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G vs R posterior correlation in bivariate MCMCglmm

3 messages · Xav Harrison, Jarrod Hadfield

Original
Xav Harrison
# 
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
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:

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Xav Harrison
    

    

      
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Hi Jarrod

Thanks for the swift and helpful reply. That makes a lot of sense. I should
have said in the original email that I have taken this approach rather than
fitting a Disease ~ Phenotype model because we have predictors that
influence both simultaneously. For example we know both mean disease
intensity and mean phenotype were lower in 2014. I was hoping to use the
bivariate model approach to estimate the posterior correlation between the
two traits once 'controlling' for the predictors and seeing what's left.
Does this sound sensible?

If you wouldn't mind elaborating further, I'm trying to work out what it
means when there is a non-zero correlation at the site level but not at the
units level, which some models have recovered. Is this a problem suggestive
of insufficient data to estimate both matrices accurately, or something
that could be biologically plausible? I'm afraid of thinking I've found the
golden egg but in fact have built myself a nice random number generator by
asking too much of the models.

Thanks again

Xav

        
On 26 October 2016 at 10:34, Jarrod Hadfield <j.hadfield at ed.ac.uk> wrote: