[R-sig-ME] MCMCglmm: Within- versus between-individual covariances

[Ned Dochtermann]

I tried to find an answer to this problem but either failed to select the
proper search terms or this topic hasn't been discussed. I apologize if the
former. I've also consulted the "class notes" for MCMCglmm but that hasn't
clarified the issue for me.

I am currently working on the analysis of some repeated measures behavioural
data and am attempting to estimate variances and covariances among
behaviours. Specifically we're looking to estimate the between-individual
(co)variance matrix and I was using a multi-response model and the Poisson
family in MCMCglmm to do so.

I had thought that the posterior modes for the (co)variance matrix (i.e.
posterior.modes(model$VCV)) was producing this on the latent scale, with the
".ID" terms being the between individual (co)variances and the ".unit" terms
being the within/residual indvidual (co)variances. This conclusion was based
on Jarrod Hadfield's appendix to the recent animal model paper published in
the Journal of Animal Ecology (2009). In the appendix discussing MCMCglmm
for repeated measures, repeatability is calculated using the .ID variance as
the between individual component and the .unit variance as the within
individual component.

The problem I've encountered is that due to the experimental design certain
aspects of the within individual covariance matrix should not be
estimatable, but estimates are nonetheless reported. This is, of course, due
to my misspecification of the model. 

To use an example provided by a colleague, consider a situation where three
distinct behaviours are measured and we're interested in their covariance.
Due to aspects of the experimental manipulation all the behaviours cannot be
measured on the same days. Thus the data might look something like:

ID	Day	Behav1	Behav2	Behav3
1	1	4		NA		10
1	2	3		NA		15
1	3	NA		5		NA
1	4	NA		4		NA
2	1	2		NA		12
2	2	1		NA		18
2	3	NA		4		NA
2	4	NA		3		NA
...		
N	1	6		NA		8
N	2	5		NA		6
N	3	NA		8		NA
N	4	NA		7		NA
[[note, I know these fictitious data aren't necessarily Poisson distributed
but the actual data are]]

In this case between individual variances and covariances can be calculated
among all three behaviours and within individual covariances should be
calculated between Behav1 and Behav3 but not for Behav2 with either 1 or 3
due to separation.

In attempting the initial analyses I specified the random statement using an
unstructured matrix:
>random=~us(trait):ID
(I'm pretty sure that's what I want and what produces the between individual
covariance matrix)

I also kept the residual covariance matrix as the default unstructured:
>rcov=~us(trait):units

However, if the within-individual/residual covariances really shouldn't be
calculated between Behav2 and the other two responses, rcov should actually
look something like:

v.B1		0		cov.B1*3
0		v.B2		0
cov.B1*3	0		v.B3

For this example "~us" is thus estimating two extra parameters that
shouldn't be estimated (for the actual dataset these elements have
credibility intervals overlapping 0).

Clearly idh wouldn't be appropriate for this either but none of the options
listed in Table 3.1 of the Class Notes for MCMCglmm seem correct. It also
doesn't look like I can specify the matrix structure directly (which is how
my colleague dealt with this concern using ASreml and which I know SAS
allows for regular mixed models).

Is there any advice on how best to deal with this issue? I'm at a loss.

Thanks a lot,
Ned Dochtermann


--
Ned Dochtermann
Department of Biology
University of Nevada, Reno

ned.dochtermann at gmail.com
http://wolfweb.unr.edu/homepage/mpeacock/Dochter/
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