Reduced Animal Models 1 Introduction In situations where many offspring can be generated from one mating as in fish poultry or swine or where only a few animals are retained for breeding the genetic evaluation of all animals may not be necessary Only animals that are candidates for becoming the parents of the next generation need to be evaluated Pollak and Quaas 1980 came up with the reduced animal model or RAM to cover this situation Consider an animal model with periods as a fixed factor and one observation per animal as in the table below Animal Model Example Data Animal Sire Dam Period Observation 5 1 3 2 250 6 1 3 2 198 7 2 4 2 245 8 2 4 2 260 9 2 4 2 235 4 - - 1 255 3 - - 1 200 2 - - 1 225 11 Usual Animal Model Analysis Assume that the ratio of residual to additive genetic variances is 2 The MME for this data would be of order 11 nine animals and two periods The left hand sides and right hand sides of the MME are 3 0 1 1 0 0 0 0 0 0 5 0 0 0 1 1 1 1 0 0 4 0 2 0 2 2 0 0 0 1 0 0 6 0 3 0 0 2 2 2 1 0 2 0 5 0 2 2 0 0 0 1 0 0 3 0 6 0 0 2 2 2 2 0 2 0 5 0 0 0 0 2 0 2 0 0 5 0 0 0 0 2 0 2 0 0 5 0 0 0 2 0 2 0 0 0 5 0 0 2 0 2 0 0 0 0 5 680 1188 0 225 200 255 250 198 245 260 235 1
and the solutions to these equations are ˆb1 ˆb2 â 1 â 2 â 3 â 4 â 5 â 6 â 7 â 8 â 9 2258641 2363366 24078 13172 102265 113172 23210 127210 67864 97864 47864 A property of these solutions is that 1 A 1 â 0 which in this case means that the sum of solutions for animals 1 through 4 is zero 12 Reduced AM RAM results in fewer equations to be solved but the solutions from RAM are exactly the same as from the full MME In a typical animal model with a as the vector of additive genetic values of animals there will be animals that have had progeny and there will be other animals that have not yet had progeny and some may never have progeny Denote animals with progeny as a p and those without progeny as a o so that a a p a o In terms of the example data a p a o a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 Genetically for any individual i the additive genetic value may be written as the average of the additive genetic values of the parents plus a Mendelian sampling effect which is the animal s specific deviation from the parent average ie a i 5a s + a d + m i Therefore a o Ta p + m 2
where T is a matrix that indicates the parents of each animal in a o and m is the vector of Mendelian sampling effects Then and a ap a o I T a p + 0 m V ara Aσa 2 I A T pp I T σ 2 a + 0 0 0 D σ 2 a where D is a diagonal matrix with diagonal elements equal to 1 25d i and d i is the number of identified parents ie 0 1 or 2 for the i th animal and V ara p A pp σ 2 a The animal model can now be written as yp Xp Zp 0 b + y o X o 0 Z o I T a p + e p e o + Z o m Note that the residual vector has two different types of residuals and that the additive genetic values of animals without progeny have been replaced with Ta p Because every individual has only one record then Z o I but Z p may have fewer rows than there are elements of a p because not all parents may have observations themselves In the example data animal 1 does not have an observation therefore Consequently Z p 0 0 0 0 0 0 e R V ar p e o + m Iσ 2 e 0 0 Iσe 2 + Dσa 2 I 0 σe 2 0 R o The mixed model equations for the reduced animal model are X p X p + X or 1 o X o X pz p + X or 1 o T Z px p + T R 1 o X o Z pz p + T R 1 o T + A 1 pp α ˆb â p 3
X p y p + X or 1 o Z py p + T R 1 o Solutions for â o are derived from the following formulas y o y o â o Tâ p + ˆm where ˆm Z oz o + D 1 α 1 y o X oˆb Tâp and Using the example data T D diag 5 0 5 0 5 0 5 0 0 5 0 5 0 5 0 5 0 5 0 5 5 5 5 5 5 then the MME with α 2 are The solutions are as before ie 3 0 1 1 0 4 8 12 8 12 0 8 24 0 4 0 1 12 0 36 0 6 1 8 4 0 34 0 1 12 0 6 0 36 ˆb1 ˆb2 â 1 â 2 â 3 â 4 680 9504 1792 521 3792 551 ˆb1 2258641 â 1-24078 â 3-102265 ˆb2 2363366 â 2 13172 â 4 113172 2 Backsolving for Omitted Animals To compute â o first calculate ˆm as: I + D 1 α 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5 4
and y o X oˆb Tâ p 250 198 245 260 235 63172 63172 63172 63172 63172 2258641 2363366 ˆm I + D 1 α 1 y o X oˆb Tâp 39961 64039 4692 34692 15308 Tâ p + ˆm 23211 127211 67864 97864 47864 The reduced animal model was originally described for models where animals had only one observation but Henderson1988 described many other possible models where this technique could be applied Generally with today s computers there is not much problem in applying regular animal models without the need to employ a reduced animal model 5
3 EXERCISES Below are data on animals with their pedigrees Animal Sire Dam Year Group Observation 1 - - 192 24 2 - - 192 12 3 - - 192 33 4 - - 1920 2 27 5 - - 1920 2 8 6 - - 1920 2 19 7 1 4 1922 3 16 8 1 4 1922 3 28 9 1 4 1922 3 30 1 5 1922 3 42 11 1 6 1922 3 37 12 1 6 1922 3 44 13 2 4 1922 4 11 14 2 4 1922 4 18 15 2 4 1922 4 23 16 2 5 1922 4 9 17 2 5 1922 4 2 18 2 6 1922 4 25 19 7 16 1924 5 14 20 7 16 1924 5 19 21 7 16 1924 5 17 22 13 1924 5 39 23 13 1924 5 43 Assume a heritability of 032 for this trait Analyze the data with both the usual animal model and the reduced animal model y ijk Y i + G j + a k + e ijk where Y i is a year effect G j is a group effect a k is an animal effect and e ijk is a residual effect The solutions to both analyses should be identical In the RAM backsolve for â 23 What about the prediction error variance for â 23? 6