MANLY'S ALPHA FOR PREDATION SELECTIVITY: MODELS, ESTIMATION AND INTERPRETATION

In the paleoecological literature, drilling frequencies on prey taxa are commonly taken to indicate a predator’s preference. Those taxon-specific drilling frequencies are often compared with one another and related to underlying prey characteristics such as cost: benefit ratio or body size. Although this approach can demonstrate a predator’s relative preference for one prey type over another, it fails to consider whether predation on any prey type is greater than would be expected by a predator without preference. Manly’s alpha, which is a well-established model in the ecological literature, provides a means to test the null model of no prey preference against an alternative of preference for one or more taxa and also identify which taxa deviate from the null.

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.