The Annals of Applied Statistics

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

Charles J. Geyer, Caroline E. Ridley, Robert G. Latta, Julie R. Etterson, and Ruth G. Shaw

Full-text: Open access


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)]). 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).

Article information

Ann. Appl. Stat., Volume 7, Number 3 (2013), 1778-1795.

First available in Project Euclid: 3 October 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Additive genetic variance approximate maximum likelihood breeding value Darwinian fitness exponential family latent variable life history analysis local adaptation missing data variance components Avena barbata Chamaecrista fasciculata Raphanus sativus


Geyer, Charles J.; Ridley, Caroline E.; Latta, Robert G.; Etterson, Julie R.; Shaw, Ruth G. Local adaptation and genetic effects on fitness: Calculations for exponential family models with random effects. Ann. Appl. Stat. 7 (2013), no. 3, 1778--1795. doi:10.1214/13-AOAS653.

Export citation


  • Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
  • Barndorff-Nielsen, O. (1978). Information and Exponential Families. Wiley, Chichester.
  • Booth, J. and Hobert, J. P. (1999). Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 265–285.
  • Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. J. Amer. Statist. Assoc. 88 9–25.
  • Crouch, E. A. C. and Spiegelman, D. (1990). The evaluation of integrals of the form $\int^{+\infty}_{-\infty}f(t)\exp(-t^{2})\,dt$: Application to logistic-normal models. J. Amer. Statist. Assoc. 85 464–469.
  • Etterson, J. R. (2004a). Evolutionary potential of Chamaecrista fasciculata in relation to climate change. I. Clinal patterns of selection along an environmental gradient in the great plains. Evolution 58 1446–1458.
  • Etterson, J. R. (2004b). Evolutionary potential of Chamaecrista fasciculata in relation to climate change. II. Genetic architecture of three populations reciprocally planted along an environmental gradient in the great plains. Evolution 58 1459–1471.
  • Etterson, J. R. and Shaw, R. G. (2001). Constraint to adaptive evolution in response to global warming. Science 294 151–154.
  • Fisher, R. A. (1918). The correlation between relatives on the supposition of Mendelian inheritance. Trans. R. Soc. Edinburgh 52 399–433.
  • Fletcher, R. (1987). Practical Methods of Optimization, 2nd ed. Wiley, Chichester.
  • Geyer, C. J. (1994). On the convergence of Monte Carlo maximum likelihood calculations. J. R. Stat. Soc. Ser. B Stat. Methodol. 56 261–274.
  • Geyer, C. J. (2009). Likelihood inference in exponential families and directions of recession. Electron. J. Stat. 3 259–289.
  • Geyer, C. J. (2012). R package aster, version 0.8-19. Available at
  • Geyer, C. J. (2013). Asymptotics of maximum likelihood without the LLN or CLT or sample size going to infinity. In Multivariate Statistics in Modern Statistical Analysis: A Festschrift for Morris L. Eaton (G. Jones and X. Shen, eds.) 10 1–24. IMS, Hayward, CA.
  • Geyer, C. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. J. R. Stat. Soc. Ser. B Stat. Methodol. 54 657–699.
  • Geyer, C. J., Wagenius, S. and Shaw, R. G. (2007). Aster models for life history analysis. Biometrika 94 415–426.
  • Geyer, C. J., Ridley, C. E., Latta, R. G., Etterson, J. R. and Shaw, R. G. (2012). Aster models with random effects via penalized likelihood. Technical Report 692, Univ. Minnesota School of Statistics. Available at
  • Hummel, R. M., Hunter, D. R. and Handcock, M. S. (2012). Improving simulation-based algorithms for fitting ERGMs. J. Comput. Graph. Statist. 21 920–939.
  • Hunter, D. R., Handcock, M. S., Butts, C. T., Goodreau, S. M. and Morris, M. (2008). ergm: A Package to fit, simulate and diagnose exponential-family models for networks. J. Stat. Softw. 24 nihpa54860.
  • Latta, R. G. (2009). Testing for local adaptation in Avena barbata, a classic example of ecotypic divergence. Molecular Ecology 18 3781–3791.
  • Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics: Some Basic Concepts, 2nd ed. Springer, New York.
  • McCulloch, C. E. (2003). Generalized Linear Mixed Models. NSF-CBMS Regional Conference Series in Probability and Statistics 7. IMS, Beachwood, OH.
  • Nocedal, J. and Wright, S. J. (1999). Numerical Optimization. Springer, New York.
  • Okabayashi, S. and Geyer, C. J. (2011). Gradient-based search for maximum likelihood in exponential families. Electron. J. Stat. 6 123–147.
  • Penttinen, A. (1984). Modelling interations in spatial point patterns: Parameter estimation by the maximum likelihood method. Jyväskylä Studies in Computer Science, Economics and Statistics No. 7, Univ. Jyväskylä.
  • Ridley, C. E. and Ellstrand, N. C. (2010). Rapid evolution of morphology and adaptive life history in the invasive California wild radish (Raphanus sativus) and the implications for management. Evolutionary Applications 3 64–76.
  • Rockafellar, R. T. and Wets, R. J. B. (2004). Variational Analysis, 2nd corrected printing. Springer, Berlin.
  • Rutter, M. T., Roles, A., Conner, J. K., Shaw, R. G., Shaw, F. H., Schneeberger, K., Ossowski, S., Weigel, D. and Fenster, C. B. (2012). Fitness of Arabidopsis thaliana mutation accumulation lines whose spontaneous mutations are known. Evolution 66 2335–2339.
  • Shaw, F. H., Geyer, C. J. and Shaw, R. G. (2002). A comprehensive model of mutations affecting fitness and inferences for Arabidopsis thaliana. Evolution 56 453–463.
  • Shaw, F. H., Promislow, D. E. L., Tatar, M., Hughes, K. A. and Geyer, C. J. (1999). Towards reconciling inferences concerning genetic variation in senescence. Genetics 152 553–566.
  • Shaw, R. G., Geyer, C. J., Wagenius, S., Hangelbroek, H. H. and Etterson, J. R. (2008). Unifying life-history analyses for inference of fitness and population growth. Am. Nat. 172 E35–E47.
  • Stanton-Geddes, J., Shaw, R. G. and Tiffin, P. (2012). Interactions between soil habitat and geographic range location affect plant fitness. PLoS ONE 7 e36015.
  • Sung, Y. J. and Geyer, C. J. (2007). Monte Carlo likelihood inference for missing data models. Ann. Statist. 35 990–1011.
  • R Core Team (2012). R: A language and environment for statistical computing. R foundation for statistical computing, Vienna, Austria. Available at
  • Thompson, E. A. and Guo, S. W. (1991). Evaluation of likelihood ratios for complex genetic models. IMA J. Math. Appl. Med. Biol. 8 149–169.
  • Travisano, M. and Shaw, R. G. (2013). Lost in the map. Evolution 67 305–314.