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sampled at all in a focal bin (Ns), taxa sampled in a bin but not immediately before or after (one-timers, or 1t), taxa sampled immediately before and within the ith bin (two-timers, or 2ti) or within and immediately after the ith bin (2ti+ 1), taxa sampled in three consecutive bins (three-timers, or 3t), and taxa sampled before and after but not within a bin (part-timers, or Pt). The overall sampling probability Ps is just 3t/(3t + pt), where 3t and Pt are summed across the entire dataset.

The measures primarily used in this chapter are called three-timer rates (Alroy, in press). The three-timer extinction rate μ is log(2ti/3t) + log(Ps), which expresses the exponential decay rate of a cohort crossing the base of a bin and continuing to its top, corrected for the fact that members of this cohort may be present but not sampled in the following (third) bin. The corresponding origination rate λ is log(2ti+ 1/3t) + log(Ps). The same counts can be rearranged to compute a three-timer-based estimate of the extinction proportion, 1 − 3t/(Ps2ti). Turnover rates for the first and last intervals in the time series cannot be computed because of the structure of these equations.

These expressions assume that sampling standardization has succeeded, so Ps is uniform across all intervals, and that Ps is not systematically correlated with μ or λ. It might be if high turnover makes it harder to sample taxa in a cohort that actually originated in the immediately preceding bin (for μ) or succeeding bin (for λ). However, it can be shown by simulation that this problem is not substantial over a reasonable range of turnover rates.

Nonetheless, Ps is never completely uniform. Therefore, it is better to use separately computed values for the relevant bins. To obtain μ one uses the sampling probability for the third bin (Ps,i +1), and to obtain λ one uses the probability for the first bin (Ps,I− 1). The corrected formulas log(2ti/3t) + log(Ps,i+ 1) and log(2ti+1/3t) + log(Ps,i− 1) are used throughout the main analyses. This correction decreases the volatility of the turnover rates. Volatility can be quantified by averaging changes in rates between bins, i.e., for extinction taking the mean of abs(log[μi+1i]). The volatility of extinction drops from 0.778 to 0.707 with the correction, and that of origination drops from 0.824 to 0.511. Likewise, Ns is systematically related to Ps,i, and the similar correction psNs/Ps,i decreases the volatility of the diversity curve from 0.207 to 0.179.

ACKNOWLEDGMENTS

I thank J. Madin for helpful comments on an early draft. I thank the National Science Foundation for recognizing the importance of the Paleobiology Database and an anonymous donor for funding the database. This work was hosted by the National Center for Ecological Analysis and



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