Dear all,

I am trying to implement a MCMCglmm/lmer analysis to estimate 
narrow-sense heritability h2 and sire/maternal+environmental effects of 
a single phenotypic trait in an insect. We used a nested half-sib 
full-sib design, treating sex of the offspring as fixed effect. This 
analysis is outside my main areas of interest and I have no-one to 
discuss it with directly, so I would much appreciate any help to help me 
wrap my head around the problem. Please suggest a better place to 
migrate the question if it does not fit here.

The data (44 males x 3 females nested within each female, total ~430 
offspring) are unbalanced due to differences in fertility and offspring 
survival - many females yielded no or few offspring. I understand that 
the basic nested analysis of variance formulas provided in the Lynch & 
Walsh classics should not be trusted there.

Our raw data look like this:

head(df)

   animal sex mother father    val
1    201   m     64      5 -23.03
2    202   m     64      5 -17.43
3    203   m     64      5 -14.39
4    204   f     64      5 -17.39
5    205   f     65      5 -22.04
6    206   f     65      5 -19.84


The underlying theory of the linear model for the nested design in Lynch 
& Walsh (pp. 571-574), i.e.
val_ijk = mu + sex_ijk + father_i + mother_ij + e_ijk,
says that 4*Var(s)/Var(z) or alternatively 2*(Var(s)+Var(d))/Var(z) is 
the best estimate of h2.

Question 1: How do the quantities from Lynch & Walsh relate to the VCV 
components provided by MCMCglmm or lmer?
Question 2a: Can the animal model of Wilson et al (2010, WAMWiki) be 
applied "as-is" with the appropriate pedigree to this nested design and 
what are the pros/cons of (not) doing that?
Question 2b: Why do the results for heritability differ when I use both 
father and mother as explanatory variables as opposed to using only 
mother and supplying the pedigree (= essentially the 
animal+mother+father columns of my data including the NA rows for parents)?
Question 3: MCMCglmm course notes suggest DIC may not be meaningful to 
use when priors differ. How does one select between models that differ 
in the number of random effects then?

I have tried the following priors and models in MCMCglmm; all models had 
nitt = 65000, thin = 50, burnin = 15000, skipped for brevity:

p.var <- var(df$val, na.rm=TRUE)

prior1 <- list(G = list(G1 = list(V = matrix(p.var/2), n = 1)), R = 
list(V=matrix(p.var/2), n = 1))
prior2 <- list(R = list(V=matrix(p.var/2), nu = 0.2),
                  G = list(G1 = list(V = matrix(p.var/2), n = 1), G2 = 
list(V=matrix(p.var/2), n = 1))  )

m1 <- MCMCglmm(val ~ sex - 1, random = ~animal, data = df, pedigree = 
ped, prior = prior1)
[DIC: 1854.0]
m2 <- MCMCglmm(val ~ sex - 1, random = ~animal + mother, data = df, 
pedigree = ped, prior = prior2)
[DIC: 1926.6]
m2b <- MCMCglmm(val ~ sex - 1, random = ~father + mother, data = df, 
pedigree = ped, prior = prior2)
[DIC: 2306.1]
m2c <- MCMCglmm(val ~ sex - 1, random = ~father + mother, data = df, 
pedigree = NULL, prior = prior2)
[DIC: 2305.9]

The following heritability formulas:
h2.m1 <- m1$VCV[,"animal"] / (m1$VCV[,"animal"] + m1$VCV[,"units"])
h2.m2<-m2$VCV[,"animal"] / (m2$VCV[,"animal"] + m2$VCV[,"mother"] + 
m2$VCV[,"units"])
h2.m2b<-m2b$VCV[,"father"] / (m2b$VCV[,"father"] + m2b$VCV[,"mother"] + 
m2b$VCV[,"units"])
h2.m2c<-m2c$VCV[,"father"] / (m2c$VCV[,"father"] + m2c$VCV[,"mother"] + 
m2c$VCV[,"units"])

give the following mode of posterior distribution of heritability estimate:
h2.m1 = 0.92
h2.m2 = 0.75
h2.m2b = 0.47
h2.m2c = 0.45

So model m1 has the lowest DIC, but the heritability estimate is kind of 
high...

Many thanks,
David Boukal

-------------------------------------------------
Department of Ecosystems Biology,
Faculty of Science, University of South Bohemia
	&&
Department of Biosystematics and Ecology,
Biology Centre AS CR, Institute of Entomology
Branisovska 31
37005 Ceske Budejovice
Czech Republic

boukal@entu.cas.cz	  www.entu.cas.cz/boukal
phone +420 387 772 327    fax   +420 385 310 354

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