T h e o r e t i c a l and Applied Genetics 46, 55-61 (1975) 9 by Springer-Verlag 1975

Multi-Stage Index Selection E.P. Cunningham Animal Breeding & Genetics Department, The Agricultural Institute, Duns inea, Castleknock, Co. Dublin (Ireland) S u m m a r y . S e l e c t i o n i n d e x t h e o r y i s e x t e n d e d t o c o v e r t h e c a s e of s e l e c t i o n i n s e v e r a l s t a g e s . G e n e r a l a l g e b r a i s g i v e n f o r a d j u s t i n g i n l a t e r s t a g e s f o r t h e e f f e c t s of s e l e c t i o n i n e a r l i e r s t a g e s . In a d d i t i o n a m e t h o d i s d e v e l o p e d f o r t h e i n c o r p o r a t i o n of a n i n d e x i n t o a n i n d e x . T h i s s i m p l i f i e s t h e r e u s e of d a t a f r o m e a r l i e r s t a g e s o f s e l e c t i o n . A n u m e r i c a l e x a m p l e i s u s e d to i l l u s t r a t e t h e m e t h o d s a n d to c o m p a r e t h r e e s i n g l e - s t a g e a n d t h r e e t w o - s t a g e s e l e c t i o n p r o c e d u r e s .

Introduction

work and a data management algorithm which should make possible efficient multi-stage index selection.

When genes act additively, selection is the appropriate m e t h o d f o r c h a n g i n g t h e g e n e t i c c o n s t i t u t i o n of a p o p u l a -

D i c k e r s o n a n d H a z e l (1944) give the m e t h o d f o r d e a l ing w i t h t h e c a s e o f s e l e c t i o n f o r a s i n g l e t r a i t in two

t i o n . The m a x i m u m g a i n f r o m s e l e c t i o n i s o b t a i n e d b y

stages.

u s i n g a s e l e c t i o n i n d e x ( H a z e l , 1943; H e n d e r s o n , 1 9 6 3 ) .

( 1 9 5 1 ) . J a i n a n d A m b l e (1962) e x t e n d t h i s t r e a t m e n t to

The m o r e c o m p l e x t h e s e l e c t i o n o b j e c t i v e , a n d t h e m o r e

three stages.

Its m a t h e m a t i c a l b a c k g r o u n d is given by C o c h r a n Papers by Cohen (1950), Finney (1956),

c o m p r e h e n s i v e t h e d a t a to b e c o n s i d e r e d , t h e m o r e a d -

R o b s o n ( 1 9 6 4 ) , Young ( 1 9 7 2 ) a n d W a n g ( 1 9 7 2 ) d e a l w i t h

v a n t a g e o u s it b e c o m e s t o u s e a n i n d e x . P o o r e s t i m a t e s of

a s p e c t s of the d i s t r i b u t i o n a l p r o p e r t i e s of t r u n c a t e d p o -

the current genetic structure ofthepopulation may make indexes inefficient (Heidhues and Henderson, Harris,

p u l a t i o n s a n d of t h e i r c o n s e q u e n c e s in s e l e c t i o n .

1962;

1 9 6 4 ) , but t h e s e c o n d i t i o n s a l s o u n d e r m i n e t h e

b a s i s f o r a n y a l t e r n a t i v e s e l e c t i o n p r o c e d u r e . In g e n e ral, therefore,

s e l e c t i o n i s o p t i m i s e d b y t h e u s e of an

One Stage S e l e c t i o n

index. In m a n y m o d e r n p l a n t a n d a n i m a l i m p r o v e m e n t p r o grammes,

selection is a continuous process,

and sere-

r a l s u c c e s s i v e s c r e e n i n g s m a y b e a p p l i e d in a s i n g l e g e n e r a t i o n . F o r e x a m p l e , in the s e l e c t i o n of d u a l - p u r -

The i n f o r m a t i o n r e q u i r e d i n c o n s t r u c t i n g a s e l e c t i o n i n dex can be specified inthefollowing 4 vectors and3 matrices.

Y = YI' ..... ' Y m

i s a v e c t o r of a d d i t i v e g e n e t i c v a l -

p o s e b u l l s f o r u s e in a r t i f i c i a l i n s e m i n a t i o n i n s e v e r a l

u e s f o r the m t r a i t s i n c l u d e d in

European countries,

the aggregate genotype.

the bulls are first selected using

information about their parents;

they are subsequently

v = Vl,

.....,

v

m

s c r e e n e d f o r t h e i r own g r o w t h p o t e n t i a l i n a p e r f o r m a n c e

representing the relative economic

t e s t s t a t i o n , a n d f i n a l l y s e l e c t e d on t h e b a s i s of a c o m p r e h e n s i v e d a i r y p r o g e n y t e s t of t h e i r d a u g h t e r s . The

i s a v e c t o r of c o n s t a n t s , u s u a l l y

v a l u e s of the m X = X 1,

.....,

X

n

t r a i t s in Y .

i s a v e c t o r of p h e n o t y p i c m e a s u r e s

s e l e c t i o n o b j e c t i v e r e m a i n s c o n s t a n t , but d i f f e r e n t i n f o r -

f o r the n v a r i a b l e s o r s o u r c e s of

mation is used, and different selection intensity is ap-

information

p l i e d at e a c h s t a g e . T h e f i n a l e v a l u a t i o n o f a b u l l m a y b e

index.

u s e d as i n f o r m a t i o n in the s e l e c t i o n of the next g e n e r a -

b= bl,

t i o n . T h e r e is thus a c o n t i n u o u s c a s c a d e of i n f o r m a t i o n t h r o u g h t i m e . W i t h m o d e r n c o m p u t i n g e q u i p m e n t , it

b

n

i s a v e c t o r o f w e i g h t i n g f a c t o r s to be u s e d in the i n d e x .

p

i s a n n • n m a t r i x of p h e n o t y p i c covariances between the n variab-

s h o u l d b e p o s s i b l e t o u s e all t h i s i n f o r m a t i o n . In p r a c t i c e , e a c h s t a g e i s m o s t o f t e n t r e a t e d a s an e n t i r e l y s e p a r a t e o p e r a t i o n b e c a u s e of t h e t h e o r e t i c a l a n d p r a c t i c a l

...o.,

b e i n c l u d e d in the

l e s in X . G

is an n x n m a t r i x of p h e n o t y p i c

p r o b l e m of d e a l i n g w i t h s e l e c t i o n at s e v e r a l l e v e l s . The

covariances betweenthe n variables

p u r p o s e of t h i s p a p e r i s to p r o v i d e a t h e o r e c t i c a l f r a m e -

in X and the m t r a i t s in Y.

56

E. P. Cunningham : Multi-stage Index Selection

C

is an m • m

m a t r i x of g e n o t y p i c

cedures 4,5 and 6 require that all the variances and

covariances between the m traits in Y.

covariances linking X I with X 2 and Y__ be adjusted for the effects of selection on X I . Alternatives 3 and 6 require the incorporation of an index into an index. In the

The a g g r e g a t e g e n o t y p e o r b r e e d i n g v a l u e i s d e f i n e d a s

sections which follow, the general algebra for these two T =v'Y

=VlY 1 + v2Y 2 + ...

requirements is developed. A numerical example is then

VmY m .

+

used to illustrate the methods and to c o m p a r e the selecS i n c e T i s not m e a s u r a b l e ,

it c a n n o t b e s e l e c t e d

tion alternatives.

f o r d i r e c t l y . I m p r o v e m e n t in T is b r o u g h t about by s e l e c t i o n on an index I = b 'X=

or selection

criterion:

I n c o r p o r a t i o n of a n i n d e x i n t o an i n d e x The v a r i a n e e - c o v a r i a n c e

blX l + bzX 2 + ... + bnX n-

m a t r i x f o r t h e full s e t o f v a r i -

ares and genotypic values can be represented by the The index Pb

weighting

factors

in I are

obtained

by solving

the

equations.

following supermatrix. M= E(X,Y)'

= Gv

(X,Y) =

to give b = p-IGv The v a r i a n c e of the index, the v a r i a n c e of the a g g r e gate genotype and the covariance of index and aggregate

genotype are ,el

G 2 oI =b'Pb

2 ~T = v ' C v

C;Ti = b ' P b

(1) The genetic change, in economic units, resulting from one round of selection is {'(~I'w h e r e {" is the selection differential achieved on a standardised distribution corresponding to the distribution of index values. G'

C

m

T w o Stage Selection Let the variables X I = X 1 , ..., X r be available for the

If f i r s t s t a g e s e l e c t i o n i s to t a k e p l a c e o n X 1 = X i ,

first stage of selection. Let the additional variables

...,

X 2 = Xr+l, .... X n b e c o m e available for the second

M=

stage.

E ( X l , X 2, Y ) ' ( X t , X z, Y) =

X r this matrix can be re-labelled as follows:

Selection can then be done in one or two stages, and the data can be used in several ways: 1.

O n e - s t a g e s e l e c t i o n on _X1 a n d X 2 ,

2.

O n e - s t a g e s e l e c t i o n o n X 1'

3.

O n e - s t a g e s e l e c t i o n o n I 1 and_~2 9 w h e r e I 1 i s t h e

I I I

$

121 l I i I I I I I i i I I

index that would have been used for option 2 above, 4.

T w o - s t a g e s e l e c t i o n , u s i n g X 1 in the f i r s t s t a g e

a n d X 1 a n d X 2 in t h e s e c o n d , 5.

Two-stage selection,

u s i n g _X1 in t h e f i r s t s t a g e 121'

a n d X 2 in t h e s e c o n d , 6.

T w o - s t a g e s e l e c t i o n , u s i n g X 1 in the f i r s t s t a g e

and I 1 and X 2 Procedures

in the s e c o n d .

1 and 2 are straightforward applications

of a s e l e c t i o n index on v a r i a t e s

X 1, o r X 1 a n d X 2. P r o -

----

fl-r

R I I I I i I I i I

1

fll

(2)

E .P.

Cunningham

, If v a r i a t e s b_ 1 X I '

: Multi-stage

Index Selection

X 1 are now replaced

then a new reduced

to contain the variances and genotypic

values

matrix

by thefr

57

index

Il =

can be constructed

and covariances

involved in stage

the variance of the prior index, ai21 be w = s/ai21= s/ b'lSb 1" In the case of two-stage selection with the

of the variates 2.

second stage based on 11 and X 2 (i.e. procedure 6) define a vector,

Mr E(_I1,_X2, Y)

' ( _ I 1 , _X2, Y_) = t h a t i s d e f i n e it to b e t h e f i r s t

!

b~Sb,

~o

(3).

I I t I I I I I

The variances

adjusted

for prior

Iv~~ = M --r --r

-T'T

row of the matrix

M in --r in M can then be --r on I I by calculating

and covariances selection

.n-r

(3)

RI I I I I

This matrix ,m

I i i i I i

w.

(5)

can now be repartitioned

to give the

input matrices

for a selection

index based

9

prior

h a s t a k e n p l a c e o n I 1.

X n where

selection

on I 1' X r+l'

It b e c o m e s M* = --r

Note that the section covariance

(R) of the original

supermatrix

involving

ed by collapsing

X 1 into an index.

this new matrix

is the variance

der of the first

row and column

o f I1 w i t h

the elements

variance-

The first

element

contain

to give the input matrices

is for selection

procedure

for prior

If s e l e c t i o n

has taken place

account

If t h e r e

whose mutual

truncation

selection

variances

adjusted

tained from (1951).

covariances lection on z differential

3 normally

for the effects

that

be

the co-

z i , zj , and Zk, and their truncation

t giving a standardised

1 {'. T h e s e c a n b e c o m b i n e d s = [({" - t ) . after selection

This can be used

to modify

expression

procedures

4 and

5, define

a vector T =b~

(S,Q)

that is, define it to be the product rows

of the supermatrix

covariances

in M

M

of bl

and

the first

in (2). The variances

can then be adjusted

for prior

r

and selec-

tion on 11 by calculating M e =M-

T'T

w.

This supermatrix

is then

(7) can now

justed input matrices

be repartitioned

corresponding

to give ad-

to those

in (I).

(4) Numerical

a whole

for the effects of selection

follows 9

se-

selection

into a single se-

A general

of selection

can be ob-

(10) of Cochran

-1 ~k = ajk - aij aiR aii s

ances

C*

and if

of selection

Assume

In the case

totake

distributed

are known,

of formula

be aij, aik and ajk.

for a covariance

(6)

to c a l c u -

takes place on one of them,

a generalisation

lection parameter

can be

Xn'

must be modified

are any

at a point

,n-r§

on I 1

covariances

Let the variates

G ~,

on I 1 ' then the variances

in this matrix

of this.

"'" '

pC

G'~'

and covariances

variates

needed

I 1' Xr+l'

selection

:

selec-

3.

Adjustment

Y)

of

the covariances

tion has taken place on I 1 ' then this supermatrix repartitioned

X 2 , Y_) ' ( I 1 , _ X 2 ,

of I 1 , and the remain-

o f X 2 a n d _Y. If n o p r i o r

late an index on the variates

E*(I1,

X 2 and Y is unaffect-

matrix

of covari-

on any one variate

Let the ratio of the selection

parameter,

as

s , to

Example

The data used in this example (1943)

classic

objective

paper

or aggregate

are taken from

on selection genotype

indexes.

Hazel's

The selection

in a swine selection

58

E. P.

scheme

consists

market

score

(Y3) . Their The

of the three

traits

(Y2) and number relative

information

weights

on which

comprises

the five variates

X 1 = pig's

own market

weight

X 2 = pig's

own market

score

weight (Y1) ,

X 3 = productivity

of pigs born per litter

economic

available

market

are v = ( 1/3 1 2).

to base

selection

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

market

weight of pig and littermates

market

score

From

the variances,

.

.

.

of pig and littermates.

heritabilities

it i s p o s s i b l e

and correlations

to c o n s t r u c t

for the full system

given

the P, G and C ma-

(1) as follows

M : E (_X,_Y) (x,__Y): ,

-7.4323 -3.7634 94.4784 -7.4323 -3.7634

.

of dam

X 5 = average

by Hazel,

93.5066 22.8484 -3.7634 41.2218 8.2985 .

Index Selection

X 4 = average

trices

1015.0596 93.5066 -7.4323 457.9949 41.2218

Cunningham : Multi-stage

.

.

.

.

.

.

.

.

457.9949 41.2218 -7.4323 457.9949 41.2218 .

.

.

.

.

.

.

.

.

.

.

.

.

I

41.2218 8.2985 -3.7634 41.2218 8.2985 .

.

.

.

.

.

.

.

.

.

.

.

1 302.5893 1 13.5093 I 0.0 I 181.5536 t 8.1056 .

.

.

L

.

.

.

.

.

.

.

.

.

13.5093 2.2391 0.0 8.1056 1.3435 .

.

.

.

.

.

-- .

.

.

.

.

0.0 0.0 7.6339 0.0 0.0 .

.

.

.

.

.

.

.

.

.

(8)

.

.

.

.

I

302.5893 13.5093 0.0

13.5093 2.2391 0.0

0.0 0.0 7.6339

181.5536 8.1056 0.0

8 . 1056 1.3435 0.0

I i I

302.5893 13.5093 0.0

13.5093 2.2391 0.0

0.0 0.0 15.2677

I

One-Stage

Selection

If selection operation rectly

is to take

place

(procedure

to give

measures

on all variates

I), this system

the index

weighting

of the effectiveness

ing factors b'

If s e l e c t i o n priate

as a single

can

be solved

factors

ing this system.

di-

The

weight-

factors

X 2 in

solv-

to use are

b I = S-1G 1 v

- 0.1654

0.1620

0.0867

- 0.1892)

where

variance

of the index

is 17.6863

and

its accuracy

G 1 consists

numerical

is

values

The variance

0.4087. If information

on the first three

sooner

the vector and

The weighting

2)the appro-

ignoring

are

= (0.0976

available

only X 1(procedure

and various

of the index.

m

The

involves

index can be found by simply

than

of variates

X 2 = (X4X

information

variates

on the remaining

can then be divided

5) , and

the supermatrix

accuracy

becomes

are

b~

= (0. 1350 - 0.2309

with the aggregate

As specified

genotype

in selection

procedure

is

formula

3, single-stage

could be done using all five variates

placing the first three by their

(X1,_x 2,X) :

0. 1630).

of this index is b ~ S b 1 = 16. 3646, and its

or correlation

selection

M : E ( X 1 , x 2,Y)

rows of G. Their

0.3932.

two,

into X = (XIX2X 3 ) --I can be relabelled

(2) accordingly:

of the first three

(3),

the variance

, but re-

i n d e x I 1 = b ~ X 1. U s i n g

- covariance

matrix

13.5093 2.2391 0.0

0.0 0.0 7.6339

of all

I

1015.0596 93.5066 -7.4323

93.5066 22.8484 -3.7634

-7.4323 -3.7643 94.4784

457.9949 41.2218 -7.4323

41.2218 8.2985 -3.7634

I 302.5893 [ 13.5093 ] 0.0

41.2218 8.2985

t I I

I

457.9949 41.2218

41.2218 8.2985

-7.4323 -3.7634

457.9949 41.2218

181.5536 8.1056

8.1056 1.3435

302.5893 13.5093 0.0

13.5093 2.2391 0.0

0.0 0.0

] I

302.5893 13.5093 0.0

13.5093 2.2391 0.0

0.0 0,0 7.6339

181.5536 8.1056 0.0

8.1056 1.3435 0.0

I I I

0.0 0.0 15.2677

(9)

E . P . Cunningham : Multi-stage Index Selection variates

and traits involved

59

becomes

M r =E ( I i , X2,_Y) ' (I1, _X2,Y) = I 16.3646

51.0997

51.0997 3.0353

457.9949 41.2218

37.7433 1.3075 1.2437

I

3.0353

I t

]

41.2218 8.2985

181.5536 8.1056 0.0

t I

8.1056 1.3435 0.0

If no s e l e c t i o n h a s t a k e n p l a c e o n

[ [ I

37.7433

1.3075

181.5536 8.1056

8.1056 1.3435

1.2437

0.0 0.0

(lO) 302.5893 13.5093 0.0

13.5093 2.2391 0.0

0.0 0.0 15.2677

This m a t r i x can be u s e d to c a l c u l a t e a s e c o n d s t a g e

_X1 ~ t h i s s y s t e m

c a n b e s o l v e d t o g i v e a n i n d e x in w h i c h t h e f i r s t t h r e e

index basedonvariates

v a r i a t e s a r e r e p l a c e d b y t h e i r i n d e x I 1" I n d i v i d u a l s w o u l d It h a s t h e be selected using the criterion

words,

12 = 0 . 7 8 8 6 I 1 + 0 . 0 7 9 3 X 4 - 0 . 1 9 5 1 X 5.

I1, X 4 a n d X 5 ( i . e . p r o c e d u r e

6).

s a m e w e i g h t i n g f a c t o r s a s 12 a b o v e . I n o t h e r

the actual index is the same

selection

on

variances

and covariances

I I has taken

The a c c u r a c y of t h i s i n d e x i s 0 . 4 0 7 0 , a n d i t s v a r i a n c e

ced by selection

is 17.5504.

Its accuracy,

place.

whether

However

or not prior because

of its constituents

on 11, its variance or correlation

is reduced

the

are reduto 6. 1787.

with the aggregate

genotype

is 0.2557. For Two-Stage

Selection

In order

to deal with the effect of stage

is necessary Assume

to specify the amount

that the upper

This gives

a selection

truncation

at a point

distribution

0.695

and

of selection

38 % of pigs on differential t = 0.305

1 selection,

The selection

of i" = 1.0

on a standard

w = s/~l

parameter

it

involved.

and implies normal

= 0.0425.

Applying

matrix

to the matrix

of variances

fects of selection M* =E* --r

as described

and covariances on

(I 1 ' X 2 '-Y)

4.9912

(10)

adjusted

in (5) the for the ef-

11, is ' (I 1, X -- 2 , Y--)

15.5854

I

I

could consider available

but retaining

and covariances

11 (procedure

the ad-

which

5).This

the first row

It has a variance

To calculate variates

a second

individually M

must

allow for

index

is cal-

and column

of 3.5661

of ma-

and an accuracy

stage index which

(procedure

first be adjusted

I 1 by using formula

stage index. are the same

uses

4), the original for prior

7. This adjusted

can then be repartitioned

produces

0.9257

on

by discarding

the second =

(X 4 and X5),

to variances

we

becoming

of 0. 1943.

matrix these values

stage

selection

trix (11). of

is s = [(r - t) =

- 0.695/16.3646

justments

culated

with this index,

solely on the information

at the second

prior

11 are selected.

or + O. 305 ~i1 on the actual distribution

index values.

comparison

selecting

all five super-

selection

supermatrix

to give the input matrices The weighting as where

factors

no prior

which

selection

11.5117

0.3988

0.3793

99.6432 3.2402

5.2681 1.1750

-2.6990 -0.1603

242.1881 11.4135 -1.9935

11.4135 2.1666 -0.0690

-1.9935 -0.0690 15.2021

on M *~ for it has

I

I

15.5854 0.9257

347.0989 34.6346

34.6346 7.9032

I II I

(11)

l

11.5117 0.3988 0.3793

99.6432 5.2681 -2.6990

3.2402 1.1750 -0.1603

t l I I

60

E. P. Cunningham

taken

place

(option

I). However,

dex and its accuracy

the variance

are reduced

of the in-

to 6. 3231 and 0.2308

the effects of these

lection procedures,

it is necessary

same

of selection.

case

final intensity of one-stage

selection,

rential of two standard in the case

Discussion

selection is straightforward. to compare

se-

that they all havethe Let this be

6 ~0. Inthe

this gives a selection

deviations.

of a two-stage

different

Equivalent

procedure

Index Selection

The e x t e n s i o n o f t h e m e t h o d to m o r e t h a n t w o s t a g e s o f

respectively. In order

9 Multi-stage

diffe-

selection

can be achievedby

The s u p e r m a t r i c e s

M~ o r

Iv~~ a d j u s t e d f o r p r i o r s e l e c t i o n on I 1 c o u l d b e r e g a r d e d ~ r a s t h e s t a r t i n g p o i n t (1) f o r a two s t a g e s e l e c t i o n , t h u s g i v i n g t h r e e s t a g e s in a l l . T h i s c a n b e r e p e a t e d a s o f t e n a s r e q u i r e d . The m a i n r e q u i r e m e n t i s t h a t t h e full v a r i a n c e - c o v a r i a n c e m a t r i x of all v a r i a t e s to b e u s e d a n d t r a i t s to b e s e l e c t e d f o r m u s t b e a v a i l a b l e at t h e b e g i n -

t a k i n g t h e b e s t 38 ~0 o n t h e f i r s t s t a g e a n d t h e b e s t 16 %

ning.

T a b l e 1. R e l a t i v e e f f e c t i v e n e s s of t h e s i x s e l e c t i o n p r o c e d u r e s

Selection

1.

2.

3.

4.

5.

6.

Gain from Absolute

Procedure

One s t a g e , v a r i a t e s X 1X2X3X4X 5

2.0

4.2055

8.4110

One stage, variates XIX2X

2.0

4.0453

8.0906

96.2

One stage, variates IIX4X 5

2.0

4.1893

8.3786

99.6

Two s t a g e , v a r i a t e s X 1 X 2 X 3 in s t a g e 1

1.0

4.0453 7.8675

93.5

v a r i a t e s X 1X2X3X4X5 in s t a g e 2

1.52

2.5146

Two s t a g e , v a r i a t e s X1X2X 3 in s t a g e 1

1.0

4.0453

6.9157

82.2

variates

1.52

1.8884

Two s t a g e , v a r i a t e s X I X 2 X 3 in s t a g e 1

1.0

4.0453 7.8236

93.0

variates

1.52

2.4857

X4X

3

5 in stage 2

IIX4X

5 in stage

2

(i.e.

6 / 3 8 ) of t h e s e r e m a i n i n g o n t h e s e c o n d s t a g e .

gives

a selection

for stage one,

differential

and

is [cr I where

therefore

effectiveness

deviations

on any index,

f is the standardised

rential and crI is the standard relative

of one standard

1.52 standard

The net effect of selection units,

selection Relative

deviation

be given as follows

(Table

for stage two. in economic selection

of the index.

of the six options I)9

This

deviation

considered

diffeThe can

100.0

One convenient r e s u l t is that the actual index w e i g h t ing factors course

coefficient variate.

are not affected by prior selection.

is the same

thing as saying

is unaffected

In practice

multi-stage

by selection

this simplifies

index selection,

be calculated sities of prior

at different selection.

on the independent the application

since new

stages

This of

that a regression

indexes

of

need

and for different

not

inten-

E. P.

Cunningham

: Multi-stage

61

Index Selection

The c a l c u l a t i o n o f a s e l e c t i o n i n d e x d o e s n o t d e p e n d

the first stage of selection

on the f o r m of m u l t i v a r i a t e d i s t r i b u t i o n linking the v a -

Howeve

r, if much

for reuse

the same

r i a t e s a n d t r a i t s i n v o l v e d . H o w e v e r , t h e c a l c u l a t i o n of

rely by using the index values

t h e e f f e c t s of s e l e c t i o n o n t h e i n d e x i s d i s t r i b u t i o n d e -

stage as input variates

p e n d e n t . The m e t h o d s g i v e n in t h i s p a p e r a s s u m e a n

practical

i n i t i a l m u l t i v a r i a t e n o r m a l d i s t r i b u t i o n . The r e s u l t s up

lection are greatly

possibilities

in the second

result can be achieved calculated

stage. me-

for the first-

for the second-stage, of efficient multi-stage

then the index se-

increased.

t o a n d i n c l u d i n g t h e i n d e x to b e u s e d i n t h e s e c o n d s t a g e s h o u l d t h e r e f o r e b e e x a c t . The e s t i m a t e d g a i n f r o m s e cond s t a g e s e l e c t i o n will t e n d to be o v e r e s t i m a t e d ,

Acknowledgements

since

t h e s e l e c t i o n d i f f e r e n t i a l u s e d r e l a t e s to a n o r m a l d i s t r i b u t i o n , w h e r e a s t h e d i s t r i b u t i o n of i n d e x v a l u e s c a n b e

I w i s h to t h a n k P r o f . A . L . R a e , M r . J . Brascamp,

Prof. D.J.

Connolly, Mr.P.

Finne~r a n d P r o f . A l a n R o b e r t -

son for their useful comments.

e x p e c t e d to d e p a r t f r o m n o r m a l i t y . A t t h e t h i r d s t a g e , both the index and its estimated effects will be biased b e c a u s e the input m a t r i c e s

a r e a d j u s t e d on the a s s u m p -

Literature

t i o n t h a t t h e y w e r e m u l t i v a r i a t e n o r m a l at s t a g e 2 . T h e r e

C o c h r a n , W . G . : I m p r o v e m e n t b y m e a n s of s e l e c t i o n . P r o c . 2nd B e r k e l e y S y m p . on M a t h . , S t a r . a n d P r o b a b i l i t y , ed~ N e y m a n , J . 4 4 9 - 4 7 0 (1951) f o r the c a s e w h e r e the v e c t o r s of v a r i a t e s X and of t r a i t s C o h e n , J . A . C . " E s t i m a t i n g the m e a n and v a r i a n c e of __Y c a n e a c h b e r e g a r d e d a s a s i n g l e v a r i a b l e . J a i n a n d normal populations from singly truncated and doubl y t r u n c a t e d s a m p l e s . A n n a l s M a t h . S t a t . 2_j.1 , A m b l e (1962) d i s c u s s a l t e r n a t i v e m e t h o d s of d e a l i n g w i t h 557-569 (1950) i t , w h i l e Y o u n g (1972) g i v e s a m e t h o d o f n u m e r i c a l l y D i c k e r s o n , G . E . ; H a z e l , L . N . : E f f e c t i v e n e s s of s e l e c t i o n o n p r o g e n y p e r f o r m a n c e a s a s u p p l e m e n t to e v a l u a t i n g t h e m e a n a n d v a r i a n c e of s e l e c t e d p o p u l a t i o n s e a r l i e r c u l l i n g i n l i v e s t o c k . J . A g r i c . R e s . 69, f o r u p t o f o u r s t a g e s .of s e l e c t i o n . 459-475 (1944) F i n n e y , D . J . : The c o n s e q u e n c e s o f s e l e c t i o n f o r a v a The g r a d u a l d e f o r m a t i o n o f t h e i n i t i a l n o r m a l d i s t r i r i a t e s u b j e c t to e r r o r s o f m e a s u r e m e n t . R e v . I n s t . bution can be quite s e v e r e if s e l e c t i o n is i n t e n s e , if one I n t e r n . S t a t i s t i q u e 24, 1-10 ( 1 9 5 6 ) Harris, D.L. : Expected and precided progress from c a n j u d g e f r o m t h e e f f e c t s in t h e b i v a r i a t e c a s e ( C o c h i n d e x s e l e c t i o n i n v o l v i n g e s t i m a t e s of p o p u l a t i o n r a n , 1 9 5 1 ) . It m a y b e t h a t w i t h an a r r a y o f v a r i a t e s i n t h e p a r a m e t e r s . B i o m e t r i c s 20, 46-72 (1964) H a z e l , L . N . : The g e n e t i c b a s i s f o r c o n s t r u c t i n g s e l e c s e l e c t i o n c r i t e r i o n a n d a n a r r a y of t r a i t s i n t h e s e l e c t i o n t i o n i n d e x e s . G e n e t i c s 2..8.8, 4 7 6 - 4 9 0 ( 1 9 4 3 ) o b j e c t i v e , t h a t t h e n o r m a l i t y of t h e i r m u l t i v a r i a t e d i s t r i - H e i d h u e s , T . H . ; H e n d e r s o n , C . R . : B e i t r a g z u m P r o Problem des Basisindex. Z.f. Tierez. 7_~7, 291-311 bution is better protected than in the case of single vaa p p e a r s to b e no e x a c t s o l u t i o n f o r t h i s p r o b l e m ,

even

(1962)

riate selection

for a single trait objective.

be that if two stages genetic

segregation

are separated

normality

this problem

of the decay

selection

is one which

in each

have

generation.

of normality

this paper

also

by a generation,

and recombination

of recreating

It may

does

that

the effect At any rate,

under

repeated

not attempt

8, 88-109 (1962)

to

resolve. It is encouraging is apparently dure

almost

4. In practice,

to be required

Received

b~r A.

convenient

procedure

6

as good

as the full two stage proce-

it would

often by very

to reassemble

September

Communicated

that the more

25,

1974

Robertson

the original

inconvenient data used

in

Heidhues, T.H. ; Henderson, C.R. 9 Selection indexand expected genetic advance. In: Statistical genetics and plant breeding, ed. Hanson, W.D. and Robinson, H.F. Natl. Acad. Sci. Nat Res. Council Publ. 982 111-163 (1963) Jain, J.P. ; Amble, V.N. : Improvement through selection at successive stages. J.Ind. Soc. Agr. Stat. Robson, D.S. : Note on repeated selection in the normal case. Report No. BU-171-M. Cornwall University (1964) Wang, Y. Ying: Selection problems under multivariate normal distribution. Biometrics 28, 223-233 (1972) Young, J.C. : The moments of a distribution after repeated selection with error. 3. Am. Slat. Assoc. 6 , 206-210 (1972)

Prof. E.P. Cunningham Animal Breeding and Genetics Dpt. The Agricultural Institute Dunsinea Research Centre Castleknock, Co. Dublin (Ireland)