Binary response animal models in MCMCglmm

Hi Jacob,

1/ I usually go for the posterior mode for variances in which the 
posteriors are strongly skewed.

2/ Compared to family="ordinal", the DIC in threshold models is a level 
closer to what most people are after. But its still a level to high - it 
is asking how well could we predict new observations associated with the 
particular random effects (breeding values in this case) we've happened 
to sample. I don't use DIC.

3/ Because you have V=diag(2) *and* fix=1 you are fixing the residual 
covariance matrix to an identity matrix. This might be OK if your two 
responses are a male and female trait and so each row only has one value 
and one NA? Even then this makes the algorithm pretty inefficient: 
better to have a single trait indexed by sex. The intersexual genetic 
correlation is then estimated using us(sex):animal.

If this isn't the case then fixing the residual covariance to zero will 
force some of the residual correlation into the genetic term. It is 
possible then to obtain an *estimate* of zero for the genetic 
correlation if the *true* genetic correlation is positive and the 
residual covariance negative.

Much more likely though is that the prior you use is strongly 
constraining the genetic correlation to zero. I guess your motivation 
for using it is that the marginal priors for each variance have nu=1000, 
and this is what Pierre de Villemereuil suggests for probit models in 
his MEE paper (i.e. tending towards a chi-square with 1df)? However, for 
bivariate models the marginal prior for the correlation is flat when 
nu=3 (and pushed outwards towards -1/1 when nu=2). With nu=1000 it is 
pushed inwards towards zero by quite  a bit.

4/ The scale in binary models is not identifiable and so absolute values 
of the genetic variance do not have a meaning outside of the context of 
the residual variance. I've tried to explain this in pictures in the 
supp materials to this paper:

http://onlinelibrary.wiley.com/doi/10.1111/2041-210X.12354/abstract

but not sure if it helps. The other way I try and explain it is that the 
residual variance is not identifiable (Section 2.6 of the course notes). 
If there are better ways of explaining it, it would be good to know!

Cheers,

Jarrod


On 27/03/2017 05:18, Jacob Berson wrote:

Hi All

  

I have attempted to estimate the heritability of several binary traits, and
test for genetic correlations both between two binary traits, as well as
between a binary and a Gaussian trait, using the animal model in MCMCglmm.

  

Several issues have come up in my analysis that I'm hoping to get some help
with.

  

For some traits the posterior distribution of my heritability estimate is
very close to zero, resulting in a non-symmetric density plot (the left tail
is essentially cut by the y-axis). With these distributions I get very
different point estimates of heritability depending on if I use the
posterior mean or posterior mode.

  

1)      I've seen both the mean and mode used in the literature, does anyone
have a view on which is the most appropriate in the above circumstance?

  

The abovementioned distributions suggest to me that there is little, if any,
additive genetic variance present. However, for some traits the DIC value of
a model with "animal" as a random effect is lower than the null model
(though after reading
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2014q1/021875.html and
switching from using family="ordinal" to family="threshold" the difference
in DIC values was greatly reduced).

  

2)      Are differences in DIC values an appropriate tool for testing model
fit when the response is binary? If so, what level of difference is
sufficient to reject the null model (I've seen both >5 and >10)?

  

I am getting a posterior distribution centred on zero when testing for
intersexual genetic correlations between two binary traits. However, when I
look at the data (for example using sire means) it seems to me that there is
in fact a strong correlation and that my MCMCglmm results are not correct.

  

3)      Am I trying to do the impossible, i.e., is it still recommended (as
was the case a few years ago
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2009q3/002637.html) not to
fit bivariate binary models in MCMCglmm, even when using family="threshold"?

  

Finally, as I understand it, it is not appropriate to report the estimates
of the additive genetic variance from binary models because it depends on
the residual variance (which I have fixed to 1).

  

4)      Can anyone help a novice like me understand why heritability is more
appropriate to report, even though it also incorporates the fixed residual
variance?

  

Apologies for the length of this post - after much searching and reading I
haven't been able to find solutions to these questions. Any advice on the
above and/or feedback on my code below would be very much appreciated.

  

Jacob

  

  

My code:

  

#Univariate models for binary traits

  

Prior0 <- list(R = list(V = 1, fix = 1))

  

Prior1 <- list(R = list(V = 1, fix = 1),

                G = list(G1 = list(V = 1, nu = 1000, alpha.mu = 0, alpha.V =
1)))

  

Binary.Null <- MCMCglmm(bin.response ~ 1, family = "threshold", pedigree =
Ped,

                 prior = Prior0, data = Data, nitt = 1050000, burnin = 50000,
thin = 500, verbose = FALSE)

  

Binary.Va <- MCMCglmm(bin.response ~ 1, random = ~animal, family =
"threshold", pedigree = Ped,

prior = Prior1, data = Data, nitt = 1050000, burnin = 50000, thin = 500,
verbose = FALSE)

  

heritability <- Binary.Va $VCV[,"animal"] / rowSums(Binary.Va [["VCV"]])

  

  

#Genetic correlation between two binary traits (bin1 and bin2)

  

Prior1rG <- list(R = list(V = diag(2), nu = 0, fix = 1),

                  G = list(G1 = list(V = diag(2), nu = 10001, alpha.mu =
c(0,0), alpha.V = diag(2))))

  

Bin.Bin.corr <- MCMCglmm(cbind(bin1, bin2) ~ trait - 1, random =
~us(trait):animal,

      rcov = ~corg(trait):units, family = c("threshold", "threshold"),
pedigree = Ped, prior = Prior1rG,

      data = Data, nitt = 4050000, burnin = 50000, thin = 2000, verbose =
FALSE)

  

Genetic_correlation1 <- Bin.Bin.corr $VCV[, "traitbin1:traitbin2"] /
sqrt(Bin.Bin.corr$VCV[,

"traitbin1:trait bin1.animal"] * Bin.Bin.corr $VCV[,
"traitbin2:traitbin2.animal"])

  

  

#Genetic correlations between a Gaussian (gau1) and a binary (bin2) trait

  

Prior2rG <- list(R = list(V = diag(2), nu = 0, fix = 2),

                  G = list(G1 = list(V = diag(2), nu = 2, alpha.mu = c(0,0),
alpha.V = diag(2)*1000)))

  

Gau.Bin.corr <- MCMCglmm(cbind(gau1, bin2) ~ trait - 1, random =
~us(trait):animal, rcov =

~us(trait):units, family = c("gaussian", "threshold"), pedigree = Ped,


prior = Prior2rG, data = Data, nitt = 1050000, burnin = 50000, thin = 500,

verbose = FALSE)

  

Genetic_correlation2 <- Gau.Bin.corr$VCV[, "traitgau1:traitbin2"] /
sqrt(Gau.Bin.corr $VCV[, "trait

gau1:trait gau1.animal"] * Gau.Bin.corr $VCV[,
"traitbin2:traitbin2.animal"])

  

  

  

Jacob Berson BSc (Hons), PhD Candidate

Centre for Evolutionary Biology

  

School of Animal Biology (M092)

University of Western Australia

35 Stirling Highway

Crawley, WA, 6009

  

Tel: (+61 8) 6488 3425

  


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