[R-sig-ME] G vs R posterior correlation in bivariate MCMCglmm

[Jarrod Hadfield]

Hi,

If the relationship between the two is causal  - lets say phenotype 
affects disease - then the regression at the two levels will be 
identical: COV(Disease, phenotype)/VAR(phenotype) is the same at the 
site level and the units level. However, they can easily differ. Imagine 
that the amount of resource varies between sites, but within sites all 
individuals have access to the same amount of resource. If there is a 
trade-off then, for a given amount of resource, there  will be a 
positive relationship (if high values of the phenotype are 'good') 
between the two variables observed at the units level. However, imagine 
that as the amount of resource increases individuals can increase their 
phenotype but also reduce the amount of disease. As a consequence the 
between site correlation may well be negative. So, correlations at both 
levels tell you something interesting. However, it should be noted that 
if you can assume causality you are better just fitting a univariate 
model with phenotype in as a predictor: you get an increase in precision 
for your assumption.

Cheers,

Jarrod





On 26/10/16 08:46, Xav Harrison wrote:
> Hi Folks
>
> A potentially rookie question here, but here goes.
>
> I'm using MCMCglmm to fit a bivariate response model where one response is
> a Poisson count of pathogen load (Disease) and one is a Gaussian
> phentotypic measure of the host (Phenotype).
>
> I've fit a model with fixed effects that one would expect to influence each
> of these predictors, and including a random effect of site, of the form:
>
> prior1<-list(R=list(V=diag(2), nu=3),G=list(G1=list(V=diag(2),nu=3, alpha.mu
> =c(0,0),alpha.V=diag(2)*1000)))
>
> MCMCglmm(cbind(Disease,Phenotype) ~ trait-1 +
> trait:predictors,rcov=~us(trait):units,random=~us(trait):site,family=c("poisson","gaussian"),prior=prior1,verbose=F,nitt=42000,burnin=2000,thin=20)
>
> The golden egg here, and my hypothesis, is that after accounting for some
> predictors there will be a negative posterior correlation between the
> response traits, where the posterior correlation is calculated as:
>
> Cov(Disease,Phenotype) / sqrt(Var(Disease)*Var(Phenotype))
>
> My question is whether this correlation is relevant at the G structure
> level (Site random effect) or at the R structure level, which I take to be
> the residual variance at the observation (individual) level?
>
> I suspect the answer is the latter, but I'm struggling to interpret what it
> means if there is a negative correlation at the Site level? Does it mean
> that the variances of the two traits at the site level are not independent,
> in that higher values in one trait for a site tend to produce lower values
> for the other?
>
> Any help greatly appreciated.
>
> Cheers
>
> Xav
>
>
>


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