Joint Sire and Cow Evaluation for Conformation Traits
Using an Individual Animal Model
K. M E Y E R 1 and E. B. B U R N S I D E
Centre for Genetic Improvement of Livestock
Department of Animal and Poultry Science
University of Guelph
Guelph, Ontario, Canada N1G 2W1
ABSTRACT
83% when dam's index was replaced by
dam's phenotypic record.
A joint sire and cow evaluation for
selected type traits has been carried out
for the Canadian Holstein population
using an individual animal model accounting for all known relationships.
Only first lactation, first classification
linear records were utilized. Linear scores
for final class/final score (combined), feet
and legs, mammary system, and stature
were analyzed. The model of analysis
included animals' additive genetic merit
as random and herd-round-classifier subclasses as fixed effects. Calving age and
stage of lactation at classification were
fitted as linear and quadratic covariables.
There were 282,030 cows with records,
198,871 dams without records, and 8481
sires (i.e., 489,382 animals in total).
Combined with 33,968 herd-round-classifier effects in 9654 herds and four
regression coefficients, a system of
523,354 equations required simultaneous
solution. Solutions were obtained iteratively for one trait at a time. The computing strategy utilized is described, and a
detailed worked example is given. Correlations between sire proofs and the
current Holstein sire proofs were .89 to
.96 for sires with at least 20 daughters
in five or more herds. Correlations
between cow indexes and phenotypic
scores ranged from .53 to .76. The coefficient of determination in a multiple
regression of daughter's index on dam's
index and sire's and maternal grandsire's proof was 78 to 92% but only 59 to
INTRODUCTION
Received October 17, 1986.
Accepted October 13, 1987.
~lnstitute of Animal Genetics, Edinburgh University, West Mains Road, Edinburgh EH9 3JN,
Scotland.
1988 J Dairy Sci 71:1034-1049
The genetic evaluation of dairy cattle by
BLUP (3), has been concerned chiefly with the
ranking of bulls based on performance of their
female progeny. The simplest model invoked
has been the so-called "sire model" (SM). With
the SM, records on cows are utilized only to
estimate half their sires' genetic merit. This
model ignores both the dam of the cow, i.e.,
the sire's mate, and any relationships between
females. As a result, sire proofs may be biased
due to nonrandom mating or selection of cows.
More detailed models like the maternal grandsire model (MGSM) (17) or a model including
dam's record(s) or estimated genetic merit as
covariable(s), account for such bias only to a
limited extent. With such models of analysis
(SM or MGSM), cows are commonly evaluated
in a second step, incorporating sires' estimated
transmitting abilities, using for instance, a
within-herd BLUP (4) or a selection index (7)
procedure.
Conceptually, the simplest model is a
breeding value or individual animal model
(IAM) where each record provides information
about the additive genetic merit of the animal
on which the measurement has been taken. This
has been described by Henderson and Quaas (6)
initially as a model to utilize information from
relatives and for multiple traits.
The IAM accounts for all relationships
between animals. Thus, genetic evaluations for
animals without records are obtained from their
relatives records. In the dairy cattle situation,
for instance, these include all sires. Males and
females are evaluated simultaneously to adjust fully for nonrandom mating. In addition,
any selection of cows based on the trait(s)
under analysis is taken into account, provided
all records on which selection decisions were
1034
INDIVIDUAL ANIMAL MODEL FOR TYPE
based are available. In a simulation study (9),
the SM, MGSM, or SM with regression on dam's
predicted merit was 43 to 47% less accurate in
evaluating sires than the IAM for a random
mating but selected population and 65 to 67%
less accurate under assortative mating and
selection.
Although the IAM is clearly the model of
choice in genetic evaluation, its practical
application has been limited by the computational resources required. For large populations
or multiple trait evaluation the resulting set of
mixed model equations (MME) to be solved
simultaneously can be substantial. Furthermore, diagonal elements in the MME are small
compared with those under a SM or MGSM,
and slow rates of convergence have been
reported (1, 16). For cattle data, the IAM has
been used on a within-herd basis (e.g. 16, 20,
35), but it has been considered prohibitively
expensive for large data sets (9).
Westell (32) obtained genetic evaluations
using an IAM, based on first lactation milk
records, for just over 1 million cows and 6000
sires. This involved solving a system of almost
1.75 million equations. In solving, 30 rounds
of iteration, each requiring at least 4 h and
12 rain computing time, were completed,
before solutions approached convergence.
Westell concluded that "a joint evaluation is
feasible in terms of time and m o n e y " (p. 46).
To date, the IAM accounting for all relationships between animals is not used for the
routine evaluation of dairy cattle anywhere in
the world. One of the most sophisticated
systems currently in use is the Australian
so-called "all lactation cow and bull BLUP" for
production traits. Under this model, however,
relationships between females across herds are
ignored, which implies that cows changing
herds are treated as different animals (L. P.
Jones, personal communication).
Holstein Canada carries out a type classification program for its members. Herds are
visited in a 9 to 10-mo cycle (round of classification). Cows are scored for a total of 27
conformation traits on a linear scale in 9 or 18
categories. Sire proofs are obtained for each
trait by BLUP under a SM that includes joint
herd-round-classifier (HRC) effects as fixed
effects. In addition, dams' scores for nine traits
are included as linear covariables in an attempt
to account, partially at least, for assortative
1035
mating. Data are precorrected for the effects of
age at calving and stage of lactation at classification [see (21) and (23) for further details].
Records from all lactations are utilized, but for
each cow only her most recent classification is
considered. As cows may be reclassified selectively and as breed policy does not allow
cows to be downgraded upon reclassification,
this may bias sire rankings. At present, cows are
ranked phenotypicaUy only.
The objectives of this study were to investigate the feasibility of a j o i n t sire and cow
evaluation for conformation using an IAM in
the Canadian Holstein population.
MATERIAL AND METHODS
Data consisted of all type classification
records (n = 476,329) on Canadian Holsteins
collected since the introduction of the linear
scoring system in August 1982. Sequential edits
eliminated records in second and later lactations or records with missing or invalid
calving dates. Age at calving was required to be
in the range of 18 to 40 too. Stage of lactation
at classification was truncated at 395 d. Excluding reclassifications and records with
double cow identities then left a final data set
of 282,030 first lactation, first classification
records.
Due to the short period considered (less than
4 yr), only 8% of cows with records had daughters in the data (n = 22,474). Most daughterdam pairs were in the same herd (n = 18,621)
while about one sixth were split over herds (n =
3583). For 3.8% of the cows, dam identities
were u n k n o w n (n = 10,806). Including 198,871
dams without records gave a total of 480,901
females to be evaluated. Of these dams without
records, 93.9% had daughters in only one
herd (n = 186,707), and 3.7% had two daughters in different herds (n = 7347), while the
remaining 2.4% had more than two daughters in
more than one herd or were dams of bulls (n =
4817).
Pedigrees for 4231 of the 8269 sires of cows
were obtained from the Holstein pedigree file
held at the University of Guelph. A total of 415
bulls had both male and female progeny and 8
bulls had more than 100 sons each. Only
4 bulls had dams with records. Including 212
sires with sons, but not daughters in the data,
increased the n u m b e r of bulls to 8481. This
Journal of Dairy Science Vol. 71, No. 4, 1988
1036
MEYER AND BURNSIDE
yielded a total of 489,382 animal effects to be
taken into account.
The mixed model of analysis (Model 1):
Yijk = HRC i + aj + bl (Xliik - -X1) +
b2 ( X l i j k - X1) 2 + b3 (X2ijk X2) + b 4 (X2ijk - X 2 ) 2 + e i j k
[1]
included herd-round-classifiers, HRC~ as fixed
effects, and the additive genetic merit of
animals (ai) and a residual error (eijk) as random effects. Age at calving (X1 ijk) and stage of
lactation at classification (X2ijk) were taken
into account by fitting each as a linear and
quadratic covariable.
For the purpose of analysis, ratings for final
class and final score were combined as described by Schaeffer (21). All traits were then
transformed to "objective scores" on a scale
from 0 to 100, using a transformation suggested
by Snell (25) to make residuals for categorical
traits approximately normally distributed with
homogeneous variance. Its application to cattle
data has been considered by Tong et al. (29).
The BLUP sire proofs and cow indexes were
obtained for final class/final score (combined),
feet and legs, mammary system, and stature.
Ratios of residual to additive genetic variance
(= X) used were 4.89, 9.05, 5.65, and 1.38,
respectively, corresponding to heritability estimates of .17, .10, .15, and .42 (21). In addition, corresponding sire proofs were determined under a SM (Model 2) accounting for
relationships between sires and maternal
grandsires. Apart from replacing aj above by sj,
half the animal's sire's genetic merit, Model 2
was identical to Model 1 given earlier. Sire
evaluations from the two models were contrasted to the current official Holstein proofs
(July 1986).
COMPUTING STRATEGY
Choice of an efficient computing strategy
and careful programming are crucial factors
determining whether or not evaluation under
IAM is feasible for a large population. Procedures utilized earlier for BLUP evaluations
under SM usually involved forming the MME
for sires absorbing herd-year-season or similar
effects, and iterative solution using Gauss-Seidel
iteration with successive overrelaxation (SOR).
Often, this required storage of intermediate
results on auxiliary devices and consequently
Journal of Dairy Science Vol. 71, No. 4, 1988
extensive input and output operations in
numerous computing steps (19). The improvement of computational resources available has
since stimulated the development of new
strategies. Generally these demand larger
amounts of core space but less auxiliary storage
and fewer computational steps. Suggestions
range from the use of "hash-storing", or "linkedlist" procedures for in-core iteration [R. L.
Quaas, personal communication, (13)] to an
indirect approach that avoids setting up the
MME altogether (24). Alternative solution
schemes that have been investigated included
block iteration (with SOR or utilizing a triangular decomposition), the method of conjugate
gradients, various forms of Jacobi iteration, and
the use of a QR algorithm (2, 8, 12, 14, 15, 31,
32).
The computing strategy pursued in this
study has been determined by two main criteria:
1. with single records per cow connections
between herds arise only from relationships
between animals, i.e., daughter-dam pairs split
over herds. 2. herds impose a natural block
structure on the MME; nonzero elements of the
coefficient matrix tend to form clusters for
herds. The rate of convergence of a system of
equations can be improved by the use of block
iteration techniques if the equations can be
partitioned so that the dominant coefficients
are in the diagonal block (18). This has been
utilized by Westell (33), who, using GaussSeidel iteration with SOR, in each round of
iteration obtained solutions for one herd at a
time in three to five "internal" iterates.
Adopting Westell's (33) strategy, the MME
were partitioned into equations pertaining to
individual herds and other equations. Designating parents without own records to be
"within" or "across" herd animals attempted to
minimize connections between within herd
animals in different herds.
The equations for each herd comprised 1)
cows with records in the herd; 2) dams (without records) with daughters in one herd only;
3) dams (without records) with two daughters
in different herds; 4) sires with daughters in one
herd only, with parents unknown and without
sons; and 5) HRC effects. The maximum
number of records per herd was 357 and the
maximum number of equations per herd was
629. Connections between herds were then
only due to cows with daughters in other herds
INDIVIDUAL ANIMAL MODEL FOR TYPE
(n = 3850) or dams with two daughters in
different herds (n = 7347), 11,197 animals
altogether. The latter dams were each assigned
to the herd in which their first daughter was
found. There were 2977 sires of cows nested
within herds.
Across-herd animals, then, consisted of 1)
widely used sires, sires of sons, and limited use
sires with known parents; 2) clams (without
records) of cows with more than two daughters
in more than one herd; 3) darns (without
records) of bulls, a total of 10,321 animals,
5504 sires, and 4817 dams. Distinguishing
between within-herd and across-herd animals in
this way, limited-use sires were treated like
dams and, conversely, cows with many daughters (e.g., embryo transfer dams) were treated
like sires, resulting in a "unisex" model.
The MME were solved iteratively using
Gauss-Seidel iteration with SOR for one trait at
a time, performing between 30 and 60 rounds
of iteration. Starting values for solutions were
zero for all animal effects, phenotypic means
for HRC effects, and within-HRC least squares
(LSQ) estimates for regression coefficients.
Relaxation factors used were 1.00 for rounds 1
to 5 and 1.25 to 1.45 subsequently. Solutions
were obtained first for individual herds, as
suggested by Westell (33). After all herds were
processed, across-herd animals were evaluated,
and, as the last step in each round of iteration,
new estimates for the four regression coefficients were determined.
Statistics calculated in each round of iteration to monitor convergence behavior were the
sum of squared changes in solutions between
the previous and current round (SS DEV); the
square root of the ratio of SS DEV to sum of
squared solutions for the current round (CC), a
kind of standardized average change in solution;
the maximum change, sign ignored, in an
individual solution (MAX DEV); MAX DEV
divided by the respective solution (current
round) REL DEV, where solutions refer to the
complete set of equations, i.e., animal and HRC
effects and regression coefficients.
Presolution Step
To minimize input and output operations
per round of iteration and auxiliary storage
media required, only parts of the inverse of the
numerator relationship matrix (NRM) were set
1037
up and written to disk in a preiteration step.
This step, aimed at reducing operations to be
carried out in each round of iteration, included
the following calculations. 1) Type traits were
recoded to "objective scores" and expressed as
deviations from the phenotypic mean. 2) A
vector of linear and higher order covariables
(deviated from phenotypic mean) was set up. 3)
The LSQ equations for regression coefficients
(ignoring all other effects in the model) were
accumulated. In addition, estimates of regression coefficients with HRC subclasses were
obtained. 4) Animals within and across herds
were identified and animal and parental identities recoded to the respective running numbers.
A code was assigned to each record describing
its "status of connections", i.e., whether the
cow had daughters in another herd, whether her
dam was known, was in the same or a different
herd, or was an across-herd animal, and whether
her sire was known, a within-herd, or acrossherd animal. 5) The parts of the inverse of the
NRM accumulated were the diagonal coefficients for within-herd animals with connections to other herds, and the nonzero elements
of the submatrix pertaining to across-herd animals (halfstored, upper triangle only). Ignoring
inbreeding, contributions were (5) as follow.
For animal i with parent j and k: * +2 to
element (i,i); * - 1 to elements (i,j) and (i,k); *
+1/2 to elements (j,j), (j,k), and (k,k). For
animal i with parent j and the other parent
unknown: * +4/3 to element (i,i); * - 2 / 3 to
element (i,j); * +1/3 to element (j,j). For animal
i with both parents unknown: * +1 to element
(i,i).
Recoded information for each herd was
written to disk in seven vectors of lengths equal
to the number of records per herd or a multiple
thereof (factor: no. of traits or regression
coefficients). This ensured that the number of
read operations per round of iteration was low.
After all records were processed the accumulated parts of the NRM inverse, LSQ equations
for regression coefficients and information
linking recoded identities of bull dams with
own records to those of their sons and mates,
were written to disk.
Solution Step
The MME were set up for each herd in each
round of iteration. Each record was adjusted
Journal of Dairy Science Vol. 71, No. 4, 1988
1038
MEYER AND BURNSIDE
for covariables using estimates of regression
coefficients from the previous round of iteration. Contributions were as follow. To the coefficient matrix: * + 1.00 to the diagonal element
for the cow; * + 1.00 to the diagonal element
for the HRC in which the record was made; * +
1.00 to the element linking cow and HRC. To
the right-hand side (RHS): * + adjusted record to the RHS for the cow; * + adjusted
record to the RHS for the HRC pertaining to
the record. The RHS for animals without
records was initially set to zero.
Coeffients of the NRM inverse between
animals in the herd were obtained as described,
utilizing the recoded pedigree information. For
animals with connections to other herds the
diagonal coefficients accumulated previously
were substituted. After multiplying with the
variance ratio X, coefficients were added to the
MME for the herd. Only the nonzero elements
of the upper triangle of the coefficient matrix
were stored, using a "linked-list" procedure
(R. L. Quaas, 1984; personal communication).
Let within-herd animal i have sire j and dam
k, and let ~j and ~k denote their respective
solutions. These were solutions from the
previous round if j and k were across animals or
within animals in a herd not yet processed in
this round, and from the current round otherwise (previous solutions for the first round of
iteration were the starting values). To account
for relationships to animals not in the herd then
required the following adjustments to RHS
for within-herd animals (all animals with known
dams had sires identified). For sire j, an acrossherd animal: * + ~ ~j to RHS for animal i, if
dam k was known; * + 2/3 X ~j to RHS for
animal i, if dam k was u n k n o w n ; * - 1 / 2
;k ]j to RHS for dam k, if dam k belonged to
the herd. For dam k, not belonging to the herd:
• + X ~tk to RHS for animal i (sire j known); * 1/2 X ak to RHS for sire j, if sire j was assigned
to the herd. For dam k, belonging to the
herd: * + X ai' to RHS for dam k, if dam k
(without record) had daughter i' in another
herd. For cow i (with record) having daughter(s) in other herd(s) or son(s); * + weighted
sum of progeny solutions to RHS for cow i.
The necessary adjustments could be determined
efficiently utilizing the previously assigned
codes on "connection status".
Herd equations were solved iterating in-core
for five (internal) rounds, utilizing solutions
Journal of Dairy Science Vol. 71, No. 4, 1988
from the previous round, stored on disk, as
starting values. Adjustments for the new
solutions to other parts of the MME were then
accumulated and the new solutions written to
disk. Only estimates for within-herd animals
connected to other herds were saved in core.
Adjustments for the new solutions were
strictly corresponding to those described above
but in reverse direction; solutions for cows with
records and for HRC effects contributed to the
vector of adjustments to the RHS for regression
coefficients. For across-herd parents, adjustments were accumulated as the RHS of the
respective equations, whereas for within-herd
animals connected to animals not in the herd, a
separate vector was maintained.
After processing all herds, solutions for
across-herd animals were obtained, again
iterating in core (15 to 20 rounds) and utilizing
solutions from the previous round (held in
core). Before solving, the RHS for sites (and
sires of sires) were adjusted for solutions of bull
dams with records. Corresponding adjustments
to the RHS for cows with records and sons
were accumulated after obtaining the new
across solutions. The RHS for regression
coefficients were then adjusted for the accumulated cow and HRC effects, and new
estimates were obtained using the direct inverse
of the respective coefficient matrix.
Further details can be seen from the worked
example given in the Appendix.
RESULTS A N D DISCUSSION
The convergence behavior of the system of
equations is illustrated in Table 1 for mammary
system, using an arbitrary relaxation factor of
1.25 from round 5 onward. As in Westell's (33)
study, concerns about instability of the MME
(18) could not be confirmed, although convergence was stow. Using zero starting values
for animal effects, 20 to 25 rounds of iteration
were required for the average change in solution
(CC) to drop to 1% or less. For the maximum
change in an individual solution to reach the
same accuracy needed about another 10 rounds.
Calculations were carried out on the IBM
4341 of the Centre for Genetic Improvement of
Livestock at the University of Guelph, running
under CMS. Computing (CPU) time required
was 30 min for the presolution step (all traits)
and 12 min for each round of iteration (one
trait). Hence, about 5 h CPU time were de-
INDIVIDUAL ANIMAL MODEL FOR TYPE
m a n d e d t o o b t a i n s o l u t i o n s for each trait. In
practical terms, this m e a n t t h a t o n e trait could
be run per night. F o r f u t u r e runs, use o f t h e
c u r r e n t s o l u t i o n s as starting values is e x p e c t e d
t o r e d u c e t h e n u m b e r o f iterations r e q u i r e d
c o n s i d e r a b l y , so t h a t use o f t h e IAM s h o u l d
remain c o m p u t a t i o n a l l y feasible, even w i t h an
increasing a m o u n t o f data and size o f t h e MME
t o be solved.
Sire Proofs
S o l u t i o n s f o r 1098 " r e p o r t a b l e " sires w i t h at
least 20 d a u g h t e r s in five or m o r e h e r d s w e r e
e x t r a c t e d and s t a n d a r d i z e d to a m e a n o f zero
and variance o f 25, as is c u r r e n t p r a c t i c e f o r
H o l s t e i n C a n a d a ' s sire p r o o f s (HOL). Table
2 s h o w s t h e f r e q u e n c y d i s t r i b u t i o n o f changes
in p r o o f s b e t w e e n d i f f e r e n t analyses for final
1039
score class and stature. T a b l e 3 s u m m a r i z e s
c o r r e l a t i o n s b e t w e e n d i f f e r e n t p r o o f s and
m e a n s and s t a n d a r d deviations o f changes.
C o r r e l a t i o n s b e t w e e n evaluations derived
f r o m t h e s a m e d a t a set (i.e., first l a c t a t i o n , first
classification c o n f o r m a t i o n scores) u n d e r diff e r e n t m o d e l s (IAM and SM) w e r e close to
u n i t y (.97 to .99). T h e change in m o d e l did n o t
a f f e c t t h e r a n k i n g o f sires t o a g r e a t e x t e n t . F o r
final score/class, f o r instance, for 95% o f sires
d i f f e r e n c e s in p r o o f s (IAM-SM) w e r e in t h e
range - 2 t o +2, i.e., - . 4 t o .4 s t a n d a r d deviations.
C o r r e l a t i o n s b e t w e e n SM and H O L p r o o f s
ranged f r o m .92 t o .97, suggesting t h a t t h e
m a j o r p a r t o f changes b e t w e e n IAM and H O L
p r o o f s was d u e t o d i f f e r e n c e s in t h e d a t a
utilized. D u e t o a n o n z e r o m e a n (see T a b l e 3)
TABLE 1. Convergence behavior of mixed model equations for mammary system (variance ratio 5.65, relaxation factor 1.25 from round 5).
Round
SS DEV l
CC ~
MAX DEV a
1
2
3
4
5
6
7
8
9
10
270,828.0
5971.2
2427.3
1714.7
1391.3
1159.5
984.9
844.0
727.4
629.6
51.50
7.56
4.79
4.00
3.58
3.24
2.97
2.73
2.51
2.32
5.049
1.865
-.841
-.733
--.640
-.559
-.490
--.436
--.399
--.365
100.00
68.32
28.76
20.03
14.89
11.52
9.16
8.98
7.59
6.50
15
20
25
30
35
40
45
50
55
60
320.0
172.6
98.5
59.9
39.0
27.3
20.5
16.2
13.5
11.5
1.60
1.15
.85
.65
.52
.43
.37
.33
.30
.27
--.233
-.148
.102
.076
.058
.044
.038
.032
.028
.025
3.32
1.87
1.72
1.20
.86
.64
.81
.68
.58
.50
(%)
REL DEV 4
(%)
SS DEV = Sum of squared deviations in changes in solutions between current and previous round of iteration.
2 CC = Square root of ratio SS DEV to sum of squared solutions (current round).
3MAX DEV = Maximum change in individual solution (maximum referring to absolute value).
4 REL DEV = MAX DEV divided by pertaining solution.
Journal of Dairy Science Vol. 71, No. 4, 1988
1040
M E Y E R AND BURNSIDE
TABLE 2. F r e q u e n c y d i s t r i b u t i o n of differences in sire proofs f r o m different analyses, for 1098 sires with at
least 20 daughters in at least five herds.
Final score/class
SM--HOL 2
IAM--SM l
-9
-8
-7
-6
A3
Stature
IAM--HOL
B
A
SM--HOL
B
IAM--SM
A
IAM--HOL
B
A
B
2
3
6
1
1
4
1
2
4
5
1
2
6
1
1
1
1
16
101
259
11
38
75
151
261
10
18
54
I02
202
19
51
104
180
189
8
25
71
120
200
7
206
3
15
30
127
358
3
13
26
102
319
5
12
36
133
349
5
11
30
119
314
0
362
274
259
211
197
668
365
378
339
362
1
2
3
4
5
243
83
5
1
152
73
26
13
7
235
124
55
17
10
156
90
51
11
16
210
131
72
30
12
203
11
2
1
134
45
17
2
157
61
26
10
1
149
54
15
4
1
148
70
28
7
2
4
2
4
2
5
1
1
1
10
1
1
-5
-4
-3
-2
-1
2
6
7
8
9
10
11
1
1
1
1
i IAM = Proofs under an individual animal model; SM = proofs under a sire model.
2 HOL = Official Holstein Canada sire proofs from J u l y 1986.
3A = Difference using HOL proofs as extracted; B = difference adjusting subset o f HOL proofs to mean 0
and variance 25.
TABLE 3. Means and standard deviations (SD) of differences b e t w e e n sire proofs from different analyses, and
correlations (r) b etween proofs.
Final
score
Feet and
legs
Ma mma ry
system
Stature
IAM--SM 1
SM--HOL 2
IAM--HOL
HOL
--.007
--.515
--.521
.514
--.014
--.084
--.097
.102
--.015
--.451
--.436
.440
.014
--.450
--.436
.448
SD
IAM--SM
SM--HOL
IAM--HOL
1.271
1.899
2.223
1.233
2.042
2.386
1.235
1.865
2.197
.685
1.281
1.329
.968
.929
.904
.970
.920
.891
.970
.932
.905
.991
.967
.964
r
IAM, SM
SM, HOL
IAM, HOL
1 IAM = Proofs under an individual animal model; SM = proofs u n d e r a sire model.
2 HOL = Official Holstein Canada sire proofs from July 1986.
Journal of Dairy Science Vol. 71, No. 4, 1988
INDIVIDUAL ANIMAL MODEL FOR TYPE
a n d variance slightly h i g h e r t h a n 25 o f H O L
p r o o f s as e x t r a c t e d , t h e d i s t r i b u t i o n o f changes
b e t w e e n H O L a n d SM ( a n d c o n s e q u e n t l y , also
H O L a n d IAM) was s k e w e d t o t h e negative side
(see c o l u m n s A in T a b l e 2). S t a n d a r d i z i n g H O L
p r o o f s in t h e s u b s e t (to m e a n 0 a n d variance
2 5 ) r e m o v e d this s k e w n e s s (see c o l u m n s B in
T a b l e 2). C h a n g e s b e t w e e n H O L a n d SM p r o o f s
e n c o m p a s s e d b o t h r a n d o m changes d u e to a
r e d u c t i o n in t h e n u m b e r o f r e c o r d s p e r sire a n d
s y s t e m a t i c d i f f e r e n c e s w h i c h m a y b e associated
w i t h selection.
C o r r e l a t i o n s b e t w e e n IAM a n d H O L p r o o f s
were o f t h e o r d e r o f .9 (higher f o r s t a t u r e , see
T a b l e 3), i.e., t h e m a j o r i t y o f sires r a n k e d
similarly u n d e r b o t h systems. In c o m p a r i s o n ,
Westell (33) r e p o r t e d a c o r r e l a t i o n o f .78
b e t w e e n IAM sire p r o o f s for m i l k yield and
c o r r e s p o n d i n g N o r t h e a s t e r n Artificial I n s e m i n a t i o n Sire C o m p a r i s o n s ( N E A I S C ) u n d e r a M G S
m o d e l . F o r final score/class, 70% of d i f f e r e n c e s
( I A M - H O L ) w e r e in t h e range of --2 t o +2 a n d
97% were in t h e range - 5 t o +5; average
d i f f e r e n c e ( a f t e r rescaling H O L p r o o f s in t h e
s u b s e t ) was close to zero.
T a b l e 4 gives an e x a m p l e o f IAM sire p r o o f s
w i t h t h e c o r r e s p o n d i n g H O L values (as ex-
1041
t r a c t e d ) . T h e first five lines s h o w sires w i t h few
changes in proofs. Bulls listed in lines 6 to 11,
h o w e v e r , m a y give rise t o c o n c e r n . In spite of
high t o very high n u m b e r s of d a u g h t e r s , high
c u r r e n t p r o o f s ( H O L ) f o r t h e s e bulls d r o p p e d
b y u p t o 4 p o i n t s u n d e r t h e IAM. S o m e o f
t h e s e bulls are " k n o w n " t o have b e e n m a t e d
n o n r a n d o m l y w i t h s u b s e q u e n t selective reclassification o f daughters. Hence, t h e app a r e n t l y s y s t e m a t i c c h a n g e in p r o o f s m a y ,
partially at least, reflect t h e r e m o v a l of bias
u n d e r t h e IAM.
Cow Indexes
S o l u t i o n s for f e m a l e s were e x p r e s s e d o n t h e
s a m e scale as t h e s o l u t i o n s f o r " r e p o r t a b l e "
sires. T h e resulting cow i n d e x e s r a n g e d for - 1 6
( - 1 9 f o r s t a t u r e ) to +16. S t a n d a r d d e v i a t i o n s
w e r e 2 . 5 6 3 , 2 . 3 6 7 , 2 . 4 7 6 , a n d 2 . 8 1 4 for final
score/class, f e e t a n d legs, m a m m a r y s y s t e m , a n d
s t a t u r e , respectively. T h e d i f f e r e n c e in variabilit y o f sire p r o o f s ( w i t h s t a n d a r d d e v i a t i o n
5 . 0 0 0 ) r e f l e c t e d d i f f e r e n c e s in t h e a m o u n t o f
i n f o r m a t i o n available a n d utilized in e v a l u a t i n g
each animal. C o r r e s p o n d i n g m e a n s were 1.395,
.681, 1.099, a n d .579.
TABLE 4. Example of sire solutions.
Individual animal model
Bull
325144
337778
348080
356420
364769
305376
310451
290725
1491007
341536
1629391
Holstein Canada
ND 1
NH 2
FS 3
FL 4
MS s
ST 6
FS
FL
MS
ST
335
295
71
53
25
344
95
84
362
1175
2390
257
102
54
45
23
257
31
50
190
524
985
--9
12
--2
7
--11
10
--3
7
--1
12
2
--3
3
4
0
2
4
-9
12
--5
8
-12
11
--4
7
3
2
3
13
5
--7
7
5
6
8
9
11
2
--i
5
4
6
5
3
11
7
--9
5
4
7
6
8
3
2
2
--1
10
3
--4
11
12
10
0
--1
7
12
10
7
-6
10
11
10
3
--2
5
4
i0
11
i0
1
11
12
6
7
9
11
2
1 ND = No. of daughters (first lactation, first class).
2 NH = No. of herds.
3 FS = Final score/class.
4 FL = Feet and legs.
s MS = Mammary system.
6 ST = Stature.
Journal of Dairy Science Vol. 71, No. 4, 1988
1042
MEYER AND BURNSIDE
TABLE 5. Example of cow indexes together with the cows' phenotypic records and their sires' proof.
Cow
Cow no.
A
Days in milk
Calving age, mo
20
Final score/class
Type classification
Cow index
Sire proof
Feet and legs
Type classification
Cow index
Sire proof
Mammary system
Type classification
Cow index
Sire proof
B
3737216
C
3400515
D
3603593
E
3411125
27.7
173
26.0
130
31.4
33
24.4
295
26.8
82/GP 1
4
5
85/VG 2
10
12
79/G
-2
-4
82/GP
10
12
80/GP
-7
- 11
GP--1
2
6
GP-2
6
10
GP-2
3
5
VG--1
8
10
GP-2
0
- 1
GP-3
4
3
VG-2
11
12
GP-2
-1
-5
GP-3
10
12
GP-1
-7
-1
7
3
3
-7
4
2
6
-2
4
-1
4
-1
3774147
Stature
Type classification
Cow index
Sire proof
7
0
-1
1Good plus classification.
2Very good classification.
Correlations b e t w e e n p h e n o t y p i c t y p e scores
and cow indexes were .62, .53, .61, and .76 for
the four traits. Table 5 gives an e x a m p l e of
indexes together with the corresponding
p h e n o t y p i c records and proofs for the cows'
sires. Indexes are heavily influenced by sires'
proofs. Cows A and D, for instance, b o t h with a
final score of 82, had indexes of 4 and 10, the
difference largely being explicable by the
difference in the respective sire proofs of 5 and
12.
The relationship b e t w e e n parental evaluations and cow indexes is quantified in Table 6.
Standardized partial regression coefficients of
daughter's index or p h e n o t y p i c record on dam's
index or p h e n o t y p i c record and sire's and
MGS's p r o o f were calculated for all cows with
dams with records in the data. Clearly, dam's
index was a m u c h b e t t e r p r e d i c t o r of b o t h
daughter's genetic merit and p h e n o t y p i c record
than dam's p h e n o t y p e . N o t including MGS's
proof, the c o e f f i c i e n t of d e t e r m i n a t i o n (R 2) for
d a u g h t e r ' s genetic merit increased by 18.8 to
23.4% (absolute difference, e.g., 66.8% to
87.8% = 21.0% for final score/class) w h e n
d a m ' s t y p e classification was replaced by d a m ' s
Journal of Dairy Science Vol. 71, No. 4, 1988
genetic index. If MGS proofs were considered,
the corresponding increase was 8.9 to 18.5%,
giving R 2 of 77.8 to 91.5%. Westell and Van
Vleck (34) r e p o r t e d partial regression coefficients of daughter's estimated transmitting
ability (ETA) for first lactation milk yield on
d a m ' s E T A (based on first lactation records
only) and sire's and MGS's N E A I S C of .755,
.484, and - . 1 3 5 , respectively, with R 2 of 84%
(for a heritability of .25).
CONCLUSIONS
Simultaneous genetic evaluation o f cows and
sires for c o n f o r m a t i o n traits in the Canadian
Holstein population is feasible and should be
i m p l e m e n t e d . Use of first lactation, first
classification records u n d e r an IAM gives
estimates of genetic merit unbiased by nonr a n d o m mating or selective reclassification of
cows. F o r m a x i m u m accuracy with such a
scheme, breed policy should aim at classifying
all females during the first lactation. The
p r o p o r t i o n of a sire's daughters classified to
calved (excluding sales for dairy purposes) and
t h e p r o p o r t i o n reclassified should be monitored.
INDIVIDUAL ANIMAL MODEL FOR TYPE
1043
TABLE 6. Standardized partial regression coefficients and coefficient of determination (R 2), predicting daughters' cow index (C1) or phenotypic type classification (TY) from their dams' CI or TY, and sires' and maternal
grandsires' (MGS) proofs (n = 23,664).
CI daughter
Dam
Sire
TYdaughter
MGS
R2
Dam
Sire
MGS
(%)
Finalscore~lass
TY l
.283
.222
C1
.541
.651
.718
.711
.715
Feetandlegs
TY
.229
.179
.491
.606
.813
.823
.818
.815
Mammary sy~em
TY
.306
.232
C1
.574
.700
.728
.709
.705
.707
Stature
TV
CI
.373
.315
.597
.677
.747
.623
.604
.601
.609
.318
-.143
.330
--.141
.361
-.157
.210
--.128
R2
(%)
66.8
76.2
87.8
88.6
.171
.161
.288
.557
.215
.211
.198
.207
72.0
82.6
90.8
91.5
.113
.106
.255
.637
.182
.183
.184
.173
64.6
77.0
88.0
88.9
.158
.148
.285
.635
.205
.202
.194
.198
55.3
59.3
76.8
77.8
.251
.233
.386
.503
.282
.276
.268
.279
.047
-.350
8.1
8.3
13.4
18.4
.042
4.6
4.8
9.9
--.465
16.9
.049
-.433
.064
--.187
7.0
7.2
12.6
19.1
15.0
15.4
23.5
25.6
1 Information on dam.
C o m p u t i n g costs are essentially p r o p o r t i o n a l
t o t h e n u m b e r o f traits evaluated. U n d e r t h e
IAM it a p p e a r s n e c e s s a r y t o c h o o s e a n u m b e r o f
k e y traits t o b e evaluated. G e n e t i c p a r a m e t e r s
for t h e s e s h o u l d be e s t i m a t e d b a s e d o n first
l a c t a t i o n , first classification r e c o r d s only.
Literature
evidence
[(11)
and references
t h e r e i n ; ( 3 0 ) ] suggests t h a t h e r i t a b i l i t i e s m a y
b e s o m e w h a t h i g h e r t h a n all l a c t a t i o n , latest
classification values r e p o r t e d b y S c h a e f f e r (21).
Use o f the l a t t e r e s t i m a t e s in t h e c u r r e n t s t u d y ,
however, is n o t e x p e c t e d to have a f f e c t e d
r a n k i n g s s i g n i f i c a n t l y ; u n i v a r i a t e e v a l u a t i o n s are
r o b u s t against small errors in t h e variance r a t i o
used (27).
F u t u r e e v a l u a t i o n s s h o u l d e m p l o y a multivariate p r o c e d u r e . W i t h all traits r e c o r d e d o n
all animals at t h e same time, c a n o n i c a l transf o r m a t i o n can be used to r e d u c e t h e multivariate analysis t o a series o f c o r r e s p o n d i n g
u n i v a r i a t e analyses [e.g., (26, 1 0 ) ] . T h i s implies
t h a t t h e a d d i t i o n a l c o m p u t a t i o n a l resources
required for the multivariate rather than a
u n i v a r i a t e a p p r o a c h are trivial. T h e m u l t i v a r i a t e
evaluation, however, requires appropriate and
a c c u r a t e e s t i m a t e s o f t h e g e n e t i c covariance
s t r u c t u r e [e.g., (22, 2 8 ) ] .
M a n y N o r t h A m e r i c a n AI o r g a n i z a t i o n s o f f e r
c o m p u t e r m a t i n g s y s t e m s t h a t , for c o n f o r m a t i o n , utilize t h e c o w ' s p h e n o t y p i c score t o
d e t e r m i n e a bull for a c o r r e c t i v e mating.
R e s u l t s o f this s t u d y i n d i c a t e t h a t a m a r k e d
i m p r o v e m e n t in a c c u r a c y o f p r e d i c t i o n o f t h e
daughter's
conformation
can be achieved
t h r o u g h use o f h e r d a m ' s i n d e x ( E T A ) . Furt h e r m o r e , use o f d a m s ' g e n e t i c i n d e x e s f r o m
t h e I A M instead o f d a m s ' p h e n o t y p i c r e c o r d s
will p r o v i d e a c o n s i d e r a b l e increase in a c c u r a c y
o f p r e d i c t i n g t h e g e n e t i c m e r i t for c o n f o r m a t i o n of y o u n g sires, t h u s i m p r o v i n g t h e scope
f o r s e l e c t i o n b a s e d o n pedigree i n f o r m a t i o n .
ACKNOWLEDGMENTS
This
Canada,
s t u d y was s u p p o r t e d b y H o l s t e i n
B r a n t f o r d , O n t a r i o , a n d t h e AgriJournal of Dairy Science Vol. 71, No. 4, 1988
1044
MEYER AND BURNSIDE
cultural and Food Research Council (UK).
D i s c u s s i o n s w i t h M. S. H u n t a r e a c k n o w l e d g e d .
REFERENCES
1 Blair, H. T., and E. J. Pollak. 1984. Comparison of
an animal m o d e l and an equivalent reduced animal
model for c o m p u t a t i o n a l efficiency using mixed
m o d e l methodology. J. Anita. Sci. 58:1090.
2 Bonaiti, B., a n d M. Briend. 1986. C o m p u t i n g
algorithm for dairy sire evaluation on several
lactations considered as t h e same trait. Genet. Sel.
Evol. 18:41.
3 Henderson, C. R. 1973. Sire evaluation and genetic
trends. Page 10 in Proc. Anim. Breed. Genet.
Symp. in Honor of Dr. J. L. Lush. A m . Dairy Sei.
Assoc., A m . Soc. Anim. Sci., Champaign, IL.
4 Henderson, C. R. 1975. Use o f all relatives in
intra-herd prediction of breeding values and
producing abilities. J. Dairy Sci. 58:1910.
5 Henderson, C. R. 1976. A simple m e t h o d for
c o m p u t i n g the inverse o f a n u m e r a t o r relationship
matrix used in prediction of breeding values.
Biometrics 32:69.
6 Henderson, C. R., and R. L. Quaas. 1976. Multiple
trait evaluation using relatives records. J. Dairy Sci.
43:1188.
7 Hill, W. G., and G.J.T. Swanson. 1983. A selection
index for dairy cows. Anim. Prod. 37:59.
8 Hudson, G.F.S. 1986. C o m p u t i n g genetic evaluations through application of generalized least
squares to an animal model. Genet. Sel. Evol.
18:31.
9 Hudson, G.F.S. and L. R. Schaeffer. 1984. Monte
Carlo Comparison of Sire Evaluation Models in
Populations subject to selection and n o n r a n d o m
mating. J. Dairy Sci. 67:1264.
10 Lee, A. J. 1979. Mixed model multiple trait
evaluation of related sires w h e n all traits are
recorded. J. Anita. Sci. 4 8 : 1 0 7 9 .
11 Meyer, K., E. B. Burnside, K. H a m m o n d , and A. E.
McClintock. 1985. Evaluating dairy sires for
conformation of their daughters: use of first
classification records. Aust. J. Agric. Res. 36:509.
12 Misztal, I. 1986. Survey of s o m e c o m p u t i n g
m e t h o d s in BLUP sire evaluation. Mixed models applications and analysis, a workshop on the
interface of practice and theory, April 6 - 8 , 1986,
Irsee, FRG.
13 Misztal, I., and H. Haussman. 1985. Comparison of
alternative c o m p u t a t i o n a l procedures in BLUP sire
evaluation. Z. Tierz. Zuchtung. 102:143.
14 Poivet, J. P. 1986. Methode simplifiee de calcul des
valeurs genetiques des femelles t e n a n t compte de
toutes les parentes. Genet. Sel. Evol. 18:321.
15 Poivet, J. P., and J. M. Elsen. 1986. Some advantageous aspects of block iterative m e t h o d s for
solving the m i x e d model equations. Proc. 3rd
World Congr. Genet. App1. Livest. Prod., Lincoln,
NE XII:403.
16 Pollak, E. J., and R. L. Quaas. 1981. Monte Carlo
s t u d y of within-herd multiple trait evaluation of
Journal o f Dairy Science Vol. 71, No. 4, 1988
beef cattle growth traits. J. Anim. Sci. 52:248.
17 Quaas, R. L., R. W. Everett, and A. E. McCllntock.
1979. Maternal grandsire model for dairy sire
evaluation. J. Dairy Sci. 62:1648.
18 Quaas, R. L., and E. J. Pollak. 1980. Mixed model
m e t h o d o l o g y for farm and ranch beef cattle testing
programs. J. A n i m . Sci. 51:1277.
19 Schaeffer, L. R. 1975. Dairy sire evaluation for
milk and fat production. Mimeo, Univ. of Guelph,
Guelph, Ont.
20 Schaeffer, L. R. 1983. Effectiveness of model for
cow evaluation intraherd. J. Dairy Sci. 66:874.
21 Schaeffer, L. R. 1983. Estimates of variance
c o m p o n e n t s for Holstein-type traits. Can. J. Anim.
Sci. 63:763.
22 Schaeffer, L. R. 1984. Sire and cow evaluations
under multiple trait models. J. Dairy Sci. 67:1567.
23 Schaeffer, L. R., M. S. Hunt, and E. B. Burnside.
1978. Evaluation of Holstein-Friesian dairy sires
for c o n f o r m a t i o n o f their daughters. Can. J. Anim.
Sci. 58:409.
24 Schaeffer, L. R., and B. W. Kennedy. 1986. Computing strategy for solving m i x e d model equations.
J. Dairy Sci. 69:575.
25 Snell, E. J. 1964. A scaling procedure for ordered
categorical data. Biometrics 20:592.
26 T h o m p s o n , R. 1976. Estimation of quantitative
genetic parameters. Page 639 in Proc. Int. Conf.
Quant. Genet., Ames, IA.
27 T h o m p s o n , R. 1979. Sire evaluation. Biometrics
35:339.
28 T h o m p s o n , R. a n d Meyer, K. 1986. A review of
theoretical aspects in the estimation of breeding
values for multi-trait selection. Livest. Prod. Sci.
15:299.
29 Tong, A.K.W., J. W. Wilton, and L. R. Schaeffer.
1977. Application o f a scoring procedure and
transformations to dairy type classifications and
beef ease of calving categorical data. Can. J. Anim.
Sci. 57:1.
30 Van Doormaal, B. J., and E. B. Burnside. 1987.
Impact of selection on c o m p o n e n t s of variance and
heritability in Canadian Holstein conformation
traits. J. Dairy Sci. 70:1452.
31 Van Vleck, L. D. and D. Dwyer. 1985. Successive
overrelaxation, block iteration and m e t h o d o f
conjugate gradients for solving equations for sire
evaluation. J. Dairy Sci. 68:760.
32 Van Vleck, L. D., and D. Dwyer. 1985. Comparison of iterative procedures for solving equations for sire evaluation. J. Dairy Sci. 68:1006.
33 Westell, R. A. 1984. Simultaneous genetic evaluation of sires and cows for a large population of
dairy cattle. Ph.D. Thesis, Cornell Univ., Ithaca,
NY.
34 Westell, R. A., and L. D. Van Vleck. 1985. Prediction of heifer transmitting abilities from genetic
evaluations of sire, d a m and maternal grandsire. J.
Dairy Sci. 68:1432.
35 Wilson, D. E., R. L. Wilham, and P. J. Berger.
1985. Mixed m o d e l m e t h o d o l o g y for unifying
within-herd and national beef sire evaluation. J.
A n i m . Sci. 61:814.
INDIVIDUAL ANIMAL MODEL FOR TYPE
1045
APPENDIX
Consider the following records for 19 cows in 3 herds, and pedigrees for 5 sires of cows:
Cow
Sire
Dam
Stage
Age
Type
HRC
Herd
100
101
102
103
104
105
106
107
108
109
110
119
111
112
113
114
115
116
117
200
201
202
203
204
200
200
0
201
202
204
201
203
200
203
20!
203
204
204
200
202
201
200
0
205
205
206
0
200
0
300
0
100
301
116
300
302
302
303
301
304
300
111
0
106
304
305
0
109
304
0
0
306
110
96
121
85
122
135
99
100
116
104
111
109
102
105
114
107
97
104
98
30
28
31
24
35
22
23
25
29
28
33
30
34
31
33
34
32
22
29
15
12
11
9
7
14
10
13
8
16
15
12
14
6
9
11
17
18
15
1
1
1
1
1
1
1
1
2
2
2
2
1
1
2
2
3
3
3
1111
1111
1111
1111
1111
1111
2222
2222
2222
2222
2222
2222
3333
3333
3333
3333
3333
3333
3333
There are a total of 26 females and 7 males,
33 animals to be evaluated. Together with 6
HRC effects and 4 regression coefficients to
be estimated, this gives a system of MME
of order 43. There are 3 cows with records
and connections to animals outside the herd,
cow 106 and cow 116 with daughters in another herd and cow 109, a bull dam. Of the 7
dams without records, 3 are nested within herds
(dams 302, 303, 305), dam 301 has two daughters in different herds and is assigned to herd
1111 where the first daughter is found, dam
300 has three daughters in more than one
herd, and dams 304 and 306 are bull dams.
The latter three are treated as across herd
animals. Only bull 203 is nested within a herd;
the other six augment the number of acrossherd animals to 9.
Journal of Dairy Science Vol. 71, No. 4, 1988
1046
MEYER AND BURNSIDE
Recoding the data in the presolution step then gives:
Cow
Cow
Sire
Dam
100
101
102
103
104
105
106
107
108
109
110
119
111
112
113
114
115
116
117
1
2
3
4
5
6
1
2
3
2
5
6
1
2
3
4
5
3
7
1
1
0
2
3
4
2
9
1
9
2
9
4
4
1
3
2
1
0
0
5
0
1
4
3
5
7
7
8
4
6
5
1
0
1
6
8
0
Stage
linear
Stage
quadratic
Age
linear
Age
quadratic
2.895
--11.105
13.895
--22.105
14.895
27.895
--8.105
--7.105
8.895
-3.105
3.895
1.895
--5.105
-2.105
8.380
123.327
193.064
488.643
221.853
778.116
65.695
50.485
79.116
9.643
15.169
3.590
26.064
4.432
47.537
.011
102.116
9.643
82.906
.895
--1.105
1.895
--5.105
5.895
--7.105
--6.105
--4.105
--.105
-1.105
3.895
.895
4.895
1.895
3.895
4.895
2.895
--7.105
-.105
.801
1.222
3.590
26.064
34.748
50.485
37.274
16.853
.011
1.222
15.169
.801
23.958
3.590
15.169
23.958
8.380
50.485
.011
6.895
--.105
--10.105
--3.105
--9.105
w h e r e t h e first t w o letters o f t h e c o d e d e s c r i b e
t h e t y p e o f sire (SO, SW, and SA for sire unk n o w n , w i t h i n - h e r d or across-herd animal,
respectively), t h e latter t w o d e n o t e t h e t y p e
o f d a m (DO, DW, DX, DC and D A for d a m
unknown, nested within-herd, within-herd with
o n e d a u g h t e r in a n o t h e r h e r d , c o w w i t h record
Type
Code
2.789
--.211
--1.211
--3.211
--5.211
1.789
--2.211
.789
--4.211
3.789
2.789
--.211
1.789
-6.211
--3.211
--1.211
SADW
SADA
SOD0
SADW
SADX
SADC
*SADA
SWDW
SADW
*SWDW
SADX
SWDA
SADA
SADW
SAD0
SADC
SADA
*SADW
SOD0
4.789
5.789
2.789
in a n o t h e r herd and " a c r o s s " animal, respectively), and an asterisk m a r k s a cow w i t h
r e c o r d s c o n n e c t e d t o animals o u t s i d e t h e herd.
T h e markers, c o d e d a p p r o p i a t e l y for c o m p u t a tional purposes, are s u b s e q u e n t l y used to
d e t e r m i n e w h e t h e r r e c o d e d animal identities
r e f e r to r u n n i n g n u m b e r s w i t h i n or across
herds.
The NRM inverse for the 9 across-herd animals is then:
300
2.50
0
0
0
.50
0
.50
0
0
304
0
2.50
0
0
-.50
0
0
.50
0
306
0
0
1.50
.50
0
0
-1.00
0
0
200
0
0
.50
4.67
0
0
-1.00
--1.00
0
201
.50
--.50
0
0
4.00
0
0
--1.00
0
202
0
0
0
0
0
2.33
0
0
-.67
204
.50
0
--1.00
--1.00
0
0
3.50
0
0
205
0
.50
0
-1.00
-1.00
0
0
2.00
0
206
0
0
0
0
0
-.67
0
0
1.33
and the vector of diagonal elements for the four within animals with progeny outside the herd,
106, 109, 116, and 301, has elements 2.50, 2.50, 2.50, and 1.00, respectively.
T h e L S Q e q u a t i o n s for covariables, ignoring all o t h e r e f f e c t s in t h e m o d e l , are:
[12309.8
3,762.6
132.8
754.0
13,762.6
979.821.7
6504.1
66,828.6
132.8
-6504.1
313.8
- 5 5 3.1
Journal of Dairy Science Vol. 71, No. 4, 1988
754.0"
[bl]
66,828.6
b2
--55 3.1
10,3 66.0
b3
b4
-99.4]
-1286.4|
--58.4]
76.2]
INDIVIDUAL ANIMAL MODEL FOR TYPE
1047
dam 301, and one HRC. Using the estimates
o f the w i t h i n - H R C regression coefficients
as starting values, gives t h e RHS as:
Estimates of the regression coefficients f r o m
these equations are - . 0 1 2 2 6 3 ,
-.004258,
- . 2 2 7 7 5 4 , and .023541, while corresponding
values absorbing H R C effects are .006499,
- - . 0 0 4 3 5 5 , --.142212, and .030815.
T h e MME for herd 1111 are of o r d e r 8,
comprising effects for 6 cows (100 to 105),
[2.9097
.2039
--.3012
2.4308 0 - 1 . 7 9 8 3 ]
--2.4680
--4.5736
With zero as initial values for all animal solutions, there are effectively no adjustments for relationships outside the herd in the first round of iteration. F o r a variance ratio of 4.00, the coefficient m a t r i x is:
m
8.33
0
0
-4.00
0
0
0
1.00
0
9.00
0
0
0
0
0
1.00
0
0
5.00
0
0
0
0
1.00
--4.00
0
0
9.00
0
0
0
1.00
0
0
0
0
9.00
0
--4.00
1.00
0
0
0
0
0
9.00
0
1.00
0
0
0
0
--4.00
0
8.00
0
1.001.00
1.00
1.00
1.00
1.00
0
6.00
D
With the HRC mean,
{.34355
--.2997, as starting value, solutions after 5 iterates are:
.05616
.00007
--.08803
--.60981
.30359
-.30490
--.30064}
Adjusting the RHS for across-herd animals ( R H S A C R ) for solutions of progeny in the herd gives
l e | m e n t s of R H S A C R :
A n i m a l no.
300:
304:
306:
200:
201:
202:
204:
205:
206
.2246 = 0 + 4.00
0
0
1.1408 = 0 + 2.67
--1.0392=0+4.00×
- - 1 . 8 2 9 4 = 0 + 4.00
1.2144 = 0 + 4.00
0
0
× .05616
× .34355 + 4.00 x .05616
(--.08803)--2.00x
.34355
z (--.60981) -- 2.00 × (--.30490)
× .30359 -- 2.00 × 0
The vector of a c c u m u l a t e d adjustments to the
R H S for regression coefficients ( A D J C O V ) is:
[1.7033
67.7452
-5.0569
-7.8132]
after accounting for cows (adjustments are
cow solution × respective covariable term,
s u m m e d over cows in the herd; e.g., 1.7033 =
.34355 x 2.895 + . 0 5 6 1 6 x ( - 1 1 . 1 0 5 ) + . . . +
.30359 x 27.895), and:
[--6.2240 --477.4253
--3.6645 --42.9603]
after adjusting for HRC effects (adjustments
are H R C solutions x sum o f respective covariable terms in the subclass, s u m m e d over
H R C in the herd; e.g., - - 6 . 2 2 4 0 = 1.7033 +
( - . 3 0 0 6 4 ) x 26.3684).
F o r herd 2222, the RHS for cow 119
is adjusted for the solution of her dam 301
estimated in the previous herd. The RHS
after adjusting for covariables (RHS 1)i adj u s t m e n t s due to relationships (ADJ), final
Journal of Dairy Science Vol. 71, No. 4, 1988
1048
MEYER AND BURNSIDE
RHS (RHS 2), starting values (START), and
solutions (SOLN) after 5 rounds of iteration
are:
Animal no.
Rhs 1
ADJ
RHS 2
START
106
107
108
109
110
119
102
303
203
HRC 1
HRC 2
--3.8886
--.0476
--3.9391
3.6568
2.9167
--.1046
0
0
0
--3.9362
2.5298
0
0
0
0
--1.2196
0
0
0
0
0
0
--3.8886
--.0476
--3.9391
3.6568
1.6971
--.1046
0
0
0
--3.9362
2.5298
0
0
0
0
0
0
0
0
0
--1.9681
.6324
Accumulating adjustments for the
solutions gives RHSACR and ADJCOV:
new
[--.4632 --.4144 0 --.8197 --.6450 --1.8294
1.2144 0 0]
SOLN
--.17197
.23255
--.62625
.46024
.11808
.05195
--.27225
.20629
.31109
1.99841
.63144
[24.7847 --684.3979 19.1591 --140.3699]
and 4.00 × .11808 = .47232 is stored as an
adjustment to the RHS for dam 301 (for
daughter 110) in the next round of iteration.
Similarly for herd 3333 :
Animal no.
RHS 1
ADJ
RHS 2
START
SOLN
111
112
113
114
115
116
117
305
HRC 1
HRC 2
HRC 3
1.8940
--6.0187
--2.9619
--1.2520
5.4533
3.2855
3.1944
0
--4.1248
--4.2139
11.9332
0
0
0
-.6879
0
1.2144
0
0
0
0
0
1.8940
-6.0187
-2.9619
-1.9399
5.4533
4.4999
3.1944
0
-4.1248
-4.2139
11.9332
0
0
0
0
0
0
0
0
-2.0624
-2.1069
3.9777
.23071
-.34304
-.14561
-.01109
.16695
.06579
-.15127
.04386
-2.00621
-2.03967
3.95058
RHSACR:
[ . 4 5 9 6 . 2 5 3 4 0 - 1 . 0 3 2 5 .0228 --1.4411 .3036 0 0]
and ADJCOV:
[--64.7327 --70.8109 - 2 9 . 4 4 1 3 --35.7816]
Journal of Dairy Science Vol. 71, No. 4, 1988
INDIVIDUAL ANIMAL MODEL FOR TYPE
1049
Bull 2 0 0 w i t h p a r e n t s 205 a n d 109 has a d a m
w i t h record. Hence, t h e RHS for 2 0 0 a n d 205
are a d j u s t e d for t h e s o l u t i o n for 109, w h i c h
gives R H S A C R :
n e x t r o u n d o f i t e r a t i o n is t h e n 4 . 0 0 × . 0 1 2 7 2
- 2 . 0 0 x ( - - . 1 3 6 5 5 ) = . 3 2 3 9 8 . T h e R H S for
covariables a d j u s t e d for cow a n d H R C solut i o n s in r o u n d 1 is t h e n :
[.45961 .25342 0 .80841
. 3 0 3 6 2 - - . 9 2 0 4 7 0]
[-34.6683
.02284--1.44112
T w e n t y r o u n d s of i t e r a t i o n ( w i t h zero s t a r t i n g
values) t h e n gives s o l u t i o n s for t h e " a c r o s s "
animals:
[.04560
--.18014
,04612
.01025
,02174 --.13655
.01272 --.03264
--.09007]
T h e a d j u s t m e n t to t h e R H S for " 1 0 9 "
A n i m a l no.
RHS 1
100
101
102
103
104
105
301
HRC 1
2.8942
.2368
--.3472
--2.3803
--4.6821
2.4408
0
--1.8378
in t h e
--1215.5741
--28.9798 111.9755]
w h i c h gives new e s t i m a t e s for t h e regression
coefficients:
[.008296
--.004365
--.131094
.031347]
In t h e s e c o n d r o u n d o f i t e r a t i o n , t h e r e are
a d j u s t m e n t s to t h e R H S for w i t h i n herd a n i m a l s
for p a r e n t s or p r o g e n y o u t s i d e t h e herd. F o r
herd 1111:
ADJ
.0992
.2333
0
--.1306
--.7206
.3501
.8326
0
RHS 2
START
SOLN
2.9934
.4701
--.3472
--2.5109
--5.4027
2.7909
.8326
--1.8378
.34355
.05616
.00007
--.08803
--.60981
.30359
--.30490
--.30064
.35459
.08667
--.00744
--.08695
--.66817
.34455
--.23001
--.31017
T h e a d j u s t m e n t for a n i m a l 101, for i n s t a n c e ,
along with the adjustments for the daughter's
f o r s o l u t i o n s o f sire 2 0 0 a n d d a m 300 is . 2 3 3 3
s o l u t i o n , a n d w h e n t h i s was d o n e t h e c u r r e n t
= (4.00 × .04560) + (4.00 × .01272). For dam
s o l u t i o n s for t h e " a c r o s s " a n i m a l s w e r e still
301, it is m a d e up o f a t e r m for d a u g h t e r 110
e q u a l to t h e i r s t a r t i n g values o f zero. F o r
a n i m a l 100, c o n t r i b u t i o n s f r o m t h e sire ( 2 0 0 )
a c c u m u l a t e d in t h e p r e v i o u s r o u n d o f i t e r a t i o n
a n d t h e sire ( 2 0 1 ) o f d a u g h t e r 103 in t h e same
a n d a t e r m for t h e sire ( 2 0 2 ) o f d a u g h t e r 1 0 4
h e r d are ( 2 . 6 6 7 × . 0 1 2 7 2 ) - ( 2 . 0 0 × - . 0 3 2 6 4 )
in this herd, . 8 3 2 6 = . 4 7 2 3 2 - 2 . 0 0 ( - - . 1 8 0 1 4 ) .
= .0992.
In f u t u r e r o u n d s , t h e r e will also b e a t e r m d u e
S o l u t i o n s f o r t h e " a c r o s s " h e r d a n i m a l s in
t o t h e sire ( 2 0 1 ) o f d a u g h t e r 110. In t h e
p r e s e n t case, it is zero b e c a u s e it is a c c u m u l a t e d
s u b s e q u e n t r o u n d s o f i t e r a t i o n are:
Animal number
Round
300
304
306
200
201
202
204
205
206
.0461
.0582
.0103
.0154
.0201
.0240
.0272
.0297
.0354
,0380
.0389
.0392
.0393
.0393
.0393
.0393
.0393
.0127
.0389
.0627
.0827
--.0326
--.0382
--.0363
--.0330
--.0297
--.0268
--.0195
--,0160
--.0148
--.0144
--.0142
--.0142
--.0142
--.0142
--.0142
--.1801
--.2707
--.3161
--.3392
--.3514
--.3580
-.3665
.0217
.0426
.0615
.0774
.0905
.1011
.1265
.1386
.1426
.1439
.1444
.1445
.1446
.1446
.1446
--.1366
--.1439
--.1372
--.1292
--.1221
--.1160
--.1010
--.0938
--.0914
--.0901
--.1354
--.1581
--.1696
--.1757
--.1790
--.1833
--.1843
--.1846
--.1848
--.1848
--.1848
--.1848
--.1848
--.1848
1
.0456
2
.0660
3
4
5
.0783
.0872
.0940
6
.0993
10
15
20
25
30
35
40
50
.1120
,1180
.1201
.1208
.1210
.1211
.1211
.1211
.1212
99
.0599
.0593
.0587
.0581
.0570
.0567
.0566
.0565
.0565
.0565
.0565
.0565
.0565
.0994
.1132
.1467
.1650
.1684
.1703
.1709
.1711
.1712
.1712
.1712
--.3687
-.3693
-.3695
--.3696
--.3696
--.3696
--.3696
--.3696
--.0906
--.0903
--.0902
--.0902
--.0902
--.902
Journal of Dairy Science Vol. 71, No. 4, 1988