Abstract
It is often assumed in animal breeding theory that models used for data analysis are “correct” with respect to functional form and distributional assumptions. However, a transformation may be needed to achieve this. An extension of the Box-Cox theory of transformations to univariate mixed linear models is presented. The discussion includes estimation of the transformation and of the required variance components, including computing algorithms. An analysis of fixed effects and breeding values after the transformation involves the following steps: (1) estimate ratios of variance components and the transformation parameter from their joint posterior distribution; (2) conditionally on these values, integrate out the residual variance (σ 2e ) from the joint posterior distribution of fixed, random effects and σ 2e , and (3) complete inferences using a multivariate-t distribution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc Series B 26:211–252
Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Addison-Wesley, Reading
Box GEP, Tidwell PW (1962) Transformation of the independent variables. Technometrics 4:531–550
Brotherstone S, Hill WG (1986) Heterogeneity of variance amongst herds for milk production. Anim Prod 42:297–303
Bulmer MG (1980) The mathematical theory of quantitative genetics. Clarendon Press, Oxford
Dahlquist G, Bjorck A (1974) Numerical methods. Prentice Hall, Englewood Cliffs
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Series B 39:1–38
Everett RW, Keown JF, Taylor JF (1982) The problem of heterogeneous within-herd error variances when identifying elite cows. J Dairy Sci 65 suppl 1 (Abstr):100
Fernando RL, Gianola D (1986) Optimal properties of the conditional mean as a selection criterion. Theor Appl Genet 72:822–825
Foulley XL, Im S, Gianola D, Hoschele Ina (1987) Empirical Bayes estimation of parameters for n polygenic binary traits. Genet Sel Evol 19:197–224
Gaudry MJI, Dagenais MG (1979) Heteroscedasticity and the use of the Box-Cox transformations. Econ Letters 2:225–229
Gianola D (1986) On selection criteria and estimation of parameters when the variance is heterogeneous. Theor Appl Genet 72:671–677
Gianola D, Fernando RL (1986) Bayesian methods in animal breeding theory. J Anim Sci 63:217–244
Gianola D, Foulley JL, Fernando RL (1986) Prediction of breeding values when variances are not known. In: Dickerson GE, Johnson RK (eds) Proc 3rd World Congr Genet Appl Livest Prod. Agric Commun, Univ Nebraska, Lincoln, Nebraska XII:356–370
Goffinet B (1983) Selection on selected records. Genet Sel Evol 15:91–97
Haldane JBS (1930) A mathematical theory of natural and artificial selection. III. Selection intensity as a function of a mortality rate. Proc Camb Phil Soc Biol Sci 27:131–136
Harville DA (1974) Bayesian inference for variance components using only error contrasts. Biometrika 61:383–385
Harville DA (1977) Maximum likelihood approaches to variance component estimation and related problems. J Am Stat Assoc 72:320–338
Harville DA (1985) Decomposition of prediction error. J Am Stat Assoc 80:132–138
Henderson CR (1973) Sire evaluation and genetic trends. In: Proc Anim Breed Genet Symp in Honor of Dr. J.L. Lush. ASAS and ADSA, Champaign, Illinois, pp 10–41
Henderson CR (1975) Best linear unbiased estimation and prediction under a selection model. Biometrics 31:423–447
Henderson CR (1984) Applications of linear models in animal breeding. Univ Guelph, Guelph
Henderson CR, Quaas RL (1976) Multiple trait evaluation using relatives’ records. J Anim Sci 43:1188–1197
Hill WG (1984) On selection among groups with heterogeneous variance. Anim Prod 39:473–477
Hill WG, Edwards MR, Ahmed MKA, Thompson R (1983) Heritability of milk yield and composition at different levels and variability of production. Anim Prod 36:59–68
Hoschele I, Gianola D, Foulley JL (1987) Estimation of variance components with quasi-continuous data using Bayesian methods. Z Tierz Züchtungsbiol 104:334–349
Judge GC, Griffiths WE, Hill RC, Lutkepol H, Lee TC (1985) The theory and practice of econometrics. Wiley. New York
Kackar RN, Harville DA (1981) Unbiasedness of two-stage estimation and prediction procedures for mixed linear models. Comm Stat Theor Meth A10:1249–1261
Lofgren DL, Vinson WE, Pearson RE, Cassell BG (1984) Effect of herd mean and standard deviation on cow index. J Dairy Sci suppl 1 (Abstr) 67:185–186
Mirande SL, Van Vleck LD (1985) Trends in genetic and phenotypic variances for milk production. J Dairy Sci 68:2278–2286
Patterson HD, Thompson R (1971) Recovery of interblock information when block sizes are unequal. Biometrika 58:545–554
Pericchi LR (1981) A Bayesian approach to transformations to normality. Biometrika 68:35– 43
Poirier DL (1978) The use of the Box-Cox transformation in limited dependent variable models. J Am Stat Assoc 73:284–287
Powell RL, Norman HD, Weinland BT (1983) Cow evaluation at different milk yield of herds. J Dairy Sci 66:148–154
Rook A J (1982) The effects of different environmental correction methods on genetic parameter estimates of traits in on-farm, individually fed, pig performance tests. PhD Thesis, Wye College, Univ London
Savin NE, White KJ (1978) Estimation and testing for functional form and autocorrelation: a simultaneous approach. J Econometrics 8:1–12
Seaks TG, Layson SK (1983) Box-Cox estimation with standard econometric problems. Rev Econ Stat 65:160–164
Searle SR (1974) Prediction, mixed models and variance components. In: Proschan F and Serfling RJ (eds) Proc Conf Reliabilty and Biometry. SIAM, Philadelphia
Searle SR (1979) Notes on variance component estimation: a detailed account of maximum likelihood and kindred methodology. Paper BU-673-M, Biometrics Unit, Cornell Univ, Ithaca.
Solomon PJ (1985) Transformations for components of variance and covariance. Biometrika 72:233–239
Spitzer JJ (1982) A primer on Box-Cox estimation. Rev Econ Stat 64:307–313
Thompson R (1979) Sire evaluation. Biometrics 35:339–353
Wright S (1921) Systems of mating. I. The biometric relations between parent and offspring. Genetics 6:111–123
Wright S (1968) Evolution and the genetics of populations. Vol 1 Univ Chicago Press, Chicago
Zellner A (1971) An introduction to Bayesian inference in econometrics. Wiley, New York
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gianola, D., Im, S., Fernando, R.L., Foulley, J.L. (1990). Mixed Model Methodology and the Box-Cox Theory of Transformations: A Bayesian Approach. In: Gianola, D., Hammond, K. (eds) Advances in Statistical Methods for Genetic Improvement of Livestock. Advanced Series in Agricultural Sciences, vol 18. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-74487-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-74487-7_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-74489-1
Online ISBN: 978-3-642-74487-7
eBook Packages: Springer Book Archive