### “A BLUP derivation of the multivariate breeder's equation, with an elucidation of errors in BLUP variance estimates, and a prediction method for inbred populations”

**Authors:**Rolf Ergon,

**Affiliation:**University of South-Eastern Norway

**Reference:**2022, Vol 43, No 4, pp. 131-140.

**Keywords:**BLUP, Multivariate breeder's equation, Robertson-Price identity, Price equation, Variance estimation

**Abstract:**The multivariate breeder's equation, Lande (1979), was derived from the Price equation Price (1970,1972). Here, I present a derivation based on the BLUP (best linear unbiased predictions) equations in matrix form, first given in summation form by Henderson (1950). The derivation makes use of a comparison with the known form of the multivariate breeder's equation, and it is therefore not an independent derivation. The alternative derivation does, however, clarify why and to which extent the variances of BLUP predictions of random effects are underestimated. The BLUP random effects can in fact be used for prediction of phenotypic responses by use of the Robertson-Price identity, Robertson (1966), Price (1970,1972). The BLUP derivation also leads to a prediction method for populations with inbreeding, i.e., where the additive genetic relationship matrix departs from a unity matrix, and where the multivariate breeder's equation therefore will fail.

PDF (481 Kb) DOI: 10.4173/mic.2022.4.2

**References:**

[1] Bonamour, S., Teplitsky, C., Charmantier, A., Crochet, P.-A., and Chevin, L.-M. (2017). Selection on skewed characters and the paradox of stasis, Evolution. 71:11:2703--2713. doi:10.1111/evo.13368

[2] Ergon, R. (2019). Quantitative genetics state-space modeling of phenotypic plasticity and evolution, Modeling, Identification and Control. 12:51--69. doi:10.4173/mic.2019.1.5

[3] Ergon, R. (2022). The important choice of reference environment in microevolutionary climate response predictions, Ecology and Evolution, 2022. 12:e8836. doi:10.1002/ece3.8836

[4] Ergon, R. (2022). Microevolutionary system identification and climate response predictions, Modeling, Identification and Control, 2022. 43:3:91--99. doi:10.4173/mic.2022.3.1

[5] Ergon, T. and Ergon, R. (2017). When three traits make a line: evolution of phenotypic plasticity and genetic assimilation through linear reaction norms in stochastic environments, J. Evol. Biol.. 30:486--500. doi:10.1111/jeb.13003

[6] Hadfield, J. (2010). The misuse of blup in ecology and evolution, Am. Nat.. 175:116--125. doi:10.1086/648604

[7] Henderson, C. (1950). Estimation of genetic parameters, Annals of Mathematical Statistics. 21:309--310.

[8] Lande, R. (1979). Quantitative genetic analysis of multivariate evolution, applied to brain:body size allometry, Evolution. 33:402--416. doi:10.1111/j.1558-5646.1979.tb04694.x

[9] Lande, R. (2009). Adaptation to an extraordinary environment by evolution of phenotypic plasticity and genetic assimilation, J. Evol. Biol.. 22:1435--1446. doi:10.1111/j.1420-9101.2009.01754.x

[10] Lande, R. and Arnold, S. (1983). Measurement of selection on correlated characters, Evolution. 37:1210--1226. doi:10.1111/j.1558-5646.1983.tb00236.x

[11] Lush, J. (1937). Animal Breeding Plans, Iowa State College Press.

[12] LynchM, W.B. (1998). Genetics and Analysis of Quantitative Traits, Sinauer Associates.

[13] Morrissey, M., Kruuk, L., and Wilson, A. (2010). The danger of applying the breederâ€™s equation in observational studies of natural populations, J. Evol. Biol.. 23:2277--2288. doi:10.1111/j.1420-9101.2010.02084.x

[14] Pick, J., Lemon, H., Thomson, C., and Hadfield, J. (2022). Decomposing phenotypic skew and its effects on the predicted response to strong selection, Nature Ecology & Evolution. 6:774--785. doi:10.1038/s41559-022-01694-2

[15] Price, G. (1970). Selection and covariance, Nature. 227:520--521. doi:10.1038/227520a0

[16] Price, G. (1972). Extension of covariance selection mathematics, Annals of Human Genetics. 35:485--490. doi:10.1111/j.1469-1809.1957.tb01874.x

[17] Robertson, A. (1966). A mathematical model of the culling process in dairy cattle, Animal Science. 8:95--108. doi:10.1017/S0003356100037752

[18] Robinson, G. (1991). That blup is a good thing: The estimation of random effects, Statistical Science. 6:15--32. doi:10.1214/ss/1177011926

[19] Walsh, B. and Lynch, M. (2018). Evolution and Selection of Quantitative Traits, Oxford University Press.

**BibTeX:**

@article{MIC-2022-4-2,

title={{A BLUP derivation of the multivariate breeder's equation, with an elucidation of errors in BLUP variance estimates, and a prediction method for inbred populations}},

author={Ergon, Rolf},

journal={Modeling, Identification and Control},

volume={43},

number={4},

pages={131--140},

year={2022},

doi={10.4173/mic.2022.4.2},

publisher={Norwegian Society of Automatic Control}

};