MANLY'S ALPHA FOR PREDATION SELECTIVITY: MODELS, ESTIMATION AND INTERPRETATION
Given a sample of specimens of m taxa, some with and some without drill holes, the probability model of the drill hole counts per taxon is a multinomial distribution where the probability of selection is parameterized by a vector of m alphas that are greater than or equal to zero and sum to 1. The null hypothesis is that each alpha is equal to 1/m.
Statistical tests can be derived to test the null hypothesis. There is a closed-form maximum likelihood estimate that can be used to form a likelihood ratio test (LRT). The literature on alpha suggests that, owing to a high proportion of undrilled taxa and low specimen counts, a classical LRT is not appropriate. The accepted alternative is a randomization test. In addition, a Bayesian formulation enables a true probability assessment of the alpha for each taxon by assessing the probability of being less than or equal to 1/m for the estimated posterior distribution of each alpha. This approach provides a less cumbersome interpretation of drilling data and can be computed in a few lines of code.
Here, we describe Manly’s alpha probability model and the estimation approaches. We demonstrate the application of this approach on several data sets.