Paleobiologists frequently hypothesize that a taxon's duration (i.e., the true span from origination to extinction) exceeds its stratigraphic range (i.e., the span from first appearance to last appearance in the fossil record). One can test hypothesized duration by assessing the plausibility of the implicitly hypothesized gaps between origination and first appearance and/or between last appearance and extinction. Several tests assess the probability of not finding a taxon over some stratigraphic gap. Because the likelihood of a hypothesis given data reflects the probability of the data given that hypothesis, these probabilities also give the likelihood of a hypothesized duration. However, many probability/likelihood tests require simplifying assumptions about unknown sampling parameters such as the consistency of sampling over time, sampling intensities for unknown ancestors, and actual sampling intensities themselves.
This paper examines the effects of sampling parameters on probability/likelihood tests and presents methods for testing hypotheses about these unknowns while testing hypotheses about true durations. Two data sets are used here as examples. One analysis tests the origination times among Paleozoic gastropods implied by phylogenetic inferences. The other analysis tests the extinction times among Maastrichtian ammonites implied by different numbers of extinction events. In both cases, hypotheses positing many gaps in the fossil record become more likely after accommodating uncertainty about sampling. However, the increased likelihoods are insufficient to prevent these hypotheses from being rejected in favor of hypotheses positing fewer gaps. In both cases, the conclusions are identical to those derived by simple methods using simple models for unknown sampling parameters. Although numerous factors can exaggerate the implausibility of gaps, making these factors parts of testable hypotheses is possible. Thus, excessive assumptions about sampling parameters need not hinder empirical testing of hypothesized durations.