On the Pleistocene extinctions of Alaskan mammoths and horses
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Edited by C. Vance Haynes, University of Arizona, Tucson, AZ, and approved March 17, 2006 (received for review October 31, 2005)
Abstract
The fossil record has been used to shed light on the late Pleistocene megafaunal extinctions in North America and elsewhere. It is therefore important to account for variability due to the incompleteness of the fossil record and error in dating fossil remains. Here, a joint confidence region for the extinction times of horses and mammoths in Alaska is constructed. The results suggest that a prior claim that the extinction of horses preceded the arrival of humans cannot be made with confidence.
The end of the Pleistocene was a period of largescale megafaunal extinctions in North America and elsewhere (1). The factors responsible for these extinctions remain open to question, with climate change and hunting by humans among the leading candidates (2). According to Guthrie (3), “details of extinction timing are crucial in assessing competing theories about Pleistocene extinctions.” In using the fossil record for this purpose, it is common practice to identify the time of extinction of each taxon with the radiometric (or other) age of the most recent set of fossil remains. As discussed in more detail below, Guthrie (3) took this approach in analyzing the late Pleistocene fossil record of mammoths and caballoid horses in Alaska. He concluded that the extinction of the horse predated that of the mammoth by >1,000 years and, moreover, predated the arrival of humans, thereby ruling out the possibility that it was caused by hunting.
The age of the most recent remains is only a rough estimate of extinction time, and statistical methods are needed to determine how well extinction time is constrained by the fossil record. Such methods have been discussed in the paleobiological literature (4–6). Existing methods assume that the errors associated with dating fossil remains are negligible in relation to other sources of variability in the fossil record. Although this may be reasonable in some cases, it is not reasonable for late Pleistocene records, where dating error can be comparable in magnitude to the intervals between the ages of different sets of remains. The purpose of this paper is to describe a method for inference about the extinction times of two taxa that accounts for dating error in the fossil record and to apply this method to the data of Guthrie (3, 7). The results indicate that the possibility that the horse survived beyond the arrival of humans cannot be confidently ruled out.
Results
Understanding the cause or causes of the late Pleistocene megafaunal extinctions in Alaska is particularly difficult. The difficulty arises from the confluence of two potentially important factors, a period of climate cooling known as the Younger Dryas and the arrival of humans from Asia. In an attempt to unravel the contributions of these factors, Guthrie (3) used radiocarbon ages of late Pleistocene fossil remains of mammoths (Mammuthus primigenius) and caballoid horses (Equus ferus/caballus) in Alaska to study the timing of the extinctions of these taxa. The most recent ages for mammoths and horses are 11.50 thousand radiocarbon years (ka) and 12.48 ka before present (ka BP), respectively. Because these ages straddle the arrival time of humans in Alaska at around 12.00 ka BP, Guthrie (3) concluded that the “present Alaska data, showing that mammoths … survived more than a millennium after horses runs counter to extinction scenarios based on … human overhunting.”
Here, we apply the method described below to construct an approximate 0.95 joint confidence region for the extinction times β_{1} and β_{2} of mammoths and horses, respectively. The 25 most recent ages for each taxa, along with the standard deviations σ_{j} of the corresponding radiocarbon age errors, are reported in Table 1. These data were extracted from refs. 3 and 7. The oldest of these ages for horses is somewhat anomalous, lying almost 1.0 ka earlier than the nextoldest age, so it was omitted from the analysis. This omission had a negligible effect on the results. Thus, in this application, the numbers of samples are m = 25 and n = 24. The maximum likelihood (ML) estimate of β_{1} is 11.51 ka BP. The corresponding ML estimate of the lower endpoint γ_{1} for mammoths is 14.09 ka BP, the ML estimate of β_{2} is 12.36 ka BP, and the corresponding ML estimate of the lower endpoint γ_{2} for horses is 16.68 ka BP. The joint 0.95 confidence region for β_{1} and β_{2} based on these data is shown in Fig. 1.
Although the confidence region in Fig. 1 contains no value for which β_{1} > β_{2}, it does contains values for which the difference between β_{2} and β_{1} is as small as ≈0.2 ka. More importantly, a substantial part of this confidence region covers values for which β_{2} < 12.0 ka BP. These results are not sensitive to the choice of m and n. Thus, the possibility that the horse survived the arrival of humans cannot be confidently ruled out. The result that the confidence region covers values for which the extinction time of horses postdates the arrival of humans would also hold if calibrated radiocarbon ages had been used.
The method on which these results are based assumes that, for each taxon, the true ages of the fossil remains are distributed uniformly between lower and upper endpoints (which corresponds to extinction time). The uniformity assumption is a strong one and should be checked. Because the true ages are inaccessible, it is not possible to check this assumption directly. Instead, the adequacy of the fitted models for the radiocarbon ages was checked through the following simulation procedure. For the first taxon, m true ages were distributed uniformly over the estimated range (β^_{1}, γ^_{1}). Let the ordered ages be U*_{1} < U*_{2} < … U*_{m}. For each j, a simulated radiocarbon age was formed according to X*_{j} = U*_{j} = ε*_{j}, where ε*_{j} is a simulated normal dating error, with mean 0 and standard deviation σ_{j} given in Table 1. The radiocarbon ages simulated in this way were ordered from smallest to largest, and the procedure was repeated 1,000 times. In Fig. 2 a, the actual ordered values of radiocarbon ages in Table 1 are plotted against the expected order statistics of the radiocarbon ages estimated from these simulations along with the upper and lower 0.025 quantiles of the distribution of simulated ordered values. The analogous plot for the second taxon is shown in Fig. 2 b. If the fitted models are correct, then the actual ordered radiocarbon ages should lie close to the 45° line and should exhibit no systematic excursion beyond the upper or lower simulations bands. In neither case is there a clear departure from this behavior.
Discussion
In drawing inferences from the fossil record, it is important to account for variability due to the underlying sampling process that produced the record and also for errors in locating fossil remains temporally or stratigraphically. This paper has focused on the general question: What can be inferred about extinction times based on a collection of radiocarbon dates? In other situations, such questions can be addressed from, or even settled by, the stratigraphic context of fossil remains. The statistical model described here incorporates both sampling variability and radiocarbon dating error. A basic assumption of the model is that the true ages of fossil remains are distributed uniformly over an unknown interval whose lower bound corresponds to the extinction time. This assumption is a strong one. Fortunately, it can be checked. Turning to the application to Pleistocene extinction in Alaska, the point estimates of the extinction times of mammoths and horses in Alaska straddle the arrival of humans ≈12.0 ka BP. However, the joint 0.95 confidence interval contains values for which both extinction times postdate this event. It is, therefore, not possible to conclude confidently on the basis of these data that hunting pressure by humans was not a contributing factor to the extinction of both taxa.
Methods
Let X _{1}, X _{2}, …, X _{m} be the radiocarbon ages of the remains of the a taxon with: where U _{j} is the true age, and ε_{j} is a normal radiocarbon dating error with mean 0 and known variance σ_{j} ^{2}. Assume that the true ages U _{1}, U _{2}, …, U _{m} are independent and uniformly distributed over the interval (β_{1}, γ_{1}), where the lower bound β_{1} corresponds to the time of extinction of the taxon. The uniform assumption is a strong one and should be checked. In general, it will not be valid over the entire fossil record and, in the next section, we will adopt it only for the most recent part of the record. It is straightforward to show that, under this model, the probability density function of X _{j} is: where Φ is the standard normal distribution function. This model has been used for statistical inference about human settlement time based on the radiocarbon ages of archaeological remains (8). The presence of the normal dating error ensures that the regularity conditions underlying the basic likelihood theory hold. This is not the case in the absence of dating error.
Interest here goes beyond a single taxon to the relationship between the extinction times of two taxa. Accordingly, let Y _{1}, Y _{2}, …, Y _{n} be the radiocarbon ages of the remains of a second taxon, with where V _{j} is the true age, and η_{j} is a normal dating error with mean 0 and known variance τ_{j} ^{2}. As with the first taxon, assume that the true ages are independent and uniformly distributed over the interval (β_{2}, γ_{2}), where β_{2} is the extinction time of the second taxon. The probability density function of Y _{j} is:
The twotaxon model can be used to address a variety of questions about the relationship between extinction times. Here, we focus on constructing a joint confidence region for β_{1} and β_{2}. As illustrated in the next section, this confidence interval is useful in understanding the relative timing of the two extinctions and their relationship to other events. The loglikelihood function for the twotaxon model is where x _{j} and y _{j} are the observed values of X _{j} and Y _{j}, respectively. The two terms on the righthand side of (5) are the singletaxon log likelihoods for the first and second taxons, respectively. When, as here, interest centers on β_{1} and β_{2}, inference can be based on the profile loglikelihood function: where γ^_{1}(β_{1}) and γ^_{2}(β_{2}), respectively, are the ML estimates of γ_{1} and γ_{2} found by maximizing Eq. 5 with β_{1} and β_{2} fixed.
Let β^_{1} and β^_{2} be the ML estimates of the two extinction times found by maximizing the profile log likelihood in Eq. 6 . An approximate joint 1−α confidence region for β_{1} and β_{2} is given by the set of values for which where χ_{2} ^{2} is the upper α quantile of the χ^{2} distribution with 2° of freedom.
Acknowledgments
We acknowledge with gratitude the helpful comments of two anonymous reviewers.
Footnotes
 ^{†}To whom correspondence should be addressed. Email: asolow{at}shoi.edu

Author contributions: D.L.R. and K.M.R. designed research; A.R.S. analyzed data; and A.R.S. wrote the paper.

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.
 Abbreviations:
 ka,
 thousand radiocarbon years;
 BP,
 ka before present;
 ML,
 maximum likelihood.
Abbreviations:
 © 2006 by The National Academy of Sciences of the USA
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