Pagel's transformations

• Pagel developed three transformations that modify the variance-covariance matrix $\mathbf{C}$ to test for deviations from Brownian motion.
• Each transformation has a parameter that, when equal to 1, recovers the original BM tree.
• The three transformations:
  • $\lambda$ (lambda) — phylogenetic signal
  • $\delta$ (delta) — tempo of evolution
  • $\kappa$ (kappa) — speciation vs. gradual change

Pagel's $\lambda$ (lambda)

• Multiplies all off-diagonal (shared branch) elements of $\mathbf{C}$ by a value from 0 to 1.
• This compresses the internal (deeper) branches; tips remain unaffected.
• Interpretation:
  • $\lambda = 1$: no change — trait evolves as expected under BM on the true tree.
  • $\lambda = 0$: star phylogeny — every tip is effectively independent (no phylogenetic signal).
• Often used to measure phylogenetic signal: how much does the phylogeny predict trait similarity?

Pagel's $\delta$ (delta)

• All elements of $\mathbf{C}$ are raised to the power $\delta$. This changes node heights.
• Interpretation:
  • $\delta = 1$: unchanged BM tree.
  • $\delta < 1$: node heights compressed — deep branches reduced less than shallow ones.
  • $\delta > 1$: shallower branches stretched more than deeper ones.
• Designed to capture variation in rates of evolution through time:
  • $\delta < 1$ suggests trait evolution was faster early in the phylogeny.
  • $\delta > 1$ suggests trait evolution accelerated more recently.

Pagel's $\kappa$ (kappa)

• Raises all branch lengths to the power $\kappa$.
• Effect depends on $\kappa$ and the number of branches between root and MRCA of each pair of tips.
• Interpretation:
  • $\kappa = 1$: standard BM tree.
  • $\kappa = 0$: all branch lengths become 1 — only topology matters.
• Interpreted as a model where characters mostly change during speciation events (punctuated evolution) rather than gradually along branches.

Testing for different rates across clades

• BM assumes $\sigma^2$ is constant across the whole tree. What if this is wrong?
• These methods ask whether specific clades have different $\sigma^2$.
• Three approaches:
  1. PIC-based rate test: compare magnitudes of squared contrasts between clades.
  2. ML + AIC: fit multi-rate BM models with separate $\sigma^2$ for each clade; compare via AIC.
  3. Bayesian MCMC: each branch can have its own rate, with posteriors identifying rate shifts.
• Note: methods typically require pre-defined groups suspected of having different rates.

Ornstein-Uhlenbeck (OU) model

• A Brownian-like model where the trait is drawn towards a fitness optimum $\theta$.
• Analogous to stabilizing selection — but more precisely, models tracking the movement of an adaptive optimum.
• Adds a pull parameter $\alpha$ that determines how strongly the trait is attracted to $\theta$: $$dX = \alpha(\theta - X)\,dt + \sigma\,dW$$
• Key properties:
  • $\alpha = 0$: pure Brownian motion.
  • Large $\alpha$: trait tightly constrained near $\theta$.
  • Variance reaches a stationary value $\sigma^2/(2\alpha)$ rather than growing indefinitely.
• Can be used within ML or Bayesian frameworks to test whether stabilizing selection fits better than BM.

Early burst models

• Models of adaptive radiation: some phenomenon creates an opportunity (migration, evolution of a new trait, extinction of competitors).
• Taxa rapidly evolve and diversify to fill available niches, then slow down as niches fill up.
• These models start from BM and attach a decay parameter that slows the evolutionary rate over time: $$\sigma^2(t) = \sigma^2_0 \, e^{rt}$$ where $r < 0$ produces the "early burst" pattern.
• Evolution starts rapidly near the root and gradually slows to the rate at the tips.
• Can be compared to BM using AIC or Bayes Factors.

Peak shift models

• Models of punctuated evolution: long periods of relative stasis punctuated by brief periods of intense change.
• These periods of change are often associated with adaptive radiations or shifts in selective regime.
• The punctuated periods can follow one of three regimes:
  • Random
  • Fixed interval
  • Associated with changes in other traits on the tree
• Can identify parts of the tree evolving differently from the rest.
• Approach is similar to the multi-rate methods described earlier.

The Mk model

• Discrete traits have fixed states (e.g., 0/1, red/yellow/blue, legs/no legs) — these do not work with continuous-trait methods like BM.
• The Mk model (Lewis, 2001) provides a general approach to modelling discrete character evolution on phylogenies.
• Properties:
  • Unordered: traits can change to any other state without intermediate steps.
  • Transitions are independent across branches.
  • All transition rates are equal (in the simplest version).
• Extended versions relax the equal-rates assumption.
• Analogous to the Jukes-Cantor model for DNA, but for arbitrary discrete characters.

Simulating Mk on a tree

• The simulation process:
  1. Assign a state to the root (equal probability of all states, or from observed proportions).
  2. Assign that value to all daughter branches.
  3. For each branch, "roll the dice" — determine if a transition occurs (probability depends on branch length and rate).
  4. Repeat down the tree to the tips.
• These simulated trees allow us to calculate likelihoods for comparative data.
• But uncertainty makes exact computation difficult. Felsenstein's pruning algorithm works backwards from tips to root, making likelihood calculation tractable.

State-dependent speciation and extinction (SSE) models

• SSE models test correlations between trait evolution and speciation/extinction rates.
• An "alphabet soup" of methods:
  • BiSSE: Binary State Speciation and Extinction
  • HiSSE: Hidden State SSE — allows for hidden (unmeasured) states
  • MuSSE: Multiple State SSE
• When you are interested in whether a particular trait affects the rate of diversification and/or extinction, SSE models provide a formal framework.
• Caution: these models can be sensitive to model violations and have known issues with false positives.

Summary of models beyond BM

Discussion

• If you do not know a priori which clade has a different evolutionary rate, what are the obstacles and risks of testing every clade?
• Do you think it is a good idea to link punctuated evolution of one trait to changes in other traits?
• For what reasons might Pagel's statistics be less popular today than in the past?
• Under the early burst model, why would we expect evolutionary rates to slow down over time?

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