T. rex size distributions

I ought to be spending this weekend working on a risk intelligence thing for work, but I’ve stumbled across something that brings my inner child immense joy and such things cannot be ignored. It is this 2019 paper “An Older and Exceptionally Large Adult Specimen of Tyrannosaurus rex” by Persons et al. First it’s cool to see paleontology in action, instead of filtered through talking head documentaries. Second, the diligent authors have personally measured 12 T. rex specimens, which is a decent chunk of the 29 or-so specimens discovered to-date, and so we have a cool dataset to play with.

library(tidyverse)
library(knitr)

t.rex <- read_tsv("t.rex-persons.tsv", col_types = "ccnnnnnnnnnnnnnnn") |>
  rename(Specimen = Specimens) |>
  mutate(Morph = factor(Morph, levels = c("Robust", "Gracile")))

kable(t.rex, format.args = list(decimal.mark = ".", big.mark = ","))

Table 1: T. rex bone measurements in millimetres (Table 1 from Persons et al 2019).

Specimen	Morph	Dentary tooth row length	Scapula blade width	Manual p. I-1 length	Ilium length	Femur length	Femur circumference	Proximal femur width	Tibia length	Tibia shaft width	Fibula length	Fibula shaft width	Astragalus height	Astragalus width	Pedal p. IV-1 length	Body mass (kg)
RSM P2523.8	Robust	595	74.8	98.5	1,545	1,333	590	426	1,140	184	995	62	328	310	184	8,870
CM 9380 (AMNH 973)	Robust	508	73.0	NA	1,540	1,269	534	399	1,166	150	1,025	NA	NA	NA	145	6,740
FMNH PR2081	Robust	585	68.0	78.0	1,525	1,321	580	380	1,140	160	1,030	59	NA	NA	154	8,462
MOR 1125	Robust	492	65.7	NA	NA	1,150	515	370	1,060	150	915	53	280	295	170	6,100
RTMP 81.12.1, NMC 9950	Robust	NA	NA	NA	NA	NA	495	NA	1,095	155	985	65	NA	NA	140	5,469
BHI 3033	Gracile	575	NA	NA	1,540	1,350	505	350	1,065	158	945	55	325	280	NA	5,779
MOR 555	Gracile	635	65.0	89.0	1,470	1,280	520	370	1,150	170	1,035	56	NA	235	176	6,264
MOR 980	Gracile	546	NA	NA	1,397	1,232	483	NA	NA	NA	NA	NA	NA	NA	NA	5,112
RTMP 81.6.1	Gracile	530	NA	NA	NA	1,210	460	270	1,030	160	NA	52	NA	NA	107	4,469
BM R8040 (AMNH 5881)	NA	NA	NA	NA	NA	NA	480	330	NA	NA	NA	NA	NA	NA	124	5,025
MOR 009	NA	NA	NA	90.0	1,180	1,100	469	NA	1,105	140	930	NA	NA	NA	NA	4,714
USNM 6183	NA	NA	NA	NA	NA	1,040	426	NA	910	NA	NA	NA	NA	NA	NA	3,617

We can also add “Goliath”, the exciting 2024 specimen. Bear in mind that Persons didn’t personally measure this fossil, so that’s a source of potential error. I took the measurements from the few social media posts about the fossil. The lack of information on this specimen seems strange to me, I even wonder if it’s a hoax. It wouldn’t be the first time.

campione.body.mass.kg <- function(min.femoral.circ.mm) 10**((2.754 * log10(min.femoral.circ.mm) - 0.683) - 3)

goliath <- read_tsv("t.rex-goliath.tsv", col_types = "ccnnnnnnnnnnnnnnn") |>
  mutate(
    Morph = factor(Morph, levels = c("Robust", "Gracile")),
    `Body mass (kg)` = campione.body.mass.kg(`Femur circumference`)
  )

t.rex <- bind_rows(t.rex, goliath)
kable(select(goliath, Specimen, `Femur length`, `Femur circumference`, `Body mass (kg)`))

Table 2: Reported measurements for the Goliath specimen.

Specimen	Femur length	Femur circumference	Body mass (kg)
Goliath	1371	648	11483.59

The most interesting bone is the femur, being as it’s more likely to be present and is used to estimate body mass. The equation used is from Campione et al 2014. Bear in mind they suggest a 25% prediction error.

An interesting thing about T. rex is that there are two morphs, gracile and robust, which I understand to be layman’s terms for “elven” and “chonky”. Let’s see if that’s visible in the femur measurements.

ggplot(t.rex) +
  aes(x = `Femur length`, y = `Femur circumference`, colour = Morph, label = Specimen) +
  geom_label(nudge_y = 10, na.rm = TRUE) +
  geom_point(na.rm = TRUE) +
  scale_x_continuous(limits = c(1000, 1400)) +
  scale_y_continuous(limits = c(400, 700)) +
  theme_minimal()

Figure 1: Femur circumference and length for T. rex specimens.

Looking at this scatterplot, there’s a linear relationship between length and circumference, and Goliath sits neatly on it. (Too neatly? Hmm. Should it even be linear? Wouldn’t the square-cube law suggest femurs should get thicker faster than they get longer?) However it’s not obvious if Goliath should be gracile or robust. In fact, it’s not particularly clear from the femurs alone why some are gracile and some robust; CM 9380 (Holotype) and MOR 555 (Wankel) are very close, for example. We need more features. Unfortunately, the T. rex data are very sparse, so we need to impute missing values.

Tibia length is relatively complete, can we use that?

ggplot(t.rex) +
  aes(x = `Femur length`, y = `Tibia length`, colour = Morph) +
  geom_point() +
  scale_x_continuous(limits = c(1000, 1400)) +
  scale_y_continuous(limits = c(900, 1200)) +
  theme_minimal()

Figure 2: Femur length and tibia length for T. rex specimens.

The relationship between fibia length and tibia length is surprisingly not very informative.

ggplot(t.rex) +
  aes(x = `Femur circumference`, y = `Tibia length`, colour = Morph) +
  geom_point() +
  scale_x_continuous(limits = c(400, 700)) +
  scale_y_continuous(limits = c(900, 1200)) +
  theme_minimal()

Figure 3: Femur circumference and tibia length for T. rex specimens.

Femur circumference relates a little better. Let’s try a dead simple linear model to predict the missing femur lengths. We can include body mass, which is a function of femur circumference, to try and capture some of the knowledge from that model.

reg <- lm(`Femur length` ~ `Femur circumference` * `Body mass (kg)` + `Tibia length`, data = t.rex, subset = !is.na(t.rex$`Tibia length`) & !is.na(`Femur length`))
summary(reg)

Call:
lm(formula = `Femur length` ~ `Femur circumference` * `Body mass (kg)` + 
    `Tibia length`, data = t.rex, subset = !is.na(t.rex$`Tibia length`) & 
    !is.na(`Femur length`))

Residuals:
       1        2        3        4        6        7        9       11 
  -2.758   10.381    3.116 -116.353   97.051   30.241   49.109  -69.132 
      12 
  -1.655 

Coefficients:
                                         Estimate Std. Error t value Pr(>|t|)
(Intercept)                            -1.136e+04  2.430e+04  -0.467    0.665
`Femur circumference`                   5.020e+01  1.037e+02   0.484    0.654
`Body mass (kg)`                       -3.832e+00  8.868e+00  -0.432    0.688
`Tibia length`                         -2.585e-01  8.777e-01  -0.294    0.783
`Femur circumference`:`Body mass (kg)`  3.317e-03  8.054e-03   0.412    0.702

Residual standard error: 88.3 on 4 degrees of freedom
Multiple R-squared:  0.6787,    Adjusted R-squared:  0.3574 
F-statistic: 2.112 on 4 and 4 DF,  p-value: 0.2433

It’s not great, though slightly better than the other combinations I tried and probably the best we could do with such a small dataset. The additional imputed femur length (RTMP 81.12.1) is feasible enough.

preds <- predict(reg, newdata = t.rex)

t.rex |>
  mutate(`Femur length` = coalesce(`Femur length`, preds)) |>
  select(`Specimen`, `Morph`, `Femur length`, `Femur circumference`, `Tibia length`, `Body mass (kg)`) |>
  kable()

Table 3: Imputed femur length for RTMP 81.12.1.

Specimen	Morph	Femur length	Femur circumference	Tibia length	Body mass (kg)
RSM P2523.8	Robust	1333.000	590	1140	8870.00
CM 9380 (AMNH 973)	Robust	1269.000	534	1166	6740.00
FMNH PR2081	Robust	1321.000	580	1140	8462.00
MOR 1125	Robust	1150.000	515	1060	6100.00
RTMP 81.12.1, NMC 9950	Robust	1230.292	495	1095	5469.00
BHI 3033	Gracile	1350.000	505	1065	5779.00
MOR 555	Gracile	1280.000	520	1150	6264.00
MOR 980	Gracile	1232.000	483	NA	5112.00
RTMP 81.6.1	Gracile	1210.000	460	1030	4469.00
BM R8040 (AMNH 5881)	NA	NA	480	NA	5025.00
MOR 009	NA	1100.000	469	1105	4714.00
USNM 6183	NA	1040.000	426	910	3617.00
Goliath	NA	1371.000	648	NA	11483.59

I’m not sure what else to do with this. Must be tough being a paleontologist, working with so little direct data!