Parameter Estimates of a Random Regression Test Day Model for First Three Lactation Somatic Cell Scores

Transcription

1 Parameter Estmates of a Random Regresson Test Day Model for Frst Three actaton Somatc Cell Scores Z. u, F. Renhardt and R. Reents Unted Datasystems for Anmal Producton (VIT), Hedeweg 1, D Verden, Germany Introducton Somatc cell counts have beng routnely recorded n most dary countres. The avalablty of the data, unform trat defnton and reasonable connectedness across countres make t feasble to evaluate anmals on nternatonal level (Mark et al., 2000). For natonal genetc evaluatons both sngle trat repeatablty model appled to lactaton averages of somatc cell scores (SCS) as well as multple trat test day models have beng used. For estmatng parameters of test day yelds or SCS, the covarance functon approach ncorporated wth an teratve twostep algorthm has been proven to be an effcent way, as t enables analysng very large data set (u et al. 2000). The objectves of ths study were 1) to estmate parameters of frst three lactaton test day SCS va the covarance functon approach, and 2) to derve parameters of lactaton SCS usng the test day parameter estmates. Materal and Methods Orgnal data from August 2000 German Holsten genetc evaluaton were selected matchng the followng crtera: herd-test-dateparty (HTD) classes wth at least fve records, supervsed monthly testng wth two tmes mlkngs only, and calvng years for frst three lactatons no earler than 1993, 1994 and 1995 respectvely. One test day record was randomly chosen n case of multple records wthn each of the sx lactaton stages defned on days n mlk (DIM): 5-50, , , , , Only full lactatons were kept for estmatng parameters. Cows were allowed to have later lactaton records mssng n order to remove the bas caused by selecton. Sres wth fewer than 30 daughters were dscarded to acheve a better data structure. The orgnal pedgree fle from the routne genetc evaluaton was used to extract pedgree nformaton for all ancestors of cow sres. Table 1 shows the structure of the fnal test day data set and sre pedgree fle used n parameter estmaton. For each of the three lactatons, 60 fxed lactaton curves were defned based on three calvng seasons, fve classes of age at calvng and four breed-regon classes. Of the total number of test day records, 49%, 32% and 19% belong to frst, second and thrd lactatons, respectvely. Table 1. Descrpton of the fnal data set and sre pedgree fle for parameter estmaton Factors Cows Sres of cow Test day records n total HTD of all lactatons Fxed lactaton curves n total Anmals n sre pedgree fle Sze 1,727,682 5,415 17,161,866 3,336, ,645 A mult-trat sre model was appled to the frst three lactaton test day SCS to estmate (co)varance components of the sx lactaton stages: y = µ + HTD + β v + s + e jklmn lm l jlp pd p= 1 [1] 5 klm jklmn 61 y jklmn s test day SCS at lactaton stage m of lactaton l of cow n, µ lm s general mean for lactaton stage m of lactaton l, HTD l s the - th herd-test-date effect of lactaton l, v pd s the

2 p-th parameter of Al-Schaeffer functon for DIM d, β jlp s the p-th fxed regresson coeffcent for lactaton l specfc to subclass j of fxed lactaton curves, s klm s addtve genetc effect of sre k for lactaton stage m of lactaton l, and e jklmn s the resdual effect. Test day SCS from dfferent lactaton stages are treated as genetcally dstnct trats n the above model. The teratve two-step algorthm (u et al., 2000) was appled to estmate the (co)varance components of lactaton stages for three lactatons jontly. Ths approach was proven to yeld dentcal results as one-step approach (Royle and Berlner, 1999), provded the teraton process s converged. Resdual maxmum lkelhood estmates of the (co)varance components for lactaton stages were obtaned va VCE (Neumaer and Groeneveld, 1998). Due to the large number of components to be estmated for three lactatons, the estmaton task was parttoned nto seven 9-trat analyses to obtan parameter estmates of all 18 trats. The teraton process was stopped when all (co)varance components and estmated breedng values (EBV) were converged upon predefned convergence crtera. After the teraton process was completed, smple averages of the (co)varance estmates from the seven parallel runs were calculated for dervaton of (co)varances of random regresson coeffcents (RRC). Thrd order normalsed orthogonal egendre polynomal was chosen to smooth the estmated (co)varance matrces of lactaton stages n order to derve (co)varances of RRC. The extended Krkpatrck et al. s weghted least squares method (u et al., 2000) was appled to estmate (co)varances of genetc RRC and to separate the (co)varances of permanent envronmental RRC from error effects, because the extended weghted least squares method can account for dfferent accuraces of the (co)varance estmates of lactaton stages and also appears to guarantee the postve defnteness of the derved (co)varance matrces of RRC. Based on the (co)varances of RRC, genetc parameters for lactaton records of any length can be derved. The hertablty for lactaton records s calculated as follows: h 2 = h 2 gj (, ) gj (, ) + p (, j) + e (,) [2] s hertablty of a lactaton record of length of days, gj (, ) and j (, j = 1,..., ), p (, j) s permanent envronmental covarance between DIM and j, and e (,) s error varance at DIM. For lactatons of length of days, genetc correlaton between two lactatons s: r gkl (, ) = [3] j j g (, j) k g ( k, jk) g ( l, jl) l r g( k, l) s genetc correlaton between lactaton k and l of length days, g ( k, j l ) of lactaton k and DIM j of lactaton l, g ( k, j k ) and j of lactaton k, and g ( l, j l) and j of lactaton l. 62

3 Results and Dscusson Fortran 90 programs and Unx shell scrpts were developed for estmatng the parameters of the multple trat sre model. The computaton was conducted on a HP9000 K460 computer runnng HP-UX 11 and a Pentum III PC runnng nux. For the frst round of the teratve two-step algorthm, sre EBV from the routne SCS genetc evaluaton wth a fxed regresson test day model were used as startng values. Four teratve steps were requred to get both (co)varance estmates and sre EBV converged. Hertablty, genetc and resdual correlatons dd not vary at thrd decmal place between the thrd and fourth teratve steps. The whole estmaton took consderable tme and memory resources. To derve the (co)varances of RRC based on the (co)varance estmates of lactaton stages, Maple 6 programs were wrtten for Mcrosoft Wndows system. For separatng (co)varances of RRC for permanent envronmental effects from error effects usng the extended weghted least squares method, 25 rounds of teraton, wth the (co)varance matrx of genetc RRC as startng value, were conducted to acheve hgh accuracy. Derved hertablty values: Fgures 1 and 2 show derved daly hertablty values and hertablty values for lactaton records, respectvely. Farly homogeneous daly hertablty, wth an average of 0.09, was observed throughout the course of lactaton. Though second and thrd lactatons have 0.01 hgher hertablty than frst lactaton, no sgnfcant dfferences n hertablty values were found between lactatons. The average daly hertablty values agree wth those estmates usng a fxed regresson test day model (Reents et al., 1995) as well as those usng a multple trat anmal model (Reents et al., 1994), but these daly hertablty estmates are strkngly lower than those obtaned va a random regresson model approach (Jamrozk et al., 1997). In Fgure 2 t can be seen that the hertablty of lactaton record ncreases as lactaton makes progress, whch can be explaned by hgher genetc than phenotypc correlatons between DIM of the same lactaton. When a lactaton record s completed, ts hertablty reaches approxmately 0.20, that s hgher than those estmates obtaned based on lactaton models (Mark et al., 2000). It can be concluded that the use of test day models, whch account for envronmental effects specfc to each test day, leads to hgher hertablty estmates. Fgure 1. Daly hertablty values of frst three lactaton SCS Fgure 2. Hertablty values of lactaton SCS as lactaton progresses Hertablty Hertablty ength of lactaton n days Derved genetc covarance structure: at the begnnng, n the mddle, and at the end of lactaton were chosen to descrbe the genetc covarance structure of test day SCS and the results for frst and second lactatons are shown n Fgures 3 and 4, respectvely. Genetc correlatons between two ends of lactaton, e.g. between and 305 or between DIM 5 and 250, are 1 or 4 for frst lactaton SCS, whch are hgher than for test day yelds (u et al., 2000). Daly SCS of second lactaton are less correlated than daly SCS of frst lactaton, but the dfference n wthn-lactaton genetc correlatons s less evdent between lactatons than test day yelds (u et al., 2000). The genetc correlaton structure was almost dentcal between second and thrd lactatons. 63

4 Fgure 5 presents genetc correlatons between the same DIM of two lactatons. Between frst and second lactatons the genetc correlatons range from 4 at 5 to 4 at DIM 105. Compared to the correlatons between frst and second lactatons, lower genetc correlatons were observed between frst and thrd lactatons, and ths was not observed n the parameters of test day yelds (u et al., 2000). It can be seen that the mddle stages of lactaton are more hghly correlated between lactatons than the two ends of lactaton. The genetc correlatons of the same DIM between second and thrd lactaton are qute hgh, above 5, whch ndcates hgh genetc smlarty between second and thrd lactatons. Genetc correlaton Fgure 3. Genetc correlatons between a gven DIM and the remanng part of lactaton for frst lactaton SCS Genetc correlaton Fgure 4. Genetc correlatons between a gven DIM and the remanng part of lactaton for second lactaton SCS Fgure 6 dsplays the rato of genetc standard devatons of two lactatons for test day SCS. The rato curves dffer drastcally to those of test day yelds (u et al., 2000), suggestng dfference n genetc (co)varance structures between SCS and yeld trats. The rato of later to frst lactaton ncreases from early stage up to the mddle of lactaton and then decreases towards the end of lactaton. Fgure 5. Genetc correlatons between the same DIM of two lactatons for test day SCS )JXUH5DWRRIJHQHWFVWDQGDUGGHYDWRQV EHWZHHQWZRODFWDWRQVIRUWHVWGD\6& Genetc correlaton and 1. and 1. and Rato g2 / g1 g3 g3 / / g2 g1 0 Derved phenotypc covarance structure: Fgure 7 shows phenotypc varances durng the course of lactaton for test day SCS. Compared to test day yeld trats, SCS has also a decreasng phase of phenotypc varance at early stage of lactaton but no ncreasng phase at the end of lactaton. Phenotypc varances of test day SCS vary lttle from the mddle through the late stage. Fgure 8 presents derved phenotypc correlatons between DIM 30, 150 or 250 and the remanng part of lactaton for frst lactaton SCS. It should be noted that the phenotypc correlaton at the selected, 150 or 250 corresponds to the repeatablty value at the same DIM. Daly repeatablty values are about, on average, for all lactatons. The rather hgh repeatablty at may be caused partally by the 64

5 assumpton of homogeneous error varance throughout the course of lactaton (u et al., 2000). In general, permanent envronmental effects at dfferent DIM of the same lactaton are less correlated than addtve genetc effects, and consequently the wthn-lactaton phenotypc correlatons are smaller than genetc correlatons. No negatve wthn-lactaton correlatons were found for genetc, permanent envronmental and phenotypc effects. On a full lactaton bass, genetc correlatons are 5, 9 and 7 between frst and second, between frst and thrd, and between second and thrd lactatons, respectvely. These genetc correlatons between full lactatons are hgher than ther correspondng averaged daly correlatons, because daly SCS are all postvely correlated. Varance Fgure 7. Daly phenotypc varances of frst three lactaton SCS Fgure 8. Phenotypc correlatons between a gven DIM and the remanng part of lactaton for frst lactaton SCS Phenotypc correlaton Summary The covarance functon approach ncorporated wth an teratve two-step algorthm was appled to estmate genetc parameters of a random regresson test day model for frst three lactaton SCS. Daly hertablty values of SCS were found to be homogeneous throughout the course of lactaton and between lactatons, about Test day SCS has both hgher wthn-lactaton and between-lactaton genetc correlatons than test day yeld trats. For completed SCS lactaton records, hertablty can reach as hgh as 0.20, whch s larger than those obtaned from a repeatablty model appled to lactaton averages of SCS. Genetc correlatons between full lactatons were very hgh but sgnfcantly less than one. terature Cted Jamrozk, J., Schaeffer,.R., u, Z. & Jansen, G Multple trat random regresson test day model for producton trats. Interbull Bulletn No. 16, Mark, T., Fkse, W.F., Sgurdsson, A. & Phlpsson, J Feasblty of 65 nternatonal genetc evaluatons of dary sres for somatc cell count and clncal mastts. Interbull Bulletn No. 25, u, Z., Renhardt, F. & Reents, R Estmatng parameters of a random regresson test day model for frst three lactaton mlk producton trats usng the covarance functon approach. Interbull Bulletn No. 25, Neumaer, A. & Groeneveld, E Restrcted maxmum lkelhood estmaton of covarances n sparse lnear models. Genet. Sel. Evol. 30, Reents, R., Dekkers, J.C.M. & Schaeffer,.R Genetc parameters of test day somatc cell counts and producton trats. Proc. 5 th World Congr. Genet. Appl. vest. Prod., Guelph, Canada. Vol. 17, Reents, R., Jamrozk, J., Schaeffer,.R. & Dekkers, J.C.M Estmaton of genetc parameters for test day records of somatc cell score. J. Dary Sc. 78, Royle, J. A. & Berlner,. M A herarchcal approach to multvarate spatal modellng and predcton. Journal of Agrcultural, Bologcal, and Envronmental Statstcs. Vol. 1,