Short Version: OU models need to be compared against both Brownian Motion and so-called ‘white noise’ evolution (i.e. a case in which phylogeny is not important and trait value independently vary around some global mean) to infer the existence of phylogenetic signal.
The Longer Version: Based on a pursual of the literature there seems to be the temptation to claim that there is phylogenetic signal in data, and that selection is important if an OU model fits the data better than a BM model. This, however, does not work, because [a] one can not claim that there is phylogenetic signal in data without comparing likelihoods against a model in which there is no phylogenetic correlation, and [b] one can not go on to infer that selection is important just because an OU model fits better than a Brownian motion model. This second point is true for a variety of conceptual reasons (the alpha parameter in the OU measures how (un-) correlated the species trait values are to one another, and strength of stabilizing selection is only one of many many evolutionary and/or ecological forces that could causes covariance to decline). But what I to focus on here is whether phylogeny is important at all–indeed, an OU model could fit the data better than Brownian Motion because phylogeny is utterly unimportant, and in fact trait values vary due entirely to say environmental factors (through say plasticity) or measurement error.
A short illustration in R
If we simulate data under a white noise model.
trait<-rnorm(nsp, mean=0, sd=1)
wn<-fitContinuous(tree, trait, model=‘white’)
ou<-fitContinuous(tree, trait, model=‘OU’) # The fitConinuous function gives an error message signaling that things are screwed up. (But maybe as an empiricist one says “Hey, whatever? things are always giving error messages…?”)
bm<-fitContinuous(tree, trait, model=‘BM’)
## Extract likelihoods from all models
## Conduct likelihood ratio tests between the full model (OU with alpha parameter) against null models with only mean mean and sigma parameters (white noise, and BM). Note that this is side stepping the point that likelihood ratio tests of phylogenies don’t really work–see Boettiger 2012–but none the less they offer a useful approximation of true ‘significance’
pchisq(2*(lou–lbm), df=1, lower.tail=F) # Here the OU is way better than BM. If this is all one tests you might think that “Hey! Selection along the phylogeny is important!”
pchisq(2*(lou–lwn), df=1, lower.tail=F) # But wait, there’s no support for the OU model versus one in which there is simply variation around some mean. Not only is there no evidence that selection is important (based at least on trait data…) but there’s not even evidence that ‘along the phylogeny’ is relevant!
This illustrates the need to test that OU model against *BOTH* null models.
Back to the second point regarding selection
The (perhaps?) larger problem is that there is a strong temptation to interpret alpha in single peak OU models as meaning something about the importance of stabilizing selection. Often shorthanded to just ‘selection’, even though directional selection will not be captured by the alpha parameter (under standard conditions), and will be instead conflated with sigma.
lam<-fitContinuous(tree, trait, model=‘lambda’)
pchisq(2*(llam–lwn), df=1, lower.tail=F)
pchisq(2*(llam–lbm), df=1, lower.tail=F)
# In comparison to the single peak OU model, the much maligned (I would argue largely unjustifiablly) Pagel’s Lambda ‘model’ of evolution does not offer the same degree of a stumbling block. I think this is largely because lambda is (1) more easily interpretable, and (2) makes no special claims about where non-brownian motion variation is coming from (e.g. no one stricly interprets 1-lambda as ‘selection’ per se). First the interpretability–because lambda effectivly spans from 0 to 1 it has clear bounds and these bounds have meaning (0 is no importance of phylogeny and 1 is BM evolution). The single peak OU’s alpha parameter instead spans from 0 (i.e. BM) and +Inf (i.e. white noise). So in theory the user could obviously say “oh, my alpha value is really really big” and this would give a quick visual check that phylogeny is not important. In practice however a maximum likelihood estimate will not permit an actually value of +Inf (as we saw in the warning message above the alpha was at its bounds) and so the user is (unwisely) given disgression to ignore the warning, and to interpret a poorly fit model for its biological significane. In short, when lambda is 0 it’s easy to interpret that “Hey, phylogeny isn’t important!”, but alpha is never really allowed to be +Inf and therefore there is no alternative but to conduct hypothesis testing with OU models to ensure that phylogeny is necessary to explain patterns.
Despite the admirable statistical properties lambda models tend not to be favored. The reason is that interpretting its biological significance is tricky. Many will critique it for not having a good one. I actually see this as one of the unsung boons of using lambda, and heuristically the ‘proportion of the variance in a trait’s value that can be ascribed to strict brownian motion evolution’ is as far as I’m concerned a completely adequate if unwieldy and slightly vague description. Indeed an equivalent formulation is a model which has two sigma terms, one of which is multiplied by the identity matrix (for white noise evolution), and the other of which is multiplied by the phylogeny’s correlation matrix (for brownian motions). The stregth-in-vagueness of this description of lambda is how it remains agnostic about the source of the remaining variance. This variance could come from selection, or measurment error, or plasticity. Using lambda here seems infinitly favourable to me over the interpretation of an OU model, in which one is tempted to think that alpha necessarily indicates the ‘importance of selection’. Making claims based on alpha is innappopriate, because the same ensemble of (non-selection) things that depress lambda can also inflate alpha.
In short, perhaps a phenomenological model that makes no claims is preferable to a mechanistic one in which users are hell bent on reading (undeserved) meaning into its parameter estimates?