Binary response animal models in MCMCglmm

Hi Jarrod and Pierre

Thank you both very much for your responses.

Pierre, that script is really useful - thanks for making it available.

Jarrod, your answers have been a great help. In regards to 3/, yes you are
correct that my two responses were male and female traits so each row only
had one value and one NA. And yes, my motivation for using that prior for
this model was an attempt at a bivariate form of the prior suggested in de
Villemereuil et al 2013. Both running the model with nu=3, and a different
model using us(sex):animal, have given me similar estimates which are much
closer to what I would expect.

As a follow up...

In the case of a genetic correlation between two binary traits where both
traits have been measured on the same individual (i.e. each row has a value
for both responses), is there a way of fixing both of the residual variances
but estimating the residual correlation? Am I right in my understanding of
below that using V=diag(2) and fix=1 should always be avoided in this
circumstance as it will bias the genetic correlation estimate in the
direction of the residual correlation?

Thanks again for the help, I really appreciate it.

Jacob

-----Original Message-----
From: Pierre de Villemereuil [mailto:pierre.de.villemereuil at mailoo.org]
Sent: Tuesday, 28 March 2017 4:54 AM
To: r-sig-mixed-models at r-project.org; Jacob Berson
<jacob.berson at research.uwa.edu.au>
Subject: Re: [R-sig-ME] Binary response animal models in MCMCglmm

Hi Jacob, hi Jarrod,

A bit ago, I wrote a script for a friend to view the priors for
multi-response models when one response is binary (hence with fixed residual
variance). I decided to share it on GitHub, see here:
https://github.com/devillemereuil/prior-MCMCglmm

It will help you Jacob to see what Jarrod is meaning about priors I think.
It might be useful to even more people.

Cheers,
Pierre.

On Monday, 27 March 2017 20:49:39 NZDT Jarrod Hadfield wrote:

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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