## Materials And Methods

The data consist of genus counts for 48 intervals, averaging 11.0 Myr and ranging from the traditional Early Cambrian through the Neogene. These temporal bins sometimes correspond to marine stages, but often comprise sets of neighboring stages lumped to minimize variance in duration. The counts are derived from 281,491 occurrences of 18,541 genera within 42,627 fossil collections that have been sampling standardized by randomly drawing entire collections up to a quota of 15,800 specimens per bin. When the specimen count for an individual collection is not directly available, it is estimated from the occurrence count by examining rarefaction patterns for other collections in the same bin. Each collection's sampling probability is inversely weighted by its specimen count to avoid having a few large collections from a narrow range of environments and geographic areas dominate the analysis. Collections from entirely unlith-ified sediments are excluded. Details concerning the data and methods are reported in Alroy et al. (2008).

A large number of equations have been proposed to quantify origination and extinction, using paleontological data (Foote, 1994b, 2005). Traditional measures consisted of simple first and last appearance counts that sometimes were divided by some form of a diversity count to create a proportion. Proportions are biologically meaningful if they describe a sudden turnover event, but a more realistic general approach is to view turnover as an exponential decay process (Raup, 1985) and, therefore, compute instantaneous rates that are equivalent to decay constants (Alroy, 2000, in press; Foote, 2000a,b).

The problem with all existing equations is that they were developed to handle traditional compilations that only record first and last appearances. Simple range data are subject to edge effects, such as the Signor-Lipps effect and Pull of the Recent, that create systematic smearing of rates backward before a large extinction begins, smearing of rates forward after a burst of origination, and drops in extinction rates before a large sampling spike such as the Recent (Foote, 2000a). For example, backward smearing is clearly visible in family-level data on both marine and continental organisms (Benton, 1995), and the Pull of the Recent seems to amplify the downward trend in Sepkoski's genus-level extinction rates (Foote, 2000a; Peters, 2006).

Two new continuous rate equations (Alroy, in press) remove the edge effects by ignoring ranges and focusing instead on occurrence data that show which fossil taxa are actually sampled in which time intervals. These methods are only made possible by the existence of occurrence-based relational databases, and could not have been applied to the Phanero-zoic marine record before the development of the Paleobiology Database (Alroy et al., 2001). The new rates depend on five fundamental counts: taxa

224 / John Alroy 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 2t) or within and immediately after the ith bin (2tl + 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 |i is log(2t;/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 X is log(2t; + 1/3t) + log(Ps). The same counts can be rearranged to compute a three-timer-based estimate of the extinction proportion, 1 - 3t/(Ps 2t). 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 || or X. It might be if high turnover makes it harder to sample taxa in a cohort that actually originated in the immediately preceding bin (for |) or succeeding bin (for X). 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 | one uses the sampling probability for the third bin (Ps l + 1), and to obtain X one uses the probability for the first bin (PsI _ 1). The corrected formulas log(2t;/3t) + log(Ps/! + 1) and log(2t; + 1/3t) + log(Ps,; _ 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[|| + 1/|i]). 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 Psi, and the similar correction psNs/Psi decreases the volatility of the diversity curve from 0.207 to 0.179.

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