Estimation of maternal effects on birth and weaning weight of Hereford cattle
by Rodolfo Juan Carlos Cantet
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Animal Science
Montana State University
© Copyright by Rodolfo Juan Carlos Cantet (1984)
Abstract:
Evidence has been found that maternal effects are important sources of variation in mammals. In beef
cattle, evidence has been found to support the hypothesis that maternal effects can reduce the expected
response to selection for growth traits, especially preweaning growth. It is not clear whether maternal
effects in preweaning growth of beef cattle are genetic or environmental. Therefore, the present
research was conducted to study the nature and. magnitude of maternal effects in birth and weaning
weight of Hereford cattle. The data used were the records of 4,423 noncreep-fed beef calves raised at
the Northern Agricultural Research Center, Havre, Mt from 1938 to 1983. Least-squares, fitting
constants method (Henderson III) and Restricted Maximum Likelihood procedures were used to
estimate eighteen covariances between different types of relatives. The basic models included the fixed
effects of line-year, age of dam, sex of calf, age of dam by sex interaction and the regressions of birth
weight on birthdate of calf and weaning weight on weaning age of calf. The source of variation
depicting the relative relationship was considered a random effect. Multiple regression procedures were
used to obtain the nine parameters: additive genetic, dominance genetic and environmental direct and
maternal variances and the covariance between the direct and maternal source in each case.
All solutions showed a negative additive genetic correlation between additive direct effects and
additive maternal effects (rG). The results were not clear regarding to the magnitude of rG for birth
weight but the probable value for weaning weight was around -.60 to -.75. There was also evidence for
a negative environmental correlation for maternal phenotype of the dam and daughter in the case of
weaning weight. This path coefficient (fm) was calculated to be .08 for birth weight and -.10 for
weaning weight. After correcting the expectations of the covariances between relatives for fm, rG was
still present and negative for both weights. The solutions showed that dominance was also involved in
determining maternal effects in preweaning growth of beef cattle but its effects may be confounded
with epistasis. ESTIMATION OF MATERNAL. EFFECTS ON BIRTH AND
WEANING WEIGHT OF HEREFORD CATTLE
by
R o d o lfo Juan C a r lo s C an tet
A t h e s i s s u b m it t e d i n p a r t i a l f u l f i l l m e n t
o f th e r e q u ir e m e n ts f o r th e d e g r e e
of
M aster o f S c i e n c e
in
Animal S c i e n c e
MONTANA STATE UNIVERSITY
Bozeman, Montana
May, 1984
APPROVAL
o f a t h e s i s s u b m itte d by
R o d o lfo Juan C a r lo s C an tet
T h is t h e s i s has been read by each member o f th e t h e s i s c o m m itte e
and h a s b e e n f o u n d t o be s a t i s f a c t o r y r e g a r d i n g c o n t e n t , E n g l i s h
u s a g e , f o r m a t , c i t a t i o n s , b i b l i o g r a p h i c s t y l e , and c o n s i s t e n c y , and i s
read y f o r s u b m is s io n t o t h e C o l l e g e o f Graduate S t u d i e s .
S'/z sr /
______
fD D
Date
C h a ir p e r so n , Graduate Committee
Approved f o r t h e Major Department
___S
Date
.
,
-
^
Head, Major Department
Approved f o r th e C o l l e g e o f Graduate S t u d i e s
Date
Graduate Dean
ill
STATEMENT OF PERMISSION TO USE
In
p r e se n tin g
th is
th e sis
in
p a r tia l
fu lfillm e n t
of
th e
r e q u i r e m e n t s f o r a m a s t e r ’s d e g r e e a t M ontana S t a t e U n i v e r s i t y ,
I
a g r e e t h a t t h e L ib r a r y s h a l l make i t a v a i l a b l e t o b o r r o w e r s under th e
r u l e s o f t h e L ib r a r y .
B r i e f q u o t a t i o n s from t h i s t h e s i s a r e a l l o w a b l e
w ith o u t s p e c ia l p e r m issio n ,
source i s
p r o v id e d t h a t a c c u r a t e acknow ledgem ent o f
made.
P e r m i s s io n f o r e x t e n s i v e q u o t a t i o n from or r e p r o d u c t i o n o f t h i s ,
t h e s i s may be g r a n te d by my m ajor p r o f e s s o r ,
or i n h i s a b s e n c e , by th e
D i r e c t o r o f L i b r a r i e s when, i n t h e o p i n i o n o f e i t h e r ,
o f th e m a t e r i a l i s f o r s c h o l a r l y p u r p o s e s .
m a te r ia l
in
th is
th e sis
for fin a n c ia l
w i t h o u t my w r i t t e n p e r m i s s i o n .
S ign atu re
Date
Mous 2 S , I9%4
t h e proposed u se
Any c o p y in g or u s e o f th e
g a in s h a ll
n o t be a l l o w e d
To ray w i f e , t o ray mother and
t o t h e memory o f ray f a t h e r
V
VITA
R o d o l f o Ju an C a r l o s C a n t e t w as b o r n t o Mr. and Mrs. R o d o l f o
Manuel C a n te t i n Buenos A i r e s , A r g e n tin a , on March 17, 1954.
He a t t e n d e d L i c e o M i l i t a r G en eral San M artin H ig h s c h o o l and was
a d m it t e d t o th e F a c u lta d de Agronomia, U n iv e r s id a d de Buenos A ir e s i n
1972.
He g r a d u a te d a s an I n g e n i e r o Agronomo i n 1978.
He h a s w o r k e d a s a r e s e a r c h e r i n B e e f C a t t l e B r e e d i n g a t t h e
D epartam ento de Z o o t e e n ia , U n iv e r s id a d de Buenos A i r e s , from 1978 t o
p resen t.
In S eptem ber 1982, he was a d m it t e d t o Montana S t a t e U n i v e r s i t y
f o r h i s M a ste r 's d e g r e e i n Animal S c i e n c e (Animal B r e e d in g ).
He m arried P a t r i c i a P r a n d in i i n 1982.
vi
ACKNOWLEDGMENTS
I w a n t t o e x p r e s s my d e e p a p p r e c i a t i o n and a d m i r a t i o n t o Dr. D.
D.
K ress.
H is r e s e a r c h
was
th e r e a so n
d i s t a n c e from Buenos A i r e s t o Bozeman.
for
tr a v e llin g
H is a d v i c e ,
th e
lo n g
common s e n s e and
k n o w l e d g e c o u l d be t h e r e a s o n s f o r d o i n g t h a t t r a v e l s e v e r a l t i m e s
a g a in .
I a l s o w a n t t o t h a n k t h e m em b ers o f my g r a d u a t e c o m m i t t e e ,
D rs . P. J. B u r f e n i n g and M. D. H u ffm an f o r t h e i r a d v i c e and h e l p f u l
com m ents.
To Dr. R. L. B l a c k w e l l , f o r h i s k i n d n e s s a s a p e r s o n and
h i s w i s e n e s s a s an a n im a l b r e e d e r .
To Dr. R. C. C h r is t e n s o n from whom
I l e a r n e d t h e u s e f u l s k i l l s o f t h e l i n e a r m od els.
S p e c i a l t h a n k s t o D a l e T r o w b r i d g e , D i a n e Doede and my f e l l o w s
g r a d u a te s t u d e n t s f o r t h e i r f r i e n d s h i p ,
h e l p and p a t i e n c e .
A note of
g r a t i t u d e t o Juan F. Chavez f o r th e good moments we sh a r e d d i s c u s s i n g
o v e r a n im a l b r e e d in g .
To my p r o f e s s o r s and g u i d e s a t B u e n o s A i r e s , D rs . J. L op ez S e c o ,
J. G a r c i a Tobar and I n g . Agr. L. F. S a n t a C o lo m a. . They b e l i e v e d i n me
fr o m t h e b e g i n n i n g .
To Mrs. F r a n c i s c a P e r r i e r de M a g n in , t o whom I
ow e t h e f a c t o f h a v i n g s t u d i e d i n U.S.A.
To A r g e n t i n a , f o r a l l t h e
b e a u t i f u l t h i n g s my c o u n tr y h as g i v e n t o me.
F in a lly ,
t a u g h t me.
w ife,
I would l i k e t o thank my p a r e n t s b e c a u s e I am what t h e y
To my s i s t e r and au n t Leonor f o r b e in g a s t h e y a re.
P a tr ic ia ,
f o r b e in g the b e a u t i f u l p a s t ,
and t h e p r o m is in g f u t u r e .
To my
the s u p p o r t i v e p r e s e n t
v ii
TABLE OF CONTENTS
LIST OF TABLES .
. . .
LIST OF FIGURES
. . .
ABSTRACT .............................
INTRODUCTION ....................
REVIEW OF LITERATURE .
E s t im a t io n o f d i r e c t and m a te r n a l s o u r c e s o f v a r i a t i o n
Problems i n e s t i m a t i n g d i r e c t and m a te r n a l g e n e t i c
c o v a r i a n c e s ......................................................................................
1. Standard e r r o r o f th e e s t i m a t e s and non
in d e p e n d e n c e o f th e c o e f f i c i e n t s i n th e
e x p e c te d v a l u e s . . . .
.....................................................
2 . Sm all number o f r e l a t i v e s i n v o l v e d i n th e
e s t i m a t i o n i n b e e f c a t t l e f i e l d d a ta . . . ,
3 . M aternal e f f e c t s e v a l u a t i o n l e n g h t e n s th e tim e t
con d u c t t h e s t u d y ..................................................... .... •
E s t im a t e s o f d i r e c t and m a tern a l g e n e t i c v a r i a n c e s and
c o v a r i a n c e s f o r b i r t h w e ig h t and weaning w e i g h t i n
b e e f c a t t l e ...................................... ...............................................
1. B i r t h w e ig h t ........................................................................ »
2 . Weaning w e i g h t ...................................... .................................
MATERIALS AND METHODS
Weather ..........................................................
S o i l s ...............................................................
Range c o n d i t i o n s ..................................
E x p erim en tal a n im a ls ........................
Management o f the b r e e d in g herd .
S e l e c t i o n and b r e e d in g p r o c e d u r e s .
S t a t i s t i c a l P ro c ed u r es ....................
v iii
x
xi
I
4
4
18
19
21
21
22
22
24
31
31
32
32
32
33
34
36
RESULTS AND DISCUSSION .
46
SUMMARY
.78
LITERATURE CITED
82
APPENDIX
89
v iii
LIST OF TABLES
Table
1
2
Page
C o e f f i c i e n t s f o r th e d i r e c t and m a te r n a l v a r i a n c e s
and c o v a r i a n c e s i n t h e e x p e c t e d v a l u e s o f th e
c o v a r i a n c e s betw een r e l a t i v e s ............................................ . . .
11
E s t im a t e s o f d i r e c t and m a te r n a l a d d i t i v e v a r i a n c e s
and c o v a r i a n c e s on b i r t h w e i g h t o f c a t t l e .............................
23
E s t im a t e s o f d i r e c t and m a te r n a l a d d i t i v e g e n e t i c ,
v a r i a n c e s and c o v a r ia n c e o f c a l f growth through
w e a n i n g .....................................................................................
25
4
D i s t r i b u t i o n o f r e c o r d s by l i n e s and by y e a r s ....................
40
5
A n a ly s e s o f v a r ia n c e f o r s i r e - t y p e r e l a t i v e s
.....................
47
6
L e a s t - s q u a r e s means f o r age o f dam, s e x , age o f dam by
s e x s u b c l a s s e s and r e g r e s s i o n s i n t h e PHS m o d e l
.
48
7
A n a ly s e s o f v a r i a n c e f o r c o u s i n s f a m i l i e s . ...............................
49
8
A n a ly s e s o f v a r ia n c e f o r f u l l - s i b s and dam-maternal
granddam m odels ......................................
50
3.
9
10
11
12
13
14
E s t im a t e s o f h e r i t a b i l i t i e s and c o r r e l a t i o n s f o r b i r t h
w e i g h t and w eaning w e i g h t .................... . . . . . . . . . .
52
C o v a ria n ce betw een r e l a t i v e s : e s t i m a t e s , d e g r e e s o f
freedom f o r e s t i m a t i o n and e x p e c t e d v a l u e s i n term s
o f d i r e c t and m a te r n a l c o v a r i a n c e s . . . . . .
.......................
53
A sym p totic v a r i a n c e - c o v a r i a n c e m a t r i c e s f o r th e
s i r e - m a t e r n alg ra n d s i r e model ... .........................................................
60
C o e f f i c i e n t s o f c o v a r ia n c e s between r e l a t i v e s and
s o l u t i o n s f o r F a l c o n e r ' s (1 9 6 5 ) model .
62
S o l u t i o n s f o r d i r e c t and m a tern a l c o v a r ia n c e s u sed
by o t h e r w orkers .......................................
63
S o l u t i o n s f o r th e d i r e c t and m a tern a l v a r i a n c e s ,
c o v a r ia n c e s and h e r i t a b i l i t i e s
66
ix
LIST OF TABLES (CONTINUED)
Tab le
15
Page
S im p le c o r r e l a t i o n c o e f f i c i e n t s among th e
c o e f f i c i e n t s o f d i r e c t and m a te r n a l v a r i a n c e s
and c o v a r i a n c e s ............................. ..............................................................
73
LIST OF FIGURES
Path c o e f f i c i e n t diagram o f D i c k e r s o n ’ s (1947) model .
A p ath c o e f f i c i e n t diagram d e s c r i b i n g F a l c o n e r ’ s
(1 9 6 5 ) m a te r n a l e f f e c t s model
. . . . .
....................
A path c o e f f i c i e n t diagram d e s c r i b i n g Koch’s (19 7 2 )
model ..................................................... .... ....................... ....
A path c o e f f i c i e n t diagram d e s c r i b i n g th e
c o v a r ia n c e betw een m a te r n a l grand progeny
.................... .
xi
ABSTRACT
E v id en ce h a s been found t h a t m a te r n a l e f f e c t s a r e im p o r ta n t
s o u r c e s o f v a r i a t i o n i n m am m als. I n b e e f c a t t l e , e v i d e n c e h a s b e e n
fou n d t o s u p p o r t th e h y p o t h e s i s t h a t m a te r n a l e f f e c t s can r e d u c e th e
e x p ected r e sp o n se to s e l e c t i o n f o r g r o w th t r a i t s , e s p e c i a l l y
p rew ea n in g g ro w th .
I t i s n ot c le a r w h eth er m a tern a l e f f e c t s i n
p r e w e a n in g g ro w th o f b e e f c a t t l e a r e g e n e t i c or e n v ir o n m e n ta l.
T h e r e f o r e , t h e p r e s e n t r e s e a r c h was c o n d u c te d t o stu d y t h e n a tu r e and
m agnitude o f m a te r n a l e f f e c t s i n b i r t h and w e an in g w e i g h t o f H ereford
c a ttle.
The d a t a u s e d w e r e t h e r e c o r d s o f 4 , 4 2 3 n o n c r e e p - f e d b e e f
c a l v e s r a i s e d a t t h e N or th e r n A g r i c u l t u r a l R esearch C e n te r , Havre, Mt
from 1938 t o 1983. L e a s t - s q u a r e s , f i t t i n g c o n s t a n t s method (Henderson
I I I ) and R e s t r i c t e d Maximum L i k e l i h o o d p r o c e d u r e s w e r e u s e d t o
e s t i m a t e e ig h t e e n c o v a r ia n c e s b etw een d iffe r e n t types o f r e la t iv e s .
The b a s i c m o d e ls i n c l u d e d t h e f i x e d e f f e c t s o f l i n e - y e a r , age o f dam,
s e x o f c a l f , a g e o f dam by s e x i n t e r a c t i o n and t h e r e g r e s s i o n s o f
b i r t h w e i g h t on b i r t h d a t e o f c a l f and w e a n in g w e i g h t on w e a n in g age o f
c a l f . The s o u r c e o f v a r i a t i o n d e p i c t i n g t h e r e l a t i v e r e l a t i o n s h i p was
c o n s id e r e d a random e f f e c t . M u l t i p l e r e g r e s s i o n p r o c e d u r e s w ere used
t o o b t a i n t h e n in e p a r a m e te r s: a d d i t i v e g e n e t i c , dom inance g e n e t i c and
e n v ir o n m e n t a l d i r e c t and m a te r n a l v a r i a n c e s and th e c o v a r ia n c e b etw een
t h e d i r e c t and m a te r n a l s o u r c e i n each c a s e .
A ll s o l u t i o n s show ed a n e g a t i v e a d d i t i v e g e n e t i c c o r r e l a t i o n
b e t w e e n a d d i t i v e d i r e c t e f f e c t s and a d d i t i v e m a t e r n a l e f f e c t s ( r G).
The r e s u l t s w ere n o t c l e a r r e g a r d in g t o t h e m agnitude o f rg f o r b i r t h
w e i g h t but t h e p r o b a b le v a l u e f o r w ean in g w e i g h t was around - .6 0 t o
-.7 5 .
T here was a l s o e v id e n c e f o r a n e g a t iv e e n v ir o n m e n ta l
c o r r e l a t i o n f o r m a te r n a l phenotype o f t h e dam and d au gh ter i n th e c a s e
o f w e a n in g w e i g h t .
T h is path c o e f f i c i e n t (fm) was c a l c u l a t e d t o be
.08 f o r b i r t h w e i g h t and - .1 0 f o r w ean in g w e i g h t .
A fter c o r r e c tin g
th e e x p e c t a t i o n s o f th e c o v a r i a n c e s b e tw e e n r e l a t i v e s f o r fm, rg was
s t i l l p r e s e n t and n e g a t i v e f o r b o t h w e i g h t s .
The s o l u t i o n s s h o w e d
t h a t dom inance was a l s o i n v o l v e d i n d e t e r m in in g m a te r n a l e f f e c t s i n
p r e w e a n i n g g r o w t h o f b e e f c a t t l e b u t i t s e f f e c t s may be c o n f o u n d e d
w ith e pi s t a s i s .
I .
INTRODUCTION
The
success
of
c h a r a c te r istic s
of
a . b r e e d in g
farm
schem e
a n im a ls
to
im p rove
depends
o n how
p erform an ce
g e n e tic
e n v ir o n m e n t a l s o u r c e s o f v a r i a t i o n a r e ta k e n i n t o a c c o u n t.
and
L ike o t h e r
d o m e s t i c mammals, b e e f c a t t l e a r e s u b j e c t t o m a te r n a l e n viron m en t from
t h e e a r l y m o m e n ts o f g e s t a t i o n
effect
is
an
effect
i n d i v i d u a l by i t s
th rou gh w ea n in g tim e .
c o n trib u ted
dam (W illh am ,
to
1980).
the" p h e n o t y p i c
A m atern al
v a lu e
of
an
A lthough m a te r n a l e f f e c t s are
e n v i r o n m e n t a l s o f a r a s t h e i r i n f l u e n c e on o f f s p r i n g i s c o n c e r n e d ,
th e y a r e d e te r m in e d by g e n e t i c and e n v ir o n m e n t a l f a c t o r s (Koch,
B irth
w eig h t
g e s ta tio n len g th .
is
the
r e su lt
of
Weaning w e i g h t i s
g e sta tio n a l
grow th
1972).
ra te
and
t h e c o n se q u e n c e o f b i r t h w e ig h t
and g r o w t h d u r i n g t h e s u c k l i n g p e r i o d .
B o th t r a i t s a r e m e a s u r e d a s
th e p h e n o t y p ic v a lu e o f th e c a l f , b u t th e y a re com posed o f a t l e a s t
tw o com p onents,
o f f s p r i n g g e n e t i c s f o r grow th and a m a te r n a l e f f e c t
c o n trib u te d
th e
by
dam.
T h is
la tte r
in flu e n c e
is
produced
by
n u t r i e n t s p r o v id e d by t h e u t e r u s ( i n t h e c a s e o f b i r t h w e i g h t ) and th e
mammary g l a n d ( i n t h e c a s e o f w e a n i n g w e i g h t ) .
com m ents,
it
in v o lv e d
in
As R o b is o n ( 1 9 8 1 )
h a s r e c e n t l y become a p p a r e n t t h a t o t h e r f a c t o r s may be
th is
a c tio n ,
perhaps
th rou gh
c ir c u la tin g
horm ones,
c y to p la sm ,e tc .
The o t h e r c o n t r i b u t i o n o f th e dam t o t h e p h e n o ty p ic v a l u e o f th e
o f f s p r i n g i s a sa m p le h a l f o f h e r g e n e s t o th e c a l f .
th e s i r e
p a ssin g
c o n trib u te s
a
sa m p le
T h e r e fo r e , w h i l e
t o t h e p h e n o t y p i c v a l u e o f t h e o f f s p r i n g by
h a lf
of
h is
genes
to
th e
o ffsp r in g ,
th e
dam
2
c o n t r i b u t e s i n a t l e a s t tw o ways.
Willham (1980) p o i n t s o u t t h a t th e
c o n fo u n d in g o f t h e two c o n t r i b u t i o n s from th e dam and th e p o s s i b i l i t y
o f a n e g a t i v e g e n e t i c c o r r e l a t i o n b e t w e e n t h e d i r e c t and m a t e r n a l
e f f e c t c o n s t i t u t e th e b a s e s f o r
m a te r n a l e f f e c t s .
It is
th e paramount p rob lem s i n e s t i m a t i n g
c le a r th at h e r it a b i l l t i e s
(h2 ) can be b ia s e d
b e c a u s e o f th e p r e s e n c e o f m a te r n a l e f f e c t s (R o b iso n ,
The m o d e l i n g p r o c e s s
to
e stim a te
m a tern a l
1981).
effects
h a s been
i n i t i a t e d by D i c k e r s o n ( 1 9 4 7 ) .
K e m p th o r n e ( 1 9 5 5 ) w a s an i m p o r t a n t
paper
of
in
so lv in g
the
p r o b le m
th e
e stim a tio n
of
g e n e tic
e n v ir o n m e n t a l v a r i a n c e s b ased on c o v a r i a n c e s b e tw e e n r e l a t i v e s .
and
The
p r e s e n t a t i o n i n c lu d e d th e b a s i s o f th e a c t u a l th e o r y f o r m ea su r in g
m a te r n a l and d i r e c t v a r i a t i o n .
Koch and C lark (1955b) was t h e f i r s t
paper t o e s t i m a t e t h e a d d i t i v e g e n e t i c c o v a r ia n c e b e tw e e n d i r e c t and
m a te r n a l e f f e c t s
th e o r y .
F in a lly ,
( a AoAm) i n b e e f c a t t l e ,
by u s i n g path c o e f f i c i e n t s
Willham (1963) d e v e lo p e d a l i n e a r model th e o r y f o r
e s t i m a t i n g d i r e c t and m a te r n a l g e n e t i c c o v a r i a n c e s and v a r i a n c e s by an
e x t e n s i o n o f th e p r o c e d u r e s f i r s t d e v e lo p e d by Kempthorne (1955).
The u s e o f W i l l h a m ' s m e th o d a p p l i e d t o c a t t l e r e c o r d s h a s b e e n
r e p o r t e d by E v e r e t t and Magee ( 1 9 6 5 ) , H i l l ( 1 9 6 5 ) , Brow n and G a lv e z
(1 969),
V e s e l y and R o b i s o n ( 1 9 7 1 ) ,
F ish e r
and W i l l i a m s
w e ig h t.
(1978)
Koch ( 1 9 7 2 ) ,
and B u r f e n i n g e t
P h ilip sso n
al
(1981)
H i l l e t a l ( 1 9 6 5 ) , D e e s e and R o g e r ( 1 9 6 7 ) ,
B rin k s (1 9 6 7 ),
for
(1 9 7 6 ),
b ir th
H o h en b ok en and
V e s e l y and R o b is o n ( 1 9 7 1 ) , Koch ( 1 9 7 2 ) , B e l t r a n Bru
(1978) and K r e s s e t a l (1979) d e a l t w i t h w ean in g w e i g h t .
c o n c l u s i o n s from
t h e r e v ie w e d l i t e r a t u r e
The g e n e r a l
w ere t h a t h e r i t a b i l i t y
for
a d d i t i v e g e n e t i c d i r e c t e f f e c t s (h 2 o) was l a r g e r f o r b i r t h w e i g h t than
3
t h e c o n t r i b u t i o n s o f a d d i t i v e g e n e t i c m a t e r n a l e f f e c t s (hm^).
c o n v e r s e was t r u e f o r w e a n in g w e i g h t .
The
A n e g a t i v e g e n e t i c antagonism
b e tw e e n a d d i t i v e g e n e t i c d i r e c t and a d d i t i v e g e n e t i c m a te r n a l e f f e c t s
(i.e .,
n e g a tiv e v a lu e o f
m a g n itu d e o f
oAoAm) h a s been found f o r b oth t r a i t s .
The
oAoAm i s l i k e l y t o be r e l a t i v e l y g r e a t e r f o r w e a n i n g
w e i g h t than f o r b i r t h w e i g h t .
However,
th e e x a c t v a l u e o f
b e e n t h e m a t t e r o f s o m e d i s c u s s i o n (K och,
1972; B a k e r ,
V ie c k e t a l (1977) have shown t h a t t h 6 v a l u e o f
AoAm has
1980).
Van
a AoAm d e t e r m in e s th e
l o n g term r e s p o n s e t o s e l e c t i o n f o r w e an in g w e ig h t .
T otu sek (1 9 6 8 ),
(1 9 7 2 ),
Mangus and B r i n k s ( 1 9 7 1 ) , K r e s s and B u r f e n i n g
M artin e t a l (1981) and Ochoa e t a l (1981) have found t h a t th e
e n v ir o n m e n t i n w h ic h th e h e i f e r c a l f i s r a i s e d a f f e c t s h e r f u t u r e
m a te r n a l p h enotype t h a t sh e p r o v i d e s f o r h er c a l v e s .
T h is c o m p l i c a t e s
t h e m a t t e r o f t h e r e l a t i v e m a g n i t u d e o f e n v i r o n m e n t a l and g e n e t i c
s o u r c e s o f v a r i a t i o n on t h e e x p r e s s i o n o f m a t e r n a l e f f e c t s , a s Koch
(1972) and Hohenboken (1973) d i s c u s s .
The o b j e c t i v e s o f t h e p r e s e n t s tu d y w ere:
1) t o e s t i m a t e th e amount o f v a r i a t i o n due t o a d d i t i v e g e n e t i c d i r e c t
and a d d i t i v e g e n e t i c m a t e r n a l e f f e c t s i n b i r t h w e i g h t and w e a n i n g
w e i g h t o f H ereford b e e f c a l v e s ,
2) to c la r if y
t h e problem o f th e r e l a t i v e im p o r ta n c e o f g e n otyp e and
e n v ir o n m e n t f o r th e e x p r e s s i o n o f m a te r n a l e f f e c t s on b oth t r a i t s , and
3 ) t o e s t i m a t e t h e a d d i t i v e g e n e t i c c o v a r i a n c e b e t w e e n d i r e c t and
m a te r n a l e f f e c t s f o r b oth b i r t h w e i g h t and w ean in g w e i g h t o f H ereford
b e e f c a lv e s.
4
REVIEW OF LITERATURE
.EaMroatlon o f d i r e c t and m a te r n a l s o u r c e s o f v a r i a t i o n
As i n
th e o r y i s
m ost o th e r a n im a l b r e e d in g p ro b lem s,
stro n g ly r e la te d
m atern al e f f e c t s
to qu a d ra tic e s tim a tio n .
The m od els used
t o e s t i m a t e v a r i a n c e com ponents due t o m a te r n a l e f f e c t s s t a r t e d from
th e v a r i a n c e o f th e c l a s s i c m odel:
O2 P =
w here
O2E
O2 G +
(I)
a 2 P i s th e t o t a l p h e n o ty p ic v a r ia n c e ,
due t o g e n e t i c
effects,
D u rin g th e e n t i r e
in c o rp o ra te
th e
and
C 2t E i s
m o d e lin g p r o c e s s ,
gen otyp e
a 2G i s th e v a r ia n c e
th e en v iro n m e n ta l v a r ia n c e .
no a t t e m p t h a s b e e n made t o
by e n v i r o n m e n t a l
in te r a c tio n
source of
v a r i a t i o n ( a 2 GE) i n t o m a te r n a l e f f e c t s q u a d r a t i c e s t i m a t i o n .
A fu rth er p a r titio n of
O2 G =
a 2G can be made a s f o l l o w s :
O2 A +
d 2D + c £ l
a 2 A i s t h e v a r i a n c e due t o a d d i t i v e g e n e t i c e f f e c t s ,
where
th e v a r i a n c e due t o dom inance e f f e c t s
and
(2)
a2I i s
th e
v a r ia n c e
due
to
(i.e .,
a 2D i s
in tr a lo c u s in te r a c tio n )
e p ista sis
(i.e .,
in te r lo c u s
in te r a c tio n ).
D i c k e r s o n ( 1 9 4 7 ) made t h e f i r s t p a r t i t i o n o f t h e v a r i a n c e s i n
m odel
(I)
to
in c o rp o ra te
m a tern a l
effects.
He d i d
not
c o n sid e r
dom inance or e p i s t a s i s i n h i s model a s shown below :
a 2 P = O2 Ao +
O2 Am +
o Ao Am +
O2 Em +
O2 Eo
The term s o f model ( 3 ) a r e
O2 Ao
= v a r i a n c e due t o a d d i t i v e d i r e c t e f f e c t s ,
O2 Am
s v a r i a n c e due t o a d d i t i v e m a te r n a l e f f e c t s ,
(3)
5
crAoAm
= c o v a r ia n c e b e tw e e n a d d i t i v e m a te r n a l and a d d i t i v e d i r e c t
effects,
p
a Em
= v a r i a n c e due t o m a te r n a l e n v ir o n m e n t a l e f f e c t s common to
f u l l - s i b s and m a te r n a l h a l f - s i b s , and
a^Eo
= v a r i a n c e due t o random e n v iro n m e n t.
The m o d e l w a s u s e d t o e s t i m a t e m a t e r n a l and d i r e c t c o v a r i a n c e s i n
carcass
d ata
of
sw in e .
D ic k e r so n
( 19 4 7 ) d e v e l o p e d
th e
p ath
c o e f f i c i e n t diagram shown i n F ig u r e I.
The p h e n o t y p e o f X (P x ) i s
t h e r e s u l t o f i t s own g e n o t y p e f o r
d i r e c t e f f e c t s (Gox) p l u s a commmon or m a te r n a l e n v iro n m e n t component
among l i t t e r
m a te s (Pmw)s where w i n d i c a t e s th e dam o f X.
D ic k e r so n
(1947) d id n o t i n c l u d e t h e random e n v ir o n m e n t a l s o u r c e o f v a r i a t i o n i n
the
d ia g ra m
but in
th e a n a ly s is .
The t r a n s m i s s i b l e g e n o t y p e or
a d d i t i v e g e n e t i c e f f e c t s a r e s p l i t i n t o two com ponents: 0 ( d i r e c t ) and
m ( m a t e r n a l) .
The Gmx component w i l l be e x p r e s s e d o n ly i f i n d i v i d u a l
X s u b s e q u e n t l y b e c o m e s a dam ( W i l l h a m , 19 6 3 ) .
a ffected
by t h e m a t e r n a l g e n o t y p e o f W ( i . e . ,
The c o m p o n e n t Pmw i s
Gmw).
T h e r e w as no
i n t e n t i o n to r e l a t e th e m a tern a l p h en otyp e o f w w ith th e m atern al
p h e n o t y p e o f h e r dam, h e r m a t e r n a l g r a n d dam and s o on.
6
F ig u r e I . Path c o e f f i c i e n t diagram o f D i c k e r s o n ' s ( 1947) model.
7
In te r m s o f
or r e g r e s s i o n
th e path c o e f f i c i e n t s o f F ig u r e I ,
of
tra n sm itted
a b ility
the h e r i t a b i l i t y
(Gox + Gmx) on i n d i v i d u a l
p e rform an ce (X) i s
b(Go + Gmx) Px = go 2 + 3 / 2 gogm Pgogm + 1 /2 gm2
This e x p r e s s i o n i s
th e num erator f o r what i s
(4)
c a lle d h e r it a b ility
f o r t o t a l e f f e c t s w here " to ta l" s ta n d s f o r a d d i t iv e g e n e t i c d i r e c t
p l u s a d d i t i v e g e n e t i c m a te r n a l e f f e c t s .
The m a in c o n t r i b u t i o n s o f K e m p th o r n e ( 195 5 ) t o t h e t h e o r y o f
m a te r n a l e f f e c t s was t h e i n c l u s i o n o f dom inance i n th e model and th e
d e sc r ip tio n
of
th e b a s is
for
th e
c o e ffic ie n ts
of
t h e d i r e c t and
m a te r n a l c o v a r i a n c e s i n t o th e e x p e c t a t i o n o f t h e c o v a r i a n c e b e t w e e n
r e la tiv e s.
H is
th eory
assum es
th at
th e
g e n o ty p ic
v a lu e
of
an
i n d i v i d u a l i s a d d i t i v e l y d e te r m in e d by t h e j o i n t e f f e c t s o f the g e n e s
p o s s e s s e d by th e i n d i v i d u a l and i t s
h is
m odel
O2 G =
The g e n e t i c v a r ia n c e o f
is
Cf2 Ao +
w h e r e O2 Do and
d o m in a n c e
m other.
O2 Am +
a AoAm +
cr^Do +
o
+
CDoDm
(5 )
O2 Dm a r e t h e v a r i a n c e s d ue t o d o m in a n c e d i r e c t and
m a tern a l
d e v ia tio n s,
r e sp e c tiv e ly ,
and
ODoDm i s
th e
c o v a r ia n c e b e tw e e n th e dom inance e f f e c t s .
Kempthorne ( 1955) d id n o t c o n s i d e r t h e common or m a te r n a l s o u r c e
o f e n v ir o n m e n t a l v a r i a t i o n ( i . e . ,
C2 Em).
The o t h e r e x p r e s s i o n s d e r iv e d
a r e t h e c o v a r i a n c e s b e t w e e n o f f s p r i n g and s i r e ( c o v ( 0 ,S ) ) , o f f s p r i n g
and dam (co v (0 ,D )) and among f u l l - s i bs
cov ( 0 , S ) = pq C 2 Ao + 1 /2 pq
cov (0 ,D ) = pq C 2 Ao + pq
(cov(FS)) as
C AoAm,
C 2 Am + 5 / 4
C 2 AoAm +
c DoDm,
and
8
cov (FS)
= 1 /2
a2 Ao +
O2 Am + 2 pq
aAoAm + i/H
a 2Do +
O2Dm.
p and q a r e t h e ge n e f r e q u e n c i e s f o r g e n e s A and a , r e s p e c t i v e l y .
The m ost commonly u s e d model f o r e s t i m a t i n g m a te r n a l e f f e c t s o f
b ir th
w e i g h t and w e a n i n g w e i g h t i n c a t t l e w as d e r i v e d by W illh a m
(1963).
a d d in g
B a sic a lly ,
e p ista sis
he g e n e r a l i z e d
to
th e
m o d e l.
t h e w ork o f K em p th o r n e ( 1 9 5 5 )
H ow ever,
th is
in te r lo c u s
gene
e x p r e s s i o n h a s a lw a y s been assumed t o be z e r o or n e g l i g i b l e f o r both
b i r t h w e i g h t and w e a n i n g w e i g h t .
W illh a m 's
paper
was
th e
P erh ap s th e m ost u s e f u l r e s u l t o f
d e r iv a tio n
of
th e
c o e ffic ie n ts
for
th e
a d d i t i v e and dom inance c o v a r i a n c e s i n t h e e x p e c te d v a l u e s o f v a r io u s
k in d s o f r e l a t i v e s .
The
c o e ffic ie n ts
th a t
K em pthorne
(1955)
f u n c t i o n s o f W righ t’s c o e f f i c i e n t o f r e l a t i o n s h i p
had e x p r e s s e d
w ith
as
no i n b r e e d in g
o r M a l e c o t ’s " c o e f f i c i e n t de p a r e n t e " r e c e i v e d c o n c r e t e n u m e r i c a l
v a l u e s w i t h t h e work o f W illham .
W illham (1963) showed t h a t th e g e n o t y p i c c o v a r ia n c e b etw een th e
p h e n o ty p e s o f i n d i v i d u a l s X and I , w i t h r e s p e c t i v e m o th e r s W and Z, i s
equal to
co v (G x ,
Gy)
% c o v (G o x ,G o y )
+ c o v (G o x ,G m z )
+ cov(Gmw,Gmz)
+ c o v ( G m w ,G o y )
(6)
w h e r e Gox and Goy a r e t h e g e n o t y p i c v a l u e s o f X and Y f o r d i r e c t
effects,
and Gmw and Gmz a r e th e g e n o t y p i c v a l u e s o f th e dams W and Z
fo r m atern al e f f e c t s .
th e f o u r c o v a r i a n c e s i n
T h us, t h e p r o b le m w as r e d u c e d t o c a l c u l a t i n g
(6).
9
The f i r s t c o v a r ia n c e b e tw e e n X and Y f o r component o was g i v e n by
Kempthorne (1957) a s
2pxy
O2 Ao + uxy
O2Do +
(2px y ) r (ux y ) s
O2 (Ar Ds ) 0
f o r 21 r+ siN
where
O2 (Ar Ds )0 i s
the e p i s t a t i c
d i r e c t c o m p o n e n t o.
component o f g e n o t y p i c v a r i a n c e f o r
th e
For t h i s t e r m , r and s r e f e r t o t h e num ber o f
l o c i i n v o l v e d i n t h e i n t e r a c t i o n and N r e f e r s t o t h e num ber o f l o c i
s e g r e g a t i n g f o r component o.
The term pxy i s
W righ t’s c o e f f i c i e n t o f
r e l a t i o n s h i p w i t h no i n b r e e d i n g o r t w i c e M a le c o t ’s " c o e f f i e c i e n t de
p arente".
The c o e f f i c i e n t uxy i s
th e p r o b a b i l i t y
t h a t t h e two g e n e s
a t a l o c u s i n i n d i v i d u a l x a r e i d e n t i c a l by d e s c e n t w i t h t h e two g e n e s
a t t h a t l o c u s i n i n d i v i d u a l y.
The v a l u e o f uYV i s z e r o u n l e s s t h e
tw o i n d i v i d u a l s a r e r e l a t e d by tw o l i n e s o f d e s c e n t such a s f u l l s i b s
or d o u b le f i r s t
c o u sin s.
The s e c o n d and t h i r d t e r m s i n ( 6 ) a r e t h e g e n o t y p i c c o v a r i a n c e s
b e tw e e n t h e i n d i v i d u a l s and t h e i r m o th e r s f o r com p onents o e x p r e s s e d
i n x o r y. and c o m p o n e n t m e x p r e s s e d i n y o r x , r e s p e c t i v e l y .
S in ce
Mode and R obinson ( 1959) showed t h a t t h e g e n o t y p ic c o v a r i a n c e b e tw e e n
ch a ra cters,
or
in
th is
case
com p on en ts
of
a
ch a ra cter
can
be
p a r t i t i o n e d i n an a n a l o g u e s m anner t o t h e g e n o t y p i c v a r i a n c e , t h o s e
term s are
2 pxz
a AoAm + uxz
CfDoDm + r ^ s (2 Px z ) r ^uX z)8
Cf(Ar D3 ) 0 (Ar Ds ) m
2£r+s£M
and
10
2 pyw
CTAoAm + uyw CTDoDm +
(2pyw) r (uyw) s
CT(Ar Ds )0 (Ar Ds )m
2^r+s£M
Cf (Ar Ds )0 (Ar Ds ) m i s
th e e p i s t a t i c c o v a r ia n c e b etw e e n o and m„
M is
t h e number o f l o c i s e g r e g a t i n g t h a t a f f e c t both 6 and m.
A general
is
ex p ressio n fo r
th e cov(Gx,Gy) a ssu m in g
th at e p i s t a s is
equal to z e r o i s :
cov(G x,G y) = 2 Pxy
+
Cf2 Ao + ( 2 Pxz + 2 pyw)
Cf2Do +
Ux y
(U x z
+ uyw)
cfAoAm + 2 pwz
a 2Am
CfDoDm + uwz CT2Dm
(7)
Table I show s t h e c o e f f i c i e n t s 2p and u f o r s e v e r a l r e l a t i v e s and
i s b a s e d on t h e r e l a t i o n s h i p s c o n s i d e r e d by W illh a m ( 1 9 6 3 ) and Van
V le c k and Hart (1966) f o r t h e g e n e t i c com ponents o f (7) w i t h th e more
general ex p ressio n s fo r
th e e n v ir o n m e n t a l com ponents o f v a r i a t i o n a s
sh o w n i n Thom pson ( 1 9 7 6 ) .
Then, w hen x and y h a v e t h e sa m e m a t e r n a l
grand s i r e we have
2 Pxy = ( 1 / 2 )
= 1/1 6, 2 Pxz - 1/8; .2 Pyw = 1/8; 2 Pzw - ( 1 / 2 ) ^ - 1/4
S i n c e x and y a r e n o t r e l a t e d by two l i n e s o f d e s c e n t a l l u’s a r e
zero.
T h erefore,
the exp ected v a lu e o f
E( Cf2JdQg) = 2 pxy
E(
Cf 2MqS) = 1/16
a 2 ^cs i s
CT2 Ao + (2 pxz + 2 Pyw) Cf AoAm + 2 pwz Cf2 Am
Cf2Ao + 1/4
CfAoAM + 1/4
Cf2Am
Any k i n d o f r e l a t i v e r e l a t i o n s h i p c a n be e v a l u a t e d i n t h e sam e
way.
C o v a r ia n c e s i n T a b le I w ere d i v i d e d i n t o t h r e e g r o u p s .
fir st
group
sir e -ty p e
r e la tio n sh ip s
are
c o n sid e r e d ;
In t h e
hence,
th e
Paternal
Paternal
Maternal
Maternal
(MGGS)
Maternal
H alf-S ibs (PHS)
Grand S ir e - S ib s (PGS)
Grand S ir e - S ib s (MGS)
Great Grand S ir e - S ib s
Grand Dam-Sibs (MGD)
F u ll-S ib s (FS)
Maternal H alf-Sibs (MHS)
S in g le F i r s t Cousins (SFC)
PHS plus Dams MHS (PHS + MHSD)
PHS + SFC
FS plus PHS parents (FS + PHS)
PHS + PHS Dams (PHS + PHSD)
a 2C 0
0Vm
aX
S ir e-tvn e erouD
O
0
O
0
1/4 '
0
0
0
0
0
0
0
0
0
o.
0
0
0
0
0
0
° x
IA
1/16
1/16
O
O
1/4
1/64
1/16
1/16
1/4
0
0
0
0
0
0
0
0
0
0
1/2
1/2
•
Covariance erouo
1/4
O
O
1/2
5/4
0
0
0
0
I
0
0
0
0 .
0
I
0
0
1/4
5 /8
1/4
0
0
0
0
0
0
1/8
1/4
O
0
0
0
0
0
0
1/4
3/4
1/2
0
1/4
0
0
0
0
1/4
3 /4
1/2
0
1/4
0
0
0
0
I
I
1/4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
I
1/16
1/4
o . « - C= C - —
I
I
B
I
a
I
B
I
I
S
■
Offspring and S ir e (COV(O1S))
Offspring and Dam (COV(OjD))
Offspring and Maternal
Grand Dam (COV(Oj Mj D))
Maternal Grand S ir e Progeny and
Grand Offspring COV(SjMGS)
Maternal F u ll-S ib Aunt and
Offspring (COV(MAjN))
Paternal F u ll-S ib Aunt and
Offspring (COV(PAjN))
0V m
£
cf2flO
Q
R elatives
Q
ro
TABLE I . COEFFICIENTS FOR THE DIRECT AND MATERNAL VARIANCES AND COVARIANCES IN THE EXPECTED VALUES OF THE
COVARIANCES BETWEEN RELATIVES
R e lative r e la t io n s h ip s in v o lv in g dominance
1/4
0
1/2
I
I
I
0
0
1/4
I
1/8
1/2
1/2
0
5/16
3/8
5/8
5/16
1/4
1/4
1/16
1/2
1/2
I
1/4
25/64
1/64
5/4
1/4
1/8
0
0
0
0
0
0
1/4
I
0
0
0
0
I
I
12
e x p e c ta tio n s
o n ly
in v o lv e s
a d d itiv e
effects.
The
second
group
c o n t a i n s c o v a r i a n c e s b e tw e e n t h e o f f s p r i n g g e n e r a t i o n and a c l o s e l y
r e la te d
in d iv id u a l
of
th e p a r e n ta l g e n e r a tio n .
composed by t h e te r m s i n v o l v i n g dom inance.
i n w h ich t h e r e i s
used.
The l a s t
group i s
The e r r o r t e r m s f o r m o d e ls
o n l y one f a m i l y component have n o t been e x t e n s i v e l y
An u n c l e a r i n t e r p r e t a t i o n o f t h e c o m p o n e n t s i n v o l v e d i n t h e
e x p e c t a t i o n f o r t h o s e e r r o r ter m s i s t h e p o s s i b l e r e a s o n .
So f a r
it
has
been
assum ed
th ere
is
no
effect
of
th e
dam
e n v i r o n m e n t on t h e f u t u r e m a t e r n a l a b i l i t y o f t h e f e m a l e o f f s p r i n g .
T h is i s l i k e l y to o c c u r f o r w e a n in g w e i g h t i n b e e f c a t t l e ,
( 1972) p o in t e d o u t.
I f t h i s i s t r u e , m a t e r n a l e n v i r o n m e n t w i l l be
a f f e c t e d by g r a n d m a t e r n a l e n v i r o n m e n t and t h e l a t t e r
g r e a t g r a n d dam e n v i r o n m e n t and s o on .
effect i s
a s Koch
view ed as p a r t i a l l y
a ffected
by m a t e r n a l
In t h i s s e n s e , th e m a te r n a l
by. m a te r n a l e n v ir o n m e n t from
p r e v io u s gen era tio n s.
The f i r s t a t t e m p t t o i n c o r p o r a t e t h e s e e f f e c t s i n t o th e model was
made by F a lc o n e r (1965).
CT2 P =
t
C2 A
+
The v a r ia n c e o f h i s model i s :
CT2M + 2
CTAM +
CT2D +
CT2 Em +
C2t Eo
w here
CT2 A i s th e v a r i a n c e o f a d d i t i v e e f f e c t s ,
due t o
th e m a te r n a l e f f e c t s t o w h ich t h e i n d i v i d u a l i s
is
C2t M i s
th e c o v a r ia n c e b e tw e e n a d d i t i v e arid m a te r n a l e f f e c t s ;
( 8)
th e v a r i a n c e
su b ject,
C t2D i s
CT AM
th e
v a r i a n c e due t o dom inance d e v i a t i o n s ( F a lc o n e r i n c l u d e d h e r e a l s o th e
e p ista tic
to
d e v i a t i o n s i n v o l v i n g d om in an ce),
m a tern a l
or
common e n v i r o n m e n t ,
v a r i a n c e d u e t o random e n v i r o n m e n t .
d e fin e d as a l in e a r fu n c tio n ,
as
CT2 Em i s
b efore,
t h e v a r i a n c e due
and
C t2 Eo i s
The m a t e r n a l e f f e c t s
th e
(M) a r e
fm, o f t h e m o th e r ’s p h e n o t y p ic m a te r n a l
13
v a l u e , P», m easured a s a d e v i a t i o n from th e p o p u la t io n mean, s o t h a t
( 9)
M = fm P'
F a l c o n e r ( 1 9 6 5 ) a d m i t t e d t h a t t h e way M i s
r e str ic te d
because i t
c o r r e la te d w ith
m a t e r n a l i n f l u e n c e s t h a t a r e not
th e m o th e r ’s p h e n o t y p ic m a te r n a l v a l u e .
p o in ts out th a t i f
in c lu d ed
e x c lu d es a l l
d e fin e d i s ra th er
But he a l s o
t h e s e o t h e r i n f l u e n c e s a r e p r e s e n t t h e y w i l l be
w i t h t h e r e s t o f th e common or m a te r n a l e n v ir o n m e n t ( a 2 Em).
The c o e f f i c i e n t fm i s
d e f in e d t o be "a p a r t i a l r e g r e s s i o n c o e f f i c i e n t
r e l a t i n g d a u g h te r s to m o th e r s’ p h e n o ty p ic v a lu e s in th e absence of
g e n e t i c v a r i a t i o n am ong t h e m o t h e r s " .
r e a lly lin e a r i s ,
r e la tio n sh ip i s
D, Em
W hether t h e r e l a t i o n s h i p i s
o f c o u r s e , open to q u e s t io n .
e a s ie r to ev a lu a te .
O b v io u sly , a lin e a r
I t s h o u ld a l s o be n oted t h a t th e
and Eo te r m s i n th e model a r e u n c o r r e l a t e d w i t h P’.
o th er c o v a r ia n c e s in
( 8) a r e z e r o b u t
a AM.
H e n c e ,a ll
An i n t e r p r e t a t i o n o f
model ( 8) i n te r m s o f path c o e f f i c i e n t s i s shown i n F ig u r e 2.
The p r i m e i n t h e g r a p h i n d i c a t e s t h e p r e v i o u s g e n e r a t i o n .
The
a r r o w p o i n t i n g a t M’ fr o m t h e l e f t i n d i c a t e s t h e c a r r y i n g o v e r o f
m a te r n a l
effects
from
p r e v io u s g e n e r a t i o n s .
F a lc o n e r
( I 965) d e r iv e d
th e cov(0,D ) i n ter m s o f fm a s
c o v ( 0 ,D ) =
a 2 A ( 1 / ( 2 - fm )) + fm
I t i s w orth n o t i n g th e d e r i v a t i o n o f
a 2 Pm’
a AM.
By d e f i n i t i o n M = fm Pm’ = fm (A’ + M» + D» + Em’ + E o ').
S in ce
D’, Em’ and Eo' a r e n o t c o r r e l a t e d w i t h A, t h e y c a n be o m i t t e d w h i l e
d e r iv in g
a AM. T h e r e f o r e , t h e r e l e v a n t p a r t o f M can be w r i t t e n as
M = fm (A' + fm P ” )
= fm (A ’ + fm (A ”
+ fm P’ ” ) )
14
F ig u r e 2.
A p a t h c o e f f i c i e n t d ia g r a m d e s c r i b i n g F a l c o n e r ' s ( 1 9 6 5 )
m a te r n a l e f f e c t s m odel.
a d d itiv e
gen otyp e;
P is
D,
the phenotype o f th e i n d i v i d u a l ;
h is
d o m in a n c e
Em a r e
th e
e n v ir o n m e n t a l f a c t o r s common t o f u l l - s i b s and m a te r n a l h a l f - s i b s
that
a r e n ot i n c l u d e d i n t h e m a te r n a l e f f e c t ;
are o t h e r e n v ir o n m e n t a l f a c t o r s
is
d e v ia tio n ;
A, h i s
M is
p a r tic u la r
th e m a te r n a l e f f e c t ;
Eo
t o th e i n d i v i d u a l and r AM
t h e c o r r e l a t i o n b e t w e e n a d d i t i v e g e n e t i c and m a t e r n a l e f f e c t s .
P a th c o e f f i c i e n t s ( e 0 , a , d, m, e m, .5 and fm ) a r e s t a n d a r d p a r t i a l
r e g r e ssio n c o e ffic ie n ts .
15
= fm (A». + fm (.Ag 1 + fm (A ggg + . . . . e t c )
T h e r e fo r e
CAM = fm
OAM =
OAAg + fm2
Iflfm l< I
OAAggg + . . . .
O2A ( 1 / 2 fm + 1/1} fm2 + 1 /8 fm3 + . . . .
The e x p r e s s i o n i n b r a c k e t s i s
1 /2 fm.
OAAgg + fm3
t h e g e o m e t r i c s e r i e s w i t h common r a t i o
th e s e r i e s c o n v e r g e s w i t h sum
( 1/ 2 ) fm
—— — —— ——
fm
=
I - fm/ 2
2 - fm
Hence
oAM =
Thom pson ( 1 9 7 6 ) h a s
d e r iv e d
fm
O2 A -------------- '
2 - fm
th e
c o e ffic ie n ts
for
fm i n
th e
e x p e c t e d v a l u e s o f some r e l a t i v e r e l a t i o n s h i p s .
W i ll h a m ( 1 9 7 2 ) c o n s i d e r e d a m o d e l i n w h i c h t h e e f f e c t o f t h e
m a te r n a l grand dam i s
tw o r e l a t i v e s
is
p resen t.
e v a lu a te d
The g e n o t y p i c c o v a r ia n c e b etw e e n any
as in
( 6),
but w ith
n in e
c o v a r ia n c e s
i n s t e a d o f f o u r . An a d d i t i v e g e n e t i c t e r m f o r t h e v a r i a t i o n due t o
grand m a te r n a l e f f e c t s
(i.e .
O 2 An) i s
a lso
d e fin e d .
For g e n e t i c
e f f e c t s o n l y t h e v a r i a n c e o f t h i s model i s
O2 A =
where
O2 Ao +
OAoAn and
o AoAm +
OAoAn +
O2 Am +
OAmAn + O 2 An
(9 )
OAmAn a r e t h e c o v a r i a n c e s b etw e e n d i r e c t and grand
m a t e r n a l a d d i t i v e e f f e c t s and m a t e r n a l and g r a n d m a t e r n a l a d d i t i v e
effects,
r e sp e c tiv e ly .
From a t h e o r e t i c a l p o i n t o f v i e w ,
it
is
n ot c le a r w h eth er to
p r e f e r th e m a te r n a l-g r a n d m a te r n a l a d d i t i v e m odel to th e m a te r n a l
16
a d d i t i v e m odel.
U sin g t h e same r e a s o n i n g ,
it is
p o s s ib le to in c lu d e
g r e a t grand m a te r n a l e f f e c t s i n t o th e model and s o on.
A d d i t i v e g e n e t i c e f f e c t s a r e c a r r i e d by t h e dam s i d e t h r o u g h
se v e r a l g en era tio n s.
H ow ever,
t h e sa m e t h e o r y o f t h e c o v a r i a n c e
b e tw e e n r e l a t i v e s (Kempthorne, 1955) show s a h ig h l e v e l o f r e l i a b i l i t y
in
th e e v a lu a tio n
of
th e a d d it iv e
gen otyp e
or b r e e d in g v a lu e
c o n s i d e r i n g o n l y one g e n e r a t i o n p r e v io u s t o th e one c o n s id e r e d .
r e su lt of
th ese id e a s,
Koch ( 1 9 7 2 ) d e f i n e d
by
As a
th e m ost b i o l o g i c a l l y
m e a n in g fu l m odel to e v a l u a t e m a te r n a l e f f e c t s i n b e e f c a t t l e b i r t h
w e i g h t and w e a n i n g w e i g h t .
The m o d e l
c o m b in e d W i l l h a m ' s
( 196 3 )
a d d i t i v e and d o m i n a n t c o m p o n e n t s w i t h F a l c o n e r ' s ( 1 9 6 5 ) c o n c e p t o f
t r a n s m i t t e d m a te r n a l p henotype e f f e c t s .
The c o v a r ia n c e b e tw e e n random
a EoEm)
e n v ir o n m e n t a l and m a te r n a l or common e n v ir o n m e n t a l e f f e c t s (
was a l s o added.
The v a r i a n c e o f Koch's (1972) model can be e x p r e s s e d
as
a 2 p = o^ko +
a AoAm + o^Am + a 2-Do + a DoDm + a ^Dm +
+ a 2eo + a EoEm +
The o r i g i n a l
b e tw e e n
a ^Em
e x p r e ssio n in
o^jjo ando^Eo.
(10)
th e paper d o es n ot d i f f e r e n t i a t e
f o u r v a r i a n c e s a p p ear i n tw o te r m s :
The c l a s s i c a l g e n e t i c
a 2Dm and a 2 Em.
I t a l s o does not s p l i t
a 2Do +
a 2 Eo and a 2Dm +O2 Em.
th e o r y a ssu m e s no c o v a r ia n c e b e tw e e n dominance
and e n v i r o n m e n t a l d e v i a t i o n s o f any k i n d ( F a l c o n e r ,
( 19 7 2 ) j u s t i f i e d
u su a lly
h is
a v a ila b le
in
These
procedure
beef
by s a y i n g
c a ttle
d ata
th at
do
not
the
1981).
Koch
r e la tio n sh ip s
p r o v id e
c r itic a l
c o n t r a s t s f o r s e p a r a t i n g dom inance and e n v ir o n m e n t a l d e v i a t i o n s .
17
F ig u r e 3 .
A path c o e f f i c i e n t diagram d e s c r i b i n g Koch’s (1972) model.
Po i s th e p h e n o t y p ic ,
and Eo i s
exp ressed
Go i s
th e a d d i t i v e g e n e t i c ,
by i n d i v i d u a l s .
Pm,
Cm,
Dm and Em a r e
(.5,
c o e ffic ie n ts.
g,
c o r re sp o n d in g
P r i m e s on v a l u e s r e p r e s e n t p a r e n t a l v a l u e s and
d ou b le p r im e s r e p r e s e n t gra n d p a r e n t v a l u e s .
tra its
th e dominance
t h e e n v ir o n m e n t a l v a l u e f o r b i r t h w e i g h t or weaning w e ig h t
m atern al e f f e c t s .
sy m b o ls
Do i s
d,
e,
p,
fm )
are
Path c o e f f i c i e n t s b etw een
sta n d a rd
Double a r r o w s r e p r e s e n t r e s i d u a l
p a r tia l
r e g r e ssio n
co rrela tio n s
b etw een
18
F i g u r e 3 i s t h e p a t h d ia g r a m w h i c h s u m m a r i z e s t h e c o n c e p t s o f
Koch ( 1 9 7 2 ) .
In
th is
d ia g r a m
P0 ^ ,
P 1o2 and
P 1o1
represent
th e
p h en o ty p e f o r b i r t h w e ig h t or w ea n in g w e ig h t o f th e o f f s p r i n g , s i r e
and dam, r e s p e c t i v e l y .
cov(0,D ) = 1 /2
I f a l l th e p o s s i b l e p a th s e x i s t ,
cf^Ao + 5 / 4
crAoAm + 1 / 2
cf^Am +
+ (1+fm) CTEoEm + f m / ( 2 - f m ) ( 1 / 2
th en
a DoDm + f^m +
CT2 Am + 5 / 4
Cf AoAm)
(11)
N ote t h a t i f fm = 0 , t h i s e x p r e s s i o n r e d u c e s t o
cov(0,D ) = 1/2
Cf ^Ao + 5 /4
Cf AoAm + 1/2 CT^Am +
a s sh o w n i n Thom pson ( 1 9 7 6 ) .
a DoDm +
Cf EoEm
(12)
The f o r m u l a s ( 1 1 ) o r ( 1 2 ) show t h a t i f
g e n e t i c or e n v ir o n m e n t a l c o v a r ia n c e t e r m s a r e
n e g a tiv e ,
th e cov(OjD)
can be l o w e r than a n t i c i p a t e d from d i r e c t g e n e t i c or m a te r n a l g e n e t i c
e f f e c t s (K och ,1972).
T h is e x p l a i n s
part of
the d is a g r e e m e n t b etw een
t h e e s t i m a t e s o f h 2 by p a t e r n a l h a l f - s i b a n a l y s i s a s c o m p a re d t o
r e g r e s s i o n o f o f f s p r i n g on dam ( h 2 b g D = 2 c o v ( 0 , D ) / Cf 2 P, F a l c o n e r ,
1981).
The m o d e ls f o r e s t i m a t i n g d i r e c t and m a te r n a l v a r i a t i o n o f b i r t h
w e i g h t and w e a n in g w e i g h t i n b e e f c a t t l e have been c o n s id e r e d i n order
o f c o m p le x i t y .
Some d i f f i c u l t i e s a r i s e i n t h e a p p l i c a t i o n o f th e more
c o m p le x m o d e ls to b e e f c a t t l e .
T h at i s
th e s u b j e c t
of
th e
next
se ctio n .
P r o b le m s .in e s t i m a t i n g d i r e c t and m a te r n a l g e n e t i c c o v a r i a n c e s
There a r e some p ro b le m s i n e s t i m a t i n g m a te r n a l v a r i a t i o n i n any
a n im a l s p e c i e s .
A lso ,
th ere are s p e c i f i c
p r o b lem s i n e s t i m a t i n g
m a te r n a l and d i r e c t c o v a r i a n c e s i n b e e f c a t t l e . The o r d e r i n which th e
p r o b le m s a r e p r e s e n t e d i m p l i e s no o r d e r o f im p o r ta n c e .
19
1.
Standard e r r o r o f
the
estim a tes
and
non
in d e p e n d e n c e
of
the
c o e f f i c i e n t s i n th e exp ected v a lu e s
T h is i s
a general
p ro b lem .
The l a r g e
sta n d a rd e r r o r o f th e
e s t i m a t e s o f m a te r n a l c o v a r i a n c e s i s due t o th e g e n e r a l l y s m a l l number
of r e la tiv e s
used ( i.e .,
in v o lv e d
(sp e c ia lly
i n b e e f c a t t l e ) and t h e m u l t i p l i e r s
1 / 2 , 1 / 4 , 1 / 8 , 1 / 1 6 ) (K o c h , 1 9 7 2 ; W i l l h a m , 1 9 8 0 ) .
( 1972 ) p o i n t s o u t ,
As Koch
e r r o r s o f e s t i m a t e s i n one component ten d t o c a u se
o t h e r com p onents t o d i f f e r i n th e o p p o s i t e d i r e c t i o n s i n c e th e sum o f
com ponents i s f o r c e d t o eq u a l t h e w h ole.
In g e n e r a l , i t i s e x p e c t e d t h a t i n c r e a s i n g th e number o f r e l a t i v e
r e la tio n sh ip s
in v o lv e d
w o u ld
decrease
th e
sta n d a rd
errors of
e stim a tio n .
D e s ig n e d e x p e r im e n t s t h a t y i e l d s p e c i f i c u s e f u l c o v a r i a n c e s t h a t
are u n c o r r e la t e d have been s u g g e s te d
(1967).
( 19 6 3 )
Bondari e t a l
in
by W i ll h a m
( 1 9 7 8 ) u s e d t w o d e s i g n s s u g g e s t e d by W illh a m
T r i b o l i u in C-astaneum
to
in v e stig a te
i n f l u e n c e s on pupa w e i g h t and f a m i l y s i z e .
c r e a tin g
a system
of
p atern al h a lf - s ib s .
a ls o in v o lv e d .
d e sig n s.
( 1 9 6 3 ) and E i s e n
m a tin g s
g e n e tic
m atern a l
The d e s i g n s w e r e based on
betw een f a m i l i e s
of
fu ll-sib s
M atin gs b etw e e n d i f f e r e n t f u l l - s i b
and
f a m i li e s are
At l e a s t t h r e e g e n e r a t i o n s a r e r e q u i r e d i n a l l t h e
The g e n e r a t i o n i n t e r v a l i n T r ib o liu m i s 30 d a y s
(Bondari e t
a l , I 9 7 8 ); w h i l e i n b e e f c a t t l e t h e g e n e r a t i o n i n t e r v a l i s 4.3 y e a r s
(Koch e t a l ,
I 982).
The p r o b le m i s a g g r a v a t e d by t h e f a c t t h a t t h e
f e c u n d i t y r a t e i n c a t t l e i s low and r e p e a t e d m a tin g s t o produce f u l l sib
fa m ilie s
sh o u ld
take
p la c e
in
d iffe r e n t
years.
The r e s u l t
20
in c r e a s e s the g e n e r a tio n i n t e r v a l .
T h e r e fo r e ,
the d e s ig n s are b e tte r
s u i t e d f o r l a b o r a t o r y a n i m a l s than f o r b e e f c a t t l e .
In c a s e t h e r e a r e more c o v a r i a n c e s b etw e e n r e l a t i v e s than d i r e c t
and m a te r n a l c o v a r i a n c e s t o e v a l u a t e , a method t o s o l v e s i m u l t a n e o u s l y
f o r t h e p a r a m e t e r s s h o u l d n e c e s s a r i l y be u s e d .
Van V l e c k and H a rt
(1966) have u s e d I e a s t - s q u a r e s , w e i g h t i n g th e e q u a t io n s by th e numbers
o f o b s e r v a tio n s used i n th e e s t im a t e o f th e r e l a t i v e r e la t io n s h ip .
For th e d e s i g n s d i s c u s s e d i n th e p r e v io u s paragraph,
E is e n (1967) h a s
s u g g e s t e d t h e u s e o f t h e d i a g o n a l e l e m e n t s o f (X’X) ” 1 t o w e i g h t t h e
e q u a tio n s.
He h a s a l s o i n d i c a t e d
o ffic ic n t i f
t h e v a r i a n c e s o f t h e e s t i m a t e s o f th e c o v a r i a n c e s b etw een
r e l a t i v e s a r e n o t hom ogeneous.
th a t th e procedure i s
not f u lly
This i s l i k e l y t o o c c u r when d i f f e r e n t
m e t h o d s a r e u s e d t o e s t i m a t e t h e v a r i a n c e c o m p o n e n t s , and when t h e
num ber o f r e c o r d s u s e d f o r t h e e s t i m a t e a r e e n t i r e l y d i f f e r e n t ,
u s u a l l y happens w i t h b e e f c a t t l e .
To overcom e t h i s problem ,
(1976) h a s d e v e lo p e d a m o d i f i e d maximun l i k e l i h o o d
a s s u m in g t h a t
in d e p e n d e n t m a t r i c e s o f sum o f s q u a r e s i s m axim ized.
to
so lv e
the
e q u a tio n s.it e r a t iv e ly .
The c a s e
Thompson
p roced u re.
t h e o b s e r v a t i o n s a r e n o r m a lly d i s t r i b u t e d ,
A fte r
th e l o g o f
It is
w here
as
necessary
p aren ts
s u b j e c t t o c u l l i n g was a l s o c o n s id e r e d by Thompson (1 9 7 6 ).
are
However,
t h e f a c t t h a t in d e p e n d e n t sum o f s q u a r e s i s r e q u ir e d p r e c l u d e s th e u se
o f t h e p r o c e d u r e when t h e d a t a a r e n o t c o l l e c t e d u n d e r a d e s i g n e d
e x p e r im e n t.
A n o th e r p r o b le m i s
t h a t t h e m e th o d c a n n o t a v o i d t h e
e f f e c t s o f th e c o r r e l a t i o n s b e tw e e n th e c o e f f i c i e n t s
d e sig n i s
good (Thompson,
1976).
e v e n though th e
21
2. S m a ll number o f r e l a t i v e s i n v o l v e d i n t h e e s t i m a t i o n i n b e e f c a t t l e
f i e l d d a ta
T h is
se ctio n .
p ro b lem
has
p a r tia lly
been e x p la in e d
Hohenboken (1973) s a i d t h a t i t
is
in
th e
p r e c ed in g
d i f f i c u l t t o l o c a t e enough
t y p e s o f f a m i l i e s t o e q u a l t h e number o f unknowns o f i n t e r e s t ,
and i t
i s d i f f i c u l t t o a c c o u n t s t a t i s t i c a l l y f o r a l l e n v ir o n m e n t a l c a u s e s o f
lik e n e ss
b etw een
r e la tiv e s.
Koch
(1972)
used
seven
r e l a t i o n s h i p s and th e e r r o r o f m a te r n a l h a l f - s i b s m odel.
r e la tiv e
There i s no
o t h e r p a p e r u s i n g m o re r e l a t i v e r e l a t i o n s h i p s t h a n Koch( 1 9 7 2 ) .
d i r e c t consequence i s
t h a t some te r m s m ust be dropped from th e model
t o s o l v e f o r th e r e s t .
is
I f the. term dropped i s n ot z e r o ,
b i a s e d and t h e o t h e r
o v e r estim a ted .
The
com p on en ts a r e
e ith e r
the s o lu tio n
u n d erestim a ted
or
The m agnitude o f th e e r r o r depends on t h e c o r r e l a t i o n
among th e t e r m s f o r t h a t p a r t i c u l a r s e t o f r e l a t i v e s and th e s i z e o f
t h e term dropped.
3 . M aternal e f f e c t s e v a l u a t i o n l e n a h t e n s t h e tim e t o c o n d u c t t h e stu d y
i'
As W illham (1980) p o i n t s o u t m a te r n a l e f f e c t s ar e :
1)
at
le a st
a g e n e r a tio n
b e h in d
th e
direct
effects
in
their
ex p ressio n ,
2 ) s e x l i m i t e d , and
3 ) occur l a t e i n the l i f e o f the fem a le.
A ll
t h e s e f a c t o r s l e n g t h e n th e e v a l u a t i o n t im e .
a r e tak en over a lo n g p e r io d o f tim e ,
th ere i s
If
th e r e c o r d s
th e p o s s i b i l i t y o f
i n t r o d u c i n g e n v i r o n m e n t a l c o r r e l a t i o n s ( E i s e n , 1 9 6 7 ) ; y e a r by s i r e
in tera ctio n i s
a lso
p o ssib le
tq o c c u r w h ich i n tu r n t e n d s t o i n f l a t e
th e s i r e v a r i a n c e component (Koch,
1972).
22
J5aj^mat.e-a-.Pf._.dJ.r-e.Q.t-,.and_ma.ternal g e n e t i c v a r i a n c e s and c o v a r i a n c e s f o r
b i r t h w e i g h t and w ean in g w e i g h t i n b e e f c a t t l e
The
e stim a te s
r e v ie w e d
in
th is
se c tio n
are
th e
a p p l y i n g t h e W i ll h a m ( 1 9 6 3 ) and Koch ( 1 9 7 2 ) p r o c e d u r e s .
r e su lts
of
The f i r s t
approach t o t h e problem was made by Koch and C lark (1955b).
I . B i r t h w e ig h t
Table 2 su m m arizes t h e e s t i m a t e s
w eig h t.
of
p u b lis h e d r e p o r t s on b i r t h
Three e s t i m a t e s o f d a i r y c a t t l e a r e a l s o in c lu d e d .
In g e n e r a l ,
t h e r e i s a t r e n d f o r h2o t o be g r e a t e r than h2 m.
The
means s u g g e s t t h a t a d d i t i v e d i r e c t i n f l u e n c e s a r e two t i m e s th e amount
o f a d d i t i v e m a te r n a l i n f l u e n c e s .
The c o r r e l a t i o n b e t w e e n a d d i t i v e d i r e c t and a d d i t i v e m atern al
effects
(r g ) was n e g a t iv e i n
a lm o st a l l
th e
e stim a te s.
The f e w
e s t i m a t e s and t h e b i g r a n g e o f t h e v a l u e s p r o d u c e u n c e r t a i n t y a b o u t
th e r e a l m a gn itu d e o f rg.
A p o s s i b l e b i o l o g i c a l e x p l a n a t i o n g i v e n by
F i s h e r and W i l l i a m s (1978) i s
t h a t t h e two o p p o s in g a d d i t i v e - e f f e c t s
tend to p re v e n t e x c e s s e s
b ir th
in
w e ig h t
th ereb y
p r o te c tin g
the
s u r v i v a l o f both c a l f and cow a t p a r t u r i t i o n .
This h y p o t h e s i s i s s u p p o r te d by t h e f a c t t h a t th e e s t i m a t e s o f r G
for d y sto cia ,
a t r a i t s t r o n g l y d e p en d en t on b i r t h w e i g h t (B u r fe n in g e t
a l , 1978), w ere -.1 9 f o r P h ilip s s o n (1 9 7 6 ), -.5 4 f o r B u rfe n in g e t a l
(1981) and - . 3 8 ( h e i f e r s )
The f i f t h
B aker (1 9 8 0 )
a n a ly sis,
and - .2 5
(cow s) f o r
Thompson e t a l (1981).
c o lu m n i n T a b le 2 w a s i n c l u d e d s i n c e Koch ( 1 9 7 2 ) and
p o stu la te d
th a t,
when c o v ( 0 , D ) i s
e x c l u d e d fr o m
t h e v a l u e s o f OA0 Am and r G a r e c l o s e t o z e r o .
th e.
The s i m p l e
TABLE 2 . ESTIMATES OF DIRECT AND MATERNAL ADDITIVE GENETIC VARIANCES AND COVARIANCES ON BIRTH WEIGHT OF
CATTLE
H er ita b ilitie s
D irect
Maternal
Total
effects effects
effects
Ch2mJ
Ch2t J
Ch20 J
Genetic c o r r e la tio n
between a d d itiv e
d i r e c t and a d d it iv e
maternal e f f e c t s
In clu sion
o f COV
Number o f
(OsD) in
records
the s o lu tio n
used
Breed
Authors
(rG)
.19
.51
.42
.00
.27
.36
.17
-
>0
-.9 3
-.9 8
-.5 8
-.3 9
- . 5 5 to —.89
No
Noa
Nob
Yes
Yes
Yes
4,553
1,064
Hereford
H olstein
Koch and Clark (1955c)
Everett and Magee (1965)
789
932
1,962
Hereford
Angus
Hereford
Brown and Galvez (1969)
-
- .1 7 to .27
Yes
4,060
Hereford
Koch (1972)
No
No
6,724
SwedishF r ie sia n
H olstein
P h ilip sson (1976)
Vesely and Robinson
(1971)
in
O
.44
to
.40
.44
.04
.15
.30
.25
.48
to
.12
.04
to
.19
.08
k
.35
.22
.65
.56
.14
.72
”
- . 0 7 to .30
-.5 3
.44
—.36
Yes
1,534
- .2 4
No
11,552
.11
—
.40
.17
.27
&
.21
^ R elatives included: O2PHS, O2MGS, COV(S,MGS)
"R elatives included: o^PGS, oZ^GS, COV(SsMGS)
Fisher and Williams
(1978)
Sinmental Burfening e t a l . (1981)
Averages
24
mean o f t h e e s t i m a t e s w i t h o u t Cov(OilD) i s e q u a l t o - . 3 5 .
T h is v a l u e
d o e s n ot seem t o s u p p o r t t h a t h y p o t h e s i s .
The o n l y
tw o
rep o rts
in
w h ic h
d o m in a n c e
and e n v i r o n m e n t a l
m a te r n a l e f f e c t s w ere i n c l u d e d w ere Brown and G alvez (1969) and Koch
(1 972).
H ow ever, b o th p a p e r s assu m ed
a DoDm and
oEoEm t o be z e r o .
The dom inance d i r e c t e f f e c t s a c c o u n te d f o r 9 and 17 % o f
and H erefo rd ,
r e s p e c t i v e l y (Brown and G a lv e z ,
1969).
i n Angus
The e s t i m a t e o f
a ^Dm was n e g a t i v e w h ich i s p o s s i b l y s u g g e s t i n g t h a t a n e g a t i v e s o u r c e
o f v a r i a t i o n w a s l e f t o u t fr o m t h e m o d e l.
Koch ( 1 9 7 2 ) p r e s e n t e d a
a ^Em t h a t a c c o u n t e d f o r 10 % o f
t e r m f o r o^Dm +
a ^P. T h is a u t h o r
c o n c lu d e d t h a t t h e e n v ir o n m e n t a l m a te r n a l a b i l i t y o f dams d id n o t have
a sig n ific a n t
d ir e c t
effect
on m a t e r n a l
a b ility
in
th e
next
g e n era tio n .
2 . Weaning w e ig h t
E s t i m a t e s o f d i r e c t and m a t e r n a l a d d i t i v e g e n e t i c s o u r c e s o f
v a r i a t i o n on p r e w e a n in g grow th a r e shown i n Table 3.
C o n tra ry to b i r t h w e i g h t , a v e r a g e d a i l y g a i n or w e a n in g w e ig h t
have more v a r i a t i o n f o r a d d i t i v e m a te r n a l
d ir e c t e ffe c ts .
effects
than f o r a d d i t i v e
The m ean o f t h e e s t i m a t e s o f r Q i s h i g h l y n e g a t i v e .
Koch (1972) em p h a sized t h e f a c t t h a t th e e s t i m a t e s o f r^ when the cov
( 0 ,D ) w as e x c l u d e d f r o m t h e s o l u t i o n w e r e s m a l l e r and s u g g e s t e d an
o v e r e stim a tio n o f
a AoAm.
However, t h e l a s t two e s t i m a t e s i n Table 3
in d ic a t e th a t t h is i s not n e c e s s a r ily th e ca se.
that i t i s
g e n e tic
I t s h o u l d be n o t e d
v e r y d i f f i c u l t t o compare e s t i m a t e s com ing from d i f f e r e n t
m o d e ls.
The o n l y
stu d y
in
s o l u t i o n w h ich o n l y c o n t a i n s a d d i t i v e
w h ich r Q was e v a l u a t e d
effects is
by a
th e one o f Kress e t
TABLE 3« ESTIMATES OF DIRECT AMD MATERNAL ADDITIVE GEMETIC VARIANCES AND COVARIANCE OF CALF GHOHTH THROUGH
WEANING
H e r l t a M I it i e s
D irect
Maternal
Total
effects effects
effects
Ch20 )
Ch2m)
Ch2t )
Genetic c o r r e la tio n
between a d d itiv e
d i r e c t and a d d it iv e
maternal e f f e c t s
In clu sion
o f COV
Number o f
(OjD) in
records
the s o lu tio n
used
Breed
Authors
(rG)
.21
-
.12
-.65
.18
.40
.15
.46
.25.
.17
.0
-.7 3
No
Yes
Yes
4,553
725
466
No
4,060
Hereford Koch and Clark (1955c)
Brahman
Deese and Roger (1967)
Brahman x
Shorthorn
Hereford Koch (1972)
Averages
.29
.50
.54
.34
.34
.17
.08
.23
.14
.14
.20
.20
.12
.34
.64
.34
.53
.47
.05
.32
.09
.02
. “
-.2 8
- 1 .1 4
-.0 7
-.9 0
-.7 5
—.68
.41
O
CV
• -.6 5
R e latives included:
R e la tiv e s included:
CO
CVJ
.32
.31
•37
.23
a
P1 P3I
HeapifflgJfeight
iiPHSj
2PGSj
- .3 1
-.4 6
—.73 to —1.07
-.7 9
^MGSj COV(SjMGS)
2MGSj COV(SjMGS)
Yes
Yes
Yes
Yes
No
Yes
No
Yes
No
No
717^ Hereford
b
H il l (1965)
1,692
2,618
Hereford
Hereford
Vesely and Robison (1971)
Hohenboken and Brinks
(1971a)
228
3,765
Charolais Baker (1980)
Brahman
Beltran Bru (1978)
13,682
Sinsnental Kress e t a l . (1979)
Crosses
Averages
ro
Ul
26
al
(1 979).
In
th e
rem a in in g
ones,
th e
s o lu tio n s
were
o b ta in ed
a s s u m in g t h a t some o f th e d i r e c t and m a te r n a l c o v a r i a n c e s w ere z e r o .
As d i s c u s s e d b e f o r e , e r r o r s i n t h e e s t i m a t e o f o n e c o v a r i a n c e c a u s e
th e o th e r co m p on en ts to d i f f e r i n th e o p p o s i t e d i r e c t i o n , s i n c e th e
sum i s f o r c e d t o e q u a l t h e t o t a l .
so lu tio n ,
ODoDm and
te r m s a r e n o t n u l l ,
When c o v ( 0 , D ) w as i n c l u d e d i n t h e
a EoEm w ere a lw a y s assumed t o be z e r o .
I f these
th e n t h e i n c l u s i o n o f t h e cov(0,D ) i n t h e s o l u t i o n
te n d s t o o v e r e s t i m a t e t h e v a l u e o f
a AoAm
and a s a c o n s e q u e n c e ,
a lso
° f r G.
The im p o r ta n c e o f d e t e r m in in g t h e e x a c t m agnitude o f
oAoAm comes
from th e f a c t t h a t r e s p o n s e t o s e l e c t i o n f o r prew eahing g r ow th r e l i e s
m o stly
on i t s
v a lu e
as
sh o w n
r e a s o n a b l e t o s u p p o se t h a t
have t h i s s i g n (T a b le 3 ).
by Van V l e c k e t a l
a AoAm i s
(1 977).
n e g a tiv e s in c e a l l i t s
effects
in
is
estim a tes
T h i s m ean s t h a t m o s t o f t h e p r o g r e s s made
i n o n e g e n e r a t i o n due t o s e l e c t i o n f o r g r o w t h r a t e i s
m atern al
It
th e
next
g en era tio n .
A fter
l i t e r a t u r e o f s e l e c t i o n e x p e r im e n t s i n b e e f c a t t l e ,
o v e r c o m e by
r e v ie w in g
th e
Koch e t a l (1982)
c o n c lu d e d t h a t th e r e a l i z e d h2 ( th e p o r t i o n o f th e t o t a l or p h e n o ty p ic
c h a n g e due t o g e n e t i c p r o g r e s s ) w a s .4 5 f o r b i r t h w e i g h t ,
.24 f o r
w e an in g w e i g h t , .35 f o r p o s tw e a n in g g a in ; .41 f o r f i n a l w e i g h t a t the
p e rform an ce t e s t and .33 f o r g a i n e f f i c i e n c y .
be l e s s
su ccessfu l
for
Thus,
s e l e c t i o n would
w e a n i n g w e i g h t t h a n f o r o t h e r g r o w t h and
efficie n c y t r a it s .
Im p ortan t
e v id e n c e
for
th e
presence
of
m a tern a l
w e an in g w e i g h t a r e g i v e n by the c o m p arison o f m a te r n a l
g en etic
p a r a m e te r s.
effects
on
and p a t e r n a l
Koch e t a l (1982) i n d i c a t e d t h a t m a te r n a l h a l f -
27
sib co r re la tio n s
vs.
.0 7 ).
are la rg e r
than p a t e r n a l h a l f - s i b c o r r e l a t i o n s (.34
H o h e n b o k e n and B r i n k s
(1971a) r e p o r te d
a d isa g reem en t
b e t w e e n t h e c o v ( 0 , S ) ( 4 8 .7 k g 2 ) and t h e c o v (Oj D) ( 1 8 .7 k g 2 ) .
The a v e r a g e s o f s i r e
and dam c o n t r i b u t i o n s to g e n e t i c
trend s i n
w e a n i n g w e i g h t o f c o m m e r c i a l h e r d s w e r e 1 .7 4 and .2 7 kg (K ennedy and
H e n d e r s o n , 1 9 7 7 ) and 1 .30 and .8 6 ( Z o l l i n g e r and N i e l s e n ,
1984).
If
t h e s e e s t i m a t e s m easured o n l y t r a n s m i t t e d or d i r e c t e f f e c t s i n a l a r g e
c lo s e d herd over s e v e r a l y e a r s w ith c o n s i s t e n t s e l e c t i o n p r a c t ic e s
from y e a r t o y e a r ,
eq u a l.
s i r e and dam c o n t r i b u t i o n s would be e x p e c t e d t o be
T h erefore, i f
a AoAm ( o r
aEoEm) i s n e g a t i v e , t h e l o w e r dam
t r e n d w ould be e x p e c t e d a s compared t o t h e t r e n d e s t i m a t e
( Z o l l i n g e r and N i e l s e n , 1 9 8 4 ) .
for s ir e s
H o h en b ok en and B r i n k s ( 1 9 7 1 b ) f o u n d
n e g a t iv e g e n e t i c c o r r e l a t i o n s b etw een th e m atern al a b i l i t y o f b eef
h e i f e r s and g r o w t h t r a i t s o f p a t e r n a l h a l f - s i b b r o t h e r s t e e r s .
r e v ie w e d v a lu e s
for
th e c o r r e l a t i o n
b etw een m ilk y i e l d
The
o f d a ir y
h e i f e r s w ith b e e f t r a i t s o f p a te rn a l h a l f - s i b s b ro th er i s c lo s e to
z e r o (W h ite e t a l ,
w e ig h ts
w ere -.1 6 ,
r e sp e c tiv e ly ,
1981).
-.3 1
H i l l (1 9 6 5 ) h a s found t h a t r Q f o r w ea n in g
and - . 4 5 ,
w hich s u g g e s t s
at
150,
1 80 and 2 1 0 d o f
t h a t an im p o r t a n t
part o f
a s s o c i a t i o n b e tw e e n d i r e c t and m a te r n a l e f f e c t s i s
age,
th e n e g a t iv e
e n v ir o n m e n t a l and
g i v e s a p a r t i a l e x p la n a tio n fo r such a s m a ll c o r r e l a t i o n in d a ir y
c a ttle .
The a b o v e p a r a g r a p h h a s i n t r o d u c e d o n e o f t h e m o s t i m p o r t a n t
p rob lem s i n th e a s s o c i a t i o n o f d i r e c t and m a te r n a l e f f e c t s f o r w eaning
w e ig h t.
The p r o b le m
is
to
d e t e r m i n e how much o f
c o r r e l a t i o n i s g e n e t i c and how much i s e n v ir o n m e n t a l.
th is
n e g a tiv e
I
28
M atern al a b i l i t y
o f a b e e f cow i s u s u a l l y m e a s u r e d u s i n g h e r
" m o st p r o b a b l e
p ro d u cin g a b i l i t y "
e v id e n c e
en v iro n m e n ta l
th a t
(MPPA) ( L u s h ,
sources
of
1945).
v a r ia tio n
T h er e i s
a ffect
th e
r e l a t i o n s h i p b e tw e e n e a r l y grow th o f a b e e f f e m a l e and h e r su b se q u e n t
m a tern a l a b i l i t y .
b ir th year
-.1 1
For e x a m p l e , t h e p h e n o t y p i c c o r r e l a t i o n b e t w e e n
m e a n s o f t h e cow and
MPPA w a s f o u n d t o b e - . 5 2 , - . 2 0 and
and t h e c o r r e l a t i o n b e tw e e n age o f dam e f f e c t s and MPPA f o r th e
tw o t r a i t s
(1970),
was - .7 0 ,
-.6 8
and - . 1 9 ,
M angus and B r i n k s
as rep o rted
by E l l i c o t
( 1 9 7 1 ) and K r e s s and B u r f e n i n g
et
al
(1972).
T h e s e l a t t e r a u t h o r s a l s o r e p o r t e d a p o s i t i v e t r e n d (b = .0 6 + .0 3 )
f o r w e an in g w e i g h t on b i r t h d a t e and a n e g a t i v e tr e n d (b = - .1 0 ± .02)
f o r MPPA on b i r t h d a t e .
of
th e
n e g a tiv e
m a te r n a l a b i l i t y
Hence,
a sso c ia tio n
th e d a t a i n d i c a t e d t h a t a t l e a s t p a rt
betw een
grow th
ra te
and s u b s e q u e n t
(or b e tw e e n d i r e c t and m a te r n a l e f f e c t s )
was caused
by e n v ir o n m e n t a l s o u r c e s o f v a r i a t i o n .
The b a s i c c a l f f o o d d u r i n g t h e p r e w e a n i n g p e r i o d i s
produced
by h i s
c o r re la tio n
m o th er.
C a n tet
(1983)
c a lc u la te d
an
th e m ilk
average
b e t w e e n cow m i l k p r o d u c t i o n and c a l f w e a n i n g g a i n or
w e i g h t o f .5 8 f r o m 26 s t u d i e s .
T h e r e fo r e , a h ig h a v e r a g e d a i ly g a in
o f th e h e i f e r c a l f i s p o s i t i v e l y a s s o c i a t e d w ith her m o th e r 's m ilk
p r o d u c t io n and n e g a t i v e l y
Hence,
a sso cia ted
t o h e r f u t u r e m a te r n a l a b i l i t y .
t h e e x p e c t e d r e l a t i o n s h i p b e tw e e n w eaning w e i g h t o f a cow and
h e r s u b s e q u e n t m i l k p r o d u c t i o n w o u ld be n e g a t i v e .
C h ristia n e t al
(1965) found n e g a t i v e c o r r e l a t i o n s b e tw e e n dam w eaning w e i g h t and m ilk
ii
and b u t t e r f a t p r o d u c t i o n . T o t u s e k ( 1 9 6 8 ) f o u n d t h a t c o w s r a i s e d on
low p l a n e s o f n u t r i t i o n p r e v i o u s t o 240 d o f a g e ,
weaned c a l v e s t h a t
29
w e i g h e d 8 and 10 k g m o re t h a n c a l v e s w e a n e d by c o w s r a i s e d on t h e
medium and h ig h p l a n e s o f n u t r i t i o n , r e s p e c t i v e l y .
J o h n sson and Obst
(1984) found t h a t t h e r a t e o f grow th b e f o r e w eaning had more i n f l u e n c e
on s u b s e q u e n t m a t e r n a l a b i l i t y i n t h e f i r s t t h r e e l a c t a t i o n s t h a n
e i t h e r g r o w t h b e t w e e n 8 t o 14 mo o f a g e o r p r e m a t i n g l i v e w e i g h t o f
H ereford h e i f e r s .
A m ajor p a r t o f t h e d e v e lo p m e n t o f t h e mammary g la n d t a k e s p la c e
b etw e e n b i r t h and f i r s t c a l v i n g ( S e j r s e n ,
by Koch ( 1 9 7 2 ) ,
1978). Many s t u d i e s r e v ie w e d
K r e s s and B u r f e n i n g ( 1 9 7 2 ) and S e j r s e n ( 1 9 7 8 ) show
th a t h ig h l e v e l s
of
energy
in
th e r e a r in g
p e r io d
d e c r e a s e o f s u b s e q u e n t m il k y i e l d o f d a i r y h e i f e r s .
th is
phenom enon
is
r e la te d
to
a d ecrease
in
r e su lte d
in
a
The mechanism o f
g row th
horm one
c o n c e n t r a t i o n i n t h e b l o o d and an i n c r e a s e i n t h e a m o u n t s o f f a t i n
th e
udder,
e ith e r
at
th e
sam e
p h y s io lo g ic a l
or
at
th e
sam e
c h r o n o l o g i c a l a g e , a s t h e l e v e l o f en e r g y i n t h e d i e t i n c r e a s e s d u rin g
t h e r e a r i n g p e r i o d ( S e j r s e n , 1 9 7 8 ) . M a r t i n e t a l ( 1 9 8 1 ) and Ochoa e t
al
(1981) r e p o r te d t h a t c r e e p - f e e d i n g h e i f e r c a lv e s r e s u l t e d i n a
d e c r e a s e d l i f e t i m e p r o d u c t i v i t y and MPPA f o r w ean in g w e i g t h which i s
c o n s i s t e n t w i t h t h e e a r l i e r f i n d i n g s o f i n f e r i o r udder d e v elop m en t.
From t h e s t u d i e s r e v i e w e d a b o v e i t . i s c l e a r t h a t t h e c o w s t h a t
had h i g h e r w e an in g w e i g t h s ten ded t o produce c a l v e s w i t h l o w e r weaning
w eig h ts.
T h is r e s u l t was p a r t i a l l y
a s s o c ia tio n ,
i.e .,
n e g a tiv e valu e o f
due t o a n e g a t i v e
a EoEm.
e n v ir o n m e n ta l
Koch (1972)
sp e c u la te s
t h a t th e e f f e c t s o f m a te r n a l e n v ir o n m e n t f o r p r e w e a n in g grow th o f
p r e v io u s g e n e r a t i o n s ( t h e fm path i n F ig u r e I) s h o u ld h ave a v a lu e o f
-0 .1
to -0 .2 fo r
th e e n v ir o n m e n ta l c o v a r ia n c e to s a t i s f y
observed
30
c o r r e l a t i o n s and r e g r e s s i o n s . S i n c e t h e . e n v ir o n m e n t a l c o v a r ia n c e i s
a lso a fu n c tio n o f
O^Am and
OAoAm, a s Koch ( 1 9 7 2 ) s h o w e d , t h e o n l y
way t o d e t e r m in e t h e r e l a t i v e c o n t r i b u t i o n o f
p h e n o t y p ic v a r i a n c e o f w e a n in g w e i g h t i s
a ll
the co v a r ia n c e s.
(1967) found a v a l u e o f z e r o f o r
v a r ia tio n .
v a ria n ce) i s
to s o lv e sim u lta n eo u sly fo r
T h is p r o c e d u r e h a s n o t been done y e t .
R egarding dom inance e f f e c t s
(1971a) s tu d y
aAoAm. and oEoEm to th e
on w e an in g w e i g h t ,
D eese and Koger
cr2Do w h i l e i n Hohenboken and B rin k s
a 2 do was 52.4 kg2, c o r r e s p o n d in g t o 10.3 % o f th e t o t a l
T h eir
e stim a te
of
oDoDm ( - 76.7 k g 2 ? - 17.8 % o f
th e o n l y one found i n t h e l i t e r a t u r e .
a u t h o r s assum ed
However,
th e
s i n c e th e
O2Dm t o be z e r o i n o r d e r t o s o l v e t h e e q u a t io n s ,
th e
e s t i m a t e i s p o s s i b l y n o t v e r y r e l i a b l e s i n c e by t h e C a u c h y -S c h w a r z
in e q u a lity :
a ^Do .
O2Dm
2.
<3 DoDm.
31
MATERIAL AND METHODS
The d a t a
fo r
A g r ic u ltu r a l
Montana.
30'N,
th is
R esearch
stu d y
C enter
c o lle c te d
(NARC) l o c a t e d
at
th e
N orth ern
13 km SW o f
H a v re ,
The g e o g r a p h ic c o o r d i n a t e s o f th e s t a t i o n a r e : l a t i t u d e 48°
lo n g itu d e
109° 48*W.
The a p p r o x im a te a l t i t u d e i s
be d e s c r i b e d
bordered
w ere
on
as
ty p ic a l
th e
sou th
of
819 m above s e a l e v e l .
th e
n orth ern
g la c ia te d
The area can
p la in s
by t h e B e a r Paw m o u n t a i n s .
and i s
The B e a r Paw
m o u n t a i n s w e r e t h e s i t e o f t h e Rocky Boy I n d i a n R e s e r v a t i o n l e a s e
l o c a t e d 48 km s o u t h o f H avre and t h i s l e a s e w as u s e d a s t h e summer
g r a z i n g s i t e from th e l a t e s 40’s t o th e e a r l y 70’s.
Weather
Weather s t a t i s t i c s a t NARC w e r e t a k e n by i t s own c l i m a t i c s t a t i o n
f o r t h e p e r i o d 1 9 5 1 - 1 9 8 0 (USDC, 1 9 8 2 ) .
6°C .
Annual m ean t e m p e r a t u r e w as
A v e r a g e t e m p e r a t u r e f o r t h e c o l d e s t m onth ( J a n u a r y ) and t h e
h o ttest
m o n th ( J u l y )
w ere -1 0 .9 °C
and 2 0 .7 ° C ,
r e sp e c tiv e ly .
The
e x tr e m e t e m p e r a t u r e s r e g i s t e r e d w ere -4 9 .4 °C (January 1953) and 42.2°C
( J u n e 19 0 0 ) .
Mean t e m p e r a t u r e s w e r e a p p r o x i m a t e l y 4°C
l e s s a t th e
Rocky Boy s u b s t a t i o n d u r in g t h e same p e r io d .
Annual mean p r e c i p i t a t i o n was 297 mm a t NARC and 438 mm a t Rocky
Boy.
M ost o f
th e
p r e c ip ita tio n
b e tw e e n A p r il and Septem ber.
(80%)
(A nderson,
19 6 6 )
occurred
Anderson (1966) comments t h a t the snow
f a l l a t t h e m a in s t a t i o n i s l i g h t w i t h a maximum h e i g h t o f t h e snow
c o v e r o f 15 cm d u r in g m ost o f th e y e a r s .
This i s
a l s o r e l a t e d t o th e
32
w a r m in g e f f e c t s o f t h e C h in o o k w i n d s w h i c h t e n d t o r e d u c e t h e snow
cover.
A la rg e
am ount
of
e v a p o r a tio n
in
p r e c i p i t a t i o n o c c u r s a t NARC d u r in g th e summer.
r e la tio n sh ip
to
th e
The l o w e r e v a p o r a t io n
r a t e and g r e a t e r p r e c i p i t a t i o n a t R ocky Boy p r o d u c e d b e t t e r r a n g e
c o n d i t i o n s d u r in g th e summer.
S o ils
The NARC s o i l s w e r e c l a s s i f i e d a s m ixed a r i d i c A r g i b o r o l l s o f th e
s e r i e s A t t e w a n , T e l s t a d and J o p l i n (U SD A -SC S,19 8 2 ) .
and J o p l i n have f i n e - l o a m y
textu re.
S e r ie s T e lsta d
Whereas t h e A ttew an t e x t u r e i s
f i n e - l o a m y o v e r sandy or s a n d y - s k e l e t a l .
Ranee C o n d i t i o n s •
The p a s t u r e s a t NARC can be d e s c r i b e d a s mixed p r a i r e g r a s s l a n d s .
The s p e c i e s a r e n a t i v e and i n t r o d u c e d
grasses lik e
crested
w heat
( A e r o o v r o n d e s e r t o r u m ) . n e e d l e and t h r e a d ( S t i n a c o m a t a ) and b l u e
gramma (B o u t e lo a e r a c i l i s ) .
Anderson (1966) d e c r ib e d t h e summer g r a z i n g s p e c i e s a t Rocky Boy
a s b a s i c a l l y g r a s s e s s u c h a s T im o t h y ( P hleum p r a t e n s e ) , Ju n e g r a s s
(K o e le r ia c r i s t a t a ).
Idaho f e s c u e
( F e s t u c a i d a h o e n s i s l . Rough f e s c u e
( F e s t u c a s c a b r e l l a ) and Mountain brome (Bromus Sp p J .
E x p er im e n ta l Animals
B i r t h and w e an in g w e i g h t s w ere ta k e n on 4,423 c a l v e s from 1938 to
1983.
sin ce
I n f o r m a t io n on th e same w e i g h t s o f th e cows was a l s o a v a i l a b l e
1928.
There w ere t h r e e d i f f e r e n t p h a s e s o f d a ta c o l l e c t i o n and
b r e e d in g s y s t e m s .
1946.
The f i r s t one c o n s i s t s o f th e a n i m a l s r a i s e d b e f o r e
These c a t t l e w ere o r i g i n a t e d a t Havre and w ere t h e s t o c k graded
33
t o t h e o l d e s t l i n e d e v e l o p e d a t t h e F o r t K eogh , L i v e s t o c k and Range
R esearch S t a t i o n ,
M i l e s C it y ,
was b e t t e r known a s l i n e H.
Montana i n th e secon d phase.
T h is l i n e
The se co n d phase i s c h a r a c t e r i z e d by th e
c r e a t i o n and d e v e lo p m e n t o f i n b r e d l i n e s and c r o s s l i n e s o f H e r e f o r d
c a ttle.
1966.
in
The p r o c e s s was i n i t i a t e d i n 1946 and was m a in t a in e d through
I n u r e d l i n e I w a s i n i t i a t e d i n I 9 4 6 , l i n e 2 i n 1947 and l i n e 3
1948.
L in e 4 a c t e d a s a c o n t r o l l i n e .
C r o s s lin e s c a lv e s were
produced by m a tin g cow s from l i n e 4 w i t h b u l l s o f l i n e s I ,
c r e a te l i n e s 5, 6
and 7 , r e s p e c t i v e l y .
1975 and i s c o n t in u in g .
2 and 3 t o
The f i n a l p h a s e s t a r t e d i n
The f o u n d a t io n o f t h e s e l i n e 4 c a t t l e i s from
in b r e d s t o c k purchased from F o r t Keogh,
L i v e s t o c k and Range S t a t i o n ,
M il e s C ity i n 1962 and 1963. These purebred c a t t l e a r e s e l e c t e d by th e
in d e x :
1=365 d. a d j u s t e d w e i g h t - 3 . 2 a d j u s t e d b i r t h w e ig h t .
F u rth er d e s c r i p t i o n o f m anagem ental and s e l e c t i o n p r o c e d u r e s h a s
been g i v e n by F lo w e r e t a l (1963)' and Anderson (1966).
Management
o f t h e b r e e d in g herd
H e i f e r s w ere bred to c a lv e f i r s t a s 3 - y e a r - o l d s i n th e e a r l i e r
yea rs of th e p r o jec t.
y e a r -o ld s.
A f t e r 1951, h e i f e r s w ere bred to c a lv e a s 2 -
The b r e e d i n g s e a s o n l a s t e d a p p r o x im a t e ly 60 d b e g in n in g
th e f i r s t week o f June.
N atu ral
se r v ic e
w as u s e d i n
sin g le
sir e
p a s t u r e s a t t h e summer g r a z i n g l e a s e .
H e i f e r s and cows w ere f e d d u r in g t h e w i n t e r a t NARC s t a r t i n g i n
December d u r in g m ost o f t h e y e a r s .
When n a t i v e ran ge p a s t u r e s became
an i n s u f f i c i e n t s o u r c e o f fo o d s u p p ly ,
a r a t i o n c o n s i s t i n g o f a lm o s t
e q u al p a r t s o f c e r e a l s t r a w s and l e g u m e - g r a s s hay a d - l i b i t u m was
to th e cow s.
In t h e l a t e r y e a r s ,
fed
corn s i l a g e r e p la c e d th e leg u m e-
34
g r a s s hay.
Corn s i l a g e was d e c r e a s e d p r i o r t o c a l v i n g w i t h hay b e in g
in crea sed .
F i r s t c a l v i n g h e i f e r s w e r e w i n t e r e d s e p a r a t e l y fro m t h e o l d e r
cow s and f e d a d i f f e r e n t r a t i o n o f g r a i n and a l f a l f a hay.
S in c e 1970,
g r a i n h a s n o t b e e n f e d and t h e r o u g h a g e s o u r c e w a s c o r n s i l a g e and
a l f a l f a hay.
N u t r i t i o n a f t e r c a l v i n g c o n s i s t e d o f c r e s t e d w h ea t g r a s s p a s t u r e s
f o r a l l th e f e m a l e s from t h e m id d le o f A p r il through May su p p le m e n te d
by d e c r e a s e d amounts o f co rn s i l a g e r e p l a c e d by se co n d c u t t i n g a l f a l f a
hay.
Then, t h e c o w s w e r e t r u c k e d t o t h e Rocky Boy s u b s t a t i o n t o be
a l o t t e d i n t o t h e i r r e s p e c t i v e b r e e d in g h e r d s i n e a r l y June.
A f t e r th e
b r e e d i n g s e a s o n , c o w s w e r e c o m b in e d i n t o o n e h e r d w h i c h g r a z e d t h e
m ountain p a s t u r e u n t i l a p p r o x im a t e ly November.
At t h a t t i m e , t h e herd
w a s t r a i l e d b a c k t o NARC i n o r d e r t o p a s s t h e w i n t e r p e r i o d .
1975,
S in c e
th e purebred herd h a s been m a in t a in e d a t NARC d u r in g t h e summer
p r i m a r i l y on i r r i g a t e d p a s t u r e s .
S e l e c t i o n and B r e e d in g P r o ced u res
The p r o j e c t i n i t i a t e d i n 1946 was d e s ig n e d t o m easure g e n e r a l and
s p e c i f i c c o m b in in g a b i l i t y o f l i n e s
t h is g o a l,
in to
I , 2 and 3.
In o r d e r t o f u l l f i l l
a s y s te m o f s e q u e n t i a l s e l e c t i o n on t h e m ale s i d e was put
p ra ctice.
Fem ale s e l e c t i o n a c c o u n t e d f o r a v a r i a b l e
but s m a l l e r
p o r t i o n o f t h e s e l e c t i o n d i f f e r e n t i a l s o f b i r t h w e i g h t and w e a n i n g
w e i g h t (F low er e t a l ,
a t w ea n in g t im e .
1964). A f i r s t c u l l i n g o f m ale c a l v e s took p l a c e
S lo w p r e w e a n i n g g r o w t h , c o n f o r m a t i o n d e f e c t s , o r
c o l o r p a t t e r n w ere th e r e a s o n s f o r c a s t r a t i n g a p p r o x i m a t e ly 17? o f t h e
b u ll
c a lv e s.
The r e m a i n i n g
b u ll
c a lv e s
w ere
put
on a 168
d
35
p e r fo r m a n c e t e s t w i t h i n d i v i d u a l f e e d i n g .
T h e r e fo r e ,
mass s e l e c t i o n
was a p p l i e d on th e b a s i s o f g r ow th r a t e through a y e a r o f age.
The r e c u r r e n t p h a s e o f
th e s e l e c t i o n
s c h e m e w a s p r o d u c e d by
m a tin g two h i g h - g a i n i n g y e a r l i n g b u l l s from each o f th e t h r e e purebred
l i n e s t o c o w s o f t h e g r a d e t e s t e r l i n e 4 , u s i n g h e r d s o f 15 f e m a l e s
per s i r e .
c o n c lu sio n
T h is p roce d u r e was n o t a l w a y s p o s s i b l e t o a c h i e v e .
of
th e
progeny
tests,
th e
s tr a ig h tlin e
b u lls
At th e
w h ic h
p e r f o r m e d b e s t w i t h r e s p e c t t o t h e i r c r o s s l i n e p r o g e n y w e i g h t s and
g a in s w ere s e le c t e d
to s ir e
s t r a i g h t l i n e c a l v e s u n l e s s c u r r e n t herd
b u l l s had a t t a i n e d s u p e r i o r c r o s s l i n e p r o g e n y t e s t s .
w ere a b le
to
s ir e
str a ig h tlin e
c a lv e s
Proven b u l l s
as 3 -y e a r -o ld s.
S e le c tio n
p r e s s u r e a c h ie v e d .b y r e c u r r e n t s e l e c t i o n o f b u l l s was n e g l i g i b l e
(F lo w e r e t a l , 1964).
A p p r o x im a te ly o n e - t h i r d o f th e t e s t e r l i n e cow s w ere bred to
b u l l s o f t h e i r own l i n e t o p r o v id e r e p la c e m e n t h e i f e r s .
Some h e i f e r c u l l i n g was e x e r t e d a t 18 mo based on w e a n in g w e i g h t ,
140-d g a i n t e s t and 18-mo w e i g h t .
The r a t i o n f o r t h e b u l l f e e d t e s t c o n s i s t e d o f r o l l e d b a r l e y ,
(35%), d r i e d m o l a s s e s b e e t p u l p (35%), r o l l e d o a t s (20%) and w h e a t
bran (10%).
A l f a l f a hay a d ^ i i b l t u m was p r o v id e d a s a roughage so u r c e.
The h e i f e r s w ere g r o u p - f e d i n o u t s i d e l o t s w i t h good p r o t e c t i o n
from th e w e a th e r d u r in g 140 d.
The r a t i o n was made o f se co n d c u t t i n g
a l f a l f a hay p l u s .90 kg d a i l y o f r o l l e d b a r l e y i n l a t t e r y e a r s .
K re ss and B u r f e n in g (1972) r e p o r t e d means and s ta n d a r d d e v i a t i o n s
o f i n b r e e d i n g p e r c e n t a g e o f c o w s o f 1 2.6 + 7.4%, 10 .7 ± 6.5% and 11.4
±. 7.0% f o r l i n e s I , 2 and 3 , r e s p e c t i v e l y .
A sa m p le o f y e a r s 1939,
36
1 949 and 1959 had a v a l u e o f 1.2 ± 2 .6 f o r l i n e 4 c o w s . Few c o w s had
i n b r e e d i n g p e r c e n t a g e s g r e a t e r than 24 t o 28 %.
S t a t i s t i c a l procedures
The o r i g i n a l
d a ta s e t c o n t a in e d 5 ,1 7 3 o b s e r v a t i o n s .
E d itin g
p r o c e d u r e s w e r e b a s e d on d e l e t i n g i n c o m p l e t e r e c o r d s ( i . e . , l a c k o f
one o f
the w e ig h ts,
unknow n s i r e i d e n t i f i c a t i o n ) .
T h er e w e r e 75
c a l v e s w i t h e i t h e r b i r t h w e i g h t o r w e a n i n g w e i g h t m i s s i n g and 660
c a l v e s w i t h an unknown s i r e ,
Abnormal v a l u e s ( i . e . ,
from
th e
m ean)
a l m o s t a l l com ing from t h e e a r l i e r y e a r s .
w e i g h t s below or above f o u r s ta n d a r d d e v i a t i o n s
w ere
a lso
d e lete d ,
th ou gh
th e ir
num ber
was
c o m p a r a t i v e l y s m a l l (15 c a l v e s ) none o f w h ich w ere s e l e c t s a s s i r e s .
The f i n a l d a t a f i l e
f o r p a t e r n a l h a l f - s i b a n a l y s i s c o n s i s t e d on 4,423
o b serv a tio n s.
The t o t a l number o f s i r e s t h a t produced progeny was 202.
A ll th e
s i r e s r a i s e d a t NARC ( 1 0 5 ) had r e c o r d s a v a i l a b l e on t h e i r own b i r t h
w e i g h t and w e a n in g w e i g h t .
C ity
and,
th erefore,
The r e m a i n i n g b u l l s w e r e r a i s e d a t M ile s
th e y w ere r a i s e d
under d i f f e r e n t
e n v ir o n m e n ta l
co n d itio n s.
The g r e a t num ber o f y e a r s o f d a t a a l l o w e d f o r a c o n s i d e r a b l e
number o f r e p e a t e d m a tin g s .
Hence, f a m i l i e s o f a t l e a s t two dams per
s i r e and tw o or more c a l v e s per dam w ere s e l e c t e d and t h i s r e s u l t e d i n
976 f u l l - s i b c a l v e s from 402 f u l l - s i b f a m i l i e s .
W illh a m ( 1 9 6 3 ) h a s s u g g e s t e d t h a t t h e b e s t s e t o f r e l a t i v e s t o
u s e t o e s t i m a t e th e im p o r ta n c e o f g e n e t i c m a te r n a l e f f e c t s a r e s e t s o f
m a tern a l
c o u sin s
sin c e
th e ir
use
c o n t r ib u t io n to th e m a tern a l v a r ia n c e .
reduces
th e
en v iro n m e n ta l
H ence, th e c o u s in p a t t e r n s
37
p r o p o s e d . by Van V l e c k and H a r t ( 1 9 6 6 )
c o v a r ia n c e o f f u l l - s i b s
were
e v a lu a te d .
A lso
th e
w h ose p a r e n t s a r e p a t e r n a l h a l f - s i b s
and
p a t e r n a l h a l f - s i b s whose dams a r e p a t e r n a l h a l f - s i b s w e r e c o n s id e r e d .
M a te r n a l g r a n d d a m -sib s (MGD)
Dam I------------------------O ffsp rin g I
Granddam<
-Dam 2-------:----------- :
O ffsp rin g 2
Single first cousins (SFC)
Granddam
— — P=-Dain T——:-----=------------ O ffsp rin g I
G randsire
----- — ^Dam 2 — :------------------ O ffsp rin g 2
Paternal half-sibs Plus maternal half-sib dams (PHS + MHSDl
•Dam I ------- :— ;-----------O ffsp rin g I
Granddam
S ir e < ^
-Dam 2 ------ ------ ;-------- O ffsp rin g 2
Paternal half-sibs plus single first cousins (PHS + SFCl.
Dam 1 ----------------------- O ffsp rin g I
S ire«c
Dam 2 — :------------------- O ffsp rin g 2
Full-sibs Plus paternal half-sib parents (FS + PHSPl
•S ir e c—---------------- —r Off sp rin g I
G r a n d sire;
D a m ------:----------------O ffsp rin g 2
38
Eateraal half-sibs plus paternal half-sib dams CPHS + P H s m
Dam I
G ra n d sir e
O f f s p r in g I
S ire
Dam 2
As e x p e c t e d ,
O f f s p r in g 2
th e c l o s e r t h e r e l a t i v e r e l a t i o n s h i p i s ,
t h e number o f d e g r e e s o f freedom f o r t h e a n a l y s i s becomes.
th e s m a l l e r
The r e a so n
f o r u s i n g t h e s e e s t i m a t e s was t o i n c r e a s e th e number o f e q u a t io n s f o r
e s t i m a t i n g t h e n in e d i r e c t and m a te r n a l v a r i a n c e s and c o v a r i a n c e s and
t o p r o v id e more e q u a t io n s t o e s t i m a t e th e dom inance v a r i a n c e s and the
c o v a r i a n c e b e t w e e n d o m in a n c e d i r e c t and dominance m a te r n a l e f f e c t s .
Error te r m s w ere i n c lu d e d s i n c e
u s u a lly in v o lv e d i s
h igh ,
t h e num ber o f d e g r e e s o f f r e e d o m
w hich i n c r e a s e s t h e a c c u r a cy o f e s t i m a t i o n .
E x p e c t a t i o n s f o r th e e n v ir o n m e n t a l com p onents w ere o b t a in e d under th e
a s s u m p tio n t h a t e r r o r s i n th e m o d e ls i n c l u d i n g i n d i v i d u a l s r e l a t e d
through s i r e - t y p e r e l a t i o n s h i p s in v o lv e d
2 Eo,
EoEm and
s i n c e a lm o s t one f o u r t h o f th e r e c o r d s w ere f u l l - s i b s ,
2 Em
w h ic h h a v e
m a te r n a l e n v ir o n m e n t i n t h e i r e x p e c t a t i o n s .
F a m ily r e l a t i o n s h i p s w ere a n a ly z e d by l e a s t s q u a r e s p r o c e d u r e s as
o u t l i n e d by Harvey (1977).
The b a s i c m odel u sed was
y i j k l m = u + I 1 + a j + s k + ( a s ) j k + b X ij ^ m + gi;L + BjJ klmn
where
Jr1 JkIm = o b s e r v a t i o n on t h e mth b i r t h w e ig h t and w ean in g w e ig h t ,
u
I1
= g e n e r a l mean common t o a l l th e o b s e r v a t i o n s ,
= e f f e c t o f th e I th
lin e -y e a r ,
2 -5 0 , 3 - 5 0 , 4 - 5 0 , . . , 4-83»
i
= 4 - 3 8 , .......
1-50,
39
aj
= e f f e c t o f t h e j t h a g e o f dam,
j = 2 ,3 ,4 ,5 ,6 -1 0 ,1 1
or
more,
sk
= effect
of
t h e k fch s e x ,
k = I (h e ife r ),
2 (b u ll), 3
(steer),
( a s ) j k = e f f e c t o f t h e i n t e r a c t i o n b e tw e e n t h e j th age o f dam and
k fch s e x ,
bXi J k lm r r e Sr e s s i 6 n o f b i r t h w e i g h t on b i r t h d a t e or w e a n i n g
w e i g h t on age a t w e a n in g ,
£>11
= random f a m i l y e f f e c t .
sir e ,
m atern al
T h is i s a g e n e r a l term u se d f o r
g r a n d sire ,
patern al
grand
sir e
and
m a t e r n a l g r e a t g r a n d s i r e n e s t e d w i t h i n t h e i th l i n e y e a r , and
e i j k l m r nandom e r r o r .
The s o l u t i o n
of
th e le a s t - s q u a r e s
e q u a tio n s
in
t h e H a r v e y ’s
package i s o b t a in e d by im p o s i n g t h e r e s t r i c t i o n s
E
i ^i r
S
J aJ
E
=
E
k sk =
j (as)jk
=
E
Jf ( a s ) j k = 0
Table 4 sh ow s th e d i s t r i b u t i o n o f r e c o r d s per l i n e and per y e a r .
S i n c e d u r in g t h e e a r l i e r and l a t e r y e a r s l i n e fo u r c a l v e s a r e t h e o n ly
r e c o r d s a v a i l a b l e , d a t a was c l a s s i f i e d by l i n e - y e a r s u b c l a s s e s i n s t e a d
o f r e c o r d i n g t h e tw o f a c t o r s i n d i v i d u a l l y .
Due t o th e s m a l l number o f o b s e r v a t i o n s per s u b c l a s s ,
th e r e c o r d s
w e r e c o r r e c t e d f o r l i n e - y e a r e f f e c t s i n som e o f t h e m o d e l s u s i n g
c o n s t a n t s o b t a in e d from a f i x e d
t h e e f f e c t s o f a g e o f dam,
sex,
effects
lin e ,
m odel.
T h is model in c lu d e d
y e a r and t h e r e g r e s s i o n s , o f
b i r t h w e i g h t on b i r t h d a t e and w e a n in g w e i g h t on w e a n in g age.
UO
TABLE 4 . DISTRIBUTION OF RECORDS BY LINES AND BY YEARS
L in e s
Year
I
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
2
3
-
-
-
-
-
-
-
-
-
—
-
—
_
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
.17
14
20
20
8
12
13
10
14
13
13
14
22
19
22
17
27
27
18
7
8
5
-
-
-
-
15
19
18
21
25
22
16
25
19
20
27
23
18
20
21
32
28
17
6
7
4
5
22
22
14
19
16
13
14
14
6
10
11
14
14
12
15
21
22
4
5
5
-
-
-
-
-
-
4
78
89
42
38
44
61
69
85
95
70
108
79
60
53
25
33
21
20
21
29
20
20
16
27
22
25
21
14
11
28
36
25
33
28
38
5
6
«**
—
-
_
7
—
■
_
—
—
-
-
—
«■
-
-
-
-
-
■
-
-
-
-
—
7
-
-
3
I
11
18
18
21
22
23
17
22
20
14
19
11
26
17
23
27
21
18
23
32
I
-
16
20
22
17
19
20
' 18
14
18
17
18
15
22
24
21
23
18
22
15
-
_
-
-
—
I
3
14
20
20
21
22
20
17
14
16
10
3
21
18
25
30
24
20
19
21
-
(C o n tin u e s on th e n e x t page)
TABLE 4 .
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
(C on tin u ed )
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
.
-
-
-
-
-
-
-
-
44
52
55
59
61
63
70
71
79
85
133
U2
The dam c o m p o n e n t w a s e s t i m a t e d f r o m t h e f u l l - s i b m o d e l w h e r e
dam s w e r e n e s t e d w i t h i n s i r e and f r o m a m o d e l i n w h i c h dam s w e r e
n e s t e d w i t h i n m a te r n a l granddams.
The
fittin g
(H enderson,
co n sta n ts
1953) was u s e d ,
m eth od
or
H en d erso n ’s
m eth od
a s o u t l i n e d by Harvey (1977)»
III
to estim a te
t h e v a r i a n c e com ponents i n a l l t h e a n a l y s e s d e s c r i b e d above.
The c o v a r i a n c e s b e tw e e n o f f s p r i n g and dam ( c o v ( 0 , D ) ) ,
o ffsp r in g
and s i r e ( c o v ( 0 , S > ) , o f f s p r i n g and m a t e r n a l g r a n d dam ( c o v ( 0 , MGD)),
a u n t and n i e c e ( p a t t e r n D) (co v (A » N )) and a u n t ( f u l l s i b o f t h e s i r e )
and
n ephew
or
n ie ce
( c o v ( N ,A ) )
w ere
e v a lu a te d
by
sim p le
r e g r e s s i o n p r o c e d u r e s a f t e r a d j u s t i n g b oth w e i g h t s f o r a l l
lin e a r
th e f i x e d
effects.
The c o v a r i a n c e b e t w e e n m a t e r n a l g r a n d s i r e p r o g e n y and g r a n d
o ffsp r in g
( c o v ( S , MGS)) w a s o b t a i n e d f r o m
a tw o-w ay
random m o d e l
w ith o u t i n t e r a c t i o n w ith th e data c o r r e c te d fo r th e f ix e d e f f e c t s .
The e s t i m a t i o n p r o c e d u r e was R e s t r i c t e d Maximum L i k e l ih o o d ( P a t t e r s o n
and Thompson,
1971) a s o u t l i n e d by J e n n r ic h and Sampson (1976) i n th e
BMDP p a c k a g e .
S in c e co m p u ter c a p a b i l i t i e s p r e c lu d e d th e u s e o f a l l
the o b s e r v a tio n s ,
t h e d a ta u se d w e r e t h e 1,774 c a l v e s o f l i n e 4.
The f i n a l s o l u t i o n s f o r t h e m a t e r n a l g e n e t i c c o v a r i a n c e s w e r e
o b ta in e d through o r d in a r y l e a s t - s q u a r e s p ro ced u res. B a s i c a ll y , th e
e stim a te s of
th e c o v a r i a n c e s b e tw e e n r e l a t i v e s
for
b irth
w e i g h t and
w e a n in g w e i g h t w ere r e g r e s s e d on t h e c o e f f i c i e n t s f o r th e d i r e c t and
m a t e r n a l c o v a r i a n c e s i n t h e e x p e c t e d v a l u e s ( T a b l e 1 0 ).
th e
e stim a te s
of
th e
d ir e c t
and
m atern al
c o v a r ia n c e s
T h erefore,
were
th e
43
r e g r e s s i o n c o e f f i c e n t s o f a model w it h z e r o i n t e r c e p t .
The n in e ter m s
were
o 2 Ao
=
v a r i a n c e due t o a d d i t i v e d i r e c t e f f e c t s ,
a 2 Am
=
v a r i a n c e due t o a d d i t i v e m atern al e f f e c t s ,
a Ao Am
= c o v a r ia n c e b etw e e n a d d i t i v e d i r e c t and a d d i t i v e
m a te r n a l e f f e c t s ,
C 2Do
=
v a r ia n c e due t o dominance d i r e c t e f f e c t s ,
o 2Dm
=
v a r i a n c e due t o dominance m a te r n a l e f f e c t s ,
a DoDm
=
c o v a r ia n c e betw een
dom inance
d irect
and
dom inance m a te r n a l e f f e c t s ,
O2Eo
=
v a r i a n c e due t o random e n v iro n m e n ta l e f f e c t s ,
Cf2 Em
=
v a r i a n c e due t o m a te r n a l e n v ir o n m e n t a l e f f e c t s ,
a EoEm
=
c o v a r ia n c e
b e tw e e n
random
and
and
m a te r n a l
e n v ir o n m e n t e f f e c t s .
The p r o g r a m u s e d w a s t h e PIR o f t h e BMDP p a c k a g e ( D i x o n e t a l ,
1981) .
The v a l u e o f fm ( t h e e f f e c t o f t h e dam m a te r n a l p h enotype on t h e
fu tu r e d a u g h ter m a te r n a l p h enotype as d e fin e d
estim a ted
fit
by f i t t i n g
F a l c o n e r ’s
(1965)
m odel.
th e m odel was o r d i n a r y - l e a s t s q u a r e s .
by K och ,
1 9 72) w as
The p r o c e d u r e used t o
The e s t i m a t e s
of
the
c o v a r i a n c e s b e t w e e n r e l a t i v e s w e r e r e g r e s s e d on t h e c o f f i c i e n t s f o r
th e te r m s i n th e m odel:
P = A+
fm P ’ + D + C + E
where
P
= p h e n o t y t p ic v a l u e o f an i n d i v i d u a l measured a s a d e v i a t i o n
from th e p o p u l a t i o n mean,
A
= b r e e d in g v a l u e o f th e i n d i v i d u a l ,
fm
= p a r t i a l r e g r e s s io n c o e f f i c i e n t r e l a t i n g d a u g h te r s ’ to
m o th e r s’ p h e n o ty p ic
v a lu e s
in
th e
absence
of
g e n e tic
v a r i a t i o n among t h e m o th e r s,
p*
= p h e n o t y p ic v a l u e o f th e i n d i v i d u a l ’ s m other,
D
= dominance d e v i a t i o n o f t h e i n d i v i d u a l ,
C
- e n v ir o n m e n t a l
factors
common t o f u l l - s i bs t h a t a r e
n o t i n c l u d e d i n t h e m a te r n a l e f f e c t , and
E
= e n v ir o n m e n ta l f a c t o r p a r t i c u l a r t o th e i n d i v i d u a l .
For c o n v e n ie n c e o f
a n a ly sis
Thompson (1976) regrou p ed t h e components
a s i t i s s h o w n i n T a b le 12.
The a d d i t i v e v a r i a n c e w a s s p l i t i n t o a
l i n e a r ( 2 n d c o lu m n f r o m t h e r i g h t i n T a b l e 12) and a n o n l i n e a r ter m
( 3 r d c o lu m n f r o m t h e r i g h t i n T a b le 1 2 ) .
and
a 2a
E stim a te s o f
O2 D w e r e f o u n d f o r d i f f e r e n t v a l u e s o f fm : r I , - . 5 , 0 , .5 and
1 .0 , s i n c e t h e r e w e r e n o n - l i n e a r t e r m s .
Thompson ( 1 9 7 6 ) p r o p o s e d a
m o d if i e d maximum l i k e l i h o o d p r o c e d u r e b a se d on d e s ig n e d e x p e r im e n ts t o
e s t i m a t e m a t e r n a l v a r i a n c e s and c o v a r i a n c e s .
c o v a r ia n c e s
in d e p e n d e n t.
b etw een
r e la tiv e s
in
th e
The e s t i m a t e s o f t h e
present
stu d y
were not
T h e r e fo r e , t h e l e a s t - s q u a r e s m e th o d w a s u s e d i n s t e a d .
At e a c h r u n a d i f f e r e n t v a l u e o f fm ( r a n g i n g f r o m - 1 . 0 t o 1 .0 ) was
u s e d and h2 w a s c a l c u l a t e d a s t h e r a t i o
CT2 A/
c 2 p0
£ p r i o r i i t w as
d e c id e d t o s t o p t h e i t e r a t i v e p r o c e s s when t h e e s t i m a t e d h2 was equ al
t o t h e e s t i m a t e f r o m t h e PHS m e th o d u s i n g a l l r e c o r d s s i n c e
e s t i m a t e had t h e s m a l l e s t sta n d a r d e r r o r
th at
45
H e r i t a b i l i t y f o r a d d i t i v e d i r e c t e f f e c t s (h2 o) was e s t i m a t e d a s
o 2 Ao
h2 o =
o 2P
where
a2 P was d e f i n e d i n ( 1 0 ) (page 1 6 ) .
H e r i t a b i l i t y f o r a d d i t i v e m a te r n a l e f f e c t s (h2 m) was e s t i m a t e d a s
O2 Am
h2m
H e r ita b ility
for
to ta l
o2p
a d d itiv e
effects
was d e f in e d
by Willham
(1963) a s
o^Ao + 1 .5
a AoAm + .5 O2Am
h2 i =
o2p
It
is
th e
p r o p o r tio n o f
th e
p h e n o ty p ic v a r ia n c e
due
to
th e
c o n trib u tio n of a l l a d d itiv e g e n e tic sources of v a r ia tio n .
The g e n e t i c c o r r e l a t i o n b e t w e e n a d d i t i v e g e n e t i c e f f e c t s and
a d d i t i v e m a te r n a l e f f e c t s (rg ) was e s t i m a t e d by
CfAoAm
rG =
F in a lly ,
th e
O2 Ao
e n v ir o n m e n ta l
O2 Am
c o r r e la tio n
b etw een
d ir e c t
e n v ir o n m e n t a l and m a te r n a l e n v ir o n m e n t a l e f f e c t s ( r E) was e s t i m a t e d by
o EoEm
rP =
O2Eo
O2Em
46
RESULTS AND DISCUSSION
The a n a l y s e s o f v a r i a n c e f o r b i r t h and w eaning w e i g h t f o r
m o d e ls i n w h ich t h e c a l v e s a r e r e l a t e d
those
by s i r e - t y p e r e l a t i o n s h i p s a r e
p r e s e n t e d i n Table 5.
The m o d e l s f o r t h e s i r e - t y p e r e l a t i v e s a c c o u n t e d f o r 36 t o 65 %
and 56 t o 72 % o f t h e v a r i a t i o n i n b i r t h w e i g h t and w e a n i n g
r e sp ec tiv e ly .
L in e-y ea r,
w e ig h t,
age o f dam, s e x and t h e r e g r e s s i o n s o f b i r t h
w e i g h t on b i r t h d a t e o f c a l f and w e a n i n g w e i g h t on w e a n i n g a g e w e r e
sig n ific a n t
(PC.OI) s o u r c e s o f v a r i a t i o n i n a l l
th e m o d e ls.
The s i r e
d i f f e r e n c e s w e r e s i g n i f i c a n t i n a l l t h e m o d e l s e x c e p t PGS f o r b i r t h
w e ig h t.
The a g e o f dam by s e x i n t e r a c t i o n a c c o u n t e d f o r a s m a l l
p o r t i o n o f t h e t o t a l v a r i a t i o n i n b i r t h w e i g h t ( .3
and w e a n i n g
w e i g h t (.5 %) i n t h e PHS model i n w h ich a l l a v a i l a b l e r e c o r d s (4,423)
w ere u s e d .
As
th e
number
of
records
decreased
th e
age
of
i n t e r a c t i o n became l e s s i m p o r t a n t (PX05) f o r both t r a i t s
and 8 ) .
dam by s e x
(T a b le s 5,
7
L e a s t - s q u a r e s m ea n s f o r a g e o f dam by s e x s u b c l a s s e s a r e
sh o w n i n T a b l e 6.
The i n t e r a c t i o n w a s a t t r i b u t a b l e t o t h e h i g h e r
d i f f e r e n c e b e t w e e n m a l e s ( b u l l s and s t e e r s ) and h e i f e r s grow th r a t e
o u t o f o l d e r cow s as compared t o th e same d i f f e r e n c e i n younger cows.
S i m i l a r r e s u l t s w ere found by S e l l e r s e t a l (1970) and S c h a e f f e r and
W ilt o n (1974)
i n H ereford c a t t l e .
The means f o r age o f dam and s e x (T ab le 6) a g r e e w i t h th e v a l u e s
r e v ie w e d by W old e h a w a r ia t e t a l (1977) f o r H er e fo r d s.
The r e g r e s s i o n
o f b i r t h w e i g h t on b i r t h d a t e had t h e sa m e v a l u e (b = .0 2 ) a s t h e on e
TABLE 5 . ANALYSES OF VARIANCE FOR SIRE-TYPE RELATIVES
S ire
df
MS
7 2 .3 0 0
Line-Year (L-Y) 178
202
SireZL-Ya
Age of Dam (A)
5
2
Sex
10
A x Sex
Regression on
B irth Date
I
4024
E rror
R2
1837-4SG
3 0 .I fifi
2
10
S30.4fifi
12.4
I
2163
MS
34070 s
6730 a
770296 «
354558 a
Line-Year (L-Y) 178
SireZL-Y
202
Age of Dam (A)
5
2
Sex
A x Sex
10
Regression on
B irtii Date
I
E rror
4024
.56
123
36
5
46I . Sofi
2
10
15.8
1133.9efi
8 01.Ififi
17.9
I
2166
6 4 3 .Ififi
12.9
198.900
2 0 .1
.
df
59.4 0 0
52.7fi8
1 3 .Ififi
432.79S
11.9
.52
.36
df
R2
1322.760
174
1263
5
22.280
B irth Weight
P ateriB l
Grand S ire
df
Maternal
__GrancLSire____
df
MS
M aternal G reat
Grand S ire
df
MS
164
1323
5
4 3 .8 0 0
12.900
4 9 6 .3 0 0
2
10
238.3fifi
I
524.100
1232
.36
MS
174
1263
5
253500
4070 a
725900
123
36
5
1766ss
2
10
10l89fifi
457
2
10
546fifi
56517efi
20389 fifi
1343Bfi
718212fifi
376
I
2163
331l88fifi
357
I
2166
357021fifi
371
O(PC-OS)
9 6 (P<.01)
a stan d s f o r "genetic" source o f v a r ia tio n i n a l l an a ly sesb PHS+PHSD i s P a te rra l h a l f - s i bs p lu s dams PHS-
273600
.56
1 2 .2
df
2188fifi
400fis
23863®°
6816®°
750°
2
10
I
169347°°
364
1232
.72
14,498
153.9fio
189.9°°
16.9
2
. 10
377 .9 0 0
I
2162
1 1 ,8
.56
Jfi
164
1323
5
49 .2 0 0
174
1261
5
.65
Weaning Weight
df
MS
.63
2 2 .2
PHS + PHSDb
df
MS
df
Jfi
174
1261
5
247Sfifi
2
10
538700
I
201213°°
355
45300
5088°°
638
2162
.66
JJ8
TABLE 6 . LEAST-SQUARES MEANS FOR AGE OF DAM, SEX, AGE OF DAM BY
SEX SUBCLASSES AND REGRESSIONS IN THE PHS MODEL
Item
U
Age o f Dam
2
3
4
5
6 -1 0
11 or +
Sex
H e i f e r C a lv e s
B u l l C a lv e s
S t e e r C a lv e s
Age o f Dam x Sex
2 -H eifers
2 -B u lls
2 -S teers
3 -H eifers
3 -B u lls
3 -S teers
4 -H eifers
4 -B u lls
4 -S teers
5 -H eifers
5 -B u lls
5-S teers
6 t o 10 - H e i f e r s
6 t o 10 - B u l l s
6 t o 10 - S h e e r s
11 or + - H e i f e r s
11 or + - B u l l s
11 or + - S t e e r s
n
B i r t h Weight (kg)
Weaning Weight (kg)
4423
3 5 .5 ± .1 5
1 8 8 .7 ± 1 .0 2
485
751
704
572
1712
199
3 2 .3
3 5 .1
3 6 .3
3 7 .2
3 7 .0
3 5 .1
1 65.4
182.1
1 92.3
199.1
2 0 1 .0
192 .2
2409
883
1131
3 3 .9 ±. .1 6
3 6 .2 ± .21
3 6 .4 ± .2 0
181.6 ± 1 .0 8
1 9 4 .2 ± 1 .33
190.3 ± 1 .2 4
3 1 .0
3 2 .8
3 2 .9
3 3 .6
3 5 .6
3 6 .2
3 5 .1
3 6 .9
3 6 .8
3 5 .3
3 8 .0
3 8 .1
3 5 .4
3 8 .2
3 7 .4
3 2 .9
3 5 .7
3 6 .7
1 6 1 .7
1 66.3
1 6 8 .2
1 76.7
1 8 4 .8
1 84.6
1 8 3 .8
186 .6
194 .4
189 .4
2 0 9 .0
1 9 8 .9
1 9 1 .9
2 1 0 .6
2 0 0 .5
1 8 5 .9
1 95.5
195.1
257
103
125
394
180
177
404
143
157
305
127
140
955
297
460
94
33
72
R e g r e s s i o n s ( k g /d a y )
B i r t h Date (day o f y e a r )
Weaning Age
±
±
±
±
±
.2 3
.2 0
.21
.21
.1 7
± .3 2
± .2 8
.41
«38
.2 3
.3 2
.31
.2 3
.3 5
± .3 3
+ .2 5
± .3 6
± .3 4
+ .1 8
± .2 7
± «23
± .4 0
± .6 6
± .4 6
±
±.
+
±
±
±
±
.023 + .003
—
n
± 1.43
± 1 .2 5
± 1 .2 9
± 1 .33
+ 1.11
+ 1 .86
± 1 .6 5
± 2 .3 6
± 2 .1 8
± 1.41
+ 1.8 7
± 1 .8 4
± 1.41
± 2 .0 1
± 1 .9 2
± 1 .50
± 2 .1 0
± 1 .9 9
± 1.17
± 1 .5 9
± 1.41
± 2 .3 2
± 3 .6 6
± 2 .6 3
.881 ± .020
TABLE 7 .
ANALYSES OF VARIANCE FOR COUSINS FAMILIES
S o u rce o f
V a ria tio n
R ela tiv e
Age o f dam
Sex
Age o f dam x s e x
R e g r e s s i o n on
b ir th date
Error
MGD
df
MS
df
418
5
23-4BS
79
5
2
10
I
2301
11 4 2 .3 * *
10 02 . 6 BB
13.2
4 9 0 .3 * *
12.2
I
381
R ela tiv e
Age o f dam
Sex
Age o f dam x se x
R e g r e s s i o n on
b ir th date
Error
418
5
2
10
I
2301
B(PC-OS)
1 6 1 .6 * *
I
483
2 62 . 6 »*
12 .7
I
141
12.6
2
-■
MS
Weaning Weight
PHS + MHSD
df
MS
2.
10
I
381
37246»»
331
.5 6
BB(PC-OI)
df
95
5
2
10
620 »»
6100»»
2814»«
592
I
483
41161»*
359
I
141
.64
-
9 1 .1 » »
13.2
FS + PHSP
df
MS
42
4
2
584»»
1502*»
3729**
2
O-
2. 8
12.0
I
46
.5 9
FC + PHS
MS
-
1 7.3
13.1
5 .5
-
.51
.5 9
366
5
334814»»
342
.56
95
5
877»»
8652»»
2582 »»
827 »»
' 79
5
1 5.8
5 9 .9 * *
5 8 .7 * «
2 0 .5 * »
1 5 6 .7 s *
1 1 3 .7 s *
1 6.2
df
1051BB
41813*»
22580 »»
596
MS
366
5
2
10
CO
______MGD______
df
MS
df
1 9 .3 * *
2 2 0 . 2 »»
7 0 .7 s *
16.1
CTA
S o u rce o f
V a ria tio n
R*
2
10
CTa
CO
R^
MS
B i r t h Weight
PHS + MHSD
df
MS
FS + PHSP
df
MS
42
4
628 *
491
749
2
-
-
11222 »»
274
.6 7
I
46
4817»*
297
• 76
TABLE 8 . ANALYSES OF VARIANCE FOR FULL-SIBS AND DAM-MATERNAL GRAND DAM MODELS
df
Sou rce o f V a r i a t i o n
S ir es
Dams/S
Age o f dam
Sex
Age o f dam x s e x
R e g r e s s i o n on b i r t h d a t e
Error
50
402
5
B i r t h W eight
Sou rce o f V a r i a t i o n
MGD
Dams/MGD
Age o f dam
Sex
Age of"dam x s e x
R e g r e s s i o n on weaning age
Error
4 9 .1 * *
1 6 . 7 s0
53.488
102 . I gs
1 2 .7
4 5 3 .9 * *
1 2 .5
2
10
I
505
R*
df
MS
183
462
5
3 0 .2 * 0
2 1 .4 8 8
2 0 9 . Os s
- 1 4 3 .3 ° °
2 7 .0 *
6 4 .8 * *
11.7
2
10
I
650
.61
S ires
Dams/S
Age o f dam
Sex
Age o f dam x s e x
R e g r e s s i o n on b i r t h d a te
Error
"
MGD
Dams/MGD
Age o f dam
Sex
Age o f dam x sex
R e g r e s s i o n on weaning age
Error
94988
476108
600s
1 16431°°
296
2
10
I
505
a (P < .0 5 )
Weaning W eight
Sou rce o f V a r i a t i o n
I 85388
658 **
50
402
5
— —
—
.5 7
MS
df
Sou rce o f V a r i a t i o n
F
MS
>76
" ——
o o (p < .01)
—
—
—
df
MS
129088
183
462
5
72900
6153**
390386
7 17°°
6.1408*8
283
2
10
I
650
- -
—
51
c a l c u l a t e d by K re ss and B u r f e n ln g (1972) u s i n g p a r t o f th e r e c o r d s o f
th e p r e s e n t stu d y .
b ir th w eig h ts.
I t i n d i c a t e s t h a t l a t e b orn c a l v e s had h e a v i e r
The m o s t p l a u s i b l e e x p l a n a t i o n i s t h a t l a t e c a l v i n g
c o w s had a b e t t e r l e v e l o f n u t r i t i o n t h a n e a r l y c a l v i n g c o w s .
The
v a l u e o f th e r e g r e s s i o n o f w eaning w e i g h t on age a t w ean in g (b=. 88 ) i s
h i g h e r than t h e
e .,
v a l u e s r e v ie w e d by C a n te t
(1983)
in
H e r e fo r d s
(i.
.6 4 t o . 7 5 ) .
The m o d e l s f o r c o u s i n s ( T a b l e 7 ) ,
fu ll-sib s
and dam m a t e r n a l
grand dam (T ab le 8 ) showed t r e n d s s i m i l a r t o the s i r e m od els.
dam,
sex,
r e g r e ssio n s
and
th e
betw een
fa m ily
co m p o n en ts
s i g n i f i c a n t s o u r c e s o f v a r i a t i o n i n a lm o s t a l l th e m o d e ls.
number o f r e c o r d s u sed became f e w e r ,
was l e s s im p o r ta n t.
Age o f
were
As t h e
t h e i n t e r a c t i o n age o f dam by s e x
T h e r e f o r e , i t w a s e x c l u d e d i n t h e PHS + SFC and
FS + PHSP m o d e l s t o a l l o w f o r m ore d e g r e e s o f f r e e d o m i n t h e e r r o r
term .
In th e l a t t e r
m o d e l o n l y t h e b e t w e e n f a m i l y c o m p o n e n t w as
s i g n i f i c a n t f o r b o t h t r a i t s and t h e r e g r e s s i o n o f w e a n i n g w e i g h t on
w ea n in g a g e.
p resen ted
in
The e s t i m a t e s o f h e r i t a b i l i t i e s and c o r r e l a t i o n s a r e
T a b l e 9»
The e s t i m a t e s
r e l a t i v e s a r e sh o w n i n T a b le 10.
of
th e c o v a r ia n c e s b etw een
When m a t e r n a l e f f e c t s a r e p r e s e n t
f o u r t i m e s t h e s i r e v a r i a n c e component i s an u n b ia se d e s t i m a t e f o r th e
a d d it iv e g e n e t ic v a r ia n ce fo r d ir e c t e f f e c t s
(Table
I).
Most o f
th e
e s t i m a t e s i n T a b le 9 a r e c o m p a r a b l e t o t h e w e i g h t e d a v e r a g e s i n t h e
r e v ie w o f W o ld e h a w a r ia t e t a l (1977).
th e
PHS m e t h o d ,
th e
one o b ta in e d
perhaps th e m ost r e l i a b l e .
Of th e t h r e e e s t i m a t e s o f h 2 by
u sin g
a ll
th e r e c o r d s
(PHS) i s
U n e x p e c te d ly , th e e s t i m a t e o f h 2 f o r b i r t h
w e i g h t was s i m i l a r t o t h e e s t i m a t e f o r w e a n in g w e i g h t .
The l a t t e r
52
TABLE 9 . ESTIMATES OF HERITABIL IT IE S AND CORRELATIONS FOR BIRTH
WEIGHT AND WEANING WEIGHT
Weaning Weight
B i r t h Weight
In tra cla ss
C o r r e la tio n s
PHSa
b
C
MHSb
C
FS
Present
Study
L itera tu re
V a lu e s
Present
Study
L ite r a tu r e
V alu es
.0 7
.10
.0 7
.21
.21
.22
. 11®
.07
.11
.10
.3 2
.3 0
.44
.06®
H e r ita b ilitie s
PHSa
. 28+.02
b
.l»1±.15
C
FS
MGSd
C
2 bod
2 bos
.2 8
.45& .08
. 5 4 +.02
.06
. 4 5 + .0 2
.2 l£ .0 4
.27®
-
.44®
-
•42®
.3 5 h
. 28+.02
.4 7 + .1 6
.4 2
• 88+.08
. 23+.02
.36®
.3 7 f
.
.21
. 28+.02
.0 6 + .0 2
.26®
.7 4 f
. 21 s
.31®
.2 5 h
C o r r e l a t i o n s Between B i r t h and Weaning W eight
L ite r a tu r e
P r e s e n t Studv
C o r r e la tio n s
.54®
• 5 6 + .1 3
G e n e t ic
.38®
• 41
P h e n o ty p ic
.2 4 1
Environm ental
• 36
by fo r m u la e p r o v id e d by Osborne and P a t t e r s o n ( 1 9 5 2 ) .
In t h e c a s e o f
tr c a lc u la te d as
2 b o ffsp r in g -p a re n t,
sta n d a r d
e r r o r s were
c a l c u l a t e d a s 2 t i m e s th e s t a n d a r d e r r o r o f bop.
F i n a l l y , S.E. o f
t h e FS model were t a k e n from t h e program (H arvey, 1 9 7 7 )•
^ C a lc u la t e d from t h e PHS m odel.
^ C a lc u la te d from t h e FS m odel.
^ C a lc u la te d from t h e S-MGS m odel.
^ C a lc u la t e d from t h e MGS m o d e l.
e Review by W oldehaw ariat e t a l . ( 1 9 7 7 ) •
^ E stim a te d by Hamman e t a l . ( 1 9 6 3 ) •
^ E stim ated by Crow and H ow ell ( 1 9 8 2 ) .
^ E stim a ted by Koch and C lark ( 1 9 5 5 ) .
^Review by P r e s to n and W i l l i s ( 1 9 7 0 ) .
WW
0V m ■ 0X
a DcAn
a^Do
CT12Eo
UE0Em
U^ejd
I
BW
Q
. df
Q
R e la tiv e s
£
TABLE 10. COVARIANCES BETWEEN RELATIVES: ESTIMATES, DEGREES OF FREEDOM FOR ESTIMATION
AND EXPECTED VALUES IN TERMS OF DIRECT AND MATERNAL COVARIANCES
PHSa '
202
PHSb
50
PHS0
.96
W ithin PHSa
4024
PGS
36
W ith in PGS
2166
MGSd
1263
MGS0
96
W ith in MGSd
2163
MGGS
1323
W ith in MGGS
1232
MGDe
418
MGDf
183
W ithin MGDe
2301
COV(O1S)
2344
COV(O1D)
3619
COV(O1MGD)
2573
COV(S1MGS)0
96
W ithin S-MGS0 1582
COV(MAt N)
210
W ithin MAt N
208
COV(PAt N)
104
W ithin PAt N
102
FSb
402
W ithin FSb
523
mbs);
402
MHSf
462
SFC
479
W ithin SFC
381
PHS+MHSD
366
W ithin PHS+MHSD 483
PHS+SFC
95
W ithin PHS+SFC 141
FS+PHSP
30
W ith in FS+PHSP
46
PHS+PHSD
901
W ithin PHS+
PHSD
1481
29.38
.97
1.69
62.65
1.02
44.62
12.48
376.38
16.04
.26
12.92
357.93
1.88
22.83
.23
22.13
11.88
371.55
0.44
19.99
12.18
364.93
109.50
1.73
1.38
93.38
12.20
342.84
1.73
11.53
62.20
2 .8 5
48.64
1.25
-.0 0 2
-5 .3 2
12.37
357.65
- .1 7
-4 5 .3 3
348.68
13.71
-3 .0 6
6.85
14.51
509.39
3 .6 7
234.09
12.55
296.27
1.98
171.44
164.26
3-55
1.17 ' 95.94
12.62
331.01
112.92
3.36
12.72
359.29
I i03
126.86
274.75
13.27
259.69
2 .6 3
12.03
297.99
1.19
45.77
355.50
11.79
1/4
0
0
0
0
0
0
0
0
3 /4
1/16
15/16
1/16
0
0
0
1/4
0
0
0
1/4
0
0
0
0
0
0
0
0
0
0
0
0
I
0
I
0
I
0
I
0
I
0
I
0
15/16
1/64
63/64
1/16
3 /4
1/16
15/16
1/4
3 /4
1/16
15/16
1/4
0
0
0
0
0
0
0
0
0
0
0
0
I
0
I
0
I
0.
I
0
I
0
I
0
15/16
1/2
1/2
1/4
1/8
9/16
1/4
3/4
1/4.
3 /4
1/2
1/2
1/4
1/4
1/8
7 /8
5/16
11/16
3 /8
5/8
5 /8
3 /8
5/16
3 /4
1/4
5 /4
5 /8
1/4
1/2
3/4
1/4
3 /4
1/4
I
0
I
I
1/2
1/2
1/4
3/4
1/2
1/2
5/4
-1 /4
1/4
3 /4
0
1/2
1/4
0
3 /4
1/2
1/2
1/2
1/2
I
O
I
I
1/2
1/2
1/4
3/4
1/2
1/2
I
0
1/4
0
0
0
0
0
0
0
0
0
0
1/4
3/4
1/4
0
0
0
1/16
15/16
1/8
7 /8
25/64
39/64
1/64
0
0
I
0
0
0
1/4
3/4
1/4
3 /4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
I
I
1/4
3/4
I
0
1/4
3/4
I
0
.0
I
0
0
0
0
I
0
I
0
I
0
I
0
0
0
I
0
I
0
I
0
I
0
I
0
I
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
I
0
0
0
0
I
0
I
I
0
0
0
0
0
o ■
I
6
0
9/16
3 /4
3 /4
63/64
0
0
0
0
0
.
.
aE stim a te s from th e PHS m odel; ^ E stim a te s from th e FS m odel; e E stim a te s from th e S-MGS model; a E stim ates
from th e MGS m odel; e E stim a te s from th e MGD model; f E stim a te s from th e MGD-D/MGD model.
a g r e e d c l o s e l y w i t h th e l i t e r a t u r e mean e s t i m a t e o f W o ld eh aw ariat e t
a l (1 9 7 7 ).
The h 2 v a l u e s by PHS e s t i m a t e d t h r o u g h t h e f u l l - s i b f i l e
were
h i g h e r t h a n t h e v a l u e s e s t i m a t e d i n t h e PHS f i l e f o r b o t h w e i g h t s .
Hamman e t a l
(1963) a l s o found a l a r g e r s i r e component u s i n g FS d a ta
than by u s i n g PHS d a t a f o r w e an in g w e i g h t .
a sso cia ted
w ith
a sm a ller
T h e r e fo r e ,
number o f r e c o r d s f o r
s a m p lin g e r r o r
e stim a tio n i s
a
p o s s i b l e e x p l a n a t i o n f o r t h e l a r g e r s i r e component i n th e FS d ata s e t .
Koch ( 1 9 7 2 ) i n d i c a t e d
p ro g en ies,
t h a t e n v i r o n m e n t a l c o r r e l a t i o n s am ong s i r e
s i r e - y e a r i n t e r a c t i o n or a s i z e a b l e f r a c t i o n
b ein g t h r e e - q u a r t e r
s ib s
in stea d
over
an e x t e n d e d
p e r io d
of
o ffsp rin g
o f h a lf s ib s in sm a ll herds are
p o s s i b l e s o u r c e s o f b i a s e s i n PHS e s t i m a t e s .
taken
of
tim e ,
When t h e r e c o r d s a r e
changes in
m anagem ental
p r o c e d u r e s or e n v ir o n m e n t a l c o n d i t i o n s c o u ld in d u c e an o v e r e s t i m a t i o n
o f t h e s i r e component.
T h is can come about b e c a u se o f a p o s s i b l e s i r e
by l i n e o r s i r e by y e a r i n t e r a c t i o n .
n e g a t i v e e n v ir o n m e n t a l
F lo w er e t a l (1964) r e p o r te d a
tr e n d i n b i r t h and w ean in g w e i g h t i n t h e herd
under s tu d y p o s s i b l y r e l a t e d
to a d e c r e a s e i n r a i n f a l l d u rin g th e
g r a z i n g s e a s o n (March t o Septem ber) from 1952 to 1958.
S i r e by l i n e -
year in te r a c t io n s are a d i s t i n c t p o s s i b i l i t y .
I t s h o u ld be n oted t h a t
due t o r e s t r i c t i o n s
t h e s i r e component was
in
com puter c a p a b i l i t i e s
e s t i m a t e d w i t h th e d a ta c o r r e c t e d by l i n e - y e a r s t o a l l o w f o r th e dams
t o be n e s t e d w i t h i n s i r e s .
I n t h e PHS f i l e ,
e v a lu a te d in a w ith in lin e - y e a r
b a sis.
t h e s i r e c o m p o n e n t w as
In t h i s c a s e th e sta n d a r d
e r r o r o f h2 was a l m o s t t e n t i m e s l o w e r than t h e e s t i m a t e s o b t a in e d i n
th e FS m odel (T a b le 9).
Kennedy and H enderson (1975a) found t h a t th e
55
s i r e by y e a r i n t e r a c t i o n a c c o u n t e d f o r I t o 4 % o f t h e v a r i a t i o n i n
w e a n i n g w e i g h t o f Angus and H e r e f o r d c a l v e s w h e r e a s i n t h e s t u d y o f
D in k el
and B u s c h ( 1 9 7 3 ) t h i s c o n t r i b u t i o n w a s a 6 .5 % i n H e r e f o r d
though i t
was n o t s i g n i f i c a n t
The cow
com ponent
(PX 05).
(MHS) w a s e s t i m a t e d fr o m
m a te r n a l grand dam m o d e ls.
t h e FS and dam-
There was an im p o r t a n t d i f f e r e n c e i n
the
MHS e s t i m a t e s f o r b i r t h w e i g h t a s compared t o t h e v a l u e s o b t a in e d f o r
rw e a n i n g w e i g h t ( T a b l e 1 0 ).
When dam s a r e n e s t e d w i t h i n s i r e s , t h e
e x p e c ta tio n a ls o
in c lu d es
1 /4
a2Do.
The e s t i m a t e s o f r e p e a t a b i l i t y
w e r e .1 3 ± .01 and .21 + .0 3 i n t h e c a s e o f b i r t h w e i g h t and .36 + .04
and .30 ±, .03 f o r w e an in g w e i g h t .
The e s t i m a t e s f o r b i r t h w e i g h t w ere
l o w e r than th e a v e r a g e o f th e s t u d i e s r e v ie w e d by W old eh aw ariat e t a l
(1977) and t h e one e s t i m a t e d by K re ss and B u r fe n in g (1972) (.27 ± .02)
who w o r k e d w i t h t h e r e c o r d s f r o m 1933 t o 1966 o f t h e p r e s e n t s t u d y .
S i n c e 1975 t h e herd h a s been s e l e c t e d by an in d e x w hich d i s c r i m i n a t e s
a g a in s t la r g e r b ir th w eig h t.
. It is
known t h a t s e l e c t i o n r e d u c e s the
a d d i t i v e v a r i a t i o n ( B u l m e r , 1 9 71; R o b e r t s o n , 1977; F a l c o n e r , 1 9 8 1 ) .
T h rift
et
al
(1981) r e p o r te d
an i m p o r t a n t r e d u c t i o n i n
th e
sir e
v a r i a n c e component (PHS) and h^ e s t i m a t e s o f b i r t h and w e a n in g w e ig h t
of
th ree
se le c te d
H erefo rd
g en e r a tio n s o f s e le c t i o n ,
co n tro l herds.
Angus h e r d s
r e sp ec tiv e ly ,
The s e l e c t i o n
through one y e a r o f age.
and
a fter
c r i t e r i o n was b a s i c a l l y
U n fo rtu n a tely ,
T h erefore,
h2 f o r
both
herds
(see
1.25
and
I
a s compared t o th e u n s e l e c t e d
grow th r a t e
no o t h e r r e p o r t s d e a l i n g w it h
s e l e c t i o n i n b e e f c a t t l e i n w h ic h a c o n t r o l l i n e i s
rep o rted
2,
th e r e v ie w
in c lu d e d have
o f Koch e t
a l , 1982).
t h e e v i d e n c e i s n o t c o n c l u s i v e t o su p p o r t th e h y p o t h e s i s o f
56
a d r a m a tic r e d u c tio n o f a d d it iv e v a r ia t i o n i n b ir th w e ig h t in t h i s
data s e t .
E s t i m a t e s o f h2 by FS i n c a t t l e a r e n o t common i n t h e l i t e r a t u r e
d u e t o t h e f a c t t h a t t h e cow i s
b a sic a lly
commonly u se d f o r one or tw o y e a r s ,
f u ll - s i b fa m ilie s.
The f u l l - s i b
u n i p a r o u s and b u l l s a r e
d e c r e a s i n g t h e chance f o r h a v in g
e s t i m a t e f o r b i r t h w e i g h t (.45 ± .08)
was c l o s e t o th e v a l u e r e p o r t e d by L e g a u l t and Touchberry (1962) (.51)
by p o o l i n g d a ta o f d i f f e r e n t b r e e d s o f d a i r y c a t t l e i n a w i t h i n - b r e e d
estim a te
w ith 432 c a l v e s .
A v a lu e
o f .8 8 ± .0 8 w a s e s t i m a t e d f o r
w e a n in g w e i g h t through FS i n the p r e s e n t stu d y .
f o u n d a h2 o f .7 4 u s i n g 3 3 2 H e r e f o r d c a l v e s .
Hamman e t a l (1963)
The e x p e c t a t i o n s i n
Table I show t h a t th e e s t i m a t i o n o f h2 by FS i s b i a s e d when dominance
and common or m a te r n a l e n v iro n m e n t a r e p r e s e n t .
T h e r e fo r e ,
s in c e the
e s t i m a t e s o f h2 through FS a r e d i f f e r e n t from th e e s t i m a t e s u s i n g PHS
(w h ich ,
a ssu m in g p r e s e n c e o f m a tern a l
g e n e tic e x p e c ta tio n ),
m atern al e f f e c t s
effects,
co u ld
have a d if f e r e n t
be i n v o l v e d i n
th at
d i f f e r e n c e f o r both t r a i t s .
The e s t i m a t e s o f t h e cov(0,D ) w ere l a r g e r i n both t r a i t s than th e
e stim a te s
of
c o v (0 ,S )
(T ab le
10).
H e r ita b itilie s
u sin g
th ose
c o v a r i a n c e s w e r e . 4 5 + .0 2 and .21 + .0 4 f o r b i r t h w e i g h t and .2 8 + 02
and .0 6 ± .0 0 4 f o r w e a n i n g w e i g h t by u s i n g
r e sp e c tiv e ly .
c o v ( 0 , D ) and c o v ( 0 , S ) ,
The v a lu es, f o r cov(0,D ) a g r e e d w i t h th e a v e r a g e s o f th e
e s t i m a t e s r e v ie w e d by W oldehaw ariat e t a l (1977).
However, th e v a l u e s
f o r c o v ( 0 , S ) a r e l o w e r t h a n t h o s e r e p o r t e d by Koch and C l a r k ( 1955b )
( . 2 5 ) , H o h e n b o k e n and B r i n k s ( 1 9 7 1 a ) ( . 2 8 ) , and Koch ( 1 9 7 2 ) ( .2 0 ) f o r
t h e H ereford breed.
The number o f b u l l s u se d i n t h e p r e s e n t stu d y was
57
h ig h e r than t h e number o f b u l l s u sed i n
errors
of
th ose
co m p a riso n i s
e stim a te s
are
the other s tu d ie s .
not rep o rted .
Th us,
oAoAm.
w ea n in g
g ro ss e stim a te
T h is e s t i m a t e i s - .8 4 and - 1 8 8 .9 2 kg^ f o r b i r t h w e i g h t and
w eig h t,
a n ta g o n ism
m atern al
fu rth er
d iffic u lt.
The e x p e c t a t i o n o f 4 ( c o v ( 0 ,S ) - 2 PHS) p r o v i d e s a
of
an y
Standard
r e sp e c tiv e ly .
The n e g a t i v e
b etw een a d d it iv e g e n e t ic
effects
for
d ir e c t
p rew ea n in g gro w th .
Hohenboken and B r in k s (1 9 7 1 ),
v a lu e s
in d ica te
and a d d i t i v e
an
g e n e tic
U sin g th e e s t i m a t e s o f
a AoAm was - 3 8 .8 kg2 f o r w e an in g w e ig h t .
I n t h e c a s e o f Koch and C l a r k ( 1 9 5 5 a , b)
kg2 f o r b i r t h and w e a n in g w e i g h t ,
a AoAm w as o f 1.84 and - 2 6 . 7 2
r e sp e c tiv e ly .
Buchanan e t a l (1982)
f o u n d t h e c o v ( 0 , S ) t o be g r e a t e r t h a n c o v ( 0 , D ) f o r b i r t h w e i g h t and
w e an in g w e i g h t a s i n th e p r e s e n t s tu d y .
n e g a t i v e f o r w ean in g w e i g h t ( - .0 4 ) .
p r e v io u sly d isc u sse d ,
ex p e c ta tio n .
S in c e
However,
t h e h2 by 2 Uqd was
Under t h e m a te r n a l e f f e c t s m odels
th e cov(0,D ) (T a b le I) h a s a r a t h e r c o m p lic a t e d
O2 Ao and
O2Am a r e i n v o l v e d , one or more o f th e
c o v a r i a n c e s b e tw e e n d i r e c t and m a te r n a l e f f e c t s
s i g n f o r th e cov(0,D ) t o be n e g a t i v e .
s h o u ld have a minus
The c o v a r i a n c e s MGD and MGS have
th e same e x p e c t e d v a l u e s i n te r m s o f d i r e c t and m a te r n a l v a r i a n c e s , a s
T a b le I and 10 s h o w .
1 .38 k g 2 ,
for
MGD and
The e s t i m a t e s f o r b i r t h w e i g h t w e r e 1.73 and
1 .8 8 and .2 3 k g 2 f o r
MGS.
In th e
case
of
w eaning w e i g h t t h e v a l u e s w ere c o n s i s t e n t l y l a r g e r f o r MGD (109.50 and
93.38 kg2 ) a s compared t o th e e s t i m a t e s f o r MGS (22.83 and 23.13 kg2).
I f t h e r e e x i s t s an e f f e c t o f t h e m a t e r n a l p h e n o t y p e o f t h e dam on
m a t e r n a l p h e n o t y p e o f t h e d a u g h t e r (fm p a t h , f i g u r e 3 ) , t h e r e i s an
e x t r a d i f f e r e n c e i n a d d i t i v e v a r i a t i o n s i n c e a term fm ( 1 / 2 O2 Am + 1/4
58
a AoAm) s h o u ld be added t o th e MGD
ex p e c ta tio n
( s e e a p p e n d ix ).
This
can c a u s e t h e MGD and MGS e s t i m a t e s o f c o v a r ia n c e f o r w e a n in g w e ig h t
t o be d i f f e r e n t .
Crow and H o w e l l ( 1 9 8 2 ) c a l c u l a t e d
th e m atern al
g ran d sire's c o n tr ib u tio n to t h e i r d a u g h te r s ' p e rfo rm a n ce a s m oth ers
f o r w ean in g w e i g h t a s
4
h2MGS * a 2E
1/4
O2 Ao +
For t h e
OAoAm
H ereford
a 2 Ao +
+
. O2MGS
a AoAm +
O2 Am +
breed
O2 MGS
a^Am
O2 Eo +
th ese
OEoEm +
O2 Em
a u t h o r s c a l c u l a t e d Ii 2 mqs
o f . 2 0 , . 1 4 , .2 5 and . 2 3 , f o r c o w s a t t h e i r f i r s t ,
f o u r t h or more p a r i t i e s , r e s p e c t i v e l y .
second,
v a lu es
t h i r d and
T h e r e fo r e , t h e e s t i m a t e s found
i n t h e p r e s e n t s t u d y (.2 1 and . 2 3 ) a g r e e c l o s e l y .
By u s i n g t h e sam e
p r o c e d u r e w i t h t h e p u b l i s h e d v a l u e s , i t w as c a l c u l a t e d a h2 MQS f o r
b i r t h w e i g h t o f .01 i n H o l s t e i n ( E v e r e t t and M agee,
H ereford and .26 i n Angus (Brown and G a lv e z ,
19 6 5 ) ,
.25 i n
1969) and i n H ereford .20
(K och, 1 9 7 2 ) .
The e s t i m a t e d v a l u e s i n t h e H avre H e r e f o r d h e r d w e r e
.5 4 and . 0 6 .
The f i r s t e s t i m a t e w a s o b t a i n e d f r o m o v e r 3 , 6 1 9 c a l f
r e c o r d s by u s i n g H enderson's method I I I . For w eaning w e i g h t th e v a l u e s
o f t h e l i t e r a t u r e w e r e .1 9 i n B r a h m a n s and .2 4 i n c r o s s b r e d s ( D e e s e
and Roger,
1967),
.31 i n H er e fo r d s (Hohenboken and B r in k s ,
1971a) and
. 3 2 i n H e r e f o r d s (K och, I 9 7 2 ) .
The
c o v (S ,M G S )
g r a n d o ffsp r in g
(K och ,
Or
c o v a r ia n c e
19 7 2 ) i n d i c a t e s
grand
th e
sir e 's
degree
progeny
of
an d
a s so c ia tio n
59
b e tw e e n t h e d i r e c t or t r a n s m i t t e d e f f e c t s o f a s i r e f o r a g i v e n t r a i t
and t h e p e r f o r m a n c e o f h i s d a u g h t e r s a s m o t h e r s f o r t h e sa m e t r a i t .
In t h e p r e s e n t s tu d y t h e cov(S,MGS) was e s t i m a t e d from a tw o-w ay c r o s s
c l a s s i f i e d random model i n w h ich l i n e 4 c a l v e s w ere coded by s i r e and
m a te r n a l grand s i r e .
grand progeny.
A lm ost a l l b u l l s (95 o u t o f 96) had progeny and
R estricted
Maximum L i k e l i h o o d e s t i m a t i o n o f v a r ia n c e
com p on en ts, a s o u t l i n e d by J e n n r ic h and Sampson (1 9 7 6 ), was used
w it h
p r i o r e s t i m a t e s b e i n g t h e o n e s o b t a i n e d i n t h e PHS and MGS m o d e l s .
The a l g o r i t h m
w as c o n v e r g e n t f o r b o t h t r a i t s .
The
e stim a te s of
cdv(S,M G S) w e r e - . 0 0 0 2 3 and - 5 . 3 2 kg^ f o r b i r t h and w e a n i n g w e i g h t ,
r e sp e c tiv e ly
-.0 1 .
(T a b le 11).
The e s t i m a t e d c o r r e l a t i o n s w e r e -.0 0 0 0 1 and
E v e r e t t and M agee ( 1 9 6 5 ) r e p o r t e d t h a t cov(S,M G S) w a s .2 6 k g 2
fo r b irth
w eigh t.
The c o r r e l a t i o n o b t a in e d from t h e i r d a ta i s .0028.
The c o v a r i a n c e w a s o b t a i n e d by u s i n g a l l p o s s i b l e c o m b i n a t i o n s o f
m a te r n a l g r a n d o f f s p r i n g o f a common g r a n d s i r e .
Koch (1972) r e p o r te d a
c o r r e l a t i o n o f .05 f o r b i r t h w e i g h t and .02 f o r a v e r a g e d a i l y g a in t o
w ean in g.
The method u se d i s n o t r e p o r t e d i n t h e paper.
The cov(0,M G D) w a s 1 .2 5 k g 2 f o r b i r t h w e i g h t and 4 8 .6 4 k g 2 f o r
w e an in g w e i g h t .
The c o r r e l a t i o n s w e r e .10 and .09.
The c o r r e s p o n d in g
e s t i m a t e s o f Koch ( 1 9 7 2 ) w e r e .1 3 f o r b o t h t r a i t s .
He s u g g e s t e d t h e
u s e o f t h e d i f f e r e n c e Ccov(O5D) - 2 cov(0,M G D )) t o e v a l u a t e f u r t h e r
e v i d e n c e on e n v i r o n m e n t a l c o v a r i a n c e t h r o u g h m a t e r n a l a b i l i t y .
It
p r o v i d e s an e s t i m a t e o f ( 1 - 2 f m ) (oDoDm + aEoEm (l-s-frn) + fm ) i f fm i s
d i f f e r e n t f r o m z e r o , and o f ( a DoDm -KJEoEm), i f fin i s e q u a l s t o z e r o .
For b i r t h
w e ig h t th e d iff e r e n c e
is
.3 5 k g 2 s u g g e s t i n g a p o s i t i v e
d i r e c t pathw ay, p o s i t i v e dom inance c o v a r i a n c e or both.
In th e c a s e o f
60
TABLE 11. ASYMPTOTIC VARIANCE - COVARIANCE
MATERNAL GRAND SIRE MODEL
MATRICES. FOR
B i r t h W eight
Error
Error
S'
THE
SIRE-
Weaning Weight
MGS
Error
S
MGS
.1 8 1 0
-.0 0 5 7
-.0 0 4 9
1 5 9 .4 8
-9 .0 5
-1 0 .7 8
S ire
-.0 0 5 7
.1 0 3 2
-.0 0 2 3
-9 .0 5
1 1 3 .1 5
-5 .3 2
MGS
-.0 0 4 9
-.0 0 2 3
.0176
-1 0 .7 8
"5«32
55.81
61
w ean in g w e i g h t t h e d i f f e r e n c e i s - 3 8 ,0 8 kg2 s u g g e s t i n g n e g a t i v e d i r e c t
effects
(i.e .,
co v a r ia n c e s.
n e g a tiv e
v a lu e
of
fm ),
n e g a tiv e
aDoDm
or
both
The r e s u l t s a g r e e w i t h t h o s e r e p o r t e d by Koch (1972).
E stim a te s
of
th e
c o v a r ia n c e s
and
v a r ia n c e s
from
the
oth er
r e l a t i v e r e l a t i o n s h i p s have not been r e p o r te d i n th e l i t e r a t u r e fo r
b ir th
and w e a n i n g
w e ig h t in
c a ttle .
The
p h e n o ty p e s o f t h e dam and d a u g h te r (fm ,
path
b etw een
m atern al
f i g u r e 3) was e s t i m a t e d t o be
.0 8 and - . 1 0 f o r b i r t h and w e a n i n g w e i g h t , r e s p e c t i v e l y ( T a b l e 1 2 ).
The m e t h o d o f e s t i m a t i o n w a s m u l t i p l e r e g r e s s i o n l e a s t - s q u a r e s i n
w h ich
th e
regressed
c o v a r ia n c e s
on t h e
betw een
c o e ffic ie n ts
r e la tiv e s
d e r iv e d
for
b oth
w eig h ts
by Thompson ( 1 9 7 6 )
to
were
fit
F a l c o n e r ’s ( 1 9 6 5 ) m o d e l , a s e x p l a i n e d i n t h e r e v i e w o f l i t e r a t u r e .
The h2 f r o m t h e PHS f i l e w a s u s e d a s t h e c o n v e r g e n c e v a l u e s i n c e i t
was e s t i m a t e d u s i n g a l l
a v a ila b le records.
Koch (1972) s p e c u l a t e d a
v a l u e f o r fm o f - . 1 0
to - .2 0 f o r w e a n in g w e ig h t .
p o in ted
path
out
th at
a
of
c o r r e l a t i o n s and r e g r e s s i o n s .
supports h i s h y p o th e sis.
c a ttle.
th at
m a g n itu d e
Th us,
F u r t h e r m o r e , he
sa tisfie s
observed
th e v a lu e found i n t h i s stu d y
No o t h e r e s t i m a t e s o f fm w e r e found f o r b e e f
F a l c o n e r ( 1 9 6 5 ) r e p o r t e d a n e g a t i v e v a l u e o f fm f o r l i t t e r
s i z e i n m ic e and A l s i n g e t a l (1981) a n e g a t i v e v a l u e f o r l i t t e r
siz e
in p ig s.
The
so lu tio n s
fo r
c o v a r i a n c e s a r e sh o w n i n
th e
d ir e c t
and
T a b l e s 13 and 14.
m a tern a l
v a r ia n c e s
and
The f i r s t a p p r o a c h t o
e s t i m a t e t h e d i r e c t and m a te r n a l v a r i a n c e s and c o v a r i a n c e s was t o u se
s o l u t i o n s a l r e a d y t r i e d by o t h e r r e s e a r c h e r s t o compare r e s u l t s .
fir st
e s t i m a t e s w ere t h e s o l u t i o n s u t i l i z e d
The
by Hohenboken and B rin k s
62
TABLE 12.
R e la tiv es
PHS
PGS
FS
W ith in FS
COV (0 ,D )
COV ( 0 , S )
COV (MA1N)
COV ( S j MGS)
COV (PAj N)
SFC + PHS
COEFFICIENTS OF COVARIANCES BETWEEN
SOLUTIONS FOR FALCONER’ S (1 9 6 5 ) MODEL
BW
.9 7
.2 6
3 .6 7
1 2 .5 5
2 .8 5
1 .7 3
-.1 7
-.0 0 2 3
-3 .0 6
1 .0 3
WW
2 9 .3 8
1 6 .0 4
2 3 4 .0 9
2 9 6 .2 7
6 2 .2 0
11.5 3
-4 5 .3 3
-5 .3 2
6 .8 5
1 2 6.86
CT2 P
RELATIVES3
AND
m
2(2-m ) CT2 A
CT2 A
1/4
0
1
/8
O0
m2
1/2
I =m2
-1 /2
m
1/2
1
/2
o„
m^
( I +2m)/ 4
( I+2m)/ 8
0
1/4
0
0 ( 1+4(m+m2 ) ) / T 6
0
0
0
0
4
1/4
-4
-1 /4
0
I
I
0
I +4m m/4
0
0
0
I
0
0
a As d e r iv e d by Thompson (197£
b O2 R= CT2M + 2 CT AM + Cf2D + Oe■EM + CT^Eo
fm
—1.0 0
-.5 0
-.2 5
.0 0
.0 5
.0 8
.0 5
1 .0 0
CT2 P
S o l u t i o n f p r B i r t h Wei&ht
CT2 R
CTidA
6 .1 6
11.31
1 4.67
1 6.22
1 6.35
1 7 .4 0
1 3 .7 7
9 .0 1
9-3 7
1 1 .2 5
10.31
6 .1 6
5 .0 6
5 .0 2
.6 4
—.76
-1 .0 0
-.5 0
-.3 7
-.2 5
-.1 2
—.1 0
.0 0
.5 0
1 .0 0
CT2 P
S o l u t i o n f o r Weanine Weight
CT2 A
CT2 R
2 1 8 .6 6
3 9 0 .0 2
4 4 4 .1 4
4 7 9 .4 6
5 1 5 .8 4
5 4 6 .5 2
5 3 0 .3 6
4 1 7 .2 2
2 5 0 .1 8
fm (W eaning W eigh t)
h2
1.52
.9 9
.7 0
.37
.3 0
.2 8
.04
—.0 8
CO
O
fm
Il
fm ( B i r t h W eight)
7 .3 7
2 3 .9 5
2 8 .9 9
4 6 .5 2
4 9 .5 3
4 9 .0 9
3 7 .4 9
9 .4 7
2 6 0 .3 3
2 5 5 .6 9
2 3 2 .3 3
2 3 1 .0 1
1 5 3 .2 0 ,
1 5 7 .3 0
105.41
- 3 8 .3 1
-6 0 .9 6
- .1 0
-7 6 4 .3 5
168.07
4 8 3 .4 8
7 8 1 .7 0
1143.82
1 1 97.78
1413.62
1 4 03.95
5 0 5 .5 6
CT2b
h2
1.1 9
.6 5
.52
.4 8
.29
.2 8
.1 9
-.0 9
-.2 4
TABLE 13.
SOLUTION FOR DIRECT AND MATERNAL COVARIANCES USED BY OTHER WORKERS
• Hohenboken and B rinks'M Q 71) s o lu tio n s
S o lu tio n
CT^flO
OAoAn
O ^Am
O^Do
o DoDm
O ^Eo
O2Eo
rQ
h2o
Ii2Hi
h2T
.46
B ir th w eight
-1 .7 3
9.64
-5 .5 4
- .1 5
.28
.54
- 1 .0 8
-
9 .6 4
-5 .5 4
-.5 1
.28
.72
- 1 .0 8
-
9.64
-5 .5 4
- .1 5
.28
.54
.46
. I HB
3 .8 8
- .8 4
7 .3 9
.6 5
2 HB
3 .8 8
-3 .1 5
9 .7 0
3 HB
3 .8 8
- .8 4
7 .3 9
.
.29
HeaniroL-WeiRhfc
1 HB
117.52
-1 8 8 .9 2
250.86
- 2 38.20
114.16
270.46
80.12
- 1.10
.28
.61
- .0 9
2 HB
117.52
-3 6 .7 0
98.64
-1 2 4 .0 4
-
270.46
80.12
- .3 4
.28
.24
.27
3 HB
117.52
- 188.92
250.86
- 61.71
-
208.13
80.12
- 1.10
.28
.61
- .0 9
(Continues on th e next page)
TABLE 13.
(Continued)
K och's MQ72) s o lu tio n s
S o lu tio n
CT^Ao
0
AoAm
O^Am
Cf
^Dm+Em
Cf
^Do+Eo
rG
h2o
h^m
h2T
B irth w eight
I K
3.88
-.8 4
7.39
-5 .5 4
9.64
-.1 5
.26
.50
.43
2 K
3.88
-1 .9 4
8.49
-5 .5 4
9.64
-.3 3
.26
.58
.35
3 K
4.61
-2 .3 0
8.67
-5 .5 4
9.09
-.3 6
.31
.59
■ .37
4 K
3.88
-.8 4
3.92
-2 .0 7
9.64
-.2 1
.26
.27
.31
5 K
3.88
-1 .9 4
6.67
-3 .7 2
9.64
—#38
.26
.46
.29
6 K
5.42
-3 .9 2
10.08
-5 .5 4
8.48
-.5 3
.37
.69
.31
Weaning w eight
I K
117.52
- 188.92
250.86
80.12
270.46
- 1.10
.22
.47
-.0 7
2 K
117.52
-8 0 .0 4
141.98
80.12
270.46
-.6 2
.22
.26
.13
3 K.
44.93
-43.7,4
123.83
80.12
270.46
-.5 8
.08
.23
.20
I) K
117.52
-188.92
479.18
80.12
270.46
-.7 9
.22
.90
.14
5 K
117.52
-80.04
206.98
15.12
270.46
-.5 1
.22
.39
.19
6 K
16.04
14.03
73.27
80.12
346.57
.41
.03
.14
.14
(Continues on the next page)
TABLE 13.
(Continued)
R elatives used
1 HB
PHS, MHS1 PHS + E rro r
PHS1w ith in FS, MGS, c o v (0 ,S ), cov(0,D)
2 HB
PHS, MHS1 PHS + E rro r
PHS1w ith in FS, MGS, cov(0,D)
3 HB
PHS, MBS, PHS + E rro r
PHS, w ith in FS, MGS, cov(0,S)
1 K
PHS, c o v (0 ,S ), MGS, MHS
2 K
PHS1 Cov(S1MGS) 1 MGS, MHS
3 K
c o v (0 ,5 ), Cov(S1MGS) 1 MGS, MHS
2| K
PHS, c o v (0 ,S ), Cov(O1D) 1 MHS
5 K
PHS1 Cdv(S1MGS) 1 Cov(O1D) 1 MHS
6 K
COv(O1S) 1 MGS1 Cov(O1D) 1 MHS
Kreas e t a l (1979)
B irth w eight
Weaning w eight
a 2Am
O2Ao
a AoAm
4.08
-2 .0 4
1.94
178.48
-110.52
158.42
Relatives used: PHS1 MGS1 Cov(S1MjS).
h2o
h2m
h2T
-.7 2
.30
.14
.14
-.6 5
.42
.37
.22
rG
TABLE 14. SOLUTIONS FOR THE DIRECT AND MATERNAL VARIANCES3 , COVARIANCES3 AND HERITABILITIES
B irth Weight
GLS0
OLSb
Weaning Weight
GLSd
CLSb
GLS0
OLS0 and
fin corrected
GLS1 and
fin corrected
GLSd
a2A0
-B.76±1.3
■3 • 66±1 • O
4 .S it. 8
192.49±57.2
117.06*45.4
189.83*60.6
107.12*46.7
116.25*54.4
04Oab
-3,11±1.7
.-•72±1.5
-2.51i1.1
-189.08i73.3
-54.41±J56.2
-125.81*94.6
-19.32*99.4
-110.26*78.9
o2V
2. 1JIiI .8
.73±1.5
7.13*1.5
206.51*79.0
80. 16±67.6
152.02*94.3
44.02*101.1
321.90*105.5
oX
1.11±.8
•35±.7
-
47.08±37.9
16.03*31.8
38.31*39.7
16.85*32.3
-
oDoDm
2.27±.9
2.03^1.4
-
106.53*42.5
71.59*52.5
87.26*44.0
16.85*46.1
-
° X
2.10±.7
1.72±.9
-
29.38*33.3
49.88*41.3
29.81*34.0
72.18*34.6
-
-
8.98±1.2
9.63±1.0
-
137.92*51.2
216.70*46.5
152.47*51.2
217.04*46.5
-
0 6 O1V
.61±1.0
-.54±1.2
-32.92*46.1
-40.48*55.9
-32.33*35.9
-5.13*36.8
-
o X
°x
- .2 3 i.9
a ?P
.02i1.0
-
72.07*40.3
72.03*48.3
59.53*30.9
46.51*35.9
-
17.90
16.88
g
569.98
528.56
551.19
496.12
S
h2o
.21
.21
.26
.33
.22
.34
.21
.22
h2m
.13
.04
■ .42
.36
.15
.27
.08
.61
h2T
.01
.17
.25
.02
.14
.14
.20
.21
rG
-1.03
-.4 4
-.4 4
-.9 4
-.5 6
-.7 4
-.28
rE
-
-1.23
-
-.3 3
-.3 2
-.3 3
*•05
_a - r -
Jt g i i
.
■
:
dCLS = O d ln a ry L east-S quares
0GLS = G e isra llz e d L east-S q u ares.
Weighted by th e number o f re c o rd s o f e stim a tio n i n th e cov arian ces between
r e la tiv e s .
0GLS = For covariances with a d d itiv es components only
0GLS z A fter co rrectin g by fm
GLS z A fter c o rrectin g by fm
% e r i l a b i l i t i e s calcu lated by using a^P o f GLS^.
-.5 7
-
67
( 1 9 7 1 a ) (HE,
T a b le 1 3 ) .
The s e c o n d g r o u p o f s o l u t i o n s c o n s i s t s o f
t h o s e s u g g e s t e d by Koch ( 1 9 7 2 ) (K, T a b le 1 3 ) . A l m o s t a l l ( 1 8 o u t o f
19) t h e e s t i m a t e s f o r rg i n both t r a i t s w ere n e g a t i v e .
HB2 and HB3 p r o d u c e d r g e s t i m a t e s o f - . 1 5 ,
- . 5 1 and - . 1 5 f o r b i r t h
w e i g h t and - 1 . 1 0 , - . 3 4 and - 1 . 1 0 f o r w e a n i n g w e i g h t .
o p p o s ite to
r e sp e c tiv e ly .
S o lu tio n
HB2 i n c l u d e s
c o v ( 0 , S ) and d o e s n o t u s e t h e c o v ( 0 , S ) .
c o v a ria n ces.
The t r e n d w as
t h e r e s u l t s o f Hohenboken and B r in k s (1971a) f o r
w e i g h t who found v a l u e s o f - . 2 8 , - . 7 9 and - . 2 8 ,
3,
S o l u t i o n s HBI,
The eov(0,D ) was s m a l l e r
weaning
f o r s o l u t i o n s I , 2 and
th e c o v (0 ,D )
but n o t th e
S o l u t i o n HBI i n c l u d e s b o t h
than the c o v (0 ,S ) f o r w eaning
w e i g h t i n Hohenboken and B r in k s (1971a) s tu d y w h i l e t h e cov(0,D ) was
l a r g e r than th e c o v (0 ,S ) f o r both t r a i t s i n th e p r e s e n t stu d y .
A l l t h e s o l u t i o n s i n T a b l e 13 a r e b a s e d on s o l v i n g a c o n s i s t e n t
s y s te m o f e q u a t i o n s o f s i z e no g r e a t e r than 7 by 7.
The e s t i m a t e s o f
t h e v a r i a n c e s and c o v a r i a n c e s f o r d i r e c t and m a te r n a l e f f e c t s a r e th e
unknowns to
so lv e fo r.
S in c e
t h e num ber o f
c o v a r ia n c e s
b etw een
r e l a t i v e s e s t i m a t e d i n t h e s t u d i e s o f H oh en b ok en and B r i n k s ( 1 9 7 1 a )
and Koch (1972) was l e s s than n i n e ,
some o f the unknowns w ere assumed
t o be z e r o i n o r d e r f o r t h e s o l u t i o n t o be c o n s i s t e n t .
is
then c r e a te d
by w h i c h o n e d i r e c t
A dependency
or m a te r n a l v a r ia n c e
a n d /o r
c o v a r ia n c e depends m o s t l y on t h e e s t i m a t e s o f one or more c o v a r ia n c e s
b e tw e e n r e l a t i v e s .
For exam p le
a 2Ao depends on the e s t i m a t e o f PHS
whose e x p e c t a t i o n i n v o l v e s one non z e r o term ( 1 / 4
a 2 Ao) and t h e r e s t
a r e a l l z e r o e s . T h is i s d u e t o t h e f a c t t h a t by u s i n g e l e m e n t a r y row
o p e r a t i o n s i n t o t h e s y s te m o f e q u a t io n s t h e m a t r ix o f th e c o e f f i c i e n t s
i s r e d u c e d t o a l o w e r t r i a n g u l a r m a t r i x ( J o h n s o n and R i e s s ,
1 9 8 1 ).
68
The e q u a t i o n f o r
PHS i s
then m u lt ip lie d
by 4 t o o b t a i n a I i n t h e
f i r s t r o w - f i r s t c o lu m n p o s i t i o n .
The r e s t o f t h e e l e m e n t s i n
f i r s t colum n a r e r e d u c e d t o z e r o e s .
The n e x t e q u a t io n m ust have a non
z e r o term i n th e se co n d r o w - s e c o n d colum n p o s i t i o n .
th e
A ll th e r e m a in in g
e l e m e n t s i n t h e se co n d colum n a r e t h e n r e d u c e d t o z e r o .
T h e r e fo r e , i f
the secon d unknown i s oAoAm, i t s e s t i m a t e w i l l depend on a c o v a r ia n c e
b e tw e e n r e l a t i v e s
whose e x p e c t a t i o n i n v o l v e s a non z e r o term i n th e
s e c o n d r o w - s e c o n d c o lu m n p o s i t i o n and a non z e r o t e r m i n t h e s e c o n d
r o w - f i r s t c o lu m n (c^Ao i s i n v o l v e d i n a l l t h e e q u a t i o n s , T a b le I and
10).
The r e s t o f t h e t e r m s w i l l be z e r o .
a AoAm w i l l depend
Hence,
on t h e e s t i m a t e s o f c o v ( 0 , S ) o r cov(S,M G S) a n d , o f c o u r s e , PHS.
n e x t unknown (a 2 Am) w i l l depend on cov(0,MGS) or MGS, e t c ,
In c o n c l u s i o n ,
c o v a r ia n c e s
T h erefore,
and s o on.
t h e f i r s t unknown depends on o n ly one e s t i m a t e o f
b etw een
th e
The
r e la tiv e s,
in fo r m a tio n
th e
secon d
a v a ila b le
is
not
on
tw o,
e v e n ly
and
so
on.
used i n
th e
e s t i m a t i o n p r o c e s s and e r r o r s i n one e s t i m a t e o f d i r e c t and m a te r n a l
v a r ia n c e s
and
c o v a r ia n c e s
can
cause
th e
oth er
to
d iffe r
in
th e
o p p o s i t e d i r e c t i o n s i n c e th e sum o f com ponents i s f o r c e d t o equal th e
w h o l e (K och ,
1972).
The p r o b le m
is
e v e n m ore s e r i o u s when i t
e r r o n e o u s ly assum ed t h a t a com ponent i s z e r o .
e x p la n a tio n fo r the n e g a tiv e v a lu e fo r
is
T h is s e e m s t o be t h e
C 2Do,O2Dm and C2 Em i n v a r io u s
s o l u t i o n s o f the p r e s e n t stu d y and i n th e one o f Hohenboken and B rin k s
(1971a) when
OgoEm i s
assumed t o be z e r o i n t h e e x p e c t a t i o n o f th e
c o v (0 ,D ). In th e f i r s t s e c t i o n e v id e n c e h as b een r e v ie w e d (T o tu se k ,
1968;
E llic o t
B u rfe n in g ,
et
1972)
a l,
th a t
1 970;
th ere
M angus and B r i n k s ,
e x is ts
a n e g a tiv e
1971;
K ress
and
e n v ir o n m e n ta l
69
a s so c ia tio n
b e t w e e n p r e w e a n i n g g r o w t h o f t h e h e i f e r c a l f and h e r
f u t u r e m a te r n a l a b i l i t y .
was q u a n t if ie d .
In a p r e v i o u s p a r a g r a p h t h i s r e l a t i o n s h i p
The e x p e c t a t i o n s i n T a b l e s I and 10 a l s o show t h a t
EoEmis t h e o n l y p o s s i b i l i t y
s i n c e t h e o t h e r tw o c o v a r i a n c e s h a v e
a l r e a d y been c o n s id e r e d i n t h e s e s o l u t i o n s .
The s i m p l e means o f t h e t h r e e
h^o z .2 8
and . 2 0 ,
and w e a n in g w e i g h t ,
h e r ita b ilitie s
h^mz .5 4 and .4 2 and h
r e sp ec tiv e ly .
^
g
in
Table
13 a r e
and . 0 9 f o r b i r t h
They a r e s u g g e s t i n g t h a t m a te r n a l
a d d i t i v e g e n e t i c v a r i a t i o n i s l a r g e r th an d i r e c t a d d i t i v e g e n e t i c
v a r i a t i o n f o r both t r a i t s .
For b i r t h w e i g h t , t h i s r e s u l t d i s a g r e e s
w i t h t h e r e s u l t s o f E v e r e t t and Magee (1 9 6 5 ),
Koch (1 9 7 2 ),
F i s h e r and
W i l l i a m s ( 1 9 7 8 ) and t h e s o l u t i o n f o r H e r e f o r d s o f Brow n and G a lv e z
(1969).
In t h e c a s e o f w e a n in g w e ig h t th e r e s u l t s a r e s i m i l a r to
t h o s e o f D e e s e and K o g er ( 1 9 6 7 )
B r in k s
(1971a) and Koch (1 9 7 2 ).
(Brahman h e r d ),
(crossbred
h erd ),
H oh en b ok en and
The r e s u l t s o f Deese and Koger (1967)
H i l l e t a l (1965) and V e s e l y and Robison (1971) showed
a h2 o s l i g h t y g r e a t e r th an h2 m f o r w e a n in g w e ig h t or a v e r a g e d a i l y
g a i n t o w ean in g.
The
last
solution
in
Table
13 is
based
on
the independent
estimates of PHS, MGS and Cov(SjjMGS) from the sire-maternal grandsire
model by REML procedures.
expectation.
It only involves additive components in its
It is the solution to the following system of equations:
70
AoAm
3
a
CM
O
O2 Ao
PHS
1/4
O
O
c o v ( S , MGS)
1 /8
1 /4
O
MGS
1/16
1/4
1/4
The s ta n d a r d e r r o r o f t h e e s t i m a t e s o f PHS and MGS a r e t h e square
r o o t o f th e se co n d and t h i r d d ia g o n a l e l e m e n t s o f th e m a t r i x i n Table
11.
No s t a n d a r d e r r o r f o r
m o d e ls
u sin g
t h e e s t i m a t e s o f t h e m ore c o m p l i c a t e d
H en d e rso n ’s
m e th o d I I I o f e s t i m a t i o n o f v a r i a n c e
o
They presum ab ly i n v o l v e e x t e n s i v e m a tr ix
com p onents w e r e a v a i l a b l e .
m a n i p u l a t i o n ( S e a r l e , 1971? p. 4 5 1 ) .
The e s t i m a t e d v a l u e s f o r h2 o ,
h2 m and h 2 T f r o m t h e l a s t s o l u t i o n i n T a b l e 13 w e r e . 3 0 , .1 4 and .1 4
f o r b i r t h w e i g h t and . 4 2 , .3 7 and .2 2 f o r w e a n i n g w e i g h t .
same s o l u t i o n but w i t h nonindep en d en t com ponents,
U sin g th e
K ress e t a l (1979)
f o u n d a v a l u e f o r h2 o o f .1 2 and f o r h2 m o f .0 5 and r g o f - . 6 8 ,
w ean in g w e i g h t .
b ir th
In t h e p r e s e n t s tu d y ,
w e i g h t and - . 6 5
for
w ea n in g
for
t h e v a l u e s o f r Q w ere - .7 2 f o r
w e ig h t.
The r e s u l t s
of
th is
s o l u t i o n show a t o t a l l y d i f f e r e n t p i c t u r e f o r b i r t h w e i g h t a s compared
to th e r e s u l t s o b ta in e d in
th e f i r s t
and s e c o n d s e t o f s o l u t i o n s
r e g a r d i n g t o t h e r e l a t i v e i m p o r t a n c e o f a d d i t i v e d i r e c t e f f e c t s and
a d d i t i v e m a te r n a l e f f e c t s .
i n t h is f in a l so lu tio n .
The v a l u e o f rg f o r b i r t h w e i g h t i s h ig h e r
However, r Q f o r w eaning w e i g h t ( - .6 8 ) i s c l o s e
t o the s i m p l e mean o f t h e
HB and
H o h e n b o k e n and B r i n k s
w eakness
of
th e
te c h n iq u e
K
s o l u t i o n s ( - .7 2 ) .
(1971a) p o in te d
of
out
s im u lta n e o u sly
c o v a r ia n c e s to t h e ir t h e o r e t ic a l e x p e c ta tio n s i s
th a t th e g r e a te s t
e q u a tin g
observed
t h e i n a b i l i t y o f th e
e x p e c t a t io n to a c c o u n t f o r a l l c a u s e s o f e n v ir o n m e n ta l c o v a r ia n c e
71
am ong
r e la tiv e s .
In
order
to
overcom e
th is
p r o b le m
several
c o v a r i a n c e s b e t w e e n r e l a t i v e s w e r e e v a l u a t e d s o t h a t t h e number o f
e q u a t i o n s u s e d was s e v e r a l t i m e s g r e a t e r than a l l p r e v io u s r e s e a r c h on
th e s u b j e c t .
S o l u t i o n s i n T a b le Vl a r e t h e r e s u l t o f u s i n g o r d i n a r y
le a st-sq u a r e s
e stim a to r s
(OLS) and w e i g h t e d o r g e n e r a l i z e d l e a s t
s q u a r e s e s t i m a t o r s (GLS) on th e d e s i g n m a t r i x o f Table 10.
procedure i s
The b a s i c
a m u l t i p l e r e g r e s s i o n method o f r e g r e s s i n g t h e e s t i m a t e s
o f th e c o v a r i a n c e s b e tw e e n r e l a t i v e s on th e c o e f f i c i e n t s o f t h e i r
r e s p e c tiv e e x p e c ta tio n s.
The i n t e r c e p t w a s s e t e q u a l t o t h e o r i g i n
and th e r e g r e s s i o n c o e f f i c i e n t s w ere th e e s t i m a t e s o f th e d i r e c t and
m a t e r n a l v a r i a n c e s and c o v a r i a n c e s .
The w e i g h t s u s e d i n t h e GLS-
s o l u t i o n s w e r e th e numbers o f r e c o r d s i n t h e f i l e
for e stim a tin g
th e
p a r t i c u l a r c o v a r ia n c e b e tw e e n r e l a t i v e s , a s s u g g e s t e d by Van V leck and
H a r t ( 19 6 6 ) .
T hese a u t h o r s i n d i c a t e d
t h e m e th o d
sh o u ld
be v e r y
s i m i l a r t o w e i g h t i n g a c c o r d in g t o t h e i n v e r s e o f t h e v a r i a n c e s o f th e
c o v a r ia n c e s betw een r e l a t i v e s .
S tan d ard e r r o r s o f th e r e g r e s s io n
c o e f f i c i e n t s a r e i n c lu d e d to com pare t h e a c c u r a cy o f th e d i f f e r e n t
e stim a tes.
The f i r s t two colum ns o f b i r t h and w e an in g w e i g h t a r e th e
OLS and GLS s o l u t i o n s u s i n g a l l 37 e q u a t io n s .
th e
case
of
w e a n in g
co rrectin g a l l
w e ig h t are
th e
t h e e x p e c t a t i o n s by fm.
The n e x t tw o columns i n
OLS and GLS s o l u t i o n
a fter
This c o r r e c t i o n c o n s i s t e d o f
i n t r o d u c i n g th e p r o p o r t i o n a l i t y o f e n v ir o n m e n t a l c o r r e l a t i o n i n t o
c o e f f i c i e n t s f o r a AoAm, a 2 Am and
F in a lly ,
th e
e q u a tio n s
e x p e c ta tio n s
la st
(T a b le
so lu tio n
10)
w ith
in
th e
a EoEm a s s u g g e s t e d by Koch (1972).
both
o n ly
tr a its
a d d itiv e s
is
GLS u s i n g
com p on en ts i n
th e
11
th e ir
72
In g e n e r a l ,
the w e ig h tin g procedure in c r e a s e d the c o e f f i c i e n t o f
v a r i a t i o n and r e d u c e d t h e v a l u e f o r
s i m p l e sum o f
for
t h i s w as d e f i n e d a s t h e
th e n in e v a r i a n c e s and c o v a r i a n c e s .
rg were a l l n e g a t iv e .
v a lu e s
o^p.
The e s t i m a t e s f o r
The e s t i m a t e s f o r h^o w e r e g r e a t e r t h a n t h e
h2 m e x c e p t
in
th e
s o lu tio n
based
on
th e
a d d itiv e s
c o m p o n e n t s f o r b o t h t r a i t s and t h e OLS s o l u t i o n f o r w e a n i n g w e i g h t .
The s t a n d a r d d e v i a t i o n s f o r t h e GLS s o l u t i o n w e r e t h e l o w e s t o f a l l
th e e s t i m a t e s o f a d d i t i v e com ponents i n Tab le 14.
w e a n in g w e ig h t a s
th e
one e s t i m a t e d
in
th e
A v a l u e o f fm f o r
present
stu d y
(-.1 0 )
n e g a t i v e l y m o d i f i e s t e r m s l i k e th e cov(S,MGS) and th e c o e f f i c i e n t f o r
okokm i s
reduced
from
.2 5 0
(1 /4 ,
T a b le s
I and
10)
to
.2 2 5 .
The
e x p e c t e d v a l u e o f t h e cov(0,M G D) c o n t a i n s fm a s t h e c o e f f i c i e n t f o r
DoDm (K o c h , 1 9 7 2 ) .
T h erefo re, in order to take th e s e f a c t o r s in t o
a c c o u n t , a GLS s o l u t i o n f o r a d d i t i v e s com ponents a f t e r c o r r e c t i n g th e
cov(S,MGS), cov(0,MGD) and MGD by fm,
The v a l u e s
w ere
was o b t a in e d f o r w ean in g w e ig h t .
a 2 Ao: 1 2 7 .1 8 ± 5 5 . 2 ,
CT2 Am: 341.15 ± 106.31.
The
a AoAm: - 1 3 6 . 4 4 ± 80 .4
c o r r e s p o n d in g
g e n e tic
and
p a r a m e te r s w ere
h2 o= . 2 4 , h^m= . 6 5 , h 2 ^ - , i y and Tq= - . 6 5 .
E isen
(1967)
in d ic a te d
th at
th e d e le t io n
o f any i n d e p e n d e n t
v a r i a b l e w i l l l e a d to b ia s e d e s t i m a t e s o f th e c a u s a l co m p o n en ts o f
v a ria n ce
(p a rtia l
reg re ssio n c o e f f ic ie n t s )
if
there i s
a s ig n ific a n t
c o r r e l a t i o n b e t w e e n t h e d e l e t e d v a r i a b l e and an y o f t h e r e m a i n i n g
in d e p e n d e n t
v a r ia b le s .
c o e ffic ie n ts
for
th e
The
system
of
m a tr ix
of
e q u a tio n s
th e
sim p le
used,
as
c o r r e la tio n
g iv en
by t h e
program, i s shown i n Tab le 15. These c o r r e l a t i o n s c o e f f i c i e n t s m easure
th e
degree
of
a s so c ia tio n
b etw een
th e
c o e ffic ie n ts
of
th e
n in e
TABLE 15. SIMPLE CORRELATION COEFFICIENTS AMONG THE COEFFICIENTS OF DIRECT AND MATERNAL VARIANCES
AND COVARIANCES
Q
ro
s*
O 2 D,
7%
O
1.00
.27
aVm
1.00
.22
.15
.19
1.00
aDoDm
.17
.20
.09
-.1 7
1.00
a2Dm
.03
.46
.52
.16
—. 1 8
1.00
.81
-.0 7
.16
.41
.08
-.1 7
1.00
c t e O e D1
.63
.23
.22
-.0 3
.09
-.2 7
.53
1.00
ro
Bm
CO
CO
.88
Q
1.00
.53
.47
.60
.06
-.2 2
,31
.34
.66
1.00
BW
.83
-.0 3
.21
.4 1
.13
-.0 5
.97
.51
.37
WW
.84
.13
.41
.38
.12
.09
.50
.54
\
o
.
-
W
OO
VO
- X
74
p a ra m eters i n p a ir s , fo r th e p a r t ic u la r e q u a tio n s in v o lv e d in th a t
so lu tio n .
For e x a m p le , a h ig h v a l u e f o r the c o r r e l a t i o n b etw een
and C2t Eo m ea n s t h a t
c o e ffic ie n t for
if
th e c o e f f i c i e n t
f o r O2 Ao i s
la rg e ,
C2t E o w o u ld a l s o be l a r g e f o r t h a t e q u a t i o n .
g r e a t e s t c o r r e l a t i o n s i n T a b le 14 a r e t h e o n e s b e t w e e n
CT2 Eo and b e tw e e n CT2 Am and CTAoAm.
d e s i g n s o f E is e n (1 967).
stu d y
are
lo w er
th an
c o r r e l a t i o n betw een
o v e r estim a ted ,
CT2 Am
ones
and
th e
The
CT2 Ao and
The same t h i n g was o b s e r v e d i n the
In g e n e r a l ,
th e
o^-ko
in
t h e c o r r e l a t i o n s i n th e p r e s e n t
E i s e n ’s
d e s ig n s.
The h i g h
CTAoAm ( . 8 8 ) s u g g e s t s t h a t i f on e i s
th en th e o t h e r w i l l be to o .
This can be th e c a u se f o r
2
CT Am t o be h ig h i n t h e s o l u t i o n f o r a d d i t i v e com ponents.
In d e c i d i n g
c o n sid e r a tio n s
w h ich
sh o u ld
s o lu tio n
be made.
is
th e
F ir st,
m ost
r e lia b le
Seber (1977)
e f f e c t s o f u n d e r f i t t i n g or o v e r f i t t i n g a l i n e a r m odel.
several
a n a ly se d th e
He h a s c l e a r l y
shown t h a t u n d e r f i t t i n g r e s u l t s i n b i a s e d e s t i m a t e s o f t h e v e c t o r o f
p a r a m e te r s ,
th e d i r e c t and m a te r n a l v a r i a n c e s and c o v a r i a n c e s i n t h e
p resen t case.
In stea d ,
o v e r f i t t i n g r e s u l t s i n no b i a s e d e s t i m a t e s o f
th e p a r a m e te r s but l e a d s t o i n f l a t e d e x p r e s s i o n s f o r t h e v a r i a n c e s o f
th e p aram eters (S eb er,
p r e v io u sly d isc u sse d ,
1977, page 143).
T h e r e fo r e , a s E is e n (1967)
th e f i r s t two s e t s o f s o l u t i o n s i n Table 13 are
b i a s e d and t h e OLS hnd GLS s o l u t i o n s have l a r g e r sta n d a r d e r r o r s than
th e tr u e o n e s.
Second,
th e s o l u t i o n s f o r th e 9 p a r a m e te r s i n Table 14
a r e b a s e d on t h e a s s u m p t i o n t h a t t h e e r r o r t e r m s f o r t h e s i r e - t y p e
r e l a t i v e r e la t io n s h ip s c o n ta in
CTEoEm
based on th e f a c t t h a t t h o s e f i l e s
and O2 Em.
This a s s u m p tio n i s
c o n t a in e d 25 % o f f u l l - s i bs.
More
t h a n 2 / 3 o f t h e c o w s i n t h e p r e s e n t s t u d y had t w o o r m ore c a l v e s . I f
75
EoEm and
2 Em a r e
not in v o lv e d in
c o v a r ia n c e i n w h ich
(X1X) t o
w ords,
EoEm i s
be i l l - c o n d i t i o n e d
a sm a ll
so lu tio n s .
change i n
present i s
(S eb er,
th e
d ata
c o v (0 ,D ).
1977,
T h is p r o d u c e s
p. 3 1 9 - 3 2 2 ) .
provokes a b ig
th e
change in
e q u a tio n s
by t h e
th e
number o f
th e v a r i a n c e s and c o v a r i a n c e s o f
th e c o v a r ia n c e s b e tw e e n r e l a t i v e s a r e n ot hom ogeneous.
ten d ed
to
l i t e r a t u r e r e v ie w e d ,
la st
In o th e r
He s u g g e s t e d t h a t t h i s p r o c e d u r e proposed by Van V leck and
Hart (1966) i s n o t f u l l y e f f i c i e n t i f
so lu tio n s
th e o n ly
The t h i r d p o i n t r e s i d e s i n t h e o b s e r v a t i o n o f Thompson
(1976) w ith r e s p e c t to w e ig h t in g
records.
th o se e x p e c ta tio n s,
c o n sid e r a tio n
produce v a lu e s
far
b elo w
th e
e s p e c i a l l y f o r w ean in g w e i g h t .
is
r e la te d
to
th e
fa ct
th at
The GLS-
average
The
th ere
of
th e
f o u r t h and
w as
c le a r
e v id e n c e f o r t h e e x i s t e n c e o f a path b e tw e e n m a te r n a l phenotype o f th e
dam and d a u g h t e r ( fm ) i n t h i s s t u d y .
r e f le c t th a t e ff e c t.
T hus, th d e x p e c t a t i o n s s h o u l d
The OLS and GLS s o l u t i o n s t h a t w e r e c o r r e c t e d
f o r fm f o r w e a n in g w e i g h t ta k e t h a t i n t o a c c o u n t.
P u ttin g i t
a ll
togeth er,
it
s e e m s t h a t t h e OLS-fm c o r r e c t e d
s o l u t i o n f o r w e a n in g w e i g h t i s perhaps th e m o st r e l i a b l e i n the stu d y .
I t i s n o t c l e a r w h ich s o l u t i o n i s t h e c o u n t e r p a r t f o r b i r t h w e ig h t .
H ow ever,
t h e s o l u t i o n s i n T a b le 14 s e e m s t o be m ore r e l i a b l e
than
t h o s e s o l u t i o n s i n T a b l e 13. I n su m m a r y , m a t e r n a l e f f e c t s a f f e c t e d
b irth
and w e a n i n g
w e ig h t
of
beef
c a lv e s
in
th is
H erefo rd
herd.
M aternal e f f e c t s i n w ean in g w e i g h t o f b e e f c a t t l e have a g e n e t i c and
en v iro n m en ta l o r ig in .
m a te r n a l e f f e c t s i s
In b i r t h w e ig h t i t seem ed t h a t th e b a s i s f o r
m o stly g e n e tic .
were not o n ly a d d it iv e in t h i s stu d y .
It is
c le a r th at g e n e tic e f f e c t s
D o m in a n c e w as a l s o i n v o l v e d .
76
However,
the la r g e
valu e
of
aDoDm a s compared t o
s u g g e s t e d t h a t so m e e p i s t a s i s w o u ld be i n v o l v e d .
o r ig in a l
m odel
c o e ffic ie n ts
in
r e la tiv e s of
c o n ta in e d
th e
term s
exp ected
fo r
v a lu e s
a ^Do
O2Dm
I
W i l l h a m ’s ( I 96 3 )
e p ista s is .
of
th e
and
H ow ever,
c o v a r ia n c e
th e
betw een
th e dom inance com ponents a r e h i g h l y c o r r e l a t e d t o t h o s e
of e p ista sis.
In s t a t i s t i c a l
te r m s t h i s i m p l i e s
to rep a ra m eterize a
r e g r e s s i o n m o d e l w i t h 9 p a r a m e t e r s t o a m o d e l w i t h 12 p a r a m e t e r s
h i g h l y c o r r e l a t e d among t h e m s e l v e s .
Then, t h e s e 12 p a r a m e te r s have t o
be e s t i m a t e d w i t h 37 e q u a t i o n s a t m ost.
A d i r e c t path b e tw e e n t h e m a te r n a l phenotype o f t h e dam and t h e
m a t e r n a l p h e n o t y p e o f t h e d a u g h t e r a s s u g g e s t e d by Koch ( 1 9 7 2 ) w as
a l s o found f o r b oth t r a i t s .
Even t h o u g h t h e v a l u e f o r b i r t h w e i g h t
(.08) was c l o s e i n a b s o l u t e v a l u e t o t h e e s t i m a t e f o r w ean in g w e i g h t
(-.1 0 ),
its
b io lo g ic a l
m e a n in g i s
not
c le a r .
A d d itiv e
m a tern a l
e f f e c t s se em s t o be l e s s im p o r t a n t f o r b i r t h than f o r w ean in g w e ig h t .
The l a t t e r
seem s
t o be o f th e same m agnitude or perhaps even l a r g e r
than a d d i t i v e d i r e c t e f f e c t s ( s e e h2 o and h2 m i n T ab le 14).
of
th e
c o v a r ia n c e
b etw een
a d d itiv e
g e n e tic
d ir e c t
and
The s i g n
a d d itiv e
m a te r n a l e f f e c t s was c l e a r l y n e g a t i v e f o r both w e i g h t s i n t h i s stu d y.
The n a tu r e o f th e d a t a u s e d and t h e r e s t r i c t i o n s o f th e a n a l y s e s make
it
d i f f i c u l t to su g g e st a h ig h ly a c cu ra te va lu e,
w e ig h t.
H ow ever,
sm a ller
than - . 5 0 ,
e s p e c i a l ly for b irth
t h e v a l u e f o r r Q o f w e a n i n g w e i g h t s e e m s t o be
p o ssib ly
b etw een - .6 5
and - . 7 5 .
A p r a c tic a l
i m p l i c a t i o n o f t h i s v a l u e i s t h e r e d u c t i o n i n th e e x p e c t e d r e s p o n s e t o
se le c tio n
for
w e an in g w e i g h t
th e o r e tic a lly
shown by Van V le c k e t a l
77
( 197 7 ) and o b s e r v e d i n
th ose s e le c t io n
e x p e r im e n ts in b eef c a t t l e
r e v ie w e d by Koch e t a l (1982).
A f i n a l s u g g e s t i o n f o r f u t u r e r e s e a r c h on t h e s u b j e c t i s r e l a t e d
t o th e e x p e r im e n t a l d e s i g n t o stu d y m a te r n a l e f f e c t s i n b e e f c a t t l e .
Willham (1980) i n d i c a t e d t h a t r a t h e r than u s i n g f i e l d d a ta t o e s t i m a t e
th e d i r e c t and m a te r n a l v a r i a n c e s and c o v a r i a n c e s ,
d e s ig n e d p r o j e c t s
t h a t y i e l d s p e c i f i c u s e f u l c o v a r i a n c e s t h a t a r e u n c o r r e l a t e d s h o u ld be
attem p ted .
C o n s i d e r a t i o n s c o m m e n te d i n
th e l i t e r a t u r e r e v ie w
in d ic a t e th a t t h i s i s a d i f f i c u l t ta sk w ith b eef c a t t l e .
c o v a r ia n c e s l i k e
a lso
H o w e v e r,
cov(S,MGS) i n a d e s i g n i n w h ich b u l l s a r e t e s t e d a s
s i r e s and m a te r n a l grand s i r e s a r e a p p e a lin g .
The c o n d i t i o n i s
th at
o n l y t h o s e y e a r s i n w h ich t h e r e a r e progeny and grand progeny sh ou ld
be i n c l u d e d .
T h i s i s o n l y an e x a m p l e .
In th e f u t u r e ,
a p p l i c a b i l i t y o f t e c h n i q u e s l i k e embryo t r a n s f e r ,
th e g en era l
can h e l p th e a n im a l
b r e e d in g r e s e a r c h e r t o have a more c o m p r e h e n s iv e u n d e r s ta n d in g o f t h i s
i n t r i g u i n g problem o f . m a t e r n a l e f f e c t s i n b e e f c a t t l e .
78
SUMMARY
B irth
and w e an in g w e i g h t s o f it,423 n o n c r e e p - f e d H ereford c a l v e s
r a i s e d a t t h e N o r t h e r n A g r i c u l t u r a l R e s e a r c h C e n t e r , H a v r e , Mt fr o m
1 9 3 8 t o 1 983 w e r e u s e d t o s t u d y m a t e r n a l e f f e c t s o v e r b o t h t r a i t s .
C a l v e s w e r e p r o d u c e d o u t o f 2 0 2 b u l l s and 1 ,2 7 1 c o w s .
The h e r d w a s
k e p t i n r a n g e c o n d i t i o n s d u r i n g t h e summer and s u p p l e m e n t e d o n l y
d u r in g th e w i n t e r .
The b a s i c m o d e ls i n c l u d e d f i x e d e f f e c t s o f l i n e - y e a r , age o f dam,
s e x , ag e o f dam by s e x i n t e r a c t i o n and t h e r e g r e s s i o n o f b i r t h w e i g h t
on b i r t h d a t e o f c a l f and w e an in g w e i g h t on w eaning a g e o f c a l f .
r e l a t i v e r e l a t i o n s h i p s w e r e t h e random e f f e c t s .
The
The c o v a r i a n c e s
b e t w e e n r e l a t i v e s w e r e e s t i m a t e d by t h e f i t t i n g c o n s t a n t s m e th o d
( H e n d e r s o n , I 9 5 3 ) and w e r e
p a t e r n a l h a l f - s i b s , m a te r n a l grand s i r e
s ib s,
sib s,
p atern al
grand s i r e
m a te r n a l grand dam s i b s ,
c o u sin s,
fu ll-sib s,
p atern al h a lf - s ib s
m a tern a l
g r e a t grand s i r e
m a te r n a l h a l f - s i b s ,
sib s,
sin g le f i r s t
p l u s f u l l - s i b dams, p a t e r n a l h a l f - s i b s
p l u s p a t e r n a l h a l f - s i b s dam s and f u l l - s i b s p l u s p a t e r n a l h a l f - s i b
p aren ts.
The s i r e
(m atern al
grand
(p a tern a l
sir e
sib s)
h a lf-sib s)
com p on en ts
and m a t e r n a l
were
a lso
grand s i r e
e stim a te d
by
R e s t r i c t e d Maximum L i k e l i h o o d a s o u t l i n e d by J e n n r i c h and Sam pson
( 1 9 7 6 ) i n a t w o - w a y c r o s s c l a s s i f i e d random m o d e l w i t h 1 ,7 7 4 c a l v e s
o u t o f l i n e 4.
The c o v a r ia n c e b e tw e e n t h e two random e f f e c t s s e r v e d
a s an e s t i m a t e o f th e c o v a r ia n c e s i r e - m a t e r n a l grand s i r e .
(1965) model was a l s o f i t t e d
F a lc o n e r 's
t o o b t a i n an e s t i m a t e o f t h e c o r r e l a t i o n
b e t w e e n m a t e r n a l p h e n o t y p e o f t h e d a u g h t e r and dam f o r b i r t h and.
79
w eaning w e i g h t (fm p a th ,
Koch,
1972).
The p roced u re u sed was l e a s t -
squares m u lt ip le r e g r e s s io n o f th e c o e f f i c i e n t s
(fo r
th e p h en o ty p ic
v a r i a n c e , a l i n e a r and a non l i n e a r te r m f o r t h e a d d i t i v e v a r i a n c e , ,
and a term i n c l u d i n g
estim a te s
of
su g g ested
dom inance and e n v ir o n m e n t a l d e v i a t i o n s )
on th e
th e c o v a r i a n c e s b e tw e e n r e l a t i v e s f o r both w e i g h t s ,
by
Thom pson ( 1 9 7 6 ) .
The
e q u a tio n s
were
so lv e d
as
for
d i f f e r e n t v a l u e s o f fm ( - 1 . 0 , - . 5 , 0 , .5 and 1.0) and i t e r a t i o n s w e r e
made t o a c o n v e r g e n c e v a l u e o f t h e e s t i m a t e d h e r i t a b i l i t y
p a te r n a l-h a lf
sib s
for
b ir th
and w e a n i n g w e i g h t .
m a te r n a l s o u r c e s o f v a r i a t i o n t o
( h 2 ) by
The d i r e c t and
G2-Ao,
be e s t i m a t e d w ere:
a d d itiv e
g e n e t i c d i r e c t v a r i a n c e ; o 2 km, a d d i t i v e g e n e t i c m a t e r n a l v a r i a n c e ;
oAoAm,
c o v a r ia n c e betw een a d d it iv e s
effects;
O2Do,
g e n e tic
d i r e c t dom inance v a r ia n c e ;
d ir e c t
O2Dm,
and m a t e r n a l
m a te r n a l dominance
v a r i a n c e ; ODoDm, c o v a r i a n c e b e t w e e n d i r e c t and m a t e r n a l d o m in a n c e
effects;
O2 Eo, random e n v i r o n m e n t o v e r t h e d i r e c t p h e n o t y p e ; O2 Em,
m a te r n a l e n v ir o n m e n t a l v a r i a n c e or v a r i a t i o n common t o m a te r n a l h a l f s i b s and f u l l - s i b s and
and m a t e r n a l
oEoEm, e n v ir o n m e n t a l c o v a r ia n c e b e tw e e n d i r e c t
effects.
T hese n in e
p aram eters
were
e stim a te d
by
o r d i n a r y and w e i g h t e d l e a s t - s q u a r e s m u l t i p l e r e g r e s s i o n t e c h n iq u e s
w here
th e
e stim a te s
of
th e
c o v a r ia n c e s
betw een
r e la tiv e s
were
r e g r e s s e d on t h e e x p e c t e d c o e f f i c i e n t s f o r t h e d i r e c t and m a t e r n a l
v a r i a n c e s and c o v a r i a n c e s . The w e i g h t s u sed w ere th e number o f r e c o r d s
of
e stim a tio n
for
th e
p a r tic u la r
c o v a r ia n c e
b etw een
r e la tiv e s
c o n sid er e d .
H e r i t a b i l i t y e s t i m a t e s w ere .28 f o r both w e i g h t s through p a t e r n a l
h a l f - s i b s and .4 5 and .8 8 t h r o u g h f u l l - s i b s f o r b i r t h
and w e a n i n g
80
w e ig h t,
r e sp e c tiv e ly .
R e p e a ta b ility
w e i g h t and .32 f o r w e a n in g w e i g h t .
e stim a te s
w e r e .21 f o r b i r t h
H e r i t a b i l i t i e s through r e g r e s s i o n
o f o f f s p r i n g on dam and o f f s p r i n g on s i r e w e r e .4 5 and .21 f o r b i r t h
w e i g h t and .2 8 and .0 6 f o r w e a n i n g w e i g h t .
G e n e t i c , p h e n o t y p i c and
e n v i r o n m e n t a l c o r r e l a t i o n s w e r e e s t i m a t e d t o be . 5 6 , .41 and .3 6 f o r
b i r t h w e i g h t and .5 4 ,
.38 and .41 f o r w e an in g w e i g h t ,
r e sp e c tiv e ly .
A ll s o l u t i o n s show ed a n e g a t iv e c o r r e l a t i o n b e tw e e n a d d i t i v e
d i r e c t and a d d i t i v e m a t e r n a l e f f e c t s ( r G).
from -.3 6 t o - I .03
w eig h t.
E s tim a te s o f r G ranged
f o r b i r t h w e i g h t and from - .2 8 t o - . 9 4 f o r weaning
The e s t i m a t e s o f h e r i t a b i l i t y
for
d ir ec t
effects
(h2 o ),
h e r i t a b i l i t y f o r m a t e r n a l e f f e c t s ( h 2 m) and h e r i t a b i l i t y f o r t o t a l
a d d i t i v e g e n e t i c e f f e c t s ( h 2 T) w e r e .21 t o . 2 6 , .0 2 t o .4 2 and .01 t o
.2 5 f o r b i r t h
w e i g h t and
w e a n in g w e ig h t .
.2 1
to .3 4 ,
D o m in a n c e a l s o a f f e c t e d b o t h d i r e c t and m a t e r n a l
e f f e c t s on b i r t h and w e a n i n g w e i g h t .
r e s p e c t t o th e p r o d u c t o f
be a l s o i n v o l v e d .
.0 8 t o .6 1 and .0 2 t o .21 f o r
O2Do and
L a r g e r v a l u e s o f ODoDm w i t h
O 2Dm s u g g e s t t h a t e p i s t a s i s may
The e n v ir o n m e n t a l c o v a r ia n c e ( OEoEm) was c l e a r l y
n e g a t i v e f o r w e a n in g w e i g h t ( - 5 .1 3 t o - 4 0 .4 8 k g 2 ).
T h is r e s u l t was
s u p p o r t e d by a n e g a t i v e e s t i m a t e o f fm ( - . 1 0 ) f o r w e a n i n g w e i g h t .
T h is e f f e c t i n v o l v i n g m a te r n a l p h e n o ty p e s o f dam and d a u g h te r was l e s s
c le a r
for
b ir th
w eig h t
w here
fm
was p o s i t i v e
( .0 8 )
but th e
tw o
e s t i m a t e s o f OEoEm w ere n o t i n c l o s e a g r e em e n t (.61 and - . 5 6 kg2).
The r e s u l t s
o b ta in e d
in
th is
stu d y
in d ica te d
t h a t oAoAm i s
l a r g e l y n e g a t i v e f o r b o t h t r a i t s c a u s i n g g e n e t i c i m p r o v e m e n t t o be
m ore d i f f i c u l t (Van V l e c k e t a l ,
1977). R e s u lt s a l s o i n d i c a t e d t h a t
81
e n v i r o n m e n t a l c o v a r i a n c e b e t w e e n m a t e r n a l phenotype o f th e dam and
d a u g h te r was n e g a t i v e f o r w e an in g w e i g h t .
82
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and c o n t r o l b e e f
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APPENDIX
Derivation
of
th e
e x p e c t a t i o n s f o r t h e m ate r na l grand dam fMGD) and
mate rna l grand s i r e (MGS) components bv u s i n g oat h c o e f f i c i e n t s
F ig ur e 4 d e s c r i b e s th e phenotype o f two g r a n d o f f s p r i n g under the
m a te r n a l e f f e c t model (Koch,
1972).
The a b b r e v i a t i o n s a r e d e f i n e d by
Koch ( 1 9 7 2 ) . Po i s t h e p h e n o t y p i c , Go i s t h e a d d i t i v e g e n e t i c , Do i s
t h e dominance and Eo i s
th e e n v i r o n m e n t a l v a l u e f o r
weaning w e ig h t e x p ressed
c o r r e s p o n d i n g m a te r n a l
values
and d o u b l e
coefficien t
by i n d i v i d u a l s .
effects.
prim es r e p r e s e n t
co efficien ts.
Cm,
Dm and Em a r e
Primes on v a l u e s r e p r e s e n t p a r e n t a l
be tw ee n s y m b ol s (.5,
reg ressio n
Pm,
b i r t h w e i g h t or
g,
d,
D ouble
grandparent
e,
p,
valu es.
Path
fm) a r e s tan dar d p a r t i a l
arrows
represent
resid u al
c o r r e l a t i o n s betwe en t r a i t s .
P0 I and
P0 2 a r e
grandsire sib s.
(g e n o t y p e
of
eith er
m aternal
granddam
G11 and Gt2 a r e th e g e n o t y p e s o f
m aternal
grand
dam or
sib s
or
m aternal
the d a u g h te r s o f Gm
a m a te r n a l grand s i r e ) .
The
c o r r e l a t i o n be tw ee n P0 ^ and Po2 i s
r ^pOl >'p o 2 ) = V 1 6 g 02 + 1 / 4 r G g0 gm + 1 / 4 gm2 Pm2
but g D2
= O2 Ao/ O2 P8 r G g0 g m = ( OAoAm
and g^2 pm2 = o2Am/ o 2 p , (Koch, I 9 7 2 ) .
s i d e s o f ( I ) by
OAo
OAm)/( OAo
OAm O2 P)
T h e r e f o r e , on d i v i d i n g b o t h
O^ps the c o v a r i a n c e be t w e e n P0 1 i s
c o v ^ p O l » p o 2 ) = 1 / 1 6 O 2 Ao + 1 / 4 .
(I)
o A o Am + 1 / 4
C 2 Am
90
T h i s t e r m i s common f o r MGD and MGS.
I f t h e p a t h (f m ) b e t w e e n
m a t e r n a l p h e n o t y p e s o f t h e g r a n d dam ( P nIn) and o f t h e d a u g h t e r (P'm)
e x i s t s the n
r ^ o V ^oE^ = ^/ 1 6 g o 2 + 1/ 4
r G go gm pm + 1/iJ
gm2
+ pm2 fm2 +
+ 1/2 pm2 fm gm2 + 1/1} pm fm g0 r G gm
= 1 / 1 6 2 Ao + 1/1} oAoAm + 1/1} a2 Am + pm2 f m 2 +
+ 1/2 pm^ fm
a 2 Am + 1/1} pm fm a AoAm
and
c o v (P o 1 * po2) = 1/16
+ 1/2
= 1/16
(J2Ao + 1/1} crAoAm + 1 /4
O2 Am + pm2 fm2 o 2 P +
O2 Am fm + 1/4 fm oAoAm
O2 Ao + ( 1 / 4 + 1 / 4 fm) oAoAm +
+ ( 1 / 2 + 1 / 2 fm) O2 Am
I f fm = - . 1 , a s i t w as e s t i m a t e d i n t h e p r e s e n t s t u d y , f m 2 = . 0 1 .
The t e r m pm2 i s t h e v a r i a n c e o f m a t e r n a l p h e n o t y p e s i n " a b s e n c e o f
a d d itive gen etic
(1965).
effects
among t h e dams",
a s d e f i n e d by F a l c o n e r
B e c a u s e o f t h e way t h i s v a r i a n c e (pm2 ) i s d e f i n e d , i t i s n o t
e x p e c t e d t o be l a r g e .
Therefore,
th is
term i s
approximate d i f f e r e n c e b e t w e e n MGD and MGS i s
= 1/4 fm oAoAm + 1 / 2
fm O2 Am
= fm ( 1 / 4 oAoAm + 1/2 O2 Am).
n e g l i g i b l e and t h e
91
F i g u r e it.
A path c o e f f i c i e n t
between ma ternal grand progeny.
diagram
d e s c r ib in g the covariance
MONTANA STATE UNIVERSITY LIBRARIES
llllllllilllliill
3 1762 10013199 2
N378
C166 C eatet, R. j . c.
c o p. 2
E s t i m a t i o n o f m at er na l
e f f e c t s on b i r t h and . . .
WAiN Li&
«378
C166
c o p. 2