Paleobiology, 29(4), 2003, pp. 455±467 On the continuity of background and mass extinction Steve C. Wang Abstract.ÐDo mass extinctions grade continuously into the background extinctions occurring throughout the history of life, or are they a fundamentally distinct phenomenon that cannot be explained by processes responsible for background extinction? Various criteria have been proposed for addressing this question, including approaches based on physical mechanisms, ecological se- lectivity, and statistical characterizations of extinction intensities. Here I propose a framework de®ning three types of continuity of mass and background extinc- tionsÐcontinuity of cause, continuity of effect, and continuity of magnitude. I test the third type of continuity with a statistical method based on kernel density estimation. Previous statistical ap- proaches typically have examined quantitative characteristics of mass extinctions (such as metrics of extinction intensity) and compared them with the distribution of such characteristics associated with background extinctions. If mass extinctions are outliers, or are separated by a gap from back- ground extinctions, the distinctness of mass extinctions is supported. In this paper I apply Silverman's Critical Bandwidth Test to test for the continuity of mass ex- tinctions by applying kernel density estimation and bootstrap modality testing. The method im- proves on existing work based on searching for gaps in histograms, in that it does not depend on arbitrary choices of parameters (such as bin widths for histograms), and provides a direct estimate of the signi®cance of continuities or gaps in observed extinction intensities. I am thus able to test rigorously whether differences between mass extinctions and background extinctions are statisti- cally signi®cant. I apply the methodology to Sepkoski's database of Phanerozoic marine genera. I conclude that mass and background extinctions appear to be continuous at this third levelÐcontinuity of mag- nitudeÐeven though evidence suggests that they are discontinuous at the ®rst and second levels. Steve C. Wang. Department of Statistics, Harvard University, Cambridge, Massachusetts 02138 Present address: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081. E-mail: [email protected] Accepted: 21 March 2003 Introduction ers. Quinn (1983), Raup (1986, 1991a,b, 1994), Bambach and Gilinsky (1986), and McKinney Ever since the proposal that the end-Creta- (1987) argued for the continuity of mass ex- ceous mass extinction resulted from a bolide tinctions with background extinctions. On the impact (Alvarez et al. 1980), the nature of other hand, Raup and Sepkoski (1982), Gould mass extinctions has been the subject of much (1985), and Bambach and Knoll (2001) argued debate. A key question is whether mass ex- that mass extinctions are a distinct phenome- tinctions grade continuously into the back- non, and Stigler (1987) also found evidence for ground extinctions occurring throughout the this position. One reason for the debate stems history of life, or whether they constitute a from the fact that different authors have dif- fundamentally different phenomenon. In the ferent meanings for what it means for mass former view, mass extinctions represent the extinctions to be ``continuous'' with or ``dis- right tail of a continuum, separated from tinct'' from background extinctions, and their background extinctions by an arbitrary cutoff, meanings are often stated only implicitly. much as a blizzard and a ¯urry represent Here I propose a framework de®ning three varying degrees in a continuum of snowfall. types of continuityÐcontinuity of cause, con- In the latter view, mass extinctions represent a tinuity of effect, and continuity of magnitude. distinct phenomenon that cannot be explained These three types of continuity are indepen- by merely scaling up background extinctions, dent of each other, in that mass extinctions just as a hailstorm is not merely a larger ver- may be discontinuous at one level but contin- sion of a snow ¯urry. uous at the others. For the third type of con- Both sides of the debate have their support- tinuity, I propose the application of a statisti- q 2003 The Paleontological Society. All rights reserved. 0094-8373/00/2904-0001/$1.00 456 STEVE C. WANG cal method to test whether mass extinctions end-Permian mass extinction compared with are continuous with background extinctions. background extinction patterns, and Boyajian (1991) found no differences in selectivity in Types of Continuity mass extinctions with regard to taxon age af- Continuity of cause occurs when the same ter controlling for extinction size. On the other processes that are responsible for background hand, several studies have found evidence for extinctions, operating at an increased level or differential patterns of survival at mass ex- intensity, also cause mass extinctions (Miller tinctions compared with background extinc- 1998). For example, if background extinctions tion. These include studies by Anstey (1986) are caused by terrestrial factors (e.g., changes on Ordovician bryozoans, Jablonski (1986) on in sea level and climate), and these same fac- Cretaceous mollusks, Jablonski and Raup tors at a more extreme intensity also cause (1995) on Cretaceous bivalves, Johansen (1989) mass extinctions, then mass and background on Cretaceous brachiopods, and Westrop extinctions would be continuous in cause. (1989) on Cambrian trilobites; see also Stanley Continuity of cause would also be established 1987 for a general discussion. Such ®ndings if mass extinctions result primarily from com- support a discontinuity of effect. Gould (1985) petition among clades, as Briggs (1998) ar- also supported discontinuity of effect, argu- gues. If, on the other hand, mass extinctions ing that mass extinctions constitute a third are caused by factors different from those ``tier'' distinct from and irreducible to within- causing background extinctions (e.g., bolide species competition (the ®rst tier) and species- impact), this would constitute a discontinuity level selection (the second tier). of cause. In the latter case, mass extinction Continuity of magnitude exists when the dis- would differ qualitatively from background tribution of intensities of mass extinctions (as extinction and in a fundamental way. The measured by the number of extinctions per well-accepted evidence for an impact at the unit time or some other metric) grade smooth- end of the Cretaceous (Alvarez et al. 1980) ly and continuously into the intensities of supports discontinuity of cause, as does the background extinctions. Quinn (1983), Bam- newer and more controversial evidence for bach and Gilinsky (1986), McKinney (1987), impact at the ends of the Permian and Triassic and Raup (1986, 1991a,b, 1994) found that Periods (Becker et al. 2001; Olsen et al. 2002). mass extinctions are continuous at this level, Raup and Boyajian's ®nding (1988) that major and Thackeray (1990) arrived at a similar con- extinction events result from environmental clusion for nine extinction events with the ex- disturbances may also be interpreted as sup- ception of the end-Cretaceous event. On the porting discontinuity at this level. Continuity other hand, Raup and Sepkoski (1982) and of cause may be further subdivided into ulti- Bambach and Knoll (2001) argued for discon- mate cause (e.g., bolide impactÐthe trigger tinuity, with Stigler (1987) also ®nding evi- mechanism) and immediate cause (e.g., re- dence for the latter position. sulting nutrient crisisÐthe kill mechanism). A common method of determining conti- Continuity of effect is established when back- nuity of magnitude is by examining histo- ground and mass extinctions exhibit common grams of extinction intensities of Phanerozoic patterns of selectivity on taxonomic, function- stages. If mass extinctions are continuous in al, morphological, geographical, or other cri- magnitude, then such a histogram should ap- teria. In other words, continuity of effect re- pear unimodal, with no apparent gaps sepa- fers to whether the biological and ecological rating mass extinctions in the right tail of the effects of background extinction and mass ex- distribution from background extinctions tinction are similar in nature, if not in mag- (Raup 1986: Fig. 1, also cited in Jablonski 1989; nitude. For example, McKinney (1987) found Raup 1991b: Fig. 4±4; Raup 1994: Fig. 2, also that extinction rates in background and mass cited in Jablonski 2001). On the other hand, if extinctions are strongly correlated for ten ma- such a histogram appears bimodal, with mass jor marine taxa, Erwin (1989, 1990) found no extinctions forming a cluster of outliers in the difference in selectivity of gastropods at the right tail, we would infer that mass extinctions CONTINUITY OF MASS EXTINCTION 457 FIGURE 2. A hypothetical example of an extinction in- tensity curve f(x). Such a curve is a probability density function (pdf) that models the underlying process gov- erning extinction and describes how likely various in- tensities of extinctions are. ``bins.'' Raup (1994: Fig. 2) chose bins of {0± 5%, 5±10%, ...,95±100%}. The choice of bin width and bin location is arbitrary: the bins {0±10%, 10±20%, ..., 90±100%} would be as valid a selection, as would be {(-5)±5%, 5±15%, ...,95±105%}. Usually these choices are made FIGURE 1. Histograms of per-genus proportional ex- automatically by software and have a relative- tinction rate for 107 Phanerozoic stages and substages ly minor