Fossil Record, Pages 53-62, Sean R Connolly.pdf

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have focused on taxonomic richness, at least when addressing regional or global patterns in biodiversity.



I. SAMPLING EFFECTS IN THE

FOSSIL RECORD

A. The Fossil Record Is Incomplete

One obvious feature of the fossil record is that it is

incomplete: only a small fraction of individuals are fossilized; of those, very few are collected and identified.

As a result, the biological diversity recorded in the fossil

record is less than total diversity over the region and

time interval from which it is a sample. An additional

consequence of this fact is that it makes stratigraphic

ranges of taxa correspond to shorter periods of time

than their true life spans. This is because the first appearance of a taxon in the fossil record occurs sometime

after it actually originated, unless its very first representative was fossilized and subsequently sampled (a highly

unlikely prospect). Similarly, the last appearance of a

taxon occurs sometime before it actually went extinct.

Biased estimates of stratigraphic ranges are important

for the study of trends in biodiversity because they are

often used to estimate biodiversity. Thus, rather than

estimating diversity as the number of taxa that actually

occur in samples from a particular stratigraphic interval,

diversity is estimated as the number of taxa whose stratigraphic ranges encompass that interval. This will always

be equal to or greater than the number of sampled taxa.



B. The Quality of Preservation Varies

If incomplete sampling were the only problem with the

fossil record, it would still be possible to accept relative

trends through time at face value—that is, an increase

in diversity in the fossil record would indicate a real

increase in biological diversity, even if the true number

of extant taxa could not be determined. Unfortunately,

however, the degree to which the fossil record is incomplete varies in both space and time. One source of this

variation is characteristics of organisms themselves. For

instance, organisms with hard parts are much more

likely to be preserved in the fossil record than those

without them. Clams, for instance, have a better fossil

record than nematodes. One consequence of this is

obvious: the difference between apparent and true diversity tends to be greater for the latter group than

for the former. However, estimates of diversity trends

through time can be affected as well, if the proportion



of taxa with body parts that are readily fossilized does

not remain constant. Indeed, some have argued that

the ‘‘Cambrian explosion’’ represents, not an explosion

of multicellular life, but a rapid and extensive proliferation of hard parts. Recently, this hypothesis has been

fueled by molecular phylogenetic studies predicting

that the major animal phyla diverged long before the

early Cambrian. This hypothesis remains highly controversial; nevertheless, the very fact that it has received

considerable attention illustrates just how profoundly

sampling effects are believed to influence the fossil record of major events in the history of life.

Differences in abundances among taxa can also affect

their preservation in the fossil record. It is individuals

(or parts of individuals) that are fossilized; thus, more

abundant taxa are, on average, likely to have more

complete fossil records than will rare taxa. Further,

fossil diversity is likely to be higher for intervals during

which abundances were higher, on average, than during

other intervals. This, too, has important consequences

for inferences about diversity trends. For instance, several major episodes of diversification, as recorded in

the fossil record, coincide with geophysical changes

that probably increased rates of nutrient supply to the

biosphere. Some workers (e.g., Vermeij, 1995) have

argued that these geophysical changes were important

causes of the coincident biological diversification. However, if increases in rates of nutrient supply also allowed

taxa to sustain higher abundances, then increased probabilities of sampling during those sequences could be

contributing to the increased diversity of the fossil record. Thus, inferences about the causes of trends in

biodiversity, in addition to inferences about the trends

themselves, can be influenced by sampling effects.

Differences in abundances can also affect estimates

of stratigraphic ranges. One of these effects is illustrated

by the previous example: first appearances of new taxa

in the fossil record will be higher between two stratigraphic intervals when large increases in abundance

occur between them as well, even if there were, in fact,

an equal number of originations during the time periods

that correspond to those intervals. Assessing the severity of this effect is complicated by the fact that there

are sound biological reasons for increases in abundance

to facilitate originations of new taxa; to the evolutionary

biologist, a correlation between abundance and rate of

speciation may be precisely what is expected!

Yet another characteristic of organisms that can affect inferences about biodiversity trends is habitat. In

particular, individuals in some habitats are more likely

to become fossils than individuals in others. For instance, marine soft-bottom habitats are likely to provide



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more complete records of their inhabitants than will

rocky shores. In the former case, individuals will much

more readily be covered in sediment shortly after (or

even before) they die. In the latter, wave action may

render the remains of individuals unidentifiable before

currents carry them to a location where they might be

buried by sediment and preserved. Indeed, the fossil

record of rocky shore communities is among the poorest

in the marine realm. Again, the difficulties associated

with these effects has led to disagreements about key

events in the history of life. For instance, fossils record

an explosive diversification of mammals in the early

Tertiary, following the mass extinction that ended the

Cretaceous period (best known for catastrophic extinctions among dinosaurs). Most workers believe that this

reflects the true pattern, at least qualitatively. However,

others, using molecular phylogenetics, have argued that

much of this diversification occurred in the Cretaceous,

prior to this mass extinction. One explanation for this

discrepancy has been that these early Cretaceous mammals occupied habitats (such as forest interiors) for

which their probability of being preserved was much

lower than for their Tertiary descendants, who expanded into new habitats to occupy niches vacated by

the extinction of dinosaurs. Whether this discrepancy

between molecular and fossil data is primarily due to

a poor fossil record for Cretaceous mammals or to shortcomings in the molecular methods remains highly controversial. Attempts to assess the severity of these shortcomings, such as those of Foote and coworkers (1999),

should ultimately lead to a resolution.

As this discussion of the effects of ecology suggests,

the probability of an individual being preserved in the

fossil record depends, in part, on the sediments themselves. This, too, can have profound consequences on

inferences about diversity patterns. For instance, in a

classic paper, Raup (1976) noted a systematic increase

in the volume of sedimentary rock through time. The

implication was that the probability of individuals being

preserved in the fossil record becomes progressively

greater through time. Indeed, he presented a graph

of sedimentary rock volume through time that looked

strikingly similar to a graph showing biological diversity

in the fossil record through time. After removing the

effect of rock volume on diversity, he found no evidence

for a long-term increase in species diversity through

time. Although a key paper by Sepkoski et al. (1981),

along with subsequent work, has convinced most workers that the apparent long-term diversity increase is

real; the extent to which systematic increases in rock

volume exaggerate the trend remains unknown.

This long-term increase in sedimentary rock volume



has another potential effect on large-scale diversity patterns. If the probability of preservation and sampling

increases through time, then the difference between the

time of origination of a taxon and the time of its first

appearance in the fossil record should shrink. Similarly,

the difference between the time of extinction and the

time of last appearance should shrink. This effect would

lead to a progressive decrease in apparent extinction

rates through time. Indeed, a long-term trend of decreasing extinction rates has been noted in work in the

1980s by Sepkoski and Raup and by Van Valen. Knowing the extent to which this trend reflects a real decrease

in extinction rates (rather than a sampling effect as just

described), is important, because Sepkoski (1984) has

noted that such a trend, if real, could explain long-term

changes in the relative prevalence of different taxa in

the fossil record. As a result, paleobiologists (e.g., Pease,

1988, 1992) have proposed methods for assessing the

effects of sampling biases on these rates. There appears

to be an emerging consensus that the declines in evolutionary rates are real, but that they may be exaggerated

by coincident trends in the quality of preservation.

The extent and quality of the fossil record varies

over smaller timescales as well. Geophysical transitions

can dramatically affect the probability that individuals

will be preserved as fossils. When these transitions are

global in scope, or occur in regions that receive disproportionately large amounts of attention from paleontologists, then apparent changes in diversity can result.

Just such a possibility has complicated analysis of what

was probably the most extensive mass extinction in the

history of life, an event that marks the end of the Permian and the beginning of the Triassic. This event coincided with a major loss of marine benthic habitat (due

to decreasing sea level). While it is certainly reasonable

to suspect that biodiversity would decrease as available

habitat decrease, it would also reduce the probability

of sampling taxa that remained, as noted by Signor and

Lipps in an influential paper published in 1982. This

and other sampling effects associated with characterizing this extinction event and its aftermath are extensively reviewed by Erwin (1993).



C. The Extent of Sampling Varies

Decisions paleontologists make can also introduce, or

exaggerate, differences in the quality of the fossil record.

For instance, a disproportionately large amount of effort

has been devoted to sampling in the Mesozoic and

Cenozoic eras, because paleontological sampling tends

to be associated with petroleum exploration, and rocks



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FOSSIL RECORD



that date from these areas have been subject to more

extensive exploration than older rocks. Similarly, North

America has been more extensively sampled than many

other regions of the world. This reflects, in part, the

fact that many paleontologists live in North America.

The logistical problems associated with fieldwork are

generally less severe, and the cost of travel lower, when

fieldwork is conducted near a home institution. In addition, fieldwork tends to be easier in regions with a

well-developed infrastructure. As a result, more of the

preserved biodiversity is actually sampled and recorded,

and thus the apparent diversity of well-sampled regions

is higher than that of poorly sampled regions.

Similarly, the questions that one can ask of paleontological data, and the degree of confidence that can be

assigned to answers, depends in large measure on the

quality and quantity of data that can be collected. As

a result, disproportionately large numbers of paleontologists focus their efforts on taxa, stratigraphic intervals, and geographical regions where fossils are abundant and well-preserved. This tends to exaggerate the

effects of differences in preservation: where preservation is good (something which, alone, would tend to

make biodiversity appear greater), a larger proportion

of that record has been sampled. One effect of this

tendency is that more workers focus on the most recent

era of time, the Cenozoic (0–65 million years ago),

than on the previous era, the Mesozoic (65–250 million

years ago), despite the fact that the latter era was longer

in duration. Likewise, more paleontologists study the

Mesozoic than the earlier Paleozoic (250–600 million

years ago).



D. Time-Averaging

Yet another factor affecting estimates of diversity patterns is time averaging. When fossils are collected, their

location in time is generally reported according to the

particular stratum in which they were found. This corresponds to an interval, rather than a specific point, in

time. Some of these strata represent longer periods of

time than others. Thus, the number of taxa found in a

particular stratum may be greater than the number of

those taxa that were actually extant at a particular point

in time during that interval (assuming that extinctions

or originations occurred during the interval). The

longer the interval lasts, and the higher the rates of

origination and extinction, the more apparent diversity

is inflated.

The issue is further complicated by uncertainties

associated with the points in time that correspond to

the boundaries of stratigraphic intervals. Indeed, the



ages of these boundaries are continually revised. Thus,

it may not always be clear which intervals were longer

than others, and it is never clear just exactly how long

those intervals were. Unfortunately, this problem becomes more severe as intervals are more finely divided.

This makes minimizing the time-averaging problem

more complicated. That is, the time-averaging problem

should become smaller as the length of time spanned

by an interval decreases. Since stratigraphic intervals are

classified hierarchically (with some types of intervals

nested within larger ones), one could, in principle, minimize the time-averaging problem by using a low level

in this hierarchy. Unfortunately, however, the shorter

the interval, the greater the proportional uncertainty

associated with the estimated ages of the intervals’

boundaries. For instance, overestimating the age of the

lower boundary of a 10 million year interval by 1 million

years inflates the estimated duration of that interval by

10%. However, an overestimate of the same magnitude

on the duration of a 3 million year subunit of that

interval represents a 33% inflation.

In addition, it can be difficult to determine how

stratigraphic intervals in one location correspond to

intervals in other locations. If one is interested in estimating diversity or macroevolutionary rates for a region

that includes more than one sampling location, then

this can introduce further uncertainties. One source of

this uncertainty is a simple extension of the problem

of uncertainty in dating the boundaries of particular

strata: if there are errors in the estimates of the ages of

these boundaries at two different locations, then two

intervals may be assumed to be substantially coincident

when, in fact, they are not. This problem, like that

discussed in the previous paragraph, becomes more

severe as the duration of stratigraphic intervals decreases.

A second problem associated with this process of

chronostratigraphic correlation results from the fact

that direct estimates of ages are not available for all

stratigraphic boundaries in all locations. Often, correlations are based on the presence or absence of particular

indicator taxa. That is, the stratum in which a particular

taxon first appears at, say, location B is assumed to

correspond to the stratum in which it appears at location A. If the ages of the boundaries of the stratum at

location A have been estimated, but those of location

B have not, then the lower boundary of the stratum at

location B is assumed to fall within that range of ages.

However, since new taxa originate in particular regions,

then expand their ranges gradually into new regions,

part of the uncertainty associated with this effect is

based on this rate at which indicator taxa increase their



FOSSIL RECORD



geographical ranges. In addition, however, as noted

earlier, the difference between the time at which a taxon

was first present at a location and the time of its first

appearance in the fossil record for that location varies

depending on factors specific to particular locations,

such as quality of preservation and local abundance.

This further contributes to uncertainties in stratigraphic

correlation. In practice, paleontologists attempt to minimize this problem by using multiple indicator taxa.

They also emphasize those taxa that are likely to have

high probabilities of preservation in the fossil record

and whose spread is likely to have been rapid, such as

planktonic foraminifera.



II. ESTIMATING DIVERSITY TRENDS

A. Rarefaction

One way to control for these sampling effects in making

inferences about diversity trajectories is to compare

taxonomic richness among locations or strata with samples that are equivalent in extent. This technique, called

rarefaction, was developed by Sanders (1968)—and

amended by Simberloff (1972)—to compare the diversities of different habitats in present-day ecosystems.

The approach is as follows: a sample is collected from

each of a set of habitats (for instance, a certain volume

of sediment is obtained) using an identical sampling

scheme. All of the individuals in each sample are identified, and their taxonomic identity is recorded. The next

steps can be visualized by imagining placing all of the

records for a particular sample in a bowl, stirring them,

and then randomly selecting records from that bowl

until none are left. Each time a record is picked, the

taxonomic identity of that sample is noted. From this

sequence of random draws, one constructs a graph with

the number of records on the horizontal axis and the

number of distinct taxa on the vertical axis. What is

plotted on that graph are the results of that sequence

of record selection. This curve must intersect the points

(0, 0) and (1, 1). That is, before the experiment begins,

0 records have been picked from the bowl, so 0 distinct

taxa have been selected. When one record is chosen,

exactly one distinct taxon has been found. When two

records have been chosen, either one or two distinct

taxa have been found. [If the second record has the

same taxonomic identity as the first, the next point

is (2, 1); if it has a different identity, the next point is

(2, 2)]. Once all of the points have been plotted, a curve

is fit to them: this is the rarefaction curve. This process

is repeated for each sample collected, so that each sam-



57



ple has a rarefaction curve associated with it. Biodiversity in the different samples is then compared by examining the number of distinct taxa encountered for a

given number of records selected. That number must

be equal to or less than the number of records in the

sample with the fewest records. This comparison is

illustrated in Figure 1.

Figure 1 illustrates something else: rarefaction

curves sometimes cross. This means that the rank order

of diversities of a set of samples can change, depending

on sample size. One might therefore ask what rarefied

diversity means, and, further, what it means to say that

one site has a greater rarefied diversity than another.

Hurlburt (1971) ventured an answer to that question,

proposing that rarefied diversity measured the number

of distinct taxa encountered, on average, by an individual organism in a particular habitat over the course of a

particular number of encounters with other individuals

(provided that individuals are not encountered multiple

times). The fact that one habitat has a greater rarefied

diversity than another for a given number of occurrences of individuals does not mean that that habitat

actually has a greater taxonomic richness (i.e., contains

more taxa) than another.

Although rarefaction can be applied to the fossil

record in the manner just described, it is often applied

somewhat differently. When comparing diversities over

very large scales (global diversity during two strati-



FIGURE 1 Hypothetical rarefaction curves for three samples representing three different habitats, designated A (solid line), B (dashed

line), and C (dotted line). Note that habitat A, whose sample contains

the most species in its entirety (note that its rarefaction curve ends

at a higher diversity than the others), is not the most diverse habitat

when its diversity is compared with the others for a particular sample

size (X1 on the horizontal axis). Also note that the rarefaction curves

for samples B and C cross. Thus, habitat B is more diverse when

there are X1 records in the sample, but habitat C is more diverse

when there are X2 records.



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FOSSIL RECORD



graphic intervals, for instance), a sample enumerating

all of the individuals in those intervals will not be available. Rather, data will consist of a number of distinct

samples for each of the regions or intervals being compared. Further, while a list of the taxa found in each

sample may be available, information on the abundances of those taxa may not. Such data have been

rarefied in two different ways. In one approach (hereafter called rarefaction by occurrence), a rarefaction curve

is constructed for a habitat by randomly selecting occurrences of taxa from its associated samples. The rarefaction curve is then a plot of the number of distinct

taxa sampled against the number of occurrences of taxa

drawn from the available samples. This is illustrated in

Figure 2. Diversities are compared by constructing these

rarefaction curves for different regions, taxa, or stratigraphic intervals. An alternative approach (hereafter

called rarefaction by list) involves randomly selecting

the entire list of taxa present in a given sample as a

unit, rather than selecting occurrences of taxa within

those samples. In this case, the rarefaction curve for a

particular habitat, region, or stratum consists of the

number of distinct taxa detected plotted as a function

of the number of samples drawn. Note that in this

instance, the curve is not constrained to pass through

the point (1, 1), since several taxa may appear in a

single sample. However, it still must pass through the

point (0, 0). This method is illustrated in Figure 3.

Rarefaction has been applied to paleontological data

in several instances. Raup (1975) pioneered its use in

paleobiology. He examined an apparent increase in di-



FIGURE 2 A graphical illustration of rarefaction by occurrence. Occurrences of a taxa within samples are treated independently, and

these occurrences are drawn individually at random from the set of

occurrences in all samples. This can be visualized as adding each

occurrence individually (regardless of the sample from which it

comes) into a pool (arrows into the black box), then randomly choosing these occurrences from that pool (arrows out of the black box).

In this figure, the rarefied diversity after five records have been drawn

is four distinct species.



FIGURE 3 A graphical illustration of rarefaction by list. In this case,

an entire paleontological sample is drawn at random from the population of samples that constitute the data set, and rarefied diversity is

the total number of unique species contained in those lists for a given

number of samples drawn. In this figure, the rarefied diversity after

two samples have been chosen at random is six species (A, B, E, F,

G, and H).



versity of echinoid families since the Paleozoic.

Through rarefaction, he demonstrated that this increase

was qualitatively unchanged after accounting for increased sampling in younger stratigraphic intervals. In

the 1990s, Miller and coworkers applied rarefaction by

occurrence to explore taxonomic and regional patterns

in the diversification of benthic marine invertebrates

during the Ordovician Radiation. Unlike Raup, they

found that broad-scale patterns suggested by raw data

were changed by rarefaction. For instance, global rarefied diversities increased only through the mid-Ordovician, when they stabilized. This contrasts with the

trend apparent from the fossil record as a whole: increasing diversity through the late Ordovician. Similarly, large peaks in diversity at the scale of individual

paleocontinents were often reduced or eliminated by

rarefaction. By contrast, Alroy (1999) applied rarefaction by list to explore the extent to which the radiation

of mammals preceded the end-Cretaceous mass extinction (see Section I.B, ‘‘The Quality of Preservation Varies’’). His results agreed qualitatively with the trend

apparent from the ‘‘uncorrected’’ fossil record: much of

the taxonomic diversification of mammals appeared to

occur after this mass extinction.

Paleontologists have typically been very cautious in

their interpretation of rarefied diversities. There is a

reason for this caution: the biological meaning of a

rarefied diversity measurement is unclear in the paleobiological context. The interpretation offered by Hurlburt

(described earlier) is appropriate only when (a) the

sampling protocol described in the first paragraph of

this section has been followed or (b) each occurrence



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of a taxon is a truly independent, random sample from

the habitat about which one wishes to make inferences.

Generally, neither is true for large-scale paleobiological

diversity estimates, because sampling effort has not

been distributed randomly. For instance, when comparing the rarefied diversities of two stratigraphic intervals,

a disproportionately large number of samples may be

available for one habitat or region during one interval,

but not for the other interval (but see methods developed by Alroy [e.g., 1998] for minimizing these biases).

In addition, for rarefaction by occurrence, occurrences

of taxa are not independent, because some occurrences

come from the same sampling locations and others from

different sampling locations. A biological interpretation

of rarefied diversity given these sampling effects remains elusive. Nevertheless, rarefaction can provide important information. In particular, discrepancies between rarefied and overall diversity patterns indicate

where marked increases or decreases in apparent diversity are likely to be artifacts of sampling.



B. Capture-Recapture Estimates

Although rarefaction has probably been the most widely

used tool to account for the effects of variation in sampling effort, other approaches have been tried as well.

For instance, in the mid-1980s, Nichols and coworkers

proposed that models used to estimate abundances from

capture-recapture data in population biology could be

adapted, by analogy, to estimate taxonomic diversity

in the fossil record. When one conducts a capturerecapture study, individual organisms are captured during discrete sampling occasions, which can occur at

different locations, different times, or both. Each individual captured is given a unique mark, so that its

capture history can be constructed. That is, if one constructs a matrix, the columns j of which represent different sampling occasions and the rows i of which correspond to each individual captured at least once, then

each element aij in the matrix will be either 1 or 0,

indicating whether individual i was captured on occasion j. Similarly, fossil data are collected on discrete

sampling occasions, and one can simply list the taxa

sampled on each occasion. In this case, the matrix elements aij are 1 or 0 according to whether taxon i was

found in sample j. In this context, taxonomic diversity

(the total number of distinct taxa) is analogous to population size (the total number of distinct individuals).

Unlike rarefaction, these approaches estimate taxonomic diversity (rather than sampled diversity given a

particular sample size) when the models’ assumptions

are met. Unfortunately, however, these assumptions are

rarely fully met by fossil data. It is not always clear



59



which results are robust to violations of assumptions

and which are not. Therefore, as with rarefaction, workers have been cautious when interpreting these estimates.

Two different types of capture-recapture models can

be used to estimate diversity, depending on the sampling scheme. When sampling occasions occur at different times and the duration of a particular sampling

occasion is short relative to the time between sampling

occasions, open-population models can be used. These

models are designed to account for ‘‘births’’ and ‘‘deaths’’

of taxa (i.e., originations and extinctions) that may

occur between sampling occasions. In the paleobiological context, stratigraphic intervals have been treated as

sampling occasions. Thus the taxa sampled in an interval are considered as having been ‘‘captured’’ at some

point during that interval (usually its midpoint).

By contrast, when sampling occasions are sufficiently

close together that originations and extinctions are few

relative to the total number of taxa extant, then closedpopulation models can be used. These models have

been applied when multiple samples have been taken

within a stratigraphic interval. Although these models

assume that there are no originations or extinctions

between the specific points in time represented by different samples, they tend to be more robust to other

types of sampling problems (discussed later). The particular models that have been applied to paleontological

data include the Jolly-Seber open-population model and

the closed-population models of Burnham and Overton

(1979) and Chao (1987).



1. Jolly-Seber Model

The Jolly-Seber model makes four key assumptions:

1. The strata that represent different sampling occasions are short in duration relative to the time between those strata.

2. All taxa have the same probability of being sampled within a particular stratum.

3. All taxa have the same probability of going extinct

between strata j and j ϩ 1.

4. If a taxon goes extinct in the region being sampled, it does not subsequently reinvade that region from elsewhere.

Given these assumptions, diversity (Di) can be estimated for any stratum i from the following data (notation follows that used by Nichols and Pollock, 1983):

the number of taxa sampled during i (ni), the number

of ni that were also sampled during at least one earlier

interval (mi), the number of ni that were also sampled

during at least one later interval (ri), and the number



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of taxa sampled at least once before i, at least once after

i, but not during i itself (zi). The first step in estimating

diversity involves estimating the number of taxa extant,

but not sampled, during i (Mi). Note that ri / ni is the

proportion of taxa sampled during i that were sampled

again during a later interval. Similarly, zi /(Mi Ϫ mi) is

the proportion extant but not sampled during i that were

sampled again subsequently. If the model’s assumptions

hold, these two fractions will be equal, and Mi can be

estimated. The second step involves noting that mi /ni

is the proportion of taxa sampled during i that were

sampled during earlier intervals. Similarly, Mi /Di is the

proportion of taxa extant during i that were sampled

during earlier intervals. Again, if the model’s assumptions hold, these two fractions will be equal, and diversity can be estimated.

As discussed earlier, taxa differ in many characteristics that affect their probability of being sampled as

fossils (e.g., presence of hard parts, habitat, abundance).

This violates assumption (2) and causes estimated diversity to tend to be lower than true diversity. Perhaps

not surprisingly, then, when Nichols and Pollock

(1983) applied this model to late Eocene mammals from

the Big Horn Basin, Wyoming, goodness of fit statistics

indicated rejection of the model. When applied to molluscan diversity in the Middle Miocene of South Jutland,

the model provided an adequate fit. However, diversity

estimates were lower than those obtained with a method

that does not assume equal sampling probabilities (discussed later).



2. Closed-Population Models

Several alternative models can be used when multiple

samples are available for a particular time interval.

Those that have been applied to the fossil record share

one important feature in common: they are designed

to allow for the possibility that some taxa are more

likely to be sampled than others. All of these methods

utilize the frequency distribution of occurrences of the

taxa sampled. That is, the raw material for the diversity

estimate is the number of taxa occurring in only one

sample, f1, the number occurring in two samples, f2,

and so on, as well as the total number of occurrences

in all samples.

Burnham and Overton (1979) utilized a statistical

approach known as the jackknife to estimate biodiversity. The mathematics of the derivations are too complex to review here, but with this approach they obtained a series of possible estimators. The simplest of

these utilizes only the number of samples and the number of taxa occurring in only one sample:



where Dˆ1 is the first-order jackknife estimate of diversity,

Dobs is the number of distinct taxa appearing in the

sample, f1 is the number of taxa occurring in exactly

one sample, and k is the number of samples. They

developed additional estimates by incorporating the

number of taxa occurring in more than one sample (for

instance, their second-order jackknife uses the number

of taxa appearing in exactly two samples, as well as the

number appearing in just one). Nichols and Pollock

(1983) applied these models to same molluskan data

on which they used the Jolly-Seber method described

earlier. They found that, even under a relatively intensive sampling regime, sampled diversity was as much

as 30% lower than estimated diversity.

An alternative model, proposed by Chao (1987), uses

the number of taxa occurring in either exactly one or

exactly two samples:



Wing and DiMichele (1992) used this estimate, usually

termed ‘‘Chao-2,’’ to compare regional vegetation diversity in the late Paleozoic and early Cenozoic for North

American river and delta floodplains. Somewhat surprisingly, they found similar biodiversity levels during

the two periods for these regions, despite the markedly

higher apparent diversities for the latter interval at the

global level.

Like the other methods discussed, these models

make several assumptions that limit their applicability.

The jackknife and Chao estimates assume that each

taxon has an equal probability of occurring in each

sample. That is, taxon A may have a different probability

of being present in a sample than taxon B, but that

taxon-specific probability is the same for every sample

within each region. This assumption may be violated

if samples differ in extent or quality. However, even

if great care is taken to minimize this problem, the

assumption may still be violated. For instance, if abundances of particular taxa differed among sampling locations, then the associated probabilities of sampling may

differ accordingly.



C. Generalized Inverse

Gaussian-Poisson Distribution

Models other than those based on capture-recapture

theory utilize frequency distributions of occurrences to



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estimate diversity. One of these methods is known as

the Generalized Inverse Gaussian-Poisson (GIGP) distribution. This method involves fitting a statistical distribution of specified form to the observed frequency

distribution, then extrapolating from this statistical distribution to estimate the number of species that have

been sampled zero times, f0. Total biodiversity, then, is

simply this number plus the number of species that

were sampled at least once. Anderson and coworkers

(1996) applied this method to plant and insect data for

late Triassic braid-river plains in South Africa. They

used these estimates to argue that plant and insect diversity in the sampled habitats was comparable to those

of the present day, again in contrast to apparent global

diversity patterns, which record increasing biodiversity

levels through time.

The primary limitation of this approach is that it

assumes that the underlying frequency distribution of

occurrences follows a particular statistical distribution.

That is, it is a parametric method. In this respect, it

differs from the closed-population capture-recapture

methods discussed earlier, which are nonparametric.

As a general rule, parametric methods are more precise

than nonparametric analogs when the assumptions

about the underlying distribution are met. When they

are not, however, the estimates can be very inaccurate.

Another limitation, shared by the capture-recapture estimates, is that the uncertainty associated with the estimated diversity increases as the proportion of unsampled taxa increases. That is, when the probability of a

taxon appearing in a sample is low on average (or when

there are few samples), the estimates are especially

prone to error. For this reason, the fact that a method

works well as an estimate of present-day diversity does

not necessarily mean that it will work well when applied

to the fossil record, except perhaps where that record

is unusually complete.



on. Wing and DiMichele (1994) used this approach to

examine local vegetation biodiversities in the Paleozoic

and Cenozoic floodplain data set discussed previously.

They found that, on average, local diversity in the late

Paleozoic was similar to local diversity in the early

Cenozoic. This was consistent with their findings for

regional diversity using the Chao-2 estimator. However,

they did find greater variation in diversity levels among

sites in the Cenozoic; in particular, the most speciesrich Cenozoic sites were much more diverse than the

most species-rich Paleozoic sites.

A second model, developed by Chao and Lee (1992),

utilizes the entire frequency distribution of occurrences

in a set of samples, rather than just the number occurring in only one or a few samples. Anderson and

coworkers (1996) estimated diversities with this

method for the late Triassic plant and insect data mentioned in the previous section. These estimates ranged

from about 45% lower (for insects) to 55% higher (for

plants) than estimates obtained with the GIGP model

discussed previously.

These models share the limitations of the closedpopulation capture-recapture models discussed previously. In particular, the Chao-1 estimator assumes

that probabilities of sampling are identical among samples for a particular taxon. The Chao and Lee estimator

makes a similar assumption: each taxon has a constant

probability of sampling associated with it. Further, it

treats each occurrence as a separate sample. That is, it

does not account for the fact that occurrences of taxa

are grouped according to the sampling units in which

they were found.



D. Other Nonparametric Methods



To this author’s knowledge, no other diversity estimation methods have been applied to the fossil record in

published studies. However, several other methods of

estimating biodiversity exist, and some may be applied

to fossil data in the future. These include those capturerecapture estimates that assume all taxa have an equal

probability of occurring in any given sample. When this

assumption is reasonable (for instance, when studying

groups of closely related taxa preserved in very similar

sediments), the resulting estimates should be more precise than the methods discussed previously, which generally have greater uncertainties associated with them.

Other estimates involve fitting observed distributions

of occurrences to particular statistical distributions (the



While most biodiversity estimates other than the closedpopulation capture-recapture models are parametric,

some are nonparametric. Two of these models have

been applied to paleontological data. One of these,

‘‘Chao-1,’’ was formulated by Chao (1984) and is actually mathematically equivalent to the Chao-2 estimate

discussed previously. In this case, however, the number

of individuals representing each taxon in a single sample is used rather than the frequency of occurrences in

a set of multiple samples. That is, f1 is the number of

taxa represented by only one individual in a sample, f2

is the number represented by two individuals, and so



III. UNEXPLORED DIVERSITY

ESTIMATES



62



FOSSIL RECORD



GIGP method discussed previously is an example of

this). These methods have been applied to estimate

present-day diversity for particular taxonomic groups

and particular regions, sometimes with good results.

Many of these methods and applications were reviewed

by Colwell and Coddington (1994) and by Bunge and

Fitzpatrick (1993). The major barrier to their application to the fossil record is that differences among taxa

in their probability of entering the fossil record, and

subsequent loss of fossiliferous rock (at rates that may

vary among regions and through time), may make distributions of fossil occurrences very different from distributions of occurrences of living organisms.



IV. CONCLUSIONS

As we have seen, many features of the fossil record

make assessing diversity trends difficult. Some of these

difficulties can be eased, at least in principle. For instance, unequal distribution of sampling effort by paleontologists can be reduced by emphasizing undersampled regions and strata in future fieldwork. Similarly,

the expanding palette of tools for dating fossils and

their surrounding sediments, their increasing precision,

and the development of more robust statistical methods

for chronostratigraphic correlation should progressively improve the fossil record’s temporal accuracy and

precision. However, some problems are less tractable.

For instance, fossiliferous rock is progressively lost as

it ages. Thus, the fossil records of large regions (and

thus the record of biodiversity in the associated habitats) may simply not exist. This problem is particularly

acute for older time intervals. Similarly, organisms in

some habitats are simply less likely to be preserved than

organisms in others, and those habitats will thus have

poorer records of their biodiversity history.

Several approaches have been used to account for

these sampling effects in the estimation of diversity or

(in the case of rarefaction) to minimize the effects of

variation in sample size on estimates of diversity. All

of these approaches were originally designed to estimate

local diversity; thus, their application to regional and

global patterns is problematic, and workers have interpreted results cautiously. Nevertheless, these results

can sometimes be used to eliminate some explanations

for particular biodiversity trends. For instance, Raup’s

use of rarefaction confirmed that the increased diversity

of echinoids through time was not solely due to the

greater number of paleontological samples available for

younger strata. When the results of these approaches

differ from one another, or from diversity trends appar-



ent from the fossil record in its entirety, however, things

become complicated. Do the estimates indeed account

for sampling effects and reveal true diversity trends, or

does violation of model assumptions render the estimated trends even less reliable then the uncorrected

diversity trends they are intended to improve?

The future of diversity estimation will no doubt involve considerable effort on several fronts. One promising approach is to apply several methods, then identify

biodiversity trends that are robust to these alternative

methods. Another is to investigate directly how different estimates are biased when particular assumptions

are violated, then devise means of minimizing these

biases. Diversity estimation methods that have not yet

been applied to the fossil record may be incorporated

into the paleobiological research program, and new estimates will undoubtedly be forthcoming as well. Nonparametric methods based on the frequency distribution

of occurrences have been identified as a promising area

for further progress by many biostatisticians. Finally,

much recent work focuses on assessing the completeness of the fossil record of particular taxa or strata. In

the future, these tools will undoubtedly be brought to

bear on the problem of estimating diversity. For the

past two decades, quantitative approaches have been

rapidly growing in popularity and sophistication among

paleobiologists. This movement is still in its infancy,

and its future is likely to produce an increasingly clear

picture of the history of biodiversity.



See Also the Following Articles

BIODIVERSITY, EVOLUTION AND • BIODIVERSITY, ORIGIN

OF • EXTINCTION, RATES OF • MASS EXTINCTIONS, NOTABLE

EXAMPLES OF • MEASUREMENT AND ANALYSIS OF

BIODIVERSITY • PALEOECOLOGY • SPECIES-AREA

RELATIONSHIPS



Bibliography

Colwell, R. K., and Coddington, J. A. (1994). Estimating terrestrial

biodiversity through extrapolation. Phil. Trans. Royal Soc. Lond.

B 345, 101–118.

Gilinsky, N. L., and Signor, P. W. (Eds.) (1991). Analytical Paleobiology. The Paleontological Society, Knoxville.

Guex, J. (1991). Biochronological Correlations. Springer-Verlag,

Berlin.

Pollock, K. H., Nichols, J. D., Brownie, C., and Hines, J. E. (1990).

Statistical inference for capture-recapture experiments. Wildlife

Monographs 107, 1–97.

Raup, D. M. (1976). Species diversity in the Phanerozoic: An interpretation. Palebiology 2, 289–297.

Sepkoski, J. J., Jr., Bambach, R. K., Raup, D. M., and Valentine, J.

W. (1981). Phanerozoic Marine Diversity and the Fossil Record.

Valentine, J. W. (Ed.) (1985). Phanerozoic Diversity Patterns: Profiles

in Macroevolution. Princeton University Press, Princeton.