mixture models; maximum likelihood; EM algorithm; mastitis; dairy cattle
Abstract :
[en] A Gaussian mixture model with a finite number of components and correlated random effects is described. The ultimate objective is to model somatic cell count information in dairy cattle and to develop criteria for genetic selection against mastitis, an important udder disease. Parameter estimation is by maximum likelihood or by an extension of restricted maximum likelihood. A Monte Carlo expectation- maximization algorithm is used for this purpose. The expectation step is carried out using Gibbs sampling, whereas the maximization step is deterministic. Ranking rules based on the conditional probability of membership in a putative group of uninfected animals, given the somatic cell information, are discussed. Several extensions of the model are suggested.
Disciplines :
Veterinary medicine & animal health
Author, co-author :
Gianola, D.
Odegard, J.
Heringstad, B.
Klemetsdal, G.
Sorensen, D.
Madsen, P.
Jensen, J.
Detilleux, Johann ; Université de Liège - ULiège > Département de productions animales > Génétique quantitative
Language :
English
Title :
Mixture model for inferring susceptibility to mastitis in dairy cattle: a procedure for likelihood-based inference
Aitkin M., Wilson G.T., Mixture models, outliers and the EM algorithm, Technometrics 22 (1980) 325-331.
Ali A.K.A., Shook G.E., An optimum transformation for somatic cell concentration in milk, J. Dairy Sci. 63 (1980) 487-490.
Boichard D., Rupp R., Genetic analysis and genetic evaluation for somatic cell score in French dairy cattle, Interbull Bull. 15 (1997) 54-60, International Bull Evaluation Service, Uppsala, Sweden.
Booth J.M., Progress in the control of mastitis, in: Proceedings of the 3rd International Mastitis Seminar, 1995, Tel Aviv, Israel. S4.3-S4.11, International Dairy Federation, Brussels, Belgium.
Dempster A.P., Laird N.M., Rubin D.B., Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. Royal Stat. Soc. B 39 (1977) 1-38.
Detilleux J., Mathematical modeling of inflammatory reaction during bovine mastitis, in: Proceedings of the 7th World Congress on Genetics Applied to Livestock Production, Montpellier, 19-23 August 2002, Vol. 31, Inra, pp. 711-714.
Detilleux J., Leroy P.L., Application of a mixed normal mixture model to the estimation of mastitis-related parameters, J. Dairy Sci. 83 (2000) 2341-2349.
Fernández C., Steel M.F.J., On Bayesian modelling of fat tails and skewness, J. Am. Stat. Assoc. 93 (1998) 359-371.
Gianola D., Strandén I., Foulley J.L., Modelos lineales mixtos con distribuciones-t: potencial en genetica cuantitativa, in: Actas, Quinta Conferencia Española de Biometria, Sociedad Española de Biometria, pp. 3-4, Valencia, Spain.
Green P., Reversible jump MCMC computation and Bayesian model determination, Biometrika 82 (1995) 711-732.
Guo S.W., Thompson E.A., Monte Carlo estimation of mixed models for large complex pedigrees, Biometrics 50 (1994) 417-432.
Han C., Carlin B.P., Markov chain Monte Carlo methods for computing Bayes factors: a comparative review, J. Am. Stat. Assoc. 96 (2001) 1122-1132.
Harville D.A., Bayesian inference of variance components using only error contrasts, Biometrika 61 (1974) 383-385.
Hathaway R.J., A constrained formulation of maximum likelihood estimation for normal mixture distributions, Ann. Math. Stat. 13 (1985) 795-800.
Heringstad B., Klemetsdal G., Ruane J., Selection for mastitis resistance in dairy cattle: a review with focus on the situation in the Nordic countries, Livest. Prod. Sci. 64 (2000) 95-106.
Heringstad B., Rekaya R., Gianola D., Klemetsdal G., Weigel K.A., Genetic change for clinical mastitis in Norwegian cattle: a threshold model analysis, J. Dairy Sci. 86 (2003) 369-375.
Heringstad B., Chang Y.M., Gianola D., Klemetsdal G., Genetic analysis of longitudinal trajectory of clinical mastitis in first-lactation Norwegian cattle, J. Dairy Sci. 86 (2003) 2676-2683.
Hoeting J.A., Madigan D., Raftery A.E., Volinsky C.T., Bayesian model averaging: a tutorial, Stat. Sci. 14 (1999) 382-417.
Hosmer D.W., On MLE of the parameters of a mixture of two normal distributions when the sample size is small, Comput. Stat. 1 (1973) 217-227.
Kiefer J., Wolfowitz J., Consistency of the maximum likelihood estimates in the presence of infinitely many incidental parameters, Ann. Math. Stat. 27 (1956) 887-906.
Kosinski A., A procedure for the detection of multivariate outliers, Comput. Stat. Data Anal. 29 (1999) 145-161.
Land R.B., The expression of female sex-limited characters in the male, Nature 241 (1973) 208-209.
Little R.J.A., Rubin D.B., Statistical Analysis with Missing Data, 1st edn., John Wiley and Sons, New York, 1987.
Madigan D., Raftery A.E., Model selection and accounting for model uncertainty in graphical models using Occam's window, J. Am. Stat. Assoc. 89 (1994) 1535-1546.
McLachlan G., Peel D., Finite Mixture Models, John Wiley and Sons, New York, 2000.
Militino A.F., Ugarte M.D., Fean C.B., The use of mixture models for identifying high risks in disease mapping, Stat. Med. 20 (2001) 2035-2049.
Mrode R.A., Swanson G.J.T., Genetic and statistical properties of somatic cell count and its suitability as an indirect means of reducing the incidence of mastitis in dairy cattle, Anim. Breed. Abstr. 64 (1996) 847-857.
Myllys V., Asplund K., Brofeldt E., Hirvela-Koski V., Honkanen-Buzalski T., Junttila J., Kulkas L., Myllykangas O., Niskanen M., Saloniemi H., Sandholm M., Saranpaa T., Bovine mastitis in Finland in 1998 and 1995: Changes in prevalence and antibacterial resistance, Acta Vet. Scand. 39 (1998) 119-126.
Patterson H.D., Thompson R., Recovery of interblock information when block sizes are unequal, Biometrika 58 (1971) 545-554.
Pöso J., Mantysaari E.A., Relationships between clinical mastitis, somatic cell score and production for the first three lactations of Finnish Ayrshire, J. Dairy Sci. 79 (1996) 1284-1291.
Raftery A.E., Madigan D., Hoeting J.A., Model selection and accounting for model uncertainty in linear regression models, J. Am. Stat. Assoc. 92 (1997) 179-191.
Richardson S., Green P., On Bayesian analysis of mixtures with an unknown number of components (with discussion), J. Royal Stat. Soc. B 59 (1997) 731-792.
Robert C.P., Casella G., Monte Carlo Statistical Methods, Springer-Verlag, New York, 1999.
Rodriguez-Zas S.L., Gianola D., Shook G.E., Evaluation of models for somatic cell score lactation patterns in Holsteins, Livest. Prod. Sci. 67 (2000) 19-30.
Rosa G.J.M., Gianola D., Padovani C.R., Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation, Biom. J. 54 (2003) 1-18.
Schukken Y.H., Lam T.J.G.M., Barkema H.W., Biological basis for selection on udder health traits, in: Proceedings of the International Workshop on Genetic Improvement of Functional Traits in Cattle, Interbull Bull. 15 (1997) 27-33.
Shook G.E., Selection for disease resistance, J. Dairy Sci. 72 (1989) 2136-2142.
Shoukri M.M., McLachlan G.J., Parametric estimation in a genetic mixture model with application to nuclear family data, Biometrics 50 (1994) 128-139.
Sorensen D., Gianola D., Likelihood, Bayesian and MCMC Methods in Quantitative Genetics, Springer-Verlag, New York, 2002.
Strandén I.J., Robust mixed effects linear models with t-distributions and application to dairy cattle breeding, Ph.D. Thesis, University of Wisconsin-Madison, 1996.
Stranden I., Gianola D., Attenuating effects of preferential treatment with Student-t mixed linear models: a simulation study, Genet. Sel. Evol. 30 (1998) 565-583.
Tanner M.A., Tools for Statistical Inference, 1st edn., Springer-Verlag, New York, 1993.
Titterington D.M., Smith A.F.M., Makov U.E., Statistical Analysis of Finite Mixture Distributions, John Wiley and Sons, Chichester, 1985.
Wei G.C.G., Tanner M.A., A Monte Carlo implementation of the EM algorithm and the poor's man's data augmentation algorithm, J. Am. Stat. Assoc. 85 (1990) 699-704.
West M., Blanchette C., Dressman H., Huang E., Ishida S., Spang R., Zuzan H., Olson J.A. Jr., Marks J.R., Nevins J.R., Predicting the clinical status of human breast cancer by using gene expression profiles, Proc. Natl. Acad. Sci. USA 98 (2001) 11462-11467.
Willham R.L., The covariance between relatives for characters composed of components contributed by related individuals, Biometrics 19 (1963) 18-27.
