[R] Mixed-effects models / heterogeneous covariances

[Douglas Bates]

On 1/31/06, Lutz Ph. Breitling <lutz.breitling at gmail.com> wrote:
> Dear R-list,
>
> maybe someone can help me with the following mixed-effects models
> problem, as I am unable to figure it out with the 'nlme-bible'.
>
> I would like to fit (in R, obviously) a so-called animal model (google
> e. g. "Heritability and genetic constraints of life-history" by Pettay
> et al.) to estimate the variance component that is due to genetic
> effects. The covariances of the genetic random effects between
> observations are given by the different degrees of relatedness between
> the individuals examined. (I find it difficult to explain, but Pettay
> et al. describe it nicely in their methods section...)
>
> Is there any straight-forward way to fit such a model with R? I first
> thought I could handle it somehow with nlme's correlation structures,
> but these within-group structures are quite a different thing, right?

Sorry to say, yes they are quite a different thing.

I am aware of  models like the animal model and the sire model in
animal breeding.  A student in our Animal Sciences Department, Ana
Inés Vázquez Saravia, is working with me on developing extensions to
the lmer function to handle such models.  The actual calculations are
not extraordinarily difficult - the difficulty is in deciding how to
specify the model and in massaging the data to convert the model
specification to model matrices.

The model specification for an lmer model assumes that each predicted
response is affected by one and only one random effects vector
associated with each of the grouping factors.  That is, the random
effects have only an instantaneous effect and there is no "carry-over"
of random effects from other levels of the grouping factor.  This is
not the case for the animal model or for the sire model.  A given
predicted response is affects by the random effects for each of the
ancestors of the animal on which the observation is made.  The "no
carry-over" assumption is also violated in longitudinal "value-added"
models for student achievement where the effect of a teacher in a
given year can carry over to subsequent years. J.R. Lockwood and
Harold Doran are very interested in these models.

All of these are important practical models but, as I said, it is
tricky to decide how to specify the model in these cases.