Evolutionary time has a characteristic direction as demonstrated by the asymmetry of clade diversity diagrams in large statistical samples. Evolutionary groups generally concentrate diversity during their early histories, producing a preponderance of bottom-heavy clades among those that arise early in the history of a larger group. This pattern holds across taxonomic levels and across differences in anatomy and ecology (marine invertebrates, terrestrial mammals). The quantitative study of directionality in life's history (replacing vague, untestable, and culturally laden notions of "progress") should receive more attention from paleobiologists.

The well-known decline of global background extinction intensity was caused by the sorting of higher taxonomic groups. Two factors were responsible. First, probabilities of familial origination and extinction in these groups (taxonomic orders) were highly correlated. Groups whose families had high probabilities of origination and extinction tended to have highly volatile diversity paths and, consequently, short life spans. Second, orders with high probabilities of familial origination and extinction were rarely replaced by new high-turnover orders. Thus, because high-turnover orders tended to become extinct without replacement, the global background extinction intensity declined. Since familial origination and extinction probabilities are correlated, global background origination intensity inevitably declined as well. As a consequence of these processes, virtually all groups of organisms now living have low probabilities of familial origination and extinction.Simulations of branching evolution were used to obtain the expected relationships among probabilities (of origination and extinction), volatilities, and longevities for the entire range of possible probabilities, and these relationships were compared to those obtained from the empirical record. In the simulations, only the probabilities of origination and extinction were specified, so volatilities and clade longevities were determined entirely by the probabilities. The similarity between results obtained by simulation and those obtained by analysis of the empirical record further supports the inference that the observed decline of background extinction (and origination) intensity can be explained largely by the loss of high-probability groups to induced volatility.

Using Sepkoski's compendium of fossil marine families (1982a, and updates), we have analyzed the changing pace of familial origination and extinction within 55 extinct and 44 extant higher taxa of marine organisms. Eight different metrics were calculated, and least-squares regression analysis was used to identify within-taxon trends in the data. All metrics and analyses gave essentially the same results. Origination metrics decline significantly with time during the histories of higher taxa, while extinction metrics increase significantly. The number of statistically significant declines of origination metric, however, substantially and invariably exceeds the number of statistically significant increases of extinction metric for each pair of corresponding metrics analyzed. It follows, therefore, that temporal trends in the pace of origination and extinction within higher taxa are highly asymmetrical.Further analysis shows that truncating data from temporal endpoints has little effect upon the intensity of origination trends, implying that declining pace of origination is a sustained property of the long term histories of taxa. Such truncation, however, reduces the intensity of extinction trends to statistical insignificance and confirms Van Valen's (1985a) suggestion that extinction behaves largely as a stationary process. If the histories of higher taxa are characterized by substantial declines in the pace of origination while the pace of extinction remains largely stationary, it follows that declining pace of origination is an important controlling factor in long term taxic evolution.

Paleoecologists have long sought to obtain estimates of the sizes of extinct populations. However, even in ideal cases, accurate counts of individuals have been hampered by the fact that many organisms disarticulate after death and leave their remains in the form of multiple, separated parts. We here analyze the problem of estimating numbers of individuals from collections of parts by developing a general counting theory that elucidates the major contributing variables. We discover that the number of unique individuals of a particular species that are represented in a fossil collection can be described by an intricate set of relationships among (1) the number of body parts that were recovered, (2) the number of body parts that were possessed by organisms belonging to that species, and (3) the number of individuals of that species that served as the source of the parts from which the paleontological sample was obtained (the size of the “sampling domain”). The “minimum number of individuals” and “maximum number of individuals” methods currently used by paleontologists to count individuals emerge as end members in our more general counting theory. The theory shows that the numbers of individuals of a species that are represented in a sample of body parts is fully tractable, at least in a theoretical sense, in terms of the variables just mentioned. The bad news is that the size of the “sampling domain” for a species can never be known exactly, thus placing a very real limit on our ability to count individuals rigorously. The good news is that one can often make a reasonable guess regarding the size of the sampling domain, and can therefore make a more thoroughly informed choice regarding how to estimate numbers of individuals. By isolating the variables involved in determining the numbers of individuals in paleontological samples, we are led to a better appreciation of the limits, and the possibilities, that are inherent in the fossil record.

In this paper we model the process of taxonomic evolution as a Galton-Watson branching process in discrete time and, using maximum likelihood, develop methods to estimate the probabilities of origination, persistence, and extinction of fossil taxa. We use the methods to estimate the probabilities of origination, persistence, and extinction of families (1) within 135 orders of marine invertebrate organisms, (2) within 12 phyla, and (3) within all marine invertebrate life (independently of the suprafamilial classification).Most orders, including the arcoid bivalves, the dentaloid scaphopods, the orders of chitins, and many others, have relatively low probabilities of familial origination and extinction. The various ammonoid and trilobite orders, and some others, have high probabilities of origination and extinction. Among the phyla, the Archaeocyatha have the highest probabilities of familial origination and extinction, and the Annelida the lowest, with the more typical phyla of shelly organisms clustering near the high end of the probability scale. The Porifera and Protozoa also have low probabilities but not as low as the Annelida. The estimated origination and extinction probabilities for families within all marine invertebrate life are 0.470 and 0.452 per stage, respectively, values that are at the high end of the probability scale. We have also estimated the probabilities of ultimate extinction (extinction of all families) of the supertaxa.By analyzing the changes of the diversity during each stratigraphic stage separately, we have also determined the trajectories of the estimated origination and extinction probabilities for families within all marine invertebrate life. The estimated origination probability is relatively high in association with the expansion of the Cambrian and Paleozoic evolutionary faunas and declines to more normal levels for the remainder of the Phanerozoic. The trajectory of the estimated extinction probability is from nearly zero early in the Phanerozoic to more normal levels later, showing clearly defined peaks in association with the five Phanerozoic mass-extinction events. The terminal Cretaceous mass extinction is the only one of the five that was not preceded by a monotonic decline of origination probability or by a series of stages with low origination probability. It appears to have been a unique, singular event.Because the mathematical theory we employ as a model corresponds so closely to the processes of taxonomic evolution as we understand them, we believe that the theory provides a reasonable model of biological reality.