Best unbiased linear Prediction: Sire and Animal models

Transcription

1 Best unbiased linear Prediction: Sire and Animal models Raphael Mrode Training in quantitative genetics and genomics 3 th May to th June 26 ILRI, Nairobi Partner Logo Partner Logo

2 BLUP The MME of provided a framework to obtain BLUE of k'b, given that k'b is estimable BLUP of the vector of random effects. The properties of BLUP are more or less incorporated in the name: Best - means it maximises the correlation between true (a) and predicted breeding value (â) or minimises prediction error variance (PEV) (var(a - â)). Linear - predictors are linear functions of observations. Unbiased - estimation of realized values for a random variable such as animal breeding values and of estimable functions of fixed effects are unbiased (E(a = â)). Accounts for selection if all data on which selection has been based is included Prediction - involves prediction of true breeding value. 2

3 Numerator relationship matrix In lecture, we estimated the fixed effect solutions and predicted genetic merit of 3 sires. Assumed that the sires were unrelated. Usually animals or sires tended to be related and the genetic relationship among these animals is incorporated The genetic covariance among individuals is comprised of three components: the additive genetic variance the dominance variance and the epistatic variance. This lecture will address only the additive genetic relationship Use of additive genetic relationship matrix usually increases the accuracies of evaluations and should help account for previous selection decisions if all pedigrees are utilised 3

4 Numerator relationship matrix The numerator relationship matrix (A) describes the additive genetic relationship among individuals The additive genetic relationship between animals i and j is twice the probability two genes taken at random from i and j are of identical by descent. It is equal to twice the coancestry or the coefficient of kingship The matrix A is symmetric and its diagonal element for animal i (a ii ) is equal to + F i, with F i is the inbreeding coefficient off-diagonal element, a ij equals the numerator of the coefficient of relationship When multiplied with genetic variance (σ 2 u) it is equal to the covariance of breeding values. Thus var(u i ) = a ii σ 2 u = ( + F i )σ 2 u. 4

5 Recursive method for computing A Henderson (976) described method for calculating the matrix A Pedigree are coded to n and ordered such that parents precede their progeny. If both parents (s and d) of animal i are known a ji = a ij =.5(a js + a jd ) ;j = to i- a ii = +.5(a sd ) If only one parent s is known and assumed unrelated to the mate a ji = a ij =.5(a js ) ;j = to i- a ii = If both parents are unknown and are assumed unrelated a ji = a ij = ;j = to i- a ii = 5

6 Example pedigree Calf Sire Dam unknown a = + = a 2 =.5(+) = = a 2 a 22 = + = a 3 =.5(a +a 2 ) =.5(. + ) =.5 = a 3 a 23 =.5(a 2 +a 22 ) =.5( +.) =.5 = a 32 6

7 Recursive method for computing A a 66 = +.5(a 52 ) = +.5(.25) =.25 From the above calculation the inbreeding coefficient for calf 6 is.25 7

8 Decomposing the matrix A A = TDT' where T is a lower triangular matrix. Non-zero elements of T, say t ij, is the coefficient of relationship between animals i and j, if i and j are direct relatives It can easily be computed applying the following rules: For the i th animal t ii = If both parents (s and d) are known t ij =.5(t sj + t dj ) If only one parent (s) is known t ij =.5(t sj ) If neither parents is known t ij = 8

9 Decomposing the matrix A The Mendelian sampling (m) for an animal i m i = u i -.5(u s + u d ) var(m i ) = var(u i ) - var(.5u s +.5u d ) = var(u i ) - var(.5u s ) - var(.5u d ) + 2cov(.5u s,.5u d ) = + F i )σ 2 u -.25a ss σ 2 u -.25a dd σ 2 u -.5a sd σ 2 u var(m i )/σ 2 u = d ii = ( + F i ) -.25a ss -.25a dd -.5a sd Since F i =.5a sd d ii = -.25( + F s ) -.25( + F d ) d ii = (F s + F d ) 9

10 Decomposing the matrix A D is a diagonal matrix of the covariance matrix for Mendelian sampling. It is calculated as: if both parents of animal i are known, the d ii = (F s + F d ) If only one parent (s) is known diagonal element is d ii = (F s ) and if no parent is known d ii =

11 Matrix T for example pedigree The matrix T is

12 Matrix D for example pedigree D = diag(.,.,.5,.75,.5,.469) For instance, animal 4 has only the sire known which is not inbred, therefore d 44 =.75 - =.75 and d 66 = (.25 + ) =.469 because both parents are known and the sire has inbreeding coefficient of.25 2

13 Deriving rules for A inverse A - = (T - )'D - T - I - M = T -, where M is a matrix that traces genes from parents to offspring = D - = Diag(,,2,.333,2,2.33) 3

14 Deriving rules for A inverse Therefore A - = (T - )'D - T - is:

15 Rules for the inverse of A (ignoring inbreeding Henderson (976) used equation to develope rules for A - D - = 2 if both parents are known 4/3 if one parent is known if no parent is known If d i = diagonal element of D - for animal i. d i = 4/(2 + no of parents unknown) 5

16 Rules for the inverse of A (ignoring inbreeding A - = If both parents of the i th animal are known, add d i to the (i,i) element -d i /2 to the (s,i), (i,s), (d,i) and (i,d) elements d i /4 to the (s,s), (s,d), (d,s) and (d,d) elements If only one parent (s) of the i th animal is known, add d i to the (i,i) element -d i /2 to the (s,i) and (i,s) elements d i /4 to the (s,s) element Neither parents of the i th animal are known, add d i to the (i,i) element 6

17 Rules for the inverse of A The same rules can be used when accounting for inbreeding but with the elements of D - computed accounting for the inbreeding of parents. Verify this using the pedigree above 7

18 Sire model Using of the performance of progeny to evaluate the genetic merits of their sires is referred to as the sire model. For instance, the genetic merits of bulls can be predicted on the basis of the milk production Given that the covariance between the sire and his progeny is.5, this method predicts the probable transmitting ability (PTA) of the sires which is half the breeding values of the sires. 8

19 Sire model MME with the relationship matrix incorporated are XX ZX XZ bˆ Xy = Z Z+ A aˆ Zy with α = σ 2 e/σ 2 s = 4-h 2 /h 2 9

20 Example: Sire model Same data as in lecture, a sire model with A among the 3 sires incorporated. Assume that α = σ 2 e/σ 2 s = /2 and the data of recoding is as follows: Cow Herd Calving class Sire Test day milk yield (kg) The sires,2, 3 were recorded as 5,6,7 respectively and cows to recoded from 8 to 7 2

21 Example : Sire model Bull Sire dam Using the rules outlined earlier, the A - for the above pedigree is A symmetric

22 22 Example: Sire Model The matrix X is as defined in lecture. Z is now set up to included the 4 animals in the pedigree The addition of A - α to Z Z gives Z Z Z and A Z Z

23 23 Example : Sire Model The MME are: a a a a a a a b b b b sym

24 Example: Sire Model Fixed effects Herds Calving class Random sire effects

25 Accuracy and prediction error variance The accuracy (r) of predictions is the correlation between true and predicted breeding values. Dairy cattle evaluations, the accuracy of evaluations is usually expressed in terms of reliability, which is r 2. Calculation for r or r 2 require the diagonal elements of the inverse of the MME. If the coefficient matrix of the MME is represented as C: 25

26 Accuracy and prediction error variance C C C 2 C C 2 22 and a generalise d inverse Prediction error variance (PEV) = var(a-â) = C 22 σ 2 e PEV could be regarded as the fraction of additive genetic variance not accounted for by the prediction. C C C 2 C C 2 22 Therefore PEV = C 22 σ 2 e = ( - r 2 )σ 2 a or PEV = ( - r 2 )σ 2 s for a sire model with r 2 = squared correlation between the true and estimated breeding values 26

27 Accuracy and prediction error variance Thus for animal i d i σ 2 e = ( - r 2 )σ 2 a where d i is the i th diagonal element of the C 22 d i σ 2 e/σ 2 a = - r 2 r 2 = - d i α The standard error of prediction (SEP) is SEP = var(a - â) = d i σ 2 e for animal i Note that r 2 = - ( SEP 2 /σ 2 a ). ASReml gives SEP and not r 2, so compute r 2 from SEP Diagonal elements from the 3 by 3 block for the 3 sires in example were.975,.97 and.45. The corresponding reliabilities for the 3 sires therefore equals were.55,.57 and.54 respectively. 27

28 Uses of the PEV Note: v ar A v ar ˆ A PEV v ara ˆ 2 A PEV The effective number of progeny or records (n e ) also related to PEV: PEV = ( r 2 )σ 2 a = α / (n e + α)σ 2 a Therefore n e can be computed as: n e = [PEV - σ s -2 ]σ e 2 with a sire model n e = [PEV - σ A -2 ]σ E 2 with an animal model 28

29 Individual Animal model In the previous example, the genetic merit of only the sires of cows were predicted. The main advantage: number of equations are reduced since only sires are evaluated compared to all animals in the data set. Disadvantages: genetic merit of the mate (dam of progeny) is not accounted for. It is assumed that all mates are of similar genetic merit and this can result in bias in the predicted breeding values if there is preferential mating. Therefore an evaluation that predicts the genetic merit of all animals in the data set would overcome this problem. This is called the individual animal model Uses the usual MME to evaluate all animals with records and all their relatives in the pedigree. However the α term with an animal model, α = σ 2 e/σ 2 a = - h 2 /h 2 29

30 Individual Animal model Calves Pens Sire Dam WWG (kg) Pen unknown Pen Pen Pen Pen Assume = σ 2 a=2 and σ 2 e= 4 and α= σ 2 e/σ 2 a= 4/2 =2 3

31 3 Incidence matrices ' ; y = X ; Z = X Z =

32 Matrices for the MME Z X = transpose of X Z X' y Tranpose of Z' y Z'Z diag(,,,,,,,) 32

33 The A - for the example data A - =

34 Solutions to the MME Effects Solutions Pens Animals

35 Fixed effect solutions From first row of MME (X X)b = X y - (X Z)â b = (X X) - X (y - Z â) bˆ ( y aˆ ) / i j ij j ij diag i - Solution for calves in pen is b = [( ) - ( )]/ 3 =

36 Understanding animal EBV From second row of MME (Z Z+A - α)â = Z y - (Z X)b (Z Z+A - α)â = Z (y - Xb) (Z Z+A - α)â = (Z Z)YD with YD = ZZ - Z (y Xb) (Z Z+ u ii α)â i = αu ip (â s + â d ) + (Z Z)YD + ασ k u im (â anim -.5â m ) (Z Z+u ii α)â i = αu par (PA) + (Z Z)YD +.5αΣk uprog (2â anim - â m ) 36

37 Understanding animal EBV EBV is made of contributions from: Parents average (PA), its record and progeny (PC) EBV = npa + n2yd + n3 PC Where n + n2 + n3 = Numerator of: n = 2α, 4/3α or α if both, or no parents is known n2 = number of records n3 = ½α or ⅓α when mate is known or not Denominator = sum of numerators for n,n2,n3 YD = record of animal corrected for fixed effects 37

38 Understanding animal EBV Animal 8 as an example: EBV 8 = n(pa) + n2(y8 b) where n = 2 λ /5=4/5 and n2 = /5 EBV 8 = n(ebv 3 +EBV 6 )/2) + n2 ( ) EBV 8 = n(.68) + n2 ( ) =.83 38

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