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 = ","))

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)`))

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.

< details class="code-fold"> < summary>Code

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()

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?

< details class="code-fold"> < summary>Code

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()

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

< details class="code-fold"> < summary>Code

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()

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()

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