Stayability Evaluation as a Categorical Trait and by Considering Other Traits

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1 University of Nebrasa - Lincoln DigitalCoons@University of Nebrasa - Lincoln Faculty Papers and Publications in Anial Science Anial Science Departent January 1980 Stayability Evaluation as a Categorical Trait and by Considering Other Traits L. Dale Van Vlec University of Nebrasa-Lincoln, dvan-vlec1@unl.edu Follow this and additional wors at: Part of the Anial Sciences Coons Van Vlec, L. Dale, "Stayability Evaluation as a Categorical Trait and by Considering Other Traits" (1980). Faculty Papers and Publications in Anial Science This Article is brought to you for free and open access by the Anial Science Departent at DigitalCoons@University of Nebrasa - Lincoln. t has been accepted for inclusion in Faculty Papers and Publications in Anial Science by an authorized adinistrator of DigitalCoons@University of Nebrasa - Lincoln.

2 Stayability Evaluation as a Categorical Trait and by Considering Other Traits L. D. VAN VLECK Departent of Anial Science Cornell University thaca, NY ABSTRACT Correspondence between usual categorical scores where frequencies su to one and stayability of herd life easured as survival to several age periods is illustrated. The singularity of the variancecovariance atrix of categorically easured traits when each category is a separate trait also is true for stayability, since the easureent for the last period for stayability is always zero, has variance zero, and covariances zero with easureents in preceding periods. Best linear unbiased prediction for stayability of a bull's daughters is achieved by ignoring the last period. A procedure is described for predicting stayability of daughters having variable opportunity for stayability easureents that utilizes production and all available easures of stayability. NTRODUCTON Stayability of a bull's daughters is the fraction surviving to a specified age. Everett et al. (4) chose survival to ages 36, 48, 60, 72, and 84 o as separate stayability traits. Stayability currently is being evaluated at 48 o (11). The evaluation considers only daughters with opportunity to have survived 48 o. Stayabilities at various ages ae up a ajor coponent of profitability (1) when lifetie yields by daughters and their descendants are considered. n these uses, stayability is a categorically scored trait. The purposes of this paper are 1) to show the relationship between utually exclusive categories of period when survival ceases and stayability, and, ore iportantly, 2) to propose a procedure for predicting stayability using il and survival inforation on all daughters, irrespective of opportunity for survival. Received October 26, Prediction of frequencies of future progeny for traits scored in utually exclusive categories such as calving difficulty was proposed with each category a separate trait (2). Descriptions of best linear unbiased prediction (BLUP) (e.g., 6, 7, 8, 10) for single trait and ultiple trait evaluation, however, include the assuption that variance-covariance atrices of rando eleents of the odel are nonsingular. Variance-covariance atrices where easureents for each category are 0 to 1 are singular, since the su of frequencies of the categories is unity. Discussion of the proble (R. Anderson, personal counication) led to the assertion that discarding easureents in one category and using variance-covariance atrices of the reaining categories would lead to BLUP of frequencies of the categories (3). That generalized inverse of the full but singular variancecovariance atrix does satisfy the condition described by Harville (5) for BLUP. Quaas and Van Vlec (13) outlined the procedure to predict frequencies of progeny in categories when the frequencies of categories su to unity. Stayability as defined here is a categorically scored trait, but the eans do not su to one as they do for utually exclusive categories. Correspondence of Stayability Measureents and Scores for Mutually Exclusive Categories of When Survival Ceases Correspondence is shown by exaple. Let easureent of 36-o stayability be 0 if the cow fails to survive past 36 o, 1 if she survives, and s~ilarly for survival to 48 o, 60 o, 72 rno, and 0 for /> 72 too. Let easureents of failure to survive be 1 for only one of the periods < 36 too, 37 to 48 o, 49 to 60 o, 61 to 72 o, and > 72 o and 0 otherwise. Let s be the vector of observations for stayability for the five categories and p be the vector of observations for failure to survive. The correspondence is: 1980 J Dairy Sci 63:

3 TECHNCAL NOTE Condition dies ~< 36 o: 36 o < dies ~< 48 o: 48 o < dies ~< 60 o: 60 o < dies ~< 72 rno: dies > 72 o: S p S = p S = p S = p = S = p = Category ( o o o o o ) ( 1 o o o o ) ( 1 0 ) ( 0 1 ) ( 1 1 ) ( o o 10 o ) ( ) ( 1 0 ) ( o ) ( 0 1 ) where rr'= (7r1 n2 rr3 rr4 7rs) 5 Z 7ri=l, i=1 and the ean vector for the s's is 0'= ( Os) The last category of stayability always ust be zero to correspond to death in the last or final period and, thus, to atch the probability of death in the last period. Let si be the observation for survival through period i and Pi be the observation for death in period i. Then by inspection i s i = 1 -- N pj j=l where 0 5 = 1 --zrl, a2 = 1 --'/'/1 --7r2, a 3 = 1 -- ~'1 -- 7r2 -- 7/3, 0 4 = 1-7/1 -- 7/2 -- "B'3 -- 7/4, and Os = 1 --Zrl --7r2 --7r3 --7/4 --7'i" 5 =0. The ean vector for the p's are the ulti- The phenotypic variance-covariance atrix noial probabilities of the p's is well nown: 7fl (1---~'1) --'fl'l 7r 2 --Tr 17r 3 --Tfl ~4 --Tlln 5 --7rl 1r2 7r2 (1--7r2) --Tf2 fl" 3 --/r2 7"f4 --7"f2 7i'5 P-- [Pij] = --7rl 7r 3 --/r27r 3 7/'3(1--71"3) --~37r4 --g3ffs --/Tlg 4 --g2 ~/" 4 --g 3"B'4 7r4 (1--714) --7r4 ~" 5 --Trl g5 --T/" 2 7l'5 --7/'371" 5 --Tr 4 rrs rt s (1--rrs) Each row and each colun su to zero so that P is singular. The phenotypic variance-covariance atrix of the s's, V = [Vij], can be derived in ters of the Pij- For exaple: S1 = 1 -- Pl so that Var(sl) = Vii = Var(1--pl) = Pll ; Var(s2 ) = V22 = V(1-pl --P2 ) = V(p~ ) + V(p2 ) + ) = Plx + P22 + 2P12 ; and Coy(s1 s2) = V12 = Coy(l-p1,1-pl--P2 ) s2 = 1 -- Pl -- P2 so that = Pll + P12

4 1174 VAN VLECK Siilarly, V33 =Pll +P22 +P33 + 2(P12 +P13 +P23) Vt3 = Pll + P12 + P13 V23 =Pll +P22 +2P12 +P3 +P23 V44 = Pll + P22 + P33 + P44 + 2(P12 + P13 + P14 + P23 + P24 + P34) V14 =P11 +P12 +P13 +Pn V24 = Pll + P22 + 2P12 + P~a + P14 + P23 + P24 V34 =Pll +P22 +P33 + 2(P12 +P13 + P23)+Pa +P24 +P34 Vs5 = 0 - Vls = V2s -- V3s = V4s The phenotypic variance-covariance atrices can be partitioned into parts as, for exaple, the variance-covariance atrices associated with sire, herd-year-season (HYS), and residual effects. Correspondences between the p's and s's are such that predictions for either syste can be converted into the other syste. Prediction of Stayability Stayability for all periods can be predicted jointly fro records for all opportunity groups as well as fro records on other traits as, for exaple, il production by ixed odel ethods of Henderson (8) and Henderson "and Quaas (10). Correct generalized inverses of variance-covariance atrices of survival traits ust be used. f stayability traits are used, the last category, e.g., /> 72 o, usually will be ignored. f traits associated with periods of death are used, then any category can be ignored but usually will be the last. Let the odel be y = X/3+ Zu + e, where the observation vector on n cows, y, is Y' = (Y' Y~. Yn) with Yi the observations on cow i which would be other traits and the nuber of stayability observations associated with her opportunity group, X is the incidence atrix for fixed effects associated with the observations, /3 is the vector of fixed effects, Z is the incidence atrix for rando effects, which ay have soe zero coluns, u is the vector of rando effects for herd-year-seasons and sires with the traits in order within sires and herd-year-seasons, and e is a vector of rando residual effects. V E[y] = X/3 and F!' co ij = = ls*s c The variance-covariance atrices for the c traits are H c for herd-year-season effects, S c for sire effects, and h and s are identity atrices of the order of nuber of herd-year-seasons and sires, respectively. f sires are related, 1 s is replaced by A, the atrix of nuerator relationships aong the sires (9), * denotes the direct product operation which, for exaple, aes V(h) into a bloc diagonal atrix with diagonal blocs of Hc, R is a bloc diagonal atrix with diagonal blocs associated with an anial and the nuber of traits easured on the anial. For exaple, if O

5 TECHNCAL NOTE 1175 rll r12 r13 r14 rls r2 2 r23 r24 r25 Due to the nature the coefficients can (12, 13). Let of the equations, be written siply r33 r34 r35 Syetric r44 r45 r5 $ is the variance-covariance atrix associated with all five traits, then the bloc diagonal for an anial with only the first record is r n = R 1 ; for an anial with two traits, the bloc is i 1 r r22] = R 2 ; etc. The general ixed odel equations (8) are Z'R'X Z'R "Z+G-1J ~_1 LZ'R -'yj R1 = rl 1 0 R 2 = rll r12 rl 2 r22 ; etc. where Let nlij be the nuber of easureents in herd-year-season i of daughters of sire j on cows with an observation only on trait 1, n2ij be the nuber of daughters easured only on traits 1 and 2, etc. Then the equations are: ]~nl.r 0. EnhR ~ [ Enl R ~ +H -1 [ [ Nnh.R ~n..r~: (~nljr ~ j=l... s) (EnhjR ~ j=l... s) (~n.jr K j=l... s) ^ # ~ ~'R[~Yl.. J ~RYh. f ~RY.. SYMMETRC Zn.lR 0 + S -1 A S NR~:y A. ~n.sr[< ~s ER~:y.s.

6 1176 VAN VLECK where Yij is the vector of totals of daughters of sire j which have opportunity for easureents on the first traits in HYS i with other eleents equal to zero. When = 1, only the first eleent of the vector is nonzero; when = 2, the nonzero eleents will be totals for traits one and two on those cows having opportunity for just those two traits; and R~ is the generalized inverse of R obtained fro the direct inverse of the nonzero bloc augented by zeros in the reaining rows and coluns. Exaple n the exaple, herd-year-seasons will be ignored, but in practice each HYS set of equations probably would be absorbed after they are accuulated. Trait 1 will be il yield, and traits 2 through 4 will be easures of stayability to 36, 48, 60, and 72 o. The exaple will consider only the two cases where 1) all anials have an opportunity to stay in the herd ore than 72 o, and 2) all anials have an opportunity to survive ore than 48 o. Suppose R 5 = " , and SYMMETRC = SYMMETRC i Sire 1 has 10 daughters with total il production of 1500 in the first lactation (il to nearest 100 units); 8 survived to 36 too, 7 to 48 too, 4 to 60 o, 2 to 72 o, and all are dead soetie after 72 too. Sire 2 has 20 daughters with total il production of 2800; 18 survived to 36 o, 16 to 48 too, 10 to 60 too, 6 to 72 too, and all are dead soetie after 72 too. The ixed odel equations becoe:

7 TECHNCAL NOTE 1177 The solutions are: [ ~2 ~. "2 ~ "2 ~ ~! ~ ~. "Q ~ ~'.. Q ~! [ ii o o 0 o o az 7 ~ 22 ~- o~v--roe o1~ Trait ~ ~ The p p bloc is 30R~ 1, the p x sire 1 bloc 10R~ 1, the p x sire 2 bloc is 20R~ l, the sire 1 diagonal bloc is 10R51 + S~ 1, and the sire 2 bloc is 20R~ 1 + S~ 1. The right-hand sides (RHS's) for the equations are: R~ 1 ~ ~? ' o o ~ o The RHS's for the sl and s2 are equations are: R? 1 ]50 80] and R; x 2800 " "2'o f the daughters of sire 1 and sire 2 had opportunity to survive for 48 o or ore, then the total vectors for sire 1 and sire 2 would be: [1 o8[ 1 p801 L76J t-- "6~Z "='t -- Z 1 Now R3 should be used, but S reains the sae as before 1 R 3 = YMMETR1C.20350_[ The ixed odel equations now becoe:

8 1178 VAN VLECK ii!! The solutions are: o o o o o o o o ' ~ Trait ~ sl 12 ooooooool'~m~ 1 o o o o o "d~ ~ [ ~o Q " 1 ' ' 0 o o o 0 o t~ o o i To predict stayability for 60 and 72 too, outside estiates of ean frequencies are needed although differences aong sires can be predicted fro the solutions. f cobinations of opportunity groups were used, R 3 would be augented by two rows and two coluns of zeros, R2 by 3 rows and coluns, etc. ~.7 ~. -t~ea ~oo ~ Prediction of Multinoial Frequencies The procedure for prediction of categorical frequencies fro records classified according to tie of leaving the herd is alost exactly the sae as for stayability. The equations have the sae for if the generalized inverses are chosen to eliinate the last category, i.e., death later than 72 o. The totals on the RHS's for /~, sl, and 12 are: ~d,.-;,,4 h~ ~o

9 D , 1 and ] 9 3 p- J TECHNCAL NOTE ! 6 or 1500 ], 2 1 and[ 1179 ] if opportunity is to ore than 48 o. B 4 Rs = " SYMMETRC = ~ SYMMETRC The solutions for all opportunities and for opportunity to go to 48 o are: Opportunity Opportunity Opportunity past 72 o past 48 o past 72 o # ~ /~ ~ A P s / ~ /~s ~ sl S s Opportunity p~t48 o ^ s3a

10 1180 VAN VLECK These solutions can be converted to those obtained fro the stayability easureents by the conversion equations given earlier for the eans. DSCUSSON There are two ajor unanswered questions. The procedure requires estiates of variances and covariances which are unbiased by selection. Stayability, alost by definition, involves selection. For exaple, only cows with relatively high il yield in first lactation ay be allowed to have further lactations (stayability). f il yield in the first lactation can be considered a nonselected trait, then use of proper variances and covariances will result in prediction of stayability unbiased by selection (8). The procedure of Rothschild et al. (15) ight result in unbiased estiates of the variances and covariances. The coputing difficulties ay be uch ore severe with ore than two traits (14). The second question deals with inclusion of fixed effects other than a ean vector in the odel. Theoretically, if frequencies change with fixed effects, the variance-covariance structure also changes. Whether relatively sall changes have uch influence is unnown. Biases in prediction of sire effects ay occur when herdyear-season effects are correlated with sire effects. Biases can be reoved by coputing as if herd-year-season effects are fixed for continuous traits (8). Can this be done also for categorical traits or ust coputation always be with herd-year-season effects rando? ACKNOWLEDGMENTS The partial support of this research by Eastern Artificial nseination Cooperative, nc., thaca, NY, is greatly appreciated. REFERENCES 1 Baer, J. J., R. W. Everett, and L. D. Van Vlec A ethod of calculating a profitability index for sires. J. Dairy Sci. (Accepted). 2 Cady, R. A., R. D. Anderson, and L. D. Van Vlec Evaluation of sires for calving difficulty. Page 96 in Genetics research Report to Eastern A Coop., nc. 3 Cady, R. A., and L. D. Van Vlec Evaluation of bulls for calving difficulty. Page 74 in Genetics research Report to Eastern A Coop., nc. 4 Everett, R. W., J. F. Keown, and E. E. Clapp Production and stayability trends in dairy cattle. J. Dairy Sci. 59: Harville, D Extension of the Gauss-Marov theore to include the estiation of rando effects. Ann. Stat. 4: Henderson, C. R Sire evaluation and genetic trend. Page 10 in Proc. Anita. Breeding Genet. Syrup. in Honor of Dr. Jay L. Lush. A. Soc. Ani. Sci. and A. Dairy Sci. Assoc., Chapaign, L. 7 Henderson, C. R General flexibility of linear odel techniques for sire evaluation. J. Dairy Sci. 57: Henderson, C. R Best linear unbiased estiation and prediction under a selection odel. Bioetrics 31: Henderson, C. R A siple ethod for coputing the inverse of a nuerator relationship atrix used in prediction of breeding values. Bioetrics 32: Henderson, C. R., and R. L. Quaas Multiple trait evaluation using relatives' records. J. Ani. Sci. 43: Keown, J Cow stayability...how do you easure it? Eastern Artif. Breeders Coop. 36:2. 12 Lee, A. J Mixed odel, ultiple trait evaluation of related sires when all traits are recorded. J. Ani. Sci. 48: Quaas, R. L., and L. D. Van Vlec Categorical trait sire evaluation by best linear unbiased prediction of future progeny category frequencies. Bioetrics 36: Rothschild, M. F., and C. R. Henderson Maxiu lielihood estiates of paraeters of first and second lactation il records. J. Dairy Sci. 62: Rothschild, M. F., C. R. Henderson, and R. L. Quaas Effects of selection on variances and covariances of siulated first and second lactations. J. Dairy Sci. 62:996.