Dear list,

I had a basic question about how standard errors are calculated for fixed
effects when using the Laplace approximation in glmer. I am looking at the
implementation details vignette:
http://cran.r-project.org/web/packages/lme4/vignettes/Implementation.pdf for
lme4. If I look at equation 40, the log likelihood depends on beta - the
fixed effects coefficients and theta - the covariance parameters for random
effects. Since Laplace method is doing approximate ML, one could use the
standard ML recipe and compute the negative Hessian of loglik(beta,theta |
y) w.r.t beta and theta, invert this and then extract the portion relevant
to the fixed effects to get the covariance of estimates betaHat. My
questions are:

1. Does glmer compute a joint Hessian for (beta,theta) as described above?

2. Another approach could be to regard thetaHat to be equal to the true
theta and use the Hessian w.r.t beta only for computing covariance of
betaHat. Does glmer follow this approach?

3. Are the required derivatives for doing Hessian calculations computed
numerically or is there an easy to understand expression for GLMMs that
reduces to the expression for LMMs?

4. Are any of these calculations fundamentally different when using
Gauss-Hermite quadrature instead of the Laplace method?

Sorry if these questions are very basic, but I am new to mixed effects
models and am not very familiar with the literature.

Thanks in advance,

Hans Tissot

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