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Approximate accuracies of prediction from random regression
models
*
J. Jamrozik , L.R. Schaeffer, G.B. Jansen
Centre for Genetic Improvement of Livestock, Department of Animal and Poultry Science, University of Guelph, Guelph, ON, Canada N1G 2 W1
Received 29 June 1999; received in revised form 2 December 1999; accepted 8 December 1999
Abstract
A procedure for obtaining approximate reliabilities of estimated breeding values under a random regression model is presented. The method is based on a concept of an equivalent number of progeny, with subsequent selection index approximation of reliability utilising equivalent progeny information on the animal and its parents. The accuracy of the proposed approximation was tested using a multiple trait random regression test day model for dairy production traits applied to Canadian Jersey data. Gibbs sampling method was used to generate exact reliabilities of genetic evaluations for several traits derived from the genetic random regression coefficients. The approximation was shown to be relatively unbiased for both bulls and cows. The method has been implemented in the Canadian test day model for dairy production traits. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Random regression model; Prediction error variance; Reliability of estimated breeding value; Dairy cattle; Test day yields
1. Introduction is a function of prediction error variance (PEV).
Variance of prediction error can be computed from Accuracy of an estimated breeding value (EBV) is the elements of the inverse of the mixed model an important tool in a breeder’s decision making equations (MME), but this is often difficult (or process. Accuracy is usually expressed as reliability nearly impossible) in practical applications due to the of prediction, which is the squared correlation be- size of the matrix to be inverted. Several methods for tween true and estimated breeding value. Reliability approximating PEV under large animal models have been developed. The method of Meyer (1989) is based on the approximation of the diagonal elements of the MME by absorbing information through the
*Corresponding author. Tel.:11-519-824-4120; fax:1
1-519-relationship matrix. Misztal and Wiggans (1988)
767-0573.
presented an iterative algorithm for approximation of
E-mail address: [email protected] (J.
Jam-rozik). the reciprocal of prediction error variance that
bines contributions due to production records and 2. Material and methods due to relationships. The method of Gengler and
Misztal (1996) uses multiple diagonalization for 2.1. Model approximation of reliabilities in multiple trait
models. Let y5Xb1Wp1Za1e be a general form of a
Daughter equivalents were used by Van Raden and RR model, where y is a vector of observations, b is a Wiggans (1991) to combine information from ani- vector of fixed effects, p is a vector of random mal’s own records, parents and progeny adjusted for regression coefficients for permanent environment mates. One daughter equivalent was the amount of (PE), a is a vector of random regression coefficients information contributed to a parent by a daughter for animal genetic effect, and e is a vector of residual having one record, infinite number of management terms.
group contemporaries and the other parent with Also, Var(y)5WPW9 1ZGZ9 1R, where P5
perfect reliability. A concept of equivalent number of I^P , G5A^G ; A is a an additive relationship
0 0
progeny (ENP) on the animal has been used in matrix, P0 and G0 are covariance matrices for approximation of accuracy of genetic evaluation for environmental and genetic regression coefficients, a single trait animal model by Koots et al. (1997). respectively. The model can be a single trait or a The general idea behind this approach is to convert multiple trait RR model. PE and genetic regressions the number of records on an animal into a corre- can be the functions of the same class (e.g. polyno-sponding ENP that would give the same reliability if mials of the same degree) or different sets of progeny were the only information available, and to functions can be used to model these two effects. accumulate this quantity over all progeny of an Let k9a define a trait of interest from RR model
i
2
animal. Selection index approximation is then ap- for animal i, and let h be heritability of this trait. plied to the ENP of an animal, its sire and dam. A Let g5k9G k denote the variance of k9a , and
0 i
2 2
similar approach has been suggested by Graser and a 5(42h ) /h .
Tier (1997) for a variety of different models, includ-ing models for multiple traits. An equivalent method
2.2. Approximation of reliabilities of combining reliabilities of three information
sources (animal’s own records, progeny records and
The procedure for approximating reliabilities of parent average) has been recently presented by
EBV for k9a can be described as follows:
i Harris and Johnson (1998).
Step 1. Estimate ENP due to animal’s own re-Random regression (RR) models are already
ap-cords, for each animal with performance data. plied on a routine basis for genetic evaluation of
dairy cattle (Schaeffer et al., 2000). Calculation of
21 9
(a) Form the coefficient matrix C5Z R Z 1
PEV in a RR model follows the same principle as for i i i i 21
G of the subset of a MME corresponding to an a regular animal model. The problem, however, 0
animal genetic effect only. becomes more difficult computationally because of
(b) Absorb equations for p (environmental re-the increased size and re-the complicated structure of i
gression coefficients pertaining to animal i ) into the mixed model equations. No approximations have
21 21
procedure for approximating reliability of genetic (c) Calculate the PEV for an animal i as m5
i evaluations from RR models. The method was k9D k. Given no other sources of information
i
applied to the RR model for analysis of test day (pedigree data) reliability of k9a can be
approxi-i (TD) yields of dairy cattle. Accuracy of the approxi- mated as 12m /g.
i
mation was tested on a Canadian Jersey data set, by (d) Determine the ENP for the ith animal as comparing approximate reliabilities with reliabilities ENP5a ( g /m 21). This is the number of
i i
reliability level as was achieved from the animal’s This method of combining contributions is equiva-phenotypic records through C .i lent to setting up a selection index based on q ,i
q 2x and q 2x progeny records on the animal, its
s d
sire and its dam, respectively, with q taken afteri Assume that the complete data file has been
Step 2. In addition, the selection index formula used processed and the ENP from own records is stored
by Harris and Johnson (1998) to combine re-for all animals. For brevity, let q5ENP for the trait
i i
liabilities from two pieces of information could be of interest. Progeny equivalents for all animals with
simplified by converting each reliability to an ENP data and the pedigree information can now be
and then computing the combined reliability from the combined through a selection index approximation
sum of the ENP. Therefore, the current method is (Koots et al., 1997). Contributions of information
equivalent to that of Harris and Johnson (1998), from own records, ancestors or collateral relatives,
although the algorithm is slightly different. and descendants are converted to ENP for parent
average, data and progeny in a manner similar to Van
2.3. Application of the method Raden and Wiggans (1991) and Graser and Tier
(1997).
The method for approximating reliabilities was
Step 2. Progeny contributions to parents are
ac-applied to the Canadian TD multiple trait, multiple cumulated while processing the pedigree file
sequen-lactation model for dairy production traits. Data tially from youngest animal to oldest. This ensures
consisted of 147 346 TD records from the first three that contributions from all generations of
descen-lactations on 10 193 Canadian Jersey cows that dants are accounted for.
calved after January 1st 1988. This subset of the national data base was formed by randomly selecting (a) The animal’s reliability based only on its
25% of all herds. Pedigree data included 23 632 *
records and progeny is computed as R 5q /
i i
animals in total: 1656 bulls and 21 976 cows. ( qi1a).
Contemporary groups (18 509) were formed on a (b) This animal’s contribution to its parents’ ENP
basis of a herd-test date-parity class. Daily milk, fat
* *
is calculated as x5aR /(42R ) if both parents
i i
and protein yields (kg) and somatic cell score (on the
* *
are known, or x52 / 3aR /(42R ) if only one
i i
log scale) in different lactations were considered as2
parent is known.
different traits. The model equations were assumed (c) ENP of the animal’s known parents are
to be the same for all four traits in any parity. For updated as q 5q 1x and q 5q 1x.
s s d d
trait h in lactation n it was
y 5HTD 1 Sb z 1 Sa z
Step 3. Finally, contributions of all ancestors and hntijkl hni hnkm tm hnjm tm collateral relatives are accumulated by processing the 1 S
p z 1e ,
hnjm tm hntijkl pedigree file sequentially one more time.
where yhntijkl is record l on cow j made on day t *
(a) Ri and x are re-computed as in Step 2. within herd-test day effect i, for a cow belonging to (b) Reliabilities for sire and dam are computed subclass k for region–age–season of calving, without the contribution of this progeny. If the HTDhni is fixed herd-test date-parity effect, b are
*
sire is known, Rs 5( qs2x) /( qs2x1a), other- fixed regression coefficients specific to subclass k, *
wise R 50. Similarly, if the dam is known a are random regression genetic coefficients
s hnjm
* *
R 5( q 2x) /( q 2x1a), otherwise R 50. specific to cow j, p are random regression
d d d d hnjm
(c) The contribution of the parent average is coefficients for permanent environmental effect on added to the ENP of this progeny as q 5q 1 cow j, e is residual effect for each observation,
i i hntijkl
* * * *
a(R 1R ) /(4 2R 2R ). z are covariates, assumed to be the same for fixed
s d s d tm
(d) The final reliability is calculated from the total and random regressions.
Table 1
age at calving–season of calving–parity) and
de-Heritability and genetic variance of protein yield (PROT), somatic
viates for both genetic and environmental effects
cell score (SCS) and persistency (PERS) for Canadian Jersey
were modelled by Wilmink’s function (Wilmink, breed 1987), w(t)5w 1w t1w exp(20.05t). Details of
0 1 2
Heritability Genetic variance
the model, genetic parameters and computing
strate-PROT 0.39 385.0
gies for solving the mixed model equations were
SCS 0.25 0.32
described by Schaeffer et al. (2000).
PERS 0.34 3.34
EBV’s for animals’ genetic random regression coefficients were used to estimate breeding values
for three traits: PROT5average 305d protein yield, each of the combined traits. Additionally, PROT,
SCS5average daily somatic cell score, PERS5 SCS and PERS are the traits for which reliability of
average persistency of lactation defined as a differ- EBV’s are actually published for dairy bulls in the ence in milk yield between days 280 and 60 of Canadian genetic evaluation system for production
9
lactation. The vector k used to calculate the EBVi traits.
9
for trait i was defined as k 5w M , where w The effect of inbreeding on genetic variances and
i i i i
represented weighting factors for each lactation and covariances is taken into consideration in the predic-M described how the within lactation EBV’s werei tion of breeding values by Schaeffer et al. (2000) but calculated from genetic RR coefficients. Assuming it has been ignored in this study. Inbreeding could be that elements in the vector of genetic RR coefficients taken partially into account in Step 1 by multiplying were ordered by lactation number, milk, fat, protein g and G by 11F , where F is the inbreeding
0 i i
and SCS within lactation, and w , w and w within0 1 2 coefficient of animal i. each trait, the values of M and k9 for PROT, SCS
and PERS were as follows: 2.4. Validation of the approximation
9
w 5(1 / 3 1 / 3 1 / 3), M
PROT PROT Reliabilities for EBV’s of PROT, SCS and PERS 1
were estimated using the Gibbs sampling method
5 S [0 0 f9 0], with f9
PROT PROT
(Gelman et al., 1995) and for purposes of this paper
5(305 4665 19.50),
were defined as ‘exact’ reliabilities. The same data and model as described earlier were applied.
Vari-9
w 5(0.25 0.65 0.10), M
SCS SCS ances and covariances were assumed to be known 1
5 S [0 0 0 f9 ], with f9 without error. The Gibbs sampling scheme involved
SCS SCS
sampling from the multivariate normal distributions
5(1.0 153 0.064),
for all location parameters. This was conducted while performing Gauss–Seidel iteration on the
9
w 5(0.5 0.25 0.25), M
PERS PERS MME. Four independent chains each of length 5500 1
5 S [f9 0 0 0], with f9 were generated. The same starting values for all
PERS PERS
solutions were used for each chain (estimates from a
5(0 220 20.050),
national evaluation run) with different seeds for the
1
random number generator. Convergence of Gibbs whereS is the direct matrix sum operator (Searle,
iteration was established after 2000 rounds of itera-1982). See Schaeffer et al. (2000) for more details
tion for each chain based on overall averages of on the trait definitions.
posterior means for variances and reliabilities of The genetic variances and heritabilities for PROT,
EBV for PROT, SCS and PERS (Table 2). Fourteen SCS and PERS, as shown in Table 1, were estimated
thousand samples (3500 from each of four chains, as in Jamrozik et al. (1988). These three traits
combined) were used to estimate posterior variances differed in the way they were created from the
for PROT, SCS and PERS for all animals in the data random regression coefficients for animals. Different
set. Under the assumptions of normality and known amounts of information related to the shape of a
Table 2 standard deviations of reliabilities obtained by the Posterior means of average (N523 632 animals) variances of method of batch means were 1.2, 1.2, and 1.0 for EBV, and reliabilities (R), for protein yield (PROT), somatic cell
PROT, SCS and PERS, respectively. Corresponding
score (SCS) and persistency (PERS), estimated from four
differ-standard deviations of these estimates were 0.59,
ent Gibbs chains
0.62 and 0.53. This indicated that Monte Carlo errors
Chain Variance R
were small and that Gibbs estimates of reliability can
PROT SCS PERS PROT SCS PER be considered as good measures of exact reliabilities 1 249.0 0.24 2.34 35.4 26.0 30.0 for the purpose of comparisons.
2 249.6 0.24 2.34 35.3 26.5 30.1 An overall comparison of exact and approximate 3 249.2 0.24 2.35 35.4 26.7 29.9
reliabilities is given in Table 3. Average error of the
4 250.0 0.24 2.32 35.2 26.5 30.5
approximation was negative but small (less than 1 on a 1–100 scale), whereas the SD of errors ranged ances of the EBV are the variances of prediction from 2.4 to 3.3 units. The largest values of error error. Reliabilities were estimated as R5100*(12 were positive for all traits and both sexes. Closer
i
PEV /g), where PEV is the prediction error variancei i examination of animals with extreme errors of for animal i, g is the genetic variance for a given approximation revealed that they were cows with all trait. Monte Carlo variances of exact reliability TD records in single herd-test day-parity subclasses estimates were computed using the method of batch (no contemporaries) or bulls with all daughters in a means (Geyer, 1992). Four independent Gibbs single herd. Relatively large overestimation of re-chains after burn in Table 2 were defined as batches liability for these animals was caused by the fact that for this purpose. our procedure for calculating ENP does not take into account distribution of records over contemporary groups. Cows are usually assigned to many HTD 3. Results and discussion classes in the TD model which is in contrast with the lactation model. Although absorption of HTD effect The approximate reliabilities for PROT, SCS and (Step 1b) is theoretically possible, this would in-PERS were compared with exact values obtained crease significantly the computing requirements for from the Gibbs scheme. The estimates of exact the approximation method.
reliabilities used in this study are subject to Monte Correlations between exact and approximate re-Carlo errors. Average estimates of Monte re-Carlo liabilities were very high, all larger than 0.98. The approximation performed better for bulls than for cows, which was mainly reflected by lower values of
Table 3
maximal differences. Using similar approximation
Comparison of exact and approximate reliabilities of EBV for
for simpler, single trait animal model, Koots et al.
protein yield (PROT), somatic cell score (SCS) and persistency
(1997) found correlations of 0.993 and 0.970 for
(PERS)
Holstein bulls and cows, respectively, whereas Harris
PROT SCS PERS
and Johnson (1998) reported a correlation of 0.998 a
Bulls Mean error 20.72 20.08 20.42
over all animals in their study. Harris and Johnson
(N51656)
(1998) also found a smaller SD of errors (1.4 units)
SD of error 2.37 2.83 2.40
b than found in this study, possibly because their
Max error 15.7 26.5 11.5
Correlation 0.996 0.993 0.996 method also accounts for the size of contemporary groups. Comparisons with the multiple trait
approxi-Mean error 20.86 21.02 20.89
mation of Gengler and Misztal (1996) are difficult
SD of error 3.34 2.83 2.99
because their results vary widely depending on
Cows Max error 42.3 33.0 28.9
amount of missing information and method of
cor-(N521 976)
Correlation 0.989 0.986 0.988 recting for missing information. However, they also a reported SD of errors that were generally smaller
Error5(approximate reliability2exact reliability).
b
records and 1.2–2.6 for sires of cows with records in for bulls. Average error and variation of the error for their dairy data). bulls by year of first daughters with data are shown Distributions of differences between exact and in Table 7. Similar statistics for cows (by year of approximate reliabilities are shown in Table 4. Not first TD record) are presented in Table 8. No less than 94% of bulls (90% of cows) had these significant time trend was found in either sex. differences in the (25, 15) interval. The distribu- The procedure of Koots et al. (1997) assumed that
tions were not symmetric, but rather an overall off diagonal element for sire and dam in the 333
underestimation by the approximate reliabilities can selection index approach using the ENP on sire, dam be noticed for both sexes and all traits. Average and the animal in question, is equal to zero. This differences between reliabilities (approximate2 could be modified as shown by Van Raden and
exact), in classes of exact reliabilities (intervals of Wiggans (1991). The possible benefits in accuracy of 10) are shown in Tables 5 and 6 for bulls and cows, the proposed reliability approximation by accounting respectively. The amount of bias and variation of the for the mate reliability (Step 2b) needs further bias seems to be independent of the level of the exact research.
reliability. Small underestimation of exact reliability The proposed method for approximating re-was observed for lower values of exact reliabilities liabilities of EBV’s under the random regression TD for bulls and cows, whereas a slight overestimation model required modest computer resources in com-can be seen for higher levels of reliabilities (. 50) parison with the process of solving the mixed model
Table 4
Distribution of differences between approximate and exact reliabilities for protein yield (PROT), somatic cell score (SCS) and persistency (PERS) (in %)
Interval Bulls Cows
PROT SCS PERS PROT SCS PERS
(250,220) 0.0 0.0 0.0 0.0 0.0 0.0
(220,210) 0.0 0.1 0.1 0.3 0.3 0.3
(210,25) 4.5 2.6 3.4 7.5 6.8 6.9
(25,22) 36.1 32.2 34.1 38.4 41.2 37.9
(22,12) 41.1 37.9 39.7 35.1 33.4 37.1
(12,15) 16.8 23.7 21.0 16.1 16.1 15.6
(15,110) 1.3 3.0 1.6 1.6 1.4 1.5
(110,120) 0.2 0.5 0.1 0.6 0.4 0.1
(120,150) 0.0 0.1 0.0 0.3 0.1 0.2
Table 5
a
Average error in approximate reliabilities for bulls, by class of exact reliabilities (standard deviation in parentheses)
Exact PROT SCS PERS
reliability
N Error N Error N Error
0–10 626 21 (1.3) 703 21 (1.5) 662 21 (1.3)
10–20 140 22 (2.4) 170 0 (3.4) 164 21 (2.8)
20–30 223 21 (2.4) 273 21 (3.1) 247 21 (2.6)
30–40 212 22 (2.6) 199 21 (3.4) 216 22 (2.5)
40–50 136 21 (2.9) 100 1 (3.3) 103 0 (2.8)
50–60 89 0 (3.4) 69 2 (2.6) 84 1 (3.0)
60–70 67 1 (2.7) 65 4 (2.7) 59 2 (2.2)
70–80 74 1 (1.6) 45 3 (1.7) 72 2 (1.5)
80–90 65 1 (1.3) 21 4 (1.2) 33 2 (1.1)
90–100 24 1 (0.4) 11 2 (0.8) 16 2 (0.5)
a
Table 6
a
Average error in approximate reliabilities for cows, by class of exact reliabilities (standard deviation in parentheses)
Exact PROT SCS PERS
reliability
N Error N Error Error N
0 – 10 4207 21 (1.7) 4920 21 (1.6) 4529 21 (1.6)
10 – 20 1392 21 (3.5) 2106 22 (3.8) 1722 22 (3.6)
20 – 30 2871 21 (4.8) 4676 21 (3.4) 3837 21 (3.9)
30 – 40 3269 23 (4.2) 4771 22 (3.0) 4143 22 (3.5)
40 – 50 3017 21 (3.5) 4272 0 (2.3) 3522 0 (2.7)
50 – 60 3544 0 (2.5) 1229 21 (1.6) 3844 0 (1.6)
60 – 70 3637 0 (1.7) 2 22 (0.9) 328 0 (1.6)
70 – 80 39 2 (1.1)
a
Error5(approximate reliability2exact reliability).
Table 7
a
Average exact reliabilities (R) and mean error of approximate reliabilities by birth year of bull’s first daughter with TD data (standard deviations in parentheses)
Year No. PROT SCS PERS
bulls
R Error R Error R Error
No daughters 1035 12 21 (1.6) 9 0 (2.7) 11 21 (1.6)
1989 28 77 1 (3.2) 68 3 (4.2) 72 1 (2.3)
1990 103 63 0 (3.1) 52 2 (3.9) 58 1 (3.1)
1991 97 53 21 (2.9) 42 1 (3.4) 48 0 (2.9)
1992 81 49 21 (2.6) 38 0 (3.7) 43 0 (3.0)
1993 66 50 21 (2.5) 40 0 (2.5) 44 21 (2.4)
1994 53 53 21 (2.5) 44 0 (3.4) 48 0 (2.9)
1995 55 57 21 (3.8) 47 0 (3.8) 51 0 (4.1)
1996 62 52 0 (3.7) 43 0 (3.4) 45 0 (4.0)
1997 53 44 0 (2.8) 34 0 (2.1) 37 0 (2.4)
1998 23 35 23 (1.8) 28 3 (2.0) 29 23 (1.8)
a
Error5(approximate reliability2exact reliability).
Table 8
a
Average exact reliabilities (R) and mean error of approximate reliabilities by year of cow’s first TD record (standard deviations in parentheses)
Year No. PROT SCS PERS
cows
R Error R Error R Error
No TD records 11783 20 22 (2.6) 16 21 (2.4) 17 22 (2.4)
1989 106 56 2 (6.0) 42 1 (4.3) 45 0 (4.6)
1990 804 57 1 (4.4) 41 0 (3.5) 49 1 (4.0)
1991 1038 55 0 (3.5) 40 0 (3.2) 46 0 (3.5)
1992 1331 56 0 (3.0) 40 21 (2.9) 47 0 (2.7)
1993 1348 57 0 (3.1) 41 21 (2.9) 48 0 (3.0)
1994 1442 56 1 (3.8) 41 21 (3.1) 47 0 (3.2)
1995 1270 56 0 (3.4) 42 21 (3.0) 48 0 (3.0)
1996 1280 54 1 (4.0) 40 0 (3.3) 45 0 (3.5)
1997 1071 50 1 (3.5) 36 21 (3.1) 41 0 (3.1)
1998 503 44 0 (3.5) 31 22 (3.3) 34 21 (3.4)
a
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The data was kindly provided by the Canadian
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Dairy Network, Guelph, Ontario, Canada. The au- (Accepted).
thors are grateful to the Ontario Ministry of Agricul- Searle, S.R., 1982. Matrix Algebra Useful for Statistics. John
ture, Food and Rural Affairs, the Cattle Breeding Wiley and Sons, New York.
Van Raden, P.M., Wiggans, G.R., 1991. Derivation, calculation,
Research Council of Canada, and the Natural
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financial support. We also thank Dr. Daniel Gianola Wilmink, J.B.M., 1987. Adjustment of test day milk, fat and for his comments on application of Gibbs sampling protein yield for age, season and stage of lactation. Livest.
