Dear Doug, et al.:
What would you recommend for analyzing a longitudinal abundance
survey of 22 species, when the species were not selected at random? A
prominent scientist tried to tell me that mixed-effects modeling is
inappropriate in that case because the species were selected
purposefully not at random.
My response is that even in that case, one should still use
mixed-effects modeling, because it will tend to produce more appropriate
estimates for the deviations of individual species from the average of
all species -- potentially much lower variance with slight bias -- than
naive ordinary least squares. The estimated variance components will
not represent the between-species variance for the actual population of
all hypothetical species of the particular type, but will represent the
between-species variability in a hypothetical population from which the
selected species might be considered a random sample.
Best Wishes,
Spencer Graves
p.s. I appreciate very much Doug's comment on this. I thought about
adding something like that to my reply but didn't feel I could afford
the time then.
Douglas Bates wrote:
On Wed, May 13, 2009 at 5:21 PM, <avraham.ad...@guycarp.com> wrote:
Hello.
I am trying to optimize a set of parameters using /optim/ in which the
actual function to be minimized contains matrix multiplication and is of
the form:
SUM ((A%*%X - B)^2)
where A is a matrix and X and B are vectors, with X as parameter vector.
As Spencer Graves pointed out, what you are describing here is a
linear least squares problem, which has a direct (i.e. non-iterative)
solution. A comparison of the speed of various ways of solving such a
system is given in one of the vignettes in the Matrix package.
This has worked well so far. Recently, I was given a data set A of size
360440 x 1173, which could not be handled as a normal matrix. I brought it
into 'R' as a sparse matrix (dgCMatrix - using sparseMatrix from the Matrix
package), and the formulæ and gradient work, but /optim/ returns an error
of the form "no method for coercing this S4 class to a vector".
If you just want the least squares solution X then
X <- solve(crossprod(A), crossprod(A, B))
will likely be the fastest method where A is the sparse matrix.
I do feel obligated to point out that the least squares solution for
such large systems is rarely a sensible solution to the underlying
problem. If you have over 1000 columns in A and it is very sparse
then likely at least parts of A are based on indicator columns for a
categorical variable. In such situations a model with random effects
for the category is often preferable to the fixed-effects model you
are fitting.
After briefly looking into methods and classes, I realize I am in way over
my head. Is there any way I could use /optim/ or another optimization
algorithm, on sparse matrices?
Thank you very much,
--Avraham Adler
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