Use of Mixed Model Methodology in Analysis of Designed Experiments

  • B. W. Kennedy
Part of the Advanced Series in Agricultural Sciences book series (AGRICULTURAL, volume 18)


Applications of mixed model methods in the analysis of designed experiments are illustrated and discussed for purposes of: (1) increasing rate of selection response to create genetically diverse lines rapidly or to demonstrate feasibility of selection, (2) estimation of genetic parameters free of bias from selection and inbreeding, (3) estimation of response to selection with or without controls, and (4) verification of experimental design prior to the experiment. For traits controlled by a large number of additive loci, use of the numerator relationship matrix in the mixed model equations accounts for changes in additive genetic variance due to inbreeding, assortative mating and gametic disequilibrium resulting from selection. If the number of loci is small, non-normality of the genotypic distribution and changes in variance due to gene frequency changes (including fixation) are not accounted for but these seem to be of small consequence, at least for short-term selection. Use of mixed model methods do not require prior knowledge of base population heritability which can be estimated from the data unaltered by selection. If dominance effects are important, properties of the dominance relationship matrix and use of mixed model methods are not yet well understood in inbred and selected populations. Simulation results indicate that use of mixed model methods can be effective in randomly mated populations, even if the number of loci is small. There is evidence of bias in selected populations, however, particularly when gene frequencies are extreme. Properties of mixed model methods under dominance and other non-additive genetic models need more study.


Selection Response Additive Genetic Variance Good Linear Unbiased Predictor Good Linear Unbiased Estimator Additive Genetic Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Avalos E, Smith C (1987) Genetic improvement of litter size in pigs. Anim Prod 44:153–164CrossRefGoogle Scholar
  2. Belonsky GM (1984) Genetic evaluation of a swine herd under selection. MSc Thesis, Univ Guelph, GuelphGoogle Scholar
  3. Belonsky GM, Kennedy BW (1984) Within-herd genetic evaluation of swine. J Anim Sci 59 suppl(Abstr) 1:164Google Scholar
  4. Belonsky GM, Kennedy BW (1988) Selection on individual phenotype and best linear unbiased prediction of breeding value in a closed swine herd. J Anim Sci 66:1124– 1131PubMedGoogle Scholar
  5. Bichard M, David PJ (1985) Effectiveness of genetic selection for prolificacy in pigs. J Reprod Fert suppl 33:127–138Google Scholar
  6. Blair HT, Pollak EJ (1984) Estimation of genetic trend in a selected population with and without the use of a control population. J Anim Sci 58:878–886PubMedGoogle Scholar
  7. Bulmer MG (1971) The effect of selection on genetic variability. Am Nat 105:201–211CrossRefGoogle Scholar
  8. Bulmer MG (1980) The mathematical theory of quantitative genetics. Clarendon Press, OxfordGoogle Scholar
  9. Cockerham CC (1954) An extension of the concept of partitioning hereditary variance for analysis of covariances among relatives when epistasis is present. Genetics 39:859– 882PubMedGoogle Scholar
  10. Falconer DS (1981) Introduction to quantitative genetics. 2nd edn Longman, LondonGoogle Scholar
  11. Felsenstein J (1965) The effect of linkage on directional selection. Genetics 42:349–363Google Scholar
  12. 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:356–370Google Scholar
  13. Henderson CR (1949) Estimation of changes in herd environment. J Dairy Sci (Abstr) 32:706Google Scholar
  14. Henderson CR (1953) Estimation of variance and covariance components. Biometrics 9:226– 252CrossRefGoogle Scholar
  15. Henderson CR (1963) Selection index and expected genetic advance. In: Statistical genetics and plant breeding. NAS-NRC 982, Washington DC pp 141–163Google Scholar
  16. 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 10–41Google Scholar
  17. Henderson CR (1975) Best linear unbiased estimation and prediction under a selection model. Biometrics 31:423–447PubMedCrossRefGoogle Scholar
  18. Henderson CR (1976) A simple method for computing the inverse of a numerator relationship matrix used in prediction of breeding values. Biometrics 32:69–83CrossRefGoogle Scholar
  19. Henderson CR (1977) Prediction of merits of a single cross. Theor Appl Genet 49:273–282CrossRefGoogle Scholar
  20. Henderson CR (1984) Applications of linear models in animal breeding. Univ Guelph Press, GuelphGoogle Scholar
  21. Henderson CR (1985a) Best linear unbiased prediction of nonadditive genetic merits in noninbred populations. J Anim Sci 60:111–117Google Scholar
  22. Henderson CR (1985b) MIVQUE and REML estimation of additive and nonadditive genetic variances. J Anim Sci 61:113–121Google Scholar
  23. Henderson CR, Kempthorne O, Searle SR, von Krosigk CM (1959) The estimation of environmental and genetic trends from records subject to culling. Biometrics 15:192– 218CrossRefGoogle Scholar
  24. Hill WG (1970) Design of experiments to estimate heritability by the regression of offspring on selected parents. Biometrics 26:566–571PubMedCrossRefGoogle Scholar
  25. Hill WG (1971) Design and efficiency of selection experiments for estimating genetic parameters. Biometrics 27:293–311PubMedCrossRefGoogle Scholar
  26. Hill WG (1972) Estimation of genetic change. I. General theory and design of control populations. Anim Breed Abstr 40:1–15Google Scholar
  27. Hill WG, Nicholas FW (1974) Estimation of heritability by both regression of offspring on parent and intra-class correlation of sibs in one experiment. Biometrics 30:447–468PubMedCrossRefGoogle Scholar
  28. Hough JD, Benyshek LL (1986) Estimates of genetic trend using the reduced animal model versus least-squares using a control population. J Anim Sci 63 suppl (Abstr) 1:178Google Scholar
  29. Kackar RN, Harville DA (1981) Unbiasedness of two-stage estimation and prediction procedures for mixed linear models. Commun Stat Theor Meth A10:1249–1261CrossRefGoogle Scholar
  30. Kemp RA (1985) The effects of positive assortative mating and preferential treatment of progeny on the estimation of breeding values. PhD Thesis, Univ Guelph,Google Scholar
  31. Guelph Legault C (1985) Selection of breeds, strains and individual pigs for prolificacy. J Reprod Fertsuppl 33:151–166Google Scholar
  32. Maki-Tanila A, Kennedy BW (1986) Mixed model methodology under genetic models with a small number of additive and non-additive loci. In: Dickerson GE, Johnson RK (eds) Proc 3rd World Congr Genet Appl Livest Prod. Agric Commun, Univ Nebraska, Lincoln, Nebraska XII:443–448Google Scholar
  33. Patterson HD, Thompson R (1971) Recovery of inter-block information when block sizes are unequal. Biometrika 58:545–554CrossRefGoogle Scholar
  34. Pearson K (1903) Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs. Phil Trans R Soc London A 200:1–66CrossRefGoogle Scholar
  35. Quaas RL, Pollak EJ (1980) Mixed model methodology for farm and ranch beef cattle testing programs. J Anim Sci 51:1277–1287Google Scholar
  36. Rao CR (1971) Minimum variance quadratic unbiased estimation of variance components. J Mult Anal 1:445–456CrossRefGoogle Scholar
  37. Robertson A (1977) The effect of selection on the estimation of genetic parameters. Z Tierz Züchtungsbiol 94:131–135CrossRefGoogle Scholar
  38. Schaeffer LR (1986) Pseudo-expectation approach to variance component estimation. J Dairy Sci 69:2884–2889CrossRefGoogle Scholar
  39. Smith C (1977) Use of stored frozen semen and embryos to measure genetic trends in farm livestock. Z Tierz Züchtungsbiol 94:119–127CrossRefGoogle Scholar
  40. Smith SP (1984) Dominance relationship matrix and inverse for an inbred population. Unpublished mimeo, Department of Dairy Science, Ohio State Univ, ColumbusGoogle Scholar
  41. Smith SP, Allaire FR (1985) Efficient selection rules to increase non-linear merit: application in mate selection. Genet Sel Evol 17:387–406CrossRefGoogle Scholar
  42. Sorensen DA, Kennedy BW (1982) Estimation of genetic variances in control and selected populations. In: Proc 2nd World Congr Genet Appl Livest Prod. Garsi, Madrid Vπ:220–225Google Scholar
  43. Sorensen DA, Kennedy BW (1983) The use of the relationship matrix to account for genetic drift variance in the analysis of genetic experiments. Theor Appl Genet 66:217–220CrossRefGoogle Scholar
  44. Sorensen DA, Kennedy BW (1984a) Estimation of response to selection using least-squares and mixed model methodology. J Anim Sci 58:1097–1106Google Scholar
  45. Sorensen DA, Kennedy BW (1984b) Estimation of genetic variances from unselected and selected populations. J Anim Sci 59:1213–1223Google Scholar
  46. Sorensen DA, Kennedy BW (1986) Analysis of selection experiments using mixed model methodology. J Anim Sci 68:245–258Google Scholar
  47. Styan GPH (1973) Hadamard products and multivariate statistical analysis. Linear Algebra and Its Applications 6:217–240CrossRefGoogle Scholar
  48. Thompson R (1976) Designs of experiments to estimate heritability when observations are available on parents and offspring. Biometrics 32:283–304PubMedCrossRefGoogle Scholar
  49. Thompson R (1977) The estimation of heritability with unbalanced data. II. Data available on more than two generations. Biometrics 33:497–504CrossRefGoogle Scholar
  50. Thompson R, Cameron ND (1986) Estimation of genetic parameters. In: Dickerson GE, Johnson RK (eds) Proc 3rd World Congr Genet Appl Livest Prod. Agric Commun, Univ Nebraska, Lincoln, Nebraska XII:371–381Google Scholar
  51. Wright S (1922) Coefficients of inbreeding and relationship. Am Nat 56:330–338CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • B. W. Kennedy
    • 1
  1. 1.Centre for Genetic Improvement of LivestockUniversity of GuelphGuelphCanada

Personalised recommendations