[R-sig-ME] Connectedness of rows/columns of a matrix

[Kevin Wright]

Suppose I have data with two factors A and B that each have two
levels.  If all combinations of the factors are observed, then an
incidence matrix showing combinations that are observed would look
like this:
   b1 b2
a1  1  1
a2  1  1
If I then fit a (mixed) model like y ~ A + B, the effects are
generally estimable.

If, however, some combinations are not observed, so that the incidence
matrix looks like this:
   b1 b2
a1  1  0
a2  0  1
then the A and B effects are aliased and non-estimable.

This is an extremely simple example, but sometimes when I am fitting
models which fail to converge, I dig into the data and find there is
not a connected path through the rows and columns of the incidence
matrix and (I assume) this is the cause of the non-estimability.

Two questions:
1. Is there a method to detect if the rows/columns of a matrix are
connected by at least one path through the matrix?
2. More generally, is there a measure of "connectedness" of the rows/columns?
3. Even more generally, what techniques do people use to diagnose
non-estimability?

Hope this is understandable and interesting.

Kevin Wright


-- 
Kevin Wright