Copyright 02000, The Paleontological Society
Origination and extinction components of taxonomic diversity:
general problems
Mike Foote
Abstract. Mathematical modeling of cladogenesis and fossil preservation is used to explore the
expected behavior of commonly used measures of taxonomic diversity and taxonomic rates with
respect to interval length, quality of preservation, position of interval in a stratigraphic succession,
and taxonomic rates themselves. Particular attention is focused on the independent estimation of
origination and extinction rates. Modeling supports intuitive and empirical arguments that singleinterval
taxa, being especially sensitive to variation in preservation and interval length, produce
many undesirable distortions of the fossil record. It may generally be preferable to base diversity
and rate measures on estimated numbers of taxa extant at single points in time rather than to adjust
conventional intervalbased measures by discarding singleinterval taxa.
A combination of modeling and empirical analysis of fossil genera supports two major trends in
marine animal evolution.(1)The Phanerozoic decline in taxonomic rates is unlikely to be an artifact
of secular improvement in the quality of the fossil record, a point that has been argued before on
different grounds. (2) The postPaleozoic rise in diversity may be exaggerated by the essentially
complete knowledge of the living fauna, but this bias is not the principal cause of the pattern. The
pattern may partly reflect a secular increase in preservation nevertheless.
Apparent temporal variation in taxonomic rates can be produced artificially by variation in preservation
rate. Some empirical arguments suggest, however, that much of the shortterm variation
in taxonomic rates observed in the fossil record is real. (1)For marine animals as a whole, the quality
of the fossil record of a higher taxon is not a good predictor of its apparent variability in taxonomic
rates. (2) For a sample data set covering a crosssection of higher taxa in the Ordovician,
most of the apparent variation in origination and extinction rates is not statistically attributable to
independently measured variation in preservation rates. (3)Previous work has shown that standardized
sampling to remove effects of variable preservation and sampling yields abundant temporal
variation in estimated taxonomic rates. While modeling suggests which rate measures are
likely to be most accurate in principle, the question of how best to capture true variation in taxonomic
rates remains open.
Mike Foote. Department of the Geoph!ysical Sciences, Unioersity of Chicago, Chicago, Illinols 60637. Email:
mfoote@midzua!y.uch~cago.edu
Accepted: 21 June 2000
Interval Estimates of Diversity and sure origination and extinction rates indepenTaxonomic
Rates dently? This paper focuses on measuring diversity
and rate changes over discrete stratiDo
changes in taxonomic diversity tend to graphic or temporal intervals, rather than in
be associated preferentially with changes in calculating longterm average rates (cf. Foote
origination rate or with changes in extinction in press a). rate? This rather simple empirical question, Using mathematical modeling of cladogenwhich
I address in a companion paper (Foote esis and fossil preservation, I will explore the
2000), involves a number of methodological behavior of several interval measures of diproblems
that I will consider here: (1)How are versity and taxonomic rates. This exercise
interval estimates of diversity and taxonomic points to a number of undesirable properties
rates affected by interval length? (2) How do of singleinterval taxa. The difficulties distrue
rates affect our ability to estimate rates? cussed here stand in addition to those that
(3) How are diversity and rate estimates af arise from differences in species abundance
fected by incompleteness of the fossil record? and, consequently, in preservation probability
(4) How are they affected by the finite strati (Buzas et al. 1982).Taxa confined to single ingraphic
extent of the fossilrecord? (5)How do tervals are problematic even if preservation is
the foregoing factors affect our ability to mea complete or if all taxa and time intervals are
O 2000 The Paleontological Society. All rights reserved.
DIVERSITY A N D RATE ESTIMATION 75
Four fundamental classes of taxa
Time interval of interest
FIGURE 1. Illustration o f four fundamental classes o f
taxa present during a stratigraphic interval. N,,is the
numbe r o f taxa confined t o the interval, N,,is the n u m 
ber that cross the b o tt om boundar y only, N,,is the n u m 
ber that cross the t o p boundary only, and N,,is the n u m 
ber that cross b o t h boundaries.
characterized by the same quality of preservation.
It is therefore advantageous to base
measures on taxa that cross between stratigraphic
or temporal intervals. Simply modifying
conventional measures by discarding
singleinterval taxa may not be the best approach,
however. Instead, it is preferable under
a wide range of circumstances to use measures
for which singleinterval are simply irrelevant
by the very nature of the measures.
Four Fundamental Classes of Taxa
Any taxon known or inferred from a stratigraphic
interval can be classified into one of
four mutually exclusivecategories (Fig. 1)(see
Barry et al. 1995 for a similar classification):
(1) taxa confined to the interval, i.e., taxa
whose first and last appearance are both within
the interval; (2) taxa that cross the bottom
boundary and make their last appearance
during the interval; (3) taxa that make their
first appearance during the interval and cross
the top boundary; and (4) taxa that range
through the entire interval, crossing both the
top and bottom boundaries. Using b and t to
refer to crossing the bottom and top boundaries
of an interval and using F and L to refer
to first and last appearance within the interval,
I will denote the numbers of taxa in the
four categories N,,, N,,, N,,, and N,,. The term
singleton is commonly used to refer to species
represented by a single specimen (Buzas and
Culver 1994, 1998).I will denote as a singleton
any taxon that is confined to a single stratigraphic
interval at the given level of resolution
(any member of the FL category). Although I
will refer to boundaries, these need not be
times of major biotic turnover; any recognizable
temporal or stratigraphic division can
serve as a point of reference. As I will discuss
below, there are many useful combinations of
these categories of taxa. Two that are especially
important are the total number of taxa
crossing the bottom boundary, N, (=
N,,+N,,), and the total number crossing the
top boundary, N, (= N,,+N,,).
All measures of diversity and taxonomic
rates for an interval are derived from combinations
of the numbers of taxa in the four fundamental
classes. The numbers denoted by N
with subscripts are the true numbers of taxa
with the given properties. Although these are
useful for modeling, in the fossilrecord we always
deal with observed numbers, which I
will denote by X with the corresponding subscripts(Foote1999:Appendix
7).The relationships
between true and observed numbers are
central to what will be developed below (see
Appendix for details). Of course,taxa can shift
categories when filtered through incomplete
preservation. For example, a taxon that truly
belongs to the bt category can contribute to
X,,, X,,, X,,, or X,,, depending on how much
of itsrange is truncated.It should also be kept
in mind that observed taxa (X) in the first
three categories must necessarily be found
within the interval, while the last category includes
taxa that are found before and after, but
not necessarily during, the interval.
Measures of Diversity and Rate
Numerous measures of diversity and rate
have been applied to paleontological data,
and their intuitive advantages and disadvantages
have been discussed at length (Gingerich
1987; Gilinsky 1991; Foote 1994; Harper
1996; Sepkoski and Koch 1996).Rather than
review all of them, I will explore the properties
of some commonly used measures that
have been proposed to solve particular problems
or to take advantage of particular aspects
76 MIKE FOOTE
TABLE1. Definitions of taxonomic diversity measures and taxonomic rate metrics for intervals of length At. Measures
are expressed in terms of numbers belonging to the four fundamental classes of taxa, N,,, N,,, N,,, and N,,
(Fig. I), or combinations derived from them (Appendix).
Diversity measures
Measure Definition
Total diversity, N,,,, NFL + NhL + NF,+ Nh, Total diversity minus singletons N,L + NF,+ Nht
Bottomboundary crossers, N, N,, + N,, Topboundary crossers, N, NF,+ Nh, Number of originations, No NFL+ N F ~ Number of extinctions, N, NFL+ N ~ L
Estimated mean standing diversity (N, + N,) 1 2
= (N,, + N,, + 2N,,)l 2
= (N,,, N 0 / 2 N, / 2 )
Rate measures
Measure
Pertaxon rate
Van Valen metric
Van Valen metric without singletons
Estimated percapita rate, and
of data. I will not discuss the wellknown
problems associated with poor constraints on
interval length (Gilinsky 1991; McGhee 1996;
Sepkoski and Koch 1996). Longterm averages,
appropriately calculated, are not very sensitive
to errors in interval length (Foote 1999:
Appendix 7). It is often shortterm variation
that is of interest, however, and the fundamental
distortion isthat an overestimate of interval
length yields an underestimate of taxonomic
rates, and conversely for an underestimate
of interval length. Thus, while dating
error may produce spurious extremes in taxonomic
rates, it is unlikely to produce longterm
secular patterns. The diversity measures
differ in whether they seek an estimate of
standing diversity at a point in time or of the
total number of taxa that exist during any part
of an interval. Most rate metrics start with a
tabulation of the number of events within an
interval and normalize this number by some
measure of diversity and by the length of the
interval. The goal of these normalizationsisto
obtain an estimate of the instantaneous, percapita
rates of origination and extinction, p
and q, per lineagemillionyears (Lmy)(Raup
1985). The percapita rate estimates advocated
here, and q^ (Table I), are derived directly
Definition
Origination: (N,, + N,,)l (N,,,,)lAt
Extinction: (N,, + NhL)l(N,,,,)lAt
Origination: (N,, + N,,)/ [(N,+ N , ) / 2 ]1At
Extinction: (N,, + N,,)I [(N,+ N , ) / 2 ]l A t
Origination: (N,,)I [(N,+ N,)1211At
Extinction: (N,,)/ [(N,.+ N,)/ 21 1At
p: ln(NhtlN,)lA t
9 : l n (Nh ,lN,)lA t
from branching theory rather than as a normalization
of the number of events observed
within an interval.
Table 1 gives definitions of diversity and
rate measures in terms of the four fundamental
classes of taxa and the interval length, At.
The expected values of these measures under
various conditions of completeness are derived
in the Appendix. Expressing these expectations
in terms of fundamental parameters
such as origination and extinction rates,
rather than counts of number of events, helps
to make sense of their behavior. Since the Appendix
contains exact expressions for a number
of quantities that have sometimesbeen derived
less directly with Monte Carlo methods,
it is hoped that it will be of some use to the
paleontologist. Table 2 provides a "road map"
to the Appendix.
The first two sections that follow will discuss
measures of diversity and rate in terms
of true numbers of taxa (N) in order to emphasize
problems that would exist even if the
fossil record were complete. The discussion
will then switch to observed numbers (X) in
order to emphasize problems related to paleontological
incompleteness.
DIVERSITY A N D RATE ESTIMATION 77
Effects of Interval Length
For the problems discussed here, it is natural
and convenient to measure time in multiples
of 11q, the average taxon duration. This
yields in effect a dimensionless expression of
time. Similarly, expressing origination and
preservation rates relative to q yields dimensionless
rate measures. Thus, an increase in
interval length with taxonomic rates held constant
has the same effect as an increase in taxonomic
rates with interval length held constant.The
number of top or bottomboundary
crossers may be larger or smaller depending
on where we choose our time lines, and if origination
is concentrated shortly after boundaries
and extinction shortly before them, average
standing diversity may be underestimated
(Raup 1991; Alroy 1992; Foote 1994).
Nevertheless, the number of boundary crossers
is not systematically affected by interval
length (Fig. 2) (Bambach 1999).This isjust one
potential advantage of measuring diversity Total minus singletons
using boundary crossers (see below). , 'Ol
The pertaxon rates have been used as a way
to normalize the number of origination or extinction
events by total diversity and interval
length. As interval length increases, a progressively
larger proportion of total diversity
consists of singletons (Fig. 2). Because these
taxa first appear and last appear within the
same interval, proportional origination and
extinction asymptotically approach unity as
At increases, and the pertaxon rates conse 3'01 Boundary crossers
quently decline as interval length increases
(Fig. 3; Appendix, section 2) (Gingerich 1987;
Foote 1994).
Van Valen (1984) used a rate metric designed
to normalize the number of originations
and extinctions by estimated average
standing diversity, (N, + Nt)/2, since this
number rather than total diversity better expresses
the number of taxa susceptible to orig Interval length (multiples of 1Iq)
ination or extinction at an instant in time FIGURE2. Eff e c t o f interval length o n numbe r o f taxa (Harper 1975; Van Valen 1984).This normali within a n interval. Rates are timehomogeneous and
zation implicitly assumes a linear change in fossil record is complete. Interval length is expressed as
standing diversity, a change that is expected multiples o f 119. Solid lines, p = 9; dashed lines, p =
1.59; dotted lines, p = 0.59. In this and subsequent figto
be exponential if rates are constant within ures, diversity at the start o f the interval is unit y . On l y
an interval. As At increases, the linear ap the numbe r o f boundary crossers is independent o f inproximation
becomes progressively worse, terval length.
and (N,+ N,)/ 2 overestimates mean standing
u
T
A
BI. ~2.
Guide to equations in Appendix. a
Quantity of ~nterest Preservdtion Edge effect? Equat~ons*
1A. Total diversity Complete  9
1B. 'Total diversity Constant No 24c,25c,2629,34,35e
1C. Total diversity Constant Yes 24b,25b,2629,34,35e
ID. Total diversity Variable
Yes 24a,25a,2629,34,35e
2A.
Taxa crossing bottom boundary only Complete  4
2B. Taxa crossing bottom boundary only Constant No 24~,25~,27,3lb,32a,34b,37d
2C. Taxa crossing bottom boundary only Constant Yes 24b,25b,27,31b,32a,34b
2D. Taxa crossing bottom boundary only Variable
Yes 24a,25a,27,31b,32a,34b
3A.
Taxa crossing top boundary only Complete  5
38. Taxa crossing top boundary only Constant No 24c,25~,28,31~,33a,34~,37e
3C.
Taxa crossing top boundary only Constant Yes 24b,25b,28,31~,33a,34c
3D. Taxa crossing top boundary only Variable
Yes 24a,25a,28,31c,33a,34c
4A. Total taxa crossing bottom boundary Complete  1
4B. Total taxa crossing bottom boundary Constant No 24c,25c,30c,35a
4C. Total taxa crossing bottom boundary Constant Yes 24b,25b,30b,35a
4D. Total taxa crossing bottom boundary Variable
Yes 24a,25a,30a,35a
5A. Total taxa crossing top boundary Complete  2
58. Total taxa crossing top boundary Constant No 24c,25c,30c,35b
5C. Total taxa crossing top boundary Constant Yes 24b,25b,30b,35b
511. Total taxa crossing top boundary Variable
Yes 24a,25a,30a,35b
6A. Taxa crossing both boundaries Complete  3
68. Taxa crossing both boundaries Constant No 24c,25c,30c,31a,34a,37c
6C.
Taxa crossing both boundaries Constant Yes 24b,25b,30b,31a,34a,36c
6D. Taxa crossing both boundaries Variable
Yes 24a,25a,30a,31a,34a
7A. Singletons Complete  6
7B. Singletons Constant No 24c,25c,29,31 d,32b,33b,34d,38b
7C. Singletons Constant Yes 24b,25b,29,31d,32b,33b,34d
7D. Singletons Variable
Yes 24a,25a,29,31d,32b,33b,34d
8A.
Number of originations or extinctions Complete  7,s
8B. Number of originations or extinctions Constant No 35c; lines 2B,38,78
8C.
Number of originations or extinctions Constant Yes 35c; lines 2C,3C,7C
8D. Number of originations or extinctions Variable
Yes 3512; lines 2D,3D,7D
9A. Proportional origination or extinction Complete  10,ll
9B. Proportional origination or extinction Constant No 10,ll; line 88
9C. Proportional origination or extinction Constant Yes 10,ll; line 8C
9E. Proportional origination or extinction Variable
Yes 10,ll; line 8D
10A. Per taxon rate metric Complete  12,13
10B. Per taxon rate metric Constant No 12,13; line 8 8
10C. Per taxon rate metric Constant Yes 12,13; line 8C
1 0 . Per taxon rate metric Variable
Yes 12,13; line 8D
11A.
Van Valen rate metric Complete  14,15,16,17
llB. Van Valen rate metric Constant No 39
11C.
Van Valen rate metric Constant Yes 14a; lines 4C,5C,X
C
11D.Van Valen rate metric Variable
Yes 14a; lines 4Dr5D,8D
T
A
B
L
E2.
Continued.
Quant~tyot rnterest I'reservatlon Edge effect? Equat~ons'
12A.
Van Valen rate metric (minus singletons)
128. Van Valen rate metric (minus singletons)
12C.
Van Valen rate metric (minus singletons)
12D.
Van Valen rate metric (minus singletons)
13A. Estimated percapita rates
138. Estimated percapita rates
13C. Estimated percapita rates
13D. Estimated percapita rates
14A.Probability of preservation before boundary
148. Probability of preservation before boundary
14C. Probability of preservation before boundary
15A. Probability of preservation after boundary
158. Probability of preservation after boundary
15C. Probability of preservation after boundary
16A.Probability of preservation during interval
168. Probability of being observed to cross a boundary
16C. Probability of being observed to cross a boundary
17A. Probability of being observed to cross a boundary
18. Probability that singleton is observed
19A. Probability that bL taxon is observed as such
19B. Probability that bL taxon is observed as such
19C. Probability that bL taxon is observed as such
20A. Probability that bL taxon is observed as singleton
20B. Probability that bL taxon is observed as singleton
20C. Probability that bL taxon is observed as singleton
21A. Probability that Ft taxon is observed as such
218. Probability that Ft taxon is observed as such
21C. Probability that Ft taxon is observed as such
22A. Probability that Ft taxon is observed as singleton
228. Probability that Ft taxon is observed as singleton
22C. Probability that Ft taxon is observed as singleton
23A. Probability that bt taxon is observed as such
238. Probability that bt taxon is observed as such
23C. Probability that bt taxon is observed as such
24A. Probability that bt taxon is observed as bL taxon
248. Probability that bt taxon is observed as bL taxon
24C. Probability that ht taxon is observed as hL taxon
25A. Probability that bt taxon is observed as Ft taxon
258. Probability that bt taxon is observed as Ft taxon
25C. Probability that bt taxon is observed as Ft taxon
26A. Probability that bt taxon is observed as singleton
26B. Probability that bt taxon is observed as singleton
27C. Probability that bt taxon is observed as singleton
Lines in l~stsof equations refer to numbered lines in this table
Complete
Constant
Constant
C
Variable
omplete
Constant
Constant
C
Variable
onstant
Constant
C
Variable
onstant
Constant
C
Variable
onstant during interval
Constant
Constant
C
Variable
onstant within interval
Constant
Constant
C
Variable
onstant
Constant
C
Variable
onstant
Constant
C
Variable
onstant
Constant
C
Variable
onstant
Constant
C
Variable
onstant
Constant
C
Variable
onstant
Constant
C
Variable
onstant
Constant
Variable
N

o
Yes
Yes
Y
No

es
Yes
No
Yes
Yes
No
Yes
Yes
Y
No

es
Yes
N

o
Yes
Yes
Y
No
es
Yes
Y
No
es
Yes
No
Yes
Yes
Y
No
es
Yes
Y
No
es
Yes
Y
No
es
Yes
No
Yes
Yes
1
18,19,20,21
8a; lines 28,38,4B,5B
18a; lines 2C,3C,4C,5C
18a; lines 2D,3D,4D,5D
2
22,23
2,23; lines 4B,58,6B
22,23; lines 4C,5C,6C
22,23; lines 4D,5D,6D
24c
2
24a
24b
5c
2
25a
25b
629
3
30c
0b
2
30a
9
2
24a,25a,26,31b
24b,25b,26,31b
24c,25c,26,31b
24a,25a,31a
24b,25b,31a
24c,25c,31a
25a,28,33b
25b,28,33b
25c,28,33b
25a,28,33a
25b,28,33a
25c,28,33a
24a,27,32b
24b,27,32b
24c,27,32b
24a,27,32a
24b,27,32a
24c,27,32a
4c,25c,26,31c
24a,25a,26,31d
24b,25b,26,31d
24c,25c,26,31d
24a,25a,26,31c
24b,25b,26,31c
'Q
V
80 MIKE FOOTE
b
a Pertaxon rate
0.0
0.0 0.2 0.4 0.6 0.8 1.0
7

.!:::I , , , , , C
m Van Valen metric (without singletons)
0.0
0.0 0.2 0.4 0.6 0.8 1.0
.0
 0.2) 4
a Estima ted percapita rate
0.01
0.0 0.2 0.4 0.6 0.8 1.0
Interval length (multiples of 1lq)
FIGUR3E. Effect of interval length on extinctionmetrics
with timehomogeneous rates and complete record.
Similar effects result for origination metrics. See Figure
2 for explanation. Dotted and dashed lines coincide for
the Van Valen metric. The percapita rate estimated from
numbers of boundary crossers is independent of interval
length. This is also the only metric for which origination
rate does not affect the estimation of extinction
rate and vice versa.
diversity when p Z q. The Van Valen metric
therefore underestimates true rates more as At
increases (Fig. 3), although the error is relatively
small if p and q are not very different.
Recognizing potential problems with singletons,
Harper (1996)suggested a variation on
Van Valen's metric in which singletons are
omitted. As interval length increases, more of
the extinctions and originations are singletons.
Discarding them completely therefore
leads to an underestimate of taxonomic rates
that becomes worse as the interval length increases
(Fig. 3).
The estimates of percapita rates advocated
here, $ and I j , are unaffected by interval
length. N,,/ N, gives the proportion of lineages
extant at the start of the interval that survive
to the end, and N,,/N, gives the proportion of
lineages extant at the end that were already
extant at the start. These ratios decay exponentially
with time if rates are constant within
the interval; thus the logarithm of each ratio
declines linearly with time. The magnitude of
the slope of this decline is exactly equal to the
extinction rate in the first case and the origination
rate in the second case. In fact, even if
rates are not constant, $ and I j provide unbiased
estimates of the mean rate within an interval
(Appendix, section 2).
Alroy (1996b)independently proposed rate
metrics similar to Van Valen's and Harper's,
normalizing by N, rather than (N, + N,)/2.
When p = q, Alroy's metrics are expected to
be identical to those of Van Valen and Harper
(since N, = N, in this case). Moreover, if p =
q, if pAt and qAt are relatively low, and if singletons
are ignored, Alroy's metric and Van
Valen's metric are approximately equal to $
and I j (see Appendix, section 2, and Alroy et
al. 2000).
In some cases, for example those involving
deepsea microfossils or Neogene macroinvertebrates,
temporal resolution may be sufficiently
fine that equallength intervals can be
established relatively easily (Wei and Kennett
1983, 1986; Pearson 1992, 1996; Budd et al.
1994; Budd and Johnson 1999). Alroy (1992,
1996b, 1998, 1999) has developed an alternative
approach to circumventing the problem
of unequal interval lengths. Using an exhaustive
compilation of faunal lists, he has ordi
DIVERSITY AND RATE ESTIMATION 81
nated first and last appearance data of North
American Cenozoic mammals and has used
this ordination to interpolate between wellconstrained
absolute ages. This has allowed
an arbitrarily fine, equalinterval temporal
grid to be superimposed on the data. Although
Alroy's approach is laborintensive, it
holds great promise for avoiding problems associated
with uneven and uncertain interval
lengths.
In general, the only measures of diversity
and taxonomic rates that are expected to be independent
of interval length are those based
exclusively on boundary crossers. This suggests
that, if interval length varies substantially,
it is a good idea to measure diversity as
N, or N, and to measure taxonomic rates as $
and I j . It is nevertheless often the case empirically
that there is ample true variation in diversity
and rates, with the result that most of
the apparent variation in rates is not attributable
to variation in interval length (Raup 1986;
Gingerich 1987;Collins 1989;Foote 1994;Patzkowsky
and Holland 1997).
Effects of Rates on Rate Estimation
For certain rate metrics, true rates of taxonomic
evolution affect our ability to estimate
rates in two principal ways, concerning the
magnitude of rates and the difference between
origination and extinction rates.
Because an increase in taxonomic rates has
the same effect as an increase in interval length,
the pertaxon rate and the Van Valen metric,
with or without singletons, become less accurate
as taxonomic rates increase. The only exception
to this is that the Van Valen metric is
insensitive to interval length and taxonomic
rates if p = 9 and if singletons are included.
The greater the difference between p and 9,
the greater the discrepancy between the true
taxonomic rates and the estimates given by the
pertaxon rate and the Van Valen metric, with
or without singletons (Fig. 3). This is largely
because of the increasing discrepancy between
an exponential diversity change within
an interval and the linear change implicit in
the normalization by (N, + N,)/ 2. The pertaxon
rate and the singletonfree Van Valen metric
exhibit a more serious problem. Suppose
that extinction rate is held fixed. As origination
rate increases (from the dotted to the solid
to the dashed lines in Fig. 3), the estimate
of extinction rate decreases when either of
these rate metrics is used. (Likewise, if origination
rate is constant and extinction rate
changes, the estimate of origination rate
changes in the opposite direction.) The reason
the pertaxon rate behaves this way can be
seen by inspecting equations (8a) and (9a)in
the Appendix, which give the numerator and
denominator of the rate metric. The normalization
makes sense only if p = 9. The reason
for the bias in the singletonfree Van Valen
metric is different. Equation (21) shows that
this metric approximates the extinction rate
only when p = 9 and when both rates are low.
With these two rate metrics, true variation in
rates will therefore contribute to a spurious
negative correlation between origination and
extinction. This will complicate the independent
measurement of origination and extinction
rates.
In contrast to the other rate metrics, $ is unaffected
by extinction rate, while I j is unaffected
by origination rate. This makes $ and I j
especially useful if one desires independent
estimates of origination and extinction rates.
Incomplete Preservation
Preservation can be modeled in a number of
realistic ways that include variation in time
and space (Shaw 1964;Koch and Morgan 1988;
Marshall 1994; Holland 1995; Holland and
Patzkowsky 1999; Weiss and Marshall 1999).
As a heuristic tool for understanding the behavior
of diversity and rate measures, it is
convenient to focus on the temporal aspect
and to start by assuming timehomogeneous
fossil preservation at a constant percapita rate
r per Lmy (Paul 1982,1988;Pease 1985;Strauss
and Sadler 1989; Marshall 1990; Foote and
Raup 1996; Solow and Smith 1997; Foote
1997).This simple assumption will be relaxed
below.In the timehomogeneous case, the proportion
of lineages preserved is equal to rl (9
+ r) if p = q and if the fossil record is of effectively
infinite length (Pease 1985; Solow
and Smith 1997)(see Edge Effects, below). It
is therefore natural for many problems to express
preservation rate as a multiple of 9.
Throughout this discussion I will assume tax
82 MIKE FOOTE
onomic homogeneity of taxonomic and preservational
rates. For modeling, this assumption
can easily be relaxed by performing calculations
for an arbitrary number of rate classes
and combining the results (see Buzas et al.
1982, Koch and Morgan 1988, Holland 1995,
Holland and Patzkowsky 1999, and Weiss and
Marshall 1999 for explicit treatments of taxonomic
heterogeneity of preservation).
As preservation rate decreases, there is a
regular decrease in observed numbersin nearly
all categories of taxa within an interval. The
sole exception is singletons, whose behavior is
especially problematic. This is shown in Figure
4, which portrays the observed number of
taxa relative to the true number. As preservation
rate decreases, singletons constitute an
ever greater proportion of observed taxa. For
a certain range of values of preservation rate
and interval length, the observed number of
singletons is not only relatively high but also
absolutely greater than the true number. In
general, the observed excess of singletons
causes the pertaxon rate and the Van Valen
metric to increase as preservation rate decreases
(Fig. 5). Because all categories of
boundary crossers are diminished in the same
proportion by incomplete, timehomogenous
preservation, however, p, 4, and the Van Valen
metric without singletons are unaffected by
incomplete preservation, as are other metrics
based only on boundary crossers (Alroy
1996b).This result is related to the fact that the
observed age distribution of taxa, exclusiveof
singletons, is expected to be identical to the
true age distribution (Foote and Raup 1996;
Foote 1997; Solow and Smith 1997).Without
independent estimates of preservation rate, it
may be difficult to distinguish a truly high
number of singletons (reflecting high taxonomic
rates) from a preservational artifact.
I have been unable to develop a measure of
absolute diversity that is insensitive to incomplete
preservation. It is nevertheless possible
to estimate changes in diversity accurately if
we accept the point that a proportional or logarithmic
scale is a natural one with which to
measure diversity (Sepkoski 1991).The quantity
ln(N,IN,) gives the proportional change
in diversity through an interval, i.e., the fundamental
growth rate (p  q) times the inter
& 1Total minus singletons E,
& , O ] Boundary crossers
0.0 2.0 4.0 6.0 8.0 10.0
Preservation rate per Lmy (multiples of q)
4.
bers of taxa within an interval. Preservation is timehomogeneous
and there are no edge effects.Interval length
is fixed at 0.5/q, and preservation rate Y is expressed as
multiples of extinction rate q. Solid, dashed, and dotted
lines are as in Figure 2. Most categories of observed taxa
increase monotonically with preservation rate, but the
number of observed singletons is disproportionately
large when preservation rate is low.
FIGURE Effect of preservation rate on observed num
DIVERSITY AND RATE ESTIMATION 83
,
o
 0.4 C
rr" Pertaxon rate 0.0
0.0 2.0 4.0 6.0 8.0 10.0
.....................................................................................
Van Valen metric (without singletonsl 7
m
PC Estimated percapita rate 0.0
0.0 2.0 4.0 6.0 8.0 10.0
Preservation rate per Lrny (multiples of q)
FIGURE5. Effects of preservation rate on extinction
metrics. Similar effects obtain for origination metrics.
See Figure 4 for explanation. Because of the inclusion of
singletons, the pertaxon rate and the Van Valen metric
are strongly influenced by preservation. If singletons are
excluded, the Van Valen metric is independent of preservation
rate, but it is sensitive to the difference between
origination and extinction rates.
val length At. In the timehomogeneous case,
this quantity is unaffected by incomplete preservation.
If one has estimates of preservation rate and
is interested in true levels of diversity, then it
is a relatively straightforward matter to adjust
observed numbers of boundary crossers in the
timehomogeneous case: the true number is
equal to the observed number times [(p + r)(q
+ Y)/ ]Y* (see Appendix, section 5).The case of
variable rates is more realistic, however. How
to adjust diversity measures in the face of variable
preservation remains an important problem.Rarefaction
and other approachesinvolving
standardized resampling and subsampling
of data have been used extensively (Hessler
and Sanders 1967; Sanders 1968;Jackson
et al. 1993;Rex et al. 1993,1997;Raymond and
Metz 1995; Alroy 1996a,b, 1998, 1999; Miller
and Foote 1996; Markwick 1998; Marshall et
al. 1999), as have methods that rely on an estimated
phylogeny to identify and fill gaps
(Norell 1992;Benton 1994;Johnson 1998).The
relative performance of these various approaches
under different conditionsstillneeds
further exploration.
In summary, many effects of incompleteness
can be overcome in the case of timehomogeneous
preservation if taxonomic rates and
changes in diversity are measured on the basis
of relative numbers of boundary crossers.
Edge Effects
The fossil record as a whole, or any part of
it we investigate, has a discrete beginning and
end. During any interval of measurement,
whether coarse or fine, the presence of a taxon
can be inferred because the taxon is actually
preserved during the interval or because it is
preserved before and after the interval; this is
the standard rangethrough approach. As the
interval in question falls toward either edge of
the record, our ability to infer the presence of
taxa by the rangethrough method diminishes.
This creates a series of related edge effects
(Figs. 6, 7). Apparent diversity declines. The
number of first appearances is high toward
the lower edge (astaxa that truly extend below
the beginning of the record make their first
appearance) and the number of last appearances
is high toward the top. The number of
MIKE FOOTE
Edge effects
True diversity constant
0.4 Total minus singletons 1
0.3 Boundary crossers m
m 4
0 2 4 6 8 10
Time (increments of 1lq)
FIGUR6E. Edge effects on taxonomic diversity. Figure
shows number of observed taxa in relation to distance
of the interval from beginning or end of fossil record. In
all cases p = q, r = q, and interval length is equal to 0.51
q. The height of curves depends on the value of r. The
effect decays exponentially and the distance from the
edge at which the curves level off is a function of the
taxonomic rates (see text for further explanation).
singletons increases, as taxa that would have
been observed to cross a boundary are now
lesslikely to be found before the interval (near
the lower edge) or after the interval (near the
upper edge). The singletons contribute heavily
to counts of origination and extinction. As
a result of these edge effects, the pertaxon
rate and the Van Valen metric with singletons
are inflated toward the edges. Note that estimates
of both origination and extinction rate
for these metrics are inflated near both edges.
The increase in first and last appearances
relative to total numbers also means that @ and
the Van Valen origination metric without singletons
are inflated toward the beginning of
the record, while the corresponding extinction
metrics are inflated toward the end. Note that
the total number of last appearances is low
near the beginning of the record and that the
apparent change in standing diversity (the
difference between X, and X,) can be substantial
within a single interval near an edge. Because
the Van Valen metric without singletons
compares a reduced number of events to an
estimate of average standing diversity that is
changing substantially near the edges, the extinction
metric is also depressed near the beginning
of the record while the origination
metric is also depressed near the end. In contrast
to the three other metrics, @ and i j are affected
only toward the beginning or end of the
record, respectively.
Edge effects are significant only to the extent
that a taxon extant at some point in time
is likely to intersect either edge. At a distance
t from the bottom edge, the edge is no longer
felt when eqt is acceptably small; near the top
edge, the relevant quantity is epf.For example,
suppose p = q and the interval is placed near
the bottom edge. If the interval is separated
from the edge by 2.3 times the mean taxon duration
(2.3/q),then estis about 0.10, and @, ij,
X,,, X,,, X,,, X,, X,, and X,, are within about
1% of the values they would have in the absence
of edge effects. Practically speaking,
then, an edge is no longer felt within about
two or so average taxon lengths.
I have presented edge effects that result
from incomplete preservation within the window
of observation as if they were created
only by the termination of the fossil record at
DIVERSITY AND RATE ESTIMATION 85
Edge effects
True rates constant
.
,
0
2 0,0 Pertaxon rate
0 2 4 6 8 10
Estimated percapita rate 0,
0 2 4 6 8 10
Time (increments of 1iq)
FIGURE7. Edge effects on taxonomic rate metrics. In all
cases p = q, r = q, and interval length is equal to 0.519.
Solid line, extinction metric; dotted line, origination
metric. All rate metrics except estimated percapita rate
are affected at both edges.
either end. Qualitatively different but mathematically
identical edge effects are also created
by sudden drops in preservation rate
(Holland 1995) and by major changes in evolutionary
rates such as mass extinctions and
evolutionary radiations (Signor and Lipps
1982). Gilinsky and Bambach (1987) discuss
an edge effect that results from the definition
of certain rate metrics even if preservation is
complete; proportional origination and extinction
(number of events divided by total diversity)
must be unity in the first and last intervals,
respectively. Similarly, $ and I j are undefined
in the first and last intervals.
A different kind of edge effect is created by
a singular increase in preservation rate. This is
obviously relevant to the effects of Recent taxa
on patterns of diversity and taxonomic rates
(Raup 1972, 1979; Pease 1988a,b, 1992). Because
the Recent fauna of skeletonized marine
animals is very well known, taxa that lack a
late Cenozoic fossil record can have their
ranges pulled forward, with the result that apparent
diversity is likely to be inflated toward
the Recent and apparent extinction rate islikely
to decline (Figs. 8, 9). Whether apparent
origination rate increases, decreases, or is unaffected
depends on which rate metric is used
(Fig. 9).
For fossil marine animal genera, a substantial
number of taxa extend from the Recent
back into the early to mid Tertiary. It is thus
conceivable that the rise in diversity and decline
in background rates seen during the
Phanerozoic (Raup and Sepkoski 1982; Van
Valen 1984; Sepkoski 1996, 1997, 1998) are
partly artificial (Raup 1972; Pease 1985,
1988a,b, 1992). As Sepkoski (1997) showed,
however, the increase in genus diversity, measured
as total nonsingleton genera in an interval,
persists even when Recent genera are
included only if they are known to have a PlioPleistocene
fossil record. Figure 10 shows the
same result for boundarycrossing diversity.
(For the sake of completeness, Figures 10 and
11 also show the effect of removing all genera
that extend to the Recent, a culling that is unreasonably
extreme since many of these genera
have a fossil record near the Recent.) The
decline in extinction rate is seen in both the
Paleozoic and postPaleozoic even when Re
MIKE FOOTE
Complete preservation in "Recent"
True diversity constant
Complete preservation in "Recent"
True rates constant
0.6 ]Total minus singletons
Pertaxon rate
, ,  4
O.6
0.4.
0.2.
0 . 0 ~
.5] Van Valen metric (without singletons)
Total 1.5 al4
T E
a2 1.0. 0
4
.0
4LE
0.5. L
.
0
4
" 0.0,
Van Valen metric
;#l:L ,
0.2 ~ingletons 4.0.
0.4' Boundary crossers 2.0 Estimated percapita rate
C
E
......,
0.2
0.1
0.0, " 0.0.
0 5 10 15 20 25 0 5 10 15 20 25
Time (increments of 1lq) Time (increments of 1Iql
z 4
r
i3 n
E
2 0.1
I
f
al
8
FIGUR9. E Effects of complete preservation at end of FIGURE8. Effects of complete preservation at end of fossil record on taxonomic rate metrics. See Figure 8 for fossil record on observed numbers of taxa. Interval explanation. Solid line, extinction metric; dotted line, length is equal to 0.519, p = q, and r = q. As with other origination metric, edge effects, the edge is no longer relevant beyond about
L= 4
1.0
.
0
C
m
0.01
two taxon lengths.
OC 0.04
0 5 10 15 20 25 0 5 10 15 20 25
DIVERSITY AND RATE ESTIMATION
600 500 400 300 200 100
Geologic time (Ma)
FIGURE10. Genus diversity through the Phanerozoic. Data are from Sepkoski's unpublished compendium (Sepkoski
1996).Diversity curves show number of boundary crossers based on 25,049 fossil genera whose first and last
appearances are fully resolved to one of 107 stratigraphic intervals of about 5.1 m.y. average duration (Foote and
Sepkoski 1999; ages based mainly on Harland et al. 1990, Tucker and McKerrow 1995, and Bowring and Erwin
1998).Solid curve, all fossil genera included. Dashed curve, 3759 fossil genera still extant today are excluded. This
is an overly extreme culling, since many extant genera do in fact have a fossil record near the Recent. Dotted curve,
1630 extant genera that are known to have a PlioPleistocene fossil record are included (Sepkoski 1997). Dotted
curve is an absolute minimum, since it is unknown to what extent Sepkoski had documented PlioPleistocene occurrences.
Curves have similar shapes except for the Tertiary portions. The Cenozoic rise in diversity may be exaggerated
by nearly complete knowledge of the Recent fauna, but it is unlikely to be a complete artifact of this bias.
Similar results hold for total diversity (Sepkoski 1997).
cent genera lacking a PlioPleistocene record val (Fig. 12). (An increase in origination rate
are omitted (Fig. 11).Moreover, the decline in yields converse results.) Because of incomorigination
rate is seen, at least in the Paleo plete preservation, the last appearances are
zoic, with f?,a metric that is not expected to smeared back in time (Signor and Lipps 1982;
feel the upper edge of the fossil record (Fig. Raup 1989; Meldahl 1990; Koch 1991; Stanley
ll), and with the singletonfree Van Valen and Yang 1994;Rampino and Adler 1998).The
metric (data not presented), which is expected estimate of extinction rate in the interval is
to increase as a result of essentially complete therefore lower than it should be, while it is
preservation in the Recent. The early Paleo higher than it should be in earlier intervals. As
zoic decline in origination rate may be exag discussed above, changes in one rate affect esgerated
by the lefthand edge effect, but the timates of both rates with the pertaxon metdecline
continues far beyond the point where ric, the Van Valen metric, and the singletonthis
edge has a substantial influence.Thesere free Van Valen metric. Incompleteness and the
sults suggest that the Cenozoic increase in di correlation of rate estimates combine to proversity
and the Phanerozoic decline in taxo duce complicated signals in apparent originomic
rates seen in marine animals are not ar nation and extinction rates even if only one
tifacts of our relatively complete knowledge of rate varies. With p? and 4, however, a change in
the Recent fauna. one rate leaves the estimate of the other unaffected,
even in the case of incomplete but hoTemporal
Variation in Taxonomic Rates mogeneous preservation. This property, lackConsider
the effect of an increase in extinc ing in other rate metrics, should lead us to
tion rate that lasts for one stratigraphic inter favor p? and q, especially in cases where inde
0
MIKE FOOTE
600 500 400 300 200 100
Geologic time (Ma)
600 500 400 300 200 100
Geologic time (Ma)
FIGURE11. Percapita origination rate (A) and extinction rate (B) for genus data portrayed in Figure 10. Solid,
dashed, and dotted curves as in Figure 10. Complete culling of extant taxa yields extinction rates that rise toward
the Recent (exhibiting the edge effect of Figure 7), but this culling is unreasonably extreme (see Fig. 10).The similarity
of solid and dotted curves suggests that the Phanerozoic decline in rates is not a consequence of nearly
complete knowledge of the living fauna.
pendent estimates of origination and extinc main constant, the number of singletons will
tion are desired. gradually change in the opposite direction.
Thus, the pertaxon rate and the Van Valen
Temporal Variation in Rate of Preservation metric will show spurious secular changes
If preservation rate increases or decreases (Fig.13).Partly for this reason, Pease (1988a,b,
gradually over time while taxonomic rates re 1992) argued that the Phanerozoic decline in
0
DIVERSITY A N D RATE ESTIMATION 89
Extinction pulse
Y
I
0.5
.
0 
Y Per taxon rate
"
m
0.07
( Van Valen metric
.0
Y
m Van Valen metric (without singletons1
0.0
0 5 10 15 20 25
IEstima ted percapita rate
0 5 10 15 20 25
Time (increments of 1/q)
F I C ~ R E12. Eff e c t o f shortlived increase i n extinction
rate o n taxonomic rate metrics. Solid line, extinction
metric; dotted line, origination metric. Extinction rate q
is constant except for a fivefold increase that lasts for
one interval (indicated b y the cross). Interval length is
equal t o 0.519, and r = y = 9. Because the record is incomplete,
extinction rate appears lower t h a n it should
taxonomic rates may be an artifact of increasing
completeness of the fossil record. If the
change in preservation rate is smooth, however,
then top and bottomboundary crossers
will not experience significantly different
preservational histories. Thus, rate metrics
based on boundary crossers should barely be
affected by longterm secular changes in preservation
rate.The fact that the Phanerozoic decline
in rates is seen in the singletonfree Van
Valen metric and in $ and q^ (Fig. 11) suggests
that this decline is not an artifact of a secular
increase in the quality of the fossilrecord. Holman
(1985) also argued that the decline in
rates is real, since the frequency of gaps, estimated
from the stratigraphic ranges of lowerlevel
taxa within the ranges of the higher
taxa containing them, does not show an obvious
decrease through the Phanerozoic.
Figures 14 and 15 illustrate the more complicated
effects of a sudden increase in preservation
rate. (A sudden decrease has converse
effects.) All categories of taxa of course increase.
Top and bottomboundary crossers
are affected equally only if origination and extinction
rates are equal. When diversity is truly
increasing, the magnitude of the increase is
exaggerated, and conversely when diversity is
decreasing (Appendix, section 5). Taxa that
would have been singletons in adjacent intervals
had preservation been homogeneous now
extend into the interval in question; the number
of singletons in adjacent intervals therefore
declines. The number of first and last appearances
increases with the increase in preservation
rate, so that both the origination and
extinction rates appear to increase. In preceding
intervals, the number of taxa that would
have made their last appearance is reduced
because they now appear last in the interval
with better preservation. The same is true of
i n the interval o f the extinction spike, and the extra extinctions
are spread backward i n time. W i t h all metrics
except estimated percapita rate, the estimate o f origination
rate is also aff e c t ed (see Fig. 3). A s wi t h related
edge e ff e c ts(Fig. 7),the e ff e c t o f a transient rate pulse
decays exponentially until it is n o longer detectable a fter
about two taxon lengths. A transient increase i n origination
rate has converse e ff e c ts,whi ch propagate i n the
opposite direction i n time.
90 MIKE FOOTE
Exponential increase in preservation rate
2 , 5 1 ~ a nValen metric
Van Valen metric (without singletons)
0.8 ...................................................................................
.2 Estimated percapita rate
3
E 1 0.................................................................................... al
0.8 0
4
.0 0.6. w
E 0.4 L
0
.0
3
0.2
CL: 0.07 I
0 5 10 15 20 25
Time (increments of 1Iq)
FIGURE13. Effect of secular trend in preservation rate
on taxonomic rate metrics. Interval length is equal to
0 . 5 / q , and p = q. Dotted line, origination; solid line, extinction.
Preservation rate increases exponentially from
q to 29. There are no edge effects. Results for linear
change in preservation rate are similar. Metrics that disregard
singletons are barely affected by a smooth
change in preservation rate.
first appearances in succeeding intervals. As a
result, all metrics underestimate extinction
rate before the interval of increased preservation
and underestimate origination rate
afterwards.In addition, the pertaxon rate and
the Van Valen metric underestimate origination
beforehand, while they underestimate extinction
afterwards. This is because they include
singletons in the origination and extinction
counts. The singletonfree Van Valen met 
ric overestimates origination before the pulse
in preservation and overestimates extinction
afterwards. A strength of the proposed estimates
of percapita rates is that p is affected
only during and after the interval of unusually
high (or low) preservation, and I j is affected
only during and before this interval.
It is often observed that certain intervals of
time appear to have unusually high rates of
both origination and extinction (Allmon et al.
1993).Because apparent rates increase when
preservation increases, abrupt variation in
preservation rates can induce a spurious positive
correlation between origination and extinction
metrics even if the two rates are in fact
independent. This effect is especially strong
for the pertaxon rate and the Van Valen metric,
since both rates also apparently decrease
before and after an interval with better preservation.
The effect is quite evident empirically
as well. Mark and Flessa (1977)and Alroy
(1996b, 1998)showed that apparent origination
and extinction rates are less strongly correlated
if singletons are removed. For Phanerozoic
marine animal genera, the pertaxon
rate and the Van Valen metric tend to show
higher correlations than do the singletonfree
Van Valen metric and p? and I j (Table 3). The
correlations tend to be lowest for the singletonfree
Van Valen metric, but this may partly
reflect the fact that true, independentvariation
in origination and extinction rates tends to
yield a spurious negative correlation between
the apparent rates with this metric (Fig. 3).
More generally, variation in preservation
rate can artificially create patterns that resemble
true temporal variation in origination and
extinction rates. This is a fundamental problem
that has long plagued paleontology. The
extreme case of Lagerstatten is relatively easy
DIVERSITY AND RATE ESTIMATION
Pulse in preservation
0.8 0.3
0
Total minus singletons Taxa crossing top only
0
0.0
.24
0 5 10 15 20 2
+
5
:
0.0
.#
0
1
:#
5
~
10 15 20
,
25
11
:
0.6
#
0.31
::::
:: Bottom boundary crossers Taxa crossing both boundaries
3
L
n
5
0 ; 00,4 .2
n
0
0.0 0.0, I
0 5 10 15 20 25 0 5 10 15 20 25
Time (increments of 1lq) Time (increments of 1lq)
FIGURE14. Effect of shortlived increase in preservation rate on number of observed taxa. Preservation rate r is
constant except for a fivefold increase that lasts for one interval (indicated by the cross).Interval length is equal to
0.519, and r = p = q. Transient decrease in preservation rate has opposite effects.
92 MIKE FOOTE
Pulse in preservation
1.51
Pertaxon rate 0.0
0 5 10 15 20 25
Van Valen metric 0.0
0 5 10 15 20 25
Van Valen metric (without singletons) 0.0
0 5 10 15 20 25
.0 
4
Estimated percapita rate [r 0.0
0 5 10 15 20 25
Time (increments of 1lq)
FIGURE15. Effect of shortlived increase in preservation
rate on taxonomic rate metrics. Solid line, extinction
metric; dotted line, origination metric. Preservation rate
r is constant except for a fivefold increase that lasts for
one interval (indicated by the cross). Interval length is
equal to 0.5Iq and r = p = q. All metrics increase artificially
as a result of spike in preservation. Because singletons
are depressed on both sides of the preservationto
identify, but more subtle variation in preservation
rate is likely to be less obvious.
Distinguishing Variation in Taxonomic
Rates from Variation in Rates of
Preservation
To what extent is apparent variation in origination
and extinction rates real, and to what
extent is it an artifact of variation in preservation
rate? This question must be addressed
case by case (Koch and Morgan 1988; Koch
1991;Alroy 1996b). I will offertwo arguments,
suggestive but certainly not conclusive, that
there is a real signal of variation in taxonomic
rates for marine animal genera as a whole.
First, if the apparent variation in taxonomic
rates were an artifact of variable preservation,
groups with lower preservation potential
should have more apparent variation in taxonomic
rates. This is because the effect of
changes in preservation rate is greater when
preservation rate on average is lower (Appendix,
section 3). There are diminishing returns,
such that an increment in preservation rate has
a substantial effect if the record is poor but a
negligible effect if the record is good. Figure
16comparesthe estimated probability of preservation
per genus per stratigraphic interval,
measured as the FreqRat (Foote and Raup
1996; Wagner 1997; Cheetham and Jackson
1998; Foote and Sepkoski 1999),with the variability
in taxonomic rates, measured as the
median absolute difference in log rate between
adjacent stratigraphic intervals, for a
number of higher taxa of animals. The data are
those used in Figures 10 and 11.Contrary to
the expectations of the artifact hypothesis, the
correlations between preservability and rate
variability are in fact positive (thoughnot significantly
so). Clearly, overall quality of preservation
is not a good predictor of apparent
variation in taxonomic rates.
a1 spike (Fig. 14), rate metrics that include singletons are
depressed on both sides of the spike. Note that with the
estimated percapita rate, apparent origination is affected
only during and after the preservational spike,
while extinction is affected only before and during the
spike. With other metrics, both rates are affected in both
directions. Transient decrease in preservation rate has
opposite effects.
DIVERSITY A N D RATE ESTIMATION
TABLE3. Correlation b e twe e n change i n origination
metric and change i n extinction metric for Phanerozoic
marine animal genera (unpublished data fr o m Sepkoski;
see Sepkos k i 1996, 1997, and Figs. 10 and 11 o f this paper).
First diff e r enc e s are us ed i n order t o detrend t h e
data. Correlation coefficient is Kendall's 7.
Lletrlc Correlation
Pertaxon rate 0.599
Va n Val en metric 0.669
Va n Val en metric (wit h o u t singletpns) 0.256
Estimated percapita rate ( p and q ) 0.381
The second argument concerns the proportion
of variation in apparent taxonomic rates
that is potentially attributable to variation in
preservation rates. This last quantity is most
directly estimated not with first and last appearance
data alone, but rather with occurrences
within stratigraphic ranges. I have
used data on occurrences of Ordovician brachiopods,
mollusks, and trilobites kindly supplied
by Arnold I. Miller of the University of
Cincinnati. The data are part of an ongoing effort
to analyze temporal, geographic, and environmental
patterns of diversity, origination,
and extinction through the Ordovician (Miller
and Mao 1995, 1998; Miller and Foote 1996;
Miller 1997a,b, 1998).At the time of writing,
the data mainly cover Laurentia, China, Baltica,
East Avalonia, Bohemia, Australia, and
South America (including the Precordillera). I
analyzed occurrences and ranges at the subseries
level of resolution and ignored occurrences
that could not be adequately resolved.
All told, I included 1075 genera and 7461 occurrences.
I first used gap analysis (Paul 1982,1998) to
estimate the preservation probability, R,, for
each Ordovician subseries. I tallied the number
of genera, X,,, known both before and after
the interval in question and the number of
these genera actually sampled during the interval,
X ,,,,, and estimated R, as X ,,,,, /X,,.
(Note that first and last occurrences of genera
are necessarily discarded by this approach. To
include them, as Paul did, would bias the estimate
of R, upward [see Holman 1985;Maas
et al. 1995; Foote and Raup 1996; Markwick
1998; Foote in press b].)As is usually the case,
R, is the joint probability that a taxon is preserved,
collected, published, and entered into
1.2 Extinction
1 I 8
FIGURE16. Qualit y o f preservation v e rsus apparent
variability i n percapita origination and extinction rates
for genera wi t h i n higher taxa. T h e FreqRat (Foote and
Raup 1996)is us ed t o estimate probability o f genus preservation
wit h i n stratigraphic intervals o f about 5.1 m . y .
duration (Foote and Sepkos k i 1999).Data are fr o m Sepkos
k i (s e e Figs. 10, 11). To measur e abrupt rather t h a n
longt e rm variation, variability is measur ed as t h e m e 
d i a n absolute diff e r enc e i n log taxonomic rates b e twe e n
adjacent stratigraphic intervals. Quality o f preservation
is n o t a good predictor o f apparent variability i n rates,
suggesting that this variability m a y b e at least partly
real. Taxa analyzed are Ammo n o i d e a , Anthozoa, Asterozoa,
Bivalvia, Blastozoa, Brachiopoda, Bryozoa, Ce p h 
alopoda, Chondrichthyes, Conodonta, Crinoidea, Echinoidea,
Gastropoda, Graptolithina, Malacostraca, Nautiloidea,
Osteichthyes, Ostracoda, Polychaeta, Porifera,
and Trilobita. Not e that Cephalopoda contains A m m o 
noidea and Nautiloidea; t h u s n o t all points are independent.
the database. Since R, is expected to be equal
to 1  exp(r,At), the preservation rate r, is estimated
as ln(l  R ,)/ At (Appendix: eq. 26).
Because Miller's data cover the Ordovician
only, it would be impossible to estimate preservation
probability in this way for the first
and last intervals of the Ordovician. I there
94 MIKE FOOTE
fore also used Sepkoski's genus data to tabulate
Cambrian first occurrences and postOrdovician
last occurrences of genera present in
Miller's data. Even though I have used data
outside the Ordovician, it is still likely that
there are edge effects. Of 1187total genera in
Miller's data, 108(9.1%) are not found at all in
Sepkoski's data.
If variation in apparent taxonomic rates
were dominated by variation in preservation
rates, then correlations between r and 6 and
between r and I j should both be large and positive.
In fact, the first is negative and the second
is positive, and neither is statistically significant
(productmoment correlation coefficients:
rr,,i = 0.10; r,, = 0.36). (Results are
similar if the Kendall's T is used, except that
T , is ~ small and positive rather than small and
negative.) It is possible that an effect of preservation
rates is obscured because all three
rates show temporal trends (Fig. 17). If we
take first differences, there is a weak negative
effect of change in preservation rate on change
in apparent origination rate, while the effect of
change in preservation rate on change in apparent
extinction rate is significant and positive
(r,,,= 0.16; r,, = 0.66).Even though
this last correlation is statistically significant
(p < 0.05), the proportion of variation in apparent
extinction rate that can be explained by
variation in preservation rate is less than 40%.
These results certainly argue against taking
all variation in taxonomic rates at face value,
but at the same time they suggest that there is
substantial variation in estimated origination
and extinction rates that is not an artifact of
variation in preservation rate.
Discussion
Using boundary crossers to estimate origination
and extinction rates is relatively insensitive
to secular trends in the quality of preservation.
Except very near the Recent, this approach
is also affected but little by the nearly
complete knowledge of the living fauna. Why
isit then that Pease (1988a,b, 1992)interpreted
the Phanerozoic decline in taxonomic rates as
an artifact of improving preservation and of
the Pull of the Recent? There seem to be at
least two reasons. First, he used origination
and extinction metrics, such as the pertaxon
4 9 0 4 8 0 4 7 0 4 6 0 4 5 0 4 4 0
Geologic time (Ma)
FIGURE17. Estimated preservation rate (A),percapita
origination rate (B), and percapita extinction rate (C)
during the Ordovician. Data are from A . I. Miller, supplemented
from Sepkoski for pre and postOrdovician
occurrences of genera in Miller's data set. Probability of
preservation R, for each interval is estimated as inverse
of ratio of number of genera known both before and after
the interval to the number of these genera actually
occurring within the interval. R, is converted to a rate
per Lmy: r, = In(l  R,)lAt.See text for discussion.
rate, that are sensitive to secular changes in
preservability. Second, some of his arguments
were based on analysis of bivalve families. Because
families tend to be longlived taxa in
general, and bivalve taxa are especially longlived,
the edge effect of the Recent extends far
back in time in this case.
Because variation in preservation can mimic
DIVERSITY AND RATE ESTIMATION 95
variation in taxonomic rates, it is important to
consider how much true variation in taxonomic
rates is contained in stratigraphic range
data and how this variation may be distorted
(Koch and Morgan 1988).For the Phanerozoic
as a whole, the magnitude of apparent variation
in taxonomic rates is not correlated with
a group's preservation potential. This suggests
that there is substantial, shortterm variation
in taxonomic rates that is potentially
measurable. The same conclusion is indicated
by the correlations among preservation rate
and apparent rates of origination and extinction
in Ordovician invertebrates. If the results
for marine animals are typical, then apparent
variation in origination and extinction may be
largely real. But knowing that the variation is
largely real in a statistical sense is not the
same as knowing the exact pattern of this variation.
There are at least two obvious approaches
to uncovering this pattern: (1) We
can exploit occurrence data to adjust first and
last appearances in a way that takes variable
preservation and sampling into consideration.
For example, by basing origination and extinction
metrics on samplingstandardized
first and last appearances, Alroy (199613,1998)
has estimated the pattern of origination and
extinction in North American Cenozoic mammals
that one would likely have observed if
preservation and sampling had been uniform
through time. This approach has shown considerable
variation in taxonomic rates that is
not easily attributable to variation in the quality
of the record or our knowledge of it. Extensive
data of the sort that Miller has collected
for the Ordovician and Alroy for the Cenozoic
have not yet been compiled for most of
geologic time, so we are far from being able to
follow the approach of afterthefact, standardized
resampling of the entire fossil record.
Such a comprehensive compilation has
been started for Phanerozoic marine animals,
however (Marshall et al. 1999), and preliminary
results suggest that there are some similarities
between genus origination and extinction
curves based on raw data and those
based on samplingstandardized data (J. Alroy
et al. unpublished). (2)We can seek to develop
methods that enable preservation rates
to be estimated and first and last appearances
to be correspondingly adjusted given only the
more readily available data on range endpoints,
with no information on occurrences
within ranges. I hope to report on one such
method in a future contribution to this journal.
Conclusion
Incomplete preservation and variation in interval
length cause most diversity measures
and origination and extinction metrics to be
inaccurate. Moreover, some metrics of taxonomic
rate by their very nature preclude the
independent estimation of origination and extinction
rates even under the ideal assumption
of a complete record. Modeling of cladogenesis
and preservation supports previous intuitive
and empirical arguments that diversity
and rates are best estimated if singleinterval
taxa are disregarded (Sepkoski 1990, 1993;
Buzas and Culver 1994, 1998; Raymond and
Metz 1995; Alroy 1996b, 1998, 1999; Harper
1996;Sepkoski and Koch 1996;Bambach 1999;
see also Pease 1985). Using measures for
which singletons are simply irrelevant is preferable
to adapting conventional measures by
discarding singletons. Thus it appears advantageous
in principle to measure relative
changes in diversity using the proportional
difference between the number of taxa crossing
into an interval and the number crossing
out, and to measure taxonomic rates using the
number of taxa that range completely through
the interval relative to the total number that
cross into or out of the interval. Although the
fossil record is incomplete to an extent that
varies substantially over time, past and current
developments aimed at coping with this
variability suggest that it may be possible to
extract true signals of origination rate, extinction
rate, and taxonomic diversity through
time, provided that one avoids the dual pitfalls
of taking the record at face value and assuming
that it is so distorted as to be uninformative.
The fact that methods for uncovering
these signals are still being developed attests
to the vibrancy of paleontology today.
Acknowledgments
I am especially grateful to R. H. Foote for
years of discussing this work and for sug
96 MIKE FOOTE
gesting substantial improvements in analysis ical America. Pp. 227242 in S. K. Donovan and C. R. C. Paul,
eds. The adequacy of the fossil record. Wiley, Chichester, Eng and presentation. This work also benefited land.
from discussions with J. Alroy,D.Jablonski, A. Collins, L. S. 1989. Evolutionary rates of a rapid radiation: the
I. Miller, and the late J. J. Sepkoski Jr.A. I. Mill Paleogene planktic foraminifera. Palaios 4:251263.
Foote, M. 1994. Temporal variation in extinction risk and tem er and J. J. Sepkoski Jr. generously provided poral scaling of extinction metrics. Paleobiology 20:424444.
data. J. Alroy, R. H. Foote, A. I. Miller, and M.  . 1997. Estimating taxonomic durations and preservation
Newman read and improved the manuscript. probability. Paleobiology 23:278300.
This research was supported by the National . 1999. Morphological diversity in the evolutionary radiation
of Paleozoic and postPaleozoic crinoids. Paleobiology
Science Foundation (grant EAR 9506568).
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Appendix
Equations Regarding True and Apparent Taxonomic Rates
and Diversity
This appendix develops equations for observed measures of
diversity and taxonomic rates given true taxonomic rates and
rates of fossil preservation. The basic equations for branching
theory are taken mainly from Kendall (1948), Raup (1985), and
Foote (in press a), while those for preservation are based upon
Foote (1997) and simplifications of Pease (1985).A header file in
C, containing library functions for all relevant calculations, is
available from the author.
A time window over which observations can be made extends
from time T = 0 to T = u! In cases where a discrete interval of
time is of interest, it has bottom and top boundaries T = t, and
T = t, and duration At = t,  t,. p(T), q(T),and r(T)are the timespecific,
percapita rates of origination, extinction, and preservation
at an instant in time T. For simplicity, I will assume that
rates are constant within a discrete interval, although they may
vary among intervals. Interval rates will be denoted p,, q,, and
r,. This assumption greatly reduces computational time for the
timeheterogeneous numerical integrations, but it can easily be
relaxed by allowing intervals to be arbitrarily short.
1.True Numbers of Taxa i n a Stratigraphic Interval
There are four fundamental, exclusive kinds of lineages that
can exist during an interval (see Barry et al. 1995 for a similar
classification): (1)those that have both first and last appearances
within the interval, i.e., singletons; (2) those that cross into the
interval and last appear within it; (3) those that first appear
within the interval and extend beyond it; and (4)those that cross
into the interval from below and extend beyond the top of the
interval. Let the corresponding numbers be denoted N,,, N,,
N,,, and N,,, where the subscripts refer tofirstlast, bottomlast,
firsttop, and bottomtop. These numbers can be combined to yield
several composite groups: (5) all lineages that cross into the interval,
N, = N,, + N,,; (6) all lineages that cross out of the interval,
N, = N,, + N,,; (7) lineages that become extinct during
the interval, N, = N,, + N,,; (8) lineages that originate during
the interval, N, = N,, + N,,; and (9) all lineages, N,,, = N,, + N,,
+ Nrt + NF,.
Let p(t) be the accumulated difference between origination
and extinction from T = 0 until T = t:
Let N(t) be the expected diversity at time t. Then
N(t) = N(O)enit1.
For simplicity, I will take N(0) to be equal to unity. Thus, the
number of taxa extant at the start of the interval is given by
In the special case where rates are constant, this is equal to
Similarly,
The probability that a lineage entering the interval will still
be extant at the end is equal to eqA1L. ikewise, the probability
that a lineage leaving the interval was already extant at the start
of the interval is equal to epa"'. Thus,
N,, = N,e'iAl = N,ea.~#, (3)
N,, = N,(1  e@), and (4)
The number of lineages confined to the interval is found by integrating
the expected number of originations at any time during
the interval (i.e., origination rate times standing diversity),
multiplied by the probability of not surviving from the time of
origination to the end of the interval. Thus,
N,, = N,
I'
p,el~q,li[l eulA+Tl]dT,
(60)
This is equal to
N,, = N,(eq"' + p,At  1) if p, = q,, and (6b)
The number of originations during the interval is the integral of
the origination rate times the standing diversity. Thus
No = .1'
p,elPq] dT, (70)
which is equal to
~f p, = q, and
P,  '7, lf p, # q,.
Similarly,
which is equal to
[ ~ , q , A f if p, = q, and
Finally, the total number of lineages within the interval (the total
progeny of Kendall 1948)is equal to
which is equal to
DIVERSITY AND RATE ESTIMATION 99
Nb(1 + p,At) ~f p, = 9, and
(9b)
If P2 + q, P# 9,
2. Taxonomic Rate Metrics for an Interval, Assuming
Complete Preservation
Various metrics have been devised to measure taxonomic
rates over some extended interval of time, usually by counting
originations and extinctions and normalizing by diversity and /
or interval length. One problem that has been noted (Gilinsky
1991) is that some normalizations implicitly assume constancy
of rates within an interval. Inspection of the foregoing equations
shows that No,N,, N,,, and N,,, are affected by variation in rates
within an interval, not just by mean rates over the interval. In
contrast, N,, N,, N,,, N,, (= N,  N,,), and N,, (= N,  N,,) depend
only on the average rates within the interval. Therefore, measures
of taxonomic rate that are based only on N,, N,, and N,,
are, at least in theory, insensitive to rate variation within an interval
and capable of accurately estimating average rates.
Proportional origination is the ratio of number of originations
to total diversity. Substituting into the equations for N, and N,,,
yields
Similarly, for proportional extinction
Normalizing these expressions by interval length yields the socalled
pertaxon rates:
and
98
1 + q,At
if p, = q, and
(13)
Clearly, P, and P, increase nonlinearly with At, whereas P,,,,,
and P,,,,decrease nonlinearly with At. This nonlinear dependence
on At, which cannot be eliminated by a simple normalization,
complicates the use of proportional origination and extinction
when interval length varies (Gilinsky 1991; Foote 1994).
P , A ~ if p, = q, and
Z ~ , [ ~ ~ P11 Y W (14b)
(p,  9,)[1+ e'pQ,Ui] if P, # q,.
The timenormalized origination metric is thus equal to
PJ if p, = q, and
2p,[ei?4PI  1 (15) I i f p , + q , . (p,  q,)[l + eip,q,lAr]At
The corresponding extinction metrics are given by
[%At if p, = q, and
2q,[e@ 4 pr 11
(p,  q,)[l + el? Q ui] if P, # q,,
and
if p, = q, and
V,/," = (17)
Thus, if origination and extinction rates are equal, Van Valen's
metric provides an accurate estimate of these rates that is independent
of interval length. The Van Valen metric progressively
underestimates origination and extinction rates as the
true difference between these rates increases.
Harper (1996) used a number of measures of origination and
extinction that disregard singletons. Harper's modification of
the Van Valen metric is given by
No  NFL  NF~
V' = (N, + N,)/2 (N, + ~ , ) / 2 ' (lsa)
which is equal to
(1  e  ~ ~ t if p, = q, and
Thus, when origination and extinction rates are equal, the singletonfree
Van Valen metric before timenormalization gives
the probability that a lineage present at either boundary willextend
all the way through the interval. (Alroy [1996b] also used
a similar approach, normalizing by N, rather than by (N, + N,)/
2. The two metrics are equivalent if p, = q,.)Normalizing by interval
length yields
if p, = q, and
At
(19)
The corresponding extinction metrics are given by
1  eqA1 if p, = q, and
V: = I 2(1  e  ~ ~ t ) (20)
1 + e i r q , l A 1 lf q"
and
Van Valen proposed a measure of taxonomic rates that normalizes
the observed number of events by the estimated standing
diversity within the interval, which is simply the mean of
N, and N, (algebraically identical to N,,,  N,/2  N,/2; [seeTable
1 and Harper 1975, 19961). The corresponding origination
metric is given by
1 if p, = q, and
In the general case, V*,,, and V*,,,,, do not provide accurate
rate estimates. They decrease nonlinearly with interval length,
and they deviate more from p, and q, as the true rates increase
which is equal to and as the difference between them increases. Because 1  e'
100 MIKE FOOTE
= x for small x, however, V',,,, = V*,,,, ;.p, = q, in the special
case where p, = q, and p,, q,, and At are not too large. The same
is true of Alroy's (1996b) metrics when singletons are disregarded
(see Alroy et al. 2000).
The number of lineages extending through the entire interval
depends only on the mean taxonomic rates for the interval. Rearranging
the simple equivalence Nh, = N,e48"' = N,ep'3' yields
estimates of percapita rates:
p? and rj reduce to p, and q,, but, in contrast to the singletonfree
Van Valen metric, this is true regardless of the magnitude of the
rates. If origination and extinction rates are unequal, then, in
contrast to the Van Valen metrics, with or without singletons,
one rate is not expected to affect the estimation of the other.
Note that p? and rj do not rely on counting events within the interval
and normalizing by total diversity or estimated standing
diversity. Also, as with the singletonfree Van Valen metric, singleinterval
taxa play no role. This will be important when the
incompleteness of the fossil record is taken into consideration.
3. Fundamental Preservation Probabilities
A lineage with timespecific preservation rate r(T) per m.y.
and duration t from t, to t, has a net probability of preservation
equal to 1  elli 'IT) ", In the case of constant preservation rates,
this is equal to 1  ec'l. This relationship and the probability
distribution of taxon durations are used to calculate probabilities
of preservation for taxa spanning all or part of relevant intervals
of time.
Probability of Preservatloi? before a Polnt In Time.The duration
of a lineage before any arbitrary point in time depends on prior
rates of origination and on the span of time. The probability of
preservation before time t, assuming that the lineage is in fact
extant at time t (not assuming that we know this fact), is therefore
equal to
In the special case where p and r are constant, this reduces to
In the special case where p and r are constant and the time span
t is effectively infinite (i.e., the probability that a lineageextends
from T = 0 to T = t is approximately nil), this reduces to
The corresponding probability of preservation after an arbitrary
point in time, given that the lineage is extant at that point, depends
on the extinction and preservation rates and the span of
tlme after that point. Thus
If q and r are constant and the time span (r c ,  t) is effectively
infinite, this reduces to
Equations (24c) and (25c) are the backward and forward preservation
probabilities of Pease (1985).
Probability of Preserrlation during an 1nterual.The probability
of preservation depends on whether a lineage (1)spans the entire
interval, (2) crosses into it from below and terminates within
it, (3)originates within it and extends beyond it, or (4) originates
and becomes extinct during the interval. For lineages that
span the entire interval, we have
For lineages that originate before the interval and terminate
within it, the probability of preservation depends on how far
into the interval they extend, which is a function of the extinction
rate. Thus
q,eqtT(l  PTdT )
P"lbL = 1  e4 A1 (274
The denominator in this equation is a normalization reflecting
the probability of extinction during the interval if the lineage is
extant at the start. This equation reduces to
For lineages that originate during the interval and extend beyond
it, the preservation probability is analogous to the foregoing,
except that the origination rate is the relevant parameter:
This reduces to
[r, + p,e(P,+')"]
Pnlrr =  ( 1  e ) . (28,)
P, + r,
For lineages that originate and become extinct during the
same interval, we need to consider the density of origination at
any point during the interval (which is uniform only if p, = q,).
This is obtained by multiplying the origination rate p, by the
standing diversity at time T within the interval, Nhe@'qlT, and
normalizing by the total number of singleinterval lineages, NFL.
The density of origination is then weighted by the density of a
given duration, which depends on the extinction rate q,, and the
probability of preservation given that duration. Thus
If p, = q, this is equal to
If p, # q, this is equal to
where, as above, u)is the upper bound of the stratigraphic window
over which observations can be made. If q and r are constant,
this reduces to
DIVERSITY AND RATE ESTIMATION 101
In equations (29b) and (29c), N, and N,, are from equations (lb)
and (6b).
4. Compound Probabilities of Preservation
Probabilities of preservation before, during, and after an interval
can be combined to yield probabilities of observed ranges
given true durations. Because I am interested in the relationship
between true and observed durations and rates, I will not present
equations expressing the probability of not being preserved
at all. These are easily derived from the fundamental probabilities
(see Pease 1985; Foote and Raup 1996; Foote 1997; Solow
and Smith 1997).The probabilities depend on a particular span
of time. To avoid ambiguities, I will use notations such as P,(t)
to indicate, for example, the probability of preservation before
a point In time when there is a time span t over which preservation
can occur. Let P, denote the probability that a lineage
is observed to cross a particular time line, given that the lineage
truly crosses it. This probability is simply the product of the
probabilities of preservation before and after:
If the rates are constant and t, and t, are the time spans before
and after the time line, then this is equal to
If the rates are constant and t, and t, are effectively infinite, then
this reduces to
In the timehomogeneous case, the foregoing expression gives
the ratio of observed to true standing diversity at any point in
time. Note that this is substantially smaller than the proportion
of taxa preserved (see below).
Lineages that truly span an entire interval can appear a number
of different ways. They can appear to span the entire interval,
to cross the bottom boundary only, to cross the top boundary
only, or to be confined to the interval. The corresponding
probabilities are
P,.l, = [ l P,(t,)]~P,l,,~P,(rc~ t,), and (31c)
Note that the relevant time span for POI,i,s At in all cases. The
probabilities that a lineage will appear to cross the bottom
boundary (whether or not it also crosses the top) or that it will
appear to cross the top boundary (whether or not it also crosses
the bottom) are given by
PbI,= P,(t,).P,(w  t, + At) and (314
= P,(t, + At).P,(w  t,). (31f
Lineages that truly cross only the bottom boundary of an interval
can appear to cross that boundary or to be confined to the
interval. The corresponding probabilities are
Similarly, lineages that truly cross only the top boundary can
appear to cross that boundaryor to be confined to the interval.
The probabilities are
PF~I= F,P,I,,.P,(zr  t,), and (330)
Lineages truly confined to an interval can be preserved only in
that interval. The corresponding preservation probability was
given above as PDI,,(eq. 29).
5. Observed Numbers of Taxa in an Interval
Given the expected true numbers of taxa from section 1 above
and the probabilities of preservation from section 4, it is easy to
derive expressions for the number of taxa in various categories
that are observed in an interval. For a taxon to appear to span
the entire interval, it must have done so in reality. The number
of taxa that are observed to cross both top and bottom boundaries
is thus equal to
A taxon that appears to cross the bottom boundary only may
have crossed only that boundary in reality, or it may have
spanned the entire interval. The number of taxa observed to
cross the bottom boundary only is thus equal to
Similarly, the number of taxa observed to cross only the top
boundary is equal to
Taxa that are observed to be confined to the interval may truly
have been confined to the interval, or they may in fact have
crossed either or both boundaries. The number of observed singleinterval
taxa is thus equal to
The four fundamental classes of observed lineages can be
combined in a number of ways to yield the total observed taxa
crossing the bottom and top boundaries, the number of taxa first
appearing (apparently originating) within the interval, the
number last appearing (apparently becoming extinct) within
the interval, and the total number of taxa known from the interval
(including those preserved before and after but not during
the interval).Thus,
X, = X,, + X,,. (35a.i)
Note that this is also equal to
X, = N,P,(t,).P,(rc,  t, + At). (35a.11)
Similarly,
which is equal to
X, = N,P,(t, + At).P,(rc,  t,). (35b.1i)
Finally,
x , = x , , + x , , , ( 3 5 ~ )
X, = X,, + X,,, and (35d
Section 2 discussed methods for estimating taxonomic rates
that use only boundarycrossing lineages. The observed numbers
of boundary crossers, X,, X,, and X,,, are given by simple
expressions in the case of constant rates:
102 MIKE FOOTE
and
As the interval in question gets farther from the beginning or
end of the window of observation, the exponential terms in
these equations become less important. When the interval is far
enough from the edges that t, and u) t, are effectively infinite,
the exponential terms vanish and we have
r X, = N, and (37b) ( p + r)(q + 0'
From these it follows that
r2
X,L= N,L and (37d
(p + r)(q + 11'
If we add the further constraint that p = q, then we have simple
relationships for X, X,, and X,, as well:
r X,= X,= N h p h t  L = N,qAt and (38a)
p + r q + r
Note that [r/(p + r)] (= [r/(q + r)]) in this case is simply the
proportion of lineages preserved (Solow and Smith 1997).
Equations (37a) through (37e) have important implications.
The ratios of observed numbers of boundary crossers are identical
to the ratios of actual numbers of boundary crossers. If
preservation rate is constant and the rates are calculated for an
interval that is far from the beginning or end of the window of
observation, the rate estimates of equations (22) and (23)are unaffected
by incompleteness of the fossil record. This result is related
to fact that the observed age distribution of taxa exclusive
of singletons is identical to the true age distribution in the timehomogeneous
case (Foote and Raup 1996; Foote 1997).Several
authors have advocated excluding singleinterval taxa from
measures of taxonomic rates (Pease 1985; Sepkoski 1990; Alroy
1996b, 1998, 1999; Harper 1996), and many have advocated
measuring diversity as the number of taxa crossing time lines
rather than the number accumulated over an interval (Raymond
and Metz 1995; Alroy 1996b, 1998,1999; Bambach 1999).Whether
the exclusion of singletons is seen as an adjustment of a conventional
metric (Harper 1996)or part of a less conventional approach
to rate estimation, the practice has much to recommend
it, at least in the timehomogeneous case. The text explores behavior
of the proposed metric and other metrics when taxonomic
rates and preservation rates vary and when the interval is
close enough to the beginning or end of the window of observation
to experience a noticeable edge effect.
The singletonfree Van Valen metrics discussed above are also
insensitive to rate of preservation in the timehomogeneouscase
when edge effects are absent. As shown earlier, however, these
metrics yield inaccurate estimates of origination and extinction
rates, and the degree of inaccuracy increases as the magnitude
of rates or the difference between them increases.
Because the Van Valen metric and the pertaxon rate include
singletons in the count of originations and extinctions, these
metrics increase as the rate of preservation decreases. Combining
equations (37a), (37b), and (38a) yields a simple and striking
result. When origination and extinction rates are constant and
equal and when there are no edge effects, the normallzed Van
Valen metric is equal to
Since r l (p + r) = r/ (q + r) is the proportion of lineages preserved,
the observed Van Valen metric in this special case is simply
the true taxonomic rate divided by proportion of lineages
preserved. Thus, in practice, the average ratio of P to V,,,,,,or of
$ to V,,,,,,may provide a measure of the completeness of the fossil
record. To my knowledge, this possibility has not yet been
explored in detail.