Fossil Record there are few samples), the estimates are especially prone to error For this reason, the fact that a method works well as an estimate of present-day diversity does not necessarily mean that it will work well when applied to the fossil record, except perhaps where that record is unusually complete In ecology, parametric models have been used to infer local and regional diversity from observed distributions of occurrences or abundance, sometimes with good results One potential problem with the application of such methods to the fossil record is that differences among taxa in their probability of entering the fossil record, and subsequent loss of fossiliferous rock (at rates that may vary among regions and through time), may make distributions of fossil occurrences very different from distributions of occurrences of living organisms However, to date, models that are commonly used by ecologists, such as the lognormal and the gamma distribution, have yet to be applied to the fossil record Turnover-Based Estimates Paleontologists typically break taxa down into five categories relative to their occurrence before, in, and after the time interval in question: (1) those sampled before, in and after; (2) those sampled before and after (but not in); (3) those sampled before and in (but not after); (4) those sampled in and after (but not before); (5) those sampled only in the interval (not before or after) In traditional stratigraphic range based estimates of diversity, all of these taxa contribute to the diversity in the time interval, although taxa only occurring in a single time interval are often removed from paleontological analyses, because such ‘‘singletons’’ are particularly sensitive to between-interval variation in sampling intensity An alternative, known as ‘‘boundary crosser’’ estimates of diversity, count diversity at interval boundaries, rather than within intervals For instance, diversity at the bottom of an interval includes all of the taxa crossing the bottom boundary (found sometime before and after the boundary, including those not actually sampled in the interval itself) Both traditional and boundary-crosser approaches have been widely used in paleontological analyses, and several other counting methods have been used occasionally as well It has been suggested that better estimates of diversity (along with rates of origination and extinction) can be obtained by only counting taxa that occur in two consecutive intervals (two timers), three consecutive intervals (three timers), or the first and third but not the second interval (part timers) Although ad hoc, these data selection criteria appear to be relatively robust to the biases typically associated with fossil data Thus, while these counting methods underestimate diversity even more than standard range-through approaches, they are likely to be more robust as measures of relative diversity across multiple time intervals In the mid-1980s, Nichols and Pollock (1983) proposed that the information contained in presence–absence sequences in the fossil record should be used more systematically to estimate diversity Specifically, they proposed that ‘‘open-population’’ statistical methods used to estimate abundances (along with birth and death rates) from 543 capture–recapture data in population biology could be adapted, by analogy, to estimate taxonomic diversity (along with origination and extinction rates – births and deaths of taxa) in the fossil record When one conducts an ‘‘openpopulation’’ capture–recapture study, individual organisms are captured during discrete sampling occasions over a period of time Each individual captured is given a unique mark, so that its capture history can be constructed That is, if one constructs a matrix, the columns j of which represent different sampling occasions and the rows i of which correspond to each individual captured at least once, then each element aij in the matrix will be either or 0, indicating whether individual i was captured on occasion j Similarly, fossil data are collected during discrete sampling periods, and one can simply list the taxa sampled in each interval In this case, the matrix elements aij are or according to whether taxon i was found in interval j In this context, taxonomic diversity (the total number of distinct taxa) is analogous to population size (the total number of distinct individuals) Thus, each species’ sampling history is a sequence of presences and absences This set of sampling histories can be used to estimate probabilities of sampling, origination, and extinction, and to estimate total taxonomic diversity, using standard statistical theory Like nonparametric and parametric diversity estimates that utilize replicate samples within an interval of time, these turnover-based methods make assumptions that are rarely fully met by fossil data, and it is not always clear which results are robust to violations of assumptions and which are not However, a major advantage of these approaches over sampling standardization approaches is that it is possible to test for, and estimate the magnitude of, violations of those assumptions When Nichols applied a turnover-based estimate to late Eocene (37–34 Ma) mammals from the Big Horn Basin, Wyoming, goodness of fit statistics indicated rejection of the model However, when applied to Middle Miocene molluscan diversity, the model provided an adequate fit Similarly, goodness of fit testing was used by Connolly and Miller (2001) to show that violation of assumptions was unlikely to have severe effects on estimates of origination and extinction (in other words, that the uncertainty in the estimates was likely to be fairly large, relative to any biases due to assumption violations) Some of the most questionable assumptions in turnoverbased methods involve the assumption that probabilities of origination, extinction, and sampling are constant for all taxa that are grouped together in an analysis To minimize the potential biases associated with violations of these assumptions, researchers have adopted several approaches One is to divide taxa into subgroups believed to have different sampling or evolutionary turnover rates, and estimate parameters for them separately An alternative is to relax that assumption, and use characteristics of individual taxa believed to be positively or negatively correlated with sampling, origination, or extinction rates, and to estimate the parameters of that relationship For instance, Connolly and Miller (2001) allowed sampling probabilities to differ among Ordovician (488– 461 Ma) marine invertebrate genera according to differences in the apparent geographical and ecological distribution of those genera