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

Local adaptation and genetic effects on fitness: Calculations for exponential family models with random effects