Investigation of Biases for Variance Components on Multiple Traits with Varying Number of Categories in Threshold Models Using Bayesian Inferences

  • Lee, D.H. (Department of Animal Life and Resources, Hankyong National University)
  • Received : 2001.09.13
  • Accepted : 2002.02.15
  • Published : 2002.07.01


Gibbs sampling algorithms were implemented to the multi-trait threshold animal models with any combinations of multiple binary, ordered categorical, and linear traits and investigate the amount of bias on these models with two kinds of parameterization and algorithms for generating underlying liabilities. Statistical models which included additive genetic and residual effects as random and contemporary group effects as fixed were considered on the models using simulated data. The fully conditional posterior means of heritabilities and genetic (residual) correlations were calculated from 1,000 samples retained every 10th samples after 15,000 samples discarded as "burn-in" period. Under the models considered, several combinations of three traits with binary, multiple ordered categories, and continuous were analyzed. Five replicates were carried out. Estimates for heritabilities and genetic (residual) correlations as the posterior means were unbiased when underlying liabilities for a categorical trait were generated given by underlying liabilities of the other traits and threshold estimates were rescaled. Otherwise, when parameterizing threshold of zero and residual variance of one for binary traits, heritability estimates were inflated 7-10% upward. Genetic correlation estimates were biased upward if positively correlated and downward if negatively correlated when underling liabilities were generated without accounting for correlated traits on prior information. Residual correlation estimates were, consequently, much biased downward if positively correlated and upward if negatively correlated in that case. The more categorical trait had categories, the better mixing rate was shown.


Categorical Traits;Bayesian;Variance Components;Biases


  1. Gianola, D. 1982. Theory and analysis of threshold characters. J. Anim. Sci. 54:1079-1096.
  2. Hoeschele, I. and B. Tier. 1995. Estimation of variance components of threshold characters by marginal posterior modes and means via Gibbs sampling. Genet. Sel. Evol. 27:519-540.
  3. Gianola, D. and J. L. Foulley. 1983. Sire evaluation for ordered categorical data with a threshold model. Genet. Sel. Evol. 15:201-244.
  4. Luo, M. F., P. J. Boettcher, L. R. Schaeffer and J. C. M. Dekkers. 2001. Bayesian inference for categorical traits with an application to variance component estimation. J. Dairy Sci. 84:694-704.
  5. Wang, C. S., R. L. Quaas and E. J. Pollak. 1997. Bayesian analysis of calving ease scores and birth weights. Genet. Sel. Evol. 29:117-143.
  6. Wright, S. 1934. An analysis of variability in number of digits in an inbred strain of guinea pigs. Genetics 19:506-536.
  7. Gianola, D. and R. L. Fernando. 1986. Bayesian methods in animal breeding theory. J. Anim. Sci. 63:217-244.
  8. Gelman, A., J. B. Carlin, H. S. Stern and D. B. Robin. 1995. Bayesian data analysis. Chapman and Hall/CRC. pp. 52-57.
  9. Geyer, C. J. 1992. Practical Markov Chain Monte Carlo (with discussion). Stat. Sci. 7:467-511.
  10. Moreno, C., D. Sorensen, L. A. Garcia-Cortes, L. Varona and J. Altarriba. 1997. On biased inferences about variance components in the binary threshold model. Genet. Sel. Evol. 29:145-160.
  11. Sorensen, D. 1996. Gibbs sampling in quantitative genetics. Internal report no. 82. Danish institute of animal science. Denmark.
  12. Korsgaard, I. R., A. H. Andersen and D. Sorensen. 1999. A useful reparameterisation to obtain samples from conditional inverse Wishart distributions. Genet. Sel. Evol. 31:177-181.
  13. Geman, S. and D. Geman. 1984. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6:721-741.
  14. Harville, D. A. and R. W. Mee. 1984. A mixed-model procedure for analyzing ordered categorical data. Biometrics. 40:393-408.
  15. Sorensen, D. A., S. Andersen, D. Gianola and I. Korsgaard. 1995. Bayesian inference in threshold models using Gibbs sampling. Genet. Sel. Evol. 27:229-249.
  16. VanTassell, C. P., L. D. Van Vleck and K. E. Gregory. 1998. Bayesian analysis of twinning and ovulation rates using a multiple-trait threshold model and Gibbs sampling. J. Anim. Sci. 76:2048-2061.
  17. Foulley, J. L., D. Gianola and R. Thompson. 1983. Prediction of genetic merit from data on binary and quantitative variates with an application to calving difficulty, birth weight and pelvic opening. Genet. Sel. Evol. 15:401-424.