[R-sig-ME] Specifying a model with a link function in MCMCglmm
Jarrod Hadfield
j.hadfield at ed.ac.uk
Fri Feb 19 09:09:03 CET 2016
Hi Sam,
Sorry, I should have said nu=k+1 (not nu=k+2) probably makes the
marginal prior for the correlation close to uniform over the -1/1
interval. This was a guess given the properties of the inverse-Wishart
(see P72 of the CourseNotes) but it seems to hold .... below is a
function for simulating from a PX prior:
V2<-rpx(10000, V=diag(2), nu=2, alpha.mu=rep(0,2), alpha.V=diag(2)*1000)
V3<-rpx(10000, V=diag(2), nu=2+1, alpha.mu=rep(0,2), alpha.V=diag(2)*1000)
par(mfrow=c(1,2))
hist(V2[,2]/sqrt(V2[,1]*V2[,4]))
hist(V3[,2]/sqrt(V3[,1]*V3[,4]))
Note that in multiparameter models having marginal priors that are flat
does NOT mean that they will have no influence on the marginal posterior.
Some simulation/analytic work on the multivariate PX prior would be a
nice thing for someone to do!
Cheers,
Jarrod
rpx<-function(n=1, V, nu, alpha.mu, alpha.V){
k<-nrow(V)
alpha<-MASS::mvrnorm(n, alpha.mu, alpha.V)
Valpha<-MCMCglmm::rIW(V, nu=nu, n=n)
Vp<-sapply(1:n,
FUN=function(i){diag(alpha[i,])%*%matrix(Valpha[i,],k,k)%*%diag(alpha[i,])})
if(k==1){
return(Vp)
}else{
return(t(Vp))
}
}
On 19/02/2016 07:27, Jarrod Hadfield wrote:
> 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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>>
>>
>>
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>> Scotland, with registration number SC005336.
>>
>>
>
>
>
> The University of Edinburgh is a charitable body, registered in
> Scotland, with registration number SC005336.
>
>
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