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CHAPTER 19
Estimating extinction with the fossil
record
Peter J. Wagner and S. Kathleen Lyons
19.1 Introduction
Many ecological and palaeontological studies focus
on extinction. The fossil record is particularly
important for studying long-term patterns in
extinction: although analyses of extant phyloge-
nies can estimate extinction rates (e.g. Alfaro et al.
2009) and even suggest mass extinctions (e.g. Crisp
& Cook 2009), they cannot imply trilobites ever
existed or that sphenodonts (now represented only
by the tuatara) were once as diverse as lepi-
dosaurs (lizards and snakes). However, workers
also use the fossil record to test ideas about the
pace of major extinction events that use meth-
ods similar to those that conservation biology
might use. Here we will review current palaeobi-
ological methods for inferring extinction patterns,
spanning ?traditional? methods using stratigraphic
ranges to methods using more exact information
about distributions of finds within stratigraphic
ranges.
Palaeontological studies of diversity patterns also
focus extensively on standing richness and origi-
nation rates. We do not focus on either of these
parameters for their own sake. However, stand-
ing richness (usually referred to simply as ?diver-
sity? in palaeontological studies) is an important
parameter when discussing extinction as extinc-
tion metrics necessarily rely on changes in stand-
ing richness. Origination rates also can be an
important parameter as the distribution of origi-
nations within time spans (i.e. evenly throughout
the interval vs concentrated in the beginning) have
small but important effects on the predictions of
extinction rate hypotheses. Moreover, palaeonto-
logical techniques for estimating origination rates
are essentially identical to those for estimating
extinction, save that they are done ?in reverse?.
Thus, discussions of how one estimates extinction
rates using fossil data can almost double for dis-
cussion of how one estimates origination from fossil
data.
The other parameter that will be important
throughout this review is preservation (i.e. sam-
pling) rate. Just as ecologists know that incom-
plete sampling affects sampled richness (e.g. Hurl-
bert 1971) and implied extinction (e.g. Solow
1993a), palaeontologists know that incomplete sam-
pling affects the first and last appearances of fos-
sils, both in local sedimentary sections (Signor &
Lipps 1982) and globally (Sepkoski 1975). Work-
ers have expressed concern that major extinctions
are exaggerated by or possibly even illusions of
intervals of poor preservation (Raup 1979; Smith
et al. 2001; Peters & Foote 2002). Because sam-
pling intensity is of interest to both fields, we shall
also discuss how palaeontological studies address
sampling when estimating diversity and diversity
dynamics.
19.2 State of the field
19.2.1 Basic metrics
Palaeontological studies of richness and diversity
dynamics (i.e. extinction and origination) date back
to the 19th century (Phillips 1860). Traditional
palaeontological studies use synoptic databases (e.g.
Sepkoski 1982, 2002), which catalogue the first
and last appearances of taxa. These almost always
are binned into chronostratigraphic units: typically
stages and substages, but sometimes as fine as
265
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266 BIOLOGICAL DIVERS ITY
Ti
m
e/
St
ra
ti
gr
ap
h
y
I
t3
t2
t1
II V
IVIII
Figure 19.1 Chronostratigraphic ranges of five hypothetical taxa. Each
t gives a separate chronostratigraphic unit, with t1 the oldest. Dashed
vertical line for V indicates that taxon V is not actually found in unit t2.
Modified from Foote (2000), Fig. 1.
faunal zones. (We use ?intervals? to refer chronos-
tratigraphic units because of the correspondence
between units of time and strata; see Grad-
stein et al. (2005)). For numerous reasons, synop-
tic studies typically use supraspecific taxa, with
genera being the most common taxon since the
mid 1980s.
Fig. 19.1 presents a simple example. We use
last appearances in each unit t to infer rates of
extinction. Two types of metrics commonly are
used: per taxon (pt) and per capita (pc; see Foote
(2000)). Per-taxon metrics measure the propor-
tion of standing richness, S, that have their last
appearance in an interval. Per-taxon extinction
thus is:
?pt =
SL
S
(19.1)
where SL is the number of taxa last appearing in an
interval. In Fig. 19.1, taxa III and IV last appear in
t2, so SL = 2. S is not the sum of taxa sampled in
an interval, but the sum of taxa with chronostrati-
graphic ranges spanning the interval. Taxon V in
t2 illustrates the difference: because it first appears
before t2 and last appears after t2, we assume that
it existed in t2 even though V is not sampled in
t2. This assumption is unsound only if taxon V is
polyphyletic. As the synoptic data for taxa I and V
are identical, we need a more detailed database to
recognize cases like taxon V. Thus, St2 = 5 regardless
of how many ?range-through? taxa (e.g. I and V)
actually are sampled in t2, and ?pt = 0.40. (Per-taxon
origination, ?pt, simply replaces SL with SF, the taxa
first appearing in t2).
Per-capita extinction rate measures the survivor-
ship of taxa present at the outset of an interval.
This is:
?pc = ln
(
Sa
Saz
)
(19.2)
where Sa is the number of taxa crossing the bound-
ary from the prior interval and Saz is the number
of the taxa also crossing through both the base
and the top (i.e. the range-through taxa). Note that
?per capita? here refers not to individuals within a
species, but to taxa as operational units. For t2 in
Fig. 19.1, Sa is 3 (taxa I, III, and V) and Saz is 2
(taxa I and V), giving ?pc = 0.41. The most likely
probability of a taxon becoming extinct in an inter-
val is now 1 ? e?pc . However, this is identical to
?pc if we restrict per-taxon rates to only boundary
crossers.
Chronostratigraphic intervals vary considerably
in temporal length (Gradstein et al. 2005). Accord-
ingly, workers often divide rates by interval length
to get rates per million years instead of rates per
interval. This is appropriate if extinction is dis-
tributed throughout an interval. One expectation
of continuous turnover rates is that longer inter-
vals should have higher extinction rates. However,
Foote (1994) showed that ?raw? extinction metrics
are random with respect to the length of chronos-
tratigraphic units, whereas per million year rates
show a negative correlation. We will return to this
issue below when we summarize tests of pulsed vs
continuous rates.
The distribution of extinctions and originations
plays a role in whether we should prefer per-taxon
or per-capita extinction rates. If extinction and orig-
ination happen throughout an interval, then per-
taxon rates overestimate immediate extinction risk
as well as exaggerating S from any one slice of
time. ?Singletons,? that is those taxa known from
only one interval, such as taxon IV in Fig. 19.1,
exacerbate this problem. Singletons often reflect
differences in research effort and/or available fos-
sils (Raup & Boyajian 1988) and excluding single-
tons greatly reduces the volatility of per-taxon rates
(Alroy 1996). Although per-capita rates are excel-
lent for describing continuous rates, they still can
accurately reflected pulsed extinction.
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EST IMATING EXTINCTION WITH THE FOSSIL RECORD 267
19.2.2 Survivorship curves
Per-capita rates lead conceptually to survivorship
curves. The per-capita rate for any one interval
reflects the proportion of taxa expected to survive
the entire interval into the next interval. Survivor-
ship curves reflect the proportion of taxa that sur-
vive multiple successive intervals. Some survivor-
ship analyses contrast rates among higher taxa in
order to assess whether extinction rates in some
groups differ markedly from those in other groups
(e.g. Simpson 1944, 1953) or whether there are com-
mon patterns among higher taxa (Van Valen 1973,
1979; Raup 1991). Alternatively, cohort analyses
contrast sets of taxa that originate in the same inter-
val (i.e. Raup 1978; Foote 1988) to examine whether
extinction rates change markedly over time. Given
a per-taxon extinction rate, we expect the pro-
portion of taxa that survive N intervals, f (N),
to be:
f (N) = ?pc = e?N?pc (19.3)
Given ?pc = ? ln( 32 ) from our example above, we
expect 66.7% of the taxa to survive 1+ intervals,
44.4% to survive 2+ intervals, etc., and thus 33.3%
of the taxa to have durations of 1 interval, 22.2%
to have durations of 2 intervals, etc. Of course, in
reality this will vary even if extinction rates are con-
stant simply by chance. However, the slope of the
logged frequency of taxa with durations of 1, 2, 3,
etc. intervals plotted against duration approximates
the average ?pc.
Foote (1988) uses Monte Carlo simulations to
demonstrate that survivorship slopes for cohorts of
Cambrian trilobites are significantly steeper than
those for cohorts of Ordovician trilobites. (Subse-
quent revisions to the timescale greatly reduced the
average longevity of Cambrian trilobites, making
the pattern more pronounced than Foote reports;
Bowring et al. 1993). Raup (1991) also uses this
approach to demonstrate that survivorship slopes
for Phanerozoic cohorts fit the expectations of a
general ?kill curve? of extinction rates.
19.2.3 The importance of sampling
For three reasons, equation 19.3 will not predict the
distributions of stratigraphic ranges even if ?pc is
constant over time and among taxa. First, if we
measure stratigraphic ranges in discrete bins (as is
almost always done), then f (N = 1) = e??pc only if
all taxa originate at the very base of intervals. If
origination is continuous throughout intervals, then
we expect (for example) half of the taxa lasting one
half an interval to span from tx to tx+1. Thus, only
the slope from taxa with ranges of 2+ intervals will
reflect ?pc under continuous diversification. Second,
extinction rates do not make predictions about the
ages of extant taxa: instead, the age distribution of
contemporaneous taxa (i.e. a ?backwards survivor-
ship? curve sensu Pease (1988) or a ?prenascence?
curve sensu Foote (2001b)) reflects origination rates
(Foote 2001b).
The third reason is the most critical: extinc-
tion rates make predictions about temporal dura-
tions, but we can observe only stratigraphic ranges.
As preservation rates strongly affect stratigraphic
ranges, ?survivorship? curves using fossil ranges
actually reflect both preservation and extinction
(Sepkoski 1975; Pease 1988; Foote & Raup 1996).
Consider a simple example of two clades where
origination and extinction occur at the beginning
and end of each interval, and for which ??pc = 0.69
(Fig. 19.2). For one clade, the probability of recov-
2 3 4 5
Duration (mpc = 0.69)
Range (R = 0.50)
Range (R = 0.05)
6
Intervals
N
u
m
be
r 
of
 ta
xa
7 8 9 10 111
100
101
102
103
Figure 19.2 The relationship between durations (time from birth to
death) and stratigraphic ranges (= number of chronostratigraphic units
from first to last appearances). This simulation of 5000 taxa uses
?pc = 0.69, which predicts half of the taxa present at the outset of an
interval to go extinct in that interval. Sampled stratigraphic ranges reflect
sampling intensity (R) of 0.5 and 0.05 per interval. Following
palaeontological studies, both durations and ranges are binned into
discrete intervals.
OUP CORRECTED PROOF ? FINAL, 18/10/2010, SPi
268 BIOLOGICAL DIVERS ITY
ery, R, is 0.5 per interval; for the other, R = 0.05
per interval. These illustrate two patterns empha-
sized by Foote and Raup (1996). First, the slopes
of the logged survivorship curves after the first
interval parallel each other quite well. Second,
the proportion of taxa sampled from 1 interval
to those sampled over 2+ intervals increases as R
decreases. At sufficiently low R, all taxa are known
from single finds (and thus single intervals) only.
From this, Foote and Raup derived preservation
intensity as:
RFR =
S2sr=2
Ssr=1 ? Ssr=3
(19.4)
where Ssr=x is the number of taxa with stratigraphic
ranges of x intervals. In our example here, RFR =
0.53 when true R = 0.50 and RFR = 0.09 when true
R = 0.05.
The ability to contrast preservation rates over
time or among taxa obviously interests palaeontolo-
gists. However, subsequent work embraced the fact
that joint hypotheses of preservation and extinction
make exact predictions about distributions of strati-
graphic ranges. This led to a shift from contrasting
slopes of survivorship curves to contrasting the
likelihoods of hypothesized extinction and preser-
vation rates (Solow & Smith 1997; Foote 1997).
Consider a simple system where origination and
extinction occur at the beginning and end of inter-
vals. Extinction here is a per-taxon metric giving
the proportion of taxa expected to become extinct
at the end of each interval. The expected proportion
of taxa with a stratigraphic range (sr) of 1 interval
is:
f [sr = 1] =
w
?
d=sr
[
P[?](1 ? P[?])(d?1) ? d(1 ? R)(d?sr ) R
]
1 ?
w
?
d=1
[
P[?](1 ? P[?])(d?1) ? (1 ? R)d
]
(19.5a)
while the expected proportion of taxa known from
x = 2, . . . , w intervals (where w is the maximum
possible) is:
f [sr ]=
w
?
d=sr
[
P[?](1?P[?])(d?1)?(1+ d? sr )(1?R)(d?sr )
?
sr?2
?
n=0
(
sr ? 2
n
)
(1 ? R)(sr?2?n) R(n+2)
] /
1?
w
?
d=1
[
P[?](1 ? P[?])(d?1)?(1 ? R)d
]
(19.5b)
where d is the true duration and n is the number
of intervals between the first and last interval in
which a taxon might be found (Foote 1997). The
three important recurring terms here are:
?(1 ? ?)d?1: the probability of avoiding extinction
for d ? 1 intervals and becoming extinct in
interval d , where ? is equivalent to ?pc for
boundary crossers only;
(1 ? R)x : the probability of being unsampled for x
intervals;
Ry: the probability of being sampled in y intervals.
The denominator in both equations estimates the
proportion of taxa that are sampled as one minus
the proportion that are unsampled, i.e. the prob-
ability of having duration d times the probability
of going unsampled for d intervals summed over
the entire range of possible durations (w). This is
because we can test hypotheses based only on their
predictions about sampled taxa. The numerator also
sums over the range of possible durations. The
second part of both numerators reflects the num-
ber of ways a stratigraphic range of sr fits within
a duration of d , which is simply 1 + (d ? sr ). For
example, there is only one way to fit sr = 4 within
d = 4(t1, . . . , t4) and two ways to fit sr = 4 within
d = 5(t1, . . . , t4 and t2, . . . , t5), etc. The final term
in equation 19.5a,
sr?2
?
n=0
(
sr ? 2
n
)
(1 ? R)(sr?2?n) R(n+2),
sums the probabilities given R of all of the different
stratigraphic records that match a synoptic range
of sr. All taxa with sr = 4 are found in at least two
intervals (the first and last). However, there are four
possible stratigraphic records consistent with that
synoptic range (Fig. 19.3), therefore the likelihood
function must sum the probabilities of all four pos-
sibilities given any R.
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EST IMATING EXTINCTION WITH THE FOSSIL RECORD 269
Ti
m
e/
st
ra
ti
gr
ap
h
y
I
t3
t4
t2
t1
II IVIII
Figure 19.3 Four exact stratigraphic distributions resulting in a
synoptic range of four intervals.
Foote (1997) presents more complex variations of
equation 19.5 for continuous stratigraphic ranges.
As most palaeontological studies use discrete bins,
we will not cover this in detail. However, we will
return to the effects of continuous ? and ? (origi-
nation) on the expectations for stratigraphic ranges
below. At this point, equation 19.5 gives us an
expected observed survivorship curve given any
combination of ? and R. Multinomial probability
then gives the probability of the data given that
combination. However, this obviously is not wholly
satisfactory for several reasons. In particular, this
approach is useful for contrasting general differ-
ences in average extinction rates between taxa while
controlling for the possibility of different average
preservation rates. Most of the interesting palaeoe-
cological questions concern differences in extinction
rates in particular intervals. We also have ample
evidence that sampling varies over time (e.g. Peters
2006; McGowan & Smith 2008), which might affect
our estimates of extinction.
19.2.4 Relevant studies
Foote (2001c, 2003) modifies the approach outlined
above to estimate interval-by-interval rates of ?, ?
and R. One applies variations of equation 19.5 on
?birth? and ?death? cohorts, that is, how long do taxa
first appearing in an interval last and how old are
the taxa that last appear in an interval? In other
words, instead of analysing a vector of ages at last
appearance, one now analyses a matrix X where
each cell Xi j gives the number of taxa first appear-
ing in ti and last appearing in tj . Now, one can apply
separate ?i and Ri (and ?i ) at each ti on variations of
equation 19.5. The expected number of taxa origi-
nating in ti with d = j now is affected by variation in
? from i to j , and the probability of any given sr ? d
is affected by variation in R from i to j . Because the
best Ri for any ti must accommodate both the back-
wards survivorship curves as well as the forwards
survivorship curve, ?i is relevant for estimating ?i .
The algorithms to find the collection of parameters
maximizing the probability of the observed matrix
are computationally intensive. Foote?s (2003) analy-
sis of Sepkoski?s (2002) compendium of marine gen-
era shows, for example, that the major loss of taxa
at the Big Five mass extinction events (Raup & Sep-
koski 1982) and the high turnover of the Cambrian
taxa (Sepkoski 1979) require high rates of extinction
even in those cases where preservation rates were
low.
Finally, estimates of extinction and preservation
from equation 19.5 effectively assume that taxa
extant are present throughout all intervals within
their durations. This can be true only if both orig-
ination and extinction are pulsed, that is, concen-
trated in some particular interval. However, if they
are continuous, then true durations will span frac-
tions of the first and last intervals for most taxa.
This has two effects (Foote 2005). One, if origina-
tion probabilities are uniform throughout intervals,
then there is a 70% chance that a taxon with a true
duration of 2.7 intervals exists in all or part of four
intervals. Two, the probability of sampling a taxon
in its first or last interval is not R, but 1 ? (1 ? R)x ,
where x is the fraction of the first/last interval in
which the taxon occurs. Given this subtle differ-
ence, Foote (2005) found strong support for pulsed
extinctions and origination in most intervals. One
implication of this is that the tradition of standard-
izing extinction rates by millions of years often is
highly misleading.
19.2.5 Occurrence-based diversity estimates
We require approaches such as outlined in the
prior section when synoptic ranges are our only
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270 BIOLOGICAL DIVERS ITY
A
21
20
19
18
17
16
15
14
13
12
H
or
iz
on 11
10
9
8
7
6
5
4
3
2
1
B C D E F G H I J K L
Figure 19.4 Hypothetical distribution of finds in a section, based
on a single simulation. ?Horizons? represent fossiliferous beds. Black
circles indicate that the species is sampled. White circles indicate that
it is unsampled. Hatched circles represent the first fossiliferous
horizon following the extinction event. Gray ellipses give the observed
stratigraphic ranges. Although all 12 species are extant throughout
the time represented, they appear to go extinct (and originate)
gradually.
data. However, in many cases we have data con-
cerning the distributions of finds within strati-
graphic ranges, such as the Paleobiology Data-
base (http://pbdb.org; see, for example, Alroy et al.
(2001)). This provides us with an independent
means of estimating Ri using range-through taxa
for any ti . In Fig. 19.4, four taxa are known to be
extant during t2 and t3 because they were sampled
before and after those intervals. This leads to a sim-
ple estimate (Connolly & Miller 2001b):
Ri =
Si,obs
Si,az
(19.6)
where Si,az is the number of range-through taxa for
ti (see Fig. 19.1) and Si,obs is the subset of those taxa
that we actually observe. If one uses empirical esti-
mates of Ri instead of solving for Ri at the same
time as ?i (and ?i ), then it becomes much easier to
solve for the diversity parameters.
Foote (2007a) uses the Paleobiology Database to
measure R with range-through taxa and then assess
the effect of preservation on recalibrated extinc-
tion rates. This suggested that extinction is less
continuous through the Phanerozoic than previ-
ously thought, as immediate Ri explains the last
appearances in many low extinction intervals. A
corresponding result is that extinction rates of sub-
sequent intervals increase, with the end result being
that intervals with high extinction given ?face value?
data have even higher extinction after optimizing
for R. In particular, Foote?s analysis suggested that
extinction rates in the intervals preceding the end-
Permian, end-Triassic, end-Ordovician, and end-
Devonian were lower than suggested by the ?face
value? fossil record. Thus, accommodating preser-
vation elevates rather than erases mass extinction
episodes. Nevertheless, and as Foote emphasizes in
that work, the ?face value? data leads to the same
conclusions, for example estimates of discontinu-
ity of extinction rates separating a few intervals of
mass extinctions from other intervals remains (see
Wang 2003).
Connolly and Miller (2001b) use the distribu-
tion of genus finds over time to estimate preserva-
tion, extinction, and origination likelihoods using
a capture?mark?recapture (CMR) approach. This
represents a modification of the CMR methods used
to estimate the same parameters for Recent taxa
given conservation data (Pradel 1996). One obvi-
ous advantage of this sort of approach is that one
can estimate diversity parameters for separate envi-
ronments or biogeographic units from the same
stages (see, for example, Miller & Foote (1996)
and Connolly & Miller (2001a)). Alroy (2000) uses
likelihood estimates of these parameters to con-
struct faunal ordinations that maximize the prob-
ability of observed species lists. The further devel-
opment of such techniques in conjunction with the
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EST IMATING EXTINCTION WITH THE FOSSIL RECORD 271
expansion of large occurrence-based databases (e.g.
the PBDB) represents an important future direction
for palaeobiological studies (see Prospectus section
below).
19.2.6 Gap analyses
The use of exact numbers of finds within strati-
graphic ranges and the issue of continuous ver-
sus pulsed extinction rates points the way to the
most exact methods for evaluating the timing of
extinctions. This is perhaps best known in associ-
ation with Alvarez et al.?s (1980) proposal that the
end-Cretaceous extinction is a sudden event trig-
gered by a major asteroid impact. However, the
idea that extinction events are pulsed rather than
gradual is not new (see, for example, Erwin (2006)).
Some theory suggests that lower-level turnovers
often are pulsed (Vrba 1985; Morris et al. 1995)
and many of our concepts about stratigraphic zones
tacitly assume pulsed turnovers (e.g. Shaw 1964).
Foote?s (2005) study argues against ?gradual? extinc-
tion mechanisms that distribute extinctions through
entire stages. However, chronostratigraphic stages
and substages typically are millions of years long
whereas the ?gradual? extinction hypotheses often
predict extinction over hundreds of thousands of
years. Thus, although the ?pulsed? hypotheses of
extinctions predict extinctions over tens of thou-
sands of years or less, both ?pulsed? and ?gradual?
might make the same predictions about synoptic
data. We therefore must focus on the distributions
of finds within chronostratigraphic units to find
different predictions between gradual and pulsed
extinction hypotheses.
Preservation again is very important when ascer-
taining the expectations of pulsed and gradual
extinction hypotheses. The probabilistic nature of
preservation and sampling means that we expect
gradual disappearances of taxa from the fossil
record even if all of the taxa share the same time
of extinction (Signor & Lipps 1982). Simple simu-
lations illustrate the Signor?Lipps effect (Fig. 19.4).
Consider the first and last occurrences of 12 taxa
over 21 fossiliferous beds (horizons), with the 12
taxa forming a Zipf abundance distribution (? = 2.0)
and having equal preservation probability per spec-
imen. Although all 12 taxa are present throughout
the section, they appear to phase out gradually sim-
ply because of sampling failure: although the com-
mon taxa are sampled up to the top of the section,
the less common taxa have highly truncated ranges.
Solow (1993a) notes a similar problem for conser-
vation biologists: it is easy to miss rare species for
5+ years by chance alone, so failing to sample one
for a long period of time is weak evidence that it is
extinct; conversely, it is difficult to miss a common
species for any length of time, so failing to sample
one is strong evidence that it is extinct.
One advantage that palaeontologists have that
conservation biologists do not is that they can con-
tinue to sample all of the time planes. Ultimately,
elevated sampling should reveal the rare taxa.
However, this is too labour intensive and expen-
sive to be done at many sites. Thus, we need solu-
tions using the available data. This, in turn, leads
to gap analyses, which attempt to use estimates of
expected gaps between samples to evaluate both
extinction and the Signor?Lipps effect.
Strauss and Sadler (1989; see also Marshall
(1990)) use a Dirichlet distribution to assess possible
gaps between last appearance and true extinction at
any given confidence interval (CI) as:
g = sr ?
(
[1 ? C1]?
1
H?1
? 1
)
(19.7)
where g is the hypothesized gap, sr is the observed
stratigraphic range, H is the number of horizons,
and C1 is the CI on one end of the stratigraphic
range. Note that if we use continuous meters (e.g.
Strauss & Sadler 1989) or time (e.g. Marshall 1990),
then the stratigraphic range would be the last hori-
zon minus the first horizon and single samples
have no range; however, with binned horizons, the
minimum range is 1. We use horizons here simply
because the non-random distribution of horizons
within units of strata or time require further mod-
ifications to these approaches (Marshall 1994, 1997;
but see Prospectus section below). Thus, for taxon
E, g50% is 18 horizons ?([1 ? 0.5]?
1
7?1
? 1) = 2.2 hori-
zons beyond the last appearance. In other words,
there is a 50% chance that the true extinction hap-
pened between the last appearance and this point
and a 50% chance that the true extinction happened
after this point.
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272 BIOLOGICAL DIVERS ITY
Marshall (1995) proposes using 50% confidence
intervals to test the null hypothesis of simultaneous
extinction. Marshall uses the range that is above
g50% for half of the species as the optimal extinction
zone. In Fig. 19.4, half of the taxa have g50% ? 2.2
and half have g50% ? 3.1. Thus, the 22nd horizon
would be the optimum extinction horizon. As that
follows the last appearance of all of the taxa, we
cannot reject a single extinction. Using metres of
sediment rather than fossiliferous horizons, Mar-
shall (1995) shows that Cretaceous ammonoids not
only fail to reject the simultaneous extinctions, but
that the g50% values do an excellent job of predict-
ing the iridium layer terminating the Cretaceous.
Conversely, Marshall and Ward (1996) demonstrate
that the 50% line for Late Cretaceous inoceramid
bivalves precedes the last appearance of nearly half
the species. As many species clearly existed after the
optimal extinction time for half the species, there
almost certainly are multiple extinction events for
inoceramids.
Marshall?s basic approach lends itself well to like-
lihood tests of numbers of extinction events. Build-
ing on a model initially presented by Solow (1996),
Wang and Everson (2007) consider the likelihood of
the hypothesized extinctions, ?1...S, of S taxa, noting
that:
L[?1...S|sr1...S, H1...S] ?
S
?
i=1
(
sri
?i ? F Ai + 1
)Hi
(19.8a)
where F Ai is the first appearance of taxon i , and
therefore:
?i ? F Ai + 1 = gi + sri (19.8b)
(i.e. the observed stratigraphic range, sr, plus the
gap, g, implicit to the hypothesized extinction ?).
Again, the ?+1? is necessary only when stratigraphic
ranges are in discrete rather than continuous units.
Note also that the likelihood is 0 rather than a nega-
tive number if g < 0 as this is an impossible hypoth-
esis. For any taxon i , the likelihood decreases as
gi increases. The number of parameters is equal to
the number of different extinction events. Thus, the
simplest (minimum varying parameter) hypothesis
has all taxa sharing the same extinction events and
the most likely single-parameter (single-extinction)
hypothesis sets all ?i to the latest last appearance
(e.g. horizon 20 in Fig. 19.4). Conversely, the most
complex possible hypothesis posits a separate
extinction for each taxon. The single most likely
version of this basic model is one in which gi =
0 for all taxa, and thus can have as many differ-
ing ? values (and thus as many different parame-
ters) as there are taxa. The support (log-likelihood
rescaled to the maximum) for this hypothesis is 0.
There are a large number of alternatives of inter-
mediate complexity, for example one can examine a
two-parameter hypothesis that posits two separate
extinctions giving ?A ... F at horizon 20 and ?G ... L at
horizon 19.
Log-likelihood ratio tests (e.g. Edwards 1992)
now can assess whether the more complex hypoth-
esis predicts the data significantly better than
the simpler hypothesis. In Fig. 19.4, the sup-
port (i.e. the log-likelihood rescaled to the most
likely hypothesis) for ?A ... L = ?7.34. Whether we
can reject this in favour of continuous extinction
depends on how many parameters we ascribe
to the best ?continuous? hypothesis. Wang and
Everson (2007) use S on the grounds that there
is one extinction for each of the S taxa. How-
ever, in this example there are only seven last
appearance beds. This means that we need only
seven extinction events to eliminate all 12 taxa,
and that the degrees of freedom should be 6
(7 ? 1) rather than 11. This is important because
we would not reject the null hypothesis of a sin-
gle extinction if we assume that the overall most
likely hypothesis requires 12 parameters (p = 0.20)
whereas we would if we assume that the over-
all most-likely hypothesis requires seven parame-
ters (p = 0.02). However, a two-parameter hypoth-
esis (?A ... C = 20, ?D ... L = 19) represents a signifi-
cant improvement over the one-parameter hypoth-
esis (support = ?4.45; p = 0.02) while withstanding
rejection from the overall most likely hypothesis
given the assumption of either 12 parameters (p =
0.54) or seven parameters (p = 0.11) or from the
most-likely three-parameter hypothesis (?A ... C = 20,
?D ... F = 19, ?G ... L = 18; support = ?3.13, p = 0.10).
This clearly is erroneous given that all 12 taxa share
the same time of extinction in the simulations, but
restricting all 12 extinctions to the final two beds
strongly contradicts the idea of a prolonged extinc-
tion event.
OUP CORRECTED PROOF ? FINAL, 18/10/2010, SPi
EST IMATING EXTINCTION WITH THE FOSSIL RECORD 273
Information theory criteria give an alternative
means for assessing the hypothesis. An appealing
aspect is that these criteria can lead one to conclude
that the simpler hypothesis is superior to the more
complex hypothesis rather than simply unrejected
(Burnham & Anderson 2002). Here, Akaike?s infor-
mation criteria (AIC; Akaike 1973), modified for
sample size (Burnham & Anderson 2002), is:
AICc = ?2 ln L[?A...L] + 2K
(
S
S ? K ? 1
)
where K is the number of parameters and
the taxon richness, S = 12, is the number of
data points. Because the AICc[?A ... L = 20] = 17.1
whereas AICc[7 separate ?i] = 42, we now prefer
the single extinction to the mostly-likely seven-
extinction hypothesis. However, we still prefer
the best two-parameter hypothesis (AICc[?A ... C =
20, ?D ... L = 19] = 14.2) to the best single-extinction
hypothesis. On one hand, one might fault the
method for failing to corroborate the correct
hypothesis that all of the extinctions happened after
the last bed. However, it is reassuring that the
favoured hypothesis is a near miss that concentrates
all of the extinctions in the last two beds.
Huelsenbeck and Rannala (1997) estimate the
probability of zero finds over some interval given
preservation rate, R. Although this is intuitively
reasonable, R itself is an unknown. R is not explic-
itly addressed in the Wang and Everson approach,
but uncertainty about R should be a concern there
too. An intuitive estimate of Ris simply:
R =
H
sr
(19.9a)
As Fig. 19.4 illustrates, this is biased towards overe-
stimating preservation rates because the denomina-
tor under-estimates the true duration (Wagner 2000;
Foote 2001a). Instead, an unbiased estimate is given
by the finds within the known range, i.e.:
R =
H ? 2
sr ? 2
(19.9b)
Even so, we should not estimate the likelihood of
hypothesized extinction as:
L[?|H, sr, R] =
(
1 ?
H ? 2
sr ? 2
)g
First, the uncertainty around the most likely R can
be very large, especially for taxa sampled from a
few beds over a short duration. Although the prob-
ability of the stratigraphic data is maximized with
equation 19.7b, the joint probability of the observed
data and the hypothesized gap (and thus the joint
likelihood of ? and R; Wagner 2000) is maximized
at:
R? =
H ? 1
g + sr ? 1
(19.10)
giving the maximum:
max L[?, R|sr, H] ? (1 ? R)g ? ([1 ? R]sr?H ? RH?1)
(19.11a)
? (1 ? R)g+sr?H ? RH?1 (19.11b)
where the first term in equation 19.9a is the proba-
bility of zero finds over the gap and the second term
is the sufficient statistic from the binomial probabil-
ity of H ? 1 finds over the observed stratigraphic
range after the first find. (The binomial combina-
toric is constant for all hypotheses.) Note that we
now subtract only one horizon as one end of the
?true? duration is implicit to the hypothesis and we
truncate only the base.
We now can estimate the likelihood of one or
more extinction events using the product of equa-
tion 19.11 for all species with H ? 2. Returning to
our example in Fig. 19.4, the log-likelihood of a
single extinction at horizon 20 for the 10 species
with Hi ? 2 is ?18.70. The log-likelihood of each
species going extinct at its last appearance increases
only to ?14.65, which is insufficient to reject the
single ? hypothesis even when assuming only
seven-parameters (p = 0.23) and which fares far
worse given information theory (AICc[?A ... L = 20] =
39.8 vs AICc[?A ... L = LAA ... L] = 71.3). However, a
hypothesis of extinctions at horizons 19 and 20
still is preferable (ln L = 16.25) given either log-
likelihood ratio tests (p = 0.03) or AIC (AICc[?A...C =
20, ?D...L = 19] = 37.8).
The potential advantage of this approach over
the approach of Solow/Wang and Everson is that
it allows one to assess whether preservation rates
change within a stratigraphic range. If so, then
change in R can be explicitly accommodated (see
Solow 1993b; Wagner 2000). For example, a sim-
OUP CORRECTED PROOF ? FINAL, 18/10/2010, SPi
274 BIOLOGICAL DIVERS ITY
ple test of linearly increasing/decreasing R over
horizons would simply test the exact sequence of
finds and misses with changing probabilities of
finds and misses. Alternatively, one might posit
a sudden shift in preservation rate. If the fre-
quency of misses increases/decreases, then the
gap between last appearance and extinction will
increase/decrease. For example, taxon I in Fig. 19.4
shows an increasing recovery rate over time: it is
found once over 10 horizons (2?11) then twice over
the next five horizons (12?16). The most likely sin-
gle R, 5?217?2 = 0.2 (equation 19.7b) has a support of
?7.51(= ln[0.23 ? 0.812]). The best gradually shift-
ing R, with R2 = 0.05 ? R16 = 0.34, increases the
support only to ?7.14. The best two-phase R,
R2...11 = 0.1 and R12...16 = 0.4, increases the support
only to ?6.86. In neither case would we reject the
idea that sampling was homogeneous throughout
taxon I?s range and thus we have no particular
reason to distrust the likelihood given sampling
over those 20 horizons. However, Wagner (2000)
presents empirical examples where this approach
does reject homogeneous R and thus leads to dif-
ferent likelihoods on hypothesized extinctions.
19.3 Prospectus
The study of extinction can proceed in many direc-
tions, many of which workers have tentatively
explored already. In particular, the use of databases
such as the PBDB to ascertain more exact tests
of extinction pacing and preservation rates within
taxa represents the logical next steps to the meth-
ods described above. For example, using observed
range-through taxa to estimate R provides an aver-
age sampling intensity within an interval. This
obviously will not represent all genera. With higher
taxa in particular, species richness within a genus or
family will strongly affect preservation rate, as will
other factors, such as the geographic distributions
of the species within the taxa. Foote (2007b) shows
that the occurrences of genera typically follow a bell
curve over time, that is, occurrences are low both
in the early and late intervals of a genus? range
and typically peak in the middle. Because genera
originate throughout time, this means that different
genera will be at different points in their bell curves
in any given interval. Moreover, the peaks of these
distributions (i.e. the maximum numbers of occur-
rences) differ among genera. These two factors com-
bined means that it should be possible to derive a
distribution of preservation intensities within each
interval rather than a single R ?per genus?. What
is most intriguing is that we might use the bell
curves themselves in a Bayesian context to assess
extinction risk (see, for example, Solow (1993b)). In
other words, extinction risk should be relatively low
at the peak of an occurrence distribution, when a
genus typically is widespread and fairly speciose
(see, for example, Foote et al. (2008)). Conversely,
the extinction risk should be relatively high near
the tail. Whether or not we wish to assume normal
curves for occurrences over time requires further
testing; however, data such as the PBDB make it
possible to assess whether ?commonness? of genera
is linked to extinction risk.
Part of the reason why we should expect common
genera to be less extinction prone than uncom-
mon genera is that we expect geographically wide-
ranging genera to be more common in the fos-
sil record than contemporaneous geographically
narrow-ranging genera (e.g. Jablonski 2000). How-
ever, because sampling is not randomly distributed
over geography, we cannot simply equate occur-
rences with geographic range. Instead, we need
to actually examine occurrences in particular geo-
graphic units (see, for example, Foote et al. (2008)).
Alroy (in Alroy et al. (2008)) presents methods for
accommodating geographic dispersal in sampling
standardized estimates of richness over time. We
should adopt similar approaches for summarizing
geographic distributions when examining the pre-
dictions of alternative extinction models.
Even occurrences within temporal and geo-
graphic units obscure important information about
abundances within localities. In general, there is a
good correlation between abundances and occur-
rences, in that taxa that are abundant anywhere
within their geographic range tend to have many
occurrences (Buzas et al. 1982; Alroy et al. 2008).
However, in gap analyses of particular sections,
variation in abundances of specimens has an impor-
tant effect on the plausibility of missing an extant
species (see, for example, Hurlbert (1971)). The sim-
ulations sample equal numbers of ?specimens? per
bed. However, in reality, all beds are not equal: due
OUP CORRECTED PROOF ? FINAL, 18/10/2010, SPi
EST IMATING EXTINCTION WITH THE FOSSIL RECORD 275
to access, taphonomy, weathering, etc., some beds
in the same section will yield many more speci-
mens than others. Weiss and Marshall (1999) explic-
itly accommodate this when examining extinction
boundaries for late Cambrian trilobites. Similarly,
McElwain et al. (2009) examine the end-Triassic
extinction in plants not in loss of richness, but in
shifts on most likely relative abundance distribu-
tions. A combination of the two approaches would
assess the likelihoods of hypothesized abundances
over time/strata, with extinction being an abun-
dance of 0. The logic of shifting R over a section
would apply here, except that we now would ask
whether we could sample X specimens without
finding a species given that it is known from n of
N specimens prior to that.
Another promising avenue of research involves
the use of ordination techniques to describe
fossil turnover without assuming a particular
chronostratigraphic scale (e.g. Alroy 1992,
1994, 2000; Sadler and Cooper 2003). Although
presented as techniques for repeatable, high-
resolution chronostratigraphy, these methods also
describe fossil distributions in the same manner
needed for confidence interval studies, but at
the geographic and geologic scales of synoptic
studies.
Finally, one might ask whether gap analyses
of extinction events frame the basic hypotheses
properly. As done above, the ?complexity? of the
hypotheses is the number of extinction events. A
single-extinction event thus is the null because it is
only one event. Statistically, this is also appropriate
because a single event necessarily is less likely than
(as in Fig. 19.4) seven events, as it maximizes the
minimum number of necessary gaps. However, this
is not quite the appropriate null hypothesis if we
look at extinction rates. A single event requires two
rates: a ?background? rate of no extinction and a
single pulsed rate of (nearly) 100% extinction. Thus,
we can ask, what is the probability of X extinctions
from Y remaining taxa given a hypothesized extinc-
tion rate per bed? As Weiss and Marshall (1999)
note, the extinction rate provides a prior probability
of a lineage existing 1, 2, 3, etc., beds after its last
find. This multiplied by likelihood of the gap gives
a posterior probability of the taxon existing 1, 2, 3,
etc., beds after its last finds. The sum of these poste-
rior probabilities is what we would use as the count
of the number of extinctions per bed. For example,
in Fig. 19.4, we would have numerous extinctions
in the last two beds given a hypothesis of uni-
form extinction rates and the sampling data. This
would be strong evidence against a gradual hypoth-
esis. However, if last appearances were distributed
more evenly throughout the section, then it would
be much more difficult to extend the durations to
the top of the section as not only would they
become increasingly improbable, but the plausibil-
ity of the taxon still being extant also would become
increasingly improbable.
19.4 Key points
1. The fossil record provides data capable of testing
a wide variety of extinction hypotheses.
2. This can be done at very broad temporal scales
as well as over intervals closer to ?ecological
time?.
3. Accommodating preservation is critical at all
temporal scopes of palaeontological analyses of
extinction.
OUP CORRECTED PROOF ? FINAL, 18/10/2010, SPi
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