USDA-DHIA ANIMAL MODEL GENETIC EVALUATIONS

DAIRY HERD IMPROVEMENT
1989

WIGGANS, G. R. AND P. M. VANRADEN

The United States Department of Agriculture (USDA) promotes genetic
improvement of the national dairy cattle population by computing
genetic evaluations of bulls and cows from data provided by Dairy
Herd Improvement Associations (DHIA's) across the United States.
Since July 1989, a statistical technique called the animal model
has been used to compute these evaluations. The animal model
predicts genetic merit of each animal in a population from the
animal's own production records (if available) and the production
records of all related animals.

The animal model replaced the Modified Contemporary Comparison
(MCC) procedure used since 1974. Major benefits of the animal
model are use of all relatives rather than just certain classes of
relatives and use of exact statistical procedures (best linear
unbiased prediction) rather than approximations. This fact sheet
describes how the animal model works, how it differs from MCC, and
what benefits can be expected.(1)

Model

For genetic evaluation of dairy cattle, lactation records are
described by a statistical model. This mathematical description
includes 1) identification of the factors that contribute to the
amount of milk produced in a particular lactation, 2) an indication
of how much of the variation between records is contributed by each
factor, and 3) an indication of how closely factors are related to
each other. The model is the blue print for the evaluations. It
determines how the data will be translated into rankings.

The USDA-DHIA animal model describes a cow's lactation record as
the sum of the effects of her management group (m), genetic merit
(animal effect, a), permanent environment (p), interaction of her
herd and sire (c), and unexplained residual (e). If cow "kl"
(daughter "l" of sire "k") had a lactation in management group "ij"
(year-season, parity, and registry group "j" in herd "i"), her
lactation yield (y) would be represented as:

E1 Yijkl = mij + akl + pkl + cik eijkl

The model does not include effects of age, length of lactation, and
number of milkings per day, because lactation records are adjusted
for these factors prior to genetic evaluation.

Management groups identify which lactations are compared with each
other. They are determined by herd, month of calving, and
lactation number. For Holsteins and Red & Whites, registry status
also is considered. Initially, 2-month seasons are defined for
first and later-lactation groups. If a management group has fewer
than five lactation records included, groups are combined in the
following order: 2 months to 4 months, registered and grade
together, 4 months to 6 months, first and later lactations
together, 6 months to 12 months in steps of 2 months. A group with
three or four lactation records is not combined with another group
if first and later lactations would have to be in the same group.
Lactation records are not included if they cannot be compared with
at least one other record. A management group composed only of
daughters of a single sire does not contribute to that sire's
evaluation.

An animal's genetic merit is the effect of all its genes (breeding
value), not the effect of just the half that progeny receive
(transmitting ability). Each offspring receives a different sample
of genes from its parents, but the average genetic merit of progeny
equals the average of parents' genetic merits. Animals that share
the most genes provide the most information about each other's
genetic merit. Expected fraction of genes that any two animals
share is determined from pedigrees by a method published by USDA
geneticist Sewall Wright in 1922.(2) Figure 1;(small version,
large version) provides an example.

Example pedigrees with expected fractions of genes in common

Pedigrees are traced back as far as 1950 for the animal model.
Every ancestral path eventually ends with unknown parents. Unknown
parents are grouped, and their average merit is used in predictions
for descendants. Groups are formed so that parents expected to have
similar genetic merit are in the same group. Because of genetic
improvement over time, more recent unknown parents have higher
genetic merit. Therefore, unknown parents are grouped by birth
year of their progeny, and several birth years may be included in
the same group to insure that estimates of unknown-parent group
effects are stable. Unknown parents also are grouped according to
sex of the parent and sex of the animal itself. Separation by sex
is necessary because the average merit of bulls (sires) usually is
greater than that of cows (dams). For Holsteins, separate groups
are defined for animals of U.S. and Canadian origin.

For most breeds, sires of cows as well as sires and dams of bulls
usually are known. Lactation records of cows with an unknown sire
are eliminated in initial editing. Parents without yield records
and not related to at least two animals with yield records also are
assigned to unknown-parent groups because they do not contribute
any information to predictions. These unknown-parent groups are
most important for grade animals because grades often are missing
pedigree data.

The model includes two effects for each cow: genetic merit and
permanent environment. The analysis is able to differentiate
between these effects because the animal's genetic merit is
correlated with its relatives' genetic merit, whereas its permanent
environmental effect is assumed to be uncorrelated with those of
its relatives. Effects of permanent environment, herd-sire
interaction, and unexplained residual are assumed to be mutually
independent and also independent of a cow's genetic ability. If
cows are given special treatment because of high genetic
evaluations, the assumption of independence will not be true, and
subsequent genetic evaluations could be biased.

Heritability (h2) is the proportion of differences between records
due to genetics. For the animal model, h2 has been set at .25, a
compromise between the higher effective h2 used for computing MCC
sire evaluations and the lower h2 for MCC cow evaluations. The
environmental correlation between daughters of a sire in the same
herd (the proportion of variation due to herd-sire interaction or
c2) is .14 as in the MCC. Accounting for c2 limits the magnitude
of an evaluation for bulls with daughters in only a few herds or
unequal daughter distribution among herds. The correlation between
repeated records of the same cow (repeatability or r) has been set
at .55 as compared with .50 for the MCC. This repeatability is the
sum of h2, c2, and also p2 (.16), the proportion of variation due
to permanent environment. The same h2 c2 and p2 are used for milk,
fat, and protein.

Data

Data for animal model evaluations include lactation yield
information (milk, fat, and protein) and pedigree information.
Lactations back to 1960 calvings and pedigree data back to 1950 are
included. Ancestors must be evaluated to account for selection.
Lactation records for cows with a missing first-lactation record
are excluded from evaluations that affect relatives to reduce
selection bias. A bias could occur if daughters with missing first-
lactation records were genetically superior to the average of the
bull's daughters as the result of culling on first-lactation
performance. Lactations after fifth are excluded because of their
reduced value in estimating genetic merit. Relatively few animals
in a herd have more than five lactations; therefore, few
contemporaries of similar age are available for valid comparisons.
In addition, the influence of environmental effects on the record
increases with each lactation.

Solutions

Solutions are obtained by a process of repeated calculations called
iteration. Initially, ali estimates of model effects are 0 or their
value from the previous evaluations. Values computed in the first
round of iteration are used in the second round; those from the
second in the third; etc. Iteration continues until the differences
in solutions between rounds becomes acceptably small. Iteration
allows the contribution of each animal to be passed on to all its
relatives.

Management group effect is estimated as the weighted average of
differences between lactation yield and other effects in the model
over all lactations in the management group:

E2 m = [E][wlen(y -a - p -c)][E]wlen
where a [/\ indicates an estimate of the effect, [E] indicates summation, and
the lactation length weights (w ) depend on number of days in milk, type of
test, and parity.

Permanent environmental effect is predicted as the weighted sum of the
differences between a cow's lactations and the other effects in the model
divided by the sum of the weights plus a variance ratio, (1 - r)/p2= 2.8, that
tends to reduce the magnitude of the estimate:

p = [E][wlen(y -m -a -c)]/([E]wlen + 2.8)

Herd-sire interaction effect is predicted as the weighted sum of differences
between lactation yield and the other effects in the model for all the lacta-
tions of all a bull's daughters in a herd divided by the sum of the weights
plus a variance ratio, (1 - r)/c2 = 3.2:

c = [E][wlen(y - m -a - p)]/([E]wlen + 3.2)

An animal's predicted genetic merit (a) is computed as a weighted combination
of three sources of information: 1) average of its sire's (a) and dam's (ad)
predicted merits, 2) its yield deviation (YD) where:

YD = [E][wlen(y - m - p - c)]/[E]wlen

and 3) average of contributions from progeny. A progeny contribution is twice
the progeny's predicted merit (a) minus the mate's predicted merit (a );
estimates of unknown-parent group effects are substituted for unknown parents
or mates. Mathematically, a can be represented as:

a = w [(a + a)/2] + w(YD) + w3(2a -a)

where the w's are weighting factors in fractional form that sum to 1. For wi,
the numerator is 2 if both parents are evaluated, 4/3 if only one parent is
evaluated, or 1 if neither parent is evaluated. For w2, the numerator is ([E]w
)[h2/(1-r)]. For w3, the numerator is half the number of progeny, but progeny
of unknown mates count only 2/3. The three w's have the same denominator,
which is the sum of their numerators. Predicted merit for animals without
records (for example, bulls) is computed the same way except that w2 is 0.

Although only parents and progeny appear to be included in an
animal's predicted genetic merit, all relatives do contribute.
Information from more distant relatives is included through the
animal,s parents and progeny because the evaluation of each parent
or progeny includes its parents and progeny. This is an application
of the method developed by Dr. C.R. Henderson for including all
relatives when inbreeding is ignored.(3)

Predictions for the Holstein and Red & White breeds are computed
jointly so that the many relationships across the breeds can be
included and the predictions compared across breed.

Computation of Final Evaluations

To compute final evaluations expressed as transmitting ability,
predictions of genetic merit for each breed first are adjusted so
that cows born in 1985 average 0. The YD is adjusted by the same
amount. This adjustment imposes a genetic base, a reference point
for comparison of animals. The base is labeled by breed and year
that the base was changed. In anticipation of future base changes
on years evenly divisible by 5, the genetic base for the first
implementation of the animal was designated as "90" rather than
"89." For example, A90 denotes the genetic base for Ayrshires, for
which the average evaluation of Ayrshire cows born 5 years
previously (1985) is set to 0. For the combined evaluations of
Holsteins and Red & Whites, the base is labeled "HW."

Adjusted predicted genetic merit (breeding value) is divided by 2
to obtain predicted transmitting ability (PTA):

E3 PTA = [a]/2
= w1(PA) + w2(YD/2) + w3(2PTAp-PTA)m

where PA = (PTAs + PTAd)/2, the average transmitting ability of the sire
(PTA) and the dam (PTAd) or parent average; PTA is PTA of a progeny; and PTAm
is PTA of a mate (the progeny's other parent). This formula differs from the
formula for a in that YD is divided by 2. For the other terms, division by 2
is not necessary because they already are expressed as transmitting abilities.
The term PTA is used for both cows and bulls. Comparison of PA with PTA
indicates the impact of progeny and records on an animal's evaluation

TABLE 1. Averages of standardized yield traits for cows
born in 1985 by breed.

Breed Milk Fat Fat Protein Protein
(lb) (lb) (%) (lb) (%)
Ayrshire 13848 535 3.86 454 3.28
Brown Swi 15907 613 3.85 550 3.45
Guernsey 12715 574 4.51 441 3.47
Holstein 19004 681 3.58 589 3.1
Red & White
Jersey 12855 606 4.71 473 3.68
Milking S 13673 493 3.61 448 3.27

The PTA's for fat and protein percentages are derived from yield
evaluations combined with first-lactation, mature-equivalent yields
of cows born in 1985 for the appropriate breed (Table 1). The PTA
for protein yield is calculated as a function of the PTA for
protein percentage (computed only from records with protein
information) and PTA for milk yield computed from all records so
that PTA's for milk and protein yields have a similar basis. If an
animal does not have protein information, its PA for protein
percentage is used to estimate a PTA for protein yield.
Frequently, if an animal does not have a protein evaluation,
neither does its dam; therefore, the PA for protein percentage
would include an unknown-parent group estimate. For bulls, this
process is particularly important because the subset of daughters
with protein information could be quite different from his complete
set of daughters.

The requirement for first-lactation data is relaxed for protein
evaluations. The problem of selection bias is not expected to be as
great for protein as for milk and fat. Widespread protein testing
was introduced relatively recently; therefore, selection emphasis
on protein has been less. In addition, cows are required to have
first-lactation milk and fat data for their protein data to be
included in computing evaluations for protein percentage.

Economic indexes called PTA dollars (PTA$) are computed from PTA's.
Separate PTA$ are calculated for milk and fat; milk, fat, and
protein; and cheese yield. Percentiles are based on the PTA$ that
includes milk, fat, and protein. For bulls, rankings are based on
bulls that were in active artificial-insemination service following
the previous evaluation. For cows, rankings are based on cows with
recent lactations.

Predicted producing ability (PPA) includes predictions of c, p, and
a. The PPA minus twice PTA is the sum of estimates for c and p.
Thus, PPA is useful as an indicator of future production of a cow
and for determining estimates of other effects in the model.

Daughter yield deviation (DYD) is the weighted average YD of a
bull's daughters adjusted for merit of their dams. This adjustment
for merit of mates is not included in YD. The DYD provides an
indication of the performance of the bull's daughters without
consideration of his parents or sons. The animal model's YD is
similar to MCC's Modified Contemporary Deviation. Table 2 lists
animal model information that is distributed to the dairy industry.

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PLEASE NOTE: THE FOLLOWING TABLE IS WIDER THAN THE SCREEN. USE THE
RIGHT ARROW KEY TO VIEW THE RIGHT SIDE OF THE TABLE. TO
PRINT THE ENTIRE TABLE YOU MUST FIRST EXPORT THE TABLE
AND USE A WORD PROCESSOR OUTSIDE THE RETRIEVAL SYSTEM.
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TABLE 2. Information generated from the USDA-DHIA animal model.

Animal model information Description
Predicted transmitting ability (PTA) One-half breeding value; adjusted so
that cows born in 1985 average 0
Average standardized yield Averaged over lactations for cows and
over daughters for bulls
Predicted producing ability (PPA) Prediction of a cow's performance in
future lactations; sum of predictions of
breeding value, permanent environmental,
and herd-sire interaction effects
Yield deviation (YD) Weighted average yield adjusted for
management group, permanent
environmental, and herd-sire interaction
effects
Daughter yield deviation (DYD) Weighted average of YD's of a bull's
daughters adjusted for merit of their
dams
Parent average (PA) Average PTA of parents; if either parent
is unknown, unknown-parent group effect
is substituted
Reliability (REL) A measure of amount of information in
the evaluation; same value for milk and
fat PTA
RELPA Amount of information in PA; calculated
as one-fourth of sum of parents' REL's
PTA dollars (PTA$) Economic index combining evaluations for
milk and components weighted by product
value; calculated as for Modified
Contemporary Comparison
Percentile Ranking based on PTA$ that includes
milk, fat, and protein

RELPA Amount of information in PA; calculated
as one-fourth of sum of parents' REL's

PTA dollars (PTA$) Economic index combining evaluations for
milk and components weighted by product
value; calculated as for Modified
Contemporary Comparison

Percentile Ranking based on PTA$ that includes
milk, fat, and protein

Indication of Accuracy

The measure of amount of information in an animal,s evaluation is
called reliability (REL). The method for computing REL is an
extension of the MCC procedure for Repeatability (RPT). In addition
to sources of information used in MCC RPT, REL includes
contributions from parents and sons for bulls and from progeny for
cows. Unknown-parent groups do not contribute to REL.

Including progeny for cows means that cows producing many progeny
through embryo transfer now can attain high REL. The name was
changed because (1) REL generally is higher than RPT as a result of
contributions from additional relatives and (2) confusion existed
between "Repeatability" for measuring accuracy and "repeatability"
for defining similarity between repeated records.

For an animal with no records or progeny information, REL is
one-fourth the sum of parent REL's, which also is REL of PA.

For animals with more information available than just that from
parents, REL is computed from daughter equivalents. Daughter
equivalents provide a common unit for measuring the amount of
information contributed by an animal's parents, its own records,
and its progeny.

The amount of information that a sire receives from any one herd is
limited because the model includes an effect for interaction of
herd and sire. Table 3 shows an example of number of daughter
equivalents that would be contributed to a sire by daughters in the
same herd. This example assumes that each daughter has one record,
a dam with a known PTA, and a large number of management group
mates that does not include paternal half-sibs. For these
conditions, daughter equivalents contributed to the sire are
calculated as 1/[.16 + (.84/d)], where d is the number of sire's
daughters within the herd.

If any daughter has more than one record, d is replaced by the sum
of

E4 1/[.39 + (.61/[E]wlen)].

TABLE 3. Example daughter equivalents contributed to
sire by daughters in the same herd.

Daughter
Number of equivalents Daughter
daughters contributed equivalents
in herd to sire per daughter
1 1 1
2 1.6 0.81
5 2.9 0.58
10 4 0.4
25 5.1 0.2
50 5.6 0.11
100 5.9 0.06

Table 4 shows example daughter equivalents contributed to a cow by
various relatives. As REL of a relative's evaluation increases, so
do the number of daughter equivalents that it contributes. An
animal's REL can be calculated by summing the daughter equivalents
(n) from all sources (parents, own records, and progeny) and then
applying the formula REL = n/(n+14).

Comparison of Animal Model and MCC Procedures

The animal model evaluation system has similarities to MCC.
Later-lactation records are included as is a herd-sire interaction
effect. provision is made for comparing a lactation with
appropriate first or later lactations of contemporaries. The
method for computing the measure of the amount of information in
animal model evaluations is an extension of the MCC method for RPT;
however, contributions from more relatives are included in the
animal model, and the name has been changed to REL.

Differences between the two methods are in Table 5. The primary
difference is that the animal model uses additional information and
employs many rounds of iteration to improve accuracy and to insure
that information from each animal is included in evaluations of all
its relatives.

The changes between the MCC and animal model genetic bases are in
Table 6 by breed and yield trait.

TABLE 4. Example daughter equivalents contributed to cow reliability (REL) by
various sources of information.

Relative Information available Daughter equivalents
Parents Sire with 70% REL and dam with 30% REL 4.7
Sire with 99% REL and dam with 50% REL 8.3
Sire with 99% REL and dam with 99% REL 14
Self 1 lactation record 4.7
3 lactation records 7.8
5 lactation records 9
Daughter 1 lactation record 1
3 lactation records 1.5
5 lactation records 1.7
Son 1 daughter with 1 lactation record 0.2
10 daughters in 10 herds, each with 1 lactation 1.8
50 daughters in 50 herds, each with 1 lactation 4.4
100 daughters in 100 herds, each with 1 lactation 5.4
Evaluation with 99% REL 7

TABLE 5. Differences between animal model (AM) and Modified Contemporary
Comparison (MCC) evaluations.

Characteristic AM MCC
Animals evaluated All (simultaneously) Recent only (bulls, then cows)
Merit of mates considered Yes No
Dams contribute to sons Yes No (ancestor merit includes
maternal grandsire)
Sons contribute to parents Yes No
Daughters contribute to dams Yes No
Base definition Cows born in 1985 Bulls weighted by number of
daughters first calving
in 1982
Environmental group Management group Contemporary group (5 mo, cen-
definition (registered-grade, 2 mo, tered, some contribution of later
first-later lactation, lactations to first, cow excluded)
groups combined to
include 5 lactations, cow
included)
First lactation required Yes (cows without first No (cows without first lactation
lactation records records receive less weight)
evaluate separately)
Lactations included 1-5 1-15
Later herd lactations Yes (in supplemental Yes
included evaluation)
Reliability components:
Parents for males Yes No
Daughters for females Yes No
Sons Yes No

Supplemental Evaluation

Lactation records for cows without a first-lactation record are not
included in the main evaluations so that selection bias can be
minimized in relatives' evaluations. However, these records are
used in calculating a supplemental evaluation for the cows so that
they can have the most accurate evaluation possible (unless the
records are unrepresentative, perhaps because of preferential
treatment). A cow with a missing first-lactation record may also
have a main evaluation if she has progeny, but this evaluation
would not include any of her lactation records.

For supplemental evaluations, a, p, and c are computed with
management group estimates and predicted genetic merit of relatives
from the main evaluation. For cows that change herds, supplemental
evaluations also are computed to calculate a common p across herds.
A common p is not predicted in the main evaluation because of
computational complexity.

TABLE 6. Changes between the genetic bases for
the Modified Contemporary Comparison
(PD82) and the animal model.

Breed Milk Fat Protein
(lb) (lb) (lb)
Ayrshire 89 1.1 0.4
Brown Swiss 170 8.3 3.7
Guernsey 196 7.7 3
Holstein 200 6.8 3.1
Jersey 224 10 3.7
Milking Shorthorn 352 10.3 12.1
Red & White 742 25.6 19.8

Supplemental evaluations are restricted to cows born in the
preceding 10 years (that is, those that might add lactation
records). Older cows still being milked probably would have
exceeded the five-lactation limit and, therefore, would not add
information.

Summary

The animal model allows simultaneous genetic evaluation of bulls
and cows with all relationships included. Previously computing
constraints limited this approach to evaluation within herd.
Recent advances in animal breeding theory and increased computer
capacity made animal model evaluation computationally feasible for
national data sets.

The USDA's Animal Improvement programs Laboratory developed an
implementation of the animal model that promises to improve the
accuracy of evaluations of U.S. dairy cattle. Supplemental
information is provided to assist in tracing the source of an
individual evaluation.

The REL reflects amount of information included in an animal's PTA
but not the quality of that information. No known system or
measure of accuracy can account for manipulation or
misrepresentation. Computers can aid greatly in breeding
decisions, but subjective judgment on credibility of original data
still is required. For data that follow the assumptions of the
model, evaluations computed with the animal model offer the best
predictions of future performance.
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%f TITLE;USDA-DHIA ANIMAL MODEL GENETIC EVALUATIONS
%f COLLECTION;DAIRY HERD IMPROVEMENT
%f ORIGIN;Iowa
%f DATE_INCLUDED;June 1992