Tuesday, November 27, 2012

A new method for analyzing growth in extinct animals (dissertation summary 1)

The last year and a half was a whirlwind, and so I never got around to blogging about the fruits of my dissertation: Mandibular growth in Australopithecus robustus... Sorry! So this post will be the first installment of my description of the outcome of the project. The A. robustus age-series of jaws allowed me to address three questions: [1] Can we statistically analyze patterns of size change in a fossil hominid; [2] how ancient is the human pattern of subadult growth, a key aspect of our life history;  and [3] how does postnatal growth contribute to anatomical differences between species? This post will look at question [1] and the "zeta test," new method I devised to answer it.

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:
1. Randomly grab a fossil (A) and a human (B) in one dental stage ('younger'), then a fossil and a human in a different dental stage ('older'). 2. Using only traits they all share, calculate relative size change in each species (older/younger): the zeta test statistic describes the difference in size change between species. 3. Calculate as many zetas as you can, creating a distribution giving an idea of how similar/different species' growth is.
The zeta statistic is the absolute difference between two ratios - so positive values mean species A  grew more than species B, while negative values mean the opposite. If 0 (zero, no difference) is within the great majority of resampled statistics, you cannot reject the hypothesis that the two species follow the same pattern of growth. During each resampling, the procedure records the identity and age of each specimen, as well as the number of traits they share in common. This allows patterns of similarity and difference to be explored in more detail. It also makes the program run for a very long time. I wrote the program for the zeta test in the 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!

I used the zeta test to compare mandibular growth in a sample of 13 A. robustus and 122 recent humans. I first showed that the method behaves as expected by using it to compare the human sample with itself, resampling 2 pairs of humans rather than a pair of humans and a pair of A. robustus. The green distribution in the graph to the left shows zeta statistics for all possible pairwise comparisons of humans. Just as expected, that it's strongly centered at zero: only one pattern of growth should be detected in a single sample. (Note, however, the range of variation in the green zetas, the result of individual variation in a cross-sectional sample)

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.

We can also examine how zeta statistics are distributed between different age groups (above). I had broken my sample into five age groups based on stage of dental eruption - the plots above show the distribution of zeta statistics between subsequent eruption stages, the human-only comparison on the left and the human-A. robustus comparison on the right. As expected, the human-only statistics center around zero (red dashed line) across ontogeny, while the human-A. robustus statistics deviate from zero markedly between dental stages 1-2 and 3-4. I'll explain the significance of this in the next post. What's important here is that the zeta test seems to be working - it fails to detect a difference when there isn't one (human-only comparisons). Even better, it detects a difference between humans and A. robustus, which makes sense when you look at the fossils, but had never been demonstrated before.

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

Monday, November 19, 2012

Osteology everywhere: Muffin tops

It's become challengingly chilly here in Astana and my days of running outdoors are fading into memories redshifting into oblivion, so last weekend I went ice skating instead. Pulling off certifiably Scott Hamiltonian moves, I espy my silhouette and what hominid face is staring back?
That's, right, Australopithecus boisei (right). Of course they're not identical, but then they don't really have to be when you see Osteology Everywhere.

But then again, when you've been doing this too long, you start to see Paleontology Everywhere, too. The shadow also reminded me of a time a few years ago, when we were picking through bags of backdirt at Dmanisi, foraging for micromammals, passing pachmelia and time with trivia. Someone posed the riddle, "What did one muffin say to the other muffin?" To which I responded:

Tuesday, November 13, 2012

Osteology Everywhere: Zubi

We're going over bone biology and bioarchaeology this week in my Intro to Bio class, and so I thought I'd open the unit with my patent-pending Osteology Everywhere series. I showed the students the various real-life objects from the series, and they kicked buttocks at seeing the bones in quotidian things. They even got this new one:
That yellow pepper is a ringer for a premolar crown, which hopefully was not as yellow. So I'm very proud of my students. I figure if I can make people see bones everywhere they look, well then I've done my job. But hopefully they don't get as bad as me: a few months ago my friend bought one of those Kinder chocolate eggs with a prize inside. Shaking it, you could hear something rattling in there. It's disconcerting that my mind immediately guessed, "Legos, or teeth." At least legos came before teeth.

Also "zubi," from the title, is the Croatian word for 'teeth' (and apparently also slang for 'breasts').

Tuesday, November 6, 2012

The beardless White House: Part I

Something's been bothering me about this election. No, it's not the silence from both major parties on climate change. It's the fact that neither Obama nor Romney (I accidentally just typed "RMoney"... accidentally?) sports facial hair. A friend and I were talking about this the other day, and a quick google search showed us there hasn't been an appreciable furface sleeping at 1600 Pennsylvania Ave. since the mustachioed WH Taft (of butter and bathtub fame), 100 years ago. That is, unless any of these recent presidents was a closet homosexual (different meaning of "beard").

This is hairy dearth is deplorable. Just look at this pic of portraits of past presidents:

You're probably thinking, "Where's all the virile scruff?" Well, no, you're probably thinking, "There's a lot of dudes / white ppl there." But your next thought is probably, "Where's all the virile scruff?" However, from Abe Lincoln through Bill Taft there's a fairly flagrant concentration of beards, mustaches and whatever you call the thing hiding Chester A. Arthur's charming smile (squared off in red); only W McKinley and A Johnson dared rain on this badass parade. Yes, there are some audacious sideburns on John Q. Adams and Martin Van Buren, but otherwise all Executive facial hair is concentrated between 1860 and 1913. What gives?

It looks like there's a fairly clear pattern: voters loathed and distrusted facial hair for the first nearly 100 years of American history, followed by a brief period in which facial hair was loved and trusted, which may then have been ruined by Taft and after which there's been nary a stache nor goat sitting in the oval office to the present day. Is this a real pattern, or could some other random process produce this same distribution of scruff? (for simplicity's sake, we'll pretend no president served more than 1 term...) Could random sampling of 43 (mostly white) men give us a clump of 9/13 with facial hair? (side burns don't count) If there's a 50/50 chance of a man growing facial hair, is 9/43 Prezes unusually high or low? I'll let you know after I write and run some tests!