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n-j if p ≠ q The last equation is a correction of equation (A18) from (9). Let P(≥ n, t, a) be the probability that diversity is greater than or equal to n lineages at time t, given that it is equal to a lineages at t = 0. Then (equation presented) Let t and T be two points in time such that t < T, and let the group have diversity equal to one lineage at t = 0. Let P(n, t, s, T) be the probability that diversity is exactly equal to n lineages at time t, given that the group survives at least until time T. Let P(n,t,N,T) be the probability that diversity is exactly equal to n lineages at time t, given that diversity is exactly equal to N lineages at time T. And let P(n,t ≥ N, T) be the probability that diversity is exactly equal to n lineages at time t, given that diversity is greater than or equal to N lineages at time T. Then, by the rules of conditional probability, P(n,t,s,T) = P(n,t,1) · P(s,T - t,n)/P(s,T,1); P(n.t.N,T) = P(n,t,1) · P(N,T - t,n)/P(N,T,1); and P(n,t, ≥ N,T) = P(n,t,1) · P(≥ N,T - t,n)/P(≥ N,T,1) The corresponding expected (mean) diversities at time t are given by (equation presented) and (equation presented)
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39
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note
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We simplify the problem using the well-known positive correlation between origination and extinction rates and constrain the net growth rate (p - q) to be equal to ln(N)/T, thus forcing p to be equal to q + ln(N)/T. For discussion of this correlation, see S. M. Stanley, in Causes of Evolution, R. M. Ross and W. D. Allmon, Eds. (Univ. of Chicago Press, Chicago, IL, 1990), pp. 103-127.
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0 (all standard errors based on bootstrap resampling of these 41 species). Because of the small number of species and the uncertainty in absolute dating, there is uncertainty in our estimate of extinction rate. But because the relation between estimated extinction rate and estimated preservation rate is linear, the value of q would have to be more than an order of magnitude lower than our estimate in order to affect our conclusions substantially. Taxonomic rates this Low are incompatible with rates estimated for Cenozoic mammals (14, 24, 25) and for even the most slowly evolving groups of animals (25).
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Because of the difference between Cenozoic and Cretaceous preservation rates, we do not recommend using Cenozoic occurrences to place confidence limits (15) on stratigraphic ranges for mammal taxa that may extend into the Cretaceous [see also (10)]. Moreover, confidence limits are more difficult to estimate for higher taxa than for single species, because the probability of group preservation per unit time is potentially greatly affected by changes in diversity [R. Bleiweiss, Geology 26, 323 (1998); C. R. Marshall and R. Bleiweiss, ibid. 27, 95 (1999) ].
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Because of the difference between Cenozoic and Cretaceous preservation rates, we do not recommend using Cenozoic occurrences to place confidence limits (15) on stratigraphic ranges for mammal taxa that may extend into the Cretaceous [see also (10)]. Moreover, confidence limits are more difficult to estimate for higher taxa than for single species, because the probability of group preservation per unit time is potentially greatly affected by changes in diversity [R. Bleiweiss, Geology 26, 323 (1998); C. R. Marshall and R. Bleiweiss, ibid. 27, 95 (1999) ].
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It is also likely that preservation rate fluctuates over time, because sea level and other factors change. Modeling of time-heterogeneous preservation (14, 24), however, shows that fluctuating preservation rate is not likely to distort substantially either the overall probability of species preservation or our estimates of preservation rate.
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If molecular rates are faster during evolutionary radiations, this may reflect two end-member mechanisms that have been suggested previously: (i) Adaptive change within species is faster during the rapid occupation of new adaptive zones (41), and many molecular substitutions are selective rather than neutral (35, 36); or (ii) a punctuational model applies to molecular evolution, with change concentrated at lineage splitting; thus, rapid speciation during evolutionary radiation (41) causes faster molecular rates (37). These mechanisms are testable with data from extant groups for which fossil diversity, genealogical relationships, and fossil divergence times are reliably known: Each pair of living species is separated by a divergence time, a number of cladogenetic events (patristic distance), a morphological distance, and a molecular distance. If morphological divergence provides a measure of adaptive change, the first hypothesis implies a positive partial correlation between molecular and morphological distance, with divergence time and patristic distance held statistically constant. The second hypothesis implies a positive partial correlation between molecular and patristic distance, with divergence time and morphological distance held constant.
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Occurrence data are from a compilation of faunal lists and synonymies, extensively supplemented with observations by J.P.H. This database is heavily influenced by the rich fossil record of North America, but also includes the known faunas of South America, Europe, Asia (Mongolia, India, and western Asia), and Madagascar. Biostratigraphic correlation for species outside North America was based largely on information in the references consulted for faunal lists. For North American species, these sources were supplemented by the following: J. A. Lillegraven, Contrib. Sci. Los Angeles Co. Mus. 232, 1 (1972); _ and L. M. Ostresh Jr., Geol. Soc. Am. Spec. Pap. 243, 1 (1990); M. B. Goodwin and A. L. Deino, Can. J. Earth Sci. 26, 1384 (1989); R. R. Rogers, C. C. Swisher III, J. R. Homer, ibid. 30, 1066 (1993); C. C. Swisher III, L. Dingus, R. F. Butler, ibid., p. 1981; F. M. Gradstein et al., in Geochronology, Time Scales and Global Stratigraphic Correlation, W. A. Berggren, D. V. Kent, M.-P. Aubry, J. Hardenbol, Eds. (Society for Sedimentary Geology (SEPM), Tulsa, OK], pp. 95-126. To determine whether a species is known from a single horizon, it is generally necessary to have only a relative biochronology. We did not base our relative chronology solely on faunal associations of mammals, but also used local and regional stratigraphic relations, radiometric dates, and correlations with marine sequences based largely on regional strandline stratigraphy, as derived from sources cited above. North American localities were arranged into 14 resolvable stratigraphic intervals, and the localities of other continents were resolved to between 1 and 3 intervals, depending on the region. We treated each of these intervals operationally as a horizon. This approach is conservative insofar as it lumps distinct horizons into one unit, overestimating the proportion of species confined to single horizons and thus underestimating preservation rate. Our approach is also conservative insofar as we generally considered species with uncertain stratigraphic ranges as single-horizon taxa. To estimate average stratigraphic ranges of North American species, for which the biochronology is best understood, we assigned absolute ages to the 14 stratigraphic intervals, mainly using correlations with ammonite zones and radiometric dates. See supplemental data at www.sciencemag.org/ feature/data/985988.shl.
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Occurrence data are from a compilation of faunal lists and synonymies, extensively supplemented with observations by J.P.H. This database is heavily influenced by the rich fossil record of North America, but also includes the known faunas of South America, Europe, Asia (Mongolia, India, and western Asia), and Madagascar. Biostratigraphic correlation for species outside North America was based largely on information in the references consulted for faunal lists. For North American species, these sources were supplemented by the following: J. A. Lillegraven, Contrib. Sci. Los Angeles Co. Mus. 232, 1 (1972); _ and L. M. Ostresh Jr., Geol. Soc. Am. Spec. Pap. 243, 1 (1990); M. B. Goodwin and A. L. Deino, Can. J. Earth Sci. 26, 1384 (1989); R. R. Rogers, C. C. Swisher III, J. R. Homer, ibid. 30, 1066 (1993); C. C. Swisher III, L. Dingus, R. F. Butler, ibid., p. 1981; F. M. Gradstein et al., in Geochronology, Time Scales and Global Stratigraphic Correlation, W. A. Berggren, D. V. Kent, M.-P. Aubry, J. Hardenbol, Eds. (Society for Sedimentary Geology (SEPM), Tulsa, OK], pp. 95-126. To determine whether a species is known from a single horizon, it is generally necessary to have only a relative biochronology. We did not base our relative chronology solely on faunal associations of mammals, but also used local and regional stratigraphic relations, radiometric dates, and correlations with marine sequences based largely on regional strandline stratigraphy, as derived from sources cited above. North American localities were arranged into 14 resolvable stratigraphic intervals, and the localities of other continents were resolved to between 1 and 3 intervals, depending on the region. We treated each of these intervals operationally as a horizon. This approach is conservative insofar as it lumps distinct horizons into one unit, overestimating the proportion of species confined to single horizons and thus underestimating preservation rate. Our approach is also conservative insofar as we generally considered species with uncertain stratigraphic ranges as single-horizon taxa. To estimate average stratigraphic ranges of North American species, for which the biochronology is best understood, we assigned absolute ages to the 14 stratigraphic intervals, mainly using correlations with ammonite zones and radiometric dates. See supplemental data at www.sciencemag.org/ feature/data/985988.shl.
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Occurrence data are from a compilation of faunal lists and synonymies, extensively supplemented with observations by J.P.H. This database is heavily influenced by the rich fossil record of North America, but also includes the known faunas of South America, Europe, Asia (Mongolia, India, and western Asia), and Madagascar. Biostratigraphic correlation for species outside North America was based largely on information in the references consulted for faunal lists. For North American species, these sources were supplemented by the following: J. A. Lillegraven, Contrib. Sci. Los Angeles Co. Mus. 232, 1 (1972); _ and L. M. Ostresh Jr., Geol. Soc. Am. Spec. Pap. 243, 1 (1990); M. B. Goodwin and A. L. Deino, Can. J. Earth Sci. 26, 1384 (1989); R. R. Rogers, C. C. Swisher III, J. R. Homer, ibid. 30, 1066 (1993); C. C. Swisher III, L. Dingus, R. F. Butler, ibid., p. 1981; F. M. Gradstein et al., in Geochronology, Time Scales and Global Stratigraphic Correlation, W. A. Berggren, D. V. Kent, M.-P. Aubry, J. Hardenbol, Eds. (Society for Sedimentary Geology (SEPM), Tulsa, OK], pp. 95-126. To determine whether a species is known from a single horizon, it is generally necessary to have only a relative biochronology. We did not base our relative chronology solely on faunal associations of mammals, but also used local and regional stratigraphic relations, radiometric dates, and correlations with marine sequences based largely on regional strandline stratigraphy, as derived from sources cited above. North American localities were arranged into 14 resolvable stratigraphic intervals, and the localities of other continents were resolved to between 1 and 3 intervals, depending on the region. We treated each of these intervals operationally as a horizon. This approach is conservative insofar as it lumps distinct horizons into one unit, overestimating the proportion of species confined to single horizons and thus underestimating preservation rate. Our approach is also conservative insofar as we generally considered species with uncertain stratigraphic ranges as single-horizon taxa. To estimate average stratigraphic ranges of North American species, for which the biochronology is best understood, we assigned absolute ages to the 14 stratigraphic intervals, mainly using correlations with ammonite zones and radiometric dates. See supplemental data at www.sciencemag.org/ feature/data/985988.shl.
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Occurrence data are from a compilation of faunal lists and synonymies, extensively supplemented with observations by J.P.H. This database is heavily influenced by the rich fossil record of North America, but also includes the known faunas of South America, Europe, Asia (Mongolia, India, and western Asia), and Madagascar. Biostratigraphic correlation for species outside North America was based largely on information in the references consulted for faunal lists. For North American species, these sources were supplemented by the following: J. A. Lillegraven, Contrib. Sci. Los Angeles Co. Mus. 232, 1 (1972); _ and L. M. Ostresh Jr., Geol. Soc. Am. Spec. Pap. 243, 1 (1990); M. B. Goodwin and A. L. Deino, Can. J. Earth Sci. 26, 1384 (1989); R. R. Rogers, C. C. Swisher III, J. R. Homer, ibid. 30, 1066 (1993); C. C. Swisher III, L. Dingus, R. F. Butler, ibid., p. 1981; F. M. Gradstein et al., in Geochronology, Time Scales and Global Stratigraphic Correlation, W. A. Berggren, D. V. Kent, M.-P. Aubry, J. Hardenbol, Eds. (Society for Sedimentary Geology (SEPM), Tulsa, OK], pp. 95-126. To determine whether a species is known from a single horizon, it is generally necessary to have only a relative biochronology. We did not base our relative chronology solely on faunal associations of mammals, but also used local and regional stratigraphic relations, radiometric dates, and correlations with marine sequences based largely on regional strandline stratigraphy, as derived from sources cited above. North American localities were arranged into 14 resolvable stratigraphic intervals, and the localities of other continents were resolved to between 1 and 3 intervals, depending on the region. We treated each of these intervals operationally as a horizon. This approach is conservative insofar as it lumps distinct horizons into one unit, overestimating the proportion of species confined to single horizons and thus underestimating preservation rate. Our approach is also conservative insofar as we generally considered species with uncertain stratigraphic ranges as single-horizon taxa. To estimate average stratigraphic ranges of North American species, for which the biochronology is best understood, we assigned absolute ages to the 14 stratigraphic intervals, mainly using correlations with ammonite zones and radiometric dates. See supplemental data at www.sciencemag.org/ feature/data/985988.shl.
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Occurrence data are from a compilation of faunal lists and synonymies, extensively supplemented with observations by J.P.H. This database is heavily influenced by the rich fossil record of North America, but also includes the known faunas of South America, Europe, Asia (Mongolia, India, and western Asia), and Madagascar. Biostratigraphic correlation for species outside North America was based largely on information in the references consulted for faunal lists. For North American species, these sources were supplemented by the following: J. A. Lillegraven, Contrib. Sci. Los Angeles Co. Mus. 232, 1 (1972); _ and L. M. Ostresh Jr., Geol. Soc. Am. Spec. Pap. 243, 1 (1990); M. B. Goodwin and A. L. Deino, Can. J. Earth Sci. 26, 1384 (1989); R. R. Rogers, C. C. Swisher III, J. R. Homer, ibid. 30, 1066 (1993); C. C. Swisher III, L. Dingus, R. F. Butler, ibid., p. 1981; F. M. Gradstein et al., in Geochronology, Time Scales and Global Stratigraphic Correlation, W. A. Berggren, D. V. Kent, M.-P. Aubry, J. Hardenbol, Eds. (Society for Sedimentary Geology (SEPM), Tulsa, OK], pp. 95-126. To determine whether a species is known from a single horizon, it is generally necessary to have only a relative biochronology. We did not base our relative chronology solely on faunal associations of mammals, but also used local and regional stratigraphic relations, radiometric dates, and correlations with marine sequences based largely on regional strandline stratigraphy, as derived from sources cited above. North American localities were arranged into 14 resolvable stratigraphic intervals, and the localities of other continents were resolved to between 1 and 3 intervals, depending on the region. We treated each of these intervals operationally as a horizon. This approach is conservative insofar as it lumps distinct horizons into one unit, overestimating the proportion of species confined to single horizons and thus underestimating preservation rate. Our approach is also conservative insofar as we generally considered species with uncertain stratigraphic ranges as single-horizon taxa. To estimate average stratigraphic ranges of North American species, for which the biochronology is best understood, we assigned absolute ages to the 14 stratigraphic intervals, mainly using correlations with ammonite zones and radiometric dates. See supplemental data at www.sciencemag.org/ feature/data/985988.shl.
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Occurrence data are from a compilation of faunal lists and synonymies, extensively supplemented with observations by J.P.H. This database is heavily influenced by the rich fossil record of North America, but also includes the known faunas of South America, Europe, Asia (Mongolia, India, and western Asia), and Madagascar. Biostratigraphic correlation for species outside North America was based largely on information in the references consulted for faunal lists. For North American species, these sources were supplemented by the following: J. A. Lillegraven, Contrib. Sci. Los Angeles Co. Mus. 232, 1 (1972); _ and L. M. Ostresh Jr., Geol. Soc. Am. Spec. Pap. 243, 1 (1990); M. B. Goodwin and A. L. Deino, Can. J. Earth Sci. 26, 1384 (1989); R. R. Rogers, C. C. Swisher III, J. R. Homer, ibid. 30, 1066 (1993); C. C. Swisher III, L. Dingus, R. F. Butler, ibid., p. 1981; F. M. Gradstein et al., in Geochronology, Time Scales and Global Stratigraphic Correlation, W. A. Berggren, D. V. Kent, M.-P. Aubry, J. Hardenbol, Eds. (Society for Sedimentary Geology (SEPM), Tulsa, OK], pp. 95-126. To determine whether a species is known from a single horizon, it is generally necessary to have only a relative biochronology. We did not base our relative chronology solely on faunal associations of mammals, but also used local and regional stratigraphic relations, radiometric dates, and correlations with marine sequences based largely on regional strandline stratigraphy, as derived from sources cited above. North American localities were arranged into 14 resolvable stratigraphic intervals, and the localities of other continents were resolved to between 1 and 3 intervals, depending on the region. We treated each of these intervals operationally as a horizon. This approach is conservative insofar as it lumps distinct horizons into one unit, overestimating the proportion of species confined to single horizons and thus underestimating preservation rate. Our approach is also conservative insofar as we generally considered species with uncertain stratigraphic ranges as single-horizon taxa. To estimate average stratigraphic ranges of North American species, for which the biochronology is best understood, we assigned absolute ages to the 14 stratigraphic intervals, mainly using correlations with ammonite zones and radiometric dates. See supplemental data at www.sciencemag.org/ feature/data/985988.shl.
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Geochronology, Time Scales and Global Stratigraphic Correlation
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We thank J. Alroy, R. J. Asher, J. Flynn, D. Jablonski, C. R. Marshall, R. R. Rogers, P. J. Wagner, and J. R. Wible for discussion, and R. H. De Simone, D. Jablonski, C. R. Marshall, D. M. Raup, and two anonymous referees for reviews. This research was supported by NSF (grant EAR-9506568) and NASA (grant NAGW-1693).
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