[R-sig-ME] Specifying a model with a link function in MCMCglmm
Jarrod Hadfield
j.hadfield at ed.ac.uk
Fri Feb 19 08:27:53 CET 2016
Hi Sam,
Priors for covariance matrices are tricky but having nu=k (where k is
the dimension of the matrix) is not a good idea. I tend to use parameter
expanded priors of the form
list(V=diag(k), nu=k, alpha.mu=rep(0,k), alpha.V=diag(k)*1000)
The marginal priors for the variances are scaled F(1,1), but I'm not
sure what they are for the covariances/correlation. For correlations I
think it would be flatish but with peaks close to -1 and 1. Upping
nu=k+2 probably makes the marginal prior for the correlation close to
uniform over the -1/1 interval. Unfortunatley there seems to be little
or no theoretical work on this prior in a multivariate context: I use it
because it *seems* to have nice properties. For exampe, if I fitted a
set of random effects which in reality are all zero, then it gives
posterior support for the correlation across the range -1/1.
I would also add that I don't attempt to fit k>2 covariance matrices
unless my data are up to the task.
Cheers,
Jarrod
k<-2
Ve<-rIW(diag(k), 10)
y<-MASS::mvrnorm(100, rep(0,k), Ve)
rfac<-gl(25,4)
# the data are simply y~MVN(0, Ve)
my_data<-data.frame(y1=y[,1], y2=y[,2], rfac=rfac)
prior1<-list(R=list(V=diag(k), nu=0), G=list(G1=list(V=diag(k), nu=k,
alpha.mu=rep(0,k), alpha.V=diag(k)*1000)))
m1<-MCMCglmm(cbind(y1,y2)~trait-1, random=~us(trait):rfac,
rcov=~us(trait):units, data=my_data, prior=prior1,family=rep("gaussian", k))
hist(m1$VCV[,2]/sqrt(m1$VCV[,1]*m1$VCV[,4]), breaks=30)
# posterior correlation of the random effects: nice
prior2<-list(R=list(V=diag(k), nu=0), G=list(G1=list(V=diag(k), nu=k)))
m2<-MCMCglmm(cbind(y1,y2)~trait-1, random=~us(trait):rfac,
rcov=~us(trait):units, data=my_data, prior=prior2,family=rep("gaussian", k))
hist(m2$VCV[,2]/sqrt(m2$VCV[,1]*m2$VCV[,4]), breaks=30)
# bad - look at the variances too!
On 18/02/2016 17:57, Sam H wrote:
> Thanks for your response Jarrod! Regarding the prior specification, I
> forgot to mention the reason I did that. Originally I specified the
> model with no prior, but it kept failing just before the 1000th
> iteration with the error message "ill-conditioned G/R structure". I
> was not very sure what to do since I don't really have any estimates
> to use for a prior, so I figured I would just use estimates produced
> by lmer. Should I just set the G structure to an identity matrix, or
> does it not really matter since nu is high?
>
> On Thu, Feb 18, 2016 at 12:48 AM, Jarrod Hadfield <j.hadfield at ed.ac.uk
> <mailto:j.hadfield at ed.ac.uk>> wrote:
>
> Hi Sam,
>
> Each distribution has a fixed link function in MCMCglmm, and an
> inverse link function for a Gaussian response would be hard to
> implement.
>
> Also, using a REML estimate as a prior, and then having quite a
> strong degree of belief parameter is a bit odd. Many, including
> myself, would consider this double-dipping.
>
> Cheers,
>
> Jarrod
>
>
>
> On 17/02/2016 22:26, Sam H wrote:
>
> Hello all,
>
>
> Before jumping into my question, let me first briefly explain
> my model to
> give context. Here is how I currently have specified the model
> using
> MCMCglmm, first specifying a model in lmer and extracting the
> variance
> estimates for my G prior:
>
> full_lmer <- lmer(RT ~ AgeGroup*cue*cong + (cue + cong|PID),
> data =
> vsdataset)
>
> sigma2 <- sigma(full_lmer)^2
>
> Lambda <- getME(full_lmer, "Lambda")
>
> Sigma <- sigma2*tcrossprod(Lambda)
>
> Gmer <- Sigma[1:4,1:4] #Extracting the VCV parameters from the
> block
> diagonalized Sigma
>
> full_mcmc <- MCMCglmm(RT ~ AgeGroup*cue*cong, random = ~us(1 +
> cue +
> cong):PID, thin = 10, nitt = 20000, data = vsdataset, prior =
> list(G =
> list(G1 = list(V = diag(diag(Gmer)), nu = 5))))
>
> Note: I used V = diag(diag(Gmer)) due to the fact that
> Sigma/Gmer was not
> positive definite, which MCMCglmm would not accept.
>
>
> Quick explanation of model terms:
>
> RT --> response time in msec, very positively skewed even
> after outlier
> removal, inverse transform seems to center it
>
> AgeGroup --> 2 level factorial var (Old and Young,
> between-subjects)
>
> cue --> 3 level factorial var (within-subjects)
>
> cong --> 2 level factorial var (within-subjects)
>
> PID --> participant ID number
>
>
> I want to specify this model with an inverse/reciprocal link
> function
> (Gaussian family). However, I can't figure out how to specify
> the link
> function. In the help section for the MCMCglmm function, they
> mention a
> "linking.function" for the random effects terms, but it
> doesn't seem to
> have anything to with specifying a link function for the
> response variable.
> According to the course notes from the MCMCglmm package,
> "there are many
> different types of distribution and link functions and those
> supported by
> MCMCglmm can be found in Table 7.1." However, Table 7.1 seems
> to just list
> the families and their PDFs, there's no column listing
> "supported" link
> functions.
>
> So, how do you specify a link function using MCMCglmm? If you
> can't
> directly specify a link function, is there something else I
> need to do such
> as specifying the prior a certain way, or is it valid to just
> specify the
> model as having a gaussian response and leave the mode as I've
> specified
> it? After plotting the model, I noticed that several of the
> parameter
> distributions were extremely skewed (some left, some right).
>
>
> As a side note, I originally tried two alternatives:
>
> 1) using lmer with an inverse transform
>
> 2) using glmer with family = gaussian(link = "inverse") and
> family =
> inverse.gaussian(link = "identity")
>
>
> #1 seems problematic due to the fact that I need to convert
> the response
> variable units back to the original units, which not only
> flips any
> confidence intervals but also makes them uneven. I'm not sure
> if converting
> these CI's is even appropriate as they were computed with
> different
> units/distribution. I also don't know of any way to validly
> convert the
> standard error back since that is certainly not valid once I
> back-transform.
>
> #2 gave me some issues: first, I had to scale down RT by a
> factor of 1000
> (from ms to s) when using gaussian(link = "inverse") otherwise
> I would get
> an error about the downdated VtV not being positive definite.
> But after
> dividing RT by 1000, it was able to continue, but the model
> did not fully
> converge (I think the max abs gradient was approximately .02).
> I decided to
> rerun the model after changing the contrasts on my variables
> from the
> default dummy coding to effect coding (using contr.sum). The
> same thing
> happened, except this time the max gradient was a little
> higher (about
> .0375) and in addition, I got the "model is nearly
> unidentifiable" warning
> due to a large eigenvalue. When I ran the model with
> inverse.gaussian(link
> = "identity"), it worked without scaling down RT by 1000 but I
> a bunch of
> optimizer warnings so I scaled it down and this time it wasn't
> able to
> converge because the max abs gradient value was about .0247.
>
>
> Any help on this would be greatly appreciated!
>
>
> - Sam
>
> [[alternative HTML version deleted]]
>
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