# [R-sig-ME] Another MCMCglmm question - fixing correlations

Jarrod Hadfield j.hadfield at ed.ac.uk
Sat Dec 4 22:11:50 CET 2010

Hi Malcolm,

I meant replace the sir function in the R directory and rebuild +
reinstall MCMCglmm. However, I later realised that for your problem
there is a simpler way:

N <- 50 # set sample size
W <- matrix(rbinom(N^2, 1, 0.25), N, N)*upper.tri(matrix(1, N, N))
W <- W + t(W) # symmetrical connectivity matrix
rho <- 0.2 # set autocorrelation coefficient
L <- diag(N) - rho*W  # my L is your M^{-1}
X1 <- runif(N, min=-5, max=5)
Xbe <- X1+rnorm(N)
y <- solve(L, Xbe) # generate lagged y
dat <- data.frame(X1=X1, y=y, W=W)

prior1 = list(R=list(V=1, nu=0.002))
MC2 <- MCMCglmm(y ~ X1+sir(~W, ~units), data=dat, prior=prior1, verbose=F)

This works because ~units sets up an identity matrix and W%*%t(I) = W.

However, I could not get sensible results from MCMCglmm with rho=0.3
and wasn't sure why (rho=0.03 for example gives sense). The problem is
associated with a change in the sign of the determinant of L (or M)
(i.e the Jacobian), which is perhaps not that surprising since if W=I
then rho=0.3 and rho = 1.7 should be indiscriminable even with a fixed
residual variance:

y = Xbe/(1-0.3) = -Xbe/(1-1.7)

I would have to think about this harder than I have to offer a
solution (this is why the sir models are undocumented!), but perhaps
you have some insight?

Also, I don't think MC3 results are meaningless, the estimate for the
sir parameter overlaps zero as expected.

Cheers,

Jarrod

Quoting Malcolm Fairbrother <m.fairbrother at bristol.ac.uk>:

> Hi Jarrod,
>
> Thanks very much for the suggestion. However, in trying what you
> said, I'm not getting very far (see code below). I redefine the
> function "sir", and then re-install the MCMCglmm package. But the
> MCMCglmm function still seems to use the old "sir" function, not the
> new one, with the result that when I try calling sir in the middle
> of a call to MCMCglmm, I get an error message telling me that my
> call to sir is problematic ("Error in sir(W) : formula not passed to
> formula1 in sir"). Can you please clarify what I'm doing wrong?
>
> Thanks again for any assistance.
> - Malcolm
>
>
> sir <- function(W) {W}
> install.packages("MCMCglmm")
> library(MCMCglmm)
>
> N <- 50 # set sample size
> W <- matrix(rbinom(N^2, 1, 0.25), N, N)*upper.tri(matrix(1, N, N))
> W <- W + t(W) # symmetrical connectivity matrix
> rho <- 0.2 # set autocorrelation coefficient
> M <- solve(diag(N) - rho*W)
> X1 <- runif(N, min=-5, max=5)
> Xbe <- X1+rnorm(N)
> y <- M %*% Xbe # generate lagged y
> dat <- data.frame(X1=X1, y=y)
>
> prior1 = list(R=list(V=1, nu=0.002), G=list(G1=list(V=1, nu=0.002)))
> MC1 <- MCMCglmm(y ~ X1, data=dat, prior=prior1, verbose=F)
> # works OK, compare with summary(lm(dat$y~dat$X1))
>
> identical(sir(W), W) # TRUE
> MC2 <- MCMCglmm(y ~ X1 + sir(W), data=dat, prior=prior1, verbose=F)
> # doesn't work
>
> fac2 <- fac1 < -factor(sample(letters, N, TRUE), levels=letters)
> MC3 <- MCMCglmm(y ~ X1 + sir(~fac1, ~fac2), data=dat, prior=prior1,
> verbose=F)
> # works--results are meaningless, but doesn't return any error
>
>
>
>
>
> On 2 Dec 2010, at 11:20, Jarrod Hadfield wrote:
>
>> Hi Malcolm,
>>
>> The suggestion of using SIR models for time series was a throw away
>> remark, but I can't see anything technically incorrect with it. For
>> example the model y[t] = \lambda*y[t-1]+..... can be treated as a
>> SIR model, although there are certainly better ways of fitting it.
>> I'm not familiar with spatial models and connectivity matrices but
>> if W[i,j] is a way of specifying y[i] = \lambda*y[j] + ... then it
>> is possible to fit it as a SIR model. You could modify the sir code
>> to do what you would like (and then reinstall MCMCglmm) , for
>> example:
>>
>> sir<-function(W){W}
>>
>> and then place W in the data.frame.
>>
>> MCMCglmm will associate design matrices with recursive/simultaneous
>> structures if the design matrix is returned from a function called
>> "sir".
>>
>> Cheers,
>>
>> Jarrod
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> On 1 Dec 2010, at 20:10, Malcolm Fairbrother wrote:
>>
>>> Dear Jarrod,
>>>
>>> I'm very interested in this SIR feature of MCMCglmm, which I
>>> hadn't been aware of until your e-mail earlier today. Do I
>>> understand correctly that this should allow for feedback effects
>>> between units, reflecting the fact that y(i) and y(not-i) affect
>>> each other? If so, this could be a useful way of fitting, for
>>> example, spatial autoregressive models, where some units may
>>> affect each other by virtue of being neighbours, and we would like
>>> to know the magnitude of these effects. (You mentioned time
>>> series, but some web searches haven't turned up much on MCMCglmm
>>> and time series. And time only flows in one direction, whereas
>>> units in space can have two-way effects.)
>>>
>>> However, in looking at the documentation and code, I've been
>>> struggling a bit with the "sir(~XX, ~XX)" part. Is there a way to
>>> specify a "connectivity" matrix directly, rather than using the
>>> "sir" function? Looking at the MCMCglmm code, as I understand it,
>>> "sir" is both a way of generating a connectivity matrix AND a way
>>> of telling MCMCglmm to treat the sir(~XX, ~XX) differently than
>>> other covariates.
>>>
>>> In the case of spatial autoregressive models, each element on the
>>> main diagonal of the connectivity matrix is 0, and rows are often
>>> standardised to sum to 1. Such a connectivity matrix might, for
>>> example, look like:
>>>
>>> N <- 100
>>> W <- matrix(rbinom(N^2, 1, 0.1), N, N)*upper.tri(matrix(1, N, N))
>>> # chance of being neighbours is 0.1
>>> W <- W + t(W) # matrix is symmetrical, with zeros on the main diagonal
>>>
>>> But I'm having a hard time figuring out how to use "sir" to
>>> specify such a matrix in the middle of a call to MCMCglmm.
>>>
>>> If you see what I mean, can you offer any suggestions? I might
>>> take a stab at modifying MCMCglmm to make this possible, but
>>> before I do that (and I'm not certain I'll have the know-how) I
>>> wanted to check whether you had any ideas.
>>>
>>> Many thanks for any assistance or clarification you can provide.
>>> - Malcolm
>>>
>>>
>>>
>>>> Message: 5
>>>> Date: Wed, 1 Dec 2010 10:28:43 +0000
>>>> From: Jarrod Hadfield <j.hadfield at ed.ac.uk>
>>>> To: Szymek Drobniak <geralttee at gmail.com>
>>>> Cc: r-sig-mixed-models at r-project.org
>>>> Subject: Re: [R-sig-ME] Another MCMCglmm question - fixing
>>>> 	correlations
>>>> Message-ID: <0AF131D6-557C-4324-A65A-A175E41DBA8F at ed.ac.uk>
>>>> Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes
>>>>
>>>> Hi,
>>>>
>>>> There is a way, but its quite involved and you may find it easier to
>>>> fit this type of model in something like ASReml if you can get a
>>>> license. If not ......
>>>>
>>>> I have implemented simultaneous-recursive (SIR) mixed models  in
>>>> MCMCglmm, although currently they are not well tested or documented
>>>> (See Chapter 9 of the CourseNotes) and cannot be used with non-
>>>> Gaussian data or certain patterns of missing data. Nevertheless, in
>>>> this example the existing code should work fine.  A simple SIR model
>>>> has the form:
>>>>
>>>> y_{i} = mu + \lambda*y_{j} + u_{i} + e_{i}
>>>>
>>>> where mu is the intercept, u is a random effect (e.g. breeding value
>>>> in your case) and e is a residual. The new part of the model is the
>>>> structural parameter \lambda multiplied by y_{j}, the jth element of
>>>> the response vector, which can be useful for modelling the effects of
>>>> behavioural interactions or time series etc.  However, placing y's on
>>>> the LHS and RHS complicates the likelihood, but MCMCglmm deals with
>>>> this.
>>>>
>>>> If we set i=j then the above equation can be rearranged:
>>>>
>>>> y_{i}  - \lambda*y_{i} = mu + u_{i} + e_{i}
>>>> y_{i}(1  - \lambda) = mu + u_{i} + e_{i}
>>>> y_{i} = (mu + u_{i} + e_{i})/(1  - \lambda)
>>>>
>>>> from this it can be seen that mu, u, e, and \lambda cannot be uniquely
>>>> estimated without restrictions.  To illustrate we'll simulate some data:
>>>>
>>>> fac<-gl(25,4)
>>>> y<-rnorm(25)[fac]+rnorm(100, -1, sqrt(2))
>>>>
>>>> # mu = -1, VAR(u)=1, VAR(e) = 2, \lambda = 0
>>>>
>>>> dat<-data.frame(y=y, fac=fac)
>>>>
>>>> prior1 = list(R=list(V=1, nu=0.002), G=list(G1=list(V=1, nu=0.002)))
>>>>
>>>> m1<-MCMCglmm(y~1, random=~fac,  data=dat, prior=prior1)
>>>>
>>>> # fitting a non-SIR model gives estimates consistent with the true
>>>> values, which you can verify through summary(m1). Now for the SIR
>>>> model. Because we know all parameters cannot be uniquely estimated
>>>> I've set the residual variance to one arbitrarily.
>>>>
>>>> prior2 = list(R=list(V=1, fix=1), G=list(G1=list(V=1, nu=0.002)))
>>>>
>>>> m2<-MCMCglmm(y~1+sir(~units, ~units), random=~fac,  data=dat,
>>>> prior=prior2)   # sir(~units, ~units) sets i=j.
>>>>
>>>> # The estimates do not look consistent with the real values.  However,
>>>> we can rescale them by 1-\lambda and the estimates  are close to being
>>>> identical up to Monte Carlo error (the prior will have slightly
>>>> different effects)
>>>>
>>>> HPDinterval(m1$VCV[,1]) >>>> HPDinterval(m2$VCV[,1]/(1-m2$Lambda)^2) >>>> >>>> HPDinterval(m1$VCV[,2])
>>>> HPDinterval(m2$VCV[,2]/(1-m2$Lambda)^2)
>>>>
>>>> HPDinterval(m1$Sol) >>>> HPDinterval(m2$Sol/(1-m2\$Lambda))
>>>
>>>
>>> 	[[alternative HTML version deleted]]
>>>
>>> _______________________________________________
>>> R-sig-mixed-models at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>>
>>
>>
>> --
>> The University of Edinburgh is a charitable body, registered in
>> Scotland, with registration number SC005336.
>>
>
>
>

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