Z. Liu, R. Reents, F. Reinhardt, and K. Kuwan United Datasystems for Animal Production (VIT), Heideweg 1, D Verden, Germany

Approaches to Estimating Daily Yield from Single Milk Testing Schemes and Use of a.m.-p.m. Records in Test-Day Model Genetic Evaluation in Dairy Cattle Z. Liu, R. Reents, F. Reinhardt, and K. Kuwan United Datasystems for Animal Production (VIT), Heideweg 1, D-27283 Verden, Germany ABSTRACT Statistical models were presented to estimate daily yields from either morning or evening test results. The 64,451 test-day records from 10,392 lactations of 8800 cows were available for analysis from experiments that were designed to investigate the accuracy of an alternate morning and evening four-weekly milk-testing scheme. The experiments were conducted in 152 herds from six German states and covered a span from 1994 to 1998. Milk yield, fat, and protein percentage were recorded for all of the morning and evening milkings. Seven statistical models were fitted to the data to derive formulas for estimating daily yields from morning or evening yields. In general, use of evening milkings less accurately estimated yields than did use of morning milkings. Among the three yield traits the lowest accuracy of estimation of daily yield was found for fat yield. Although the models do not differ much in the correlation between estimated and true daily yields, systematic under- and overestimation of daily yield at the beginning and end of lactation were observed in all models with the exception of model 6, which accounted for heterogeneous variances by parity class, milking interval class, and lactation stage by fitting separate regression formulas within each combination of the three factors. A study to validate the models showed that model 6 is also robust for the analyzed populations. Smoothing model 6 regression formulas across lactation stages caused a systematic pattern of estimation error, although loss in accuracy was minimal by fitting far fewer parameters in the regression formulas. Differences in the accuracy of alternate milking schemes to predict daily yields were found between traits, between morning and evening milkings, and between parity classes. Compared with true daily yields from different lactation stages, variances and correlations of the estimated yields were reduced, which must be accounted for in genetic evaluation. The use of estimated daily Received January 14, 2000. Accepted May 27, 2000. Corresponding author: Z. Liu; e-mail: zliu@vit.du. yields from morning or evening milkings has a smaller impact on estimated breeding values of bulls than cows. As a result of lower heritability and repeatability of estimated daily yields than true daily yields, the weight on own test-day records for estimating cows breeding values is lower when cows are in a.m. p.m. than conventional monthly testing schemes. However, the difference in the weights between estimated and true daily yields decreases as lactation progresses. Use of estimated daily yields is less reliable for estimating breeding value than use of true daily yields. (Key words: alternate a.m. p.m. testing scheme, daily yield, genetic evaluation, test-day model) Abbreviation key: a.m. p.m. = supervised alternate a.m. p.m. testing scheme, A4 = supervised monthly testing scheme, MI = milking interval, MIC = milking interval class. INTRODUCTION Milk recording is essential for herd management and genetic improvement in dairy cattle. Under increasing pressure to reduce costs, numerous milk testing schemes have been developed in many countries in the last decades (5, 14, 21, 22) to supplement the standard supervised four-weekly testing scheme (A4). The alternate morning and evening (a.m. p.m.) testing scheme was designed to achieve reasonable accuracy in estimating daily yields at a lower cost. Between 1989 and 1999, milk testing schemes in Germany comprised 2.2% a.m. p.m. and 97.8% monthly testings. The proportion of a.m. p.m. testing increased to 17.8% in 1999 with variable popularity of a.m. p.m. scheme in different regions of Germany. Many studies have been conducted to investigate the accuracy of a.m. p.m. scheme in comparison with the conventional A4 scheme (1, 2, 3, 9, 11, 21, 22). For estimating daily yield from single milkings, DeLorenzo and Wiggans model (3) has been widely used, which takes heterogeneous means and variances between milking intervals (MI) into account by fitting separate regression models to each milking interval class (MIC). Additionally, a linear regression on DIM was fitted in 2000 J Dairy Sci 83:2672 2682 2672

ESTIMATING DAILY YIELDS 2673 the modified formulas to remove the effect of the shape of lactation curve from residuals. However, parity and lactation stages are not considered in this model. Most research work (2, 6, 12, 16) has been focused on estimating 305-d lactation yields from single milkings. Some systematic biases, e.g., underestimation of daily yield at the beginning and overestimation at the end of lactation, can be canceled out, to a large extent, in the calculation of 305-d lactation yields, except for short lactations. Thus, such systematic bias does not impose serious problems in the estimation of 305-d lactation yields. However, as genetic evaluations that use testday data become more popular in dairy industries (13, 17), a thorough examination of the impacts of various models on daily yield estimates and estimated breeding values is needed. The objectives of this study are 1) to compare statistical models for estimating daily yields, and 2) to investigate the impact of the use of a.m. p.m. data on testday model genetic evaluation. Data MATERIALS AND METHODS Data stem from milk recording experiments designed to assess the accuracy of the a.m. p.m. scheme in six German states: Meklenburg-Western Pomerania, Lower Saxony, Rhineland-Palatinate, Saxony-Anhalt, Schleswig-Holstein and Thuringia, from 1994 to 1998. The 64,451 test-day records of 10,392 lactations of 8800 cows were available from 1055 milk tests in 152 herds. After screening records for unreasonable calving date, DIM, MI, lactation number, and cow registration number, 62,459 test-day records of Holstein cows remained. The MI were grouped into four classes of 30-min intervals for both morning and evening milking records, because intervals shorter or longer than 30 min were considered to be impractical for recording the information. From the preliminary study, it was found that fat and protein yields from single milkings can be more accurately adjusted to a 24-h basis than can fat and protein content. Thus, we decided to estimate regression formulas only for yield traits that are subsequently used to compute fat and protein content on a daily basis. Table 1 shows structure and some descriptive statistics of the analyzed data. It is obvious that variable means and heterogeneous variances exist between MIC, which indicates the necessity of modeling the heterogeneity of MIC variances. Compared to first lactations, later-parity records have higher means and much larger variances. Thus it is also important to consider the variance difference among parity classes. Statistical models that neglect the effects of parity and lactation stages in the estimation of daily yields would lead to biased daily yield estimates. Higher means and large variances are observed for milk yield from morning milkings than evening milkings. Table 2 shows correlations among morning, evening, and daily yields for five production traits. Evening milkings have lower correlations with daily yields than morning milkings. Among all traits, fat content from single milkings are least correlated with its daily measurements, which decreases the accuracy of estimated daily records from single milkings for fat content and consequently for fat yield. The highest correlation among all the traits is observed between morning and evening protein yield. Statistical Models for Estimating Daily Yield from Single Milkings A few statistical models have been proposed to estimate daily yield (y A4 ) from partial yield (y AT ) from single milkings (2, 3, 4, 9). In this study, seven models have been applied to the same data set to compare their accuracy. These models were selected because they were, and some cases still are, used to estimate daily yields by milking recording agencies in various countries. A demonstration of the deficiency of some models is necessary, particularly for the purpose of genetic evaluation with test-day models. Model 0. Original DeLorenzo factors without consideration of DIM (3), which had been de facto the standard method for estimating daily yield in Germany. Heterogeneous means and variances of partial or daily yields from different MIC are modeled by fitting a separate regression line within each MIC. Model 1. Doubling method: y A4 = 2y AT [1] This model assumes that daily yield is expected to be twice the average of morning and evening milkings. No information from actual milk testing experiments is used in deriving conversion formulas from partial to daily yields. Model 2. Single regression: y A4 = b 0 + b 1 y AT [2] Daily yield is regressed on morning or evening partial yield. Only one regression formula is fitted to the whole data set. Model 3. Single regression plus MI as a covariate: y A4 = b 0 + b 1 y AT + b 2 MI [3]

2674 LIU ET AL. Table 1. Description of first (in the first row) and later (in the second row) parity data from the alternate milking testing scheme experiments. <13 h* or 13 h 13.5 h or 13.5 h 14hor 14hor 11.5 h 11 h 11.5 h 10.5 h 11 h <10 h Item + µ σ 2 µ σ 2 µ σ 2 µ σ 2 AM No. records 1,046 10,250 4,116 1,559 5,702 22,701 10,626 6,438 Milk, kg 12.8 11.7 10.8 9.9 11.1 8.9 12.3 12.1 13.8 20.6 13.0 20.2 13.1 19.6 13.4 20.9 Fat 100, kg 52.5 168.5 45.8 145.8 45.6 128.9 48.7 156.8 56.5 312.6 54.2 320.9 53.9 289.5 53.6 310.6 Protein 100 kg 42.8 117.2 37.9 87.5 38.1 76.1 40.6 116.4 46.3 176.7 44.8 178.7 45.2 170.3 45.0 197.4 PM No. records 810 6,265 5,389 4,508 3,690 16,352 13,090 12,338 Milk, kg 11.1 9.0 9.8 8.4 8.8 7.7 8.8 7.7 12.7 18.1 11.8 17.5 11.1 15.8 10.3 14.8 Fat 100, kg 48.1 164.1 45.7 134.8 38.9 136.2 40.9 141.3 55.0 349.2 54.0 326.7 49.3 308.5 49.1 323.5 Protein 100, kg 36.6 72.8 35.0 72.3 30.3 68.9 30.3 70.9 42.6 153.1 40.9 155.2 37.4 137.1 35.5 136.4 DMY No. records 16,975 45,469 µ σ 2 Milk, kg 20.3 34.4 24.4 69.5 Fat 100, kg 88.6 501.8 105.6 1113.0 Protein 100, kg 70.6 301.8 83.5 605.8 + AM = a.m. milking, PM = p.m. milking, DMY = daily milking. *Length of preceding milking interval in hours for morning milkings. Length of preceding milking interval in hours for evening milkings. µ sample mean of trait. σ 2 sample variance of records. Variances have squared units of corresponding traits. Actual trait values of fat and protein yield multiplied by 100. Compared to model 2, an additional regression coefficient on MI is added. Thus the average effect of MI on daily yield is taken into account. Model 4. Separate regression for MIC i: y A4 [i] = b 0 [i] + b 1 [i] y AT [i] [4] A separate regression formula is fitted for each MIC to account for heterogeneous means and variances of partial or daily yields. Models 0 and 4 are equivalent Table 2. Correlations among morning, evening and daily records. Trait AM* PM AM DMY PM DMY Milk, kg 0.908 0.979 0.975 Fat, kg 0.839 0.927 0.922 Protein, kg 0.988 0.971 0.963 Fat, % 0.590 0.864 0.858 Protein, % 0.911 0.955 0.954 *AM = a.m. milking, PM = p.m. milking, DMY = daily milking. if the intercept term for every MIC is equal to zero. Model 4 makes a less restrictive assumption on the regression formulas than model 0. Model 5. Modified DeLorenzo and Wiggans model: y A4 [i] = b 0 [i] + b 1 [i] y AT [i] + b 2 [i] (DIM 158) [5] Similar to model 4, each MIC i has one regression. Additionally, a regression on DIM is included to remove the effect of lactation stages on the residuals (3). The reference DIM is set to 158, representing the DIM at the middle point of lactation. Model 6. Separate regression for every combination of parity i, MIC j, and lactation stage k: y A4 [ijk] = b 0 [ijk] + b 1 [ijk] y AT [ijk] [6] Because different parity and lactation stages have variable means and heterogeneous variances for both partial and daily milking yields, the regression coeffi-

ESTIMATING DAILY YIELDS 2675 cients of daily yield on partial yield are likely to be heterogeneous among parity class or lactation stages. Therefore, a separate regression is fitted for each of the 96 levels (2 parity class 4 MIC 12 lactation stages). As models become complex, more information is utilized in analysis, and this results in improved accuracy in the estimation of daily yields. Assumptions Underlying the Widely Used DeLorenzo and Wiggans Model In contrast to other models, DeLorenzo and Wiggans model (3) uses the ratio of daily to partial yield from single milkings, instead of daily yield itself, as the dependent variable in regression analysis. Thus model 0 implicitly assumes that the regression lines go through point zero, y A4 = 0ify AT = 0, which may be unrealistic, because the lower limit of yields is always greater than zero. Because no regression factors for evening milkings were provided in DeLorenzo and Wiggans study, these factors are usually derived mathematically based on those for morning milkings without analyzing actual evening milking data. As shown later, such derived factors for evening milkings lead to large error in daily yield estimates. As models 1 to 5, model 0 does not consider the effect of parity. Due to lack of individual cow information on sample day fat or protein content as well as preceding MI for single milkings, the authors (3) had to use tank percentages for fat and protein and herd average MI to derive factors to estimate daily yield of fat and protein. As shown in a study by Averdunk et al. (1), using MI information on individual cows can improve the accuracy of estimation. Because of heterogeneous variances during the course of lactation for both partial and daily yields, fitting a single regression line to test day yields from various lactation stages in model 0 as well as models 1 to 5 may not be optimal. Smoothing Model 6 Regression Formulas of Different Lactation Stages For a given parity by MI class, 12 regression formulas are derived with model 6 over 12 mo of lactation. Because the intercept and slope of individual regression formulas are estimated by ignoring test-day record information from neighboring lactation stages, the resulting regression formulas may not look smooth across lactation stages. Because intercept and slope estimates of regression formula are correlated within as well as across lactation stages, it is inappropriate to smooth the intercept or slope estimates separately. We attempted to smooth the intercept and slope jointly by considering the correlations of yields between different lactation stages. Two possible approaches could be im- plemented for smoothing the regression formulas: one is an analogue to a covariance function (8) and the other to Schaeffer and Jamrozik s multiple trait prediction (16) or random regression test-day models (7, 17). The covariance function approach models regression coefficient estimates directly without analyzing actual testday records, but requires the knowledge of (co)variance matrix of intercept and slope estimates that can only be obtained with joint analyses of at least two lactation stages. Because covariances of regression coefficient estimates between lactation stages are unknown, the covariance function approach was not implemented in this study. Instead, the second approach was fitted to actual test-day data. For a given combination of parity class and MIC, the following model is fitted to the data: y A4 = c 0 + c 1 *l + c 2 *l 2 + c 3 *l 3 + c 4 *l*y AT [7] + c 5 *l 2 *y AT + c 6 *l 3 *y AT + c 7 *y AT + e where l is lactation stage in months (l = 1, 2,, 12) or weeks (l = 1, 2,, 52). The quadratic and cubic terms l 2 and l 3 allow that the relation between partial and daily yields during the course of lactation are nonmonotone, because the terms c 0, c 1, c 2, c 3 or c 4, c 5, c 6, c 7 can have different signs. The number of parameters to be estimated for a given combination of parity and MIC is reduced from 24, as in model 6, to 8. The model [7] is more parsimonious than model 6. Once the regression coefficients of the smoothing model [7] are estimated with actual data, regression formula for a given lactation stage, i, can be obtained: ŷ [i] A4 = [ĉ 0 + ĉ 1 *i + ĉ 2 *i 2 + ĉ 3 *i 3 ] + [ĉ 4 *i + ĉ 5 *i 2 [8] + ĉ 6 *i 3 + ĉ 7 ]*y AT = b 0 [i] + b 1 [i] *y AT To study the impact of the presented models on (co)- variance structure of daily yields, a fixed effects model is used to analyze both true and estimated daily yields: y ijk = HTDP i + b 1j DIM + b 2j DIM 2 + e ijk [9] where y ijk is true or estimated daily yield of cow k, HTDP i is the i-th herd-test-date-parity fixed effect, b 1j,b 2j are regression coefficients of daily yield from j- th class of calving season and parity on DIM, and e ijk is residual effect. Note that the above model contains the fixed effects of the test-day model proposed by Reents et al. (13). A separate lactation curve is fitted for each class of calving season parity effects. Instead of assuming a complex (co)variance structure as in Reents et al. (13), residual effects are assumed to be uncorrelated between records

2676 LIU ET AL. of the same lactation. Deviations of true as well as estimated daily yields were calculated: DEV ijk = y ijk HTDPˆ i + bˆ 1jDIM + bˆ 2jDIM 2 [10] Accuracy of a.m. p.m. Testing Scheme The accuracy of a.m. p.m. milk testing scheme for estimating daily yield is defined as (10): R 2 = σ 2 /(σ 2 + MSE) [11] Where σ 2 is phenotypic variance of true daily yield, and MSE is mean squared error that is equal to variance of the difference between true and estimated daily yields. For yields on 24-h basis, formula [11] gives R 2 = 1. Phenotypic variance of true daily yield (σ 2 ) was estimated using REML method (15) under the model [9] and results are presented in Table 3. The a.m. p.m. scheme for estimating lactation yield with multiple test-day records is expected to be more accurate than estimating daily yield because of the high repeatability of the yield traits. Use of Single Milking Data in Test-Day Model Genetic Evaluation Estimated and true daily yields could be treated as genetically different traits in genetic evaluations. This so-called multiple trait approach is theoretically optimal, but technically infeasible for routine genetic evaluation for large dairy populations, e.g., the German Holstein population, as a result of high computing cost of this approach. Therefore, the conventional approach (19, 20) is modified to jointly analyze estimated and true daily yields in test-day model genetic evaluation, in which estimated daily yields are assigned to larger error variance based on the accuracy of a.m. p.m. scheme than are true daily yields. This approach leads to a much smaller increase in computing costs than the aforementioned multiple-trait approach. Because estimated daily yields from single milkings have smaller phenotypic variance than true daily yields, the estimated yields are to be first expanded by dividing the estimated yields with R 2, so that the expanded estimated yields have equal genetic and permanent Table 3. REML estimates of phenotypic variances for daily yield traits. Trait First parity Later parities All parities Milk, kg 19.34 34.42 31.95 Fat 100, kg 311.37 594.15 547.61 Protein 100, kg 173.97 314.53 291.42 environmental variances as true yields but larger error variance than true yields (18, 19, 20). Because no genetic (co)variance parameters for the estimated daily yields are available, this expansion method is based on phenotypic information only, and it assumes a genetic correlation of one between the expanded and true daily yields. The expanded yields contain exactly the same effects as the true yields plus an additional, independent error effects with variance being approximately MSE that was obtained from analyzing a.m. p.m. data. Compared to true daily yields, expanded daily yield estimates have lower heritability R 2 h 2 and repeatability R 2 t, where h 2 and t are heritability and repeatability of true daily yields, respectively. The ratio of error variances between true and expanded daily yields is (1 t)/ 1 R 2 t. For lactation with n monthly tests the accuracy of a.m. p.m. testing scheme for estimating lactation yield can be derived with the formula [12]: R 2 L = 1 + (n 1)t 1 + (n 1)tR 2R2 [12] Note that R 2 L > R 2 with n > 1. Depending on computational feasibility, it should be decided whether parity specific accuracy needs to be considered in genetic evaluation, in addition to different R 2 for each of the yield traits as well as for morning or evening milkings. RESULTS AND DISCUSSION Goodness of Fit of the Models Table 4 shows correlations between true and estimated daily yields (r ya4,ŷ A4 ), MSE, and standard deviations of daily yield estimates (σ ŷa4 ) from morning or evening milkings. The model with the smallest MSE and the highest correlation gives the best fit to the data. Standard deviation σ ŷa4 should be close to standard deviation of true daily yield, but it must not be greater. Though the correlations improve along with the complexity of the models, with model 6 achieving the highest correlations, differences between the models are small, except that model 0 leads to low correlation for evening milk yield. The poor fit of model 0 could be partially explained by the genetic and environmental differences between the US Holstein population, for which model 0 was developed (3), and the German Holstein population since the creation of Model 0. Relatively significant differences in the correlations were observed among the three production traits. The low

ESTIMATING DAILY YIELDS 2677 Table 4. Correlations (r ya4,ŷ ) between true and estimated daily yields, mean squared errors (MSE) and A4 standard deviations (σ ŷa4 ) of daily yield estimates from morning or evening milkings. Milk, kg Fat 100, kg Protein 100, kg Item + Model r ya4,ŷ A4 σ ŷa4 MSE r ya4,ŷ A4 σ ŷa4 MSE r ya4,ŷ A4 σ ŷa4 MSE AM 0 97.6 8.050* 1.764 92.6 35.29 13.30 N/A 1 97.2 8.550 2.033 93.9 35.09 12.08 96.5 25.69 6.80 2 97.2 7.812 1.895 93.9 30.03 10.97 96.5 23.23 6.34 3 97.6 7.844 1.757 93.9 30.04 10.96 97.0 23.35 5.89 4 97.6 7.847 1.746 94.3 30.05 10.93 97.0 23.36 5.86 5 97.6 7.849 1.736 94.1 30.10 10.79 97.0 23.37 5.83 6 97.7 7.852 1.720 94.3 30.16 10.60 97.1 23.38 5.77 PM 0 95.8 9.355 2.842 93.3 31.88 11.67 1 96.8 8.025 2.033 93.2 33.24 12.08 96.1 24.33 6.80 2 96.8 7.781 2.019 93.3 29.83 11.51 96.1 23.13 6.69 3 97.1 7.805 1.925 93.4 29.86 11.43 96.4 23.21 6.42 4 97.1 7.806 1.919 93.4 29.87 11.42 96.4 23.22 6.40 5 97.2 7.814 1.889 93.7 29.95 11.19 96.5 23.23 6.33 6 97.4 7.826 1.838 94.0 30.06 10.90 96.6 23.27 6.18 + AM = a.m. milking, PM = p.m. milking. Correlations multiplied by 100. *Standard deviations of daily yield estimates that are greater than those of true daily yield, 8.039, 31.97 and 24.08 for milk, fat and protein yield respectively, are underscored. As no estimates of Model 0 for protein yield were available for this study, no analysis was conducted. correlation for fat yield indicates that there may be factors influencing fat yield or content that were not accounted for in the models. Evening milkings give slightly lower correlations than morning ones. All models, except models 0 and 1, have smaller σ ŷa4 than the true value. Overestimation of σ ya4 indicates that models 0 and 1 are inappropriate for estimating daily yield. By definition, MSE measures both unbiasedness and variance of estimates, thus is the most appropriate statistic for ranking models. Based on MSE, model 6 achieves the best fit, whereas models 0 and 1 the worst. Figures 1 and 2 show averaged estimation errors: y A4 ŷ A4, in daily milk yield estimated from morning or evening milkings, respectively. Because model 6 fits an individual regression formula for each lactation stage, the average estimation error by lactation stage is expected to be zero. For the remaining models, daily milk yield is underestimated at the beginning of lactation and overestimated at the end of lactation. This indicates that the ratio of partial to daily yields varies over the course of lactation. This systematic pattern of estimation error does not have a significant effect on the estimation of 305-d lactation yields, except for short lactations, because the over- and underestimation are canceled out, to a large extent. However, for estimating yield on daily basis, this systematic pattern of estimation error cannot be eliminated by alternating morning and evening testings. The impacts of season of test or calving, herd production level, parity as well as MIC were also studied. All the models tend to overestimate daily yields from low producing test or calving seasons from morning milkings and to underestimate yields from low producing test or calving seasons from evening milkings. By alternating morning and evening testings, this estimation error in different test or calving seasons can be canceled out, to a great extent. This was also observed for parity and MIC. Because none of the seven models accounts for herd production level, which is partially confounding with MIC, yields from high producing herds were on Figure 1. Averaged error in estimated daily milk yield from morning milkings. Solid line with symbol for model 2, solid line with symbol for model 3, solid line with symbol for model 4, dotted line with symbol for model 5, dashed line with symbol for model 6, dashed line with symbol for model 0. Due to the large value of estimation errors, model 1 is not shown.

2678 LIU ET AL. of the validation of the models. In general, all of the models seem to be robust for different data sets in respect to the correlation between true and estimated daily yields. Because of the higher production level in the state from which the regression formulas were derived, daily milk yields from the rest of the states were slightly overestimated. The ranking of the models based on MSE remains very similar to the ranking based on data from all states. Model 6 is proven to be the best of all in this validation study. The regression formulas of all models were also used to analyze a data set of German Simmentals, and once again the superiority of model 6 was confirmed. Figure 2. Averaged error in estimated daily milk yield from evening milkings. Solid line with symbol for model 2, solid line with symbol for model 3, solid line with symbol for model 4, dotted line with symbol for model 5, dashed line with symbol for model 6, dashed line with symbol for model 0. Due to the large value of estimation errors, models 0 and 1 are not shown. average slightly underestimated, whereas yields from low producing herds were overestimated. As mentioned before, the estimation error by herd level can be partially removed by alternating morning and evening milkings. Because only model 6 allows for variable means and variances at different parities by fitting separate regression lines within parity, the remaining models tend to over- and underestimate daily yields from first and later parities, respectively, though the magnitude of the estimation error is small. Validation of the Models The robustness of the models was examined by applying regression formulas estimated from one subset to the rest of the data set. Table 5 shows the results Smoothing Methods The results of smoothing model 6 regression formulas are shown in Table 6 and in Figure 3. Based on r ya4,ŷ A4, σ ŷa4 and MSE, the four smoothing methods do not cause a significant loss in goodness of fit even with a much smaller number of fitted parameters. Changing the unit of lactation stage from months to weeks leads to small loss in goodness of fit, which may be explained by some inaccurate test or calving dates in the field data. In spite of the good agreement of r ya4,ŷ A4, σ ŷa4 and MSE between the original and smoothed model 6, there is an over- or underestimation of daily yield by the four smoothed models. Smoothed models with cubic function of lactation stages lead to smaller estimation error than those with quadratic function, but the regression formulas estimated using the models with cubic function do not look smooth, as is also the case for the original model 6. Therefore, it is not recommended to use smoothed regression formulas for estimating daily yields in practice. Table 5. Results of the validation study. Milk, kg Item + Model r ya4,ŷ A4 averaged error σ ŷa4 MSE AM 1 97.6 1.316 7.567 1.777 2 97.6 0.472 6.962 1.685 3 97.7 0.051 6.962 1.647 4 97.7 0.005 7.066 1.644 5 97.7 0.002 7.072 1.655 6 97.7 0.079 7.037 1.657 PM 1 97.4 1.316 7.337 1.777 2 97.4 0.680 7.280 1.770 3 97.4 0.273 7.315 1.756 4 97.4 0.265 7.293 1.758 5 97.6 0.253 7.281 1.713 6 97.6 0.300 7.310 1.713 + AM = a.m. milking, PM = p.m. milking. Correlations multiplied by 100.

ESTIMATING DAILY YIELDS 2679 Besides the smallest MSE and no systematic bias during the course of lactation and robustness with respect to populations, model 6 is simple for milk recording agents to use. With the information of milking interval and parity and lactation stage, the original morning or evening yields can be retrieved using estimated daily yields, if needed. It may not be necessary to fit one regression line for every month of lactation. For example, the months at the middle of lactation might be clustered together to derive a joint regression formula for estimating daily yields, due to similar means and variances between the lactation months. However, the large amount of the analyzed data made it possible to derive reliable regression formulas for every month of lactation. As shown above, alternating morning and evening testing is essential to reduce the influences of some systematic effects, e.g., herd production levels, on estimation of daily yields; thus, it is important to check whether a.m. p.m. testings were properly alternated before being provided to routine genetic evaluations. Milk recording organizations should conduct new experiments on a regular basis to improve the regression formulas for estimating daily yields. Accuracy of the a.m. p.m. Scheme Results of a.m. p.m. scheme accuracy for estimating daily yield are summarized in Table 7. Model 6 has the highest accuracy for all traits. A difference of about 2% in R 2 exists between morning and evening milkings for the yield traits. Differences in R 2 were found among the traits, with milk yield reaching the highest R 2 value at 91%, and fat yield the lowest at 83%. The R 2 for later parities is about 2% higher than that for first parity. From formula [12], it is clear that the accuracy of a.m. p.m. scheme for estimating daily yield is lower than for estimating lactation yield as given by VanRaden (18). Figure 4 shows the increase in accuracy of the a.m. p.m. scheme for estimating lactation yield with an increasing number of effective test-day records. For completed lactations with more than five effective testday records, the accuracy of a.m. p.m. scheme shown in this figure is comparable to the values published by VanRaden (18). It can also be seen in Figure 4 that, as lactation progresses, the difference between the accuracies for estimating lactation milk and fat yields becomes smaller. Variances and Correlations of Test Day Yields by Lactation Stages The models for estimating daily yield should not only give minimum MSE but also keep (co)variance structure of true daily yields possibly unchanged. To examine the (co)variance structure changes, true and estimated milk yields from model 6 were analyzed using the same model [9]. The ratio of the variance of estimated yields to the variance of true yields is not dependent on the sample size, because the same data were used. Figure 5 shows the ratios from morning and evening milkings and their average by lactation stage. Morning milkings have a higher ratio than evening milkings, but their average is almost constant across all lactation stages. This suggests that there is no need to consider lactation stage specific reduction in phenotypic variance due to the use of estimated daily yields in genetic evaluations. The average ratio being less than one indicates that estimated daily yields need to be expanded with the factor 1/R 2 so that estimated daily yields have equal genetic as well as permanent environmental variances as true daily yields. In Figure 6, autocorrelations of true and estimated daily milk yields from single milkings are shown. The autocorrelations of daily yields decay as the time distance between two tests increases, which is an important characteristic of test-day data. The more distant two tests are, the less they are correlated. True daily yields have autocorrelation ranging from 0.14 for tests 9 mo apart to 0.83 for tests 1 mo apart. The autocorrelations of estimated daily yields show the same tendency as those of true yields, but there is a slight Table 6. Results of the smoothing model 6 regression formula. Model r ya4,ŷ A4 σ ŷa4 MSE Model 6 97.69 7.8525 1.7198 Model 7 with quadratic function of 97.66 7.8507 1.7279 lactation stage in months Model 7 with cubic function of 97.67 7.8513 1.7252 lactation stage in months Model 7 with quadratic function of 97.66 7.8506 1.7281 lactation stage in weeks Model 7 with cubic function 97.67 7.8512 1.7254 lactation stage in weeks

2680 LIU ET AL. Figure 3. Averaged error in estimated daily milk yield using model 6 and four smoothing methods from morning milkings. Dashed line with symbol for model 6 without smoothing, dotted line with symbol for model 7 with quadratic function of lactation stage in weeks, dotted line with symbol for model 7 with quadratic function of lactation stage in months, solid line with symbol for model 7 with cubic function of lactation stage in weeks, solid line with symbol for model 7 with cubic function of lactation stage in months. average drop in correlation of 0.02 for morning milkings and 0.05 for evening milkings. The reduction in autocorrelation indicates that the (co)variance structure of estimated daily yields is slightly different than that of true daily yields. Impact of Single Milking Data on Estimated Breeding Values As mentioned above, estimated daily yields from single milkings are to be expanded in test-day model genetic evaluations, so that both true and expanded daily yields have equal genetic as well as permanent environ- Figure 4. Increase in the accuracy of a.m.-p.m. testing scheme for estimating lactation yields as lactation progresses. Dotted line for first lactation estimated milk yields from morning milkings, solid line for first lactation estimated fat yields from morning milkings. mental variances. This approach results in lower heritability and repeatability for expanded than true daily yields, and consequently the weights on data and pedigree index for estimating cow s breeding values are different between true and estimated daily yields. To illustrate the impact of using a.m. p.m. data on EBV, an example cow, instead of bull, is chosen, because cow EBV are more affected by the use of a.m. p.m. data than bull EBV. It is assumed that the EBV of the sire and dam of a cow have reliabilities of 0.85 and 0.50, respectively. The size of contemporary groups for this cow is set to be 10. Furthermore it is assumed that this cow is in first lactation and has no progeny. The parental contribution to cow s EBV is assumed to be constant throughout the lactation process. The changes in relative weights on test-day records and pedigree Table 7. Accuracy (R 2 in percent) of the alternate a.m. p.m. milk testing scheme for estimating daily yields. Milk yield from parity Fat yield from parity Protein yield from parity Item + Model 1 >1 All 1 >1 All 1 >1 All AM 0 88.2 91.5 91.1 77.9 80.3 75.6 1 85.9 88.8 88.5 76.8 79.2 79.0 83.0 86.7 86.3 2 87.2 90.2 89.9 78.4 80.6 82.0 84.6 88.3 87.9 3 88.5 91.5 91.2 78.7 80.9 82.0 86.0 89.7 89.4 4 88.5 91.6 91.3 78.7 80.9 82.1 86.0 89.8 89.5 5 88.7 91.7 91.4 79.3 81.5 82.5 86.3 90.0 89.6 6 89.3 91.8 91.5 80.5 82.0 83.0 87.0 90.1 89.7 PM 0 76.8 80.2 79.8 73.5 75.8 80.1 1 85.9 88.8 88.5 76.9 79.2 79.0 83.0 86.7 86.3 2 86.4 89.0 88.7 80.8 82.1 80.5 84.0 87.0 86.7 3 87.0 89.9 89.6 80.8 82.2 80.7 84.4 88.0 87.6 4 87.1 90.0 89.7 81.0 82.2 80.8 84.6 88.1 87.7 5 87.3 90.3 90.0 80.7 82.7 81.4 84.6 88.3 87.9 6 88.8 90.6 90.4 82.3 83.0 82.2 86.2 88.6 88.4 + AM = a.m. milking, PM = p.m. milking.

ESTIMATING DAILY YIELDS 2681 Figure 5. Ratio of variance of estimated daily milk yields from morning and evening milkings using model 6 to variance of truly daily yields. Solid line with symbol for the ratio of variance of estimate daily yields from morning milkings using model 6 to variance of true daily yields, solid line with symbol for the ratio of variance of estimated daily yields from evening milkings using model 6 to variance of true daily yields, dotted line with symbol for average of the ratios from morning and evening milkings. index for estimating first lactation breeding value of fat yield for the cow are shown in Figure 7. The relative weights on test-day records and pedigree index sum to 1 at any point of lactation for both types of testing schemes. As the cow makes progress in lactation, the weight on pedigree index decreases, and in the mean time the weight on her records increases. For this particular example, pedigree index and test-day records receive equal weights for estimating cow breeding value when the cow has approximately 3.5 test-day records Figure 7. Relative weights on own test-day records and pedigree index for estimating cow s breeding value using true and estimated daily yields (trait: first lactation fat yield). Ascending thin line for relative weights on true daily fat yields of first lactation, descending thin line for relative weights on pedigree index of first lactation fat yield in case of standard A4 testing scheme, ascending thick line for relative weights on first lactation estimated daily fat yields from a.m. p.m. testing scheme, descending thick line for relative weights on pedigree index of first lactation estimated daily yields from a.m. p.m. testing scheme. from standard A4 scheme. Assuming that the very same cow was in a.m. p.m. testing program, instead of A4 scheme, the cow s own test-day records (pedigree index) receive less (more) weight for estimating her breeding value than the case for true daily yields due to lower heritability and repeatability of estimated daily fat yields. However, the differences in the weights between true and estimated daily yields diminish when the cow has undergone seven or more tests. The cow would need nearly one more test to have her own records receive a weight of 0.5 in case of a.m. p.m. than the A4 testing scheme. For traits with higher accuracy of a.m. p.m. scheme, the differences in weights on the two sources of information between estimated and true daily yields will be smaller than for fat yield. For the same amount of data information, in terms of number of lactations and number of tests per lactation, reliability of cow EBV will be lower when this cow is in a.m. p.m. than in standard A4 testing schemes. CONCLUSIONS Figure 6. Autocorrelations of true daily yields and of estimated daily yields from morning and evening milkings (trait: first lactation milk yield). Solid line with symbol for autocorrelation of true daily yields, solid line with symbol for autocorrelation of estimated daily yields from morning milkings, solid line with symbol for autocorrelation of estimated daily yields from evening milkings. Statistical models were presented for estimating daily yields from single milkings. Model 6, which accounts for heterogeneous variances due to parity, MIC, and lactation stages by fitting separate regression formula within each combination of the three factors gives the best fit to the data, in terms of MSE, correlation between estimated and true daily yields, and variance of the estimates. With a validation study, the regression

2682 LIU ET AL. formulas of model 6 were proven to be robust for different data sets. Daily yields estimated from evening milkings or first parity are less accurate than those from morning milkings or later parities, respectively. Alternating morning and evening milkings can partially remove the estimation errors caused by some factors, like test and calving seasons, MIC, herd production level or parity. However, the systematic under- or overestimation at the beginning and at the end of lactation observed for all models, except model 6, cannot be canceled out by the alternation of morning and evening milkings. Smoothing model 6 regression formulas across lactation stages results in a systematic pattern of estimation error, although there is nearly no loss in accuracy for estimating daily yields despite fitting much fewer parameters. DeLorenzo and Wiggans model, which was widely used in Germany, was shown to be less accurate than model 6, particularly for evening milking data. Because of its simplicity, model 6 is easy to implement. The accuracy of the a.m. p.m. testing scheme for estimating lactation yields increases with the progress of lactation. Because of the high repeatability of test-day records, the accuracy of a.m. p.m. scheme is higher for estimating lactation yields than for estimating daily yields. Estimated daily yields from single milkings are expanded in routine genetic evaluations, so that expanded daily yields have equal genetic and permanent environmental variances as true daily yields, but they are assigned to larger error variances than true daily yields. This approach leads to lower heritability and repeatability for expanded daily yields, and consequently for estimating cow breeding value her own records receive less weight when she is in a.m. p.m. than standard A4 testing program. Using estimated daily yields from a.m. p.m. results in a lower reliability of cow EBV than using true daily yields. 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