Predicted breeding values for litter size with an animal model used in the Danish pig breeding program
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Knanipl Sorensen
Beretning Foulum 1991
STATENS HUSDYRBRUGSFORSØG
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Dyrefysiologi og Biokemi Forsøg med Fjerkræ og Kaniner Forsøg med Kvæg og Får Forsøg med Pelsdyr
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NATIONAL INSTITUTE OF A N I M A L SCIENCE
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ISSN 0105-6883
Statens Husdyrbrugsforsøg
Report from the National Institute ofAnimal Science, Denmark
Daniel Sorensen
Predicted breeding values for
litter size with an animal model used in the Danish pig breeding program
Med dansk sammendrag
Manuskriptet afleveret august 1991
Trykt i Frederiksberg Bogtrykkeri a-s 1991
F O R O R D
Et v æ s e n t l i g t e l e m e n t i d e t d a n s k e s v i n e a v l s s y s t e m e r e n v e l o r g a n i s e r e t d a t a b a n k , d e r b l e v u d v i k l e t som et s a m a r b e j d s p r o j e k t m e l l e m S t a t e n s H u s d y r b r u g s f o r s ø g o g D a n s k e S l a g t e r i e r . D a t a b a n k e n m u l i g g ø r a n v e n d e l s e a f m o d e r n e m e t o d e r t i l b e r e g n i n g a f g e n e t i s k e p a r a m e t r e s a m t g e n n e m f ø r e l s e a f f o r s k n i n g s p r o j e k t e r b a s e r e t p å d a t a a n a l y s e .
N æ r v æ r e n d e b e r e t n i n g d o k u m e n t e r e r d e n m o d e l o g d e n m e t o d e , d e r a n v e n d e s t i l b e r e g n i n g a f a v l s v æ r d i e r f o r a n t a l f ø d t e g r i s e p r k u l d , s å d a n s o m d i s s e e r i v æ r k s a t i d e t d a n s k e s v i n e a v l s p r o g r a m . T h o r k i l d V e s t e r g a a r d h a v d e i e n å r r a k k e i h æ r d i g t a r b e j d e t m e d a t n y t t i g ø r e f o r s k n i n g s r e s u l t a t e r i a v l s p r o g r a m m e t . H a n s e n t u s i a s m e i d e d i s k u s s i o n e r , h a n d e l t o g i t i l k n y t t e t a v l s p r o g r a m m e t s u d v i k l i n g , s p i l l e d e e n v æ s e n t l i g r o l l e
M a n u s k r i p t e t e r r e n s k r e v e t o g f o r b e r e d t t i l t r y k n i n g a f A a s e S ø r e n s e n .
S t a t e n s H u s d y r b r u g s f o r s ø g , F o u l u m J u l i 1 9 9 1 .
Sammendrag 1 Summary
1. Introduction 3 2.The individual animal model with groups ... 4
3.The reduced animal model..,..,.,...,»...,.».»....,.,... 6 4.The model for prediction of breeding values for number of
born piglets... 13
5 .Assumptions of the model 14 6.Possible developments of the model ... 15
7 .An example 19 References 25
S a m m e n d r a g
I d e n n e b e r e t n i n g d o k u m e n t e r e s d e n m o d e l , s o m a n v e n d e s i det d a n s k e s v i n e a v l s p r o g r a m til b e r e g n i n g af a v l s v æ r d i e r for a n t a l f ø d t e g r i s e per k u l d . M o d e l l e n s a n v e n d e l s e i l l u s t r e r e s m e d et lille e k s e m p e l . M o d e l l e n b e t e g n e s som en e n k e l t - d y r - m o d e l m e d g e n t a g n e m å l i n g e r ( r e p e a t a b i l i t y a n i m a l m o d e l ) , og p a r a m e t r e n e e s t i m e r e s ved h j æ l p af m i k s e d e m o d e l m e t o d e r .
I d e n o p r i n d e l i g e e n k e l t - d y r - m o d e l antages d e t , at a v l s v æ r d i e r n e stammer fra en p o p u l a t i o n , hvis g e n n e m s n i t er n u l . Indenfor L a n d r a c e og Y o r k s h i r e er m a n g e dyr b l e v e t i m p o r t e r e t fra l a n d e , h v i s g e n n e m s n i t for k u l d s t ø r r e l s e a f v i g e r i n d b y r d e s , og dette m e d f ø r e r , at den o v e n n æ v n t e a n t a g e l s e ikke o p f y l d e s . For at råde bod på d e t t e i n k l u d e r e s en g r u p p e e f f e k t i m o d e l l e n . D e n a n v e n d t e m o d e l er følgende:
Yi j k l m n o = Hi + Sj + Kk l + £ xm Gk m + ai j k l m n + P i j k l m n * ei j k l m n o
h v o r Yijk l m n o r e p r æ s e n t e r e r a n t a l fødte grise fra k u l d o og so n . H£, Sj og Kk l er y s t e m a t i s k e e f f e k t e r , d e r r e p r æ s e n t e r e r h e n h o l d s v i s b e s æ t n i n g x år x b e f r u g t n i n g s m e t o d e (KS eller n a t u r l i g b e f r u g t n i n g ) , å r s t i d og endelig race x k u l d n u m m e r . E xm G ^ er g r u p p e e f f e k t e n s bidrag til a n t a l fødte g r i s e indenfor race k (I xm = 1). ai j k l m n, pi j k l n m og ei j k l m n er t i l f æ l d i g e e f f e k t e r , der r e p r æ s e n t e r e r h e n h o l d s v i s a v l s v æ r d i e r , p e r m a n e n t e m i l j ø e f f e k t e r og e n d e l i g r e s t e f f e k t e n fra kuld o . A r v e l i g h e d e n og g e n t a g e l s e s k o e ff i c i e n t e n antages at være h e n h o l d s v i s 0 , 1 0 og 0 , 1 5 . I n v e r s e n af s l æ g t s s k a b s k or r e l a t i o n s m a t r i een er b e r e g n e t efter en a l g o r i t m e afledt af H e n d e r s o n (1976).
I b e r e t n i n g e n redegøres for n o g l e m a t e m a t i s k e og E D B - m æ s s i g e o v e r v e j e l s e r t i l k n y t t e t o p b y g n i n g e n og l ø s n i n g e n af den m i k s e d e m o d e l s l i g n i n g s s y s t e m .
E n række a n t a g e l s e r , som ligger til grund for e s t i m a t i o n af a v l s v æ r d i e r , er d i s k u t e r e t . En a n t a g e l s e , som m u l i g v i s ikke er o p f y l d t , e r , at a n t a l fødte grise kun p å v i r k e s af g e n e r , som h a r en a d d i t i v v i r k n i n g . D e t t e m e d f ø r e r , at g r u p p e e f f e k t e r k o m b i n e r e s på en a d d i t i v m å d e . Det er h e n s i g t e n , at h e t e r o s e e f f e k t e r v i l blive i n d a r b e j d e t i p r o c e d u r e n i f r e m t i d e n .
S u m m a r y
A m o d e l for selection for litter size used in the D a n i s h pig breeding progra is d e s c r i b e d , and an example of its u s e is g i v e n . T h e m o d e l is based on repeatability reduced animal model w i t h groups and the p a r a m e t e r s are estimate using m i x e d model m e t h o d o l o g y .
The usual assumption of the animal m o d e l is that b r e e d i n g values are sample from a population with zero m e a n . W i t h i n the Landrace and Yorkshire b r e e d s , considerable amount of importation has t a k e n place over the y e a r s and it is clea that there are rather large differences in litter size among the countries o o r i g i n . The purpose of introducing groups in the model is to take account of thi f a c t . The m o d e l which is used is:
Yi j k l m n o = Hi + sj + Kkl + £ xm Gk m + ai j k l m n + Pijklmn + ei j k l m n o
w h e r e , Yijk l m n o is the ot h record (number of born p i g l e t s ) of sow n , H £ , S j , Kk
are fixed effects of herd x year x type of f e r t i l i s a t i o n , season and breed : parity n u m b e r , respectively, I xm G j ^ is the fixed group contribution nested within breed k (z xm = 1) and ai j k l m n. pi j k l m n, and ei j k l m n are breeding v a l u e s , permanent environmental e f f e c t s , and r e s i d u a l s , respectively (random e f f e c t s ) . Th<
h e r i t a b i l i t y and repeatability used as p r i o r s are 0.10 and 0 . 1 5 , r e s p e c t i v e l y . Thi inverse of the additive genetic relationship m a t r i x is c o m p u t e d using an algorithr from H e n d e r s o n (1976).
Some mathematical aspects associated with the d e v e l o p m e n t of the m o d e l ari d i s c u s s e d , as well as several computing details involved w i t h the building ant solution of the mixed m o d e l e q u a t i o n s .
A n u m b e r of assumptions of the m o d e l are d e s c r i b e d . One of the most contentious assumptions is that the trait is affected by genes that act a d d i t i v e l y , w i t h i n anc between loci. Group effects combine therefore in an a d d i t i v e m a n n e r . Thi;
assumption 'is hardly tenable in a trait like litter s i z e , and it is hoped thai h e t e r o s i s ' e f f e c t will be incorporated in the m o d e l in the f u t u r e .
3
lj_Introduction
It is well established that increased litter size improves the economic efficiency of pig production. This would be further enhanced if production traits such as leanness were to be approaching an optimum for a given slaughter weight, so that further genetic improvement as carcass fat declines becomes difficult to achieve (Hill and W e b b , 1982).
Reasons why litter size has not been included in the breeding goal of the Danish pig breeding program have been, firstly, that experimental evidence showing that it can be enhanced by selection has been lacking. Secondly, it has been argued that genetic improvement of litter size using traditional selection techniques would lead to very slow progress, essentially, due to the low heritability of the trait and its sex limited expression (Smith, 1964).
Recently however, some encouraging results have become available. Le Roy et al.
(1987) presented the results of a hyperprolific sow experiment carried out under farm conditions, where the estimates of realised heritability for total number of born piglets and total number of liveborn piglets were 0.14+0.05 and 0 . 1 0 + 0 . 0 5 , respectively. A l s o , Avalos and Smith (1987) have shown that use of a family index incorporating several sources of information can lead to expected annual responses of up to 0.51 pigs per litter.
With these results as a background, in 'June 1988 the National Institute of Animal Science made available to the Danish pig breeding program, predicted breeding values for total number of born piglets of breeding animals. The trait was used partly because overall reproductive efficiency can be enhanced most effectively by increasing litter size (Smith et a l , 1983) and partly, because this is the most reliable data available on reproduction traits in the Danish databank for pig production data.
The predicted breeding values were derived using mixed model techniques, and the model used is known as the repeatability animal model with groups. This evaluation procedure takes into account the heritability and repeatability of the trait, and makes use of all the records in different parities of the animal itself, and of all the records from the animal's relatives present in the data set. There can be more than one thousand records available from an animal's relatives, and all the information is combined in an optimum manner in order to evaluate its breeding value. Both males and females receive a predicted breeding /alue. These predicted breeding values are simultaneously corrected for such
effects as h e r d - y e a r , s e a s o n , parity n u m b e r , breed and type of conceptior (artificial insemination versus natural m a t i n g ) . A c c o u n t is also taken of the fact that some animals are imported from countries whose means d i f f e r . For e x a m p l e , the population of Finnish Landrace sows has a m e a n litter size that differs from the population of Danish Landrace s o w s . R a t h e r than assuming that all breeding values are sampled from one p o p u l a t i o n , the group effect in the m o d e l reflects that the sampling process involves different p o p u l a t i o n s , and even m i x t u r e of p o p u l a t i o n s , whose m e a n s for the trait in question d i f f e r .
The first years are considered an e x p l o r a t o r y period during which alternative models will be fitted to the data and the sources of v a r i a t i o n contributing to number of born piglets will be i n v e s t i g a t e d . During this initial stage, the index will not be part of the breeding g o a l .
The purpose of this publication is to describe and document the m e t h o d s followed in the development of these p r e d i c t e d breeding v a l u e s .
model with groups
The predicted breeding values for total number of born piglets are obtained using m i x e d model techniques on a repeatability reduced animal m o d e l with g r o u p s . The computer program was w r i t t e n in PL/I and it can a c c o m m o d a t e an arbitrary number of fixed and random e f f e c t s .
In the individual animal model excluding g r o u p s , the phenotypic value is written in terms of fixed effects and the contribution from the animal's additive genetic and non-genetic v a l u e s . For e x a m p l e , assuming an additive genetic m o d e l , for a given a n i m a l , j , say, that makes a record Cy£j) in herd (H) i, we write:
where a - is the additive genetic value and e£j is the residual non-genetic e f f e c t . The usual assumption in this m o d e l with respect to the breeding values is that they have null m e a n s .
In order to take into account the fact that animals o f t e n originate from populations whose means d i f f e r , a group effect is included in the m o d e l . This idea
was introduced in the context of a sire model by T h o m p s o n (1979) and extended to an individual animal m o d e l by R o b i n s o n (1986), W e s t e l l and Van Vleck (1987) and Q u a a s (1988). The concept is simple and intuitively appealing and it arises from the fact that an animal receives half of its genes from one parent and the other half from the o t h e r . If animals 1 and 2 , which are sampled from populations with m e a n s gj and g2, r e s p e c t i v e l y , are parents of animal 3 , then under the group m o d e l , the expected breeding values of 1, 2 , and 3 are gj, g2. and 1/2 (gj + g2) ,
r e s p e c t i v e l y .
A matrix representation of the individual animal m o d e l with groups is:
y = Xb + Gg + Za + e (1)
w h e r e y is the vector of o b s e r v a t i o n s , b is the vector of fixed effects excluding g r o u p s , g is the v e c t o r of group e f f e c t s , and a and e are random v e c t o r s . The matrices X , G and Z are known design m a t r i c e s . The m a t r i x G has numbers of columns equal to the number of groups, and each row sums to o n e . The elements of a row of G are the proportion with w h i c h a group effect contributes to the given i n d i v i d u a l . They can also be interpreted as the proportion of genes originating from the different g r o u p s .
The first and second moments of (1) are:
E y = Xb + Gg
a 0
e 0
and
Var y = ZAZ'o 2 a + R ZA a 2 a R (2)
a ( Z A ) ' a2 a A "2, 0
e R ' 0 R
In (2), A is the numerator relationship matrix and a2 a is the additive genetic variance in the populations from w h i c h base animals are sampled. R is usually assumed to be equal to I 0 2 e, where I is the identity matrix and 0 2 e is variance of the residual e f f e c t . T h u s , the m o d e l assumes that groups have common v a r i a n c e . The important concept in the group m o d e l is that breeding values are given by:
a
T h e assumption in (3) is that E ( a * ) = Q g ,
w h e r e ZQ = G.
Like any other fixed e f f e c t , groups may not be e s t i m a b l e . While predicted genetic values are always e s t i m a b l e , expressing them in the form of (3) m a y render them n o n - e s t i m a b l e . Differences among predicted breeding values expressed as in (3) can be estimable if group effects are not nested w i t h i n other fixed effects.
If group effects are nested w i t h i n b r e e d s , for e x a m p l e , then estimable functions of differences among breeding v a l u e s , expressed as in (3), can be m a d e within breeds o n l y .
The total number of equations that have to be solved to obtain the predicted genetic values is equal to the number of fixed effects plus the total number of animals in the system. With large data s e t s , this can be a very large n u m b e r and computing costs can be s u b s t a n t i a l .
3 . The reduced animal m o d e l w i t h g r o u p s
A n alternative formulation is to make use of M e n d e l i a n theory, and to reparameterise the m o d e l , so that the total number of breeding values is reduced from the total number of a n i m a l s , to the number of animals that have o f f s p r i n g . This is known as the reduced animal m o d e l , o r i g i n a l l y p r o p o s e d by Quaas and Pollak (1980).
Let as ( ad) be the additive genetic effect of the sire (dam). T h e n , ignoring groups for the m o m e n t , the additive genetic effect of an i n d i v i d u a l , aQ, for the case of both parents k n o w n , one parent k n o w n , the sire s a y , and none of the parents known is w r i t t e n in (4), (5) and (6):
1/2 ag + 1/2 ad + m2 (4)
1/2 as + m i (5)
mo (6)
In (4), n»2 is the M e n d e l i a n sampling deviation and its v a r i a n c e is equal to:
V a r ( m2) = 1/2 o2 a (1 - l / 2 ( Fg + Fd) ) (7)
7
(Foulley and Chevalet (1981)) where o2 a is the additive g e n e t i c variance of the population from which the breeding v a l u e s were sampled and Fs ( Fd) is the inbreeding coefficient of the sire (dam), such that V a r ( as) = a a? ( 1 + Fg). The
variance of mj , V a r ( m ^) can be shown to equal:
Var{n»|) = 3/4 a2 a( l - 1/3 Fg) (8)
where Fs is the inbreeding coefficient of the only known p a r e n t , the sire.
T h e v a r i a n c e of mQ, V a r ( mQ) is equal to
V a r ( m0) = o2 a (9)
The Mendelian sampling d e v i a t i o n , m2 in (1) can also be v i e w e d as the sum of the deviations of each parental g a m e t e , with respective contributions of 1/4 o2 a( l - Fs) and 1/4 o2 a (1 - Fd) to Var ( m2).
In the reduced animal m o d e l , animals that have no p r o g e n y are called n o n - p a r e n t r e c o r d s , and are w r i t t e n in terms of equations (4), (5), or (6), depending on the number of known p a r e n t s , and this a p p r o a c h leads to having to solve d i r e c t l y , only the breeding values of animals that have p r o g e n y , w h i c h are called parent r e c o r d s . This is particularly beneficial in a species like p i g s , where most animals contribute as non-parents and therefore savings in computing costs can be s u b s t a n t i a l .
T h e introduction of groups in the reduced animal m o d e l leads to changes in (5) and (6). W i t h one parent k n o w n , (5) becomes:
a *Q = 1/2 a *s + 1/2 gd + m{ (10)
With both parents u n i d e n t i f i e d , (6) becomes:
*o = 1 / 2 gs + 1 / 2 8d + mo ' ( 1 1 )
If both parents are identified, the effect of groups is included in the parental c o n t r i b u t i o n , and thus (4) still h o l d s , p r o v i d e d that the breeding values ire interpreted as in (3). Absence of p e d i g r e e information o n sire or dam m u s t be supplemented with information on the group w h i c h the m i s s i n g parent o r i g i n a t e s E rom.
The repeatability reduced a n i m a l model is written in m a t r i x notation as fo1lows;
yf = X b + Gf g + Zf
Go T
w h e r e ,
j£ is the vector of parental records;
Yo is t h e vector of non-parent records;
b is the vector of all fixed effects (for p a r e n t a l and nonparental records);
af i s t he vector of random parental breeding v a l u e s ; g is the vector of fixed group effects;
tn is the vector of random M e n d e l i a n sampling deviations;
p is the vector of random permanent environmental effects;
e is the vector of random pure environmental e f f e c t s .
X , Gf, G0, Z f , M and W are known design matrices and T is a m a t r i x whose number of rows is equal to the number of rows in yQ, and n u m b e r of columns equal to the number of elements in af, with row elements equal to 1/2 or 0 , relating non-parental to parental breeding v a l u e s . The v e c t o r of n o n - p a r e n t a l breeding values ( a *D) is from (3) and (12):
a* o = To a* f + ( Q0 - T0 Qf) g + B (13)
In (13), we notice that a *f = QF g + af, and that QQ - T0QF = O if the non- parent has both parents idenfified. The m a t r i c e s Q{ and Q0 result from partitioning Q in (3) such that parent records precede n o n - p a r e n t s . Q0 is of order (number of non-parental breeding values x number of g r o u p s ) , Qf has order (number of parental breeding values x number of groups) and I0 is a subset of T which associates non-parental with parental breeding values and has order (number of non-parental breeding values x number of parental breeding v a l u e s ) .
Thus:
a" f = Qf g + af Qo ao
The expected vajue of (yf yD) in (12) is assumed to be equal to
9
X b + g , and the second moments of the model are:
= D oz
= I 02
5 V a r ( af) = Af 0 a Var(m)
Var(p)
Var(e) = I o'
(14) (15) (16) (1?)
w h e r e ,
Af is the additive genetic relationship matrix among parental breeding values;
2 D is a diagonal matrix whose elements are given by (7), (8) or (9) with a a
deleted;
I is the identity matrix, o2 a is the additive genetic variance of the 2 2 population from which the base animals were conceptually sampled, a p and a e are the variance due to permanent environmental effects and due to pure environmental effects, respectively. All covariances between a£, m , p and e are zero.
The heritability ( h2) and repeatability (r) are defined as:
,2 = „2 a£_ / ( o4_ + a '
r = ( 0 % + a in) / ( a a + + °%>
(18) (19)
Letting Z ' = (Zf T )' and G ' = ( Gf GQ)' , then the mixed model equations corresponding to the model are:
r 1 (20)
X ' X X'G X'Z X'M X'W 6 = x'y
G ' X G'G G'Z G'M G'W g G'y
Z'X Z'G Z ' Z + Af _ 1k j Z'M Z'W
%
Z'yM ' X M'G M'Z M ' M + D_ 1k2 M'W m M'y W ' X W'G W'Z W'M W ' W + I k3 P
_
W ' yw h e r e , A 1 f is the inverse of the additive genetic relationship matrix among parental breeding v a l u e s ,
k! = k2 = o2e / o2a = (l-r)/h2
k3 = a2e/a2p = (l-r)/(r-h2)
(21)
(22)
Under the conditions of the m o d e l described a b o v e , the system in (20) provides best linear unbiased estimators (BLUE) of estimable functions of fixed effects and best linear unbiased predictors (BLUP) of random effects ( H e n d e r s o n , 1973).
In (20) the matrix G is dense and therefore all the blocks containing G will be costly to b u i l d . The resulting dense coefficient m a t r i x is also computationally demanding to solve iteratively. S e c o n d l y , the predicted breeding values as defined in the group model are a *f = Qfgf + af, and these are not directly obtained from (20). Quaas and Pollak (1981) have suggested a transformation that leads to a computationally less demanding system of e q u a t i o n s .
Define a m a t r i x T *, such that:
T* I O O O O O I O O O 0 Qf I 0 0 0 0 0 1 0 0 0 0 0 1
T h e n , inserting T* ]T * between the coefficient m a t r i x and the solution vector in (20), and premultiplying both sides by ( T *- 1) ' , one obtains:
X ' X X ' L X 'Z X'M
L ' X L ' L + Q 'fA 1 fQfk1 L 'Z-Q 'fA- V l L'M Z ' X Z'L-A" fQfkj Z "Z+A V i Z ' M M ' X M ' L M 'Z M ' M + D_ 1k;
W ' X W ' L W 'Z W ' M
where L = G - Z Qf = Gf - Gf = 0 G0 TQf G0 - T Qf
X ' W L'W Z'W M ' W W'W+Iki
b = X ' y g L ' y
â* f Z - y A M ' y P M ' y
(23)
The m a t r i x L is equal to 0 when non-parent records have both parents identified and thus all blocks in (23) containing L become 0 .
From (13), solution to non-parent breeding values are obtained from (24):
S* o = T 3* f + ( G0 - T Qf) g + fi " ( 2 4 )
A remarkable property of the system in (23) is that Qf does not have to be built up explicitly. Quaas (1988) shows how the structure of Qf and A- 1 can be
11
exploited so that the blocks containing Qf can be c o n s t r u c t e d as records are read i n , following a g i v e n set of rules.
T h e predicted v a l u e of i n d i v i d u a l s ' future records is given by 5* + p . where S * ' = ( â *f â *0) ' .
Inspection of the coefficient m a t r i x of (23) shows that M ' M + D_ 1k2 and W W + Ikj are diagonal m a t r i c e s . T h e r e f o r e , to reduce the size of the coefficient m a t r i x , the equations belonging to p and m can be absorbed. This absorption process leads to the system of equations (25):
X'PFL X ' P F X L ' P F X
Z ' P F X Z'PFL
L'PFL + Q ' f A 1f Q f k j -1
1f Q fki
X ' P F Z
L ' P F Z - Q ' f A ^ k j Z ' P F Z + A fKl
b X'PFy g L'PFy â*£ Z'PFy
(25)
w h e r e ,
W I - M(M'PM + I k2) 1 M'P
Solutions to non-parental breeding values and to permanent environmental effects are obtained using back-solving techniques. To solve for m and then for a*
us ing (24), we p r o c e e d as f ol lows. A b s o r b i n g the equation for p in (2 3 ) and solving for m yields:
(M'PM + D !k2) 1 (M'Py - M ' P X b - M'PLg - M ' P Z i *f) (26)
It can be shown by expansion of (26) that the it h element of m is given by:
m . = h . c. (.£ Y . . - .E x b - n.(v 1/2 g + v ,1/2 g ) -i i l j = l l j j -1 l j i s m d n
- n.((1-v ) 1/2 I* + (1-v,) 1/2 a*,)) (27) I S S Q U
where h£ = 1 if individua1 i has one or more r e c o r d s ,
= 0 if it has no record ;
C£ = k3/ (n ik 3 + d£k2 (ni + k3) )
k3 = (1 - r)/(r - h2) ;
= number of records of i n d i v i d u a l i;
di = inverse of the it h element of D , where the elements of D are given in (7), (8), and (9) with a2 a deleted;
Yjj = the jth record of individual i;
*ij = row of m a t r i x X corresponding to jth record of individual i;
6 = solution vector of fixed effects in (25);
VS (vd ) = 1 if sire (dam) of i not known or
= 0 if sire (dam) of i is known;
gm( gn) = group effect of parent m (n);
3 *s( a *d) = predicted breeding value of sire (dam) of i.
If in (23) p is not absorbed, then the solution for fi^ is as in (27), but with
ci = (ni + d^k2) 1 and with - nLp ^ included in the term in brackets.
W i t h m a v a i l a b l e , solution to n o n - p a r e n t a l breeding values are readily obtained from (24). For e x a m p l e , for the it h non-parent with parental predicted breeding values a *s and a"'d,
a *£ = ( l - vs) 1/2 a *s + ( l - vd) 1/2 t *d + vs 1/2 gn + vd 1/2 gm + (28)
W h e n the non-parent does not have a record, the estimate of its M e n d e l i a n sampling deviation is zero and its predicted breeding value is obtained from (28) with m£ d e l e t e d .
The solution for the permanent environmental effects is obtained directly from (23). The last equation in (23) is:
P = ( W W + I k3)_ 1 (W'y - W'Xfi - W' L g - W ' Z S *f - W ' M m ) (29)
By expansion of (29), it can be shown that the it h element of p is:
Pi = wi h i( . | j Y . . - .|j x . . 6 - „ . a * ) (30)
w h e r e w^ = l/(n£ + k3) and all other terms are identified in connection w i t h (27).
13
If i is a n o n - p a r e n t , a*£ is o b t a i n e d from (28).
If a n i n d i v i d u a l does not h a v e a r e c o r d , the e s t i m a t e of its p r e d i c t e d p e r m a n e n t e n v i r o n m e n t a l effect is z e r o .
F r o m a c o m p u t a t i o n a l p o i n t of v i e w , it should be s t r e s s e d that the s y s t e m of e q u a t i o n s (25) can be b u i l t d i r e c t l y , w i t h o u t h a v i n g to a c t u a l l y carry out the a b s o r p t i o n of m . T h i s is a c h i e v e d by r e p a r a m e n t e r i s i n g m o d e l (12) as f o l l o w s :
Yf
y
0Xf xo
b + Gf G0
Zf
T
(31)
w h e r e V a r If 0 0 Da
a e 2 (32)
2a / a2e + I ,
W r i t i n g the m i x e d m o d e l e q u a t i o n s for (31) and ( 3 2 ) , a p p l y i n g the rransformation s u g g e s t e d by Quaas and P o l l a k (1981) and a b s o r b i n g p leads to (25).
4 . T h e m o d e l for p r e d i c t i o n of b r e e d i n g v a l u e s for m b e L i Q g l l l i g M i •
The m o d e l that had b e e n in o p e r a t i o n during 1988/89 is:
Yi j l n o = Hi + Sj + K1 + ai j I n + P ij I n + ei j l n o ( 3 3 )
w h e r e ,
Yi j l n o : t h e ot h r e c o r d ( n u m b e r o f b o r n p i g l e t s) o f s o w n f r o m b r e e d x p a r i t y 1 , f a r r o w i n g i n s e a s o n j , b e l o n g i n g t o h e r d x y e a r x t y p e o f f e r t i l i s a t o n i;
H £: f i x e d e f f e c t o f h e r d x y e a r x t y p e o f f e r t i l i s a t i o n i ( t y p e o f
f e r t i l i s a t i o n : a r t i f i c i a l i n s e m i n a t i o n or n a t u r a l m a t i n g ) ;
Sj : fixed e f f e c t of season j (4 seasons per y e a r ) ;
Ki : fixed e f f e c t of breed x p a r i t y number 1;
ai j I n - random effect of b r e e d i n g v a l u e of a n i m a l n;
PijIn• random e f f e c t of p e r m a n e n t e n v i r o n m e n t a l e f f e c t of a n i m a l n ;
ei j l n o: random r e s i d u a l effect of r e c o r d o .
H i g h e r i n t e r a c t i o n terms that a r e l i k e l y to be b i o l o g i c a l l y a p p r o p r i a t e , like
H Si j ' c o u l d n o t b e f i t t e d b e c a u s e the n u m b e r of o b s e r v a t i o n s in a large p r o p o r t i o n of this herd x y e a r x type of f e r t i l i s a t i o n x s e a s o n i n t e r a c t i o n becomes too s m a l l . Similar c o n s i d e r a t i o n s p r e c l u d e d i n c l u s i o n of b r e e d in this i n t e r a c t i o n .
D u r i n g 1990 this m o d e l has b e e n e x t e n d e d to i n c l u d e g r o u p s . This implies that (33) is changed to (34):
Yi j k l m n o = Hi + Sj + Kk l + ¿xm C k m + ai j k l m n + P i j k l m n + « i j k l m n o ( 3 4 ) w h e r e the p a r a m e t e r s are d e f i n e d in c o n n e c t i o n w i t h (33) . The p a r a m e t e r Kk l has e x p l i c i t y two s u b s c r i p t s w h e r e k r e p r e s e n t s b r e e d and 1 r e p r e s e n t s p a r i t y n u m b e r , a n d Gjgjj, the m t h g r o u p effect is n e s t e d w i t h i n b r e e d k . T h e known c o n s t a n t xm, is t h e c o n t r i b u t i o n of the m t h g r o u p , s u c h that I xm = 1.
T h e total g e n e t i c v a l u e , is g i v e n by:
a* ij k l m n = ^ Gk m + ai j klmn ( 3 5 )
For n o n - p a r e n t r e c o r d s , ai j k l n in ( 3 4 ) is s u b s t i t u t e d by ( as + ad) / 2 + mi j k l m n, w h e r e as ( ad) is the breeding v a l u e of the sire ( d a m ) of n , and the term in m is the M e n d e l i a n sampling e f f e c t . C o l l e c t i n g t h e f i r s t three terms in (34) in a v e c t o r b , the f o u r t h in g and so o n , (34) is w r i t t e n in m a t r i x n o t a t i o n as shown in (12) w i t h first m o m e n t e q u a l to X b + G g and s e c o n d m o m e n t s g i v e n by ( 1 4 ) , (15) ( 1 6 ) , and (17). T h e h e r i t a b i l i t y and r e p e a t a b i l i t y of t o t a l n u m b e r of l i v e b o r n p i g l e t s are for the time being a s s u m e d to be 0 . 1 0 and 0 . 1 5 , r e s p e c t i v e l y . It is a s s u m e d also that these figures h o l d w i t h all p o s s i b l e c o m b i n a t i o n of fixed e f f e c t s in the m o d e l .
T h e inverse a d d i t i v e g e n e t i c r e l a t i o n s h i p m a t r i x a m o n g a l l a n i m a l s in the s y s t e m is computed f o l l o w i n g an a l g o r i t h m by H e n d e r s o n ( 1 9 7 6 ) . The p r o g r a m can e a s i l y a c c o m m o d a t e Q u a a s ' (1976) a l g o r i t h m to a l l o w for i n b r e e d i n g , but at h i g h e r c o m p u t i n g c o s t .
5 . A s s u m p t i o n s of the m o d e l
T w o sets of a s s u m p t i o n s w i l l be d i s c u s s e d . O n e set is rather g e n e r a l and is a s s o c i a t e d w i t h the m i x e d m o d e l e q u a t i o n s (20) a n d ( 2 5 ) . T h e o t h e r set is
15
p a r t i c u l a r to the m o d e l described in section 4 . T h e s e two sets w i l l be discussed in t u r n .
T h e starting point in (1) is that the v e c t o r of observations y is a function of a set of variables that act a d d i t i v e l y . The v a r i a b l e s in the v e c t o r s b and g are fixed e f f e c t s , while those in the v e c t o r s a and e are random e f f e c t s . That i s , a and e are each considered as one sample from a p o p u l a t i o n of v e c t o r s a and e w h i c h has been drawn into the sample space a s s o c i a t e d w i t h the data v e c t o r y. The random vectors a and e determine that the data v e c t o r y is also regarded as a random v a r i a b l e sampled from a conceptual p o p u l a t i o n . In repeated s a m p l i n g , the vectors a and e have m e a n zero and the v e c t o r y has a m e a n of X b + Gg and a variance given b y the first row and column of (2).
The sampling of a and e is assumed to be carried out at random - selection as it operates in a breeding program is not allowed f o r .
The v e c t o r g has parameters representing g r o u p effects. Knowledge of an animal's ancestors defines the relative c o n t r i b u t i o n of groups to its p e r f o r m a n c e . Pedigrees m u s t be complete and the parents of those animals that are not identified m u s t have an identified group of o r i g i n . These unidentified parents are assumed to be average representatives (i.e. u n s e l e c t e d ) from their groups. Groups combine additively on their effect on p e r f o r m a n c e .
M o d e l (12) does not introduce new c o n c e p t s . Here the stochastic variables are af, m , p , and e . The derivation of (20) from (12) assumes that af, m , p , and e have null m e a n and that their covariance m a t r i x is known at least to proportionality. In our c o n t e x t , this m e a n s that the heritability and repeatability are known without e r r o r .
In going from (12) to (25), no a s s u m p t i o n is m a d e about the form of the iistribution of the stochastic elements in the m o d e l . It is assumed though that the conceptual populations they are sampled from have null m e a n s . This is acceptable for n , p , and e . The v e c t o r m represents deviations due to Mende 1 ian sampling. If there is no selection acting at the gametic s t a g e , before or after fertilisation, then these deviations from the p a r e n t a l average should add up to zero.
The assumption that p and e are v e c t o r s sampled from populations with zerc e x p e c t a t i o n is not problematic. P e r m a n e n t environmental effects and residua:
effects are not likely to affect the records in any particular d i r e c t i o n , 01 a v e r a g e , especially w h e n these effects are u n s p e c i f i e d .
O t h e r set of assumptions implied in (12), (14), (15), (16), and (17) are that v a r i a n c e s of random effects are constant across all levels of fixed effects that a s t r i c t l y additive genetic model holds and that the correlation between records of the same animal is a constant (the r e p e a t a b i l i t y ) irrespective of w h e t h e r the records are adjacent or further a p a r t .
U n d e r these a s s u m p t i o n s , given that the model is c o r r e c t , solution of the system (25) yields BLUE (best linear u n b i a s e d e s t i m a t o r s ) of estimable functions of fixed effects and BLUP (best linear u n b i a s e d p r e d i c t o r s ) of random e f f e c t s . If the v a r i a n c e s of the random e f f e c t s , or their r a t i o s , are not known and one substitutes estimates of them, then the resulting solution to the random effects are not BLUP, but are still unbiased (Kackar and H a r v i l l e , 1981).
The assumption that the vector of g e n e t i c v a l u e s , af is a random sample from a p o p u l a t i o n with a certain m e a n is not tenable when selection is known to have o p e r a t e d on a trait whose heritability is larger than zero. One wishes to know t h o u g h , if there exists a set of c o n d i t i o n s that if s a t i s f i e d , lead to the result that the system (25) leads to predictors of genetic m e r i t with good p r o p e r t i e s , even though the data have been generated by s e l e c t i o n . This is a difficult p r o b l e m w h i c h has not been totally solved y e t . H e n d e r s o n (1975) invoking early results by Pearson (1903) have shown that these set of conditions exist and that these are:
a) the vectors of random variables in the model follow a multivariate normal d i s t r i b u t i o n ;
b) the variances of the random v a r i a b l e s , or their ratios, are known:
c) selection does not operate across levels of fixed effects;
d) s e l e c t i o n does not operate on traits correlated with the data vector not specified in the m o d e l .
The genetic model implied by (a) is o n e in w h i c h the m e t r i c trait is d e t e r m i n e d
17
by a very large number of additive (nonepistatic) loci. An important assumption of this model is t h a t , w i t h directional s e l e c t i o n , gene frequency changes are infinites ima1ly small and changes in the additive genetic variance are due to joint disequilibrium (Bulmer, 1971) and in small p o p u l a t i o n s , due to inbreeding.
W i t h finite number of loci, p a r t i c u l a r l y with genes of large effects, the assumption of normality does not hold and with data spanning several g e n e r a t i o n s , gene frequency changes m a y be s u b s t a n t i a l . Under these c o n d i t i o n s , the predictors obtained from (25) are likely to be b i a s e d .
The condition specified in (b) requires the correct specification of the covariance m a t r i x of the random effects in the m o d e l . In the context of an animal m o d e l , this means that the relationship m a t r i x among all the individuals in the data is complete and that the heritability in the population from which the base animals were conceptually sampled is k n o w n . If the h e r i t a b i l i t y is not known, simulation results indicate that use of a REML (restricted m a x i m u m likelihood) estimator under an animal m o d e l in the solution of (25) leads to predictors of random effects with no detectable bias (Sorensen and K e n n e d y , 1986).
Condition (c) implies that the data had been adequately corrected for fixed effects prior to selection or that selection operated w i t h i n fixed effects. The introduction of groups in the model guarantees in fact that this condition is v i o l a t e d . In attempting to solve one problem - that is, acknowledging that animals originate from different populations - one introduces a new one: when breeding values are the selection criterion and these include a group e f f e c t , selection operates across fixed e f f e c t s .
Condition (d) requires that observed selection differentials were not the result of selection of a trait not specified in the m o d e l , and correlated with the vector y . Selection for daily gain within litters prior to sending pigs to a test station is an example of a v i o l a t i o n of (d).
Gianola et a l . (1988) has recently questioned the v a l i d i t y of some of the assumptions of the Pearson m o d e l in the context of genetic selection. Using a Bayesian a p p r o a c h , they arrive at a set of conditions w h i c h are not all in agreement with H e n d e r s o n ' s . Some of these points were further expanded by Im et al. (1989).
The second set of assumptions relates more specifically to the model proposed for genetic evaluation of number of liveborn piglets in (34).
The m o s t important of these assumptions are:
the trait (number of liveborn p i g l e t s ) is not influenced by the sire the sow i;
m a t e d t o .
It is fairly well established that litter size at birth is influenced by the litter's genotype and as s u c h , the assumption of no p a t e r n a l influence does not strictly h o l d . H o w e v e r , p u b l i s h e d analysis indicate that the p r o p o r t i o n of v a r i a t i o n in litter size due to sire of litter is v e r y s m a l l (Christensen,
1978; V a n der Steen and K o c k , 1987).
an additive genetic m o d e l h o l d s . This implies that the trait d o e s not exhibit inbreeding depression or h e t e r o s i s .
There is ample data showing that litter size traits e x h i b i t heterosis and inbreeding depression (Hill and W e b b , 1982). These results a r e not compatible w i t h a simple additive genetic m o d e l , but rather dominance a n d / o r various forms of epistasis must be i n v o k e d . A unified mixed m o d e l a p p r o a c h which takes account of the effects of inbreeding and dominance on the m e a n and v a r i a n c e is not yet d e v e l o p e d , although w o r k in this area has started (MSki-Tanila and K e n n e d y , 1986; Smith and M a k i - T a n i l a, 1990).
the genetic correlation of the trait in different parities is 1.
V a n g e n (1986) shows results that indicate that this h y p o t h e s i s m a y not h o l d , and that the genetic correlations decrease as the d i s t a n c e between parities i n c r e a s e s . On the other h a n d , in a recent r e v i e w , H a l e y et a l . (1988) conclude that genetic correlations between adjacent parities a r e h i g h , although less than o n e , but the estimates that they quote imply that the g e n e t i c correlation between parities 1 and 4 is considerably less than o n e . T h e y argue that the available estimates m a y be biased by selection. U n t i l a m u l t i p l e trait restricted maximum likelihood estimator of variance and c o v a r i a n c e c o m p o n e n t s , with an animal m o d e l , becomes a v a i l a b l e , the estimates m u s t be interpreted with r e s e r v a t i o n . Even t h e n , one could argue that estimates a r e b i a s e d by exclusion of non-additive gene action from the m o d e l . In view of the considerable computational difficulties a s s o c i a t e d with a multitrait e v a l u a t i o n s y s t e m , one can take refuge in some of the conclusions of Haley et a l . (1988) and justify the univariate approach as a temporary operational c o m p r o m i s e .
19
1. the heritability and repeatability are known and equal to 10% and 15%, r e s p e c t i v e l y , and these values hold for all breed x p a r i t y combinations and for all groups.
A s s o c i a t i n g these sets of assumptions w i t h what is known of the biology of the trait and with what goes on in a t y p i c a l breeding p r o g r a m m a y help towards understanding the discrepancy o f t e n observed between observed and expected selection responses!
6 . P o s s i b l e d e v e l o p m e n t s o f t h e m o d e l
Progress in the areas of b i o l o g y , m e t h o d s of estimation of genetic parameters and in computer science makes it p o s s i b l e to develop a more refined m o d e l for genetic evaluation of number of liveborn p i g l e t s . R e c e n t l y , a REML (restricted m a x i m u m likelihood) algorithm using a n i m a l models has become available (Meyer, 1989). This allows to test whether (a) and (d) h o l d . R E M L under an animal m o d e l provides estimates of genetic parameters w i t h well defined statistical p r o p e r t i e s , especially when used in selected data given that the correct m o d e l can be s p e c i f i e d . Statistical analysis could confirm w h e t h e r genetic p a r a m e t e r s are constant across b r e e d s , and yield e s t i m a t e s that could be used as priors in the solution of the m i x e d m o d e l e q u a t i o n s . Use of REML estimates obtained from the databank and used as variance ratios, w o u l d be a significant improvement over the present approach, where the h e r i t a b i l i t y and repeatability used as priors are obtained from the literature.
T h e performance of crosses among animals originating from different groups (countries) should be analysed for presence of h e t e r o s i s . Should this be c o n f i r m e d , the present model can be e x t e n d e d - at least from an operational point }f v i e w - to allow for n o n - a d d i t i v i t y . This can be accomodated including the expected proportion of heterosis in the m o d e l as a c o v a r i a t e .
1 . A n example
This example will illustrate the c o m p u t a t i o n of breeding values and h o w some of -he expressions developed in section 3 are u s e d .
Consider the records of 14 animals shown in Table 1. There are two fixed jffects, A (with 2 levels) and B (with 3 levels), and animals 9 , 10, and 13 have repeated records.
Animals 1, 2 , 3 , 4 , 11, and 14 have both parents m i s s i n g ; 7 , 8 , and 10 have one parent m i s s i n g . Even though these parents are u n k n o w n , one must assume that they originate from a given g r o u p . The groups of origin of the m i s s i n g parents are shown in table 2. In this t a b l e , the sign " - " m e a n s that the parent is not k n o w n .
Individuals 1, 2 , 3 , 4 , 5 , 6 , and 11 are p a r e n t s ; the remaining individuals, do not have offspring and are therefore nonpatents.
Assume the model for parent records is:
Yi j k l m = Ai + Bj + l xk Gk + ak l + pk l + ei j k l m (35) k
and for non-parent r e c o r d s , aijk l is replaced by 1/2 as + 1/2 ad + m i jk l, w h e r e , A and B are fixed e f f e c t s , G are fixed group e f f e c t s , a represents genetic values (random) and e is the residual peculiar to record m . For n o n - p a r e n t s , their genetic value is written in terms of their parental genetic values ( as and ad) and the Mendelian t e r m , m .
By collecting fixed effects A and B in a vector b , group effects in a vector g , the genetic values of parent records in a vector af, M e n d e l i a n terms in m and permanent environmental effects in p , and ordering the data such that parent records precede n o n - p a r e n t s , one can express (35) in the m a t r i x formulation (12).
rices T To Gf and Go a re
1/2 0 0 0 0 0 0 ; V 1/2 0 0 0 0 0 0 0 0 0 0 1/2 0 0 0 0 0 0 1/2 0 0 0 0 1/2 1/2 0 0 0 0 0 1/2 1/2 0 0 0 0 0 1/2 1/2 0 0 0 0 0 1/2 0 0 0 0 0 0 1/2 1/2 0 0 0 0 0 0 0 0 1/2 1/2 0 0 1/2 0 0 0 0 0 0 0 0 1/2 1/2 0 0 0 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/2 1/2
0 0 0 0 1/2 1/2 0 0 0 0 0 1/2 1/2 0 0 0 0 0 0 0 0
21
G'<
G' =
0 0 1 0 3 / 4 0 0 0 0 0 1 1 / 4 0 0
1 3/8 1/2 1/2 1/2 1/2 1/2 1/2 5/8 5/8 1/2 0 5/8 1/2 1/2 1/2 1/2 1/2 1/2 3/8 3/8 1/2
The second row of T corresponds to animal 8. The 1/2 in column 5 indicates that individual 5 is 8's p a r e n t. The second column of G 'Q gives the group contribution to 8 . Individual 8's unknown parent originates from group 2 - this contributes
ith 1/2 G2. The known p a r e n t, 5 is the offspring of 1 and 2. Individual 1 contributes 1/4 to G j , while 2 contributes with 1/8 to Gj and with 1/8 to C2. This
results in individual 8's group composition of 3/8 Gj and 5/8 G2. The total
»enetic value of 8 is thus:
= 3/8 G | + 5/8 G2 + 1/2 a5 + m g .
The variance of a*g is 1/4 o2a + 3/4 o2a = a2a . The elements of the diagonal latrix D are: diag D = (3/4 3/4 1/2 3/4 1/2 1/2 1 ), corresponding to the non-parental individuals.
The inverse of the additive genetic relationship matrix among the 6 parent- records is:
i-l 3/2 1/2 0 0 -1 0 0 1/2 3/2 0 0 -1 0 0 0 0 3/2 1/2 0 -1 0 0 0 1/2 3/2 0 -1 0 -1 -1 0 0 2 0 0 0 0 -1 -1 0 2 0 0 0 0 0 0 0 1
Assuming that the heritability and repeatability are h^ 0.10 and r = 0 . 1 5 , :hen ki = k-i 8.5 and k-, 17.
The coefficient m a t r i x of (25) is:
5.55 0 5.60
2.45 2.45 0. .65 0.85 0 .43 0.43 0 1.93 0.98 0.94 0 0 2.33 1.48 1, .79 0 1. . 16 0 0 0.74 0.94 1.16 1.16 0.4:
5.27 -0.38 -0. 12 0.43 0. ,37 0.43 0 0.70 0.33 1.31 0.79 0.4:
4.42 -0. 12 0.43 0, ,79 0 0 1.64 0.33 0.37 0.37 0 2. 68 0 0. ,43 0 0 0.33 1.27 0.43 0 0 21.68 4 . .46 -8.29 -4.25 -8.50 0 0 0 -4.2!
13. 54 0 -4.25 0.37 -8.50 0.21 0 -4.2!
SYMMETRIC 12.96 4.25 0 0 -8.50 0 0
12.75 0 0 -8.50 0 0 14.55 4.74 0 -8.50 0 14.18 0 - 8 . 5 0 0 18.53 0.37 0 17.58 0.21
8.71 The right hand side of (25) is:
(48.92 45.56 35.87 35.77 22.85 8.08 14.84 3.83 0 22.30 15.44 18.14 8.89 2.98)
The solution vector is:
¿1 = 4 .89
h 4 .38
H 1 .89
h 3 24
b3 4 14 Gl 1 64 G2 0 25
â* l 1 71 â *2 0 98 â *3 1 50 â *4 0 09 â *5 1 37 â*6 0 79 0. 94 The solution to 7 , C7 = 0 . 0 7 6 9 , n?
1.64) - 1 ((1/2) 1
the Mendelian sampling terms is obtained from (27). For anima
= 1 and the term in brackets is: 9 - 4,89 - 1.89 - J((1/2) -71) = 0 . 5 5 . T h e n , m? = (0.0769)(0.55) = 0 . 0 4 .
23
The remaining M e n d e l i a n sampling terms are:
itg = 0 . 1 3 = 0 . 0 3 ri1 0 = 0.02
"12 = - 0 . 0 1 m1 3 = - 0 . 0 0 m1 4 = 0 . 0 9
The solution to the total genetic values are obtained from (28). For animal 7 , noting that vs = 0 and vd = 1, we have:
a *7 = (1/2) 1.71 + (1/2) 1.64 + 0.04 = 1.72.
The remaining solution to the total genetic values are:
1 *8 = 0 . 9 4 a *9 = 0 . 8 3 a *1 0 = 0 . 9 0
!*j 2 = 0.86 a *1 3 = 1.08 a *1 4 = 1.04
The solution to the total genetic values can also be o b t a i n e d from (13):
1 .72 0.94 0.83 0.90 0.86 1 . 0 8 1 .04
F i n a l l y , the vector p is obtained from (29) or (30). For a n i m a l 7 , w7 = 0 . 0 5 5 , :he term in brackets in (30) is (9 - 4.89 - 1 . 8 9 - 1 (1.72)) = 0.5 and therefore
>7 = w7 (0.5) = 0 . 0 2 7 . The remaining 9 permanent environmental iffect solutions are:
>3 = -0.09 p4 = -0.09 p5 = -0.01 i8 = 0.08 p9 = 0.03 p1 0 = 0.02
»12 = -0.01 p1 3 = -0.00 p1 4 = 0.05
a *7 = 1/2 0 0 0 0 0 0 1 71 + 1/2 0 I .64 + 0 04 =
a *8 0 0 0 0 1/2 0 0 0 98 0 1/2 0.25 0 13 0 0 1/2 1/2 0 0 0 1 50 0 0 0 03 3*10 0 0 1/2 0 0 0 0 0 09 0 1/2 0 02
a* 1 2 0 0 0 0 0 1/2 1/2 1 37 0 0 -0 01
s* 1 3 0 0 0 0 1/2 1/2 0 0 79 0 0 -0 00 3*14 0 0 0 0 0 0 0 0 94 1/2 1/2 0 09
TABLE 1.
E x a m p l e . Records of 14 animals.
ID SIRE DAM A B M e a s u r e m e n t 1
2
_
- -_ _
3 -
_
1 2 84
_
. 2 3 75 1 2 1 1 8
6 3 4 - -
_
7 1 - 1 1 9
8 - 5 2 3 11
9 3 4 1 1 8
9 3 4 1 2 9
9 3 4 1 3 10
10 3 - 2 1 7
10 3 - 2 2 9
11 - - - -
_
12 6 11 2 1 7
13 6 5 2 1 7
13 6 5 2 2 9
14 - - 1 2 10
TABLE 2 . Group of origin of the u n k n o w n parents
I S Paternal group M a t e r n a l group
1 1 1
2 i 2
3 1 1 4 2 2 7 - 1 8 2
1 0 - 2
11 l 2
14 2 1
25
R e f e r e n c e s
å v a l o s , E; Smith, C . (1987). G e n e t i c improvement of litter size in p i g s . Animal Production 4 4 , 153-164.
B u l m e r, M . G . (1971). The effect of selection on genetic v a r i a b i l i t y . A m e r . N a t . 1 0 5 , 201-211.
C h r i s t e n s e n , A . (1978). Statistiske undersøgelser vedrørende kuldstørrelse hos D a n s k Landrace. M a t e m a t i s k I n s t i t u t , K V L .
F o u l l e y , J.L.; C h e v a l e t , C . (1981). M e t h o d e de prise en compte de la consanguinité dans un modéle simple de simulation de per- f o r m a n c e. A n n . G é n é t . S é l . A n i m . 13, 189-196.
G i a n o l a , D . , Im, S. and R . L . Fernando (1988). Prediction of breeding value under Henderson's selection model: a r e v i s i t a t i o n . J . Dairy S c i . 71, 2 7 9 0 - 2 7 9 8 .
H a l e y , C.S.; A v a l o s , E.; Smith, C . (1988). Selection for litter size in the p i g . A n i m . Breed. A b s t r . 56, 3 1 7 - 3 3 2 .
ienderson, C . (1973). Sire evaluation and genetic trends. P r o c . A n i m . Breed.
G e n e t . S ymp. Honor of D r . J . L . L u s h , A m . Soc. A n i m . S c i . , A m . Dairy Sci.
A s s o c . , C h a m p a i g n , II.
lenderson, C . R . (1975). Best linear unbiased estimation and prediction under a selection m o d e l . Biometrics 3 1 , 4 2 3 - 4 4 9 .
lenderson, C.R. (1976). A simple m e t h o d for computing the inverse of a numerator relationship matrix used in p r e d i c t i o n of breeding v a l u e s . Biometrics 32, 6 9 - 8 3 .
Ii 11, W . G . ; W e b b , A . J . (1982). Genetics of reproduction in the p i g . In: Control of Pig Reproduction. E d . D.J.A. Cole and G . R . Foxcroft. London, U . K . Butterworths.
Cm, S ., R . L . Fernando and D . Gianola (1989). Likelihood inferences in animal breeding under selection: a m i s s i n g - d a t a theory view p o i n t . G e n e t . S e l . Evol.
2 1 , 399-414.
K a c k a r, R.N. and D.A. Harville (1981). Unbiasedness of two-stage estimation an prediction procedures for mixed linear m o d e l s . Commun. Statist. Theor. Math A 1 0 , 1249-1268.
Le R o y , P.; Legault, C. ; Gruand, J.; Ollivier, L . (1987). Realised heritabilit;
for litter size in selection of "hyperprolif ic" sows. G6n6t. Sel. £vol. 19 351-363.
M e y e r , K. ( 1989). Restricted maximum likelihood to estimate variance component!
for anima1 models with several random effects using a derivative-fre<
algorithm. Genet. Sel. Evol. 21, 317-340.
Maki-Tanila, A . ; Kennedy, B.W. (1986) . Mixed model methodology under genetii models with small number of additive and non-additive loci. Proc. 3r d World Congr. Genet. Appl. Livest. Prod. XII: 443-448.
Pearson, K. (1903). On the influence of natural selection on the variability anc correlation of organs. Phil. Trans. Roy. Soc. Lond. A 2 0 0 , 1-66.
Q u a a s , R.L. (1976). Computing the diagonal elements and inverse of a large numerator relationship matrix. Biometrics 32, 949-953.
Q u a a s , R.L. (1988). Additive genetic model with groups and relationships. J . Dair;
Sci. 71, 1338-1345.
Q u a a s , R.L.; Pollak, E.J. (1980). Mixed model methodology for farm and ranch bee cattle testing programs. J . A n i m . Sci. 51, 1277-1287.
Q u a a s , R.L. and E.J. Pollak (1981). Modified equations for sire models wit groups. J . Dairy Sci. 64, 1868-1872.
Robinson, G.K. (1986). Group effects and computing strategies for models fo estimating breeding values. J . Dairy Sci. 69, 3106-3111 .
S m i t h , C . (1964). The use of specialised sire and dam lines in selection for mea production. Animal Production 6 , 337-344.
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smith, C. ; Dickerson, G.E.; T e s s , M . W . ; Bennett, G . L . (1983). Expected relative responses to selection for alternative measures of life cycle economic efficiency of pork production. J . Anim. Sci. 5 6 , 1306-1314.
Smith, S.P. and A . MSki-Tanila (1990). Genotypic covariance matrices and their inverses for models allowing dominance and inbreeding. G e n e t . Sel. Evol. 2 2 , 65-91.
Jorensen, D . A . and B.W. Kennedy (1986). Analysis of selection experiments using mixed model methodology. J . A n i m . Sci. 6 3 , 245-258.
rhompson, R . (1979) Sire Evaluation. Biometrics 3 5 , 339-353.
Iran der Steen, H.A.M.; Kock, M . G . M . (1987). Sire of litter and sire of dam effects on categorical and normally distributed litter size traits in p i g s . 3 8t h Annual Meeting of the European Association for Animal Production. L i s b o n , Portugal.
rangen, 0 . (1986). Genetic control of reproduction in pigs from parturition to puberty. Proc. 3r d World Congr. G e n e t . A p p l. Livest. Prod. V I I . 288-292.
lestell, R . A . , Quaas, R.L. and L.D. Van Vleck (1988). Genetic groups in an animal model. J . Dairy Sci. 71 , 1310-1318.
