Méthodes du maximum de vraisemblance en modèle linéaire mixte

Foulley, Jean-Louis; Delmas, Céline; Robert-Granié, Christèle

Journal de la Société française de statistique, Tome 143 (2002) no. 1-2, pp. 5-52.

Anderson R.L., Bancroft T.A. (1952), Statistical theory in research. Mc Graw-Hill, New-York. | Zbl

Berger J.O., Liseo B., Wolpert R.L. (1999), Integrated Likelihood methods for eliminating nuisance parameters, Statistical Science, 14, 1-28. | MR | Zbl

Cox D.R., Reid N. (1987), Parameter orthogonality and approximate conditional inference, Journal of the Royal Statistical Society B, 49, 1-39. | MR | Zbl

Cox D.R., Hinkley D.V. (1974), Theoretical statistics, Chapman & Hall, London. | MR | Zbl

Cressie N., Lahiri S.N. (1993), The asymptotic distribution of REML estimators, Journal of Multivariate Analysis, 45, 217-233. | MR | Zbl

Crump S.L. (1947), The estimation of components of variance in multiple classifications, PhD thesis, Iowa State University, Ames.

Davidian M., Giltinian D.M. (1995), Nonlinear Models for Repeated Measurement Data, Chapman and Hall, London.

Dawid A.P. (1980), A Bayesian look at nuisance parameters, Proceedings of the first international meeting held in Valencia (Bernardo J.M., DeGroot M.H., Lindley D.V., Smith A.F.M., eds), University Press, Valencia, Spain, 167-184. | MR | Zbl

Delmas C., Foulley J.L., Robert-Granié C. (2002), Further insights into tests of variance components and model selection, Proceedings of the 7th World Congress of Genetics applied to Livestock Production, Montpellier, France, 19-23 August 2002.

Dempster A., Laird N., Rubin R. (1977), Maximum likelihood estimation from incomplete data via the EM algorithm, Journal of the Royal Statistical Society B, 39,1-20. | MR | Zbl

Diggle P.J., Liang K.Y., Zeger S.L. (1994), Analysis of longitudinal data, Oxford Science Publications, Clarendon Press, Oxford

Edwards A.W.F. (1972), Likelihood, Cambridge University Press, Cambridge | MR | Zbl

Eisenhart C. (1947), The assumptions underlying the analysis of variance, Biometrics, 3, 1-21. | MR

Fisher R.A. (1922), On the mathematical foundations of theoretical statistics, Philosophical Transactions of the Royal Society of London, Séries A 222, 309-368. | JFM

Fisher R.A. (1925), Statistical methods for research workers, Oliver and Boyd. Edinburgh and London. | JFM

Foulley J. (1993), A simple argument showing how to derive restricted maximum likelihood, Journal of Dairy Science, 76, 2320-2324.

Foulley J.L., Quaas R.L. (1995), Heterogeneous variances in Gaussian linear mixed models, Genetics Selection Evolution, 27, 211-228.

Foulley J.L., Van Dyk D.A. (2000), The PX EM algorithm for fast fitting of Henderson's mixed model, Genetics Selection Evolution, 32, 143-163.

Foulley J., Jaffrezic F., Robert-Granié C. (2000), EM-REML estimation of covariance parameters in Gaussian mixed models for longitudinal data analysis, Genetics Selection Evolution, 32, 129-141.

Gianola D., Foulley J., Fernando R. (1986), Prediction of breeding values when variances are not known, Genetics Selection Evolution, 18, 485-498.

Gilmour A.R., Thompson R., Cullis B.R. (1995), An efficient algorithm for REML estimation in linear mixed models, Biometrics, 51, 1440-1450. | Zbl

Goffinet B. (1983), Risque quadratique et sélection : quelques résultats appliqués à la sélection animale et végétale, Thèse de Docteur Ingénieur, Université Paul Sabatier, Toulouse.

Gourieroux C. , Montfort A. (1989), Statistique et modèles économétriques, Economica, France.

Hartley H.O., Rao J.N.K. (1967), Maximum likelihood estimation for the mixed analysis of variance model, Biometrika, 54, 93-108. | MR | Zbl

Harvey W.R. (1970), Estimation of variance and covariance components in the mixed model, Biometrics, 61, 485-504.

Harville D.A. (1974), Bayesian inference for variance components using only error contrasts, Biometrika, 61, 383-385. | MR | Zbl

Harville D.A. (1977), Maximum likelihood approaches to variance component estimation and to related problems, Journal of the American Statistical Association, 72, 320-340. | MR | Zbl

Harville D.A. (1997), Matrix algebra from a statistician's perspective, Springer, Berlin. | MR | Zbl

Harville D.A., Callanan T.P. (1990), Computational aspects of likelihood based inference for variance components, Advances in Statistical Methods for Genetic Improvement of Livestock (Gianola D, Hammond K, eds), Springer Verlag, 136-176.

Henderson C.R. (1953), Estimation of variance and covariance components, Biometrics, 9, 226-252. | MR

Henderson C.R. (1973), Sire evaluation and genetic trends, In : Proceedings of the animal breeding and genetics symposium in honor of Dr J Lush, American Society Animal Science-American Dairy Science Association, 10-41, Champaign, IL.

Henderson C.R. (1984), Applications of linear models in animal breeding, University of Guelph, Guelph.

Henderson C.R., Kempthorne O., Searle S.R., Von Krosigk C.N. (1959), Estimation of environmental and genetic trends from records subject to culling, Biometrics, 13, 192-218. | Zbl

Kackar A.N., Harville D.A. (1984), Approximation for standard errors of estimators of fixed and random effects in mixed linear models, Journal of the American Statistical Association, 79, 853-862. | MR | Zbl

Kalbfleisch J.D., Sprott D.A. (1970), Application of the likelihood methods to models involving large numbers of parameters, Journal of the Royal Statistical Society B, 32, 175-208. | MR | Zbl

Kenward M.G., Roger J.H. (1997), Small sample inference for fixed effects from restricted maximum likelihood, Biometrics, 53, 983-997. | Zbl

Laird N.M., Ware J.H. (1982), Random effects models for longitudinal data, Biometrics, 38 963-974. | Zbl

Lamotte L.R. (1970), A class of estimators of variance components, Technical report 10, Department of Statistics, University of Kentucky, Lexington, KE.

Lamotte L.R. (1973), Quadratic estimation of variance components, Biometrics, 29, 311-330. | MR

Leonard T., Hsu J.S.J. (1999), Bayesian methods, an analysis for statisticians and interdisciplinary researchers, Cambridge University Press, Cambridge, UK. | MR | Zbl

Liang K.Y., Zeger S.L. (1986), Longitudinal data analysis using generalized linear models, Biometrika, 73, 13-22. | MR | Zbl

Lindley D.V., Smith A.F.M. (1972), Bayes Estimates for the Linear Model, Journal of the Royal Statistical Society B, 34, 1-41. | MR | Zbl

Liu C. , Rubin D.B., Wu Y.N. (1998), Parameter expansion to accelerate EM : the PX- EM algorithm, Biometrika, 85, 755-770. | MR | Zbl

Liu C., Wu Y.N. (1999), Parameter expansion scheme for data augmentation, Journal of the American Statistical Association, 94, 1264-1274. | MR | Zbl

Mardia K.V., Marshall R.J. (1985), Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika, 71, 135-146. | MR | Zbl

Mccullagh P., Nelder J. (1989), Generalized linear models, 2nd edition, Chapman and Hall, London. | MR | Zbl

Mclachlan G.J., Krishnan T. (1997), The EM algorithm and extensions, John Wiley & Sons, New York. | MR | Zbl

Meng X.L., Van Dyk D.A. (1998), Fast EM-type implementations for mixed effects models, Journal of the Royal Statistical Society B, 60, 559-578. | MR | Zbl

Monod H. (2000), On the efficiency of generally balanced designs analysed by restricted maximum likelihood, Proceedings of Optimum Design, Cardiff, 12-14 april 2000, Kluwer Academic. | MR

Mood A.M., Graybill F.A., Boes D.C. (1974), Introduction to the theory of statistics, Third edition, International student edition. | Zbl

Morrell C.H. (1998), Likelihood ratio of variance components in the linear mixed-effects model using restricted maximum likelihood, Biometrics, 54, 1560-1568. | Zbl

Patterson H.D., Thompson R. (1971), Recovery of inter-block information when block sizes are unequal, Biometrika, 58, 545-554. | MR | Zbl

Quaas R.L. (1992), REML Notebook, Mimeo, Cornell University, Ithaca, New York.

Rao C.R. ( 1971a), Estimation of variance components-Minque theory, Journal of Multivariate Analysis, 1, 257-275. | MR | Zbl

Rao C.R. ( 1971b), Minimum variance quadratic unbiased estimation of variance components, Journal of Multivariate Analysis, 1, 445-456. | MR | Zbl

Rao C.R. (1973), Linear Statistical Inference and its Applications, 2nd edition. Wiley, New-York. | MR | Zbl

Rao C.R. (1979), MIQE theory and its relation to ML and MML estimation of variance components, Sankhya B, 41, 138-153. | MR | Zbl

Rao C.R., Kleffe J. (1988), Estimation of variance components and applications, North Holland series in statistics and probability, Elsevier, Amsterdam. | MR | Zbl

Robinson G.K. (1991), The estimation of random effects, Statistical Science, 6, 15-51. | MR | Zbl

San Cristobal M., Robert-Granie C. , Foulley J.L. (2002), Hétéroscédasticité et modèles linéaires mixtes : théorie et applications en génétique quantitative, Journal de la Société Française de Statistique, 143, 1-2, 155-165. | EuDML

Searle S.R. (1979), Notes on variance component estimation. A detailed account of maximum likelihood and kindred methodology, Paper BU-673-M, Cornell University, Ithaca. | MR

Searle S.R. (1982), Matrix algebra useful for statistics, J. Wiley and Sons, New York. | MR | Zbl

Searle S.R. (1989), Variance components - some history and a summary account of estimation methods, Journal of Animal Breeding and Genetics, 106, 1-29.

Searle S.R., Casella G., Mc Culloch C.E. (1992), Variance components, J. Wiley and Sons, New-York. | MR | Zbl

Self S.G., Liang K.Y. (1987), Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under non-standard conditions, Journal of the American Statistical Association, 82, 605-610. | MR | Zbl

Stram D.O., Lee J.W. (1994), Variance components testing in the longitudinal mixed effects model, Biometrics, 50, 1171-1177. | Zbl

Stram D.O., Lee J.W. (1995), Correction to "Variance components testing in the longitudinal mixed effects model", Biometrics, 51, 1196.

Sweeting T.J. (1980), Uniform asymptotic normality of the maximum likelihood estimator, The Annals of Statistics, 8, 1375-1381. | MR | Zbl

Thompson W.A. (1962), The problem of negative estimates of variance components, Annals of Mathematical Statistics, 33, 273-289. | MR | Zbl

Thompson R. (1989), REML, Biometric Bulletin, 6, (3), 4-5.

Thompson R. (2002), A review of genetic parameter estimation, Proceedings of the 7th World Congress of Genetics applied to Livestock Production, Montpellier, France, 19-23 August 2002.

Van Dyk D.A. (2000), Fitting mixed-effects models using efficient EM-type algorithmes, Journal of Computational and Graphical Statistics, 9, 78-98. | MR

Van Dyk D.A., Meng X.L. (2001), The art of data augmentation, Journal of Computational and Graphical Statistics, 10, 1-50. | MR

Verbeke G., Molenberghs G. (2000), Linear mixed models for longitudinal data, Springer Verlag, New York. | MR | Zbl

Verbyla A.P., Cullis B.R., Kenward M.G., Welham S.J. (1999), The analysis of designed experiments and longitudinal data by using smoothing splines (with discussion), Applied Statistics, 48, 269-311. | Zbl

Welham S.J., Thompson R. (1997), A likelihood ratio test for fixed model ternis using residual maximum likelihood, Journal of the Royal Statistical Society B, 59, 701- 714. | MR | Zbl

Yates F. (1934), The analysis of multiple classifications with unequal numbers in the different classes, Journal of the American Statistical Association, 29, 51-66. | JFM