Local and global abundance associated with extinction risk in late Paleozoic and early Mesozoic gastropods

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1 Paleobiology, 37(4), 2011, pp Local and global abundance associated with extinction risk in late Paleozoic and early Mesozoic gastropods Jonathan L. Payne, Sarah Truebe, Alexander Nützel, and Ellen T. Chang Abstract. Ecological theory predicts an inverse association between population size and extinction risk, but most previous paleontological studies have failed to confirm this relationship. The reasons for this discrepancy between theory and observation remain poorly understood. In this study, we compiled a global database of gastropod occurrences and collection-level abundances spanning the Early Permian through Early Jurassic (Pliensbachian). Globally, the database contains 5469 occurrences of 496 genera and 2156 species from 839 localities. Within the database, 30 collections distributed across seven stages contain at least 75 specimens and ten genera our minimum criteria for within-collection analysis of extinction selectivity. We use logistic regression analysis, based on global and local measures of population size and stage-level extinction patterns in Early Permian through Early Jurassic marine gastropods, to assess the relationship between abundance and extinction risk. We find that global genus occurrence frequency is inversely associated with extinction risk (i.e., positively associated with survival) in 15 of 16 stages examined, statistically significantly so in five stages. Although correlation between geographic range and occurrence frequency may account for some of this association, results from multivariable regression analysis suggest that the association between occurrence frequency and extinction risk is largely independent of geographic range. Within local assemblages, abundance (number of individuals) is also inversely associated with extinction risk. The strength of association is consistent across time and modes of fossil preservation. Effect strength is poorly constrained, particularly in analyses of local collections. In addition to limited power due to small sample size, this poor constraint may result from confounding by ecological variables not controlled for in the analyses, by taphonomic or collection biases, or from non-monotonic relationships between abundance and extinction risk. Two factors are likely to account for the difference between our results and those of most previous studies. First, many previous studies focused on the end-cretaceous mass extinction event; the extent to which these results can be generalized to other intervals remains unclear. Second, previous findings of nonselective extinction could result from insufficient statistical power rather than the absence of an underlying effect, because nonselective extinction is generally used as the null hypothesis for statistical convenience. Survivorship patterns in late Paleozoic and early Mesozoic gastropods suggest that abundance has been a more important influence on extinction risk through the Phanerozoic than previously appreciated. Jonathan L. Payne and Sarah Truebe.* Department of Geological and Environmental Sciences, Stanford University, Stanford, California jlpayne@stanford.edu. *Present address: Department of Geosciences, University of Arizona, 1040 East Fourth Street, Tucson, Arizona Alexander Nützel. Bayerische Staatssammlung für Paläontologie und Geologie, Ludwig-Maximilians- University Munich, Department für Geo- und Umweltwissenschaften, Sektion für Paläontologie, Geobiocenter LMU, Richard Wagner Strasse 10, Munich 80333, Germany Ellen T. Chang. Division of Epidemiology, Department of Health Research and Policy, Stanford University School of Medicine, Stanford, California and Cancer Prevention Institute of California, 2201 Walnut Avenue, Suite 300, Fremont, California Accepted: 4 March 2011 Supplemental materials deposited at Dryad: doi: /dryad.8330 Introduction Population size is predicted to scale inversely with extinction risk under a wide variety of extinction scenarios (Pimm et al. 1988; Lande 1993; Hubbell 2001). However, the fossil record provides at best mixed support for this prediction. Four paleontological studies used specimen counts to test explicitly for an association between population size and extinction risk for identified extinction events and none found a significant association (McClure and Bohonak 1995; Hotton 2002; Lockwood 2003; Leighton and Schneider 2008). Simpson and Harnik (2009) assessed the relationship between longevity and average abundance in marine bivalves, finding a significant but nonlinear relationship between abundance and longevity: longevity increased with abundance among the very rare to common taxa, but then decreased again among the proportionally most abundant taxa. Several other 2011 The Paleontological Society. All rights reserved /11/ /$1.00

2 ABUNDANCE AND EXTINCTION RISK 617 studies used occurrence frequency (i.e., the number of distinct localities from which a taxon has been reported) as a proxy for abundance, with mixed results. Kiessling and Baron-Szabo (2004) found no association between occurrence frequency and survivorship for Maastrichtian coral genera, and Powell (2008) found no association for Late Mississippian brachiopod genera. In contrast, Kiessling and Aberhan (2007) found a significant inverse association between occurrence frequency and genus extinction risk during most stages of the Late Triassic and Early Jurassic in a combined analysis of marine genera from several invertebrate phyla. Wilf and Johnson (2004) reported the preferential survival of rare plant species across the Cretaceous/Paleogene boundary, although they did not present an explicit statistical test of this association. Stanley (1986), on the other hand, argued that population size was the key determinant of specieslevel extinction in Pacific Neogene bivalves, using a variety of qualitative proxies for abundance. Overall, the small number of previous studies, differences among them in terms of temporal and taxonomic coverage and statistical methodology, and disproportionate focus on episodes of mass extinction limit any attempt at generalization. Clarifying the relationship between population size and extinction risk is essential for assessing the extent to which simple ecological models can be scaled to geological time scales, over which rare events associated with rapid environmental change appear to be the primary causes of extinction (Raup 1991b; Foote 2005, 2007). Moreover, extinction selectivity may vary depending upon the nature or scale of the causal process (Jablonski 1986, 2005; Payne and Finnegan 2007), highlighting the need for data from multiple time intervals to assess the persistence of selectivity patterns through geological time. Rare events could select upon characteristics unrelated to success during quiescent times, leaving traits such as abundance poor predictors of extinction risk on macroevolutionary time scales or during episodes of rapid environmental change Raup s (1991a) Wanton Destruction extinction mode. In this study, we assess the association between abundance and extinction risk for Early Permian through Early Jurassic marine gastropods, using both global occurrence frequency and local population density as measures of population size. This interval spans three major extinction events (Middle Permian, end-permian, end-triassic) and their respective recoveries as well as the intervening stages. It thus provides the opportunity to assess the degree of similarity in extinction selectivity across a range of macroevolutionary conditions. Gastropods are especially suitable for such an analysis because they represent one of the most diverse and abundant clades of marine animals during this interval. They were not marginalized by any of the extinction events, so a within-group analysis of extinction patterns is especially promising. Data and Methods In this study, we use both local and global data to assess the relationship between abundance and extinction risk in genera. We compiled gastropod occurrences from the published taxonomic literature, including collection-level specimen counts wherever possible. Below we describe the database size and structure, our treatment of the data fields, and our statistical approach to the analysis of extinction selectivity. Databases Global Database. We compiled occurrence and abundance data from published monographs of gastropod assemblages of Early Permian through Early Jurassic (Pliensbachian) age ( Ma). The resultant database includes taxonomic assignments and specimen counts as well as summary information regarding the locality, preservation, collection mode, and depositional environment. The standardized global data set includes 5469 occurrences of 496 genera and 2156 species from 839 localities spanning 19 geological stages. We merged the Asselian with the Sakmarian, the Wordian with the Capitanian, and the Induan with the Olenekian for purposes of analysis, yielding 16 discrete time intervals that we refer to below as stages for simplicity. The mean stage length is 7.25 Myr and the stages range in duration from 2.6 Myr (Roadian) to 14.6 Myr (Asselian Sakmarian). The data set analyzed in this study is archived

3 618 JONATHAN L. PAYNE ET AL. at Dryad ( 8330). Local Collections. Analyses of individual collections were restricted to those containing 75 or more specimens representing ten or more genera, of which at least four were extinction victims and at least four survived to the following stage. We required a minimum number of victims and survivors because regression analysis requires variation in both the predictor and outcome variables and power is limited by the number of observations with the less common outcome; the performance of logistic regression can become unreliable when the number of instances of the less common outcome per variable is extremely small (Peduzzi et al. 1996). In total, 30 collections distributed among seven stages met our sample-size criteria. Thirteen of the collections examined were from bulk rock material, six were from hand-picked surface collections, and 11 were mixed or unknown. The bulk-rock collections were preserved primarily via silicification, whereas the handpicked and mixed collections were preserved primarily as original shells or via calcite replacement (Table 1). Data Fields Taxonomy. Taxonomic assignments were standardized on the basis of recent taxonomic literature and Alexander Nützel s unpublished taxonomic database. Subgenera were elevated to genus rank. Only specimens classified to species level in the primary literature were included in the analysis because taxonomic standardization is often impossible for specimens assigned only at the genus level (see Wagner et al. 2007). Most specimens assigned to extant genera were removed from the data set because these fossil genera frequently are polyphyletic form genera (i.e., garbage bin genera), often reflecting basic shell shapes, rather than monophyletic (or even paraphyletic) groups (Nützel 2005; Plotnick and Wagner 2006). For example, Turritella has been used simply to refer to high-spired shells and Patella to many limpets. Specimens assigned to Emarginula were included in the analysis because we consider this genus to be a phylogenetically meaningful entity in Permian Triassic samples, even if it is not the same clade represented by morphologically similar living species. Monophyly of genera cannot be ascertained with certainty for the late Paleozoic and early Mesozoic. However, comparisons of molecular and morphological phylogenies show that morphologically based genera can serve as good proxies for closely related molluscan species (Jablonski and Finarelli 2009). Survivorship. From our database we determined minimum stage-level stratigraphic ranges of genera and subgenera. If younger last occurrences were reported in Sepkoski s (2002) compendium of the stratigraphic ranges of marine animal genera, then the times of extinction were adjusted accordingly. Only 22 genera (4.4%) were affected by this adjustment in time of extinction. We used Sepkoski s compendium because it provides stable estimates of global origination and extinction time, in contrast to potentially variable estimates from the Paleobiology Database. Close similarity in the overall diversity histories of the Sepkoski database and the Paleobiology Database (Alroy et al. 2008) and the small number of genera affected by adjustments in time of extinction suggest that the choice of compendium is unlikely to substantially influence the results. Geographic Range. We determined geographic range for each genus in each stage, and thus treated it as a dynamic variable rather than a taxon trait. This approach avoids complications introduced by the bidirectional relationship between taxon longevity and maximum geographic range (Foote et al. 2008). Geographic range was quantified as the maximum great circle distance between any two occurrences of a genus within a given stage. To calculate the paleolatitude and paleolongitude of each locality, we used paleographic reconstructions embedded in Chris Scotese s Point Tracker software (Scotese 2007). Global Occurrence Frequency. Occurrence frequency has been shown to correlate with population density for a wide range of extant and fossil taxa (Buzas et al. 1982; Brown 1984). In this study we use global occurrence frequency as a proxy for global population size. We calculate occurrence frequency as the

4 ABUNDANCE AND EXTINCTION RISK 619 TABLE 1. Summary information and logistic regression results for gastropod collections analyzed using specimen counts. Coll. no. Stage Reference Preservation Collection method No. of specimens No. of genera No. of victims Abundance log-odds (b1) with 95% C.I. (univariate) Abundance log-odds (b 1 ) 6 95% confidence interval (multivariate) Range log-odds (b 1 ) 6 95% confidence interval (univariate) Range log-odds (b 1 ) 6 95% confidence interval (multivariate) 73 Carnian Bandel 1994 original mixed Carnian Blaschke 1905 original surface Rhaetian Haas 1953 silicified bulk-rock Rhaetian Haas 1953 silicified bulk-rock Rhaetian Haas 1953 silicified bulk-rock Rhaetian Haas 1953 silicified bulk-rock Carnian Kittl 1891 calcite replacement surface and museum 571 Carnian Kittl 1891 calcite replacement surface and museum Carnian Kittl 1912 unknown Carnian Koken 1897 calcite replacement surface Norian Koken 1897 calcite replacement surface Norian Koken 1897 calcite replacement surface Norian Kutassy 1927 calcite replacement surface Carnian Leonardi and Fiscon Norian Nützel and Erwin Carnian Sachariewa- Kowatschewa 1961 calcite replacement surface silicified bulk-rock calcite replacement probably surface Wordian Yochelson 1956 silicified bulk-rock Wordian Yochelson 1956 silicified bulk-rock Carnian Zardini 1978 original mixed Carnian Zardini 1978 original mixed Carnian Zardini 1978 original mixed Carnian Zardini 1978 original mixed Carnian Zardini 1978 original mixed Carnian Zardini 1978 original mixed 13, Pliens. Dubar 1948 silicified bulk-rock Pliens. Dubar 1948 silicified bulk-rock Pliens. Dubar 1948 silicified bulk-rock Pliens. Dubar 1948 silicified bulk-rock Kungur. Batten 1972, 1979, Roadian Yochelson 1956; Erwin 1988a,b,c original bulk-rock silicified bulk-rock

5 620 JONATHAN L. PAYNE ET AL. number of localities at which a given genus has been reported. Each locality is counted as one occurrence regardless of the number of individuals or species within the genus reported from that locality. Local Specimen Counts. In our analyses of local collections, we used the number of reported specimens as a proxy for local population size. Live-dead comparisons suggest that dead shell abundance is correlated with the size of the living population for benthic mollusks (Kidwell 2001, 2002). Specimen counts were log 10 -transformed prior to analysis. Statistical Analysis Analytical Method. We used logistic regression to measure the association between predictor variables and extinction risk. Logistic regression is a special case of a generalized linear model in which the link function is the logit: ln(p/[12p]). In other words, the model assumes p= ð1{pþ~1= ½1zexpðb 0 zb 1 xþš ð1þ or ln½p= ð1{pþš~b 0 zb 1 x ð2þ and can be generalized to include multiple predictor variables ln½p= ð1{pþš~b 0 zb 1 x 1 zb 2 x 2 z...zb n x n ð3þ where p is the probability of the outcome of interest, x i is a predictor variable, b 0 is a constant, and b i (where i. 0) is a coefficient of association. Relative risk estimated from logistic regression is conventionally expressed as an odds ratio, exp(b i ), which describes the change in the odds (p/[12p]) as a function of change of one unit the predictor variable. Thus, an odds ratio of 0.5 indicates a halving of extinction risk per unit change in the predictor, whereas an odds ratio of 2 indicates a doubling. An odds ratio of 1 indicates no association. To preserve symmetry, we simply report the b i values below and refer to these as log-odds to simplify terminology, following previous paleobiological studies (Payne and Finnegan 2007; Finnegan et al. 2008). In contrast to these previous studies, we present the natural logarithm of the odds ratio rather than the common logarithm (base 10) because it is the most natural expression of b i values and the default output of nearly all statistical packages. Logistic regression is applied in cases where the response variable is binary (dichotomous) rather than continuous, such as extinction versus survival. The model is used to estimate the probability that a given observation will exhibit one outcome versus the other at a given value of the explanatory variable(s). The logit function is widely used because of its favorable mathematical properties and easily interpreted results. The approach assumes a monotonic relationship between p and the explanatory variable(s) and a linear relationship between the logit and the explanatory variable(s). Estimation of model parameters does not follow the least-squares approach used in standard linear regression because variance does not remain constant for all levels of the explanatory variable(s). Instead, a maximum likelihood approach is used. Figure 1 illustrates two examples from this study: Figure 1A illustrates the relationship between abundance and extinction risk for a Norian collection from Austria and Figure 1B illustrates the relationship between global occurrence frequency and extinction risk for the Kungurian stage (Early Permian). We refer readers to Hosmer and Lemeshow (2000) for a more detailed description of logistic regression and its applications. Logistic regression offers several advantages relative to simple parametric or nonparametric comparisons of means or distributions between victims and survivors. First, it allows for estimation of effect strength separately from statistical significance. This separation differs from comparisons of mean values or distributions (e.g., t-test, Mann-Whitney, Kolmogorov-Smirnov), which can be used only to assess statistical significance. The coefficient of association (analogous to slope in linear regression) is a measure of effect strength. The confidence bounds on the coefficient (as well as the associated p-value) can be used to assess statistical significance. Second, like any regression analysis, the effects of other correlates of extinction risk can be controlled for, and different models compared. Third, model weights derived from Akaike s Information Criterion corrected for small sample size (AICc)

6 ABUNDANCE AND EXTINCTION RISK 621 We also explored the extent to which any observed association can be explained by the influence of geographic range on extinction risk and the correlation between abundance and geographic range. Extinction victims tend to be less common than surviving genera both in terms of global occurrence frequency and in terms of specimen abundance in local collections. This tendency remains even after accounting for the association between geographic range and extinction risk. FIGURE 1. Examples of logistic regression. Circles represent individual observations of extinction (0) or survival (1), with diameters scaled to the number of observations with the same value along the ordinate. Gray horizontal lines represent aggregate probabilities of genus survival for observations within a given range along the ordinate. For local abundance, aggregate probabilities are calculated binning all observations between integer values. For global occurrence frequency, aggregate probabilities are calculated for each value of occurrence frequency. Black curves represent best-fit logistic curves to the raw data. A, Graph of probability of survival versus log-transformed abundance for collection 638 (see Table 1 for additional information). B, Graph of probability of genus survival versus occurrence frequency for the Kungurian stage (see Table 2 for additional information). can be used to evaluate relative support for extinction selectivity models without privileging any particular model as the null hypothesis (Johnson and Omland 2004). Such model weights sum to one and indicate the proportional distribution of support among the models considered (Johnson and Omland 2004). In this analysis, we interpret statistical significance with type I error set at the conventional value of 0.05 (i.e., p # 0.05 is considered as statistically significant). All analyses were run using SAS version 9.2. Results We explored the relationship between population size and extinction risk, using both global and local metrics of abundance. Global Occurrence Frequency Single Regression Globally. Occurrence frequency is inversely associated with genus extinction risk in 15 out of 16 stages, significantly so in five stages (Fig. 2). The coefficients of association for the intervals with statistically significant associations are not systematically greater than those for the intervals with nonsignificant associations. Rather, the intervals with non-significant results differ from those with significant results by having fewer occurrences and genera and more poorly constrained coefficients (Table 2). Intervals with significant associations contain on average nearly three times as many occurrences (means: 610 versus 220; Mann-Whitney p ) and more than twice as many genera (means: 108 versus 51; Mann-Whitney p ) as those without. These differences in sample size suggest that many instances of apparently nonselective extinction may result from type II error. The only interval to exhibit a positive association between occurrence frequency and extinction risk is the Scythian (Early Triassic), and this effect is not statistically significant. Controlling for Geographic Range. Occurrence frequency tends to correlate with geographic range (Brown 1984) as well as total population size (Buzas et al. 1982; Brown 1984). Geographic range and occurrence frequency are correlated in our data set as well (Table 3). Because geographic range is associated with extinction risk (Jablonski 1986; Brown 1995; McKinney 1997; Payne and Finnegan 2007), the association between occurrence frequency and extinction risk could result from an independent correlation between occurrence frequency and geographic range rather than from a direct

7 622 JONATHAN L. PAYNE ET AL. FIGURE 2. Association between genus occurrence frequency and survival into the subsequent stage. The association is expressed as the common logarithm (i.e., base 10) of the odds ratio. Positive values indicate preferential survival of abundant genera; negative values indicate preferential survival of rare genera. A, Log-odds of survival and 95% confidence intervals presented by geological stage, showing positive associations in 15 of 16 stages. B, Frequency distribution of log-odds from Figure 2A. causal link between occurrence frequency and extinction risk. To estimate the associations of occurrence frequency and geographic range with extinction risk independently of one another, we conducted a multivariable logistic regression analysis modeling extinction as a function of both geographic range and occurrence frequency. Geographic range was measured as the maximum great circle distance between occurrences of a given genus in a given stage. Other measures of geographic range, such as the number of tectonic plates occupied, provide qualitatively similar results. In the multivariable logistic regression analysis, occurrence frequency is inversely associated with extinction risk in 13 of 16 stages, and significantly so in one (Table 2). TABLE 2. Summary information and logistic regression results for the analysis of extinction selectivity by stage. Stage Occurrences Genera Victims Abundance log-odds (b 1 ) with 95% C.I. (univariate) Abundance log-odds (b 1 ) 6 95% confidence interval (multivariate) Range log-odds (b 1 ) 6 95% confidence interval (univariate) Range log-odds (b 1 ) 6 95% confidence interval (multivariate) Assel. Sak Artinskian Kungurian Roadian Word. Cap Wuchiap Changhs Scythian Anisian Ladinian Carnian Norian Rhaetian Hettangian Sinemur Pliensb

8 ABUNDANCE AND EXTINCTION RISK 623 TABLE 3. Correlation coefficients (Pearson s r) between occurrence frequency and geographic range by stage and associated p-values. Stage r p Asselian Sakmarian Artinskian 0.57, Kungurian 0.56, Roadian 0.65, Wordian Capitanian 0.55, Wuchiapingian 0.61, Changhsingian 0.67, Scythian 0.90, Anisian 0.59, Ladinian Carnian 0.44, Norian 0.52, Rhaetian Hettangian Sinemurian 0.75, Pliensbachian 0.62, We compared weights for models of nonselective extinction, selectivity based upon range only, selectivity based upon occurrence frequency only, and selectivity based upon both range and occurrence frequency using weights calculated from Akaike s Information Criterion corrected for small sample size (AICc) (Johnson and Omland 2004; Hunt 2006). There is comparatively little support for nonselective extinction; support is relatively evenly divided among the full model and models including either range or occurrence frequency (Table 4). The full model is the best-supported model in many of the best-sampled intervals overwhelmingly so in the best-represented stage (Carnian). Total support for the two models including abundance as a predictor of extinction risk ranges from 0.28 (Anisian) to 0.99 (Carnian), with a mean among stages of The absence of convincing support for a single model in many intervals likely reflects insufficient statistical power due to the size of the data set. Local Abundance Single Regression by Collection. To assess the relationship between population size and extinction risk further, we used 30 large collections representing seven stages in aggregate. More than 90% of these collections (27 of 30) exhibit an inverse association between local abundance and the probability of genus extinction into the next geological stage (Table 1, Fig. 3). Although no collection exhibits a statistically significant association, the overwhelming tendency toward positive log-odds cannot be explained by chance alone. In the case of no underlying association, half of the log-odds should fall above zero and half below. The probability of obtaining 27 or more positive log-odds in 30 collections by chance alone (in the absence of any underlying association) is less than Temporal Patterns. The strength of association between abundance and extinction risk TABLE 4. Model support calculated from Akaike s Information Criterion weighted for small sample size (AICc) for models of genus survival. Stage Nonselective Full model Range only Occurrence frequency only Total support for models including frequency Asselian Sakmarian Artinskian Kungurian Roadian Wordian Capitanian Wuchiapingian Changhsingian Scythian Anisian Ladinian Carnian Norian Rhaetian Hettangian Sinemurian Pliensbachian

9 624 JONATHAN L. PAYNE ET AL. FIGURE 3. Association between collection-level abundance and survival into the subsequent stage in marine gastropod genera. The association is expressed as the common logarithm (i.e., base 10) of the odds ratio. Positive values indicate preferential survival of abundant genera; negative values indicate preferential survival of rare genera. A, Log-odds of survival and 95% confidence intervals presented by geological stage, showing positive associations in 27 of 30 stages. Collections are ordered by the width of the 95% confidence interval on the log-odds. B, Frequency distribution of logodds from Figure 3A. is generally consistent among geological stages. Figure 4 illustrates the log-odds of extinction for individual collections by stage, demonstrating a consistent tendency toward positive log-odds of similar magnitude. There is no evidence of a secular trend in extinction selectivity or of substantial variation in extinction selectivity among stages. There is no strong relationship between stage duration and extinction selectivity, although the shortest stages may have been somewhat more selective than average (Roadian, Changhsingian, Hettangian) (Fig. 5). Thus, the association between local abundance and global extinction pattern appears pervasive within the study interval and does not simply reflect analysis of collections from a particular stage or region with unusual biological or taphonomic properties. FIGURE 4. Log-odds for individual collections plotted against geological age, illustrating consistency of association between abundance and extinction risk through geological time and across modes of preservation. Confidence bounds have been omitted for clarity. FIGURE 5. Cross-plot of extinction selectivity versus stage duration, illustrating consistency of selectivity as a function of stage duration.

10 ABUNDANCE AND EXTINCTION RISK 625 TABLE 5. Correlation between log-transformed local abundance and global geographic range, by collection. Collection No. of genera r p FIGURE 6. Relationship between collection size and measures of selectivity. A, Log-odds versus number of genera in collection, illustrating greater variability in collections containing fewer genera but no simple trend in log-odds with respect to collection size. B, Width of the 95% confidence interval versus the number of genera in the collection, illustrating greater uncertainty in estimated log-odds in smaller collections. Preservation Type. The apparent relationship between abundance and extinction risk may differ depending upon preservation type, as different modes of preservation may present different taphonomic filters altering both abundance distributions and the apparent presence or absence of species in different ways. The collections studied here are dominated by three different modes of preservation: original shell material (largely from the St. Cassian Formation of Italy), calcite replacement, and silicification (largely from the Permian of West Texas, but also from several other localities and ages) (Table 1). The distributions of log-odds are similar among preservation modes, although silicified assemblages exhibit slightly more positive logodds on average (Fig. 4). Thus, although these different modes of preservation undoubtedly result in altered abundance distributions relative to living populations, and likely do so in different ways, the tendency for abundance to be associated with extinction risk is not confined to a single preservation mode. Effect of Sample Size. Associations between abundance and survivorship are strongly inclined toward positive log-odds, but not significantly so within individual collections. This pattern appears to reflect limited statistical power rather than the absence of an underlying association. As expected when sample size is the primary influence, collections containing fewer genera exhibit more variable and more poorly constrained logodds (Fig. 6A). The 21 most diverse collections all exhibit positive log-odds (Fig. 6B). Multivariable Regression. To control for global geographic range, which is significantly correlated with local abundance in some collections (Table 5), we conducted a multivariable regression analysis of extinction as a function of both local abundance and global

11 626 JONATHAN L. PAYNE ET AL. TABLE 6. Model support calculated from Akaike s Information Criterion weighted for small sample size (AICc) for regression models of extinction risk within local collections. Collection Nonselective Full model Abundance only Range only Support for abundance alone or full model geographic range. In the multivariable regression, local abundance is associated with extinction risk in 25 out of the 30 collections (Table 1). The weighted mean log-odds of association between local abundance and extinction risk are similar in the univariable and multivariable regression models ( versus , respectively, for each log 10 -unit increase in local abundance), suggesting that geographic range does not account for most of the observed association between local abundance and global extinction risk. AICc weights are roughly equal among models with nonselective extinction, selection on local abundance, selection on global geographic range, and selection on both local abundance and geographic range (Table 6). Proportional support for models including local abundance as a predictor of extinction ranges from 0.28 to greater than 0.92 using AICc weights. Support for models including some form of selectivity on local abundance, geographic range, or both, ranges from 0.54 to greater than 0.99, with a mean value of Thus, genus extinction is inversely associated with local abundance, but the data are insufficient to determine with high confidence whether this reflects an effect of population size itself or correlation of local abundance with global geographic range and/ or other variables causally associated with extinction risk. Assessment of Statistical Power The consistent association between abundance and extinction risk in our regression analysis suggests an underlying association, even if we are not able to identify this association with 95% confidence in many cases. Because knowledge of the type II error rate (i.e., the probability of failing to reject the null hypothesis when it is false) is critical to interpreting our data, we used a simulation approach to investigate the power of logistic regression for data sets similar to ours. Power, or the probability of correctly rejecting the null hypothesis when it is false (i.e., the

12 ABUNDANCE AND EXTINCTION RISK 627 FIGURE 7. Statistical power of logistic regression for data sets similar to those analyzed in this study, illustrating the limited power associated with intervals and collections containing fewer than 50 genera. A, Power calculated using the occurrence structure of the Carnian global data and assuming that extinction risk is described by the logistic function that best fits our data. B, Power calculated using the abundance structure of collection number 1356 and assuming that extinction risk is described by the logistic function that best fits our data. complement of the probability of type II error), is a function of effect strength and sample size. Using the distribution of occurrence frequency from the Carnian and of local abundance from collection 1356, we created 500 simulated data sets each for samples containing 10, 20, 30, 50, 75, 100, 150, and 200 taxa by bootstrap sampling. We assumed that the extinction risk versus abundance was described by a logistic function and assigned survival status for each genus probabilistically. Then, using logistic regression, we tested for an association between abundance and extinction risk. We varied b 0 values with b 1 values to preserve a constant expected rate of extinction across values of b 1. (Assuming a constant value for b 0 would cause the total extinction rate in the simulated collection to vary as a function of selectivity.) Figure 7 illustrates statistical power as a function of sample size and effect strength for these two scenarios. In general, power is extremely limited in data sets smaller than 50 taxa and for effects magnitudes smaller than 0.2. Half of the studied stages and 87% (26/30) of collections contain fewer than 50 genera. For most intervals and collections, we lack the power to consistently detect effects of the magnitude that are likely to exist in the data even when the data match the model perfectly. Our power may be further limited by other factors, such as mismatch between our statistical model and the true structure of the relationship between abundance and extinction risk. Discussion The preferential extinction of less frequently occurring genera globally and less abundant genera within local collections suggests an inverse association between population size and extinction risk, even if the association is complex. This interpretation is further supported by the fact that the association is observed across a broad span of geological time and modes of fossil preservation. Some of the observed association between abundance and extinction risk can be explained by the association between abundance and geographic range, which is a strong predictor of extinction risk in the fossil record (Jablonski 2005; Kiessling and Aberhan 2007; Payne and Finnegan 2007), and therefore need not reflect a direct causal link between abundance and

13 628 JONATHAN L. PAYNE ET AL. extinction risk. However, we find evidence of an association between abundance and extinction risk even in multivariable regression analyses that control for the effects of geographic range. Occurrence frequency and geographic range are both significantly associated with extinction risk in the best-sampled stages, with log-odds similar to those for more poorly sampled stages. Therefore, the fact that the association with abundance is not statistically significant in most time intervals or collections likely reflects insufficient statistical power given the moderate magnitude of the effect rather than the absence of an association. Our results are thus consistent with the predicted relationship between population size and extinction risk (Pimm et al. 1988; Lande 1993; Hubbell 2001). The estimated association between abundance and survivorship may be confounded by a variety of biogeographic, ecological, or physiological factors. Variability in extinction risk among geographic regions (e.g., Clapham et al. 2009) could obscure the relationship between occurrence frequency and survivorship. The abundance-survivorship relationship could also be confounded by ecological or physiological differences among taxa that were not controlled for in the analysis. The autecology of late Paleozoic and early Mesozoic gastropods is largely unknown. However, abundant taxa likely were more basal in the food chain, on average, than rarer taxa, owing to fundamental constraints from trophic energy transfer and the fact that predators are generally larger than their prey (Cohen et al. 2003). Ecologically and physiologically selective extinction patterns are known from both background and mass extinction events (Jablonski and Raup 1995; Knoll et al. 1996; McKinney 1997; Smith and Jeffery 1998; Jablonski 2005; Knoll et al. 2007; Payne and Finnegan 2007; Friedman 2009). The substantial ecological diversity of marine gastropods suggests that they would be particularly susceptible to confounding from selectivity on traits such as trophic mode or environmental distribution. Several taphonomic and sampling factors could also confound the relationship between abundance and survivorship as observed in the fossil record. At the global scale, uneven geographic distribution of sampling may result in disproportionately high occurrence frequencies for genera inhabiting well-sampled regions and disproportionately low occurrence frequencies for those inhabiting poorly sampled regions. In some intervals, many occurrences derive from a few relatively small regions such as the Early and Middle Permian of West Texas (Yochelson 1956, 1960; Batten 1958, 1989; Erwin 1988a,b,c, 1989), the Carnian of northern Italy (Leonardi and Fiscon 1959; Zardini 1978), and the Norian and Rhaetian of Peru (Haas 1953). At local and global scales, taphonomic factors may also play an important role in confounding the abundance-survivorship association. In particular, species with smaller and thinner shells may be preferentially lost from dead shell accumulations relative to living communities because they are more susceptible to chemical and mechanical destruction (Flessa and Brown 1983; Kosnik et al. 2009). Smaller shells also tend to be more difficult to recover and identify even when they are not destroyed (Cooper et al. 2006; Hendy 2009; Sessa et al. 2009). On the other hand, silicification may favor the preservation of smaller individuals (Daley and Boyd 1996; Pan and Erwin 2002). The similarities among log-odds and among the widths of their 95% confidence intervals across modes of collection and preservation (Table 1, Fig. 4) suggest that either taphonomic noise is introduced prior to burial or the amount of taphonomic noise does not differ substantially between preservation modes. At present, it is not possible to determine how much of the noise in our data sets may reflect biological versus taphonomic processes. However, recent taphonomic studies suggest that the taphonomic contribution of simple shell destruction may not be as large as once feared. Rank-order abundance patterns in living molluscan communities appear to be well preserved in associated dead shell assemblages (Kidwell 2001). On the other hand, the preferential undersampling of smaller species is increasingly well documented (Cooper et al. 2006; Hendy 2009; Sessa et al. 2009) and difficult to control for at the global scale, especially when analyzing older data sets for

14 ABUNDANCE AND EXTINCTION RISK 629 which explicit sampling protocols cannot be determined. The discordance between our results and those of many previous studies likely reflects two factors, one inherent in the choice of statistical methods and the other involving selection bias in the choice of study intervals. We discuss each of these factors below. Nonselective extinction is commonly used as the null hypothesis because it is the scenario most easily tested with simple parametric or nonparametric approaches. Although nonselective extinction is a statistically convenient null hypothesis, it does not reflect expectations from simple demographic models. Lande (1993) showed that mean time to extinction scales exponentially with population size (i.e., carrying capacity) when governed by demographic stochasticity alone and as a power function of population size when stochastic variation in environmental quality and sudden catastrophes are also considered. Thus, when nonselective extinction is used as the null hypothesis, a finding of nonselective extinction may reflect type II error due to insufficient statistical power to detect a true selective effect, rather than a clear demonstration that population size was decoupled from extinction risk. The likelihood of type II error is higher for nonparametric tests, which have been used in nearly all previous studies (McClure and Bohonak 1995; Lockwood 2003; Kiessling and Baron-Szabo 2004; Kiessling and Aberhan 2007). Nearly all individual collections in our study exhibit no statistically significant association between abundance and extinction risk. However, rare genera are more likely to go extinct in nearly all of these collections and the best-sampled stages are the ones that tend to exhibit statistically significant extinction selectivity. An assessment of statistical power by numerical simulation shows that we have very limited power to reject the null hypothesis given our sample sizes and apparent effect magnitudes. Most previous studies have focused on mass extinction events, including all previous studies using specimen counts (McClure and Bohonak 1995; Hotton 2002; Lockwood 2003; Leighton and Schneider 2008), whereas our study and most others that have reported an inverse association between abundance and extinction risk (Stanley 1986; Kiessling and Aberhan 2007) primarily address background intervals. Thus, the contrasting results could reflect genuine differences in extinction mode between background intervals and (at least some) mass extinctions. On the other hand, several lines of evidence suggest that even mass extinction events have been selective to some extent. Kiessling and Aberhan (2007) found that the end-triassic event was not selective when all marine invertebrate genera were assessed simultaneously, but that it was selective when bivalve occurrence frequencies were analyzed alone. In light of the result for bivalves, the nonselective extinction pattern for invertebrates as a whole could be interpreted to reflect confounding from ecological or physiological differences among phyla and classes (see Wang and Bush 2008). Our results for gastropods are consistent with Kiessling and Aberhan s (2007) results for bivalves; moreover, they indicate that extinction selectivity with respect to abundance was not markedly different during the Rhaetian than during Permian and Triassic background stages. Although Lockwood (2003) did not find a significant association between abundance and extinction risk for western Atlantic bivalves during the end-cretaceous mass extinction, she did observe an approximately twofold difference in mean abundance and proportional abundance between victims and survivors. Our data provide little constraint on the relationship between abundance and extinction risk during the end-permian mass extinction. No Changhsingian collections met our minimum criteria for sample size. Global occurrence frequency is associated with genus extinction risk, but this finding could be accounted for by correlation between occurrence frequency and geographic range. Leighton and Schneider (2008) found no significant association between abundance and extinction risk, but interpretation of their results is complicated by the long time gap between the age of the samples (Late Pennsylvanian and Early Permian) and the timing of the extinction event (end-permian). If the association between abundance and extinction risk does not simply reflect correlation between abundance and geographic range,