Modelling distributions of fossil sampling rates over time, space and taxa: assessment and...

UNIFYINGFOSSILSANDPHYLOGENIES FORCOMPARATIVEANALYSESOFDIVERSIFICATIONANDTRAIT EVOLUTION

Modelling distributions of fossil sampling rates over

time, space and taxa: assessment and implications

formacroevolutionary studies

Peter J.Wagner1* and JonathanD.Marcot2

1Department of Paleobiology, NationalMuseumof Natural History, Smithsonian Institution,MRC121, POBox 37012,

Washington, DC, 20013-7012, USA; and 2Department of Animal Biology, University of Illinois, 515Morrill Hall, 505 S.Goodwin

Ave., Urbana, IL, 61801, USA

Summary

1. Observed patterns in the fossil record reflect not just macroevolutionary dynamics, but preservation patterns.

Sampling rates themselves vary not simply over time or among major taxonomic groups, but within time inter-

vals over geography and environment, and among species within clades. Large databases of presences of taxa in

fossil-bearing collections allow us to quantify variation in per-collection sampling rates among species within a

clade.We do this separately not just for different time/stratigraphic intervals, but also for different geographic or

ecologic units within time/stratigraphic intervals. We then re-assess per-million-year sampling rates given the

distributions of per-collection sampling rates

2. We use simple distribution models (geometric and lognormal) to assess general models of per-locality sam-

pling rate distributions given occurrences among appropriate fossiliferous localities. We break these down not

simply by time period, but by general biogeographic units in order to accommodate variation over space as well

as among species.

3. We apply these methods to occurrence data forMeso-Cenozoic mammals drawn from the Paleobiology Dat-

abase and the New and Old Worlds fossil mammal database. We find that all models of distributed rates do

vastly better than the best uniform sampling rates and that the lognormal in particular does an excellent job of

summarizing sampling rates. We also show that the lognormal distributions vary fairly substantially among

biogeographic units of the same age.

4. As an example of the utility of these rates, we assess the most likely divergence times for basal (Eocene–Oligo-

cene) carnivoramorphanmammals fromNorthAmerica andEurasia using both stratigraphic andmorphological

data. The results allow for unsampled taxa or unsampled portions of sampled lineages to be in either continent

and also allow for the variation in sampling rates among species.We contrast fivemodels using stratigraphic like-

lihoods in differentways to summarize how theymight affectmacroevolutionary inferences.

Key-words: evolutionary biology, macroevolution, phylogenetics, systematics

Introduction

A concern expressed in even the oldest studies of evolution

using fossil data is that inconsistent sampling might distort

evolutionary patterns (Darwin 1859). Inconsistent sampling

over time, geography and among taxa affects our perceptions

of a wide range of macroevolutionary issues: from more

general to more specific, these span from differences in overall

richness (Raup 1972; Alroy et al. 2001), to extinction and orig-

ination rates (Sepkoski 1975; Foote 1997, 2001; Alroy 2000)

and further down to specific ideas about timings of extinctions

(Signor & Lipps 1982; Marshall 1995a) and originations

(Wagner 1995a, 2000a; Huelsenbeck & Rannala 1997). These

issues in turn spill over into other macroevolutionary issues

such as whether apparent patterns of punctuated or continu-

ous morphological change might reflect sampling (Marshall

1995b) or whether apparent shifts in rates of morphological

change reflect differences in sampling affecting howmuch time

lineagesmight have had to accumulate change (Wagner 1995b,

1997). Thus, being able tomodel variation in the rates at which

we sample taxa from the fossil record transcends simple

interest in sampling itself.

A key point is that there is no such thing as ‘the quality’ of

the fossil record: the probabilities of sampling taxa, either*Correspondence author. E-mail: wagnerpj@si.edu

Published 2013. This article is a US Government work and is in the public domain in the USA.

Methods in Ecology and Evolution 2013, 4, 703–713 doi: 10.1111/2041-210X.12088

per-stage or per-million years, vary enormously over time

(Alroy 1999; Foote 2001), among taxonomic groups (Foote &

Sepkoski 1999), across geography and environment (Smith

2001), and among species within clades (Wagner 2000a). Foote

(1997; also Foote & Raup 1996) presents methods that can

assess the first two issues, that is, variation in rates at which we

sample taxa over time (Foote 2001) and among major clades

(Foote & Sepkoski 1999). Thesemethods require only synoptic

compilations of first and last occurrences such as provided by

Sepkoski (2002). However, they provide only single ‘average’

sampling rates for whole taxonomic groups and/or strati-

graphic intervals. These rates themselves reflect two factors:

sampling at the finest stratigraphic levels (i.e. individual collec-

tions of fossils from particular rock layers) and the number of

collections within a stratigraphic or million-year interval.

Assessing variation in per-stage or per-million-year sampling

rates among taxa in the same interval (e.g. different species in a

clade, or species from different habitats or geographic areas)

therefore requires that we assess per-collection sampling rates.

Fortunately,palaeontologistshaveassembled largedatabases

of fossil occurrences anddistributions of sedimentary rock such

as the Paleobiology Database (http://paleodb.org). These pro-

vide information about numbers of finds, numbers of sampling

opportunities, and where those finds and opportunities exist

geographically and environmentally. This opens the door to

modelling sampling as distributions of per-collection rates

rather than single ‘average’ per-stage or per-million-year rates

and then extrapolating distributions of per-stage or per-

million-year sampling rates fromper-collection sampling rates.

The issue of rate variation is hardly specific to sampling rates

from fossil record. Phylogeneticists deal with an analogous

problem when accommodating variations in rates of character

change. Instead of deriving specific rates for individual charac-

ters, phylogeneticists assume that rates are drawn from model

distributions such as the gamma (Yang 1994). Instead of

assuming that single gammas fit all data partitions (e.g. differ-

ent genes), they examine whether different data partitions fit

different gammadistributions (Yang 1996).We adopt the same

approach by first assessing whether model distributions for

sampling rates such as geometrics or lognormals better predict

distributions of fossil occurrences than do single sampling rate

models. We further adapt this by breaking the distributions up

by time and geography in order to better model suspected vari-

ation in the fossil record. Finally, we apply these results to a

particular issue – the likelihood of stratigraphic gaps associ-

ated with divergence times implicit to a hypothesized phylog-

eny – and provide cursory discussion of how these results

might affect macroevolutionary inferences.

Data andmethods

DISTRIBUTIONS OF SAMPLING RATES

Some basic issues concerning the data

Our tests rely on occurrence (=incidence) data of fossil species.In particular, we are interested in how well distributions of

sampling rates predict frequencies of occurrences per-sampling

opportunity (i.e. relevant fossil-bearing collection). This intro-

duces onemajor difference between our goal and the conceptu-

ally similar goal of modelling abundance distributions within

communities (May 1975; Gray 1987). In those studies, the

number of specimens sets the limits on the possible specimens

that might go to Species A, B, C, etc. Thus, if there are 100

specimens, then only one species can have 100 individuals. The

number of possible occurrences is the number of collections.

Thus, if there are 100 collections, then theoretically all species

could have 100 occurrences.

‘Per-sampling opportunity’ leads to our next basic issue:

exactly which fossiliferous collections count as sampling

opportunities? We define ‘sampling opportunities’ as collec-

tions from which a species of interest could have been sampled

had they been present. For example, we do not expect to sam-

ple terrestrial vertebrates from marine sediments; thus, a col-

lection of marine invertebrates is not a sampling opportunity

for terrestrialmammals. This goes beyond basic environmental

differences. For example, a Cenozoic locality preserving only

terrestrial plants almost certainly captures an environment that

hosted mammals, too. However, taphonomic processes (e.g.

factors causing fossilization; Behrensmeyer & Kidwell 1985)

can exclude basic preservational groups. Thus, environmental

and taphonomic controls (Bottjer & Jablonski 1988) are criti-

cal for assessing sampling opportunities: a collection is an

opportunity only if it shows that it could have contained the

species of interest.

Assessedmodels

The first model that we assess is not a distribution, but a single

rate. This represents the simplest (one parameter) model and

thus is the null relative to all others.Many studies estimate glo-

bal sampling rates per-chronostratigraphic unit (e.g. stage or

substage; Foote & Raup 1996). Given a per-stage sampling

rate Rs and N collections in that stage, we can estimate the

per-collection sampling rate to be:

Rc ¼ 1� elnð1�RsÞ

Thus, if one estimatesRs = 0�333 for an interval withN = 100

collections, then one would estimate an ‘average’Rc = 0�004.For numerous reasons, we do not expect a singleRc to sum-

marize all taxa and all collections. Buzas et al. (1982) consider

two models when looking at the distribution of occurrences:

the log-series (Fisher, Corbet &Williams 1943) and the lognor-

mal (Preston 1948). Their primary justification for doing this

was that abundances of fossil taxa within collections often fit

these two distributions well. Although we also are using occur-

rence data, we are modelling underlying sampling rates rather

than occurrence frequencies. Thus, the log-series is unusable as

it models distributions of discrete variables (e.g. taxa with 1, 2,

etc. finds) rather than fractional variables such as rates.

However, we can use geometric distributions (Motomura

1932) as an alternative. Like the log-series, the geometric distri-

bution assumes that the relevant variables follow a uniform

704 P. J. Wagner & J. D. Marcot

exponential distribution. We might expect geometric distribu-

tions of sampling rates if there is no cohesion among the

numerous processes underlying sampling rates (e.g. geographic

ranges, relative abundance and sample size from collections,

local preservation potential and ease of identification and

recovery). Conversely, if there are central tendencies to those

processes that are associated with particular taxa, then we

expect lognormal distributions (Montroll & Shlesinger 1982).

Wagner & Marcot (2010) show that sampling rates among

some Ordovician–Silurian gastropods follow a lognormal dis-

tribution.

Both the geometric and lognormal models have a basic

sampling rate, r, as one parameter. For the geometric distri-

bution, there is an additional ‘decay’ parameter, d, giving

how many times lower the next sampling rate is. For the

taxon with the ith highest sampling rate, the per-collection

sampling rate Rci is

Rci ¼ r1dði�1Þ

where r1 is the sampling rate of the most easily found taxon.

Note that these rates are per-collection, so r1 cannot exceed

1�0. The uniform is a special case of the geometric in which

d = 1, that is, there is no decay in rates over taxa.

For the lognormal, the basic rate r represents the geometric

mean of the rates. The distribution is determined by two more

parameters. One is a magnitude parameter,m, which gives one

standard deviation in magnitude of change around the mean.

The second is true richness (S). For the taxon with the ith high-

est sampling rate,Rci is

Rci ¼ rmnorminvð½Sþ1�i�=½Sþ1�Þ

where norminv(x) gives the number of standard deviations

away from themean for which x is the area under the bell curve

to the left of that point. The latter parameter illustrates the

importance of S. At S = 50 taxa, Rc1 is proportional to m to

the power norminv(50/51) = 2�06; however, at S = 100 taxa,

Rc1 is proportional to m to the power of norminv(100/

101) = 2�33. Note that the uniform is a special case of the log-

normal wherem = 1�0 and S = ∞.There are several other distribution models (e.g. gamma,

Zipf, etc.) that onemight consider. However, we found none of

them to perform as well as the best models considered here,

and there are no particular theoretical reasons to expect these

distributions. Therefore, we do not discuss them here. We do

include the likelihoods of saturated models (Sanderson 2002)

where the expected number of taxa with 1…N finds equals the

observed. Saturated models provide the maximum possible

likelihood of any hypothesis derived from these sorts ofmodels

and thus provide a useful benchmark for evaluating the perfor-

mance of the simplemodels.

Model assessment

We assess hypotheses under any particular model by deriving

the expected frequencies of taxa with 1…N finds and then

using multinomial probability to assess the likelihood of the

rate distribution given occurrence data. For any distribution,

the expected frequencies givenRc are

fðkÞ ¼Psi¼1

� �� Rcki � ð1� RciÞðN�kÞ

� �Psi¼1

1� 1� Rci½ �N� � :

The numerator sums the binomial probability of k occurrences

given N collections and sampling rate Rc. The denominator

sums binomial probabilities of finding the taxon at all (i.e. one

minus the probability of 0 finds). This conditions ƒ(k) on the

taxa being found, which is appropriate because we can tally

taxa with 1, 2, etc. occurrences, but not those with 0 occur-

rences. The lognormal is the onlymodel for whichS is an expli-

cit parameter (Wagner, Kosnik &Lidgard 2006). In the case of

the geometric, we sum up to the i where until Rci becomes so

low that it no longer elevates the denominator past the 4th dec-

imal point. In the case of the uniform, the summation is unnec-

essary asRci = r for any taxon i; thus, the summations in both

the numerator and denominator can be eliminated (for the sat-

urated model, we forgo this equation and simply set ƒ(x) = Sx/

Sobs where Sx = observed number of taxa with x occurrences

and Sobs is the observed number of taxa).

For any particular Rc from any given model given the data,

the sufficient statistic for the likelihood is

L½Rcjdata� ¼YNx¼1

fðxÞnðxÞ

where n(x) gives the number of taxa with x = 1…N occur-

rences (see Figs S1–S4). Although the uniform represents a

special case of the geometric and the lognormal, the geometric

is not a special case of the lognormal. Therefore, we use

Akaike’s modified information criterion (Sugiura 1978) to

compare the best representatives of each model. We include

comparisons with the saturated model simply to inform the

readers’ intuitions regarding how close simple models come to

maximally explaining the data.

We use occurrences and collections of terrestrial mammals

from the Campanian through the Pliocene. The bulk of the

data come from the PALEODB (http://paleodb.org), downloaded

on 29 November 2012. We augment these data with the New

and Old Worlds database (formerly Neogene Old World;

http://www.helsinki.fi/science/now/), after vetting the data to

remove duplicate localities and occurrences. Uhen et al. (2013)

review both databases and numerous macroevolutionary stud-

ies that use these data (Alroy 1996, 1998, 1999; Fortelius et al.

1996, 2002; Raia et al. 2012). We include only occurrences

identified to the species level and thus only those collections

with such occurrences. We used species from the Lepidosauro-

morpha (e.g. lizards, snakes and relatives) as a taphonomic

control group. Our justification for this is that terrestrial

localities from which workers can identify lepidosauromorpha

species level have the potential to preserve mammal specimens

Fossil sampling rate distributions 705

that can be identified to the same level. Conversely, our analy-

ses exclude Cetacea and other exclusively marine mammals, as

we do not expect to find terrestrial mammals in fossil beds

yielding those taxa. In total, we use 46612 occurrences of 8129

species from 7871 localities (Table 1) and binned these into

standard stages and substages of the Mesozoic and Cenozoic.

PaleoDBdata represent 5587 references.

Mammalian localities are not evenly distributed over time

(Fig. 1). In particular, the Miocene and Pliocene have more

localities than expected given their durations, and the Paleo-

cene and Cretaceous have fewer localities than expected. This

pattern becomes more pronounced when we subdivide collec-

tions by continent. Although large proportions of Miocene

and Pliocene collections come from Eurasia, a much smaller

proportion of pre-Miocene collections are from Eurasia.

African sampling not only is less than North American or

Eurasian, but simply quite poor prior to the Miocene. (Both

results likely reflect the NOW database beginning as the Neo-

gene OldWorld database, leading it to still be better populated

with post-Oligocene data than pre-Miocene data.) The bulk of

Campanian–Oligocene localities in our pooled data set come

fromNorthAmerica.

Using the localities per-stratigraphic unit shown in Fig. 1,

we evaluate the different basic models for per-collection sam-

pling rates both globally and then by individual continent.

Results

For every interval considered, the lognormal distribution out-

performs the geometric and (especially) the uniform model

(Table 2; see Table S1 for the parameters of the best represen-

tative of each model). This is not simply due to extra parame-

ters: the lognormal does much better than the geometric and

uniform given AICc scores (Table S2). These simple models

do very good jobs of summarizing the data. The difference in

log-likelihoods between the best uniform rate hypothesis and

the saturated model (i.e. the best possible hypothesis) repre-

sents the maximum improvement in log-likelihood that a

model can offer. If we scale this difference to 1�0, then we find

that in nearly all intervals, lognormals provide over 90% (and

frequently over 95%) of the possible improvement in log-likeli-

hoods over the uniformdistribution (Fig. 2). Thus, lognormals

do not leave huge room for improvement by still more complex

models.

Lognormal sampling rates change perceptions of overall

sampling markedly. Here, we summarize the lognormal using

the midpoint rates associated with four equal-area partitions,

that is, the rates at which 12�5%, 37�5%, 62�5% and 87�5% of

taxa have higher rates (Yang 1994). The most likely uniform

rate frequently is higher than the 4th quartile rate (Fig. 3) sim-

ply because the few commonly occurring taxa are less probable

given low sampling rates than many infrequently occurring

Table 1. General summary of analysed data

Stage/

Substage(s)

(Millions of

YearsAgo)

Sampled

taxa Occurrences Collections

Campanian +83�5 89 314 133

Maastrichtian +70�6 86 728 198

Danian +65�5 306 1396 253

Selandian–Thanetian

+61�7 363 1668 296

Ypresian +55�8 553 6458 998

Lutetian +48�6 445 1605 328

Bartonian–Priabonian

+40�4 762 2326 374

Rupelian +33�9 484 1778 489

Chattian +28�4 397 871 202

Aquitanian–Burdigalian

+23�0 1429 6875 1082

Langhian–Serravallian

+16�0 1284 7890 1230

Tortonian–Messinian

+11�6 1667 7326 1165

Zanclean +5�3 831 2131 369

Piacenzian–Gelasian

+3�6 1309 5246 754

‘Onset’ gives the beginning of the stratigraphic units in question in mil-

lions of years as per Gradstein, Ogg & Smith (2005). Sampled taxa give

numbers of species sampled.

Fig. 1. Chronology and stratigraphy for Campanian – Pliocene mam-

mals. Time scale is modified from Gradstein, Ogg & Smith (2005).

‘Coll.’ gives the number of collections (taphonomically controlled fos-

siliferous localities) within each stratigraphic unit for the globe, Africa,

Eurasia and North America. Data are from the PALEODB and NOW.

Solid lines divide the stages and substages used in these analyses;

dashed lines separate substages lumped into single units.

taxa are given high sampling rates. The 4th quartile rates often

are similar to average per-million-year sampling rates esti-

mated by range data for North American species alone (0�34 inFoote & Raup 1996; 0�48 in Foote 1997). Thus, both methods

of ‘uniform’ sampling rates accommodate the common taxa

before the rare ones.

We approximate sampling rates per-million years within

intervals, Rm, using the number of localities per-million years

(Fig. 4). For any interval i, we estimateRm as:

Rmi ¼ 1� ð1� RciÞNi=ti

where Ni is the number of collections and ti is the duration of

the interval inmillions of years.

We can improve overall likelihoods still further not by

employing more complicated mathematical models, but by

applying separate lognormals to different data partitions.Here,

we partition sampling rate distributions among the major

biogeographic regions in our pooled data set (Fig. 5; Fig. S5

givesper-collection sampling rates;Table S3gives details about

each lognormal). We would get different sampling rates

per-millionyears for these geographicpartitions simplybecause

of the different numbers of sampling opportunities per-million

years in these regions (Fig. 1). However, the best model per-

collection rate distributions often are very different for different

regions from the same interval. The most obvious case is the

earlyEocene (approximately 52�5 Ma),where the single, global

lognormal (Fig. 4) apparently represents a bad compromise

Table 2. Log-likelihoods for the best representatives of the three rate

distributionmodels considered

Stage/Substage(s) Uniform Geometric Lognormal Saturated

Campanian �253�7 �192�2 �183�6 �172�4Maastrichtian �902�3 �232�3 �216�1 �181�9Danian �1281�9 �725�3 �696�5 �665�6Selandian–Thanetian

�1929�9 �931�4 �790�3 �755�0

Ypresian �8684�0 �1664�8 �1496�5 �1389�3Lutetian �1778�6 �944�3 �816�6 �789�6Bartonian–Priabonian

�2076�8 �1486�6 �1399�6 �1375�5

Rupelian �2186�3 �1090�3 �868�4 �836�6Chattian �752�8 �617�8 �569�6 �556�6Aquitanian–Burdigalian

�7055�8 �3549�3 �3195�5 �3122�5

Langhian–Serravallian

�8853�7 �3398�1 �3094�3 �3024�2

Tortonian–Messinian

�7938�9 �3906�0 �3486�6 �3438�1

Zanclean �1988�2 �1473�4 �1318�6 �1295�4Piacenzian–Gelasian

�5450�9 �2948�8 �2654�4 �2605�7

‘Saturated’ gives the maximum possible log-likelihood from a theoreti-

cal hypothesis predicting the E[ƒn] = observed fn for all n = 1…N

occurrences.We use log-likelihoods to evaluate howwell the geometric

and lognormal perform relative to themaximumpossible performance.

Fig. 2. The performance of the geometric and lognormal models rela-

tive to the best possible performance. The X-axis represents the differ-

ence between the best uniform rate model and a ‘saturated’ model that

predicts the observed frequencies of taxa with 1, 2, … N occurrences.

The latter represents the maximum possible likelihood for models such

as the geometric or lognormal.

Fig. 3. Per-collection sampling probabilities in different stratigraphic

intervals for Meso-Cenozoic mammals. X gives the uniform rate maxi-

mizing the probability of the observed fossil record. We summarize the

lognormal using the medians of four equal-area partitions of the log-

normal distribution; thus, the highest rate is the 87�5& rate, whereas

the lowest rate is 12�5& rate.

Fig. 4. Estimated per-collection sampling rates per-million years in dif-

ferent stratigraphic intervals for Meso-Cenozoic mammals. Each per-

collection sampling rates from Fig. 3 is rescaled to 1� ð1� RcÞNi=ti

whereNi is the number of collections and ti the duration (in millions of

years) for interval i.

between different regional lognormals (Fig. 5). However, sev-

eral other intervals show marked differences in both geometric

means andvariancesof per-collection sampling rates.

Discussion

APPLICATION OF RESULTS: D IVERGENCE TIMES AMONG

CARNIVORAMORPHAN MAMMALS

In lieu of a traditional discussion, we present an applied

example using mammalian sampling rates in a phylogenetic

context. Phylogenies are particularly useful for our discussion

for two basic reasons. First, phylogenies allow reconstruction

of ancestral geographic distributions (Ree 2005; Ree & Smith

2008), which in turn let us take advantage of different sam-

pling rates for different regions. Second, there is some branch

duration that will maximize the likelihood of any branch on a

phylogeny given some distribution of character state and

some hypothesized rate(s) of change (Felsenstein 1973). This

latter point is particular critical because alternate macroevolu-

tionary hypotheses often predict different rates of morpholog-

ical change over particular time intervals and thus are

optimized at different branch durations. Branch duration

therefore is a critical nuisance parameter for hypotheses rang-

ing from punctuated versus continuous morphological change

(Marshall 1995b) to ‘big bangs’ in morphologic evolution

early in clade histories (Wagner 1995b, 1997; Ruta, Wagner

& Coates 2006). If we can indefinitely extend branch dura-

tions (and thus the time to accumulate change), then we often

can elevate the likelihood of hypotheses of constant rates

when a literal reading of the fossil record suggests rate

decreases. However, the likelihoods of these durations might

be strongly reduced whether the stratigraphic gaps implicit to

those durations are improbable given sampling rates (Huel-

senbeck & Rannala 1997; Wagner 2000a). ‘Molecular clock’

studies reverse the emphasis on the same parameters: charac-

ter rate variation is the nuisance parameter, and branch dura-

tions are the inference (Huelsenbeck, Larget & Swofford

2000; Drummond et al. 2006). Although this traditionally

was restricted to molecular characters, recent divergence-date

studies have extended this to morphological characters when

dating branches leading to fossil taxa (Pyron 2010, 2011;

Ronquist et al. 2012). Marshall (2008) addresses bracketing

divergence times with limited occurrence data; here, we

approach the same general problem using a much broader

range of information from occurrence data.

To explore how our approach might alter inferences, we

calibrate branch durations for early (Eocene – Oligocene) car-

nivoramorphan mammals from North America and Eurasia

under five different models using morphological, biogeograph-

ic and stratigraphic data. Here, we will discuss how different

approaches in general and how branch duration calibrations in

particular might affect conclusions drawn fromWesley-Hunt’s

(2005) analyses of carnivoramorphan dental disparity.

We use character data and a corresponding parsimony tree

from Wesley-Hunt & Flynn (2005). We estimate morphologi-

cal likelihoods using an Mk model assuming continuous

change through time (Lewis 2001), with morphological rates

assumed to follow a lognormal distribution estimated from

compatibility tests (Wagner 2012) and an initial estimate of un-

sampled lineages over which change could accrue based on an

global lognormal preservation rates (Foote 1996). We also use

an Mk model to estimate likelihoods of geographic distribu-

tions (Ree & Smith 2008), with lineages assumed to occupy

either North America or Eurasia, and with the probability of

an unsampled lineage being in North America calculated

simply as:

P½North America�¼ L½North America�=ðL½North America� þ L½Eurasia�Þ:

Given this particular tree we use, ancestral geographies have

very high probabilities for either North America or Eurasia

save for a very few branches near a likely North American ?Eurasian incursion (Fig. S6; see also Table S4).

The stratigraphic likelihood of any branch duration is the

probability of zero finds overXmillion years based on per-col-

lection sampling rates illustrated in Figs 3–5 and collections

per-million years illustrated in Fig. 1:

L½DivergencejStratigraphy�

¼XAreasA¼1

P½A� �YSTs¼st

X4q¼1

1� RmA�s�q� �t

where A is a possible ancestral region, st and ST are the first

and last stages (or other chronostratigraphic units) over which

a branch spans, t is the duration within that stage that the

branch spans, and q is one of four quartiles within the lognor-

mal distribution (Wagner &Marcot 2010). Thus,RmA•s•q gives

the per-million-year sampling rate for Area A in stage s from

lognormal quartile q, and (1 � RmA•s•q)t gives the probability

of zero finds over t million years. Branch likelihoods are now

L½DivergencejData�¼ L½DivergencejStratigraphy� � L½DivergencejMorphology�:

Fig. 5. Estimated collection sampling probabilities per-million years in

different stratigraphic intervals for Meso-Cenozoic mammals now

divided into different biogeographic units. Note that intervals with too

few localities are excluded. See Fig. 3 for additional explanation.

We then estimate divergence times and branch durations

under five different models. In reverse order of complexity,

these are

Model 1: Separate regional lognormal sampling rates from

Fig. 5 (Fig. 6);

Model 2: Global lognormal sampling rates from Fig. 4

(Fig. S7a);

Model 3: Global uniform sampling rates from Fig. 4

(Fig. S7b);

Model 4: Branch durations optimized solely to fit Mk

model with sampling rates ignored (hereafter, Model 4;

Fig. S7c);

Model 5:Minimumdivergence times determined as the old-

est occurrences of descendant taxa, with both sampling

rates andMk likelihoods ignored (Fig. S7d).

Themodels become progressively simpler by assuming fewer

variable terms in the stratigraphic likelihood. Model 1 allows

for a separate RmA•s•q for each quartile in each region in each

stage s. Model 2 simplifies this by assuming that

Rm1•s•q = Rm2•s•q = … = RmA•s•q for all A regions (i.e. global

lognormal sampling rates for each stage s). This eliminates the

first summation. Model 3 simplifies still further by assuming

that RmA•s•1 = RmA•s•2 = RmA•s•3 = RmA•s•4 for all A regions

(i.e. a global uniform sampling rate for each stage s). This effec-

tively reduces the second summation to (1 � Rms)t. Finally,

Models 4 and 5 effectively eliminate this last term by assuming

Rms = 0�0 and thus the probability of any gap = 1�0. In both

cases, stratigraphic data are used only to set minimum diver-

gence times: that is, the appearance on the oldest taxon in a

clade if we assume no sampled ancestors. Within each clade,

Model 4 uses the Mk model to calibrate some divergence time

preceding the oldest taxon’s first appearance (e.g. Viverravus

gracilis in theV. minutus + V. gracilis pair). However,Model

5 ignores even this andwill essentially ascribe branch durations

of 0 to species such as V. gracilis. Note that Model 5 therefore

necessarily ascribes the minimum possible divergence times

(unless we add ancestor–descendant hypotheses; see Wagner

2000b,c) because we use no information to infer that diver-

gences might be older than absolutely necessary. Model 4

ascribes the maximum possible divergence times because

stratigraphic data cannot gainsaymorphological data.

Table 3 summarizes the total time that each model allots

for character change. Figure 7 contrasts the differences in

Fig. 6. Phylogeny of early carnivoramorphan mammals with branch durations estimated from both morphological and stratigraphic data. Branch

widths reflect stratigraphic likelihoods assuming different lognormal sampling rates for North America and Eurasia (Model 1 of text; see Fig. 5),

with the probability of unsampled lineages occupying North America or Eurasia estimated based on illustrated distributions and anMkmodel with

expectation of 0�011 geographic shifts per-myr (see Fig. S6). Morphological likelihoods use an Mk model (Lewis 2001) assuming single lognormal

distribution for rates of morphological change, which has a geometric mean of 0�028 changes per-myr that changes 3�4 times every standard devia-

tion. Data and phylogenymodified fromWesley-Hunt&Flynn (2005). See Fig. S7 for trees estimated using differentmodels.

Table 3. Sumof branch durations inferred under differentmodels

Model ∑Branch durations (myr)

1: Regional lognormal

sampling rates + Mk

131�8

2: Global lognormal

sampling rates + Mk

140�1

3: Global uniform

sampling rates + Mk

121�6

4:Mk only 141�85:Minimumdivergence times 101�5

This gives the total amount of time allotted for character change under

Mk (or other)models. myr,million years.

individual branch durations, using the Model 1 (regional log-

normal sampling rates) tree as a benchmark. These make a

very important point: although our intuition might be that

accounting for variation in the fossil record should increase the

probability of gaps, in some cases, it will have the opposite

effect. Here, assuming global lognormal sampling rates makes

many gaps more probable rather than less probable (and thus

deep divergences more likely rather than less likely; Fig. 8a)

than when we account for geographic variation in sampling

rates (i.e. Model 2 vs. Model 1 trees). This is because so many

of the necessary gaps are for lineages that likely resided in the

well-sampled North American realms. However, allowing for

variation in sampling rates among taxa yields the intuitive

result: not only do global uniform sampling rates imply much

shorter branch durations than regional lognormals (Fig. 7),

but uniform sampling rates make these shorter branch times

less likely than regional lognormal sampling rates make longer

branch durations (Fig. 8b).

Carnivoramorphan dental characters show high disparity

despite low taxonomic diversity very early in clade history

(Wesley-Hunt 2005), which is consistent with elevated early

rates of change (Foote 1993). However, dental characters are

only a subset of the characters that we use (Wesley-Hunt &

Flynn 2005). We therefore can ask whether disparity patterns

among dental characters predict rates of change among all

characters. The Model 4 tree obviously will contradict ideas of

elevated early rates because it calibrates the both early and late

branch durations assuming the same rates of change. How-

ever, stratigraphic likelihood assuming lognormal distribu-

tions of sampling rates (either global or regional) also allow for

comparably deep divergences (and thus reasonably continuous

rates of change) at the base of the clade. In contrast, theModel

3 (global uniform) tree would be much more disposed to

favour elevated early rates of change, as it allots far shorter

basal branch durations under the same model of morphologi-

cal change. Finally, the Model 5 tree basically leaves the ques-

tion open, as it provides no means for implying deeper

divergences.

Dental disparity did not increase following the near extinc-

tion of likely competitors (the Creodonta) during the late

Eocene, which suggests that increased ecological opportuni-

ties for carnivoramorphans did not elevate their rates of

dental change. Notably, all four models using the Mk model

for all characters are consistent with this: although those

using stratigraphic data shorten many of the branch dura-

tions relative to the Mk model alone, they still allot substan-

tial time for change. Again, the Model 5 tree will be more

disposed towards favouring elevated Late Eocene rates

simply because that tree allots no time for many nodal

branches to accumulate change.

Although paleontological rate studies frequently ask

whether rates are elevated early in clade history or in associa-

tion with some major event, none have explored the idea of

local rate variation among branches (Huelsenbeck, Larget &

Swofford 2000; Drummond et al. 2006). This is where the

Model 1 tree might well yield more tangible differences from

the Model 4 tree: allowing some probability of elevated ‘local’

Fig. 7. Differences in branch durations between the Model 1 tree

(using regional lognormal sampling rates to calculate stratigraphic like-

lihoods) andModel 2 (global lognormal sampling rates), Model 3 (glo-

bal uniform sampling rates),Model 4 (stratigraphic likelihood ignored)

and Model 5 (minimum divergence time) trees. Positive numbers indi-

cate that the branch duration is greater on theModel 1 tree than on the

contrasted tree.

(a) (b)

Fig. 8. Log-likelihoods (lnL) of gaps implied by branch durations in carnivoramorphan phylogeny (Fig. 6). Note that although each plotted branch

links the same taxa, the durations sometimes are different (Fig. 7). (a) Regional lognormal sampling rates versus global lognormal sampling rates.

(b) Regional lognormal sampling rates versus global uniform sampling rates.

rates would elevate the total likelihood of the branch durations

reduced by stratigraphic data (Fig. 7). However, the Model 1

tree would be less prone to doing this than the Model 3 simply

because regional lognormal sampling rates reduce the likeli-

hoods of gaps much less than global uniform rates do. Note

that any stratigraphic likelihood model would be less biased

towards supporting local rate heterogeneity than the Model 5

(minimum divergences) tree. This is not just because all strati-

graphic likelihood trees extend many branches with near-zero

durations given minimum divergences, but also because the

stratigraphic likelihoods reduce the durations of branches link-

ing clades in some cases.

FUTURE DIRECTIONS

Relaxing assumptions about continuous distributions of

localities over time

When assessing sampling rates per-million years, our approach

currently assumes that localities are continuously distributed

throughout an interval. This is rarely, if ever, true. However,

biochronological techniques for ordinating localities based on

constituent species combined with some absolute dates offer

the potential for very high resolution biochronological place-

ment of localities (Alroy 1994; Sadler, Kemple & Kooser

2003). This can allow us to generate different per-myr and even

per-collection sampling rates within intervals.

Origination and extinction

Another advantage to ordinating collections within strati-

graphic units is that these results will often show that species

durations are less than that of whole stratigraphic intervals

(Alroy 1996). This is important because our estimates of

sampling rates do not take into account turnover within strati-

graphic intervals: instead, we assume that any taxon present in

an interval was present throughout the entire interval. For taxa

with true durations less than that of the entire interval, many

collections currently tallied as ‘gaps’ actually come from before

or after the species’ lifetimes: and this biases our method

towards underestimating sampling rates. An obvious next step

in this sort of approach is to add origination and extinction

parameters (Weiss & Marshall 1999). However, we stress that

the approach as done here should be ‘conservative’ with

respect to rejecting null hypotheses because of long strati-

graphic gaps.

Conclusions

The existence of large databases of fossil occurrences such as

the PaleoDB andNOWallows us to assess sampling rates over

time and space in greater detail than ever before. Here, we

show that in the case of fossil mammals at least, lognormal dis-

tributions of sampling rates among taxa prevail. Moreover,

these distributions vary considerably over time and among

contemporaneous geographic areas. Many interesting macro-

evolutionary hypotheses concerning rates of morphological

change, speciation patterns and turnover events differ in the

gaps they require between observed stratigraphic ranges and

either divergence times or extinction times. Combining these

models and these data should allow evolutionary biologists to

more fully exploit the fossil record as a tool for corroborating

or contradicting these hypotheses while at the same time allow-

ing for the uncertainties inherent to the fossil record.

Acknowledgements

We thank the special issue editors for their invitation and their subsequent for-

bearance.We also thank D. Bapst and P. D. Polly for very insightful reviews that

(hopefully) led to clarification of our primary goals and concepts. For discussions

about the appropriate distributions to model sampling rates, we thank J. Alroy.

This represents PaleoDB Publication No. 182. For those data, we thank in

particular J. Alroy,K. Behrensmeyer,M.Uhen,A.Turner, L. v. d.HoekOstende

andM. Carrano. Occurrence data and a C program for estimating per-collection

sampling rates are available at the Dryad Data repository (http://datadryad.org;

doi:10.5061/dryad.3b87j).

References

Alroy, J. (1994) Appearance event ordination: a new biochronologic method.

Paleobiology, 20, 191–207.Alroy, J. (1996) Constant extinction, constrained diversification, and uncoordi-

nated stasis in North American mammals. Palaeogeography, Palaeoclimatolo-

gy, Palaeoecology, 127, 285–311.Alroy, J. (1998) Diachrony of mammalian appearance events: implications for

biochronology.Geology, 26, 23–26.Alroy, J. (1999) The fossil record of North American mammals: evidence for a

Paleocene evolutionary radiation. Systematic Biology, 48, 107–118.Alroy, J. (2000) New methods for quantifying macroevolutionary patterns and

processes.Paleobiology, 26, 707–733.Alroy, J.,Marshall, C.R., Bambach, R.K., Bezusko, K., Foote,M., F€ursich, F.T.

et al. (2001) Effects of sampling standardization on estimates of Phanerozoic

marine diversity.Proceedings of the National Academy of Sciences of the United

States of America, 98, 6261–6266.Behrensmeyer, A.K. & Kidwell, S. (1985) Taphonomy’s contribution to paleobi-

ology.Paleobiology, 11, 105–119.Bottjer, D.J. & Jablonski, D. (1988) Paleoenvironmental patterns in the evolution

of post-Paleozoic benthicmarine invertebrates.Palaios, 3, 540–560.Buzas, M.A., Koch, C.F., Culver, S.J. & Sohl, N.F. (1982) On the distribution of

species occurrence.Paleobiology, 8, 143–150.Darwin,C. (1859)TheOrigin of Species, 6th edn. JohnMurray, London.

Drummond, A.J., Ho, S.Y.W., Phillips, M.J. & Rambaut, A. (2006) Relaxed

phylogenetics and datingwith confidence.PLoSBiology, 4, e88.

Felsenstein, J. (1973) Maximum-likelihood and minimum-steps methods for esti-

mating evolutionary trees from data on discrete characters. Systematic Zool-

ogy, 22, 240–249.Fisher, R.A., Corbet, A.S. & Williams, C.B. (1943) The relation between the

number of species and the number of individuals in a randomsample of an ani-

mal population. Journal of Animal Ecology, 12, 42–48.Foote, M. (1993) Discordance and concordance between morphological and tax-

onomic diversity.Paleobiology, 19, 185–204.Foote, M. (1996) On the probability of ancestors in the fossil record. Paleobiol-

ogy, 22, 141–151.Foote, M. (1997) Estimating taxonomic durations and preservation probability.

Paleobiology, 23, 278–300.Foote, M. (2001) Inferring temporal patterns of preservation, origination,

and extinction from taxonomic survivorship analysis. Paleobiology, 27,

602–630.Foote, M. & Raup, D.M. (1996) Fossil preservation and the stratigraphic ranges

of taxa.Paleobiology, 22, 121–140.Foote, M. & Sepkoski, J.J. Jr (1999) Absolute measures of the completeness of

the fossil record.Nature, 398, 415–417.Fortelius, M., Werdelin, L., Andrews, P., Bernor, R.L., Gentry, A., Humphrey,

L., Mittmann, W. & Viranta, S. (1996) Provinciality, diversity, turnover and

paleoecology in land mammal faunas of the later Miocene of western Eurasia.

TheEvolution ofWestern EurasianNeogeneMammal Faunas (edsR.L. Bernor,

V. Fahlbusch&W.Mittmann), pp. 414–448. ColumbiaUniversity Press, New

York,NewYork,USA.

Fortelius, M., Eronen, J., Jernvall, J., Liu, L., Pushkina, D., Rinne, J. et al.

(2002) Fossil mammals resolve regional patterns of Eurasian climate change

over 20million years.Evolutionary Ecology Research, 4, 1005–1016.Gradstein, F., Ogg, J. & Smith, A. (2005) A Geological Times Scale 2004. Cam-

bridgeUniversity Press, Cambridge.

Gray, J.S. (1987) Species-abundance patterns. Organization of Communities Past

and Present (eds J.H.R.Gee&P.S. Gillier), pp. 53–67. Blackwell, Oxford.Huelsenbeck, J.P., Larget, B. & Swofford, D. (2000) A compound Poisson Pro-

cess for relaxing themolecular clock.Genetics, 154, 1879–1892.Huelsenbeck, J.P. & Rannala, B. (1997) Maximum likelihood estimation of

topology and node times using stratigraphic data. Paleobiology, 23, 174–180.

Lewis, P.O. (2001) A likelihood approach to estimating phylogeny from discrete

morphological character data.Systematic Biology, 50, 913–925.Marshall, C.R. (1995a)Distinguishing between sudden and gradual extinctions in

the fossil record: predicting the position of the iridium anomaly using the

ammonite fossil record on Seymour Island, Antarctica.Geology, 23, 731–734.Marshall, C.R. (1995b) Stratigraphy, the true order of species’ originations and

extinctions, and testing ancestor-descendant hypotheses among Caribbean

bryozoans. New Approaches to Studying Speciation in the Fossil Record (eds

D.H. Erwin & R.L. Anstey), pp. 208–236. Columbia University Press, New

York,NewYork,USA.

Marshall, C.R. (2008) A simple method for bracketing absolute divergence times

on molecular phylogenies using multiple fossil calibration points. The Ameri-

canNaturalist, 171, 726–742.May, R.M. (1975) Patterns of species abundance and diversity. Ecology and Evo-

lution of Communities (edsM.L. Cody& J.M.Diamond), pp. 87–120. The Bel-knap Press ofHarvardUniversity Press, Cambridge.

Montroll, E.W. & Shlesinger, M.F. (1982) On 1/f noise and other distributions

with long tails. Proceedings of the National Academy of Sciences, 79, 3380–3383.

Motomura, I. (1932) A statistical treatment of associations.ZoologicalMagazine,

Tokyo, 44, 379–383.Preston, W.H. (1948) The commonness and rarity of species. Ecology, 29, 254–

Pyron, R.A. (2010) A likelihood method for assessing molecular divergence time

estimates and the placement of fossil calibrations. Systematic Biology, 59, 185–194.

Pyron, R.A. (2011) Divergence time estimation using fossils as terminal taxa and

the origins of Lissamphibia.Systematic Biology, 60, 466–481.Raia, P., Carotenuto, F., Passaro, F., Fulgione, D. & Fortelius, M. (2012) Eco-

logical specialization in fossil mammals explains Cope’s Rule. The American

Naturalist, 179, 328–337.Raup, D.M. (1972) Taxonomic diversity during the Phanerozoic. Science, 177,

1065–1071.Ree, R.H. (2005) Detecting the historical signature of key innovations using sto-

chasticmodelsof character evolutionandcladogenesis.Evolution,59, 257–265.Ree, R.H. & Smith, S.A. (2008) Maximum likelihood inference of geographic

range evolution by dispersal, local extinction, and cladogenesis. Systematic

Biology, 57, 4–14.Ronquist, F., Klopfstein, S., Vilhelmsen, L., Schulmeister, S., Murray, D.L. &

Rasnitsyn, A.P. (2012) A Total-Evidence approach to dating with fossils,

applied to the early radiation of the Hymenoptera. Systematic Biology, 61,

973–999.Ruta, M., Wagner, P.J. & Coates, M.I. (2006) Evolutionary patterns in early tet-

rapods. I. Rapid initial diversification by decrease in rates of character change.

Proceedings of the Royal Society of London, Series B. Biological Sciences, 273,

2107–2111.Sadler, P.M., Kemple, W.G. & Kooser, M.A. (2003) CONOP9 Programs for

Solving the Stratigraphic Correlation and Seriation Problems as Constrained

Optimization. High-Resolution Stratigraphic Approaches in Paleontology (ed.

P.Harries), pp. 461–462. PlenumPress, NewYork,NewYork, USA.

Sanderson, M.J. (2002) Estimating absolute rates of molecular evolution and

divergence times: a penalized likelihood approach.Molecular Biology and Evo-

lution, 19, 101–109.Sepkoski, J.J. Jr (1975) Stratigraphic biases in the analysis of taxonomic survivor-

ship.Paleobiology, 1, 343–355.Sepkoski, J.J. Jr (2002) A compendium of fossil marine animal genera. Bulletins

of American Paleontology, 363, 1–563.Signor, P.W. & Lipps, J.H. (1982) Sampling bias, gradual extinction patterns and

catastrophes in the fossil record. Geological Society of America Special Paper,

190, 291–296.Smith, A.B. (2001) Large-scale heterogeneity of the fossil record: implications for

Phanerozoic biodiversity studies. Philosophical Transactions of the Royal Soci-

ety of London Series B, 356, 351–367.

Sugiura, N. (1978) Further analysis of the data by Akaike’s information criterion

and the finite corrections. Communications in Statistics – Theory andMethods,

A7, 13–26.Uhen, M.D., Barnosky, A.D., Bills, B., Blois, J., Carrano, M.T., Carrasco, M.A.

et al. (2013) From card catalogs to computers: databases in vertebrate paleon-

tology. Journal of Vertebrate Paleontology, 33, 13–28.Wagner, P.J. (1995a) Stratigraphic tests of cladistic hypotheses. Paleobiology, 21,

153–178.Wagner, P.J. (1995b) Testing evolutionary constraint hypotheses with early

Paleozoic gastropods.Paleobiology, 21, 248–272.Wagner, P.J. (1997) Patterns of morphologic diversification among the Rostro-

conchia.Paleobiology, 23, 115–150.Wagner, P.J. (2000a) Likelihood tests of hypothesized durations: determining

and accommodating biasing factors.Paleobiology, 26, 431–449.Wagner, P.J. (2000b) Phylogenetic analyses and the fossil record: tests and infer-

ences, hypotheses and models. Paleobiology Memoir (eds D.H. Erwin & S.L.

Wing), pp. 341–371. Paleontological Society, Deep time – Paleobiology’s

perspective.

Wagner, P.J. (2000c) The quality of the fossil record and the accuracy of

phylogenetic inferences about sampling and diversity. Systematic Biology, 49,

65–86.Wagner, P.J. (2012) Modelling rate distributions using character compatibility:

implications for morphological evolution among fossil invertebrates. Biology

Letters, 8, 143–146.Wagner, P.J., Kosnik,M.A. & Lidgard, S. (2006) Abundance distributions imply

elevated complexity of post-Paleozoic marine ecosystems. Science, 314, 1289–1292.

Wagner, P.J. & Marcot, J.D. (2010) Probabilistic phylogenetic inference in the

fossil record: current and future applications.QuantitativeMethods in Paleobi-

ology (eds J. Alroy & G. Hunt), pp. 195–217. Paleontological Society, New

Haven, Connecticut, USA.

Weiss, R.E. & Marshall, C.R. (1999) The uncertainty in the true end point of a

fossil’s stratigraphic ranges when stratigraphic sections are sampled discretely.

Mathematical Geology, 31, 435–453.Wesley-Hunt, G.D. (2005) The morphological diversification of carnivores in

NorthAmerica.Paleobiology, 31, 35–55.Wesley-Hunt, G.D. & Flynn, J.J. (2005) Phylogeny of the Carnivora: basal rela-

tionships among the carnivoramorphans, and assessment of the position of

“Miacoidea” relative to crown-clade Carnivora. Journal of Systematic Palae-

ontology, 3, 1–28.Yang, Z. (1994) Maximum likelihood phylogenetic estimation from DNA

sequences with variable rates over sites: approximate methods. Journal of

Molecular Evolution, 39, 306–314.Yang, Z. (1996) Maximum-likelihood models for combined analyses of multiple

sequence data. Journal ofMolecular Evolution, 42, 587–596

Received 25 January 2013; accepted 10 June 2013

Handling Editor: GrahamSlater

Supporting Information

Additional Supporting Information may be found in the online version

of this article.

Appendix S1.Additional results andmethodological discussion.

Fig. S1. Lognormal distribution of per-collection sampling rates (Rc)

for 100 taxa.

Fig. S2. Probabilities of X occurrences for the four species with corre-

sponding colors in Fig. 1.

Fig. S3.Expected species withX occurrences.

Fig. S4. Expected rank order plot of numbers of occurrences against

ranked taxa.

Fig. S5.Distributions of per-collection preservation rates forMeso-Ce-

nozoicmammals, broken down by basic geographic units.

Fig. S6.Estimated probabilities that unsampled lineages resided in Eur-

asia (red) orNorthAmerica (blue).

Fig. S7. Alternate phylogenies with the same cladistic topology using

differentmethods to calibrate branch durations and divergence times.

Table S1.Parameters from the best hypothesis of eachmodel.

Table S2.AICc for the best representatives of the three rate distribution

models considered for per-collection sampling rates.

Table S3.Parameters from the best hypothesis of eachmodel.

Table S4. Stratigraphic likelihoods given gaps hypothesized by carniv-

oramorphan phylogeny (Fig. 6).

SOM: Sampling Rate Distributions S1 Wagner & Marcot

Supplementary Online Material

Assessing Model Distributions

We assess model distributions using the expected numbers of taxa with 1…N occurrences given

N collections. Here we will

describe how we generate

these expectations when

testing a hypothesized log-

normal distribution rate of

per-collection sampling

rates (Rc) for 100 species

with Rcmodal = 0.01 and the

magnitude parameter, m, =

2.5. In other words, the

geometric mean of the rates

is ln(0.1) and the standard

deviation on the log-transformed rates is ln(2.75). In our example, we will focus on four taxa

shown in separate colors: the three species with the highest Rc and the species with the lowest

Now, assume that there are 200 collections from which these species might be sampled. The

single most probable number of occurrences for any species i is the integer closest to 200 x Rci.

However, there is considerable variation simply due to binomial error. Thus, although it is most

probable that we will sample Species 1 (Rc1 = 0.1056) in 21 collections, the exact probability of

this outcome is reasonably low (p = 0.092; Fig. S2). The probabilities of 20 or 21 finds are

Fig. S1. Lognormal distribution of per-collection sampling rates (Rc) for 100 taxa. Species 1, 2, 3 and 100 are set in separate colors to match Fig. S2.

nearly as great, and there is

even a very remote probability

of 0 occurrences (p = 2x10-10).

As Rci decreases, the

probability curves shift to the

left. Thus, when we look at

Species 100 (Rc = 0.0009), the

single most probable outcome

is that we fail to sample it

altogether (p=0.82).

Another crucial point is the

heavy overlap among

probability curves. Thus, the probability that Species 2 might actually have more occurrences

than Species 1 is non-trivial: and the probability of sampling taxa out-of-order increases as we

approach the mode of the distribution simply because the difference between Rci and Rci+1

becomes very small. What is more relevant to our purposes is that the expected number (and

later expected frequency) of species with 21 occurrences is not based simply on having one

taxon with 21 expected occurrences, but is instead the sum of probabilities of 21 occurrences

over all 100 species. Thus, we do not assume that observed ranks of occurrences match the

“true” ranks of sampling rates. Doing this now gives us an expected number of species with n

occurrences (Figure S3). We then can transform this to a probability density function by setting

the area under the curve = 1.0 (i.e., by dividing by the expected number of sample species). We

Fig. S2. Probabilities of X occurrences for the 4 species with corresponding colors in Figure 1. The expected number of taxa with X finds is the sum of the height of each curve at X.

now can evaluate the exact

probability of the observed

numbers of taxa with N

occurrences using multinomial

probability (e.g., Wagner, Kosnik

& Lidgard 2006).

Finally, it is worth noting that

this approach explicitly accounts

for the tendency of sampling to

distort the shapes of distributions

(McGill 2003). Given 200

collections, a rank-order plot of

numbers of occurrences (Fig. 4S)

does not look like the original

lognormal distribution: indeed, it

looks more like a geometric or a

log-series! This simply relates to

sampling error. The most

obvious is that with only 200

collections, we simply do not

expect to sample all 100 species,

and thus we do not have the

chance to see the distinctive tail

Fig. S3. Expected species with X occurrences. This sums the probability curves for the four species in Fig. S2 as well as unillustrated probability curves for 96 other species.

Fig. S4. Expected rank order plot of numbers of occurrences against ranked taxa. Note the “distortion” relative to Fig. S1 due both to sample size and the fact that occurrences must be integer values.

of a lognormal. The second is that occurrences are integers, resulting in substantial numbers of

taxa with the same number of occurrences. Indeed, if we replot Figure S3 as log-occurrences

(i.e., log the X-axis), then one begins to see a normal distribution that is truncated in the middle

(see, e.g., Preston 1948).

Additional Results

Best Distributions of Sampling Rates over Time

Each model is represented by the particular hypothesis maximizing the probability of the

observed data (Table S1). For each of these, we calculated the Akaike’s Modified Information

criterion (Sugiura 1978). This is:

AICc = !2 lnL + 2"b" Sn(Sn ! b!1)

Table S1. Parameters from the best hypothesis of each model. Rc gives per-collection sampling

rate, with Rc1 given the rate for the most common species and Rcmid given the sampling rate for the

modal species (i.e., the geometric mean sampling rate). S gives hypothesized richness. δ is the

geometric decay rate, giving Rci+1/Rci. m gives one standard deviation in the magnitude of

sampling rates.

Uniform Geometric Lognormal Stage/Substage(s) Rc Rc1 δ RcMid m S Campanian 0.025703 0.105956 0.965452 0.010208 3.08459 119 Maastrichtian 0.042745 0.606912 0.858873 0.000774 14.24028 255 Danian 0.018120 0.105374 0.985332 0.003330 4.72116 484 Selandian-Thanetian 0.015373 0.309269 0.981642 0.001186 7.45185 642 Ypresian 0.011701 0.408818 0.972632 0.000003 50.63211 1133 Lutetian 0.010704 0.163142 0.974819 0.000194 9.75773 2505 Bartonian-Priabonian 0.007700 0.032842 0.995825 0.001091 4.74733 1746 Rupelian 0.007315 0.222492 0.972557 0.000220 7.75437 2225 Chattian 0.009195 0.030295 0.992255 0.001096 4.47510 1462 Aquitanian-Burdigalian 0.004416 0.038026 0.996594 0.000318 7.08827 3386 Langhian-Serravallian 0.004994 0.059207 0.994877 0.000277 8.19477 3038 Tortonian–Messinian 0.003725 0.033635 0.996997 0.000196 7.46104 4870 Zanclean 0.006269 0.026506 0.994855 0.000423 5.80550 3432 Piacenzian–Gelasian 0.005216 0.041024 0.995743 0.000316 6.98224 3776

where lnL is the log-

likelihood of the best

hypothesis for each

model, b is the number

of parameters for that

model (i.e., the number

of columns in Table S1),

and Sn is the number of

taxa (i.e., the number of

data points). Note that

this precludes

calculating AICc from our hypothetical “best” model simply because the denominator is -1 when

there are as many parameters as data points. In many cases, workers might wish to use Akaike’s

weights to assess the “significance” of these differences. Here, the differences are sufficiently

large that this simply confirms the obvious, i.e., that the lognormal is vastly superior to the

geometric or uniform distributions for modeling per-collection sampling rates.

Best Sampling Rate Distributions over Geography

We estimated the best lognormal distributions for sampling rates per collection for three basic

geographic units separately (Fig. S5). We use these to infer sampling rates per myr (Fig. 5) by

assuming that collections are uniformly distributed over time within stratigraphic units. Note

that we do not calculate distributions for intervals with fewer than 50 taphonomically controlled

collections, as these rarely provide the statistical power to distinguish among alternate

Table S2. AICc for the best representatives of the three rate distribution

models considered for per-collection sampling rates. See Table 2 for

additional information. Note that AICc cannot be calculated for the

theoretical “best” hypothesis.

Stage/Substage(s) Uniform Geometric Lognormal Campanian 509.3 388.4 373.1 Maastrichtian 1806.5 468.6 438.2 Danian 2565.8 1454.7 1399.0 Selandian–Thanetian 3861.8 1866.8 1586.5 Ypresian 17369.9 3333.6 2998.9 Lutetian 3559.3 1892.7 1639.3 Bartonian–Priabonian 4155.6 2977.2 2805.3 Rupelian 4374.7 2184.7 1742.9 Chattian 1507.5 1239.6 1145.2 Aquitanian–Burdigalian 14113.7 7102.6 6396.9 Langhian–Serravallian 17709.4 6800.2 6194.5 Tortonian–Messinian 15879.9 7815.9 6979.1 Zanclean 3978.3 2950.7 2643.2 Piacenzian–Gelasian 10903.9 5901.6 5314.7

distributions. The parameters for the relevant lognormals are given in Table S3.

Figure S5. Distributions of per-collection preservation rates for Meso-Cenozoic mammals, broken down by basic geographic units.

Table S3. Parameters from the best hypothesis of each model. RcMid gives the sampling rate for

the modal species. m gives one standard deviation in magnitude of sampling rates. Dots give

intervals with insufficient data to estimate rates.

North America Eurasia Africa Stage RcMid m S RcMid m S RcMid m S Campanian 0.01866 2.996 84 0.02064 2.412 35 • • • Maastrichtian 0.00060 15.884 271 • • • • • • Danian 0.00470 4.204 399 • • • • • • Selandian–Thanetian 0.00436 5.683 342 0.01088 2.431 251 • • • Ypresian 0.00186 8.966 398 0.00410 2.489 2077 • • • Lutetian 0.00183 6.872 395 0.00054 6.103 2390 • • • Bartonian–Priabonian 0.00363 4.426 554 0.00351 3.675 708 • • • Rupelian 0.00189 5.824 371 0.00279 3.558 964 • • • Chattian 0.00251 4.715 552 0.00278 4.032 751 • • • Aquitanian–Burdigalian 0.00152 4.768 735 0.00032 9.752 1174 0.00565 6.299 248 Langhian–Serravallian 0.00255 4.661 512 0.00018 12.420 1341 • • • Tortonian–Messinian 0.01192 2.881 271 0.00022 8.397 3716 0.00324 3.884 725 Zanclean • • • 0.00106 4.845 1711 0.00138 6.931 819 Piacenzian–Gelasian 0.00928 3.245 350 0.00042 7.034 2714 0.00005 21.757 676

Carnivoramorphan Analysis

Analysis of morphological data

We use morphological data for the Carnivoramorpha originally published by Wesley-Hunt and

Flynn (2005). Their analyses include 39 taxa and 113 characters. However, many of these later

taxa are composite codings for derived clades of the Carnivora. To keep the analyses at the

species level, we reduced it to the 21 earliest appearing species and the 86 characters that vary

among those species. We used inverse modeling tests on the compatibility structure of these

characters to establish a base rate of 2.82 changes per character for the entire tree (i.e., about

0.028 changes per myr), with those rates showing lognormal variation in which one standard

deviation equals a magnitude of 3.41 (Wagner 2012). These tests suggest that there are an

additional 17 characters that should be invariant among these species. Thus, the likelihood

analyses included 103 total characters with 17 invariant ones. We converted this to per-million-

year rates by using the sampling rates to estimate the amount of “missing” evolutionary time (see

Foote 1996).

The first appearances and total stratigraphic ranges of each taxon are based on stratigraphic

data from the Paleobiology Database and New and Old World Database. These ranges include

occurrences for species deemed to be junior synonyms in the current taxonomic tables of the

PBDB. Note that we replaced the stratigraphic data for the early mustelid Zodiolestes

daimonelixensis Riggs 1942 with that of Promartes gemmarosae Loomis 1932, the earliest

known mustelid. However, we assumed the same morphological characters for P. gemmarosae

as Welsey-Hunt and Flynn used for Z. daimonelixensis.

Biogeographic distributions of unsampled lineages

Given the phylogeny proposed by Wesley-Hunt and Flynn (2005), early carnivoramorphans were

exclusive to North America, with one derived clade later emigrating to Eurasia. Instead of

assuming this, we

allowed uncertainty

in the geographic

“state” of

unsampled lineages

in the way allow

for uncertainty in

morphological

states along those

same branches. We

did this assuming a

base rate where there is only one expected change, which means that this rate was very low (i.e.,

0.011 changes per myr). This really affected only that portion of the tree where some transition

necessarily occurred (Fig. S6). The uncertainty is greatest on the branches spanning the longest

amounts of time, but this is in keeping with the idea that there is greater opportunity for “hidden”

changes and reversals along such branches.

Note that the original estimates derive likelihoods for Eurasia or North America as an

ancestral “state.” We convert those to probabilities by dividing each likelihood by the sum of the

two likelihoods. These numbers then are used to weight the likelihoods of gaps given the North

Figure S6. Estimated probabilities that unsampled lineages resided in Eurasia (red) or North America (blue). Bars sum to 1.0.

American and Eurasian stratigraphic records and sampling rates.

Divergence Time Likelihoods for Carnivoramorphans

We use the phylogenetic relationships among carnivoramorphan species suggested by Wesley-

Hunt and Flynn (2005). For each branch on that tree, we calculate the likelihood of the gap

implicit to that branch given both the morphological data and the stratigraphic data, finding the

branch length that maximizes the probability of both data sets. For branches linking observed

species to nodes, this is the difference between the hypothesized divergence and the species’ first

occurrence. In cases of unsampled hypothetical ancestors, this is the difference between the

hypothesized divergence and the hypothesized divergences of the daughter taxa. For each

lineage, we use the probabilities of the lineage being present in either North America or Eurasia

to weight the stratigraphic likelihood given the sampling rates and numbers of collections from

both areas (Figs. 1, 5). (We do not consider the possibility that a species was present in both.)

Thus, even within clades of species known from only one area, there is some small probability In

other words, the stratigraphic likelihood is:

(P[Present in Eurasia] × L[Gap in Eurasia]) + (P[Present in N. America] × L[Gap in America])

The details for each branch are given in Table S4.

Alternate phylogenies for Carnivoramorphans

The information above corresponds to our Model 1 tree. Here we illustrate our Model 2

(Mk+global lognormal sampling rates), Model 3 (Mk+global uniform sampling rates), Model 4

(Mk only) and Model 5 (minimum divergence times) trees.

Table S4. Stratigraphic likelihoods given gaps hypothesized by carnivoramorphan phylogeny (Fig. 6). Gap likelihoods = P[0 finds] assuming a continuous distribution of pertinent localities and lognormal distributions of sampling rates (Fig. 5). Global L[Gap] = ∑(P[Present]×L[Gap]). “X+Y” = 2-taxon clade of X and Y; “X-Y” = 3+ taxon clade including X and Y. Gap Global Eurasia North America Taxon (Myr) L[Gap] P[Present] L[Gap] P[Present] L[Gap] M. cogitus - O. herpestoides 6.7 0.2493 0.0695 0.6508 0.9305 0.2193 D. vancleveae - V. minutus 1.8 0.6007 0.0206 0.8459 0.9794 0.5955 M. cogitus - V. profectus 0.4 0.7342 0.0056 0.9832 0.9944 0.7328 Oodectes herpestoides 1.2 0.5995 0.0141 0.9572 0.9859 0.5944 M. cogitus - M. parvivorus 0.4 0.7342 0.0055 0.9832 0.9944 0.7328 V. ovatus - V. profectus 1.8 0.5475 0.0215 0.9384 0.9785 0.5389 M. cogitus - M. sylvestris 3.4 0.4587 0.0371 0.8910 0.9629 0.4421 Miacis parvivorus 0.1 0.8556 0.0022 0.9932 0.9978 0.8553 M. cogitus - T. robustus 0.1 0.8557 0.0030 0.9932 0.9969 0.8553 P. wilsoni + M. sylvestris 1.0 0.7218 0.0122 0.9791 0.9878 0.7186 M. cogitus - H. antiquus 0.1 0.8572 0.0135 0.9932 0.9865 0.8553 Tapocyon robustus 5.6 0.5504 0.0589 0.9248 0.9411 0.5270 M. cogitus - P. gemmorsae 2.0 0.7236 0.0245 0.9702 0.9755 0.7175 D. felina - H. antiquus 10.1 0.4667 0.3149 0.7103 0.6851 0.3547 M. cogitus + D. lambei 4.0 0.6774 0.0433 0.9475 0.9567 0.6652 P. gemmorsae - O. macdonaldi 3.9 0.6812 0.0424 0.9487 0.9576 0.6693 Miacis cognitus 5.5 0.4607 0.0579 0.6659 0.9421 0.4480 Daphoenus lambei 0.3 0.9422 0.0044 0.9945 0.9956 0.9420 O. macdonaldi + H. gregarius 0.3 0.9422 0.0047 0.9945 0.9952 0.9420 Promartes gemmarosae 6.9 0.4058 0.0713 0.5948 0.9287 0.3913 Hesperocyon gregarius 0.1 0.9699 0.0022 0.9973 0.9978 0.9699 Otarocyon macdonaldi 15.6 0.1687 0.1459 0.3405 0.8541 0.1394 D. felina + H. mentalis 2.1 0.6756 0.0237 0.7654 0.9763 0.6734 H. antiquus - S. brevidens 5.9 0.5523 0.9380 0.5641 0.0620 0.3744 Dinictis felina 0.9 0.8133 0.0109 0.8773 0.9892 0.8125 Hoplophoneus mentalis 0.6 0.8602 0.0076 0.9108 0.9923 0.8598 P. lamandini - S. brevidens 0.1 0.9877 0.9975 0.9879 0.0025 0.9396 Herpestides antiquus 9.9 0.5068 0.9013 0.5267 0.0986 0.3252 Palaeoprionodon lamandini 0.1 0.9878 0.9978 0.9879 0.0022 0.9396 P. lemanensis + S. brevidens 10.7 0.4670 0.8943 0.4886 0.1057 0.2844 Stenogale brevidens 0.1 0.9801 0.9978 0.9801 0.0022 0.9641 Proailurus lemanensis 0.6 0.9346 0.9923 0.9350 0.0076 0.8859 Prohesperocyon wilsoni 11.2 0.3471 0.1100 0.5969 0.8900 0.3163 Miacis sylvestris 0.1 0.9699 0.0022 0.9973 0.9978 0.9699 Vulpavus ovatus 2.5 0.4990 0.0278 0.9224 0.9722 0.4869 Vulpavus profectus 3.4 0.4507 0.0370 0.9113 0.9630 0.4330 Didymictis vancleveae 6.6 0.2614 0.0685 0.7120 0.9316 0.2282 P. schaffi - V. gracilis 0.1 0.9173 0.0023 0.9819 0.9977 0.9172 Protictis schaffi 0.1 0.9174 0.0022 0.9819 0.9978 0.9172 V. minutus + V. gracilis 2.6 0.4192 0.0289 0.8151 0.9711 0.4074 Viverravus minutus 2.8 0.4861 0.0309 0.9083 0.9691 0.4727 Viverravus gracilis 0.1 0.8556 0.0022 0.9932 0.9978 0.8553

Figure S7. Alternate phylogenies with the same cladistic topology using different methods to calibrate branch durations and divergence times. A. Mk model + stratigraphic likelihood using global lognormal sampling rates (= Model 2 in main text). B. Mk model + stratigraphic likelihood using global uniform sampling rates (= Model 3 in main text).

Figure S7 (con’t). C. Mk model along (= Model 4 in main text). This is essentially a “morphological clock” that treats the probability of a gap as 1.0. D. Minimum divegence time (= Model 5 in main text). This is a traditional paleontological time-calibrated phylogeny and basically assigns branch durations of 0 preceding the oldest species in a clade or when sister taxa appear simultaneously.

References and Notes

Foote, M. (1996) On the probability of ancestors in the fossil record. Paleobiology, 22, 141 -

McGill, B.J. (2003) Does Mother Nature really prefer rare species or are log-left-skewed SADs a

sampling artefact? Ecology Letters, 6, 766 - 773.

Preston, W.H. (1948) The commonness and rarity of species. Ecology, 29, 254 - 283.

Sugiura, N. (1978) Further analysis of the data by Akaike’s information criterion and the finite

corrections. Communications in Statistics - Theory and Methods, A7, 13 - 26.

Wagner, P.J. (2012) Modelling rate distributions using character compatibility: implications for

morphological evolution among fossil invertebrates. Biology Letters, 8, 143 - 146.

Wagner, P.J., Kosnik, M.A. & Lidgard, S. (2006) Abundance distributions imply elevated

complexity of post-Paleozoic marine ecosystems. Science, 314, 1289 - 1292.

Wesley-Hunt, G.D. & Flynn, J.J. (2005) Phylogeny of the Carnivora: basal relationships among

the carnivoramorphans, and assessment of the position of “Miacoidea” relative to crown-

clade Carnivora. Journal of Systematic Palaeontology, 3, 1 - 28.

Modelling distributions of fossil sampling rates over time, space and taxa: assessment and...

Documents

Transcript of Modelling distributions of fossil sampling rates over time, space and taxa: assessment and...

Population status of five Hawaiian endemic fern taxa within ...

INSCRIÇÕES COM PAGAMENTO DE TAXA DE INSCRIÇÃO

Sawfly taxa (Hymenoptera, Symphyta) described by Edward ...

Taxa Siscomex - Sindicato dos Despachantes Aduaneiros de ...

Fossilized bioelectric wire – the trace fossil Trichichnus

A phylogeny of Ramariopsis and allied taxa

2006 Harrison - Yuanmou fossil hominoids

Two new taxa from Central Anatolia, Turkey

QUALITY EARLY HOMININ FOSSIL RECORD - CORE

Fossil Fuels - Alba Sud

JournafloffThreatenedTaxa - Journal of Threatened Taxa

Luis Mariano Vidal (1842-1922) rudist taxa revisited - RACO

With Long-Term Fossil Commitments, Colorado Electricity ...

How Fossil Fuels Effect the Environment

Best practice in seabed image analysis for determining taxa ...

SECTION 1. SYSTEMATICS, FOSSIL RECORD, AND ...

Stratigraphic use of fossil

Silica Use Through Time: Macroevolutionary Change in the Morphology of the Diatom Fustule

Feeding fossil fuels to the soil

tshinkulu.pdf - Abc Taxa