An individual tree model with additive direct and competition effects is introduced to account for competitive effects in forest genetics evaluation. The mixed linear model includes fixed effects as well as direct and competition breeding values plus permanent environmental effects. Competition effects, either additive or environmental, are identified in the phenotype of a competitor tree by means of ‘intensity of competition’ elements (IC), which are non-zero elements of the incidence matrix of the additive competition effects. The ICs are inverse function of the distance and the number of competing individuals, either row-column wise or diagonally. The ICs allow standardization of the variance of competition effects in the phenotypic variance of any individual tree, so that the model accounts for unequal number of neighbors. Expressions are obtained for the bias in estimating additive variance using the covariance between half-sibs, when ignoring competition effects for row-plot designs and for single-tree plot designs. A data set of loblolly pines on growth at breast height is used to estimate the additive variances of direct and competition effects, the covariance between both effects, and the variance of permanent environmental effects using a Bayesian method via Gibbs sampling and Restricted Maximum Likelihood procedures (REML) via the Expectation- Maximization (EM) algorithm. No problem of convergence was detected with the model and ICs used when compared to what has been reported in the animal breeding literature for such models. Posterior means (standard error) of the estimated parameters were σ̂2Ad = 12.553 (1.447), σ̂2Ac = 1.259 (0.259), σ̂AdAc = -3.126 (0.492), σ̂2 p = 1.186 (0.289), and σ̂2e = 5.819 (1.07). Leaving permanent environmental competition effects out of the model may bias the predictions of direct breeding values. Results suggest that selection for increasing direct growth while keeping a low level of competition is feasible.
If the inline PDF is not rendering correctly, you can download the PDF file here.
ARANGO, J., I. MISZTAL, S. TSURUTA, M. CULBERTSON and W. HERRING (2005): Estimation of variance components including competitive effects of Large White growing gilts. J. Anim. Sci. 83: 1241-1246.
ARORA, V. and P. LAHIRI (1997): On the superiority of the Bayesian method over the BLUP in small area estimation problems. Statistica Sinica 7: 1053-1063.
BORRALHO, N. M. G. (1995): The impact of individual tree mixed models (BLUP) in tree breeding strategies. In Proceedings of the CRC-IUFRO Conference: Eucalypts plantations: Improving fibre yield and quality. 12-24 February. 1995, Hobart, Tasmania, Australia. Edited by B. M. POTTS, N. M. G. BORRALHO, J. B. REID, R. N.
CANTET, R. J. C., D. GIANOLA, I. MISZTAL, R. L. FERNANDO (1993): Estimates of dispersion parameters and of genetic and environmental trends for weaning weight in Angus cattle using a maternal animal model with genetic grouping. Livest. Prod. Sci. 34: 203-212.
CAPPA, E. P. and R. J. C. CANTET (2006): Bayesian inference for normal multiple-trait individual-tree models with missing records via full conjugate Gibbs. Can. J. For. Res. 36: 1276-1285.
CASELLA, G. and E. I. GEORGE (1992): Explaining the Gibbs Sampler. Am. Stat. 46: 167-174.
DEMPSTER, A. P., N. M. LAIRD and D. B. RUBIN (1977): Maximum likelihood from incomplete data via EM algorithm. Journal of the Royal Statistics Society 39: 1-38.
DUANGJINDA, M., I. MISZTAL, J. K. BERTRAND, S. TSURUTA (2001): The empirical bias of estimates by restricted maximum likelihood, Bayesian method, and method R under selection for additive, maternal, and dominance models. J. Anim Sci. 79: 2991-2996.
FOSTER, G. S., R. J. ROUSSEAU and W. L. NANCE (1998): Eastern cottonwood clonal mixing study: intergenotypiccompetition effects. Forest Ecology and Management. 112: 9-22.
GELMAN, A., J. B. CARLIN, H. S. STERN and D. B. RUBIN (1995): Bayesian data analysis. Chapman and Hall. New York, USA.
GRIFFING, B. (1967): Selection in reference to biological groups. I. Individual and group selection applied to populations of unordered groups. Aust. J. Biol. Sci. 20: 127-139.
GRIFFING, B. (1968a): Selection in reference to biological groups. II. Consequences of selection in groups of one size when evaluated in groups of a different size. Aust. J. Biol. Sci. 21: 1163-1170.
GRIFFING, B. (1968b): Selection in reference to biological groups. III. Generalized results of individual and group selection in terms of parent-offspring covariances. Aust. J. Biol. Sci. 21: 1171-1178.
GWAZE, D. P. and J. A. WOOLLIAMS (2001): Making decisions about the optimal selection environment using Gibbs sampling. Theor. Appl. Genet. 103: 63-69.
HARVILLE, D. A. (1974): Bayesian inference for variance components using only error contrasts. Biometrika 61: 383-384.
HENDERSON, C. R. (1977): Best linear unbiased prediction of breeding values not in the model for records. J. Dairy Sci. 60: 783-787.
HINSON, K. and W. D. HANSON (1962): Competition studies in soybeans. Crop. Sci. 2: 117-123.
HOBERT, J. P. and G. CASELLA (1996): The effects of improper priors on Gibbs sampling in hierarchical linear models, J. Amer. Statist. 91: 1461-1473.
JENSEN, J., C. S. WANG, D. A. SORENSEN and D. GIANOLA (1994): Bayesian inference on variance and covariance components for traits influenced by maternal and direct genetic effects, using the Gibbs sampler. Acta Agric. Scand. 44: 193-201.
KASS, R. E., B. P. CARLIN, A. GELMAN and R. M. NEAL (1998): Markov chain Monte Carlo in practice: a roundtable discussion. Amer. Stat. 52: 93-100.
KEMPTHORNE, O. (1969): An Introduction to Genetic Statistics. Iowa State Univ. Press, Ames Iowa.
LYNCH, M. and B. WALSH (1998): Genetics and Analysis of Quantitative Traits. Sinauer Associates. Sunderland M/A.
MAGNUSSEN, S. (1989): Effects and adjustments of competition bias in progeny trials with single-tree plots. Forest Science 35(2): 532-547.
MAGNUSSEN, S. (1993) Bias in genetic variance estimates due to spatial autocorrelation. Theorical and Applied Genetics 86: 349-355.
MUIR, W. M. and A. SCHINCKEL (2002): Incorporation of competitive effects in breeding programs to improve productivity and animal well being. CD-ROM Communication No. 14-07 in Proc. 7th World Cong. Genet. Appl. Livest. Prod., Montpellier France.
MUIR, W. M. (2005): Incorporation of competitive effects in forest tree or animal breeding programs. Genetics 170: 1247-1259.
PATTERSON, H. D. and R. THOMPSON (1971): Recovery of inter-block information when block sizes are unequal. Biometrika 58: 545-554.
RADTKE, P. J., J. A. WESTFALL and H. E. BURKHART (2003): Conditioning a distance-dependent competition index to indicate the onset of intertree competition. Forest Ecology and Management 175: 17-30.
SILVERMAN, B. W. (1986): Density estimation for statistics and data analysis. Chapman and Hall. London.
SMITH, B. J. (2003): Bayesian Output Analysis Program (BOA) version 1.0 user’s manual. Available from http://www.public-health.uiowa.edu/boa/Home.html.
SORENSEN, D. and D. GIANOLA (2002): Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics. Springer-Verlag, New York.
SORIA, F., F. BASURCO, G. TOVAL, L. SILIÓ, M. C. RODRIGUEZ and M. TORO (1998): An application of Bayesian techniques to the genetic evaluation of growth traits in Eucalyptus globulus. Can. J. For. Res. 28: 1286-1294.
VAN TASSELL, C. P., G. CASELLA and E. J. POLLAK (1995): Effects of Selection on Estimates of Variance Components Using Gibbs Sampling and Restricted Maximum Likelihood. J. Dairy Sci. 78: 678-692.
VAN VLECK, L. D. and J. P. CASSADY (2005): Unexpected estimates of variance components with a true model containing genetic competition effects. J. Anim. Sci. 83: 68-74.
WILLHAM, R. L. (1963): The covariance between relatives for characters composed of components contributed by related individuals. Biometrics 19: 18-27.
WOLF, J. B. (2003): Genetic architecture and evolutionary constraint when the environment contains genes. Proc. Nat. Academy of Sciences 100: 4655-4660.
WRIGHT, A. J. (1986): Individual and group selection with competition. Theoretical and Applied Genetics 72: 256-263.
ZENG, W., S. K. GHOSH and B. LI (2004): A Blocking Gibbs Sampling Method to Detect Major Genes Affecting a Quantitative Trait for Diallel Mating Design. Genetical Research 84: 1-12.