Dear Santiago,
I agree that evolving traits might have all sorts of complicated relationships,
but that doesn't mean we shouldn't rule out simple relationships first. And
besides, the most basic question one can ask - really the first question to ask
- is whether there is any association at all between two variables. If we are
trying to find out if such an association exists, independent of phylogeny,
then we need a method that gives the same results regardless of whether which
variable we look at. Of course the slope of any relationship will be
different, depending on whether we are trying to predict x from y, or y from x.
But that shouldn't biologically affect the covariance between the two
variables. The covariance by definition is not a measure of x specifically from
y, or vice-versa, it is a measure of how they both covary (there is no
directionality to this). So any method that suggests one degree of confidence
in this covariance if we look at x from y, and a different degree of confidence
if we look at y from x, is simply not biologically valid for assessing
covariance.
To put it in the context of brain and group size: Is group size covarying
significantly with brain size or not? Well, if you try to predict group size
from brain size, then PGLS says the confidence we should have of this
covariance is higher than if you try to predict brain size from group size.
This makes no biological sense, and I maintain this makes PGLS invalid for
assessing the significance of covariance between two variables.
-Tom
On Jul 22, 2013, at 2:02 AM, Santiago Claramunt wrote:
> Dear Tom,
>
> If your concept of 'relationship' is a simple correlation analysis, then it
> may not make sense to get different estimates of the 'probability of the
> relationship'. But in evolutionary biology things are always more complicated
> than a simple correlation model. Things are not linear, causality is
> indirect, and, yes, observations are not independent because of phylogen (and
> space). We clearly need methods that are more sophisticated than a simple
> correlation analysis.
>
> Brain size and groups size are variables of very different nature, and their
> relationship may be the product of natural selection acting on lineages over
> evolutionary time, which form phylogenies. I don't see any problem in
> obtaining somewhat different results depending on how the relationship is
> modeled.
>
> Santiago
>
>
> On Jul 21, 2013, at 11:47 PM, Tom Schoenemann wrote:
>
>> Thanks Liam,
>>
>> A couple of questions:
>>
>> How does one do a hypothesis test on a regression, controlling for
>> phylogeny, if not using PGLS as I am doing? I realize one could use
>> independent contrasts, though I was led to believe that is equivalent to a
>> PGLS with lambda = 1.
>>
>> I take it from what you wrote that the PGLS in caper does a ML of lambda
>> only on y, when doing the regression? Isn't this patently wrong,
>> biologically speaking? Phylogenetic effects could have been operating on
>> both x and y - we can't assume that it would only be relevant to y.
>> Shouldn't phylogenetic methods account for both?
>>
>> You say you aren't sure it is a good idea to jointly optimize lambda for x &
>> y. Can you expand on this? What would be a better solution (if there is
>> one)?
>>
>> Am I wrong that it makes no evolutionary biological sense to use a method
>> that gives different estimates of the probability of a relationship based on
>> the direction in which one looks at the relationship? Doesn't the fact that
>> the method gives different answers in this way invalidate the method for
>> taking phylogeny into account when assessing relationships among biological
>> taxa? How could it be biologically meaningful for phylogeny to have a
>> greater influence when x is predicting y, than when y is predicting x?
>> Maybe I'm missing something here.
>>
>> -Tom
>>
>>
>> On Jul 21, 2013, at 8:59 PM, Liam J. Revell wrote:
>>
>>> Hi Tom.
>>>
>>> Joe pointed out that if we assume that our variables are multivariate
>>> normal, then a hypothesis test on the regression is the same as a test that
>>> cov(x,y) is different from zero.
>>>
>>> If you insist on using lambda, one logical extension to this might be to
>>> jointly optimize lambda for x & y (following Freckleton et al. 2002) and
>>> then fix the value of lambda at its joint MLE during GLS. This would at
>>> least have the property of guaranteeing that the P-values for y~x and x~y
>>> are the same
>>>
>>> I previously posted code for joint estimation of lambda on my blog here:
>>> http://blog.phytools.org/2012/09/joint-estimation-of-pagels-for-multiple.html.
>>>
>>> With this code to fit joint lambda, our analysis would then look something
>>> like this:
>>>
>>> require(phytools)
>>> require(nlme)
>>> lambda<-joint.lambda(tree,cbind(x,y))$lambda
>>> fit1<-gls(y~x,data=data.frame(x,y),correlation=corPagel(lambda,tree,f
