Harmon PCM Chapter 6 — Beyond Brownian Motion (summary)

Faithful summary generated in a clean context, license-aware (CC-BY 4.0 → short verbatim load-bearing quotes marked). Source-only; project framing lives elsewhere.

1. Citation

Harmon, Luke J. 2019. “Chapter 6: Beyond Brownian Motion.” In Phylogenetic Comparative Methods: Learning from Trees. Book text v1.0.0. Licensed CC-BY 4.0. https://lukejharmon.github.io/pcm/chapter6_beyondbm/. Accessed 2026-07-03.

2. Access note

Numbers re-verified against the raw source HTML on 2026-07-03 (an earlier summarizing-fetch pass had introduced one error, corrected below). The mammal AICc table, OU/EB estimates, and section structure are now source-confirmed. Two findings from re-verification: (i) Section 6.5 is genuinely absent — the source TOC runs 6.4 → 6.6, a real gap in Harmon’s numbering, not a conversion artifact; (ii) the earlier draft’s “carnivores vs non-carnivores, Mann-Whitney U=251, P=0.70” was wrong — the source reports a Welch t-test P=0.42 and only recommends Mann-Whitney conceptually (§5, corrected). Appendix OU-derivation equations use quantitative-genetics notation ($G$, $\gamma$); the stationary-form moments in §5 are the standard OU forms.

3. Thesis (1 sentence)

Brownian motion is often unrealistic, so the chapter surveys four families of continuous-trait models that go beyond BM — VCV transformation, rate variation across clades, stabilizing-selection (OU) constraint, and adaptive-radiation/ecological-opportunity (early-burst) models — and how to fit and compare them.

4. Problem & context

BM “corresponds to a few simple scenarios of trait evolution” (verbatim fragment, 6.1) and “hardly seems realistic for a group of finches known to be under strong and predictable directional selection” (verbatim, 6.1). Verbatim roadmap (6.1): “In this chapter, I will consider four ways that comparative methods can move beyond simple Brownian motion models: by transforming the variance-covariance matrix describing trait covariation among species, by incorporating variation in rates of evolution, by accounting for evolutionary constraints, and by modeling adaptive radiation and ecological opportunity.”

Section structure (as extracted):

• 6.1 Introduction
• 6.2 Transforming the evolutionary variance-covariance matrix
• 6.3 Variation in rates of trait evolution across clades — 6.3a contrasts; 6.3b ML and AIC; 6.3c Bayesian MCMC
• 6.4 Non-Brownian evolution under stabilizing selection
• 6.6 Early burst models
• 6.7 Peak shift models
• 6.8 Summary; 6.9 Footnotes; 6.10 Appendix (deriving OU under stabilizing selection)

5. Method / approach

Pagel’s tree transformations (6.2). Reshape the phylogenetic variance-covariance matrix C; BM is the untransformed special case.

• $\lambda$ (Pagel’s lambda): multiplies off-diagonal (covariance) elements of C by $\lambda \in [0,1]$; tip (diagonal) variances unchanged. $\lambda=1$ → standard BM; $\lambda=0$ → star phylogeny (no phylogenetic structure). Used as a measure of “phylogenetic signal.”
• $\delta$ (delta): raises node heights / elements of C to power $\delta>0$; models time-varying rate. Verbatim (6.2): “When δ is one, the tree is unchanged and one still has a constant-rate Brownian motion process; when δ is less than 1, node heights are reduced, but deeper branches in the tree are reduced less than shallower branches (Figure 6.1). This effectively represents a model where the rate of evolution slows through time.” $\delta>1$ → rates accelerate through time. Verbatim connection (6.2): “There is a close connection between the δ model, the ACDC model (Blomberg et al. 2003), and Harmon et al.’s (2010) early burst model [see also Uyeda and Harmon (2014), especially the appendix].”
• $\kappa$ (kappa): raises individual branch lengths to power $\kappa \ge 0$; intended to capture “speciational” change. $\kappa=1$ → BM; $\kappa=0$ → all branches equal length. Assumes the tree contains all/most speciation events — flagged as unrealistic given extinction/incomplete sampling; newer speciational methods (Bokma 2008; Goldberg & Igić 2012) preferred.

Rate variation across clades (6.3). Test whether $\sigma^2$ differs between clades/regimes.

• Contrasts (6.3a): compare (squared/absolute) phylogenetic independent contrasts between clades. Worked example — carnivores vs. non-carnivores (mammal body size, Garland 1992): a Welch two-sample t-test is not significant, $P=0.42$, but the absolute-value contrasts are strongly skewed → violate t-test assumptions. The chapter therefore recommends the nonparametric Mann-Whitney U-test on the absolute contrasts (rank-based, insensitive to the skew) but does not report a U/P for it in this worked example.
• ML + AIC (6.3b): build a separate C per regime and combine, $\mathbf{V}_{H_2} = \sigma_1^2\mathbf{C}_1 + \sigma_2^2\mathbf{C}_2$; single-rate BM is the nested special case, compared by likelihood-ratio test or AICc. “Censored” vs. “noncensored” handling of the shift branch (O’Meara et al. 2006). Worked example (carnivores own rate vs. rest): $\hat\sigma_c^2=0.068$, $\hat\sigma_o^2=0.01$, $\hat{\bar z}(0)=4.51$ — carnivores appear faster, but the fit is not substantially better than single-rate BM (lnL −77.6, AICc 162.3 vs. BM’s 160.4).
• Bayesian MCMC (6.3c): sample posteriors, test rate differences via composite parameters (e.g. $\sigma^2_{diff}=\sigma_1^2-\sigma_2^2$); stochastic character mapping (Revell 2013); reversible-jump MCMC to locate shifts without a priori hypotheses (Eastman et al. 2011).

Ornstein–Uhlenbeck (6.4). Trait pulled toward an optimum (stabilizing-selection interpretation). Parameters: $\bar{z}_0$ (start), $\sigma^2$ (drift), $\alpha$ (strength of pull/constraint), $\theta$ (optimum). Standard OU stationary moments (the §6.10 appendix derives these in quantitative-genetics notation using $G$ and $\gamma$; source-confirmed form): $\mu_i(t) = \theta + e^{-\alpha T_i}(\bar{z}_0 - \theta)$ $V_i(t) = \frac{\sigma^2}{2\alpha}\left(1 - e^{-2\alpha T_i}\right)$ $V_{ij}(t) = \frac{\sigma^2}{2\alpha}\,e^{-2\alpha T_i}\left(e^{2\alpha s_{ij}} - 1\right)$ with $T_i$ = root-to-tip path length and $s_{ij}$ = shared path length. BM is the $\alpha \to 0$ limit. (The HTML render of the appendix garbles a couple of sign/factor placements; the forms above are the standard OU stationary moments the derivation targets.)

Early burst / ACDC (6.6). Rate decays exponentially through time (Harmon et al. 2010): $\sigma^2(t) = \sigma_0^2 e^{bt}$, with $b<0$ for a decelerating adaptive radiation ($b=0$ recovers BM). Moments (source-confirmed): $\mu_i(t)=\bar{z}_0$; $V_i(t)=\sigma_0^2\frac{e^{bT_i}-1}{b}$; $V_{ij}(t)=\sigma_0^2\frac{e^{bs_{ij}}-1}{b}$.

Peak shift (6.7). Reversible-jump MCMC over OU regimes so lineages shift among adaptive peaks; $\sigma^2$ and/or $\alpha$ can vary across regimes (Uyeda & Harmon 2014).

Fitting/comparison. ML fits with model selection by AICc (and likelihood-ratio tests for nested cases). Single-trait mammal body-size comparison (source-verified §6.6):

Reported interpretation: all $\Delta$AICc within ~3 units of BM; BM remains the (marginally) preferred / not-outperformed model for this dataset.

6. Key claims / findings

• BM is a nested special case of every model here: $\lambda=1$, $\delta=1$, $\kappa=1$, OU as $\alpha\to0$, EB/ACDC as $b\to0$.
• $\lambda$ measures phylogenetic signal by scaling only the between-species covariances; $\lambda=0$ is a star tree (species independent), $\lambda=1$ is BM.
• $\delta<1$ = decelerating rate through time (deeper branches shrink less than shallow); $\delta>1$ = accelerating; ties to ACDC and early-burst.
• $\kappa$ interpolates between gradual ($\kappa=1$) and speciational ($\kappa=0$) change.
• OU adds a deterministic pull of strength $\alpha$ toward optimum $\theta$; larger $\alpha$ = stronger constraint / faster loss of ancestral influence.
• EB rate $\sigma^2(t)=\sigma_0^2 e^{bt}$; $b<0$ encodes the adaptive-radiation “early burst then slowdown.”
• In the worked mammal example, competing models are statistically near-indistinguishable ($\Delta$AICc ≤ ~3).

7. Load-bearing statements (verbatim — CC-BY permits)

1. (6.1, roadmap) “In this chapter, I will consider four ways that comparative methods can move beyond simple Brownian motion models: by transforming the variance-covariance matrix describing trait covariation among species, by incorporating variation in rates of evolution, by accounting for evolutionary constraints, and by modeling adaptive radiation and ecological opportunity.”
2. (6.2, on $\lambda$) “statistical estimates of λ under a ML model tend to be clustered near 0 and 1 regardless of the true value, and AIC model selection can tend to prefer models with λ ≠ 0 even when data is simulated under Brownian motion (Boettiger et al. 2012).”
3. (6.4, OU→selection caution) “even though this model can be described by OU, we cannot make inferences the other direction and claim that OU means that our population is under constant stabilizing selection.”
4. (6.2, $\delta$ definition) “When δ is one, the tree is unchanged and one still has a constant-rate Brownian motion process; when δ is less than 1, node heights are reduced, but deeper branches in the tree are reduced less than shallower branches … This effectively represents a model where the rate of evolution slows through time.”
5. (6.8, scope + best use) “The best applications of this type of approach, I think, are in testing particular biologically motivated hypotheses using comparative data.”

8. Stated scope, assumptions, limitations

• Self-limited scope (6.8, paraphrase of extracted text): the models listed are the beginnings of a much larger model set and are not meant to be comprehensive.
• $\kappa$ assumes the tree captures all/most speciation events — violated by extinction and incomplete sampling.
• OU (single-optimum) as a literal stabilizing-selection model is often unrealistic over long timescales given plausible population-genetic parameters (population size, heritability).
• Absolute contrasts violate normal/t-test assumptions → use rank-based tests.
• Shift-branch treatment (censored vs. noncensored) is a modeling choice; usually low impact (O’Meara et al. 2006).

9. Failure modes / invalidity patterns

[referee-relevant]

• $\lambda$ overfitting / boundary pile-up. Warning sign: ML $\hat\lambda$ landing at 0 or 1, or AIC favoring $\lambda\neq0$. Consequence: spurious signal/model preference even when truth is BM. Remedy: skepticism about ML $\lambda$; cited empirical basis is the Boettiger et al. (2012) simulation under known BM truth showing estimate clustering and AIC bias (pass = method recovers BM; fail = prefers transformed model).
• Reifying OU as stabilizing selection. Warning sign: reading a fitted OU (nonzero $\alpha$, an estimated $\theta$) as evidence of literal stabilizing selection toward that optimum. Consequence: unjustified biological claim — OU is one of many processes producing OU-shaped data. Remedy: the model→process inference is one-directional only (quote 3).
• Over-interpreting phylogenetic signal as “constraint.” High signal can arise from unconstrained BM and low signal from constrained OU (Revell et al. 2008), so signal is not a constraint meter.
• Weak model discrimination on real data. In the worked mammal example, $\Delta$AICc among BM/$\lambda$/$\delta$/$\kappa$/OU/EB is ≤ ~3 units — models “not easily distinguished.” Consequence: model-selection “winners” may be noise. [summarizer-inferred that this is the referee lesson; the ≤~3-unit spread is stated.]
• Skewed-contrast test misuse. Applying a t-test to absolute contrasts (skewed) inflates error; remedy = Mann-Whitney U.
• Empirical checks named: simulation-under-known-truth (Boettiger et al. 2012 for $\lambda$); nested likelihood-ratio / AICc comparison against BM as the null baseline. No explicit posterior-predictive / formal model-adequacy procedure in this chapter (confirmed).

10. What the chapter does NOT address

• No explicit, general treatment of distinguishing OU from BM being statistically hard, and no dedicated model-adequacy / posterior-predictive section in this chapter (confirmed; likely deferred to other chapters).
• No coverage of measurement error as a distinct model component in this chapter (confirmed).
• Multivariate trait evolution, discrete-trait models, and diversification/state-dependent (BiSSE-type) models are outside this chapter (only touched via stochastic mapping of a discrete state onto continuous-rate regimes).
• Explicitly non-comprehensive: does not span the full space of non-BM continuous models.

11. Open questions / ambiguities left unresolved

• Section 6.5 is genuinely skipped in the source (TOC 6.4 → 6.6) — an authorial numbering gap, now confirmed (not a conversion artifact).
• The appendix OU derivation is written in quantitative-genetics notation ($G$, $\gamma$); the exact printed sign/factor layout of a couple of appendix moment lines is garbled in the HTML render, but the §5 stationary-form OU moments are the standard forms.
• The chapter poses OU as capturing constraint but leaves unresolved how to license any biological (selection) interpretation from a good OU fit.