deviating this far from an approximately cubic relationship is conceptually unrealistic given the relatively subtle change in external proportions in sperm whales across postnatal growth (Nishiwaki et al., 1963). In Figure 2, it’s apparent that the nonlinear model’s shallow slope produces biased predictions for both the smallest and largest observations in the sample. Table 3 indicates that the downward bias in the linear regression is very minimal, predicting a mean weight that’s < 1% below the true value for the sample.

Figure 2

In short, it appears that giving larger values extra weight in the nonlinear regression undermines the underlying geometric relationship between linear dimensions and weight. While logarithmic transformation giving more weight to smaller values has been considered a disadvantageous ‘’distortion’’ (Packard et al., 2009), it appears to act more as a beneficial ‘’anchor’’ for estimating the true relationship of the allometric increase in body weight.

Table 3

Sample mean -Actual value (tonnes)	33.74
Sample mean -Nonlinear model estimate (tonnes)	33.83
Sample mean -Back-transformed linear model estimate (tonnes)	33.56

Connecting weight to growth

Figure 3

Tying this back to my last post, I’ve used the back transformed linear model to present growth curves for weight (Figure 3). This relationship can be calculated using a modified version of the von Bertalanffy formula, as used in previous works on whales (Lockyer, 1981), using the original parameters for total length that I calculated for males and those derived from stranding data for females (Evans & Hindell, 2004) . The 46% difference in asymptotic length between males in females results in a 166% difference in weight. As expected with negative allometry, this is below the expected 211% difference in weight of an isometric weight-length relationship.

Male Phase 1 (Ages 0-16): L(t)= 21.36 (1-e^{-13.85(t+3.25)})^2.582

Male Phase 2 (Ages 17 and older): L(t)= 43.62 (1-e^{-0.0741(t-0.0162)})^2.582

Female: L(t)= 16.38 (1-e^{-0.016(t-2.58)})^2.582

The linear model predicts that an exceptionally large female sperm whale measuring 12.5 m would weigh 23.8 tonnes (95% PI: 16.2-34.9) while an 18.3 m male would weight 63.6 tonnes (95% PI: 43.2-93.7). The record size of 20.7 m in modern literature corresponds to an estimated weight of 87.4 tonnes (95% PI: 59.0-129.4). That’s about twice the weight of the average male and over 3.5 times the weight of the largest female.

Parting news

As the current year ends, I wish to share some details regarding the weight analysis section of my upcoming manuscript. Despite presenting a large dataset of piecemeal weights in this post, absolutely none of it will be used for any new regressions in my review for the sperm whale. I will be using a different kind of data for that portion of my review. I truly am looking forward to sharing once the review process gets through as there is a lot that these past two posts don’t even scratch the surface of.

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