vii-viii. some case studies using branching...

GEOS 36501/EVOL 33001 27 January 2012

VII-VIII. Some case studies using branchingmodels

1 Examples of forward problems

1.1 Stochastic survivorship of single clade

1.1.1 Relevant equations: A11-A14 from Raup 1985.

1.1.2 Example: waning of trilobites from Cambrian through Permian

(Raup, 1981, Acta Geologica Hispanica 16:25-33).

• Could trilobites have stochastically drifted to extinction (“Galton extinction”) withhigh initial diversity if p = q?

• Look for conditions that yield intermediate probability of clade extinction (too low,and it is unlikely to have happened; too high, and trilobites should have becomeextinct much sooner, and most other clades should also have become extinct).

• If not, how much higher would q have to be to account for drift to extinction?

• Note importance of sensitivity analysis: explore wide range of parameters, not justyour “best guess”.


1.2 Stochastic survivorship of multiple clades

1.2.1 Relevant equations: A11-A14 from Raup 1985, plus modifications (seeappended text).

1.2.2 Example: thought experiment on multliple origins of life (Raup andValentine, 1983, PNAS 80:2981-2984).

• Look for conditions that yield intermediate probability of bioclade survival. (Too lowand it’s unlikely that life would have survived at all; too high and Phanerozoic lifeshould be polyphyletic.)

• Again note importance of exploring range of parameters.

From Raup and Valentine (1983)


From Raup and Valentine (1983)


1.3 Expected longevity of clades with variation in startingrichness reflecting temporal variation in species-level rates

1.3.1 Relevant equations: A7-A10 and A15-A18 from Raup 1985

1.3.2 Increased longevities of genera that originate right after massextinctions; may this result from high speciation rate in post-extinctionworld (Miller and Foote, 2003, Science 302:1030-1032)?

1.3.3 If specation rate higher right after mass extinction, then genera willaccumulate more species and will persist longer even if speciation andspecies extinction rates are subsequently equal.

Fig. 1. Mean longevi-ties of genus cohortsoriginating throughoutthe Phanerozoic andlatest Neoproterozoic(solid lines). (A) Maxi-ma are labeled; statisti-cal significance is indi-cated when the peakexceeds the upper 95%confidence interval(dotted line) derivedwith a randomizationprocedure (15). (B)Mean genus longevitiescompared with percapita extinction rates(dotted line). Extinctionpeaks that precede lon-gevity maxima are la-beled. (C) The patternafter culling generathat became extinctduring several post-Paleozoic mass extinc-tions (18). (D) The pat-tern after culling generathat survived to thepresent day; dotted lineas in (A) but based onlyon genera that did notsurvive to the presentday. Cm, Cambrian; O,Ordovician; S, Silurian; D,Devonian; C, Carbonifer-ous; P, Permian; Tr, Tri-assic; J, Jurassic; K, Creta-ceous; T, Tertiary.

encemag.org SCIENCE VOL 302 7 NOVEMBER 2003 1031


Miller and Foote- 3

Equations for Birth-Death Model

The probability that a genus will have n species after an elapsed time t, assuming that n >

0 and p > q, is given by p(n ,t) = (1-B)Bn-1, where B = A (p/q) and

}])[(1}/{])[({ qtqpexpptqpexpqA −−⋅−−⋅= , and p and q are the per-capita

origination and extinction rates. This approach tacitly assumes that the genera in

question originate at the start of this initial interval. Given an initial standing diversity of

n, and assuming that speciation and species-extinction rate are equal, the median genus

duration is equal to )/ 1)/(21()(1/

−=n

m

pnT . Therefore, the expected median

duration for a genus, assuming that it originates at the beginning of the initial time

interval and is still extant at the end of it, is equal to ∞ = ⋅= 1 ),()(n tnpnm

Tm

T .

We focus in this analysis on the median duration because it is analytically more tractable

than the mean, but similar reasoning holds for mean durations. For compatibility with

our empirical results, only genera that survive to the end of their initial substages are

considered.

Fig. 2. Elevation in median genus duration pro-duced during an interval in which speciation rate( p) within the genus exceeds species extinctionrate (q). The initial interval is assumed to be 4 My,about the length of a substage in our analysis.


1.4 Early origins of major biologic groups (Raup, 1983,Paleobiology 9:107-115) (See relevant pages from this paper,reproduced below.)

1.4.1 Branching structure of evolutionary trees predicts many extinct taxa,and just a few living ones; the living ones have deep roots.

1.4.2 Thus, the homogenous branching model predicts that randomly chosen,living species are likely to have a divergence time deep in the past.

From Raup (1983)


From Principles of Paleontology 3/e

1.4.3 Within- vs. between-group pairs

• Let there be g groups with Ni species, i = 1, ..., g.

• Within-group pairs:∑g

i=1 Ni(Ni − 1)/2.

• Between-group pairs:∑

i,j(i6=j) NiNj.

1.4.4 Sensitivity

• Lower rates lead to deeper divergence.

• Higher net diversification rate leads to shallower divergences.

• Thus, single large group with recent radiation (e.g. insects) can greatly skewempirical results; this would represent clear deviation from homogeneous model.


2 Further inverse problem: Estimating species-level p

and q from genus-level survivorship

2.1 Assume genus survivorship depends only on species-levelrates and elapsed time—i.e. there is no distinct genus-levelextinction process.

2.2 We can infer something about a process below our scale ofresolution based on observations at a coarser scale.

2.3 This ability comes at a cost: the assumption oftime-homogeneous, taxonomically homogeneous model.

2.4 For candidate values of p and q, predict expected genus-levelsurvivorship curves. Compare observed data to expectedcurve and find the values of p and q that maximize fit(whether by likelihood or some other criterion).

2.4.1 Let ti be time since the origin of the cohort (by whatever convention isused to place the starting point of genera whose first appearance isknown only to level of resolution of the interval) and the end of intervali.

• As with species-level cohort curves, assignment of time of origin within bin isarbitrary. Test for robustness of results with respect to protocol.

2.4.2 Let ND,ti be the number of genera in the cohort that became extinct ininterval i.

2.4.3 Let Ps,ti be the probability, for a candidate speciation and speciesextinction rate p and q, that a genus is still alive at the end of interval i.

2.4.4 Let PD,ti be the probability that the genus becomes extinct sometimeduring interval i. Note that PD,ti = Ps,ti−1

− Ps,ti.

2.4.5 Let PS,tmax be the probability that genus is still extant at the end of thetime window we observe. And let NS,tmax be the number of genera stillextant at this time.

• This is important for truncated cohorts.

• NB: The amount of elapsed time between cohort origin and end of window ofobservation (i.e. the value of tmax) will be different for each cohort.


2.4.6 Examples of ti, ND,ti and PD,ti from the trilobite data (Foote 1988)

• Take first cohort. Assign first appearances to base of Early Atdabanian.

• Thus the relevant values of t are 0.0, 4.5, 9.0, 15.5, 22.0, 28.0, 34.0, and 39.0.


• The corresponding values of ND,ti are 10, 2, 3, 0, 1, 0, and 1.


• Let us say for the sake of argument that we are interested in evaluating p = q = 0.1.Then

Ps,0 = 1.0

Ps,4.5 = 0.690

Ps,9.0 = 0.526

Ps,15.5 = 0.392

Ps,22.0 = 0.313

Ps,28.0 = 0.263

Ps,34.0 = 0.227

Ps,39.0 = 0.204

and

PD,4.5 = 0.310

PD,9.0 = 0.163

PD,15.5 = 0.134

PD,22.0 = 0.080

PD,28.0 = 0.049

PD,34.0 = 0.036

PD,39.0 = 0.023

2.4.7 The likelihood function (for untruncated cohorts) is:

L(p, q|ND,ti) =∏

i

[PD,ti ]ND,ti

and the support (log-likelihood) is:

S(p, q|ND,ti) =∑

i

ND,ti ln PD,ti

2.4.8 These functions are modified by taking into account the genera thathave not become extinct:

L′ = L ·∏

cohorts

[PS,tmax]NS,tmax

andS ′ = S +

∑cohorts

NS,tmax ln PS,tmax,

where, again, the absolute elapsed time corresponding to tmax is different for each cohort.


2.4.9 Numerically find the values of p and q that maximize the likelihood.

2.5 Back to the trilobite example

2.5.1 This, alas, did not use strict likelihood approach.

• Instead, the approach was to minimize the mean absolute log deviation betweenexpected and observed cumulative proportions.

• With this approach, it is important to try to minimize noise in the tails of cohortcurves by truncating them at some arbitrary percent survival.

• Noise in tails not such an issue for likelihood, because the number of taxa involved issmall and therefore they do not contribute much to overall measure of support.


Graphical depiction of the cohorts, separated into those originating in Cambrian vs. Ordovician


2.5.2 Superimposed cohorts and best-fit decay curves

• Note that empirical half-lives of Cambrian and Ordovician cohorts nearly identical.

• This largely reflects the fact that the majority of genera are singletons, even in thelonger-lived cohorts.

• Singletons may be real, in which case they are not very informative at this scale ofstratigraphic resolution.

• Or they may be artifacts of incomplete sampling, in which case they are potentiallymisleading.

• One approach to the “singleton problem” is to start following cohort history at theupper boundary of the interval of first appearance.


2.5.3 Shape of goodness-of-fit surface is such that many combinations of p andq fit almost equally well. (p − q) is better constrained than p or qindividually.


2.5.4 This is relevant to an additional (and, I think, largely unresolved)problem:

• Recall that p is the rate of origin of new species within the paraclade. Speciationevents spawning new paraclades are (tacitly) ignored.

• In principle, therefore, estimating p and q from family-level data should yield same qand slightly higher p compared with estimating from genus-level data.

• In fact, however, estimates of both p and q tend to be lower when families areanalyzed.

• So, in the real world, what are we actually estimating when we estimate p and q fromsurvivorship of higher taxa?


2.5.5 Assessing “significance” of Cambro-Ordovician difference

• Simulation (parametric bootstrap): Do cohorts with estimated Cambrian rates everappear like Ordovician cohorts (or vice versa)?

• Is the Cambro-Ordovician boundary a natural place to divide the data? Artificiallyvary position of C/O boundary and assess goodness of fit.


2.5.6 Error bars on survivorship proportions

• Points on survivorship curve not independent, so successive binomial error bars wouldunderestimate range of paths that survivorship curves could take.

• For example, one excursion above or below expected survivorship is more likely to befollowed by another excursion in the same direction.

• Simulation as alternative to assess range of behaviors for given p, q, and cohort size.

vii-viii. some case studies using branching...

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