CRHR’s own Dr. Selden recently published an interesting compilation of Woodland-period radiocarbon dates from East Texas (Selden 2012), facilitating additional research into possible trends in Woodland occupation. The interpretation of such dates requires careful consideration of various issues. Many archaeologists have noted that sample size effects and the calibration curve for radiocarbon dates, among other factors, may influence the chronological distribution of dated samples. Such effects may cause calibrated samples to cluster around particular dates even though the underlying “true” distribution is very different. This post illustrates these effects and shows how a relatively simple model and statistical technique can be used to distinguish real patterns of interest in the Woodland period data from East Texas. This work thus builds on Selden’s (2012) discussion of such patterns.
I start first with the effects of the radiocarbon calibration curve. This curve depicts the relationship between radiocarbon dates and calibrated dates. If the proportion of carbon isotopes in the atmosphere had stayed constant over time, the correlation between radiocarbon dates and calibrated dates would be perfect, and the calibration curve would be a straight line with a slope of one. The proportion of the carbon isotopes in the atmosphere, however, has not stayed constant over time and has not changed at a constant rate. Through analysis of material of known age, scientists have generated a curve reflecting change in the isotopic ratios through time. The consequence of these changes is to create “cliffs” and “plateaus” in the calibration curve, segments of the curve where the slope is higher or lower than one respectively. Calibrated radiocarbon dates from the “plateaus” are thus disproportionately likely to occur. Conversely, calibrated dates from the “cliffs” will be under-represented.
The effects of the calibration curve on radiocarbon dates can be illustrated with a few simulations. For the first set of simulations, I generated 85 numbers from a uniform random number generator with limits between -500 and 800. These limits are consistent with the boundaries of the Woodland period (500 BC to AD 800) assumed in Selden (2012). Under a uniform distribution, all numbers within the limits have an equal probability of being selected. The program OxCal (Bronk Ramsey 2009), which I used for all of the calibration work, includes a function (R_Simulate) that simulates the calibration of a sample given a known true date of that sample.
The following output from OxCal shows two examples of the R_Simulate function, with the “true” date in this example set at 500 AD and the standard deviation of the simulated sample set to 54. The value of the standard deviation is consistent with the typical deviations in the archaeological data analyzed by Selden (2012). Compare the next two figures. These figures show that the simulation returns different results despite having the same parameters for both runs. The probability distribution for the calibrated date varies between runs of the simulation. The area under the probability distribution (the gray area on the following two figures) between two calibrated date values on the x-axis provides the probability of the “true” value occurring. The total area for these probability distributions must sum to one. Also note that the blue lines on the figures represents a portion of the calibration curve, expressed in such a way so that a perfect correlation between the radiocarbon date and calibrated date would have a slope of negative one.
One Example of a Simulated Radiocarbon Date, with “True” Value of AD 500. The pink shaded area gives the probability distribution of the simulated uncalibrated date. The blue line represents the calibration curve. The gray shaded area is the probability distribution that results from the application of the calibration curve to the uncalibrated probability distribution. The inner lines under the calibrated probability distribution give the 68.2% confidence intervals for the calibrated date, while the outer lines give the 95.4% confidence intervals for the calibrated date.
Another Example of a Simulated Radiocarbon Date, with “True” Value of AD 500.
I thus used the uniformly-distributed random numbers as “true” dates in the OxCal calibration simulation. I generated 85 such simulated calibrated dates and then summed their individual probability distributions, using the Sum function in OxCal. A summed probability distribution essentially adds together the individual distributions of the calibrated samples and normalizes them, so the area under the curve is equal to one. The following output from OxCal shows the summed probability distribution of a simulated sample of 85 calibrated dates.
Summed Probability Distribution for 85 Random Samples from a Uniform Distribution
Notice how the summed probability distribution has a couple peaks (modes) centered around 150 BC and around AD 500, even though the sample of “true” values was generated from a uniform distribution. The simulation’s apparent bimodal distribution is likely the result of both the calibration curve and sample size effects. The calibration curve’s effects can be further evaluated with another simulation.
I generated a series of evenly-space simulated radiocarbon dates in OxCal, ranging from 500 BC to AD 790. The “true” values in the simulation are thus spaced 15 years apart. I then created another summed probability distribution from the individual calibrated samples. Notice that this simulation hints at three poorly-defined modes, peaking around 150 BC, AD 250, and AD 650. The difference between this simulation and the previous simulation is undoubtedly due to sample size effects. With a relatively small sample size, the effects of the “cliffs” and “plateaus” in the calibration curve may be exacerbated. A relatively large proportion of samples, for example, may occur within a portion of the curve subject to plateau-effects by chance.
Summed Probability Distribution of Systematic Sample of 87 Simulated Radiocarbon Dates
With these considerations in mind, let’s now examine the East Texas Woodland Period radiocarbon dates. The following summed probability distribution shows a sample of 85 calibrated Woodland-period dates, created after combining together some site-specific samples that likely derived from the same component (see Selden  for details). Notice, in particular, that the distribution shows three apparent modes. The proportion of dated samples after AD 400 seems particularly marked. Selden (2012: 261) notes that this increase seems significant.
The following histogram shows the distribution of the median of each of the 85 calibrated archaeological samples.
Distribution of the Median Age of the 85 Calibrated Radiocarbon Samples from the East Texas Woodland Period
This result deserves some additional analysis. One relatively simple approach is to compare the result against the distribution of values that would be expected, assuming that true values are uniformly distributed between 500 BC and AD 800. In particular, is the proportion of values dating after AD 400 (the Late Woodland period in Selden ) a result of sample size and the calibration curve, or does it reflect some other processes? To evaluate this question, I again conducted a series of simulations. I generated 50 separate sets of 85 simulated calibrated dates. As before, the true value of each sample was created from a uniformly-distributed random number generator, with limits between -500 and 800. Each value was then calibrated using the R_simulate function in OxCal. For each of the 50 simulation runs, I then determined the number of the 85 samples that had a median radiocarbon date of AD 400 or greater. Examination of the median is quite straightforward, although it fails to account for the variation in the calibrated dates. This level of analysis is sufficient for indicating whether a more rigorous evaluation should be undertaken. The next histogram shows the simulation results.
Number of Late Woodland Dates among the 50 Simulation Runs. Each simulation run was comprised of 85 simulated calibrated dates.
For the archaeological data from the East Texas Woodland period, a total of 36 calibrated samples had “Late Woodland” dates; dates with a median age of AD 400 or greater. This number is larger than the number of “Late Woodland” dates in all but one of the 50 random simulation runs. Consequently, the null hypothesis can be rejected at a level of significance of 0.02. The archaeological data is unlikely to have been generated from a process with a uniform distribution throughout the Woodland period. This result does not provide a model that explains the observation. Sample selection, differential site preservation, changes in settlement patterns, and regional population changes could all contribute to the observation. Nevertheless, this analysis supports Selden’s observation and warrants further exploration.
Thanks to Eric Oksanen for starting me on this project and for his helpful comments during its development. Waldo Troell also shared insights about the nature of the archaeological samples. Finally, thanks to Dr. Zac Selden for his invitation to produce this piece.
About the Author
Dr. Scott Pletka is the Supervisor of the Archeological Studies Branch at the Texas Department of Transportation
**Why all of the visiting authors of late? Well, the CRHR recently purchased a new GPR, and I’ve been in the field with Dr. Chet Walker learning how to use it – more on that next Monday**