No statistical support for sudden (or gradual) extinction of dinosaurs Stuart H. Hurlbert J. David Archibald Department of Biology, San Diego State University, San Diego, California 92182 ABSTRACT Did dinosaurs decline gradually or abruptly at the Cretaceous-Tertiary boundary? An analysis of familial diversity patterns in dinosaur fossils from the Hell Creek Formation of central North America has claimed to present strong statistical evidence against the idea that dinosaurs declined gradually near the end of the Cretaceous. Examination of the quantitative methodologies used shows that these provide no basis for choosing between scenarios of abrupt extinction and gradual decline. INTRODUCTION The disappearance of dinosaurs has come to epitomize the Cretaceous-Tertiary (K-T) extinctions, although their fossil record is one of the poorest. Large terrestrial orga- nisms are almost never as well preserved and represented as are smaller or aquatic vertebrates. Furthermore, there is no global record of dinosaur extinction, in spite of pronouncements to the contrary. One view of their disappearance, based on the distribution of fossil teeth, is that dinosaurs declined gradually in the waning several million years of the Cretaceous (e.g., Sloan et al., 1986), possibly even per- sisting into the Tertiary (Rigby et al., 1987). No articulated dinosaur remains have been discovered in lowermost Terti- ary rocks, however. The other view is that dinosaur extinction was catastrophic at the K-T boundary. An- alyzing familial diversity of dinosaurs in the Hell Creek Formation, Sheehan et al. (1991, p. 835) concluded that there was ‘‘no evi- dence (probability P  0.05) of a gradual decline of dinosaurs at the end of the Cre- taceous . . . [and that their] findings are in agreement with an abrupt extinction event such as one caused by an asteroid impact.’’ This conclusion has been accepted and widely cited (e.g., Gould, 1992; Alvarez et al., 1994; Lucas, 1994). Careful review of this study finds methodological problems that undermine its conclusions. In the fol- lowing analysis, we address three issues: the appropriateness of the indices used to ad- dress the question, the sensitivity of familial diversity to generic extinctions, and the ap- propriateness of statistical procedures. Be- cause the methods reviewed are widely used in paleontology and ecology, the relevance of our remarks extends beyond the specific issue of dinosaur extinctions. APPROPRIATENESS OF THE INDICES The objective of the Hell Creek study was stated in general terms such as ‘‘to assess the general robustness of dinosaur populations’’ and how this may or may not have changed during the latter part of the Cretaceous, or to test ‘‘the hypothesis that the dinosaurian part of the ecosystem was deteriorating’’ at that time (Sheehan et al., 1991). The objec- tive became defined operationally when it was specified that ‘‘robustness’’ would be quantified by two diversity indices—the Shannon index, H', and expected species (or taxa) richness, E(S n ). A close examination of these two indices shows that they are not capable of answering the questions posed in the Hell Creek study. The Shannon index, H', was introduced by MacArthur (1955) and is calculated as H' = - p i log p i , (1) where p i = N i /N,N i = number of individuals in the ith taxon or category, N = N i , i = 1, 2, . . . , S, and S = the total number of taxa in the sample. E(S n ) was introduced by Sanders (1968) and Hurlbert (1971) and is the expected number of species (or taxa) in a sample of n individuals selected at random from a larger sample containing N individ- uals and S species. It is calculated as ES n  =  i  1 -  N - N i n   N n   . (2) Normally E(S n ) is calculated for a large number of values of n, and graphical plots of n vs. E(S n ) are presented and referred to as rarefaction curves. Both H' and E(S n ) in- crease in value as the number of taxa increases and as the evenness with which individuals are distributed across taxa in- creases, other things being equal. ‘‘Robustness,’’ however, as quantified by H' or E(S n ) is, at best, only weakly related to the variables of direct interest, such as ab- solute numbers of dinosaur taxa, dinosaur population densities, or dinosaur extinction rates. This is easily demonstrated. Both H' and E(S n ) could decrease in value at the same time that both the total number of di- nosaur taxa and the total number of dino- saurs were increasing. For example, compare two hypothetical assemblages, A and B, con- taining, respectively, 2251 and 14 083 dino- saurs distributed over eight families as fol- lows—A{0, 0, 41, 97, 142, 553, 610, 808} and B{82, 39, 90, 180, 230, 400, 12 082, 980}. In going from A to B, H' declines from 0.63 to 0.26, and E(S 50 ) declines from 5.46 to 4.43. The decline in both indices might lead some to characterize the transition from A to B as representing a decline in ‘‘robust- ness.’’ A flowering of dinosaurs, however, would seem a more apt characterization, be- cause numbers of families and of dinosaurs are both increasing. This example shows that it is equally incorrect to argue that ‘‘A decrease in ecologic diversity [= the indices] would occur . . . if one or more families de- clined in relative abundance, even if they did not become extinct’’ (Sheehan et al., 1991). Two properties of rarefaction curves re- quire especially careful interpretation. First, these curves almost invariably bend over strongly when n is plotted on an arithmetic scale, even when N is small. This apparent approach of E(S n ) to an asymptotic value has led to fruitless endeavors to determine a ‘‘sufficient sample size’’ for the estimation of species richness (e.g., Heck et al., 1975). When these curves are plotted using a log- arithmic scale for n, we often obtain a rather different idea of reality. The curves usually do not approach an asymptote and some- times have steeper slopes at their termini than in earlier parts (e.g., Fager, 1972; Kempton and Taylor, 1979; Dexter, 1992). For sufficiently high values of n, even on a Geology; October 1995; v. 23; no. 10; p. 881– 884; 3 tables. 881