Over a year ago, and exactly one year ago, I described some of the rational for my dissertation. Basically, in order to address questions [2-3] above, I had to come up with a way to analyze age-related variation in a fossil sample. A dismal fossil record means that fossil samples are small and specimens fragmentary - not ideal for statistical analysis. The A. robustus mandibular series, however, contains a number of individuals across ontogeny - more ideal than other samples. Still, though, some specimens are rather complete while most are fairly fragmentary, meaning it is impossible to make all the same observations (i.e. take the same measurements) on each individual. How can growth be understood in the face of these challenges to sample size and homology?
Because traditional parametric statistics - basically growth curves - are ill-suited for fossil samples, I devised a new technique based on resampling statistics. This method, which I ended up calling the "zeta test," rephrases the question of growth, from a descriptive to a comparative standpoint: is the amount of age-related size change (growth) in the small fossil sample likely to be found in a larger comparative sample? Because pairs of specimens are likelier to share traits in common than an entire ontogenetic series, the zeta test randomly grabs pairs of differently-aged specimens from one sample, then two similarly aged specimens from the second sample, and compares the 2 samples' size change based only on the traits those two pairs share (see subsequent posts). Pairwise comparisons maximize the number of subadults that can be compared, and further address the problem of homology. Then you repeat this random selection process a bajillion times, and you've got a distribution of test statistics describing how the two samples differ in size change between different ages. Here's a schematic:
statistical computing language, R, and the codes are freely available. (actually these are from April, and at my University of Michigan website; until we get the Nazarbayev University webpage up and running, you can email me for the updated codes)
The zeta test itself is new, but it's based on/influenced by other techniques: using resampling to compare samples with missing data was inspired by Gordon et al. (2008). The calculation of 'growth' in one sample, and the comparison between samples, is very similar to as Euclidean Distance Matrix Analysis (EDMA), devised in the 1990s by Subhash Lele and Joan Richtsmeier (e.g. Richtsmeier and Lele, 1993). But since this was a new method, I was glad to be able to show that it works!
In blue, the human-A. robustus statistics show a markedly different distribution. They are shifted to the right - positive values - indicating that for a given comparison between pairs of specimens, A. robustus increases size more than humans do on average.
So there you go, a new statistical method for assessing fossil samples. The next two installments will discuss the results of the zeta test for overall size (important for life history), and for individual traits (measurements; important for evolutionary developmental biology). Stay tuned!
Several years ago, when I first became interested in growth and development, I changed this blog's header to show this species' subadults jaws - it was only last year that I realized this would become the focus of my graduate career.
Gordon AD, Green DJ, & Richmond BG (2008). Strong postcranial size dimorphism in Australopithecus afarensis: results from two new resampling methods for multivariate data sets with missing data. American journal of physical anthropology, 135 (3), 311-28 PMID: 18044693
Richtsmeier JT, & Lele S (1993). A coordinate-free approach to the analysis of growth patterns: models and theoretical considerations. Biological Reviews, 68 (3), 381-411 PMID: 8347767