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How macroecology affects macroevolution: the interplay
between extinction intensity and trait-dependent extinction
in brachiopods
Peter D. Smits1,∗
1. University of California – Berkeley, Berkeley, California 94720;
∗Corresponding author; e-mail: psmits@berkeley.edu
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Abstract
Selection is the force behind differences in fitness, with extinction being the most extreme ex-
ample of selection. Modern experiments and observations have shown that average fitness and3
selection strength can vary over time and space. This begs the question: as average fitness in-
creases, does selection strength increase or decrease? The fossil record illustrates how extinction
rates have varied through time, with periods of both rapid and slow species turnover. Using6
Paleozoic brachiopods as a study system, I developed a model to understand how the average
taxon duration (i.e. fitness) varies over time, to estimate trait-based differences in taxon durations
(i.e. selection), and to measure the amount of correlation between taxon fitness and selection. I9
find evidence for when extinction intensity increases, selection strength on geographic range also
increases. I also find strong evidence for a non-linear relationship between environmental pref-
erence for epicontinental versus open-ocean environments and expected taxon duration, where12
taxa with intermediate preferences are expected to have greater durations than environmental
specialists. Finally, I find that taxa which appear more frequently in epicontinental environments
will have a greater expected duration than those taxa which prefer open-ocean environments.15
My analysis supports the conclusions that as extinction intensity increases and average fitness
decreases, as happens during a mass extinction, the trait-associated differences in fitness would
increase. In contrast, during periods of low extinction intensity when fitness is greater than18
average, my model predicts that selection associated with geographic range and environmental
preference would decrease and be less than average.
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Introduction21
Selection is the force behind differences in fitness, with the most extreme example of selection
being extinction. Modern experiments and paleontological analyses have demonstrated that se-
lection strength and fitness can vary over time and space. An unanswered question in macroevo-24
lution is if and how species fitness and selection covary; does the strength of selection change
as average fitness also changes? The fossil record demonstrates that extinction risk has varied
continuously over time, from periods of low average extinction rate to very high extinction rates27
(e.g. mass extinctions) (Foote, 2000a,b, 2001). Paleontological analyses have also demonstrated
trait-based differences in extinction risk among taxa (Jablonski, 2008). Conceptually, extinction
is the ultimate manifestation of selection, as we would expect a taxon with a beneficial trait to30
persist longer than a similar taxon without that trait due to selection (Jablonski, 2008; Rabosky
and McCune, 2010; Raup, 1994; Stanley, 1975). Thus, the expected duration of a species can
be conceived of as a measure of a species’ fitness (Cooper, 1984); meaning that trait-associated33
differences in species fitness are species selection (Rabosky and McCune, 2010).
In order to test for an association between extinction intensity and extinction selectivity, ex-
tinction rate and trait-based differences in extinction rate need to be estimated. Previous work36
has approached this problem by estimating the extinction intensity and selectivity at different
points in time, or for different origination cohorts independently and then comparing those es-
timates (Payne et al., 2016). I find this approach problematic for a few reasons. Modeling each39
time point or cohort independently does not use all of the information present in the data, and
those estimates are only based on the data from that time point. A hierarchical/mixed-effect
modelling approach leverages all data across time points or cohorts by partially pooling infor-42
mation across each of the time-points or cohorts. The resulting parameter estimates have better
behaved posteriors (e.g. smaller credible intervals) and limit overly optimistic parameter esti-
mates by weighing those estimates relative to the amount of data associated with each time point45
or cohort (Gelman et al., 2013). The partial pooling in hierarchical/mixed-effect models also mit-
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igates the effects of complete separation, which normally prevents parameter estimates for some
time points or cohorts (Gelman et al., 2013; Payne et al., 2016). Finally, treating each time point48
or cohort as statistically independent means that any and all post-hoc analyses are at risk of false
positive results due to multiple comparisons issues (Gelman et al., 2013; Gelman and Hill, 2007);
hierarchical/mixed-effect models ameliorate this problem as possible comparisons are modeled51
simultaneously.
Jablonski (1986) observed that for bivalves at the end-Cretaceous mass extinction event,
previous trait-associated differences in survival no longer mattered except in the case of geo-54
graphic range. Based on this evidence, Jablonski (1986) proposed the idea of ”macroevolutionary
regimes:” that mass extinction and background extinction are fundamentally different processes.
However, based on estimates of extinction rates over time, there is no evidence of there being57
two or more ”types” of extinction (Wang, 2003). Instead, extinction rates for marine invertebrates
are unimodal with continuous variation. This disconnect between the qualitative differences of
macroevolutionary modes and the observation of continuous variation in extinction rates im-60
plies the possibility of a relationship between the strength of selection (extinction intensity) and
the association of traits and differences in fitness (extinction selectivity) (Payne et al., 2016). As
extinction intensity increases, what happens to extinction selectivity? How do trait-associated63
differences in fitness change as average extinction rate changes over time?
Here I develop a statistical model describing the relationship between brachiopod taxon du-
rations and multiple functional taxon traits in order to understand the relationship between ex-66
tinction intensity and selectivity over time. Trait-dependent differences in extinction risk should
be associated with differences in taxon duration (Cooper, 1984; Rabosky and McCune, 2010). Bra-
chiopods are an ideal group for this study as they have an exceptionally complete fossil record69
(Foote, 2000b; Foote and Raup, 1996). I focus on the brachiopod record from the post-Cambrian
Paleozoic, from the start of the Ordovician, approximately 485 million years ago (Mya), through
the end Permian (approximately 252 Mya) as this represents the time of greatest global brachio-72
pod diversity and abundance (Alroy, 2010).
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The analysis of taxon durations, the time from a taxon’s origination to its extinction, falls
under the purview of survival analysis, a field of applied statistics commonly used in health-75
care and engineering (Klein and Moeschberger, 2003), that has a long history in paleontology
(Crampton et al., 2016; Simpson, 1944, 1953; Smits, 2015; Van Valen, 1973, 1979). I adopt a hierar-
chical Bayesian modeling approach (Gelman et al., 2013; Gelman and Hill, 2007) in order to unify78
the previously distinct dynamic and cohort paleontological survival approaches (Baumiller, 1993;
Crampton et al., 2016; Ezard et al., 2012; Foote, 1988; Raup, 1975, 1978; Simpson, 2006; Van Valen,
1973, 1979).81
While estimation of trait-dependent speciation and extinction rates from phylogenies of ex-
tant taxa has become routine (Fitzjohn, 2010; Goldberg et al., 2011, 2005; Maddison et al., 2007;
Rabosky et al., 2013; Stadler, 2011, 2013; Stadler and Bokma, 2013), there are two major ways84
to estimate trait-dependent extinction: analysis of phylogenies and analysis of the fossil record.
These two directions, phylogenetic comparative and paleobiological, are complementary and in-
tertwined in the field of macroevolution (Hunt and Rabosky, 2014; Jablonski, 2008; Rabosky and87
McCune, 2010). In the case of extinction, analysis of the fossil record has the distinct advantage
over phylogenies of only extant taxa because extinction is observable; this means that extinction
rates can be directly estimated (Liow et al., 2010; Quental and Marshall, 2009; Rabosky, 2010). The90
approach used here is thus complementary to the analysis of trait-dependent extinction based
phylogenetic structure.
Factors affecting brachiopod survival93
Conceptually, taxon survival can be considered an aspect of “taxon fitness” (Cooper, 1984; Palmer
and Feldman, 2012). Traits associated with taxon survival are thus examples of species (or higher-
level) selection, as differences in survival are analogous to differences in fitness. The traits an-96
alyzed here are all examples of emergent and aggregate traits (Jablonski, 2008; Rabosky and
McCune, 2010). Emergent traits are those which are not measurable at a lower level (e.g. species
are a higher level aggregate of individual organism) such as global geographic range, or fos-99
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sil sampling rate. Aggregate traits, like body size or environmental preference, represent the
average of a shared trait across all members of a lower level.
Geographic range is widely considered the most important biological trait for estimating102
differences in extinction risk at nearly all times, with large geographic range conferring low
extinction risk (Finnegan et al., 2012; Harnik et al., 2012; Jablonski, 1986, 1987, 2008; Jablonski
and Roy, 2003; Payne and Finnegan, 2007). This relationship is expected even if extinction is a105
completely random process. Because of its importance and size, geographic range was analyzed
as a covariate of extinction risk with the initial assumption that a taxon with greater than average
geographic range would have a lower than average extinction risk. The effect size of geographic108
range on extinction acts as a baseline for comparing the strength of selection associated with the
other covariates.
Epicontinental seas are a shallow-marine environment where the ocean has spread over the111
continental interior or craton with a depth typically less than 100 meters. In contrast, open-ocean
coastline environments have much greater variance in depth, do not cover the continental cra-
ton, and can persist during periods of low sea level (Miller and Foote, 2009). This hypothesis114
is that taxa which favor epicontinental seas would be at a greater extinction risk during periods
of low sea levels, such as during glacial periods, than environmental generalists or open-ocean
specialists. Epicontinental seas were widely spread globally during the Paleozoic (approximately117
541-252 Mya) but declined over the Mesozoic (approximately 252–66 My) and have nearly dis-
appeared during the Cenozoic (approximately 66–0 My) as open-ocean coastlines became the
dominant shallow-marine setting (Johnson, 1974; Miller and Foote, 2009; Peters, 2008; Sheehan,120
2001). Taxa in epicontinental environments could also have a greater extinction susceptibility
than taxa in open-ocean environments during anoxic events or other major changes to water
chemistry due to limited water circulation from the open-ocean into epicontiental seas (Peters,123
2007). Similarly, if there is a major and sudden change to water chemistry in a single epiconti-
nental sea, the sluggish water flow into and out of that sea would most likely not affect other
epicontinental seas leading to local extirpation but not global extinction.126
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Miller and Foote (2009) demonstrated that, during several mass extinctions, taxa associated
with open-ocean environments tended to have a greater extinction risk than those taxa associated
with epicontinental seas. During periods of background extinction, however, they found no129
consistent difference in extinction risk between taxa favoring either environment. Miller and
Foote (2009) hypothesize that open-ocean taxa may have a greater extinction rate because these
environments would be more strongly affected by poisoning of the environment from impact132
fallout or volcanic events because water circulates at a higher rate and in greater volume in
open-ocean environments compared to the relatively more sluggish ciruclation in epicontinental
environments. These two environment types represent the primary identifiable environmental135
dichotomy observed in ancient marine systems (Miller and Foote, 2009; Sheehan, 2001). Given
these findings, I would hypothesize that as average extinction risk increases, the difference in risk
associated with open-ocean environments versus epicontinental environments should generally138
increase.
Because environmental preference is defined here as the continuum between occurring ex-
clusively in open-ocean environments versus epicontinental environments, intermediate values141
are considered “generalists” in the sense that they favor neither end-member. A long-standing
hypothesis is that generalists or unspecialized taxa will have greater survival than specialists
(Baumiller, 1993; Liow, 2004, 2007; N ¨
urnberg and Aberhan, 2013, 2015; Simpson, 1944; Smits,144
2015). Because of this, the effect of environmental preference was modeled as a quadratic func-
tion, where a concave-down relationship between preference and expected duration indicates
that generalists are favored over specialists end-members. Importantly, this approach does not147
“force” a non-linear relationship and only allows one if the second-order term is non-zero.
Body size, measured as shell length, is also considered as a trait that may potentially influence
extinction risk (Harnik, 2011; Payne et al., 2014). Body size is a proxy for metabolic activity and150
other correlated life history traits (Payne et al., 2014). Harnik et al. (2014) analyzed the effect
of body size selectivity in Devonian brachiopods in both phylogenetic and non-phylogenetic
contexts and found that that body size was not associated with differences in taxonomic duration.153
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However, there are some bivalve subclades for which body size can be as important a factor as
geographic range size in determining extinction risk (Harnik, 2011). Given these results, I expect
that if body size has any effect on brachiopod taxonomic survival, it will be very small.156
It is well known that, given the incompleteness of the fossil record, the observed duration
of a taxon is an underestimate of that taxon’s true duration (Alroy, 2014; Foote and Raup, 1996;
Liow and Nichols, 2010; Solow and Smith, 1997; Wagner and Marcot, 2013; Wang and Marshall,159
2004). Because of this, the concern is that a taxon’s observed duration may reflect its relative
chance of being sampled and not any of the effects of the covariates of interest. In this case, for
sampling to be a confounding factor there must be consistent relationship between the quality of162
sampling of a taxon and its apparent duration (e.g. greater sampling, longer duration). If there
is no relationship between sampling and duration then interpretation can be made clearly; while
observed durations are obviously truncated true durations, a lack of a relationship would indicate165
that the amount and form of this truncation is not a major determinant of the taxon’s apparent
duration. By including sampling as a covariate in the model, this effect can be quantified and
can be taken into account when interpreting the estimates of the effects of the other covariates.168
Methods
The brachiopod dataset analyzed here was sourced from the Paleobiology Database (http://www.paleodb.org)
which was limited to Brachiopods as defined by the higher taxonomic groups Rhychonelliformea:171
Rhynchonellata, Chileata, Obolellida, Kutorginida, Strophomenida, and Spiriferida. Addition-
ally, samples were limited to those which originated after the Cambrian but before the Triassic.
Temporal, stratigraphic, and other relevant occurrence information used in this analysis was also174
downloaded from the same source. Analyzed occurrences were restricted to those with paleo-
latitude and paleolongitude coordinates, being assigned to either epicontinental or open-ocean
environment, and belonging to a genus present in the body size dataset (Payne et al., 2014). Epi-177
continental versus open-ocean assignments for each fossil occurrence are based on those from
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previous analyses (Foote and Miller, 2013; Miller and Foote, 2009; Ritterbush and Foote, 2017).
These filtering criteria are very similar to those from Foote and Miller (2013) with the additional180
constraint to only those taxa which occur in the body size data set from Payne et al. (2014). In
total, 1130 genera were included in this analysis.
Fossil occurrences were analyzed at the genus level, a common practice for paleobiological,183
macroevolutionary, and macroecological studies, and this is especially the case for marine inver-
tebrates (Alroy, 2010; Eronen et al., 2011; Foote and Miller, 2013; Harnik et al., 2012; Kiessling and
Aberhan, 2007; Miller and Foote, 2009; N¨
urnberg and Aberhan, 2013, 2015; Payne and Finnegan,186
2007; Ritterbush and Foote, 2017; Simpson and Harnik, 2009; Vilhena et al., 2013). While species
diversity dynamics are frequently of much greater interest than those of higher taxa (though
see Foote 2014; Hoehn et al. 2015), the nature of the fossil record makes accurate, precise, and189
consistent taxonomic assignments at the species level difficult for all occurrences. To ensure a
minimum level of confidence and accuracy in the data, I analyzed genera as opposed to species.
Additionally, when species and genera can be compared, they often yield similar results (Foote192
et al., 2007; Jernvall and Fortelius, 2002; Roy D. & Valentine, I. W., 1996). Importantly, it is also
possible that genera represent coherent biological units as there is evidence for congruence be-
tween morphologically and genetically defined genera of molluscs and mammals (Jablonski and195
Finarelli, 2009).
Genus duration was calculated as the number of geologic stages from first appearance to
last appearance, inclusive. Durations were based on geologic stages as opposed to millions of198
years because of the inherently discrete nature of the fossil record. Dates are not assigned to
individual fossils themselves; rather fossils are assigned to a geological interval which represents
some temporal range. In this analysis, stages are effectively irreducible temporal intervals in201
which taxa may occur. Genera with there last occurrence in or after Changhsingian stage (e.g.
the final stage of the study interval) were right-censored. Censoring in this context indicates that
the genus was observed up to a certain age, but that its ultimate time of extinction is unknown204
(Klein and Moeschberger, 2003).
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The covariates of duration included in this analysis are geographic range size (r), environ-
mental preference (v,v2), shell length (m), and sampling (s).207
The geographic range of a genus was calculated as the number of occupied grid cells from
a gridded map of all contemporaneous occurrences. First, the paleolatitude-paleolongitude co-
ordinates for all occurrences were projected onto an equal-area cylindrical map projection. Each210
occurrence was then assigned to one of the cells from a 70 ×34 regular raster grid placed on the
map. Each grid cell represents approximately 250,000 km2. The map projection and regular lattice
were made using shape files from http://www.naturalearthdata.com/ and the raster package213
for R (Hijmans, 2015). For each time interval a taxon’s geographic range is calculated from the
ratio of cells occupied by that taxon divided by the total number of cells with any occurrences.
For each taxon in each temporal bin, the relative occurrence probability of the observed taxa was216
calculated using the JADE method developed by Chao et al. (2015) which leverages the distribu-
tion of taxon occurrences to estimate their “true” geographic range. This method accounts for the
fact that taxa with an occupancy of 0 cannot be observed, which means that occupancy follows219
a truncated Binomial distribution. This correction is critical when comparing occupancies from
different times with different geographic sampling. After occurence probability is calculated for
all taxa for each temporal bin in which they occur, I calculated mean occurrence probability of222
each taxa. This final value is my proxy for the geographic range of a taxon.
Environmental preference is a descriptor of whether and by how much a taxon prefers epi-
continental to open-ocean environments. This approach presents environmental preference as a225
continuum from exclusive occurrence at the ends and equal occurrences in the middle. My mea-
sure of environmental preference is derived from the number of epicontinental or open-ocean ob-
servations of a taxon compared to the total number of epicontinenal or open-ocean observations228
that also occurred during time intervals shared with that taxon. Mathematically, environmental
preference was defined as probability of observing the ratio of epicontinental occurrences to total
occurrences (θi=ei/Ei) or greater given the background occurrence probability θ0
ias estimated231
from all other taxa occurring at the same time (e0
i/E0
i). This measure of environmental preference
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is expressed
pθ0
i
e0
i,E0
i∝Beta(e0
i,E0
i−e0
i)Beta(1, 1)
=Beta(e0
i+1, E0
i−e0
i+1),
(1)
where vis the percent of the distribution defined in equation 1 as less than or equal to θi. The234
Beta distribution is used here because it is a continuous distribution bounded at 0 and 1, which
is idea for modeling percentages.
Body size, measured as shell length, was sourced directly from Payne et al. (2014). These237
measurements were made from brachiopod taxa figured in the Treatise on Invertebrate Paleontology
(Williams et al., 2007).
The sampling probability for individual taxa, called s, was calculated using the standard gap240
statistic (Foote, 2000a; Foote and Raup, 1996). The gap statistic is calculated as the number of
stages in which the taxon was sampled exempting its first and last stages. Because taxa that
were right-censored only have a first appearance, one was subtracted from the numerator and243
denominator instead of two. The inclusion of genus-specific sampling probability as a covariate
are an attempt to mitigate the effects of the incompleteness of the fossil record on our ability
to observe genus duration. The implications of this choice are discussed further later in the246
Discussion.
The minimum duration for which a gap statistic can be calculated is three stages, so I chose
the impute the gap statistic for all observations with a duration of less than 3. Imputation is the249
“filling in” of missing observations based on the observed values (Gelman and Hill, 2007; Rubin,
1996).
Prior to analysis, geographic range was logit transformed and the number of samples was252
natural-log transformed; these transformations make these variables defined for the entire real
line. Sampling probability was transformed as (s(n−1) + 0.5)/nwhere nis the sample size as
recommended by Smithson and Verkuilen (2006); this transformation shrinks the range of the255
data so that there are no values equal to 0 or 1. All covariates except for sampling were stan-
dardized by subtracting the mean from all values and dividing by twice its standard deviation,
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which follows Gelman and Hill (2007). This standardization means that the associated regression258
coefficients are interpretable as the expected change per 1-unit change in the rescaled covariates.
Finally, Dis defined as the total number of covariates, excluding sampling, plus one for the
intercept term.261
Details of model
Hierarchical modelling is a statistical approach which explicitly takes into account the structure
of the observed data (Gelman et al., 2013; Gelman and Hill, 2007). The units of study (e.g. genera)264
each belong to a single group (e.g. origination cohort). Each group is considered a realization of
a shared probability distribution (e.g. prior) of all cohorts, observed and unobserved. The group-
level parameters, or the hyperparameters of this shared prior, are themselves given (hyper)prior267
distributions and are also estimated like the other parameters of interest (e.g. covariate effects)
(Gelman et al., 2013). The subsequent estimates are partially pooled together, where parameters
from groups with large samples or effects remain large while those of groups with small samples270
or effects are pulled towards the overall group mean. All covariate effects (regression coefficients),
as well as the intercept term (baseline extinction risk), were allowed to vary by group (origination
cohort). The covariance between covariate effects was also modeled.273
Genus durations were assumed to follow a Weibull distribution, which allows for age-dependent
extinction (Klein and Moeschberger, 2003): y∼Weibull(α,σ). The Weibull distribution has two
parameters: scale σand shape α. When α=1, σis equal to the expected duration of any taxon.276
αis a measure of the effect of age on extinction risk, where values greater than 1 indicate that
extinction risk increases with age, and values less than 1 indicate that extinction risk decreases
with age. Note that the Weibull distribution is equivalent to the exponential distribution when279
α=1.
Data censoring and truncation are conditions where the value of interest (taxon duration)
is only partially observed. There are a number of processes which can lead to either of these282
conditions: limited resolution, which leads to left-censoring or truncation; end of study interval,
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which leads to right censoring; and incomplete sampling, which can left-censor (short-lived taxa
are less likely to be preserved at all) or right-censor (durations are truncated). In the case of285
the right- and left-censored observations mentioned above, the probability of those observations
has a different calculation (Klein and Moeschberger, 2003). For right-censored observations, the
likelihood is calculated p(y|θ) = 1−F(y) = S(y), where F(y)is the cumulative distribution288
function. Taxa that existed for only a single stage were left-censored, which implies that that
taxon went extinct at any point between 0 and 1 stages. In contrast to right-censored data,
the likelihood of a left-censored observation is calculated from p(y|θ) = F(y). This censoring291
strategy improves model fit, as measured by WAIC and LOOIC, than treating these taxa as being
fully observed.
The scale parameter σwas modeled as a regression following Kleinbaum and Klein (2005)294
with varying intercept, varying slopes, and the effect of sampling. This model is expressed
σi=exp −XiBj[i]+δsi
α(2)
where iindexes across all observations from i=1, . . . , n,nis the total number of observations,
j[i]is the cohort membership of the ith observation for j=1, . . . , J,Jis the total number of297
cohorts, Xis a N×Dmatrix of covariates along with a column of ones for the intercept term, B
is a J×Dmatrix of cohort-specific regression coefficients, and δis the regression coefficient for
the effect of sampling s.δdoes not vary by cohort.300
Each of the rows of matrix Bare modeled as realizations from a multivariate normal distri-
bution with length Dlocation vector µand J×Jcovariance matrix Σ:Bj∼MVN(µ,Σ). The
covariance matrix was then decomposed into a length Jvector of scales τand a J×Jcorrelation303
matrix Ω, defined Σ=diag(τ)Ωdiag(τ)where “diag” indicates a diagonal matrix.
The elements of µwere given independent normally distributed priors. The effects of geo-
graphic range size and the breadth of environmental preference were given informative priors306
reflecting the previous findings while the other parameters were given weakly informative priors
which favor those covariates having no effect on duration. The correlation matrix Ωwas given
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an LKJ distributed prior (Lewandowski et al., 2009) that slightly favors an identity matrix as309
recommended by Team (2017). These priors are defined
µ0∼ N (0, 5)
µr∼ N (−1, 1)
µv∼ N (0, 1)
µv2∼ N (1, 1)
µm∼ N (0, 0.5)
δ∼ N (1)
τ∼C+(1)
Ω∼LKJ(2).
(3)
The log of the shape parameter αwas given a weakly informative prior log(α)∼ N (0, 1)
centered at α=1, which corresponds to the Law of Constant Extinction (Van Valen, 1973).312
Imputation of sampling probability
The vector sampling shas two parts: the observed part soand the unobserved part su. Of the 1130
total observations, 539 have a duration of 3 or more and have an observed gap statistic. The gap315
statistic for the remaining 591 observations was imputed. As stated above, the unobserved part
is then imputed, or filled in, based on the observed part so. Because sampling varies between
0 and 1, I chose to model it as a Beta regression with matrix Wbeing a N×(D−3)matrix318
of covariates (i.e. geographic range size, environmental preference, body size; no interactions).
Predicting sampling probability using the other covariates that are then included in the model
of duration is acceptable and appropriate in the case of imputation where the sample goal is321
accurate prediction (Gelman and Hill, 2007; Rubin, 1996). Not including these covariates can
lead to biased estimates of the imputed variable; if the covariates themselves are related, not
including them will bias this correlation towards zero which then leads to improper imputation324
and inference (Rubin, 1996).
14
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The Beta regression is defined
so∼Beta(φ=logit−1(Xoγ),λ), (4)
where γis a length Dvector of regression coefficients and Xis defined as above. The Beta327
distribution used in the regression is reparameterized in terms of a mean parameter
φ=α
α+β(5)
and total count parameter
λ=α+β(6)
where αand βare the characteristic parameters of the Beta distribution (Gelman et al., 2013).330
The next step is to estimate su|so,Xo,Xu,γwhich is used as a covariate of taxon duration
(Eq. 2). All the elements of γ,δ(Eq. 2), and λ(Eq. 4) were given weakly informative priors as
recommended by Team (2017):333
γ∼ N (0, 1)
δ∼ N (0, 1)
λ∼Pareto(0.1, 1.5).
(7)
The imputed values are estimated simultaneously and in the same manner as all other pa-
rameters; this ensures that all uncertainty surrounding these unobservable covariate values is
propagated through to all estimates.336
Posterior inference and posterior predictive checks
The joint posterior was approximated using a Markov-chain Monte Carlo routine that is a vari-
ant of Hamiltonian Monte Carlo called the No-U-Turn Sampler (Hoffman and Gelman, 2014) as339
implemented in the probabilistic programming language Stan (Stan Development Team, 2014).
The posterior distribution was approximated from four parallel chains run for 40,000 steps, split
half warm-up and half sampling and thinned to every 20th sample for a total of 4000 posterior342
samples. Starting conditions for sampling were left at the CmdStan defaults for interface except
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for the following changes: adapt delta was set 0.95 to ensure no divergent samples, and initial
value was set to 0 which allows for stable initial samples. Posterior convergence was assessed345
using both standard MCMC and HMC specific diagnostics: scale reduction factor ˆ
R, effective
sample size or ESS, energy (target >0.2), presence and number of divergent samples, and num-
ber of samples that saturated the maximum trajectory length. For further explanation of these348
diagnostic criteria, see the Stan Manual (Team, 2017).
After the model was fitted to the data, 100 datasets were simulated from the posterior predic-
tive distribution of the model. These simulations were used to test for adequacy of model fit as351
described below.
Survival analysis is complicated by censored observations, where the ultimate time of extinc-
tion for some taxa could not be fully observed during the study window. Posterior predictive354
simulations for these observations must be similarly censored. To accomplish this, posterior
predictive simulated durations for right-censored observations where the minimum of its final
observed duration and the simulated duration. For left-censored individuals, their simulated357
duration was set to a minimum of one stage.
Model adequacy was evaluated using a series of posterior predictive checks. Posterior pre-
dictive checks are a means for understanding model fit or adequacy. Model adequacy means360
that if our model is an adequate descriptor of the data, then data simulated from the posterior
predictive distribution should be similar to the observed given the same covariates, etc. (Gel-
man et al., 2013). Posterior predictive checks generally compare some property of the empirical363
data to that property estimated from each of the simulated datasets. Additionally, for structured
datasets like the one analyzed here, the fit of the model to different parts of the data can be
assessed, which in turn can reveal a great deal if the model has good fit to some aspects of data366
but not others; this is when when knowledge of the biological, geological, or paleoenvironmental
context of the data becomes important in order to explain what unmodeled processes might lead
to these discrepancies between our data and the model (Gelman et al., 2013).369
The types of posterior predictive tests used in this analysis fall into two categories: compari-
16
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son of observed mean and median genus duration to a distribution of mean and median genus
duration estimates from the posterior simulations; and comparison of a non-parametric estimate372
of the survival function from the observed data to estimates of that same survival function from
the simulations. These posterior predictive tests were done for the entire dataset as a whole and
for each of the origination cohorts individually.375
The survival function describes the probability of a taxon persisting given that it has survived
up to time t; this is expressed P(T≥t)because Tis the true extinction time of the species and tis
some arbitrary time of observation and we are estimating that probability that tis less than T. It378
is important to note, however, that the survival function does not reflect density of observations
unlike e.g. histograms. Instead, this posterior predictive check reflects the model’s ability to
predict genus survival.381
All code necessary to reproduce this analysis are available through an archived Zenodo repos-
itory DOI https://zenodo.org/record/1402252. Additionally, this project is hosted at https://github.com/psmits/preserve.
Results384
I first present the results of the multiple posterior predictive checks for the whole dataset as well
as each of the origination cohorts. I next present the parameter posterior estimates and their
interpretations.387
Comparisons between the observed distribution of durations to the distributions of 100 sim-
ulated datasets reveals the relatively good but heterogeneous fit of the model to the data (Fig. 1).
The two major aspects of possible misfit that are observable are at durations of 2-3 stages. The390
model slightly under-estimates the number of observations with duration of 2 or 3 stages. The
goal of this model is estimating the expected duration of a genus given its covariate information.
While the model estimates are not exact, it is possible that our model fits the bulk of our data393
well but fits poorly towards the extreme values.
The similarity of the empirical data and from 100 simulated datasets provides a more nuanced
17
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0 10 20 30
Duration (geological stages)
Density
y
yrep
Figure 1: Comparison of the distribution of the observed data (black) to 100 simulated distribu-
tions (blue). This is a close-up view of the bulk of the distribution which shows the more subtle
aspects of (mis)fit between the data and the model.
picture of model adequacy (Fig. 2). The survival curves of the 100 simulated datasets are very396
similar to the survival function estimated from the empirical data. The major points of misfit
between the model and the data are taxa with duration 1 stage, and taxa with a duration at least
10-13 stages. The major divergence between the observed and the estimated applies to taxa with399
a less than 15% probability of continuing to survive.
Model adequacy at the total data level was assessed through comparison of the mean and
median of the observed data to those from simulated data sets. While the previous posterior402
predictive checks have focused on the relatively good but heterogeneous fit of the model to
the entire distribution of the data, the fitted model’s ability to predict the mean and median
18
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0.00
0.25
0.50
0.75
1.00
0 10 20 30
Duration (t)
Pr(T >= t)
Figure 2: Comparison of the empirical estimate of S(t)(blue) versus estimates from 100 posterior
predictive data sets (black). S(t)corresponds to the probability that the age of a genus tis less
than the genus’ ultimate duration T.
of the observed data appears adequate (Fig. 3, 4). Because the principle goal of this model405
is to obtain adequate prediction of a taxon’s expected duration for a given set of ecological
covariates, the seemingly adequate fit of our model to mean taxon duration is reassuring (Fig. 3).
Additionally, given the skewness of the observed taxon durations (Fig. 1), the ability for the408
model to recapitulate the median observed taxon duration points to the overall good fit of the
model to the data.
When considered together, all of the above posterior predictive checks indicate adequate411
model fit for key questions such as expected taxon duration (Fig. 3). However, there is obviously
heterogeneity in model fit because, while the model can recapitulate some aspects of the observed
19
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3.3 3.6 3.9 4.2
Duration (geological stages)
Frequency
T=mean
T(yrep)
T(y)
Figure 3: Comparison of the (A) observed mean genus duration (black vertical line) to a dis-
tribution of means estimated from 100 simulated datasets (blue). Model fit is evaluated by the
similarity between the observed and the estimated, where good fit is demonstrated by the vertical
line being “within” the simulated distribution.
data (Fig. 3, 4), there are obvious discrepancies between the model and the data (Fig. 1, 2). By414
performing the same posterior predictive tests for each of the origination cohorts, it may be
possible to get a better picture of the sources of model misfit.
When the posterior predictive tests are visualized for each of the origination cohorts, a com-417
plex picture of model fit emerges. For nearly every origination cohort, the model is able to
recapitulate the observed mean duration (Fig. 5). In comparison, the model has a much more
heterogeneous fit to each origination cohort’s median taxon duration (Fig. 6). The skewness of420
the distribution underlying figure 1 means that for some origination cohorts, median duration
20
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1.8 2.0 2.2 2.4
Duration (geological stages)
Frequency
T=median
T(yrep)
T(y)
Figure 4: Comparison of the observed median genus duration (black vertical line) to a distri-
bution of medians estimated from 100 simulated datasets (blue). Model fit is evaluated by the
similarity between the observed and the estimated, where good fit is demonstrated by the vertical
line being “within” the simulated distribution.
might be pegged at 1 stage; this means that the posterior predictive distributions for some cohorts
can be extremely skewed.423
These results indicate that this model is very good at recapitulate mean taxon duration
(Fig. 3, 5) and that it is capable of estimating overall median duration and median duration of
most origination cohorts (Fig. 4, 6). The poor model fit to some origination cohorts may indicate426
that these cohorts are undergoing a different extinction process whose aspects are unmodeled in
this analysis. For those cohorts where the model recapitulates the empirical survival function,
the model is capturing some aspect of the processes underlying taxon extinction.429
21
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(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/523811doi: bioRxiv preprint first posted online Jan. 17, 2019;
31. Capitanian
32. Wuchiapingian
33. Changhsingian
25. Asselian
26. Sakmarian
27. Artinskian
28. Kungurian
29. Roadian
30. Wordian
19. Tournaisian
20. Visean
21. Serpukhovian
22. Bashkirian
23. Moscovian
24. Stephanian
13. Pragian
14. Emsian
15. Eifelian
16. Givetian
17. Frasnian
18. Famennian
7. Hirnantian
8. Llandovery
9. Wenlock
10. Ludlow
11. Pridoli
12. Lochkovian
1. Tremadoc
2. Floian
3. Dapingian
4. Darriwilian
5. Sandbian
6. Katian
1 2 3 4 5 0.0 2.5 5.0 7.5 0.6 0.7 0.8 0.9 1.0
4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 4 6 8 10 2 4 6 8
3 5 7 5 10 2 3 4 5 0 5 10 15 20 0 5 10 15 20 25 3 6 9 12
5 10 1.5 2.0 2.5 3.0 3.5 2 3 4 5 6 2 3 4 5 2 3 4 2 3 4 5 6
1 2 3 4 5 1 2 3 4 2 3 4 5 6 7 4 8 12 161 2 3 4 5 4 5 6 7 8
2 4 6 8 4 6 8 10 12 0 5 10 15 2 3 4 5 6 7 0 10 20 30 5 10 15
Duration (geological stages)
Frequency
T=mean
T(yrep)
T(y)
Figure 5: Comparison of the observed mean genus duration (black vertical line) to a distribution
of means estimated from 100 simulated datasets (blue) for each of the origination cohorts. Model
fit is evaluated by the similarity between the observed and the estimated, where good fit is
demonstrated by the vertical line being “within” the simulated distribution.
A larger than average geographic range is expected to have a positive effect on taxon survival
(Table 1). The cohort-level estimate of the effect of geographic range size indicates that as a
taxon’s geographic range increases, that taxon’s duration is expected to increase (Table 1). Given432
the estimates of µrand τr, there is an approximately 3.7% (±4.3% SD) probability that this
relationship would be reversed (PrN(µr,τr)>0)).
Body size measured as valve length is estimated to have no effect on duration for most of the435
post-Cambrian Paleozoic (Table 1).
22
.CC-BY-NC 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/523811doi: bioRxiv preprint first posted online Jan. 17, 2019;
31. Capitanian
32. Wuchiapingian
33. Changhsingian
25. Asselian
26. Sakmarian
27. Artinskian
28. Kungurian
29. Roadian
30. Wordian
19. Tournaisian
20. Visean
21. Serpukhovian
22. Bashkirian
23. Moscovian
24. Stephanian
13. Pragian
14. Emsian
15. Eifelian
16. Givetian
17. Frasnian
18. Famennian
7. Hirnantian
8. Llandovery
9. Wenlock
10. Ludlow
11. Pridoli
12. Lochkovian
1. Tremadoc
2. Floian
3. Dapingian
4. Darriwilian
5. Sandbian
6. Katian
1 2 3 0 2 4 6 0.6 0.8 1.0
2 4 6 2 4 6 1 2 3 4 5 6 1 2 3 2 4 6 8 2 4 6
1 2 3 2 4 6 1.0 1.5 2.0 2.5 3.0 3.5 5 10 15 0 10 20 2.5 5.0 7.5 10.0
5 10 15 1.2 1.5 1.8 1.0 1.5 2.0 2.5 2 3 1.0 1.5 2.0 1 2 3
1 2 3 1 2 3 1 2 3 4 2.5 5.0 7.510.012.51 2 3 4 2 3 4 5 6 7
1 2 3 4 5 3 6 9 12 0 5 10 1 2 3 4 0 5 10 15 20 25 0 5 10 15 20
Duration (geological stages)
Frequency
T=median
T(yrep)
T(y)
Figure 6: Comparison of the observed median genus duration (black vertical line) to a distribu-
tion of medians estimated from 100 simulated datasets (blue) for each of the origination cohorts.
Model fit is evaluated by the similarity between the observed and the estimated, where good fit
is demonstrated by the vertical line being “within” the simulated distribution.
23
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The copyright holder for this preprint. http://dx.doi.org/10.1101/523811doi: bioRxiv preprint first posted online Jan. 17, 2019;
Table 1: Estimates of group-level and invariant parameter values for the fitted model analyzed
here.
Category Parameter Effect of. . . Mean SD 10% 50% 90%
Mean
µiintercept -3.04 0.19 -3.29 -3.04 -2.80
µrgeographic
range
-0.98 0.16 -1.17 -0.98 -0.78
µvenvironmental
preference
-0.76 0.18 -0.99 -0.76 -0.53
µv2environmental
preference2
3.15 0.35 2.71 3.15 3.59
µmbody size -0.02 0.12 -0.17 -0.02 0.14
Standard deviation
τiintercept 0.50 0.11 0.38 0.50 0.65
τrgeographic
range
0.49 0.16 0.29 0.49 0.70
τvenvironmental
preference
0.83 0.16 0.63 0.82 1.05
τv2environmental
preference2
1.49 0.35 1.08 1.46 1.94
τmbody size 0.47 0.12 0.32 0.46 0.63
Other
δsampling 0.90 0.15 0.71 0.89 1.08
αageing 1.36 0.04 1.30 1.36 1.42
Note: These parameters are the group-level estimates of the effects of biological traits on brachiopod generic survival,
the standard deviation of the between-cohort effects, as well as the estimates of the effect of sampling δand the Weibull
shape parameter α. The mean, standard deviation (SD), 10th, 50th, and 90th quantiles of the marginal posteriors are
presented.
24
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The copyright holder for this preprint. http://dx.doi.org/10.1101/523811doi: bioRxiv preprint first posted online Jan. 17, 2019;
The group-level relationship between effect of environmental preference and duration is esti-
mated from µvand µv2. The estimate of µvindicates that taxa which slightly prefer epicontinental438
environments to open-ocean environments are expected to have a greater duration than open-
ocean favoring taxa (Table 1). Additionally, given the estimate of between-cohort variance τv,
there is approximately 18.1% (±7.5% SD) probability that, for any given cohort, taxa which fa-441
vor open-ocean environments would have a greater expected duration than taxa which favor
epicontinental environments (Pr(N(µv,τv)>0)). The estimate of µv2indicates that the overall
relationship between environmental preference and log(σ)is concave-down (Fig. 7), with only a444
2.5% (±2.9% SD) probability that any given cohort is convex up (Pr(N(µv2,τv2)<0).
The cohort-specific relationships between environmental preference and log(σ)were calcu-
lated from the estimates of β0,βv, and βv2(Fig. 8) and reflect how these three parameters act in447
concert, not individually (Fig. 9). Because of the relationship between βvand βv2, it is important
to consider them together when drawing conclusions from the model. In many cases, the cohort-
specific estimated relationship between environmental preference and duration is approximately450
equal to the group-level average, but for 14 of the 33 analyzed origination cohorts at least one of
these three parameters are noticeably different from the group-level average.
25
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0
10
20
30
−0.5 0.0 0.5
Environmental preference
(open−ocean <−−> epicontinental)
Duration in stages
Figure 7: The overall expected relationship between environmental affinity viand a log(σ)when
r = 0 and m = 0. The 1000 semi-transparent lines corresponds to a single draw from the posterior
predictive distribution, while the highlighted line corresponds to the median of the posterior
predictive distribution. The overall relationship demonstrates a greater durations among envi-
ronmental generalists than specialists. Additionally, because the apex of is rightward from 0, taxa
favoring epicontinental environments are expected to have a slightly longer durations than those
favoring open-ocean environments. The tick marks along the bottom of the plot correspond to
the (rescaled) observed values of environmental preference.
26
.CC-BY-NC 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/523811doi: bioRxiv preprint first posted online Jan. 17, 2019;
31. Capitanian
32. Wuchiapingian
33. Changhsingian
25. Asselian
26. Sakmarian
27. Artinskian
28. Kungurian
29. Roadian
30. Wordian
19. Tournaisian
20. Visean
21. Serpukhovian
22. Bashkirian
23. Moscovian
24. Stephanian
13. Pragian
14. Emsian
15. Eifelian
16. Givetian
17. Frasnian
18. Famennian
7. Hirnantian
8. Llandovery
9. Wenlock
10. Ludlow
11. Pridoli
12. Lochkovian
1. Tremadoc
2. Floian
3. Dapingian
4. Darriwilian
5. Sandbian
6. Katian
−0.5 0.0 0.5 −0.5 0.0 0.5 −0.5 0.0 0.5
−0.5 0.0 0.5 −0.5 0.0 0.5 −0.5 0.0 0.5
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
Environmental preference (v)
Duration in stages
Figure 8: Comparison of origination cohort-specific (posterior predictive) estimates of the effect of
environmental preference on log(σ)to the mean overall estimate of the effect of environmental
preference. Cohort-specific estimates are from 100 posterior predictive simulations across the
range of (transformed and rescaled) observed values of environmental preference. The oldest
cohort is in the top-left and younger cohorts proceed left to right, with the youngest cohort being
the right-most facet of the last row. Panel names correspond to the name of the stage in which
that cohort originated.
27
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The copyright holder for this preprint. http://dx.doi.org/10.1101/523811doi: bioRxiv preprint first posted online Jan. 17, 2019;
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intensity
range
env_pref
env_curv
size
250300350400450
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−2.0
−1.5
−1.0
−0.5
0.0
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−1
0
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−1.0
−0.5
0.0
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1.0
Time (My)
Effect estimate for...
Figure 9: Comparison of cohort-specific estimates of β0, the effect of geographic range on extinc-
tion risk βr, the effect of environmental preference βvand βv2, and body size βm. Points corre-
spond to the median of the cohort-specific estimate, along with 80% credible intervals. Points
are plotted at the midpoint of the cohorts stage of origination in millions of years before present
(My). Black, horizontal lines are the overall estimates of covariate effects along with 80% credible
intervals (shaded).
28
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The copyright holder for this preprint. http://dx.doi.org/10.1101/523811doi: bioRxiv preprint first posted online Jan. 17, 2019;
There is an approximately 90.4% probability that cohort estimates of β0and βrare negatively453
correlated, with median estimate of correlation being -0.35 (Fig. 10). This result means that
for any cohort, we would expect that if extinction intensity increases (β0increases), the effect
of geographic range on duration increases (βrdecreases). This result is strong evidence for a456
relationship between intensity and selectivity with respect to geographic range size.
I estimate a 97.9% probability that the cohort-specific estimates of β0and βvare negatively
correlated, with a median correlation of -0.49 (Fig. 10). This result means that as extinction459
intensity increases it is expected that epicontinental taxa become more favored over open-ocean
environments (i.e. as β0increases, βvdecreases). This result is strong evidence for a relationship
between intensity and selectivity with respect to the linear aspect of environmental preference.462
Correlations between the non-intercept estimates reflect potential similarities in selective pres-
sures between cohorts, however there is only weak evidence of any potential cross-correlations
in cohort-specific covariate effects(Fig. 10). There is an approximate 31.2% probability that βr
465
and βvare positively correlated. This lack of cross-correlation may be due in part to the higher
between-cohort variance of the effect of environmental preference τvcompared to the very small
between-cohort variance in the effect of geographic range τr(Table 1); the effect of geographic468
range might simply not vary enough relative to environmental preference.
Conversely, there is a 74.6% probability that estimates of the effect of geographic range (βr)
and the quadratic aspect of environmental preference (βv2) are positively correlated; this is weak471
evidence of a relationship between the effects of these covariates (Fig. 10). Thus, as the effect
of geographic range increases, we might expect the peakedness of relationship between environ-
mental preference and duration to increase. However, because there is only a 74.6% probability474
of a positive correlation, this result cannot interpreted with authority. Instead, this result is an
opportunity for future research to further explore the potential relationship between geographic
range, environmental preference, and species duration.477
29
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Figure 10: Mixed graphical and numerical representation of the correlation matrix Ωof variation
in cohort-specific covariate estimates. These correlations are between the estimates of the cohort-
level effects of covariates, along with intercept/baseline extinction risk. The median estimates of
the correlations are presented numerically (upper-triangle) and as idealized ellipses representing
that much correlation (lower-triangle). The darkness of the ellipse corresponds to the magnitude
of the correlation.
30
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Sampling was found to have a negative effect (positive δ) on duration: greater sampling,
shorter duration (Table 1). While potentially counter-intuitive, this result is most likely due to
some long lived taxa only be sampled in the stages of it’s first and last appearance. Also, longer480
lived taxa have more opportunities to evade sampling than shorter lived taxa. These two factors
will lead to this result.
The Weibull shape parameter αwas found to be approximately 1.41 (±0.05 SD) with a 100%483
probability of being greater than 1. This result is not consistent with the Law of Constant Ex-
tinction (Van Valen, 1973) and is instead consistent with accelerating extinction risk with taxon
age. This result is consistent with recent empirical results and may be caused by newly orig-486
inating species having a fundamentally lower risk of extinction compared to species which
have already originated (Quental and Marshall, 2013; Smits, 2015; Wagner and Estabrook, 2014).
This result is also consistent with a recently proposed nearly-neutral evolution where competi-489
tion/selection/evolution drives whole communities to increase in average fitness over time while
still maintaining constant relative fitness across the community, thus older species are more likely
to go extinct because of having a fundamentally lower average fitness than newly originating492
species (Rosindell et al., 2015). This result, however, is not consistent with other empirical results
from the marine fossil record (Crampton et al., 2016; Finnegan et al., 2008) and could potentially
be caused by the minimum resolution of the fossil record (Sepkoski, 1975). It is thus unclear495
whether a strong biological inference can be made from this result, which means that further
work is necessary on the effect of taxon age on extinction risk.
Discussion498
The generating observation behind this study was that for bivalves at the end Cretaceous mass
extinction event, the only biological trait that was found to affect extinction risk was geographic
range, while traits that had previously been associated with difference in duration had no effect501
(Jablonski, 1986). This observation raises two linked questions: how does the effect of geographic
31
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range change with changing extinction intensity, and how to the effects of other biological traits
change with changing extinction intensity?504
I find that as intensity increases (β0increases), the magnitude of the effect of geographic range
increases (βrdecreases). I also find that as intensity increases, the difference in survival for taxa
favoring epicontinental environments over open-ocean environments is expected to decrease;507
this is consistent with the results of Miller and Foote (2009). Finally, there is no evidence for
a correlation between the effects of geographic range and environmental preference on taxon
duration.510
I find consistent support for the “survival of the unspecialized,” with respect to epicontinen-
tal versus open-ocean environmental preference, as a time-invariant generalization of brachiopod
survival (Simpson, 1944). Taxa with intermediate environmental preferences are expected to have513
lower extinction risk than taxa specializing in either epicontinental or open-ocean environments
(Fig. 7), though the curvature of the relationship varies from rather shallow to very peaked
(Fig. 8). However, this relationship is not symmetric about 0, as taxa favoring epicontinental516
environments are expected with approximately 75% probability to have a greater duration than
taxa favoring open-ocean environments. This description of environment preference is only one
major aspect of a taxon’s environmental context, with factors such as bathymetry and tempera-519
ture being further descriptors of a taxon’s adaptive zone (Harnik, 2011; Harnik et al., 2012; Heim
and Peters, 2011; N ¨
urnberg and Aberhan, 2013); inclusion of these factors in future analyses
would potentially improve our understanding of the extent and complexity of the “survival of522
the unspecialized” hypothesis as it applies to all dimensions of an adaptive zone.
Hopkins et al. (2014), in their analysis of niche conservatism and substrate preference in ma-
rine invertebrates, found that brachiopods were among the least “conservative” groups, with taxa525
found to change substrate preference on short time scales. While substrate preference is not the
same as environmental preference (as defined here), a question does arise: are there three classes
of environmental preference instead of two? These classes would be taxa with broad tolerance528
(“true” generalists), inflexible specialists (“true” specialists), and flexible specialists. A flexible
32
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taxon is one with a narrow habitat preference at one time but with preference that changes over
time with changing environmental availability. My analysis assumes that traits are constant over531
the duration of the taxon, meaning that this scenario is not detectable: taxa with broad tolerances
and flexible taxa with narrow per-stage preference end up being treated the same way. Future
work should explore how environmental preference changes over lineage duration in relation534
to environmental availability to estimate if the generalists–specialists continuum is actually a
ternary relationship.
The analysis presented in this paper is an example of how to approach the interplay between537
selection and intensity using a continuous-survival framework. An alternative framework would
be a discrete-time survival analysis (Tutz and Schmid, 2016) where species survival is tracked
at discrete intervals. An example of a discrete-time survival approach that has become increas-540
ingly popular in paleontological analysis is the Cormack-Jolly-Seber (CJS) model (Liow et al.,
2008; Liow and Nichols, 2010; Royle and Dorazio, 2008; Tomiya, 2013). Discrete-survival anal-
ysis has some advantages over continuous-time approaches, specifically the ease of including543
time-varying covariates and well known extensions for allowing incomplete sampling (e.g. CJS
model).
Something that has not been modeled in these discrete-time analysis is the effect of an age-546
based varying-intercept or covariate on duration as recommded by Tutz and Schmid (2016); this
is extremely important for estimating the effect of taxon age on survival. Those varying-intercept
estimates would then be equivalent to the hazard function when all covariates are equal to 0549
(Tutz and Schmid, 2016). A good avenue for future applied research would be a CJS-type model
with survival modeled as a multi-level regression as in this study, combined with an age-based
varying-intercept as recommended by Tutz and Schmid (2016). A major hurdle to this analysis552
would be the necessity of imputing all time-varying covariates for every taxon that is estimated
to be present in a time intervals but was not sampled.
The model used here could be improved through either increasing the number of analyzed555
traits, expanding the hierarchical structure of the model to include other major taxonomic groups
33
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of interest, or including explicit phylogenetic relationships between the taxa in the model as an
additional hierarchical effect. An example trait that may be of particular interest is the affixing558
strategy or method of interaction with the substrate of the taxon, which has been found to
be related to brachiopod survival where, for cosmopolitan taxa, taxa that are attached to the
substrate are expected to have a greater duration than those that are not (Alexander, 1977).561
It is theoretically possible to expand this model to allow for comparisons within and between
major taxonomic groups, which would better constrain the brachiopod estimates while also al-
lowing for estimation of similarities and differences in cross-taxonomic patterns. The difficulty564
with this particular model expansion is in finding a similarly well-sampled taxonomic group
that is present during the Paleozoic. Potential groups include Crinoidea, Ostracoda, and other
members of the “Paleozoic fauna” (Sepkoski, 1981).567
With significant updates, it would also be possible to compare the brachiopod record with
Moden groups such as bivalves or gastropods (Sepkoski, 1981), while remembering that the
groups may not necessarily share all cohorts with the brachiopods. This particular model ex-570
pansion would act as a test of any universal cross-taxonomic patterns in the effects of emergent
traits on extinction, as has been proposed for geographic range (Payne and Finnegan, 2007). Ad-
ditionally, this expanded model would also act as a test of the distinctness of the Sepkoski (1981)573
three-fauna hypothesis in terms of trait-dependent extinction.
Traits like environmental preference or geographic range (Hunt et al., 2005; Jablonski, 1987)
are most likely heritable. Without phylogenetic context, this analysis assumes that differences576
in extinction risk between taxa are independent of the shared evolutionary history of those taxa
(Felsenstein, 1985). In contrast, the origination cohorts only capture shared temporal context. For
example, if taxon duration is phylogenetically heritable, then closely related taxa may have more579
similar durations as well as more similar biological traits. Without taking into account phyloge-
netic similarity the effects of these biological traits would be inflated solely due to inheritance.
The inclusion of phylogenetic context as an additional individual-level hierarchical effect, inde-582
pendent of origination cohort, would allow for determining how much of the observed variability
34
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is due to shared evolutionary history versus shared temporal context versus actual differences
associated with biological traits (Smits, 2015).585
The combination and integration of the phylogenetic comparative and paleontological ap-
proaches requires both sources of data, something which is not possible for this analysis because
there is no phylogenetic hypothesis for all Paleozoic taxa, which is frequently the case for marine588
invertebrates with a good fossil record. When both data sources are available the analysis can
more fully address the questions of interest in macroevolution (Fritz et al., 2013; Harnik et al.,
2014; Raia et al., 2012, 2013; Simpson et al., 2011; Slater, 2013, 2015; Slater et al., 2012; Smits, 2015;591
Tomiya, 2013).
Conclusion
My analysis demonstrates that for post-Cambrain Paleozoic brachiopds, as extinction intensity594
increases and average fitness decreases, such as in a mass extinction, the trait-associated dif-
ferences in fitness (selection) would increase and be greater than aeverage. In contrast, during
periods of low extinction intensity when fitness is greater than average, my model predicts that597
geographic range – and environmental preference – associated with differences in fitness (i.e.
selection) would decrease and be less than average. Taken together, these results point to a
potential macroevolutionary mechanism behind differences in trait-based survival during mass600
extinctions due to a correlation between intensity and selectivity. Additionally, I find continued
support for greater survival in environmental generalists over specialists; this is further evidence
that the long standing “survival of the unspecialized” hypothesis (Baumiller, 1993; Liow, 2004,603
2007; N ¨
urnberg and Aberhan, 2013, 2015; Simpson, 1944, 1953; Smits, 2015) should be consid-
ered the default hypothesis. Overall, this analysis further refines our knowledge of brachiopod
extinction dynamics while also revealing a potential macroevolutionary mechanism behind the606
difference between so-called mass and background extinction regimes.
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