A Generalized, Bootstrap-Based Approach to the Estimation of Confidence Intervals on the Ends of Biostratigraphic Ranges

Biostratigraphic confidence intervals were developed originally to address the discrepancy widely acknowledged to exist between the observed level of a taxon’s first and/or last occurrence and the actual level of these appearances in individual stratigraphic successions. Owing to the existence of gaps between recorded occurrences throughout a taxon’s stratigraphic range, the level just below the first observed occurrence, or just above the last observed occurrence could be interpreted simply as another occurrence gap. Confidence that the observed first and/or last occurrences represent good, or poor, estimates of the actual first and/or last appearances in local successions, then, depends of the size (= length) and frequency of the occurrence gaps that characterize the taxon’s stratigraphic range. Many assumptions need to be made in order to estimate accurate confidence intervals on the ends of biostratigraphic ranges. The character of the “gap size” distribution is foremost among these. Previously, a variety of standard statistical distributions (e.g., Exponential Poisson, Dirichlet, Scaled Binomial) have been proposed as the distribution that best reflects the arrangement of occurrence gaps for all taxa in all stratigraphic successions. It can easily be shown, however, that these distributions often do not fit the actual gap size distributions of empirical paleontological data accurately. Their use has been advocated on theoretical grounds primarily. A survey of alternative statistical distributions (e.g., Log-normal, Weibull) have also proven unable to provide close fits to actual biostratigraphic gap size distributions. In order to place such estimations on a firmer and more accurate statistical footing, it is suggested that attempts to justify the use of standard statistical distributions in this context be abandoned. The stratigraphic record of taxon occurrences is simply too complex, and too subject to too many local, idiosyncratic factors to expect any single distribution to model all gap sizes, in all stratigraphic successions, for all taxa accurately. As a practical alternative, a bootstrap-style resampling procedure can be used to estimate the gap size distribution that best fits a taxon’s empirical occurrence gap record. This procedure has the advantage of being able to be used in any succession where it can be tailored to fit any taxon’s empirically determined occurrence record. The same assumptions, of course, remain necessary in the interpretation of taxon-based stratigraphic confidence intervals. However, experiments carried out to date indicate this bootstrap procedure can produce gap size distributions whose fits to empirical data are far superior to those delivered by any standard parametric statistical distribution, even for taxa with modest occurrence records. Simple modifications of this procedure can also be implemented for situations in which multiple taxa are involved and/or when collection potentials vary over the course of a stratigraphic interval. Confidence intervals estimated on the basis of bootstrapped gap-size distributions tend to be substantially smaller than those estimated by both standard and alternative parametric distributions.