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lme4 (your singularity check with getME())

Ben Bolker
On 14-10-16 09:37 AM, Farrar, David wrote:
[I'm taking the liberty of cc'ing the r-sig-mixed-models list, as this
seems to me be a question of general interest]
For this check, we haven't (yet) even tried to test for convergence
in the singular case -- we're just checking whether we *are* in the
singular case or not.  The checks we have in place aren't necessarily
sensible in the singular case, although I've found surprisingly few
examples so far where the models are singular and we would deem them to
have converged properly but where the convergence tests fail.  In other
words, in most singular cases the surface appears to be flat and
(half-)convex at the optimum, even though I don't know of any
theoretical reason it would have to be.

  It's on our (long) to-do list to try to work out what the proper
convergence tests would be in a singular case.  Since our singularities
are always simple (i.e. we've hit x_i=0 for one or more parameters
independently/box constraints), we might get away with something as
simple as dropping the singular dimensions out of the gradient and
Hessian, rather than doing something more complicated with Lagrange
multipliers.

  Last bit of information: the reason we can get away with a simple
criterion for singularity (i.e. our variance-covariance matrix is
positive semidefinite but not positive definite, or equivalently some
eigenvalues are exactly zero), which isn't always easy, is that we
parameterize the variance-covariance matrix in terms of its Cholesky
factors; this means we have a singular variance-covariance matrix if and
only if the diagonal elements are equal to zero (we don't let them go
negative).  These parameterizations are nicely explained in Pinheiro and
Bates 1996 _Stat. Comput._  http://dx.doi.org/10.1007/BF00140873 ...