Robust procedures for Canadian Test Day Model final report for the Holstein breed

Robust procedures for Canadian Test Day Model final report for the Holstein breed J. Jamrozik, J. Fatehi and L.R. Schaeffer Centre for Genetic Improvement of Livestock, University of Guelph Introduction Objective of this research was to apply the robust estimation procedures to the Canadian Test Day Model (CTDM) and to the Holstein breed in particular. Following recommendations from our previous report (Jamrozik et al., 2006), the robust method with k=2.75 was selected to be tested on the Holstein data. Material and Methods Data: November 2006 genetic evaluation run (CDN) data for the Holstein breed was used, with: 42,605,959 test-day (TD) records, 3,641,329 herd test-day classes, 2,706,031 cows (with data), 3,727,746 animals in pedigree, 23 phantom parent groups, and 190 classes for the effect of region-age-season of calving. Model: The model included multiple lactations (the first three parities) and multiple traits (milk, fat, protein and SCS) and was the same as the routine genetic evaluation model used by CDN (Schaeffer et al., 2000). Methods: The robust estimation method was as in Yang et al. (2004) with k=2.75 adapted for the multiple trait model. The process was as follows in each round of iteration: 1. Calculate residuals for all observations within each of DIM, r i = y i x i b - z i a w i p and the variance of residuals, s 2 j, for the each DIM 2. Modify y i based on the value of the residuals and the DIM class as follows y * i = y i if r i < ks m, y * i = x i b + z i a + w i p - ks m if r i < - ks m, y * i = x i b + z i a + w i p + ks m if r i > ks m. Mixed model equations were also solved (for comparison purposes) with the regular BLUP method. Estimation procedures were compared overall by: sum of squared residuals, sum of absolute residuals, average and standard deviation of residuals. Separate statistics were calculated for each trait and lactation and for each trait (lactations combined). Residuals were defined either as calculated for the original observation (y i ) or the observation used in the iteration process (y * i ). 1

Outliers for the robust method were defined as observations that were corrected during the iteration. They were quantified by: number and proportion of outliers, average, standard deviation, and minimal and maximal values of changes. Outliers were also characterized by number of outliers (1, 2, 3 or 4) for a cow on a given test-day. Distributions of outliers for cows and sires of cows by trait and parity were also calculated. Estimated breeding values for the first two regression coefficients (total yield in lactation and persistency of lactation for a given trait) were summarized by trait and parity for all animals. Correlations between the BLUP and the robust method were estimated for all animals and the first two regression coefficients, by trait and parity. Combined breeding values (first three lactations) were estimated using the official CDN weights for respective traits. Estimates of the intercept of the genetic lactation curve were used for the expression of the total yield and the average SCS in lactation. Milk lactation persistency was approximated by the linear coefficient of animal genetic lactation curve for milk yield. The lactation weights were: 0.33, 0.33, 0.33 for milk, fat and protein yields; 0.25, 0.65, 0.10 for SCS; and 0.50, 0.25, 0.25 for milk lactation persistency. Neither base correction nor scaling of lactation EBV to the same variation level was performed on animal genetic solutions of the mixed model equation. Top bulls (cows) in common, between BLUP and the robust analysis were inspected in relation to number (proportion) of their outlier records. Results Approximately 1600 rounds of iteration were needed to solve MME for both methods, with convergence criterion equal to 2.7e-7. Tables 1 and 2 present sums of squared residuals (SSR) and sums of absolute residuals (SAR) for different estimation procedures, by trait, for corrected and original records, respectively. When corrected records were used for calculating residuals, the robust method performed better than the regular BLUP in terms of both SSR and SAR. Slightly different patterns for SSR and SAR were observed when original observations were used for calculating residuals of the model. The BLUP method gave the lowest SSR for all traits. Table 3 shows sum of squared residuals (SSR), sum of absolute residuals (SAR), average (MEAN) and standard deviation (SD) of residuals for protein yield, by estimation method and parity. Residuals were calculated using the corrected observations in the iteration process. Similar statistic for residuals defined for original observations are in Table 4. Within lactation SSR and SAR statistics followed described earlier trends for traits overall (Tables 1 and 2), for both definitions of residuals. No evident differences in average residuals and their SD were noticed for different methods. Residuals for original observations, however, had the smallest means and variation for the BLUP method compared with the robust procedure. The same patterns were found for within lactation fat, protein and SCS (results not shown). Table 5 gives number (N) and proportion (%) of corrected records, and average (MEAN), standard deviation (SD), minimal (MIN) and maximal (MAX) values of corrections for protein yield from the robust method, by parity. Proportion of corrected records (= outliers) was from 2 to 3%. Average values of corrections were close to zero. Similar observations were made for the remaining traits (results not shown). 2

Table 6 contains number (N) and proportion (relative to the total number of outlier records in a given lactation, in %) of corrections (1, 2, 3 or 4) for a cow on a given TD for the robust procedure in the first parity. Majority of outliers occurred for just a single trait. Proportion of single trait outliers for milk yield in the first lactation was equal to 7%. Estimates for fat (15%) and protein (5%) yields were smaller than for SCS (23%). SCS exhibited the largest number of single outliers compared with the remaining traits. Later lactations gave similar numbers of single trait outliers. Between-trait trends for later parities were the same as those for the first lactation. Two outliers (out of possible four observations) were detected for approximately 14% of records in a given lactation. Outliers for all four traits consisted only a marginal proportion of all records (not more than 1%). Distributions of outliers by DIM (all four traits combined) were in general uniform in the interval from 10 to 305 DIM within each lactation. Average proportion of outlier records in this interval was about 2.7%. The beginning of lactation (DIM from 5 to 10) was characterized by a slightly larger proportion of detected outliers. Proportion of outliers on DIM 5 ranged from 5% (first lactation) to 7% (third lactation). Distributions of outliers for protein yield resulting from the robust method, by trait and parity, are given in Tables 7 and 8 for cows with records and sires of cows, respectively. Most cows for which outliers were detected had a single corrected record. Proportion of cows with more than four records for all traits was equal to zero. Similar observations could be made for distributions of outliers for sires with daughters. Proportions of affected sires, however, were larger than the respective statistics for cows with records. No less than 53% of all sires had at least one outlier. Distributions for milk, fat and SCS (results not shown) followed in general the trends observed for protein yield. Table 9 shows average (MEAN EBV), standard deviation (SD EBV), minimal (MIN EBV) and maximal (MAX EBV) values of estimated breeding values for protein yield lactation curve intercept for all animals (N=3,722,746) for protein yield, by parity. Average estimated breeding values and their SD were practically the same for both methods. Slightly larger difference between distributions of EBV from different estimation procedures could be noticed for linear coefficient of lactation curve (results not shown). Fat, protein and SCS followed in general the behaviour of milk yield distributions (results not shown). Correlations (x1000) between estimated breeding values from BLUP and the robust procedure for the lactation curve intercept and the linear term (all animals, N=3,722,746), by trait and parity, are in Table 10. All correlations were larger than 0.99 indicating that the rankings of animals would be very similar between methods. Table 11 give number of bulls and cows in common in the top 100 lists between BLUP and the robust analysis for combined yields, by trait. Rankings of top cows were more affected by the estimation procedures than rankings for sires. Differences reflected the overall pattern of correlation coefficients between EBV: larger discrepancies between top lists for cows than for bulls, more differences for SCS and lactation persistency compared with milk, fat and protein yields. List of the top 10 sires from the BLUP method contrasted with the respective evaluations from the robust estimation method for combined protein yield is shown in Table 12. Corresponding top 10 cow results for combined protein yield are presented in Table 13. 3

Characteristics of sires that dropped from the BLUP top 100 list for combined protein yield by using the robust estimation method are in Table 14. Table 15 describes cows with data that dropped from the BLUP 100 top list for combined protein yield by using the robust estimation. Proportions of outlier observations were in the same range as those reported for the top animals. No apparent association between the magnitude of changes in EBV and the occurrence of outliers for the selected animals could therefore be established. Discussion Two ways of calculating residuals in the model were applied in this study. The first one used the values of corrected observations while the other used original observations when calculating residuals. The robust procedure was clearly superior (sum of squared residuals and sum of absolute residuals) over the BLUP method when corrected observations were used for residuals. Model comparisons that used original observations for calculating residuals followed in general single trait model results of Yang et al. (2004) and our previous CTDM results for the Jersey breed (Jamrozik et al., 2006). The robust method gave smaller sum of absolute residuals compared with the BLUP model, overall and for all traits and lactations analysed individually. Outliers were defined in this study in an arbitrary way using residuals calculated for each DIM and the coefficient k. Outliers were therefore method dependant and they did not necessarily correspond to the usual definition (perception) of outlier observations. Distributions of outliers by cows with data did not exhibit any evident trends. The same observations were made for sires of cows with records. On a given TD, outliers were more likely associated with one trait only. SCS was the trait that provided the largest proportions of outliers compared with other traits. The model might not be able to handle elevated SCS observation in an optimal way. Similar arguments might apply to the explanation why proportion of outliers was larger at the very beginning of lactation. This period of lactation could be associated with erratic or problematic values of milk recording. Again, inability of the model to account properly for all sources of variation in this part of lactation could be partially responsible for this phenomenon. More than one outlier on a given TD occurred in smaller proportions compared to single outliers. Two or more outlier observations were usually associated with yield traits (milk, fat or protein). Larger environmental correlations between these traits could have been the reason for correlated outliers. Robust estimation methods had in general little effect on estimated breeding values of animals. Rankings for different methods, as indicated by correlation coefficient, did not differ much in comparison with the regular BLUP evaluations. This is in agreement with the results of Yang et al. (2004) for the single trait model and our previous CTDM results for the Jersey breed (Jamrozik et al., 2006). Some bulls and cows changed their position on the list of superior animals. This could not be explained, however, by number or proportion of outlier observations for these animals. Traits differed slightly in their performance by the robust method. Total yields were less affected than persistency; SCS was subject to more changes compared with BLUP than milk, fat or protein yields. 4

Conclusions Application of the robust procedure for genetic evaluation of Canadian Holsteins in CTDM for production traits gave the same overall results as observed earlier for the Jersey breed. The robust method would reduce the influence of outlier observations in the model and improve the model performance in general. Differences in rankings for animals, however, would be small compared with the regular BLUP method. References Jamrozik, J. J. Fatehi, L.R. Schaeffer. 2006. Robust procedures for Canadian Test Day Model. Research Report to the GEB, September 2006, pp. 21. Schaeffer, L.R., J. Jamrozik, G.J. Kistemaker, B.J. Van Doormall. 2000. Experience with a test-day model. J. Dairy Sci. 83: 1135-1144. Yang, R., L.R. Schaeffer, J. Jamrozik. 2004. Robust estimation of breeding values in a random regression test-day model. J. Anim. Breed. Genet. 121: 221-228. 5

Table 1: Sum of squared residuals 1 (SSR) and sum of absolute residuals (SAR) for different estimation procedures, by trait; residuals were defined for corrected observations Trait Number Method SSR SAR of records Milk 42,605,959 BLUP 186,009,920 153,835,216 63,643,616 60,632,072 Fat 42,301,201 BLUP Protein 42,302,907 BLUP SCS 38,406,234 BLUP 1 Residual = y * - E(y) 569,593 469,787 183,776 152,690 22,593,166 17,674,522 3,411,699 3,266,542 2,016,746 1,926,642 20,765,974 19,433,730 Table 2: Sum of squared residuals 1 (SSR) and sum of absolute residuals (SAR) for different estimation procedures, by trait; residuals were defined for original observations Trait Number Method SSR SAR of records Milk 42,605,959 BLUP 186,009,920 197,708,368 63,643,616 63,451,420 Fat 42,301,201 BLUP Protein 42,302,907 BLUP SCS 38,406,234 BLUP 1 Residual = y E(y) 569,593 601,794 183,776 195,094 22,593,166 24,014,146 3,411,699 3,406,423 2,016,746 2,009,682 20,765,974 20,569,862 6

Table 3: Sum of squared residuals 1 (SSR), sum of absolute residuals (SAR), average (MEAN) and standard deviation (SD) of residuals for protein yield, by estimation method and parity; residuals were defined for corrected observations Parity Number Method SSR SAR MEAN SD of records 1 20,035,249 BLUP 69,232 57,189 858,680 819,230 0.00000 0.00000 0.059 0.053 2 13,315,412 BLUP 65,085 54,159 677,195 647,337-0.00002-0.00002 0.070 0.064 3 8,952,246 BLUP 49,459 41,343 480,870 460,075 0.00003 0.00002 0.074 0.068 1 Residual = y * - E(y) Table 4: Sum of squared residuals 1 (SSR), sum of absolute residuals (SAR), average (MEAN) and standard deviation (SD) of residuals for protein yield, by estimation method and parity; residuals were defined for original observations Parity Number Method SSR SAR MEAN SD of records 1 20,035,249 BLUP 69,232 73,482 858,680 855,080 0.00000-0.00061 0.059 0.061 2 13,315,412 BLUP 65,085 68,958 677,195 674,783-0.00002-0.00074 0.070 0.072 3 8,952,246 BLUP 49,459 52,654 480,870 479,819 0.00003-0.00081 0.074 0.077 1 Residual = y - E(y) Table 5: Number (N) and proportion (%) of corrected records for the robust procedure, average (MEAN), standard deviation (SD), minimal (MIN) and maximal (MAX) values of corrections for protein yield, by parity Parity Corrected records MEAN SD MIN MAX N % 1 499,726 2-0.024 0.107-1.44 2.77 2 337,798 3-0.028 0.123-1.23 1.94 3 227,129 3-0.033 0.130-1.22 2.39 7

Table 6: Number (N) and proportion (%) of corrected records (1,2,3 or 4) from the robust procedure for a cow on a given test-day for the first parity Corrected Corrections records N % 1 1,058,038 50 2 291,655 14 3 149,011 7 4 9135 <1 Table 7: Distribution of cows with corrected records from the robust procedure for protein yield, by parity Parity Cows Corrected records (%) with records >0 >1 >4 >6 >8 1 2,631,461 15 3 0 0 0 2 1,765,326 16 3 0 0 0 3 1,200,840 15 3 0 0 0 Table 8: Distribution of sires with corrected records from the robust procedure for protein yield, by parity Parity Sires Corrected records (%) with records >0 >1 >10 >100 >1000 1 32,680 59 44 18 1 0 2 28,190 56 42 14 1 0 3 24,687 53 39 9 1 0 8

Table 9: Average (MEAN EBV), standard deviation (SD EBV), minimal (MIN EBV) and maximal (MAX EBV) values of estimated breeding values for all animals (N=3,722,746) for the intercept of lactation curve for protein yield, by estimation method and parity Parity Method MEAN SD MIN MAX 1 BLUP EBV -0.011-0.011 EBV 0.065 0.065 EBV -0.292-0.292 EBV 0.629 0.622 2 BLUP -0.015-0.015 0.083 0.082-0.401-0.398 0.837 0.665 3 BLUP -0.015-0.015 0.082 0.081-0.365-0.347 0.797 0.620 Table 10: Correlations (x1000) between estimated breeding values from BLUP and the robust procedure, for the lactation curve intercept (a 0 ) and the lactation curve linear term (a 1 ) (all animals, N=3,722,746), by trait and parity Trait Parity Correlation a 0 a 1 Milk 1 999 996 2 999 997 3 999 997 Fat 1 999 996 2 999 997 3 998 997 Protein 1 999 997 2 999 998 3 999 998 SCS 1 998 994 2 998 995 3 998 996 Table 11: Number of bulls (cows) in common in the top 100 lists between BLUP and the robust analysis for combined yields, by trait Trait Bulls Cows Milk 97 85 Fat 94 86 Protein 96 85 SCS 96 92 Milk Persistency 95 69 9

Table 12: Top 10 sires from the BLUP method for combined protein yield in comparison with the ranking from the robust estimation method Sire ID BLUP Outliers EBV Rank EBV Rank N % HOCANM8242373 4.195 1 4.133 1 21 2 HOCANM6427380 4.017 2 3.813 5 311 14 HOUSAM17013604 3.950 3 3.958 2 2571 4 HOCANM7255636 3.870 4 3.927 3 87 5 HOCANM7353778 3.815 5 3.791 6 15 3 HOCANM9292481 3.733 6 3.815 4 8 3 HOCANM9113092 3.703 7 3.747 7 54 5 HOCANM7255640 3.650 8 3.680 8 3 1 HOUSAM2102136 3.597 9 3.543 13 44 5 HONLDM839380546 3.593 10 3.576 10 2338 5 Table 13: Top 10 cows from the BLUP method for combined protein yield in comparison with the ranking from the robust estimation method Cow ID BLUP Outliers EBV Rank EBV Rank N % HOCANF6947915 7.597 1 6.297 1 18 47 HOCANF6866793 6.282 2 6.059 2 26 30 HOCANF8143479 5.928 3 5.779 5 29 32 HOCANF8284233 5.856 4 5.796 4 20 22 HOCANF7255624 5.845 5 5.857 3 11 15 HOCANF8750291 5.679 6 5.691 6 12 33 HOCANF8406836 5.633 7 5.655 7 7 13 HOCANF101515119 5.618 8 5.348 8 19 52 HOCANF101700836 5.513 9 5.328 10 7 25 HOCANF8801653 5.509 10 4.281 170 8 40 Table 14: Characteristics of sires that dropped from the BLUP 100 top list for combined protein yield by using the robust estimation method Sire ID BLUP Outliers EBV Rank EBV Rank N % HONLDM189513757 2.980 91 2.937 102 57 5 HOCANM9372082 2.978 92 2.930 105 136 5 HOUSAM132118158 2.975 94 2.889 118 76 5 HOCANM8235229 2.954 100 2.930 104 0 0 10

Table 15: Characteristics of cows that dropped from the BLUP 100 top list for combined protein yield by using the robust estimation method Cow ID BLUP Outliers EBV Rank EBV Rank N % HOCANF8801653 5.509 10 4.281 170 8 40 HOCANF8421663 5.018 31 3.853 501 18 23 HOCANF8131627 4.882 49 4.434 118 16 17 HOCANF8242354 4.865 50 4.389 132 33 36 HOCANF9041690 4.773 58 3.884 462 38 40 HOCANF8649306 4.658 75 4.419 124 7 10 HOUSAF60445304 4.639 79 4.481 104 5 8 HOCANF8353505 4.637 80 4.442 113 4 13 HOCANF8371964 4.630 82 4.188 214 11 13 HOCANF9245189 4.613 87 4.402 126 20 35 HOCANF9049353 4.588 93 4.475 107 14 15 HOCANF103010876 4.576 95 4.047 313 5 21 HOCANF8750133 4.575 96 4.483 103 3 6 HOCANF8227296 4.574 97 4.485 101 10 10 HOCANF7341431 4.574 98 4.439 114 21 28 11