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Model for international evaluation of dairy sires
Livestock Production Science, 12 ( 1 9 8 5 ) 1 0 5 - - 1 1 5
105
Elsevier Science P u b l i s h e r s B.V., A m s t e r d a m - - P r i n t e d in T h e N e t h e r l a n d s
MODEL F O R INTERNATIONAL EVALUATION OF DAIRY SIRES
L.R. S C H A E F F E R
Department of Animal and Poultry Science, University of Guelph, Guelph, Ont. (Canada) (Accepted 8 November 1984)
ABSTRACT Schaeffer, L.R., 1 9 8 5 . M o d e l for i n t e r n a t i o n a l e v a l u a t i o n o f dairy sires. Livest. Prod. Sci., 12: 1 0 5 - - 1 1 5 . A linear statistical m o d e l is p u t f o r w a r d for use in c o m p a r i n g t h e g e n e t i c level o f dairy sires b a s e d o n t h e i r p r o g e n y t e s t e v a l u a t i o n s f r o m o n e or m o r e c o u n t r i e s . A d d i t i v e g e n e t i c r e l a t i o n s h i p s a m o n g bulls are i n c l u d e d t o p r o v i d e m o r e c o n n e c t i o n s o r comp a r i s o n s a m o n g c o u n t r i e s . Several d e f i n i t i o n s o f genetic d i f f e r e n c e s a m o n g c o u n t r i e s are p r e s e n t e d . A small e x a m p l e illustrates t h e use o f this m o d e l .
INTRODUCTION
There has been an increase in the desire to measure the genetic differences among dairy cattle populations, primarily black and white breeds (Gaillard et al., 1977; Dommerholt et al., 1982). This interest resulted in the comparison o f ten strains of black and white cattle in Poland, and a similar trial for red cattle breeds in Bulgaria (Stolzman et al., 1981, 1982; Hinkovski et al., 1979). However, there are a number o f problems with these comparison trials. Each trial has only compared a small group of bulls from each country with small numbers of daughters per bull. Estimates of genetic differences among countries from these trials represent the differences in bull populations at the time when these bulls were entered into artificial insemination. Because rates of genetic change are different for each country and may not be linear with time, then the ranking of bull populations for genetic merit t o d a y may be significantly different from the trial results. In addition to breed comparison trials, m a n y North American HolsteinFriesian bulls or their sons have been progeny-tested in one or more European countries as well as in their country of birth. Imported semen is generally more expensive than domestic semen, and consequently is used to breed the better cows in the herds that can afford to buy the semen. Also, progeny of foreign sires or of any expensive bull may be treated better than
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progeny of other bulls as they grow and as t h e y make their first lactation. These actions can bias the progeny-test results of imported bulls. Statistical models with maternal grandsire effects have been proposed to offset this problem, but a joint sire and cow evaluation model could be better (Hudson and Schaeffer, 1984). Assuming that progeny test results are unbiased there are other issues t h a t could cause problems in comparing bull populations. These include the model and methods of evaluating bulls within each country. Best linear unbiased prediction (BLUP) has been proposed by Gaillard et al. (1977) as well as the factors to include in a model for various traits. Not every country can follow these recommendations, and even if t h e y did, then there could also be differences in heritability values for traits among countries. The previously discussed problem areas suggest that international comparisons o f dairy sires m a y be subject to m a n y errors, and caution should be exercised. Unfortunately, the m o n e t a r y importance of international comparisons to exporting countries may force scientists to make compromises that would otherwise not be made. The objective of this paper is to propose a linear model that may serve as a starting point for discussions on the international comparison o f dairy sires. DATA REQUIREMENTS
For each country t h a t wishes to participate in the international comparison, the data t h a t would be necessary for a successful comparison are described as follows: Ca) Estimated breeding values (EBV) for the trait or traits of interest for every bull in artificial insemination (AI) evaluated within that country. (b) The number of daughters used to calculate the EBV o f each bull, or some other indicator of reliability t h a t each country can easily provide. (c) The country of birth, year of birth, breed category, and sire and maternal grandsire identification of each bull. In some cases the country of birth may be vague, for instance when the sire and dam were born in different countries. A unique sire identification system would need to be implemented to avoid problems. The main problem would be bulls that are registered in more than one c o u n t r y with a different number in each country. Breed category allows for countries which permit crossbreeding and could include different percentages of various combinations of breeds. The traits are assumed to be standardized among countries. For example, milk production may be expressed as 305-day fat corrected milk. Each participating c o u n t r y would be responsible for expressing the EBV in the accepted standard, but within that country any other expression of EBV or ETA could be used. The m e t h o d for converting to the accepted standard from the within-country proofs would need to be documented. The reason for requiring EBV from all AI bulls is to be able to evaluate
107 d i f f e r e n c e s in t h e bases t o w h i c h sires are r e f e r e n c e d w i t h i n a c o u n t r y a n d t o m e a s u r e genetic change in t h e bull p o p u l a t i o n o f a c o u n t r y o n an a n n u a l basis. O t h e r w i s e o n e m u s t a s s u m e t h a t genetic change is linear o v e r t i m e in o r d e r t o adjust p r o o f s t o t h e s a m e base o f c o m p a r i s o n . By having all EBVs o f A I bulls f r o m e a c h c o u n t r y t h e n this a s s u m p t i o n is n o t necessary. T h e i d e n t i f i c a t i o n o f t h e sire a n d m a t e r n a l grandsire (MGS) o f each bull is n e e d e d so t h a t additive genetic r e l a t i o n s h i p s a m o n g bulls can be i n c o r p o r a t e d into t h e analysis. T h e r e l a t i o n s h i p m a t r i x w o u l d p r o v i d e m o r e c o m p a r i s o n s or c o n n e c t i o n s a m o n g c o u n t r i e s t h a n s i m p l y bulls t h a t are p r o v e n in t w o or m o r e c o u n t r i e s . In fact, c o m p a r i s o n s f r o m t h e relat i o n s h i p m a t r i x m a y b e less subject to bias p r o b l e m s t h a n c o m p a r i s o n s f r o m a p o p u l a r bull t h a t has d a u g h t e r s in t w o c o u n t r i e s . F o r this r e a s o n m u c h w o r k should be d i r e c t e d to t h e i n t e r n a t i o n a l i d e n t i f i c a t i o n o f bulls, a n d a u n i q u e r e g i s t r a t i o n n u m b e r s h o u l d b e assigned t o e a c h bull. A n o t h e r r e q u i r e m e n t t h a t is critical t o i n t e r n a t i o n a l c o m p a r i s o n s is c o n n e c t e d n e s s . E a c h p a r t i c i p a t i n g c o u n t r y m u s t h a v e bulls p r o v e n in t h e i r o w n c o u n t r y t h a t e i t h e r h a v e p r o o f s in a n o t h e r p a r t i c i p a t i n g c o u n t r y or have sons or a sire w i t h a p r o o f in a n o t h e r p a r t i c i p a t i n g c o u n t r y . T h e n u m b e r o f such c o n n e c t i o n s o r ties c o u l d b e critical t o t h e reliability o f e s t i m a t e s o f genetic d i f f e r e n c e s a m o n g c o u n t r i e s . C o u n t r i e s t h a t have n e v e r used bulls f r o m a n y o t h e r c o u n t r y c a n n o t b e i n c l u d e d in t h e s e analyses. TABLE I Example data of sire proofs from two countries Countryyear of proof
Bull number
Sire numbet
Country of birth
Year of birth
EBV (kg)
Number of daughters
(n+k)/2n Elements
A-84 A-84 A-84 A-84
1 2 4 6
77 88 77 88
A A A B
80 80 81 80
0 +280 -420 +1160
100 60 70 80
0.5750 0.6250 0.6071 0.5938
0 +175 -255 +689
B~4 B-84 B°84 B-84 B-84
1 3 5 6 7
77 82 60 88 77
A A B B B
80 81 81 80 81
+80 -940 +140 +1250 -560
20 150 100 200 60
0.8750 0.5500 0.5750 0.5375 0.6250
+70 -517 +80 +666 -350
In bulls Note born each
of y
o r d e r t o illustrate t h e m e t h o d o l o g y t o b e d e s c r i b e d , a d a t a set o f f r o m t w o c o u n t r i e s (A a n d B) is p r e s e n t e d in T a b l e I as an e x a m p l e . t h a t bulls 1 a n d 6 w e r e p r o v e n in b o t h c o u n t r i e s , a n d t h a t bull 3 was in c o u n t r y A a n d p r o v e n in c o u n t r y B. Several bulls are r e l a t e d to other.
108 A MODEL Let t h e eq u ati on o f t he model be y = Xc + ZQg + Zs + e where: y = a vector o f observations t h a t are calculated f r o m the EBV o f bulls, the reliability o f th e EBV, and k (the ratio of error to sire variances within a country of proof); c = a fixed vector o f effects for c o u n t r y of p r o o f which is included to acc o u n t for differences in reference bases among countries; g = a fixed v ec t or o f effects f or c o u n t r y and year of birth o f each bull; s = a r a n d o m vect or o f sire effects with mean vector o f null and variance-covariance matrix Aa2s where A is t he relationship matrix and o2s is the sire variance; e = a r a n d o m vector o f residual or error effects with mean vector of null and variance--covariance matrix Do2e where D is a diagonal matrix with elements equal to n~jI , where nij equals the n u m b e r o f daughters for t he j t h sire in th e ith c o u n t r y o f pr oof , and O2e is t h e error of variance. X and Z are design matrices o f zeros and ones; and Q = a matrix t hat describes the group to which a sire belongs. Discussion on the c o n s t r u c t i o n o f Q and potential problems are presented later. Let bij be an EBV of t he ]th bull in the ith c o u n t r y , and let nij be the indicators o f reliability o f bij. Ideally, nij might be t he n u m b e r of effective daughters, b u t perhaps n o t all countries c o m p u t e this n u m b e r in practice, in which case nij could be t he actual n u m b e r o f daughters. T he value of nij could be altered by a constant t hat reflects the relative accuracy o f sire evaluation m e t h o d o l o g y among countries as assessed by some indep e n d e n t study. For example, an EBV f r o m t he United States based on 100 daughters might only be as reliable as an EBV f r o m Sweden based on 80 daughters. Thus all nij f r om t he United States could be multiplied by 0.8. Burnside and H a m m o n d {personal c om m uni cat i on, 1983) have suggested h o w methodologies might be compared by simulation techniques. The elements o f y are t he n
bij(nij +
k)/2nij.
Th e fact that bij in some countries is based on group plus sire solutions a n d / o r relationships among bulls, is ignored and n o t considered i m p o r t a n t provided th at bij is an unbiased estimate o f t he t rue breeding value o f a bull and nij is the relative accuracy indicator. The above elements are readily available f r o m most countries, but m o r e complicated and statistically b e t t e r quantities could be developed if necessary. Th e observation vector, y, for t he example data would be t he last column o f Table I. Assume t h r o u g h o u t t h a t k = 15.
109
Assumptions of the model 1. The primary assumption for any successful comparison of bulls among countries is that the bulls have been randomly mated to cows within a country and that the daughters have not received any preferential treatment duing their growth or lactation performance. There may be ways of measuring or avoiding bias of this type. The sire evaluation methodology of each country is assumed to be able to provide unbiased predictions of genetic values of bulls that are free of all fixed effects important to t h a t country. A description of the methods used by each country is not enough to verify this assumption. A proposal by Burnside and H a m m o n d (personal communication, 1983} was to generate, by Monte Carlo simulation, test data sets to be submitted to each participating country for analysis by their existing sire evaluation methodology. The results would be returned and comparisons made to the true values. Hence, the accuracy of each country's methods could be assessed. 2. The sire and error variances are assumed to be c o m m o n to all countries. That is, the ratio of error to sire variances is assumed to be equal for each country. With the accumulation of data from participating countries these components could be estimated and the assumption could be tested. 3. The variance--covariance matrix o f e was assumed to be diagonal. In actual fact, there are non-zero covariances among the e within a country, particularly if BLUP has been used to estimate the EBVs of bulls. In order to account for these covariances, the inverse to the entire mixed model equations for each country would be needed. Most countries that use BLUP, however, use iteration techniques to solve the equations rather than a direct inverse. Ignoring covariances will not bias results, but could increase the prediction error variances. 4. Several interactions are assumed to be unimportant. Genotype by environment interactions, country of proof by country of birth interactions, and specific combining abilities have been ignored. An investigation of these interactions could be made with this data. Each of these assumptions has also been imposed by Dommerholt et al. (1982). COUNTRY AND YEAR OF BIRTH GROUPS Assigning bulls to groups according to country and year of birth could be more difficult than at first appears. For humans at least, the country of birth is the country of citizenship unless the parents are citizens of another country. Then an individual may decide on a country of citizenship at age 18 years. In dairy bulls the same problems exist. For example, a U.S.-born bull from U.S.-born parents may be sold to Canada and progeny-
II0 tested only in Canada. Another case is where one parent may be U.S.-born and the other may be Canadian-born and the offspring may be physically born in a third country, such as Mexico. There are several situations in which country o f birth assignment may be questionable. However, the matrix Q t h a t defines groups can contain values other than zero or one. Suppose we have three groups, say A, B, and C, and a bull whose parents were both born in country A, then the row of Q for that bull would be
(1 0
O)
gA gc
A bull with a sire born in country B and a dam born in country C, then the row of Q for t h a t bull would be (0 0.5
0.5)
gA gc
If that bull also had a son, then Q could contain values of 0.25, etc. The possibilities could become very messy. Another possibility could be to group sires by breed composition and year of birth. For example, in the Netherlands there can be Dutch Friesians, North America Holsteins, the MRY breed or combinations of crosses of these three breeds of various percentages, but all could be born within the Netherlands. Thus, breed groups within country of birth may be necessary. Other schemes could be proposed, but the final strategy should be the result of collaborative work among researchers from the participating countries. MIXED MODEL EQUATIONS The equations that must be constructed and solved are as follows:
Vxo x x o zo x o,z ioz
0zo,z
/zo,z /
L_Z'D-'X
Z'D-1ZQ
Z D-' Z+A-lk_]
Iil
= i0zo , I [_Z'D-1Y
These equations could be very large, but solutions should be possible on modern computers with five to ten megabytes of memory. The international genetic values o f bulls would be calculated as: t = Q$ + $ for ETA values, or 2t if one wants EBV values. One of the group solutions would have been set to zero and would be the base for international comparisons.
T A B L E II Mixed m o d e l e q u a t i o n s for e x a m p l e d a t a 310
0 530
symmetric
160 20 180
70 150 0 220
80 200 0 0 280
0 160 0 0 0 160
100 20 120 0 0 0 140
60 0 60 0 0 0 0 80
0 150 0 150 0 0 0 0 170
70 0 0 70 0 0 0 0 0 90
0 100 0 0 0 100 0 0 0 0 120
80 200 0 0 280 0 0 0 0 0 0 300
0 60 0 0 0 60 0 0 0 0 0 0 80
0 0 0 0 0 0 -10 0 0 10 0 0 -10 30
0 0 0 0 0 0 0 -10 0 0 0 -10 0 0 25
0 0 0 0 0 0 0 0 -10 0 0 0 0 0
0 0 0 0 0
~A ~B ~,Aso SA~ ~Bso
0
SBsl
47 44 11 --95 188 -13 = 1 10 -77 -17 8 188 -21
770 050 900 400 320 000 400 500 550 850 000 320 000
0 0 0 0 -10 0 0 0
~'1 s2 ~'3 ~'4 is §6 r,7 s77
0
0
ss8
0
20
0 20
ss2 s6o
0 0
0
$..a V-a
112
T h e decision as to which group should be t he base is political and not relevant to this discussion. To predict t h e proofs o f bulls in a particular c o u n t r y , for example Mexico, add th e c o u n t r y o f p r o o f solution for Mexico to all elements o f t. The mixed m odel equations for t he example data are shown in Table II. In order to solve these equations one restriction on t he solutions is required. Let th e c o u n t r y o f birth (A) and year o f birth (80) (i.e., equation 3, gA80) be restricted to zero. GENETIC DIFFERENCES AMONG COUNTRIES
Th e estimates of g f r om t he equations do n o t provide estimates o f genetic differences among countries due to t he inclusion o f the relationship matrix and the effects it has on $. To estimate genetic differences one needs to calculate averages o f t for various groups o f bulls. An i m p o r t a n t question is which genetic differences are to be estimated. Due to the problems m e n t i o n e d earlier with assigning bulls to groups, one needs to be very precise a b o u t stating the differences t hat are to be estimated. T h e solutions to t he equations are below. CA
=
~ASO =
0
-446.949 = -75.238 = 60.452 = -126.946 = 65.310 = -59.799 = 30.226
gASl ----
sl s2 sa s4 s77 s88
99.436
~ gBa0 gBs, ss ~
= 69. 589 = 579.342 = -203.597 = 186.093 = 15.113 = -169.470
s82 s60
= =
s6
-63.473 93.047
International sire comparisons (ETA) are f o r m e d by adding group solutions to sire solutions. -75 t l = gASO-I" $I = t2 = g A 8 o - 6 $2 "~ +60 t 3 = ~A81 + ~3= - 5 7 4 t 4 = SAS~ + g 4 = - 3 8 2 t s = $B8~ + ~S= - 1 8 t6 = gBso + §6 = +594 t7 = $88~ + ~7 = - 3 7 3 Proofs for all seven bulls could be expressed on the same basis as for bulls in c o u n t r y A by adding CA to all t-proofs. For example, bull 5 would have a p r o o f o f +81 in c o u n t r y A. A similar m e t h o d would be used to express bull proofs relative t o c o u n t r y B, where bull 5 would be +51.
113
The basis for the international proofs, t, is not clearly defined, but could be made exact if desired. As long as the same group solution is set to zero with each analysis, a fixed base would be maintained. There are several kinds of genetic differences that can be calculated, and therefore, a precise definition should accompany any reported figures. Suppose we wished to compare the genetic differences of bulls by country and year of birth. The four averages for the example data would be: A-80: (tl + t2)/2 = - 7 . 5 A-81:(t3 + t4)/2 = - 4 7 8 . 0
B-80:t6 = +594 B-81: (t~ + t7)/2 = -195.5
Country B would be 601.5 kg better than c o u n t r y A in 1980, and 282.5 kg better in 1981, but both countries decline genetically from 1980 to 1981. The above averages are subject to criticism because bull 3 was never progenytested in country A and perhaps should not be included in the average for country A. Another comparison that may be more appropriate is the genetic difference in bulls used within a particular country and year. Assume all bulls in the example were used in the last year, t h e n the averages of bulls with daughters in each country would be A: (tl + t~ + t4 + t~)/4 = +49.25 B: (tl + t3 + t5 + t6 + t7)/5 = - 8 9 . 2 0 Hence, country A would be 138.45 kg better than c o u n t r y B in this respect. The bulls in c o u n t r y A that were used by dairymen in t h a t country were better than the bulls used by dairymen in c o u n t r y B. Both countries could argue t h a t the better bulls within a c o u n t r y were used more heavily than the poorer bulls. Thus, a weighted average of bull proofs by number of daughters within a country in a specific year could be calculated. For the example, we would obtain: A: (100tl + 60t2 + 70t4 + 80t6)/310 = +54.45 B: (20t~ + 150t3 + lOOts + 200t6 + 60t7)/530 = +13.25 Now the difference between the two countries is only 41.20 kg in favour of country A. This difference is an estimate of twice the genetic differences in cow populations between the two countries rather than between bull populations. There is probably interest in knowing all of the above figures to have an overall view of different aspects of international usage of dairy sires. Annual genetic trends within c o u n t r y could be calculated and the country with the best sire testing programme could be identified. One could determine if the genetic trend within a country was either adversely or favourably influenced by foreign bulls. Many possible comparisons could be made from this t y p e o f analysis.
114 DISCUSSION
A linear model has been proposed for making international comparisons of dairy sires for various traits. Many of the same assumptions were invoked as for procedures proposed b y Dommerholt et al. (1982). However, by using all AI bull proofs from each country, assumptions a b o u t the linearity of genetic trend and a knowledge of the magnitude of genetic trend are not necessary. The use of additive genetic relationships among bulls can aid in providing more comparisons among countries and therefore result in better estimates o f international proofs. The comparisons from use of the relationship matrix could actually be more reliable than comparison of proofs of bulls used in more than one country. Arguments could be put forth that proofs of bulls coming from outside their country of birth (or origin) should be omitted. With the data available a comparison could be made to study the impact of these proofs on estimates of genetic differences among countries. Many other interesting topics could be studied with the data available for international comparisons, interactions of genotype and environment, heterosis and rates of genetic change within countries to list a few. A multiple trait model could also be applied to the same data with the assumption that the trait evaluated in each country is genetically different from that in other countries. Estimates of genetic correlations would then be necessary and these are not yet available. Such a model would allow for a country of p r o o f by country of birth interaction. Finally, the definition of the genetic difference that is being estimated should be precisely specified. Several ideas were proposed, but others may also exist. The application o f this model to actual data remains to be studied, and hopefully someone will take on this task. ACKNOWLEDGEMENTS
Thanks to Dr Spencer Hudson and Dr Erling Fimland for suggestions for improving the manuscript.
REFERENCES Dommerholt, J., Hinks, C.J.M., Lederer, J.A., Mocquot, J.C., Petersen, P.H., R~nnigen, K. and Swanson, G., 1982. General recommendations for the documentation of changes in dairy cattle populations and for estimating the expected genetic merit of bulls of various origin used on specific cow populations. Livest. Prod. Sci., 10: 519--529. Galliard, C., Dommerholt, J., Fimland, E., Christensen, L.G., Lederer, J.A., McClintock, A.E., Mocquot, J.C. and Philipsson, J., 1977. AI bull evaluation standards for dairy and dual purpose breeds. Livest. Prod. Sci., 4: 115--128. Hinkovski, T., Alexiev, A., Lindh~, B. and Hickman, G.G., 1979. The red and red-andwhite cattle breed comparison in Bulgaria. World Anim. Rev., 29: 8--12.
115 Hudson, G.F.S. and Schaeffer, L.R., 1984. Monte Carlo comparison of sire evaluation model in populations subject to selection and assortative mating. J. Dairy Sci., 67: 1264--1271. Stolzman, M., Jasiorowski, H., Reklewski, Z., Zarnecki, A. and Kalinowska, G., 1981. Friesian cattle in Poland -- preliminary results of testing different strains. World Anita. Rev., 38: 9--15. Stolzman, M., Jasiorowski, H. and Reklewski, Z., 1982. Friesian cattle in Poland. World Anita. Rev., 41: 46--47.
RESUME Schaeffer, L.R., 1985. ModUle pour une ~valuation internationale des taureaux laitiers. Livest. Prod. Sci., 12:105--115 (en anglais). O n propose un module statistique lin~aire pour comparer la valeur gdn~tique des taureaux laitiers ~ partir des r~sultats de testage obtenus dans un ou plusieurs pays. Les relations gdn~tiques additives entre taureaux sont incluses afin de fournir plus de liaisons et de comparaisons entre pays. Plusieurs d~finitions des differences g~ndtiques entre pays sont pr~sent~es. U n petit exemple illustrel'utilisationdu module.
KURZFASSUNG
Schaeffer, L.R., 1985. Modell fiir die internationale Bewertung yon Milchviehbullen. Livest. Prod. Sci., 1 2 : 1 0 5 - - 1 1 5 (auf engliseh). Es wird ein lineares statistisches Modell vorgestellt, um das genetische Niveau yon Milchviehbullen miteinander zu vergleichen, yon denen Nachkommenschaftspriifungen aus einem oder mehreren L~ndern vorliegen. Additive genetische Beziehungen zwischen Bullen wurden fiir mehr Verbindungen oder Vergleiche zwischen L~indern mitangegeben. Es werden mehrere Definitionen genetischer Unterscbiede zwischen L~ndern vorgestellt. Ein kurzes Beispiel illustriert den Gebrauch des Modells.
105
Elsevier Science P u b l i s h e r s B.V., A m s t e r d a m - - P r i n t e d in T h e N e t h e r l a n d s
MODEL F O R INTERNATIONAL EVALUATION OF DAIRY SIRES
L.R. S C H A E F F E R
Department of Animal and Poultry Science, University of Guelph, Guelph, Ont. (Canada) (Accepted 8 November 1984)
ABSTRACT Schaeffer, L.R., 1 9 8 5 . M o d e l for i n t e r n a t i o n a l e v a l u a t i o n o f dairy sires. Livest. Prod. Sci., 12: 1 0 5 - - 1 1 5 . A linear statistical m o d e l is p u t f o r w a r d for use in c o m p a r i n g t h e g e n e t i c level o f dairy sires b a s e d o n t h e i r p r o g e n y t e s t e v a l u a t i o n s f r o m o n e or m o r e c o u n t r i e s . A d d i t i v e g e n e t i c r e l a t i o n s h i p s a m o n g bulls are i n c l u d e d t o p r o v i d e m o r e c o n n e c t i o n s o r comp a r i s o n s a m o n g c o u n t r i e s . Several d e f i n i t i o n s o f genetic d i f f e r e n c e s a m o n g c o u n t r i e s are p r e s e n t e d . A small e x a m p l e illustrates t h e use o f this m o d e l .
INTRODUCTION
There has been an increase in the desire to measure the genetic differences among dairy cattle populations, primarily black and white breeds (Gaillard et al., 1977; Dommerholt et al., 1982). This interest resulted in the comparison o f ten strains of black and white cattle in Poland, and a similar trial for red cattle breeds in Bulgaria (Stolzman et al., 1981, 1982; Hinkovski et al., 1979). However, there are a number o f problems with these comparison trials. Each trial has only compared a small group of bulls from each country with small numbers of daughters per bull. Estimates of genetic differences among countries from these trials represent the differences in bull populations at the time when these bulls were entered into artificial insemination. Because rates of genetic change are different for each country and may not be linear with time, then the ranking of bull populations for genetic merit t o d a y may be significantly different from the trial results. In addition to breed comparison trials, m a n y North American HolsteinFriesian bulls or their sons have been progeny-tested in one or more European countries as well as in their country of birth. Imported semen is generally more expensive than domestic semen, and consequently is used to breed the better cows in the herds that can afford to buy the semen. Also, progeny of foreign sires or of any expensive bull may be treated better than
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106
progeny of other bulls as they grow and as t h e y make their first lactation. These actions can bias the progeny-test results of imported bulls. Statistical models with maternal grandsire effects have been proposed to offset this problem, but a joint sire and cow evaluation model could be better (Hudson and Schaeffer, 1984). Assuming that progeny test results are unbiased there are other issues t h a t could cause problems in comparing bull populations. These include the model and methods of evaluating bulls within each country. Best linear unbiased prediction (BLUP) has been proposed by Gaillard et al. (1977) as well as the factors to include in a model for various traits. Not every country can follow these recommendations, and even if t h e y did, then there could also be differences in heritability values for traits among countries. The previously discussed problem areas suggest that international comparisons o f dairy sires m a y be subject to m a n y errors, and caution should be exercised. Unfortunately, the m o n e t a r y importance of international comparisons to exporting countries may force scientists to make compromises that would otherwise not be made. The objective of this paper is to propose a linear model that may serve as a starting point for discussions on the international comparison o f dairy sires. DATA REQUIREMENTS
For each country t h a t wishes to participate in the international comparison, the data t h a t would be necessary for a successful comparison are described as follows: Ca) Estimated breeding values (EBV) for the trait or traits of interest for every bull in artificial insemination (AI) evaluated within that country. (b) The number of daughters used to calculate the EBV o f each bull, or some other indicator of reliability t h a t each country can easily provide. (c) The country of birth, year of birth, breed category, and sire and maternal grandsire identification of each bull. In some cases the country of birth may be vague, for instance when the sire and dam were born in different countries. A unique sire identification system would need to be implemented to avoid problems. The main problem would be bulls that are registered in more than one c o u n t r y with a different number in each country. Breed category allows for countries which permit crossbreeding and could include different percentages of various combinations of breeds. The traits are assumed to be standardized among countries. For example, milk production may be expressed as 305-day fat corrected milk. Each participating c o u n t r y would be responsible for expressing the EBV in the accepted standard, but within that country any other expression of EBV or ETA could be used. The m e t h o d for converting to the accepted standard from the within-country proofs would need to be documented. The reason for requiring EBV from all AI bulls is to be able to evaluate
107 d i f f e r e n c e s in t h e bases t o w h i c h sires are r e f e r e n c e d w i t h i n a c o u n t r y a n d t o m e a s u r e genetic change in t h e bull p o p u l a t i o n o f a c o u n t r y o n an a n n u a l basis. O t h e r w i s e o n e m u s t a s s u m e t h a t genetic change is linear o v e r t i m e in o r d e r t o adjust p r o o f s t o t h e s a m e base o f c o m p a r i s o n . By having all EBVs o f A I bulls f r o m e a c h c o u n t r y t h e n this a s s u m p t i o n is n o t necessary. T h e i d e n t i f i c a t i o n o f t h e sire a n d m a t e r n a l grandsire (MGS) o f each bull is n e e d e d so t h a t additive genetic r e l a t i o n s h i p s a m o n g bulls can be i n c o r p o r a t e d into t h e analysis. T h e r e l a t i o n s h i p m a t r i x w o u l d p r o v i d e m o r e c o m p a r i s o n s or c o n n e c t i o n s a m o n g c o u n t r i e s t h a n s i m p l y bulls t h a t are p r o v e n in t w o or m o r e c o u n t r i e s . In fact, c o m p a r i s o n s f r o m t h e relat i o n s h i p m a t r i x m a y b e less subject to bias p r o b l e m s t h a n c o m p a r i s o n s f r o m a p o p u l a r bull t h a t has d a u g h t e r s in t w o c o u n t r i e s . F o r this r e a s o n m u c h w o r k should be d i r e c t e d to t h e i n t e r n a t i o n a l i d e n t i f i c a t i o n o f bulls, a n d a u n i q u e r e g i s t r a t i o n n u m b e r s h o u l d b e assigned t o e a c h bull. A n o t h e r r e q u i r e m e n t t h a t is critical t o i n t e r n a t i o n a l c o m p a r i s o n s is c o n n e c t e d n e s s . E a c h p a r t i c i p a t i n g c o u n t r y m u s t h a v e bulls p r o v e n in t h e i r o w n c o u n t r y t h a t e i t h e r h a v e p r o o f s in a n o t h e r p a r t i c i p a t i n g c o u n t r y or have sons or a sire w i t h a p r o o f in a n o t h e r p a r t i c i p a t i n g c o u n t r y . T h e n u m b e r o f such c o n n e c t i o n s o r ties c o u l d b e critical t o t h e reliability o f e s t i m a t e s o f genetic d i f f e r e n c e s a m o n g c o u n t r i e s . C o u n t r i e s t h a t have n e v e r used bulls f r o m a n y o t h e r c o u n t r y c a n n o t b e i n c l u d e d in t h e s e analyses. TABLE I Example data of sire proofs from two countries Countryyear of proof
Bull number
Sire numbet
Country of birth
Year of birth
EBV (kg)
Number of daughters
(n+k)/2n Elements
A-84 A-84 A-84 A-84
1 2 4 6
77 88 77 88
A A A B
80 80 81 80
0 +280 -420 +1160
100 60 70 80
0.5750 0.6250 0.6071 0.5938
0 +175 -255 +689
B~4 B-84 B°84 B-84 B-84
1 3 5 6 7
77 82 60 88 77
A A B B B
80 81 81 80 81
+80 -940 +140 +1250 -560
20 150 100 200 60
0.8750 0.5500 0.5750 0.5375 0.6250
+70 -517 +80 +666 -350
In bulls Note born each
of y
o r d e r t o illustrate t h e m e t h o d o l o g y t o b e d e s c r i b e d , a d a t a set o f f r o m t w o c o u n t r i e s (A a n d B) is p r e s e n t e d in T a b l e I as an e x a m p l e . t h a t bulls 1 a n d 6 w e r e p r o v e n in b o t h c o u n t r i e s , a n d t h a t bull 3 was in c o u n t r y A a n d p r o v e n in c o u n t r y B. Several bulls are r e l a t e d to other.
108 A MODEL Let t h e eq u ati on o f t he model be y = Xc + ZQg + Zs + e where: y = a vector o f observations t h a t are calculated f r o m the EBV o f bulls, the reliability o f th e EBV, and k (the ratio of error to sire variances within a country of proof); c = a fixed vector o f effects for c o u n t r y of p r o o f which is included to acc o u n t for differences in reference bases among countries; g = a fixed v ec t or o f effects f or c o u n t r y and year of birth o f each bull; s = a r a n d o m vect or o f sire effects with mean vector o f null and variance-covariance matrix Aa2s where A is t he relationship matrix and o2s is the sire variance; e = a r a n d o m vector o f residual or error effects with mean vector of null and variance--covariance matrix Do2e where D is a diagonal matrix with elements equal to n~jI , where nij equals the n u m b e r o f daughters for t he j t h sire in th e ith c o u n t r y o f pr oof , and O2e is t h e error of variance. X and Z are design matrices o f zeros and ones; and Q = a matrix t hat describes the group to which a sire belongs. Discussion on the c o n s t r u c t i o n o f Q and potential problems are presented later. Let bij be an EBV of t he ]th bull in the ith c o u n t r y , and let nij be the indicators o f reliability o f bij. Ideally, nij might be t he n u m b e r of effective daughters, b u t perhaps n o t all countries c o m p u t e this n u m b e r in practice, in which case nij could be t he actual n u m b e r o f daughters. T he value of nij could be altered by a constant t hat reflects the relative accuracy o f sire evaluation m e t h o d o l o g y among countries as assessed by some indep e n d e n t study. For example, an EBV f r o m t he United States based on 100 daughters might only be as reliable as an EBV f r o m Sweden based on 80 daughters. Thus all nij f r om t he United States could be multiplied by 0.8. Burnside and H a m m o n d {personal c om m uni cat i on, 1983) have suggested h o w methodologies might be compared by simulation techniques. The elements o f y are t he n
bij(nij +
k)/2nij.
Th e fact that bij in some countries is based on group plus sire solutions a n d / o r relationships among bulls, is ignored and n o t considered i m p o r t a n t provided th at bij is an unbiased estimate o f t he t rue breeding value o f a bull and nij is the relative accuracy indicator. The above elements are readily available f r o m most countries, but m o r e complicated and statistically b e t t e r quantities could be developed if necessary. Th e observation vector, y, for t he example data would be t he last column o f Table I. Assume t h r o u g h o u t t h a t k = 15.
109
Assumptions of the model 1. The primary assumption for any successful comparison of bulls among countries is that the bulls have been randomly mated to cows within a country and that the daughters have not received any preferential treatment duing their growth or lactation performance. There may be ways of measuring or avoiding bias of this type. The sire evaluation methodology of each country is assumed to be able to provide unbiased predictions of genetic values of bulls that are free of all fixed effects important to t h a t country. A description of the methods used by each country is not enough to verify this assumption. A proposal by Burnside and H a m m o n d (personal communication, 1983} was to generate, by Monte Carlo simulation, test data sets to be submitted to each participating country for analysis by their existing sire evaluation methodology. The results would be returned and comparisons made to the true values. Hence, the accuracy of each country's methods could be assessed. 2. The sire and error variances are assumed to be c o m m o n to all countries. That is, the ratio of error to sire variances is assumed to be equal for each country. With the accumulation of data from participating countries these components could be estimated and the assumption could be tested. 3. The variance--covariance matrix o f e was assumed to be diagonal. In actual fact, there are non-zero covariances among the e within a country, particularly if BLUP has been used to estimate the EBVs of bulls. In order to account for these covariances, the inverse to the entire mixed model equations for each country would be needed. Most countries that use BLUP, however, use iteration techniques to solve the equations rather than a direct inverse. Ignoring covariances will not bias results, but could increase the prediction error variances. 4. Several interactions are assumed to be unimportant. Genotype by environment interactions, country of proof by country of birth interactions, and specific combining abilities have been ignored. An investigation of these interactions could be made with this data. Each of these assumptions has also been imposed by Dommerholt et al. (1982). COUNTRY AND YEAR OF BIRTH GROUPS Assigning bulls to groups according to country and year of birth could be more difficult than at first appears. For humans at least, the country of birth is the country of citizenship unless the parents are citizens of another country. Then an individual may decide on a country of citizenship at age 18 years. In dairy bulls the same problems exist. For example, a U.S.-born bull from U.S.-born parents may be sold to Canada and progeny-
II0 tested only in Canada. Another case is where one parent may be U.S.-born and the other may be Canadian-born and the offspring may be physically born in a third country, such as Mexico. There are several situations in which country o f birth assignment may be questionable. However, the matrix Q t h a t defines groups can contain values other than zero or one. Suppose we have three groups, say A, B, and C, and a bull whose parents were both born in country A, then the row of Q for that bull would be
(1 0
O)
gA gc
A bull with a sire born in country B and a dam born in country C, then the row of Q for t h a t bull would be (0 0.5
0.5)
gA gc
If that bull also had a son, then Q could contain values of 0.25, etc. The possibilities could become very messy. Another possibility could be to group sires by breed composition and year of birth. For example, in the Netherlands there can be Dutch Friesians, North America Holsteins, the MRY breed or combinations of crosses of these three breeds of various percentages, but all could be born within the Netherlands. Thus, breed groups within country of birth may be necessary. Other schemes could be proposed, but the final strategy should be the result of collaborative work among researchers from the participating countries. MIXED MODEL EQUATIONS The equations that must be constructed and solved are as follows:
Vxo x x o zo x o,z ioz
0zo,z
/zo,z /
L_Z'D-'X
Z'D-1ZQ
Z D-' Z+A-lk_]
Iil
= i0zo , I [_Z'D-1Y
These equations could be very large, but solutions should be possible on modern computers with five to ten megabytes of memory. The international genetic values o f bulls would be calculated as: t = Q$ + $ for ETA values, or 2t if one wants EBV values. One of the group solutions would have been set to zero and would be the base for international comparisons.
T A B L E II Mixed m o d e l e q u a t i o n s for e x a m p l e d a t a 310
0 530
symmetric
160 20 180
70 150 0 220
80 200 0 0 280
0 160 0 0 0 160
100 20 120 0 0 0 140
60 0 60 0 0 0 0 80
0 150 0 150 0 0 0 0 170
70 0 0 70 0 0 0 0 0 90
0 100 0 0 0 100 0 0 0 0 120
80 200 0 0 280 0 0 0 0 0 0 300
0 60 0 0 0 60 0 0 0 0 0 0 80
0 0 0 0 0 0 -10 0 0 10 0 0 -10 30
0 0 0 0 0 0 0 -10 0 0 0 -10 0 0 25
0 0 0 0 0 0 0 0 -10 0 0 0 0 0
0 0 0 0 0
~A ~B ~,Aso SA~ ~Bso
0
SBsl
47 44 11 --95 188 -13 = 1 10 -77 -17 8 188 -21
770 050 900 400 320 000 400 500 550 850 000 320 000
0 0 0 0 -10 0 0 0
~'1 s2 ~'3 ~'4 is §6 r,7 s77
0
0
ss8
0
20
0 20
ss2 s6o
0 0
0
$..a V-a
112
T h e decision as to which group should be t he base is political and not relevant to this discussion. To predict t h e proofs o f bulls in a particular c o u n t r y , for example Mexico, add th e c o u n t r y o f p r o o f solution for Mexico to all elements o f t. The mixed m odel equations for t he example data are shown in Table II. In order to solve these equations one restriction on t he solutions is required. Let th e c o u n t r y o f birth (A) and year o f birth (80) (i.e., equation 3, gA80) be restricted to zero. GENETIC DIFFERENCES AMONG COUNTRIES
Th e estimates of g f r om t he equations do n o t provide estimates o f genetic differences among countries due to t he inclusion o f the relationship matrix and the effects it has on $. To estimate genetic differences one needs to calculate averages o f t for various groups o f bulls. An i m p o r t a n t question is which genetic differences are to be estimated. Due to the problems m e n t i o n e d earlier with assigning bulls to groups, one needs to be very precise a b o u t stating the differences t hat are to be estimated. T h e solutions to t he equations are below. CA
=
~ASO =
0
-446.949 = -75.238 = 60.452 = -126.946 = 65.310 = -59.799 = 30.226
gASl ----
sl s2 sa s4 s77 s88
99.436
~ gBa0 gBs, ss ~
= 69. 589 = 579.342 = -203.597 = 186.093 = 15.113 = -169.470
s82 s60
= =
s6
-63.473 93.047
International sire comparisons (ETA) are f o r m e d by adding group solutions to sire solutions. -75 t l = gASO-I" $I = t2 = g A 8 o - 6 $2 "~ +60 t 3 = ~A81 + ~3= - 5 7 4 t 4 = SAS~ + g 4 = - 3 8 2 t s = $B8~ + ~S= - 1 8 t6 = gBso + §6 = +594 t7 = $88~ + ~7 = - 3 7 3 Proofs for all seven bulls could be expressed on the same basis as for bulls in c o u n t r y A by adding CA to all t-proofs. For example, bull 5 would have a p r o o f o f +81 in c o u n t r y A. A similar m e t h o d would be used to express bull proofs relative t o c o u n t r y B, where bull 5 would be +51.
113
The basis for the international proofs, t, is not clearly defined, but could be made exact if desired. As long as the same group solution is set to zero with each analysis, a fixed base would be maintained. There are several kinds of genetic differences that can be calculated, and therefore, a precise definition should accompany any reported figures. Suppose we wished to compare the genetic differences of bulls by country and year of birth. The four averages for the example data would be: A-80: (tl + t2)/2 = - 7 . 5 A-81:(t3 + t4)/2 = - 4 7 8 . 0
B-80:t6 = +594 B-81: (t~ + t7)/2 = -195.5
Country B would be 601.5 kg better than c o u n t r y A in 1980, and 282.5 kg better in 1981, but both countries decline genetically from 1980 to 1981. The above averages are subject to criticism because bull 3 was never progenytested in country A and perhaps should not be included in the average for country A. Another comparison that may be more appropriate is the genetic difference in bulls used within a particular country and year. Assume all bulls in the example were used in the last year, t h e n the averages of bulls with daughters in each country would be A: (tl + t~ + t4 + t~)/4 = +49.25 B: (tl + t3 + t5 + t6 + t7)/5 = - 8 9 . 2 0 Hence, country A would be 138.45 kg better than c o u n t r y B in this respect. The bulls in c o u n t r y A that were used by dairymen in t h a t country were better than the bulls used by dairymen in c o u n t r y B. Both countries could argue t h a t the better bulls within a c o u n t r y were used more heavily than the poorer bulls. Thus, a weighted average of bull proofs by number of daughters within a country in a specific year could be calculated. For the example, we would obtain: A: (100tl + 60t2 + 70t4 + 80t6)/310 = +54.45 B: (20t~ + 150t3 + lOOts + 200t6 + 60t7)/530 = +13.25 Now the difference between the two countries is only 41.20 kg in favour of country A. This difference is an estimate of twice the genetic differences in cow populations between the two countries rather than between bull populations. There is probably interest in knowing all of the above figures to have an overall view of different aspects of international usage of dairy sires. Annual genetic trends within c o u n t r y could be calculated and the country with the best sire testing programme could be identified. One could determine if the genetic trend within a country was either adversely or favourably influenced by foreign bulls. Many possible comparisons could be made from this t y p e o f analysis.
114 DISCUSSION
A linear model has been proposed for making international comparisons of dairy sires for various traits. Many of the same assumptions were invoked as for procedures proposed b y Dommerholt et al. (1982). However, by using all AI bull proofs from each country, assumptions a b o u t the linearity of genetic trend and a knowledge of the magnitude of genetic trend are not necessary. The use of additive genetic relationships among bulls can aid in providing more comparisons among countries and therefore result in better estimates o f international proofs. The comparisons from use of the relationship matrix could actually be more reliable than comparison of proofs of bulls used in more than one country. Arguments could be put forth that proofs of bulls coming from outside their country of birth (or origin) should be omitted. With the data available a comparison could be made to study the impact of these proofs on estimates of genetic differences among countries. Many other interesting topics could be studied with the data available for international comparisons, interactions of genotype and environment, heterosis and rates of genetic change within countries to list a few. A multiple trait model could also be applied to the same data with the assumption that the trait evaluated in each country is genetically different from that in other countries. Estimates of genetic correlations would then be necessary and these are not yet available. Such a model would allow for a country of p r o o f by country of birth interaction. Finally, the definition of the genetic difference that is being estimated should be precisely specified. Several ideas were proposed, but others may also exist. The application o f this model to actual data remains to be studied, and hopefully someone will take on this task. ACKNOWLEDGEMENTS
Thanks to Dr Spencer Hudson and Dr Erling Fimland for suggestions for improving the manuscript.
REFERENCES Dommerholt, J., Hinks, C.J.M., Lederer, J.A., Mocquot, J.C., Petersen, P.H., R~nnigen, K. and Swanson, G., 1982. General recommendations for the documentation of changes in dairy cattle populations and for estimating the expected genetic merit of bulls of various origin used on specific cow populations. Livest. Prod. Sci., 10: 519--529. Galliard, C., Dommerholt, J., Fimland, E., Christensen, L.G., Lederer, J.A., McClintock, A.E., Mocquot, J.C. and Philipsson, J., 1977. AI bull evaluation standards for dairy and dual purpose breeds. Livest. Prod. Sci., 4: 115--128. Hinkovski, T., Alexiev, A., Lindh~, B. and Hickman, G.G., 1979. The red and red-andwhite cattle breed comparison in Bulgaria. World Anim. Rev., 29: 8--12.
115 Hudson, G.F.S. and Schaeffer, L.R., 1984. Monte Carlo comparison of sire evaluation model in populations subject to selection and assortative mating. J. Dairy Sci., 67: 1264--1271. Stolzman, M., Jasiorowski, H., Reklewski, Z., Zarnecki, A. and Kalinowska, G., 1981. Friesian cattle in Poland -- preliminary results of testing different strains. World Anita. Rev., 38: 9--15. Stolzman, M., Jasiorowski, H. and Reklewski, Z., 1982. Friesian cattle in Poland. World Anita. Rev., 41: 46--47.
RESUME Schaeffer, L.R., 1985. ModUle pour une ~valuation internationale des taureaux laitiers. Livest. Prod. Sci., 12:105--115 (en anglais). O n propose un module statistique lin~aire pour comparer la valeur gdn~tique des taureaux laitiers ~ partir des r~sultats de testage obtenus dans un ou plusieurs pays. Les relations gdn~tiques additives entre taureaux sont incluses afin de fournir plus de liaisons et de comparaisons entre pays. Plusieurs d~finitions des differences g~ndtiques entre pays sont pr~sent~es. U n petit exemple illustrel'utilisationdu module.
KURZFASSUNG
Schaeffer, L.R., 1985. Modell fiir die internationale Bewertung yon Milchviehbullen. Livest. Prod. Sci., 1 2 : 1 0 5 - - 1 1 5 (auf engliseh). Es wird ein lineares statistisches Modell vorgestellt, um das genetische Niveau yon Milchviehbullen miteinander zu vergleichen, yon denen Nachkommenschaftspriifungen aus einem oder mehreren L~ndern vorliegen. Additive genetische Beziehungen zwischen Bullen wurden fiir mehr Verbindungen oder Vergleiche zwischen L~indern mitangegeben. Es werden mehrere Definitionen genetischer Unterscbiede zwischen L~ndern vorgestellt. Ein kurzes Beispiel illustriert den Gebrauch des Modells.