  Random effects are implemented for aster models using two approximations
taken from Breslow and Clayton [J. Amer. Statist. Assoc. 88 (1993) 9-25].
Random effects are analytically integrated out of the Laplace approximation to
the complete data log likelihood, giving a closed-form expression for an
approximate missing data log likelihood. Third and higher derivatives of the
complete data log likelihood with respect to the random effects are ignored,
giving a closed-form expression for second derivatives of the approximate
missing data log likelihood, hence approximate observed Fisher information.
This method is applicable to any exponential family random effects model. It is
implemented in the CRAN package aster (R Core Team [R: A Language and
Environment for Statistical Computing (2012) R Foundation for Statistical
Computing], Geyer [R package aster (2012)
http://cran.r-project.org/package=aster]). Applications are analyses of local
adaptation in the invasive California wild radish (Raphanus sativus) and the
slender wild oat (Avena barbata) and of additive genetic variance for fitness
in the partridge pea (Chamaecrista fasciculata).
