Harmon PCM Chapter 7 — Models of discrete character evolution (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. Phylogenetic Comparative Methods: Learning from Trees, Chapter 7: “Models of discrete character evolution.” Book text v1.0.0. License CC-BY 4.0. https://lukejharmon.github.io/pcm/chapter7_introdiscrete/ (PDF at /pcm/pdf/chapter7_introdiscrete.pdf). Accessed 2026-07-03.

2. Access note

Read via automated web-to-markdown conversion of the HTML chapter page, cross-checked with a second targeted pass. Prose and the numeric/parameter facts are well-attested across both passes. Boundary: equations were recovered through a converter, so equation numbering (e.g. “Eq. 7.5/7.7/7.8/7.16”) and the exact typographic form of matrices should be re-checked against the source PDF before being treated as canonical; the mathematical content (matrix entries, probability formulas) is consistent across passes and standard for the Mk model. Verbatim quotes below are transcribed from the converted text and marked; treat wording as high-confidence but re-verify against the PDF if a quote is load-bearing for a decision.

3. Thesis (1 sentence)

Discrete characters (like limblessness in squamates) require a categorical-state modeling approach — the continuous-time Markov “Mk” model and its extensions — distinct from the continuous-trait (Brownian) models of earlier chapters, and this chapter sets up that machinery (definition + simulation) before fitting is taken up next.

4. Problem & context

Motivating example (§7.1): squamate reptiles (lizards + snakes). Limbs lost repeatedly. The snake-ancestral lineage “became limbless about 170 million years ago”; some estimates put limb loss at ≥26 times in the past 240 million years (Brandley et al. 2008).
§7.2 opens by contrasting with prior chapters: “So far, we have only dealt with continuously varying characters” — establishing discrete traits as the new object, not a parallel modeling derivation.
A discrete trait is one that occupies one of a set of distinct character states; each species is in one of $k$ states (limblessness: $k=2$ , e.g. 0 = legless, 1 = limbed).
Distinguishes ordered (transitions only between adjacent states) vs unordered (any state → any other) characters; chapter focuses on unordered.
Analogy to molecular evolution models (Jukes-Cantor JC, GTR), with two differences: phenotypic characters may have any number of states, and phenotypic characters are typically analyzed independently rather than concatenated like DNA sites.

5. Method / approach

The chapter introduces the continuous-time Markov model of discrete-state evolution.

Mk model (§7.3): introduced by Pagel (1994), named/formalized by Lewis (2001); presented as a direct analogue of the Jukes-Cantor model. Two core commitments: the Markov property (next state depends only on current state) and equal transition rates among all state pairs.
Rate matrix $Q$ : $k \times k$ $k \times k$ matrix of instantaneous transition rates $q_{ij}$ $q_{ij}$ (off-diagonal, $i\neq j$ $i \neq = j$ ). Each row sums to zero; diagonal entries are $-d_i$ $- d_{i}$ with $d_i = \sum_{j\neq i} q_{ij}$ $d_{i} = \sum_{j \neq = i} q_{ij}$ . A rate $q_{ij}$ $q_{ij}$ is an instantaneous rate — the limit of the change rate over vanishingly short intervals.
- Two-state Mk ( $q_{12}=q_{21}=q$ ): $Q=\begin{bmatrix}-q & q\\ q & -q\end{bmatrix}$
- Three-state Mk: $Q=\begin{bmatrix}-2q & q & q\\ q & -2q & q\\ q & q & -2q\end{bmatrix}$
- General $k$ -state Mk (unit-rate form): diagonal $=1-k$ , all off-diagonals $=1$ .
Transition probabilities: $P(t)=e^{Qt}$ $P (t) = e^{Qt}$ — the matrix exponential (explicitly not element-wise; “calculations are not trivial,” ref. Yang 2006). $p_{ij}(t)$ $p_{ij} (t)$ = probability of ending in state $j$ $j$ after time $t$ $t$ given start in state $i$ $i$ .
- General Mk closed form: $p_{ii}(t)=\tfrac{1}{k}+\tfrac{k-1}{k}e^{-kqt}$ ; $p_{ij}(t)=\tfrac{1}{k}-\tfrac{1}{k}e^{-kqt}$ ( $i\neq j$ ).
- Two-state: $p_{ii}(t)=\dfrac{1+e^{-2qt}}{2}$ , $p_{ij}(t)=\dfrac{1-e^{-2qt}}{2}$ .
Stationary distribution $\pi$ : as $t\to\infty$ , every $p_{ij}(t)\to 1/k$ , so $\pi=[\,1/k,\dots,1/k\,]$ (for $k=2$ , $[1/2,\,1/2]$ ).
Extended Mk model (§7.4) relaxes equal rates:
- Symmetric (SYM): $q_{ij}=q_{ji}$ but pair-specific; free rate parameters $=\tfrac{k(k-1)}{2}$ . Allowing unequal stationary frequencies adds $n-1$ parameters (frequencies sum to 1), total $\tfrac{k(k-1)}{2}+(n-1)$ . $Q$ built as (rate matrix)·(diagonal $\pi$ matrix), diagonals then set so rows sum to zero. Worked two-state example (forward $=2\times$ backward, $\pi_1=0.75$ ): $Q=\begin{bmatrix}-0.25 & 0.25\\ 1.5 & -1.5\end{bmatrix}$ .
- All-rates-different (ARD): every transition type its own rate; free rate params $=k(k-1)$ , total $k(k-1)+(n-1)$ . (SYM/ARD naming per Paradis et al. 2004, APE.)
Simulating on a tree (§7.5): draw root state (from stationary dist. or uniform over $k$ ); for each branch compute $P(t)$ from its length and $Q$ ; sample the descendant state using the $P(t)$ row for the ancestral state; at speciation daughters inherit the parent state then evolve independently; record tip states. Multi-state draws use a cumulative-sum vector vs a Uniform(0,1) deviate.
Relation to earlier continuous-trait chapters: framed by contrast — prior chapters handled continuous characters; the chapter does not derive the Mk model as a formal parallel of Brownian motion. Both use rate parameters, but the connection is thematic, not an explicit structural mapping in the text.

6. Key claims / findings

$Q$ rows sum to zero; diagonal $-d_i=-\sum_{j\neq i}q_{ij}$ . (Definitional property of a valid rate matrix.)
$P(t)=e^{Qt}$ is the matrix exponential; holds for any $t\geq0$ given constant $Q$ .
Mk stationary distribution is uniform: $\pi_i=1/k$ for all $i$ (consequence of equal rates); as $t\to\infty$ start-state information is lost.
Parameter counts: ER Mk = 1 rate; SYM = $k(k-1)/2$ rates ( $+\,n-1$ freqs if unequal); ARD = $k(k-1)$ rates ( $+\,n-1$ ).
Two-state simulation instance ( $q=0.5$ , $t=3$ ): $P(3)=\begin{bmatrix}0.525 & 0.475\\ 0.475 & 0.525\end{bmatrix}$ ; from state 0, $P(\text{end }0)=0.525$ , $P(\text{end }1)=0.475$ ; assign 0 if $u<0.525$ else 1.
Squamate example: ≥26 independent limb losses in 240 My; snake-ancestral limblessness ~170 mya (Brandley et al. 2008).

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

Markov property (§7.3): “The probability of changing from one state to another depends only on the current state, and not on what has come before.” (verbatim, §7.3)
Equal-rates assumption (§7.3): the basic Mk model “assumes that every state is equally likely to change to any other state.” (verbatim, §7.3)
Extended-model motivation / caveat (§7.4): “The Mk model assumes that transitions among all possible character states occur at the same rate. However, that may not be a valid assumption.” (verbatim, §7.4)
Squamate limb loss (§7.1): “with some estimates that squamates have lost their limbs at least 26 times in the past 240 million years (Brandley et al. 2008).” (verbatim, §7.1)
Deferral to next chapter (§7.6): “In the next chapter, I will show how to fit these models to data and use them to test evolutionary hypotheses.” (verbatim, §7.6)

(Quotes transcribed from converted text — see Access note; wording high-confidence, re-verify against PDF if used as a hard citation.)

8. Stated scope, assumptions, limitations

Markov property: transition probability depends only on current state (memoryless). (§7.3, stated assumption)
Equal transition rates in the basic Mk model: all state-to-state transitions equally likely; §7.4 explicitly flags this “may not be a valid assumption” and introduces SYM/ARD to relax it.
Discrete, known, finite state space of $k$ states; each species assigned exactly one state.
Constant $Q$ : $P(t)=e^{Qt}$ presumes the rate matrix is fixed (the same process applies along branches / across the tree); the chapter does not introduce rate variation across the tree here.
Chapter scope is deliberately setup — defining the process and simulating it; parameter estimation and hypothesis testing are out of scope (deferred to next chapter).

9. Failure modes / invalidity patterns

Equal-rates misspecification (ER when SYM/ARD is true): warning sign — asymmetric or state-pair-specific transition tendencies; consequence — biased process description; remedy — the extended Mk (SYM/ARD) models, explicitly motivated in §7.4 (“that may not be a valid assumption”). (Stated as motivation for extensions.)
Unobserved changes / saturation [summarizer-inferred from the math, not framed as a “failure” in-text]: because $p_{ij}(t)\to 1/k$ as $t\to\infty$ and $p_{ii}$ includes a decaying $e^{-kqt}$ term, tip states alone under-count actual transitions over long times; the model’s probability structure accounts for this, but the chapter does not present a diagnostic for saturation here.
Violation of the Markov / memoryless assumption [summarizer-inferred]: if change depends on history beyond the current state, the $P(t)=e^{Qt}$ machinery no longer applies; the chapter states the assumption but gives no in-chapter detector.
Empirical check provided — forward simulation under known truth (§7.5): the chapter gives a concrete generative procedure (draw root, propagate $P(t)$ per branch, sample tips), i.e. simulate data under a known $Q$ . This is the reference/ground-truth mechanism the book supplies; it is presented for generating data, and (per §7.6) the fitting/recovery comparison against such known truth is deferred to the next chapter. Pass/fail criteria for estimation are not given here.

10. What the chapter does NOT address

Model fitting, likelihood estimation, and hypothesis testing — explicitly deferred to the next chapter (§7.6).
Model comparison / selection among ER, SYM, ARD (parameter counts are given, but no selection procedure).
Rate variation across the tree/time, hidden states, or covarion-type processes.
A formal structural mapping between the discrete Mk model and the earlier Brownian-motion continuous-trait models (contrast only).
Diagnostics for saturation or for testing the Markov / rate-homogeneity assumptions.
Ancestral-state reconstruction as a fitted inference.

11. Open questions / ambiguities left unresolved

Equation numbers and exact matrix typography need PDF confirmation (see Access note).
The chapter states equal-rates “may not be valid” but does not, within Ch. 7, give criteria for choosing ER vs SYM vs ARD — left to the fitting chapter.
Root-state handling in simulation offers two options (stationary distribution vs uniform over $k$ ) without stating a preferred default.
Whether unordered vs ordered character structure interacts with the SYM/ARD parameterizations is not spelled out here.