CLASSICAL CONFIDENCE INTERVALS FOR TEMPORAL RANGES

Strauss and Sadler (1989) derived a one-tailed confidence interval for a taxon's stratigraphic or temporal range that we shall use here:

p₁ = 1 - (1 + a)^{-(n - 1)} (1)

where p₁ is the confidence level (e.g., 0.95), their a is the range extension expressed as a proportion of the known stratigraphic or temporal range (not to be confused with conventional use of a to represent level of significance, see below), and n is the number of independently-sampled fossiliferous horizons. Solving for a by rearranging terms yields the required range extension as a function of the confidence level and number of samples (Marshall 1990). Derivation of equation 1 by Strauss and Sadler and discussion of this by Marshall both make it seem unduly complicated (due in part to the complexity of their notation), but the derivation is actually both simple and intuitive, as we discovered by deriving this independently (Gingerich and Uhen 1994).

Start by assuming that fossils are uniformly distributed throughout the unknown estimated temporal range or expected temporal density [ETD] of a taxon of interest (Fig. 1). This assumption of uniformity is too simplistic, and we will relax it later (when the density associated with an area or volume need no longer correspond to the temporal range). Assume that sampling is random. A uniform ETD means that fossils from different times have an equal probability of being sampled. Construct a sample of some size n by drawing n independent samples at random from ETD. We can now sort these from oldest to youngest and they define the observed temporal range or observed temporal density [OTD]. We are interested in the time of origin of Archaeoceti and do not care about their time of extinction (or conversion into Odontoceti and Mysticeti), meaning that we are interested in a one-tailed range extinction. This is, in any case, more conservative (yielding a broader interval) in allocating all of the tail probability to one tail. One sample is required to define the youngest end of the OTD (t₃, the time we are not interested in here), so n - 1 samples remain for estimation of ETD and t₁ from OTD and t₂.

We require one additional number a, the level of significance or error rate we are willing to accept. This determines the confidence interval 1 - a that we seek. By convention, a = 0.05 is the usual error rate, which corresponds to a 95% confidence interval for observed temporal density OTD.

The probability that any sample drawn from unknown ETD falls in OTD is the ratio of lengths (or areas or volumes) OTD/ETD, which cannot be greater than 1 (because ETD is greater than or equal to OTD). If the probability that one sample drawn from ETD falls in OTD is OTD/ETD, then the probability that two samples drawn independently from ETD both fall in OTD is the product of OTD/ETD times OTD/ETD or (OTD/ETD)². The probability that n - 1 samples drawn from ETD all fall in OTD is (OTD/ETD)^n-1. Setting this quantity equal to the error rate a:

a = (OTD/ETD)^n-1 (2)

which is, in simpler form, exactly the same as equation 1 [where p₁ = 1 - a and (1 + a)^-(n-1) = (OTD/OTD + (ETD-OTD)/OTD)^-(n-1) = (OTD/ETD)^n-1]. Solving for ETD yields:

ETD = OTD / a^1/(n-1) = OTD / ^(n-1)Öa (3)

With a = 0.05 and an observed range OTD based on two independent samples, meaning n - 1 = 1, the 95% confidence limit for ETD is 20 × OTD. When OTD is based on three independent samples, meaning n - 1 = 2, the 95% confidence limit for ETD is 4.47 × OTD. ETD/OTD is the inverse of the (n - 1)th root of a. This quantity converges rapidly to 1 as n increases, meaning ETD approximates OTD even for relatively small n. Further, this result is not very sensitive to a.

If we assume that the distribution of fossils representing the temporal duration of a group of organisms is uniform through time, and if we know (1) the beginning and end of the group's stratigraphic or observed temporal range (t₂ and t₃, respectively, in Fig. 1), and (2) the number of independent samples this range is based on (n), then we can estimate the time of origin of the group (t₁ in Fig. 1). The beginning and end of the observed range encompass the observed temporal density OTD. OTD is necessarily represented by n is greater than or equal to 2 samples. Estimated temporal range ETD can be calculated using equation 3, and the difference between ETD and OTD is added to the beginning of the observed temporal density. This yields a classical 1 - a confidence limit for the time of origin of the group.