strong effect of prior on residual variance

Hi Jarrod,

Thanks for the speedy response,

There is just one measure of T per cohort. T is sexual dimorphism in a 
trait, so its a cohort level value - individuals can't be sexually 
dimorphic: it was calculated using mcmcglmm to estimate 
cohort-specific posterior distributions for both male and female trait 
values, then randomising the order of the distributions and dividing 
them (Male[sample_i]/Female[sample_i]) where i is 1:1000. I then took 
the mean of the posterior for each cohort as my response variable. 
It's much the same as one could do to derive a posterior distribution 
of heritability when given posterior distributions of additive and 
phenotypic variation (also as stated on page 5 of the course notes). 
Perhaps it would be more appropriate to use the entire posterior 
distributions of SD for all 90 cohort rather than the just means (so 
there would be 1000 samples of SD per cohort)?

B is a numerical variable, if expressed as a factor it has 65 levels 
(not all years have a measure of B). Should this be converted to a factor?

Cheers,
Rob

Subject: Re: [R-sig-ME] strong effect of prior on residual variance
To: robgriffin247 at hotmail.com; r-sig-mixed-models at r-project.org
From: j.hadfield at ed.ac.uk
Date: Thu, 5 May 2016 13:47:37 +0100

Hi Rob,

1) how many observations are there per cohort on average?
2) how many levels does B have in it?
3) nu=1 is typical in a parameter expanded prior, as this is flat for
the standard deviation.

Cheers,

Jarrod



On 05/05/2016 13:34, Rob Griffin wrote:

Dear list members,
I'm using MCMCglmm to model variance among ~90 cohorts as a result 

of an environmental factor ("B" - numerical). "T" is the response 
variable, which is formed as a ratio, with mean ~1 & is normally 
distributed. The cohorts come from two different populations, where 
each cohort is defined by the place and year of birth (e.g. one cohort 
is all individuals born in one area, A1, in 2014), such that there is 
one value of T per cohort. B is measured on a larger level, so there 
is one score of B per year, regardless of population. I include Area 
as a fixed effect (factor with two levels) because in some years only 
one area is measured so it may induce sampling bias of the 
environmental effect (e.g. one year where only one area is measured 
has an extreme B score). From a biological perspective I expect B to 
have a small impact on the among-cohort variance in T so I've used 
parameter expanded priors for the random effect and inverse-wishart 
for the residual (as suggested to a previou!

s thread I started - on priors for small variance components - 

last year: 
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2015q1/023370.html).

When I used similar set up to my previous model (see code below) I 

find that, contrary to my expectation, B has a relatively large 
variance estimate compared to the residual (both mean and median of 
the posterior for B is 10x higher than residual). It seems unlikely 
that 90% of the variance in T is explained by B.

This prompted me to fiddle with the prior specification to make 

sure nothing was wrong... Settings used were nu = 0.002 or 2, & V = 1 
or 10 (for the priors in R with all four combinations of V and nu 
tested), also producing 4 independent chains with 100k iterations, 25k 
burnin, and thinning interval of 50 for each; autocorrelation is low 
(<0.1 between successive samples), convergence appears good for both B 
and Residual. The estimate of variance (both the median and mean of 
the posterior) is similar across the four chains, within each 
combination of nu and V, for both B and Residual.

In the previous thread it is pointed out that "Usually the data 

overwhelm the prior for the residual variance so you can probably be 
pretty relaxed about that." I find that estimates of B and residual 
(units) are generally insensitive to changes in any of the parameters 
in G, but highly sensitive to changes in both the belief and variance 
inputs for R (increases in nu and V both increase the estimate of 
residual, while also absorbing variance from the random effect B). 
Given the statement that data usually overwhelm the prior, should I be 
concerned that the prior is strongly affecting the estimate of 
residual (and random effects) variance in this case? How should this 
be interpreted and dealt with? The consistency across independent 
chains suggests to me that the model is able to estimate B and 
residual variances well, but is drawing too much information from the 
prior rather than the data, and therefore I'm using the wrong prior.

Thanks,Rob
####################prior1 = list(G = list( G1 = list(V = 1, 

nu=0.001, alpha.mu=0, alpha.V=1000) ), R = list(V = 1, nu=0.001)) M1A 
= MCMCglmm( T ~ 1 + Area ,random = ~ B ,data = DF3 ,nitt = nitt 
,burnin = burnin ,thin = thin ,family = "gaussian" ,prior = prior1 )

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