Significance of adjacency in correlation matrix with ordered variables

Question

I am using the Discrete module in BayesTraits. I have a set of 1-9 serially homologous traits. For each pair of traits, I am comparing the 4-parameter model of independent evolution and the 8-parameter model of dependent evolution. Thus, each analysis has two inputs: A phylogeny, or evolutionary tree, and a matrix of presence/absence data (1/0) for a pair of traits for each species in the tree.

I want to test the hypothesis that the order of the traits is important. How can I get a p-value that tells me whether adjacency among traits significantly impacts the correlation between them? For example, a p-value below 0.05 would indicate that the correlation between a trait and the traits to either side of it (e.g., 4 compared to 3 and 4 compared to 5) are higher than correlations between non-adjacent traits.

Answer 1

I think I understand what you are trying to achieve (correct me if I'm wrong). You want to test the hypothesis that your trait has ordered states (I think that is what you mean by serial homologues). That is something you can do within BayesTraits, but I think you should be using multistate, not discrete. For multistate, your data would be coded as 0, 1,...,9 (I think the digits 0-9 define the upper limit on how many states you can have, so you are lucky). Within multistate, you can then specify two models, one with ordered states and one without ordered states, and compare them with a likelihood ratio test. You will have to think about how to specify the models though.

One way to do it would be to define a model with all rates equal (ARE), in which it is assumed that transitions between all character states are equally likely (e.g., transitions from 1 to 2 are just as likely as from 1 to 9). Then you could define a second model with rates among neighboring states equal (NRE), and rates among non-neighbors equal. If the NRE model fits the data significantly better than the ARE model, and it estimates a higher transition rate among neighboring states than non-neighboring states, then that tells you the order of character states matters.