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Foote, M. 2001. Inferring Temporal Patterns of Preservation, Origination

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Foote, M. 2001. Inferring Temporal Patterns of Preservation, Origination Paleobiology, 27(4), 2001, pp. 602±630 Inferring temporal patterns of preservation, origination, and extinction from taxonomic survivorship analysis Michael Foote Abstract.ÐApparent variation in rates of origination and extinction re¯ects the true temporal pat- tern of taxonomic rates as well as the distorting effects of incomplete and variable preservation, effects that are themselves exacerbated by true variation in taxonomic rates. Here I present an ap- proach that can undo these distortions and thus permit estimates of true taxonomic rates, while providing estimates of preservation in the process. Standard survivorship probabilities are mod- i®ed to incorporate variable taxonomic rates and rates of fossil recovery. Time series of these rates are explored by numerical optimization until the set of rates that best explains the observed data is found. If internal occurrences within stratigraphic ranges are available, or if temporal patterns of fossil recovery can otherwise be assumed, these constraints can be exploited, but they are by no means necessary. In its most general form, the approach requires no data other than ®rst and last appearances. When tested against simulated data, the method is able to recover temporal patterns in rates of origination, extinction, and preservation. With empirical data, it yields estimates of pres- ervation rate that agree with those obtained independently by tabulating internal occurrences with- in stratigraphic ranges. Moreover, when empirical occurrence data are arti®cially degraded, the method detects the resulting gaps in sampling and corrects taxonomic rates. Preliminary appli- cation to data on Paleozoic marine animals suggests that some features of the apparent record, such as the forward smearing of true origination events and the backward smearing of true extinction events, can be detected and corrected. Other features, such as the end-Ordovician extinction, may be fairly accurate at face value. Michael Foote. Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois 60637. E- mail: [email protected] Accepted: 8 May 2001 Apparent versus Real Variation in origination and extinction rates, whereas the Taxonomic Rates converse is true for intervals with more de- pauperate records. Likewise, apparent extinc- Secular variation in origination and extinc- tion rates may be arti®cially high upon ap- tion rates is a major feature of the history of proach to a poorly sampled interval, with ap- life, and the documentation and interpretation parent origination rates spuriously high after- of this variation continue to motivate paleon- wards. It seems intuitively reasonable that, if tological research on many temporal and spa- we had prior knowledge of how the quality of tial scales and to inform our understanding of the fossil record varied over time, we should the dynamics of diversi®cation (Newell 1967; be able to correct the apparent origination and Raup and Sepkoski 1982, 1984; Van Valen extinction record, i.e., to obtain an estimate of 1984; Raup 1986; Miller 1997a,b, 1998; McKin- the true pattern of origination and extinction ney and Drake 1998; Sepkoski 1998; Budd and underlying the observed data (Wagner 2000). Johnson 1999; Kirchner and Weil 2000; Foote One goal of this paper is therefore to develop 2000b, 2001a). An important issue of which a general method to convert observed data on paleontologists are quite aware, but which is ®rst and last stratigraphic occurrences to es- seldom confronted directly in rate studies (but timates of true origination and extinction see Koch 1991; Alroy 1996, 1998; Johnson and rates, given some estimate or assumption of McCormick 1999; Alroy et al. 2000; Smith the temporal sequence of rates of fossil pres- 2001), is that this variation is distorted by var- ervation. iability in the quality of the fossil record and We generally do not know precisely how our knowledge of it. Stratigraphic intervals preservation rates vary over time, however. In with a more complete record than adjacent in- cases where data on fossil taxa include infor- tervals will tend to exhibit spuriously high mation on occurrences within stratigraphic q 2001 The Paleontological Society. All rights reserved. 0094-8373/01/2704-0002/$1.00 ORIGINATION, EXTINCTION, AND PRESERVATION 603 TABLE 1. Summary of the principal quantities used in this paper. Quantity Explanation p Origination rate per lineage-million-years (Lmy) q Extinction rate per Lmy r Preservation rate per Lmy PB Probability of preservation before a point in time, for taxa extant at that time PA Probability of preservation after a point in time, for taxa extant at that time P→ij Expected probability that last occurrence is in stratigraphic interval j, given that ®rst occurrence is in interval i P←ij Expected probability that ®rst occurrence is in interval i, given that last occur- rence is in interval j Xij Observed number of taxa with ®rst occurrence in interval i and last occurrence in interval j F→ij Observed proportion of taxa with last occurrence in interval j of those with ®rst occurrence in interval i F←ij Observed proportion of taxa with ®rst occurrence in interval i of those with last occurrence in interval j XbL Observed number of taxa crossing bottom interval boundary and last appearing within interval XFt Observed number of taxa ®rst appearing within interval and crossing top inter- val boundary XFL Observed number of taxa with both ®rst and last appearance within interval Xbt Observed number of taxa crossing both bottom and top interval boundaries ranges, preservation rates can be estimated variation in the quality of the record, but we empirically, but much of the compiled infor- still require a way to remove those distortions mation on stratigraphic ranges consists of that would exist even if the quality of the fossil only ®rst and last appearances. A second, and record were uniform. The approach devel- in some ways more important, goal of this pa- oped herein helps to meet this need. per is therefore to estimate the temporal pat- Although the purpose here is mainly meth- tern of preservation from data on ®rst and last odological, some preliminary results will be appearances only. The method developed to presented to illustrate the approach, to show achieve this goal will be tested with simulated cases in which the apparent record of taxo- data, with empirical data that include internal nomic rates may be misleading, and to show occurrences within stratigraphic ranges, and cases in which the record may be reliable es- with empirical data that are arti®cially de- sentially at face value. Throughout this paper graded. I will use apparent rates to refer to rates of orig- Although the need to overcome the effects ination and extinction calculated from fossil of variable preservation is obvious, true vari- data taken at face value, true rates to refer to the ation in taxonomic rates is distorted by incom- underlying parameters we would like to esti- pleteness even if the quality of the fossil re- mate, and estimated rates to refer to the values cord is uniform. Variation in extinction rate is of these parameters inferred from the modi- smeared backward in time to affect prior, ap- ®ed survivorship analysis. parent rates of extinction, and variation in origination has the same effect forward in Inferring Origination and Extinction Rates time. This is a quite general problem (Foote When Preservation Rates Are Known 2000a), of which the Signor-Lipps effect is per- All taxa known to exist during an interval haps the most familiar example (Signor and of time fall into four exclusive categories (Ta- Lipps 1982; Meldahl 1990; Rampino and Adler ble 1, Appendix 1) (Foote 2000a): (1) taxa with 1998). Equalizing the quality of the record by ®rst appearance before the interval and last subsampling a uniform quota of data from appearance within the interval; (2) taxa with each interval (Alroy 1996, 1998; Alroy et al. ®rst appearance within the interval and last 2000, 2001) may help remove distortions in appearance afterwards; (3) taxa with ®rst and apparent taxonomic rates that are caused by last appearance within the interval; (4) taxa 604 MICHAEL FOOTE with ®rst appearance before the interval and ance is in interval i; (2) P←ij (``P-backward''), last appearance afterwards. Using b and t to the probability that a taxon has ®rst appear- refer to the crossing of bottom and top inter- ance in interval i given that its last appearance val boundaries and F and L to refer to ®rst and is in interval j. First appearance and last appear- last appearance within the interval, the ob- ance refer to the intervals of ®rst and last pres- served numbers of taxa falling into these cat- ervation, not to the true times of origination egories are denoted XbL, XFt, XFL,andXbt.The and extinction and not to ®nely resolved levels relative numbers of these four classes of taxa within the interval. Let Xij be the observed are determined, in addition to interval length, number of taxa with ®rst appearance in inter- by ®ve quantities (Pease 1985; Foote 2000a): (1) val i and last appearance in interval j, and ex- the rate of origination p per lineage-million- press these numbers as the following relative years (Lmy) during the interval; (2) the rate of frequencies: F→ij (``F-forward'') is the propor- extinction q per Lmy during the interval; (3) tion of taxa having last appearance in j,of the rate of preservation r per Lmy during the those that have ®rst appearance in i;andF←ij interval; (4) the probability PB that a taxon ex- (``F-backward'') is the proportion of taxa hav- tant at the start of the interval is preserved ing ®rst appearance in i, of those that have last sometime before the interval, which is a func- appearance in j.
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