Literature

The Origin of BLUP and MME

2015-03-07T00:00:00+00:00

The origin of mixed model equations appears in two abstracts published by Henderson (1949, 1950). The formal proof that beta and u in Henderson (1949, 1950) are BLUE and BLUP is given in Henderson et al. (1959) and Henderson (1963), respectively. The latter was published in 1963 but actually given in 1961. Goldberger (1962) was the first to use the term “best linear unbiased predictor”. The acronym “BLUP” was coined by Henderson (1973). I obtained a copy of Henderson (1949, 1950, 1963) from my colleague Dr. Dale Van Vleck at UNL.

Henderson CR. (1949). Estimation of changes in herd environment. J Dairy Sci. (Abstract) 32: 706. [PDF]
Henderson CR. (1950). Estimation of genetic parameters. Ann Math Stat. (Abstract) 21: 309-310. [PDF]
Henderson CR, Kempthorne O, Searle SR, and Von Krosigk CM. (1959). The Estimation of environmental and genetic trends from records subject to culling. Biometrics 15: 192–218.
Henderson CR. (1963). Selection index and expected genetic advance. In Statistical Genetics and Plant Breeding 141-163. NAS-NRC 982, Washington, DC. [PDF]
Goldberger AS. (1962). Best linear unbiased prediction in the generalized linear regression model. J Am Statis Ass. 57: 369-375
Henderson CR. (1973). Sire evaluation and genetic trends. In Proceedings of the Animal Breeding and Genetics Symposium in Honour of Dr.Jay L. Lush 10-41. ASAS and ADSA, Champaign, Ill.

Also see this correspondence between Daniel Gianola and Andrew Gelman.

Bayesian Methods

2015-01-21T00:00:00+00:00

This page contains a collection of papers that set milestones in using a Bayesian approach in quantitative genetics. This list is by no means complete history of Bayesian analyses.

K Rønningen. (1971). Some Properties of the Selection Index derived by “Henderson’s Mixed Model Method”. J Anim Breed Genet. 129:474-487.
Pehaps this is the first attempt to investigate the connection between BLUP/Selection Index and Bayesian Inference.
DA Harville. (1974). Bayesian inference for variance components using only error contrasts. Biometrika. 61:383-385.
This paper demonstrated that when flat priors were assigned for fixed effects and variance components, REML is essentially equal to the mode of the marginal posterior density of the variance components in a Bayesian model.
D Gianola and RL Fernando. (1986). Bayesian methods in animal breeding theory. J. Anim. Sci. 63:217-244. A comprehensive review of Bayesian methods in the pre MCMC era. It offers a Bayesian view of variance component estimation, prediction of genetic values with or without selection, and non-linear models.
D Gianola and JL Foulley. (1990). Variance estimation from integrated likelihoods (VEIL). Genet Sel Evol. 22:403-417.
An attempt to perform Bayesian estimation of variance components in the pre MCMC era.
AE Gelfand and AFM Smith. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association. 85:398-409.
One of the first application of Gibbs sampling in genetics.
SW Guo and EA Thompson. (1991). Monte Carlo estimation of variance component models for large complex pedigrees. IMA J Math Appl Med Biol. 8:171-189.
The first application of Gibbs sampling in quantitative genetics.
CS Wang et al. (1993). Marginal inferences about variance components in a mixed linear model using Gibbs sampling. Genet Sel Evol. 25:41-62.
Gibbs sampling was used first in animal breeding along with simulated data sets and a sire model.
CS Wang et al. (1994). Bayesian analysis of mixed linear models via Gibbs sampling with an application to litter size in Iberian pigs. Genet Sel Evol. 26:91-115.
The first application of Gibbs sampling using real data in animal breeding. An animal model was employed.
J Jensen et al. (1994). Bayesian inference on variance and covariance components for traits influenced by maternal and direct genetic effects, using the Gibbs sampler. Acta. Agric. Scand., Sect. A, Animal Sci. 44:193-201.
A Bayesian analysis of maternal and direct genetic effects.
J Albert and S Chib. (1993). Bayesian analysis of binary and polychotomous response data. J. Am. Stat. Assoc. 88:669–679.
The first Bayesian MCMC implementation of threshold models.
DA Sorensen et al. (1995). Bayesian inference in threshold models using Gibbs sampling. Genet Sel Evol. 27:229-249.
The first Bayesian treatment of ordered categories based on Gibbs sampling in animal breeding.
CP Van Tassell and LD Van Vleck. (1996). Multiple-trait Gibbs sampler for animal models: flexible programs for Bayesian and likelihood-based (co)variance component inference. J Anim Sci. 74:2586-2597.
Development of the MTGSAM software.
LA García-Cortés and D Sorensen. (1997). On a multivariate implementation of the Gibbs sampler. Genet Sel Evol. 28:121-126.
The first Bayesian treatment of multiple-traits animal models.
J Jamrozik and LR Schaeffer. (1997). Estimates of genetic parameters for a test day model wit random regressions for yield traits of first lactation Holsteins. J Dairy Sci. 80:762-770.
The first paper that applied Gibbs sampling to a random regression model.
CP Van Tassell et al. (1998). Bayesian analysis of twinning and ovulation rates using a multiple-trait threshold model and Gibbs sampling. J Anim Sci. 76:2048-2061 .
Extension of the MTGSAM to a multiple-trait threshold model.
A Blasco. (2001). The Bayesian controversy in animal breeding. J Anim Sci. 79:2023-2046.
A more recent review of Bayesian approaches in animal breeding.
J Jamrozik. (2004). Implementation issues for Markov Chain Monte Carlo methods in random regression test-day models. J Anim Breed Genet. 121:1-13.
This paper discusses problems related to the implementation of MCMC methods in dairy cattle.
DA Sorensen. (2009). Developments in statistical analysis in quantitative genetics. Genetica 136:319-332.
This is a nice overview of the application of MCMC in quantitative genetics.

Variance Component Estimation

2015-01-19T00:00:00+00:00

This page contains a collection of papers related to estimation of variance components. The techniques developed in this area are being used to infer both pedigree and genome-based heritabilities of complex traits.

HO Hartley and JNK Rao. (1967). Maximum-likelihood estimation for the mixed analysis of variance model. Biometrika. 54:93-108.
Discussion on a consistency and asymptotic efficiency of ML in the pre REML era.
HD Patterson and R Thompson. (1971). Recovery of inter-block information when block sizes are unequal. Biometrika. 58:545-554.
This is a seminal paper describing REML. REML accounts for loss of degrees of freedom arised in estimating fixed effects.
DA Harville. (1977). Maximum likelihood approaches to variance component estimation and to related problems. Journal of the American Statistical Association. 72:320-338.
A comprehensive review of ML and REML to variance component estimation.
DL Johnson and R Thompson. (1995). Restricted Maximum Likelihood Estimation of Variance Components for Univariate Animal Models Using Sparse Matrix Techniques and Average information. J Dairy Sci. 78:449–456.
AIREML has been proposed in this paper.
AR Gilmour et al. (1995). Average Information REML: An Efficient Algorithm for Variance Parameter Estimation in Linear Mixed Models. Biometrics. 51:1440-1450.
Another paper regarding AIREML.
J Jensen et al. (1997). Residual maximum likelihood estimation of (co)variance components in multivariate mixed linear models using average information. Journal of the Indian Society of Agricultural Statistics. 49:215-236.
Another paper regarding AIREML.
A Hofer. (1998). Variance component estimation in animal breeding: a review. J Anim. Breed. 115:247-265.
A review of REML in animal breeding.
R Thompson et al. (2005). Estimation of quantitative genetic parameters. Philos Trans R Soc Lond B Biol Sci. 360:1469-1477.
A review of variance component estimation applied to the animal breeding field.
R Thompson. (2008). Estimation of quantitative genetic parameters. Philos Trans R Soc Lond B Biol Sci. 360:679-686.
An updated review of R Thompson et al. (2005).

Resources

2015-01-17T00:00:00+00:00

These are some additional reading lists.

Five books every animal breeding student should read - John Cole
The Leek group guide to genomics papers - Jeff Leek
Useful reading - Joe Pickrell