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Extending Part Lactation Milk Records by Regression Ignoring Herd Effects
EXTENDING PART LACTATION MILK RECORDS BY REGRESSION IGNORING HERD EFFECTS L. D. ~AN ¥LECK AZCDC. R. HENDERSON Department of Animal :Husbandry, Corncll University, Ithaca, New York suD/[1vfARY Regression coefficients for predicting 10-mo. milk yield from single test-day records, from cumulative test-day records of less than 10 too. duration, from bimonthly and trimonthly test-day records, and from other combinations of test-day records are estimated ignoring herd effects. The records, adjusted for age a~ calving and season of calving, used for estimating these coefficients were from 9,036 Holstein cows in five New York counties. These regression coefficients are compared with intraherd coefficients reported earlier which were estimated from the same data. The relative efficiencies of the two procedures are compared to determine which procedure is more practical. For most combinations of records regression ignoring herd effects is less than 20% less efficient and in many cases less than 10% less efficient than regression considering herd effects. It is concluded that prediction of 10-too. milk yield by regression ignoring herd effects is more practical for most situations, because of computational advantages although the accuracy of prediction is slightly less in all situations than for intra-herd prediction.
M a n y reports (3, 4, 6, 8, 9, 13, 14, and 15) have shown relatively high correlations between milk yields of part lactations of varying lengths and complete lactation yield. Most of these reports have been of total correlations which were obtained without considering herd or other effects. Recently, Madden et al. (10), also ignoring herd effects, have reported regression coefficients for extending monthly test day records and cumulative monthly test-day records to a complete yield basis. These regressions ignored herd effects but were calculated separately for less than 3 yr. of age at calving and more than 3 yr. of age at calving. Fritz et al. (5), also using Michigan D.H.I.A. records, concluded that herd effects on regression factors for extending part records were not important. This conclusion was based on an analysis of variance of regression coefficients. VanVleck and Henderson (17) reported within herd regressions of complete lactation milk yield on various combinations of age and season adjusted monthly test-day records. All such reports either considering herd effects or ignoring herd effects have emphasized the relatively large correlations between part lactation yields for 5 or more too. and 10-too. lactation yield. The accuracy of prediction on a within-herd basis (intra-herd regression) is greater than of prediction ignoring herd effects (total regression), but the difficulty of using within-herd regression coefficients is considerable for commercial or cooperative dairy records processing centers. In order to use intra-herd regression coefficients in prediction equations herd averages for total yield and for part lactation yield are necessary. These averages can not be obtained economically in m a n y cases. The herd means are also likely to be estimated with considerable error. Therefore, regression equations ignoring herd effects would be desirable in order to facilitate economical extension of part records. Estimates of population averages would be used in the prediction equations rather than herd averages. T h e former Received for publication ~Vfarch 3, 1961. 1519
1520
L. D. V A N V L E C K
AND
C. R.
HENDERSON
would be easy to obtain and would be subject to less error than would estimates of herd means. Such prediction equations, however, must be reasonably accurate. The purpose of this paper is to present regression coefficients which m a y be used to estimate lactation milk yield from various functions of monthly test-day records ignoring herd effects and to compare the efficiency of these predictors with comparable predictors obtained from regression on a within-herd basis. DATA
The records used in this study were the same as those used by VanVleck and Henderson (17) for the estimation of within-herd regression coefficients. The D.H.I.A. milk records were made b y 9,036 Holstein cows in five New York counties. Ten monthly test-day records were available for each cow's lactation beginning with the first month of lactation and ending with the tenth. Zero production was defined to be a valid test-day yield. The first of the test-day records had to be taken less than 50 days after calving. Otherwise, the lactation record was discarded as were lactations having missing test-day records. The sum of the first ten test-day records was defined to be a complete yield. Multiplication by 30.5 estimates the 305-day record. The monthly records were adjusted for age at calving and season of calving with ratio factors described by VanVleck and Henderson (16). ANALYSIS
PROCEDURE
The usual normal equations were solved to estimate the desired regression coefficients. The right- and left-hand sides of the normal equations were made up of total sums of squares and cross-products corrected for means. The reduction in total sum of squares of total yield due to regression (the multiple or simple correlation coefficient squared, R 2) was computed for each regression function. Functions were estimated for the same combinations of test-day records as those treated by Van Vleek and Henderson (17) on a within-herd basis. Clearly, a method is needed to determine the relative effectiveness of the two regression procedures (within-herd versus ignoring herds). Comparison of the multiple or simple R's does not adequately show the relative efficiency of the two procedures. The R 2 associated with within-herd regression describes the amount of variation in within-herd variance which is accounted for by within-herd regression, while the R 2 associated with regression ignoring herds is the amount of the total variance (corrected for the mean) which is accounted for by regression ignoring herd effects. Certainly, these R's can not be compared. What is of concern is the variance of the difference between actual and predicted yield. The residual variance not accounted for by total regression is ( 1 - RI)Z~,~ ~2 where R[ is the square of the multiple or simple correlation coefficient when herd N N ( Z Yi) ~ effects are ignored and d~ is the estimated variance of total yield, ~=1
N-1
N
On the other hand, the variance not accounted for by within-herd regression is
EXTENDING
PART
1521
LACTATION RECORDS
~2 where R2w is the square of the correlation coefficient on a within-herd ( 1 - - R ~ ) ze, basis and a2eis the estimated error variance of total yield after eliminating herd varini
( ~ Yjk) 2
ance, j=l k=~
y2 jk N--s
.
j=l
n~-
•
Note that~ ~ ~
k~l
N
y2
ik = ~
i=1
y2.
,,
j=l
n~. = N =
the total number of observations, and s = the number of herds• These two values are the residual variances of prediction for the two procedures. The prediction procedure which provides the smaller error variance is defined to be the better of the two procedures. Experience leads to the belief that (1 --Rw)a~ ~ 2 is usually smaller than (1 - RI)a~. 2 2 Then the relative efficiency of the two procedures as a per cent can be defined as E = 100
= 100
-
R~ )a;,- I
R~)~,-]2 ~- . The absolute
errors, of course, are x / ~ and v ' ~ , which correspond to the standard deviations of the difference between actual and predicted records for the two procedures. Even with a measure of relative efficiency defined there remains another difficulty. Suppose Method 2 is 20% more efficient than Method 1. Should Method 1 be used rather than Method 2? This problem, naturally enough, is not amenable to an easy solution. The experimenter or the user of the procedures must make the final decision. He will be guided by the importance of the additional error and by the importance of the relative difficulties of computation of the two procedures. Perhaps, in the present situation, the ease of computation of Procedure 1 would more than counterbalance an increased error variance of 20% associated with Procedure 1. Then the using agency would choose Procedure 1. Or, if the user decided that the ease of Procedure 1 is not as important as the increased inaccuracy, then Procedure 2 would be used. RESULTS AND DISCUSSION Regression coefficients. The means of the monthly test-day milk records and the means of cumulative monthly test records are shown in Table 1. These values are comparable to those reported by Madden et al. (10) for Michigan data for cows 3 yr. and older at calving, except that the New York means drop off more rapidly toward the end of the lactation period. Estimates of average monthly milk production can be obtained by multiplying the test-day means by 30.5. Regression factors ignoring herd levels for predicting total yield from a single monthly test-day record are given in Table 2. These values may be compared with the comparable within-herd regression coefficients given in Table 2 of VanVleck and Henderson (17). For each monthly test-day period the regression coefficient, b, is larger when herd effects are ignored. The b's ignoring heMs are larger than the within-herd b's. A not very precise test of significance of the difference can be accomplished. The standard error of a b is about 0.06 which makes the variance about 0.0036. The variance of (b~ - b j) is less than 2(0.0036) since any covariance between bi, total regression, and bj, intra-herd regression, in our problem is ex-
1522
L.D.
VANVLECK AND C. R. I-IENDERSON
TABLE 1 M e a n s of a g e a n d season a d j u s t e d m o n t h l y t e s t - d a y r e c o r d s a n d m e a n s of c u m u l a t i v e m o n t h l y t e s t - d a y rec ords ~¢[onth
Single months Cumulative months
1
2
3
4
5
6
57.1 57.1
59.8 116.9
56.9 173.8
51.7 22'5.5
46.7 272..2
41.5 313.7
7
8
3 ~ 5 . 2 28.3 348.9 377.2
9
10
18.9 396.1
11.8 407.9
TABLE 2 Regression factors for e s t i m a t i n g total lactation yield from a single monthly test record ~ Month of l a c t a t i o n
b sb R
1
2
3
4
5
6
7
8
9
10
5.52 0.06 0.67
5.59 0.04 0.82
5..86 0.04 0.86
6.58 0.04 0.89
7.14 0.04 0.90
7.51 0.04 0.89
7.83 0.05 0.87
7.33. 0.06 0.81
6.08 0.07 0.68
4.33 0.07 0.5.2
" b is the r e g r e s s i o n coefficient, sb is t h e s t a n d a r d e r r o r of t he r e g r e s s i o n coefficient, a n d R is the c o r r e l a t i o n coefficient b e t w e e n the p r e d i c t e d a n d a c t u a l value.
TABLE
3
Regression factors for estimating total lactation yield from sequential test-day data Sequential months 1
2 3 4 5 6 7 8 9 10
M o n t h l y r e g r e s s i o n coefficients 1 5.52 1.68 1.32 1.04 0.77 0.74 0.83 0.88 0.95 1.00
10
Ra
...................................................... 4.57 ................................................ 1.46 3.86 .......................................... 0.95 1.35 3.86 .................................... 0.90 1.09 1.5.2 3.51 ...... .................. ...... 0.89 1.07 1.11 1.36 3.11 ........................ 0.9'8 0.97 0.97 1.06 1.14 2.89 .................. 1.07 1.01 0.91 0.90 0.98 0.89 2.68 ............ 1.03 1.04 0.96 0.97 0.98 1.00 0.95 1.90 ...... 1.00 1.0'0 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2
3
4
5
6
7
8
9
0.6,7 0.84 0.89 0.92 0.95 0.96 0.97 0.99 1.00 1.00
" E is the m u l t i p l e c o r r e l a t i o n coefficient b e t w e e n t he p r e d i c t e d a n d a c t u a l value.
TABLE 4 Regression factors for estimating total lactation yield from sequential monthly test day records when first m o n t h s a r e m i s s i n g Sequential months
M o n t h l y r e g r e s s i o n coefficients 1
2
3
4
5
6
7
8
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 3 4 5 6 7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
9 10
.......................................... .................................... .............................. ........................ 3.68 .................. 3.00 1.47 ............ 2.03 1.30 1.22 ...... 1.47 1.03 1.03 1.13 1.00 1.00 1.00 1.00 1.00
4.60 1.72 1.22 1.19 1.08 1.00
5.80 1.79 1.26 0.99 0.89 0.95 1.00
6.87 1.61 1.14 0.SG 0.83 0.92 1.02 1.00
9
10
4.33 6.0~8 --0.11 0.15 0.55 0.59 0.76 0.6,8 0,.91 0.81 0.99 0.92 1.00 1.01 0.94 1.00 0.9'4 1.00 1.00
0.52 0.68 0.82 0.89 0.93 0'.96 0.98 0.99 1.0:0' 1.00
EXTENDING
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LACTATION
RECORDS
1523
peered to be positive. Tabulated t-values can be compared with our sample t statistic,
b~ - bj The degrees of freedom associated with the test can only x/2 (0.0036)"
be approximated but are probably not very critical in the present situation because of the large number of degrees of freedom associated with the variances of b~ and bi. Even though the total and intra-herd b's are significantly different, this result does not necessarily preclude the use of total regressions over all herds, as will be shown later. The total correlations between single monthly test-day records and total yield are very similar to those presented by Madden et al. (10). An example of the use of regression factors to estimate total yield from a single monthly test-day record follows: suppose the fourth test-day record is 61.7 lb. of milk. Then the predicted sum of ten test-day records is ?) = 407.9 -4- 6.58(61.7 51.7) = 473.7. Multiplication by 30.5 gives the predicted 305-day yield, 14,448 lbs. of milk. The predicted sum is given in general by ~ = p~ + bl (Xi - hl) where by is the estimate of the population average sum of test-day records, b~ is the regression coefficient associated with the i th month, and p~ is the estimate of the population average test-day record in the i th month. Multiple regression factors ignoring herd effects are given in Table 3 for the situation when the last monthly test-day records are lacking and in Table 4 for the situation when the first monthly records are missing. Relatively great accuracy of prediction can be obtained from a linear function of the first three or more monthly test-day records. The expected correlation with total yield when using the first three records is 0.89. The correlation increases with the addition of more monthly test records. The total multiple regression coefficients appear to be different from the intra-herd coefficients reported in" Tables 3 and 4 of VanVleck and Henderson (17) unless five or more monthly records are included in the function. The total regression factors for extending cumulative monthly test-day records are of more interest than the multiple factors, since records are maintained in a cumulative manner by most records-processing laboratories. These coefficients are reported in Table 5 for the situation when the last monthly records are missing, and in Table 6 when the first monthly records are not available. It is interesting to notice that the accuracy of prediction using cumulative records is only slightly less than using a linear function of the same monthly records. The corresponding intra-herd regression factors for cumulative months are given in Tables 5 and 6 of VanVleek and Henderson (17). The intra-herd and total regression coefficients are very similar for the cumulative records, especially when the last months are missing and the first four or more monthly test records are included in the cumulative sum. Bimonthly or trimonthly testing is a possibility for reducing the cost of testing for herd owners. The high correlation between bimonthly and monthly testing reported by several researchers, (1, 2, 7, 11, and 12), advances this possibility. The within-herd prediction equations based on bimonthly and trimonthly testing presented in Tables 7 and 8 of VanVleck and Henderson (17) further suggest the practicality of such testing procedures. Total regression coefficients for linear rune-
1524
L.D.
VAN VLECK AND C. R. HENDERSON
TABLE 5 R e g r e s s i o n f a c t o r s for e s t i m a t i n g t o t a l l a c t a t i o n y i e l d f r o m c u m u l a t i v e t e s t - d a y records C u m u l a t i v e m o n t h of l a c t a t i o n
b R
1
2
3
4
5
6
7
8
9
5.52 0.67
3.26 0.82
2.28 0.87
1.79 0.91
1.5.2 0.93
1.34 0.95
1.21 0.97
1.12 0.98
1.05 0,.99
TABLE 6 R e g r e s s i o n f a c t o r s f o r e s t i m a t i n g t o t a l l a c t a t i o n y i e l d f r o m c u m u l a t i v e t e s t - d a y re c ords when f i r s t - m o n t h records a r e m i s s i n g C u m u l a t i v e m o n t h of l a c t a t i o n
b R
10
9
8
7
6
5
4
3
2
4.83 0.52
2.91 0.63
2.40 0.74
2.11 0.83
1.83 0.89
1.59 0.93
1.40 0.96
1.22 0.98
1.08 1.00
TABLE 7 Regression factors for estimating total lactation yield from sequential bimonthly and t r i m o n t h l y t e s t - d a y records M o n t h of l a c t a t i o n
Monthly set
1
2
3
4
5
6
7
8
9
1 2
1.32 ......
...... 1.98
2.17 ......
..... 2.03
1.94 ......
...... 2.15
2.00 .....
...... 2.0'5
2.36 ......
1
1.58
......
4.15
2
......
3
........
........... 2.57
3.17
............
3.20
e:87
...........
............
3.34
10
R
..... 1.32
0.99 0.99
.................. 3.51 ............ .......... 2.59 ......
0.96 0.98 0.98
TABLE 8 Regression factors for estimating total lactation yield from cumulative bimonthly and t r i m o n t h l y t e s t - d a y records B i m o n t h l y set
b R
T r i m o n t h l y set
1
2
1
2
3
1.96 0.99
1.95 0.99
2.94 0.95
2.92 0.97
3.09 0.98
TABLE 9 R e l a t i v e efficiency of t o t a l a n d i n t r a - h e r d r e g r e s s i o n for p r e d i c t i n g t o t a l l a c t a t i o n y i e l d f r o m s i n g l e m o n t h l y t e s t - d a y m i l k re c ords Month
V1a V~b /~
1
2
3
4
5
6
7
8
9
10
4,585 3,774 121
2,828 2,4.25 117
2,252 1,916 118
1,802 1,599 113
1,736 1,56.0 111
1,82'8 1,612 113
2,14.5 1,775 121
2,998 2,209 13.6
4,78¢ 3,176 15.1
6,446 4,0'82 158
a V1 is the r e s i d u a l v a r i a n c e of t h e sum of t h e first t e n t e s t re c ords ( t o t a l y i e l d ) n o t a c c o u n t e d f o r b y t o t a l r e g r e s s i o n . The r e g r e s s i o n coefficients a r e shown i n T a b l e 2. b V~ is the r e s i d u a l v a r i a n c e of t o t a l y i e l d n o t a c c o u n t e d for b y i n t r a - h e r d r e g r e s s i o n . The i n t r a - h e r d r e g r e s s i o n coefficients a r e shown i n T a b l e 2 of V a n V l e c k a n d H e n d e r s o n ( 1 7 ) .
C E i s the ratio 100 ( ~ ) , w h i e h
is the relative efficiency of the two procedures.
EXTENDING PART
LACTATION RECORDS
1525
tions of bimonthly and trimonthly test-day records are shown in Table 7 and coefficients for cumulative bimonthly and trimonthly test records in Table 8. The total regression coefficients do not appear different from the intraherd coefficients. Some are almost identical. Again, it is apparent that the linear functions of bimonthly or trimonthly test-day records are only slightly better predictors of total yield than the corresponding cumulative sums of bimonthly and trimonthly test records.
Comparison of the e~ciency of the total regression and intra-herd regression equations. As stated previously, the residual variances not accounted for by regression can be used to compare the two procedures which are used to predict total yield. Procedure 1 is prediction based on regression ignoring herd effects and Procedure 2 is prediction based on within-herd regression. The relative efficiencies of the two procedures when using single test-day records are given in Table 9, together with the residual variances. The relative efficiencies are smallest for the fourth, fifth, and sixth m o n t h s - - t h e same months which provide the best estimates of total yield. The difference in variances is about 200 for these residuals, yet the differences in the average errors (the square roots of the residual variances) are only about 2 or 3 lb. of milk. For example, the square roots of the residual variances for the fourth month are: (V1)t = (1802)t = 42.5 and (V2) t = (1599) i = 40.0. This, then, shows an average error of measurement of 2.5 lb. more for Procedure 1 than Procedure 2. The average total yield from Table 1 is 407.9 lb. The percentage of additional error is thus less than 1% relative to total yield. It would appear that if prediction were based on the fourth, fifth, or sixth monthly test day record, the total regression equation could be used without compromising accuracy very much and with more computational ease. On the contrary, if the least desirable monthly test (the tenth) were used then the decision would probably be to use Procedure 2. Since the cumulative test-day records are nearly as accurate as linear functions of the same test-day records for predicting total yield, the relative efficiencies of the sequential functions will be discussed only briefly, but are shown in Table 10. For the conventional situation when the last monthly records are lacking the regression procedure ignoring herd effects can apparently be used rather than the intra-herd procedure, because the difference in accuracy is relatively small. On the other hand, when the first months are missing the differences in accuracy of the two procedures are quite large unless four or five monthly tests are available. The relative effieieneies of Procedures 1 and 2 when prediction is based on cumulative monthly test-day records are given in Table 11. These effieiencies follow the same pattern as those shown in Table 10. The effieieneies of the cumulative procedures when the last months are missing are similar, but when the first months are missing Procedure 2 is much more efficient. As was seen in Table 5, the average correlation between the yield predicted from a 5-mo. cumulative record and total yield is O.93. The increase in average error of prediction over within-herd prediction is (1176) t - (1045) '~ = 2.0 lb. Therefore, it seems that prediction of total yield ignoring herd effects based on 5-mo. records is a possibility which could be considered for earlier proving of young sires. Table 12 presents the relative efficieneies between Procedures 1 and 2 for linear functions of bimonthly and trimonthly test records. It is readily apparent that
1526
L.D.
VAN YLECK AND C. R. HENDERSON T A B L E 10
Relative efficiency of total a n d i n t r a - h e r d r e g r e s s i o n f o r p r e d i c t i n g t o t a l l a c t a t i o n yield f r o m linear f u n c t i o n s o f s e q u e n t i a l m o n t h l y t e s t - d a y milk records '~ S e q u e n t i a l m o n t h s of l a c t a t i o n - - l a s t m o n t h s m i s s i n g
V1 172 E
1
9
3
4
5
6
7
8
9
4,585 3,774 121
2,638 2,293 115
1,914 1,664 115
1,378 1,226 112
93'0 839 111
654 598 109
~¢1 4,10' 108
190 182 104
51 50 10'2
Sequential m o n t h s of l a c t a t i o n - - f i r s t m o n t h s m i s s i n g
171 17~ E
10
9
8
7
6
5
4
3
2
6,4~6 4,082 158
~,784 3,164 151
2,965 2,130 139
1,789 1,384 129
1,159 9~9 122
727 622 117
348 306 113
166 149 112
65 60 108
" T h e linear f u n c t i o n s c o r r e s p o n d i n g to V~ a r e g i v e n in T a b l e s 3 a n d 4, a n d t h o s e corres p o n d i n g to V., a r e s h o w n in T a b l e s 3 a n d 4 of Y a n V l e c k a n d H e n d e r s o n ( 1 7 ) .
T A B L E II Relative efficiency of t o t a l a n d i n t r a - h e r d r e g r e s s i o n f o r p r e d i c t i n g t o t a l l a c t a t i o n yield f r o m c u m u l a t i v e m o n t h l y t e s t - d a y milk records " C u m u l a t i v e m o n t h s of l a c t a t i o n
V1 172 E
1
2
3
4
5
4,585 3,774 121
2,82.9 2,462 115
2,080 1,820 114
1,594 1,407 113
1,176 1,045 113
last months missing 6 854 763 112
7
8
9
594 534 111
325 295 110
105 96 109
C u m u l a t i v e m o n t h s of l a c t a t i o n - - f i r s t m o n t h s m i s s i n g
171 172 E
10
9
8
7
6
5
4
3
2
6,446 ~,082 158
5,292 3,396 156
3,995 2,644 151
2,794 1,927 145
1,871 1,354 138
1,191 906 131
645 512 126
281 2'32 121
86 75 115
" T h e r e g r e s s i o n coefficients c o r r e s p o n d i n g to V1 a r e g i v e n in T a b l e s 5 a n d 6 a n d those c o r r e s p o n d i n g to 172 a r e shown in Tables 5 a n d 6 of V a n V l e c k a n d H e n d e r s o n (17).
T A B L E 12 Relative efficiency of total a n d i n t r a - h e r d r e g r e s s i o n f o r p r e d i c t i n g t o t a l l a c t a t i o n yield f r o m b i m o n t h l y a n d t r i m o n t h l y t e s t - d a y milk records ~ Sequential function B i m o n t h l y set 1 2 V1 V~ E
166 163 102
171 16,3 105
1
Cumulative
T r i m o n t h l y set 2 3
678 644 105
4,26 414 103
398 370 108
B i m o n t h l y set ] 2 214 207 104
213 194 110
1
T r i m o n t h l y set 2 3
904 851 106
457 444 103
421 384 110
T h e r e g r e s s i o n coefficients a s s o c i a t e d w i t h V1 a r e given in T a b l e s 7 a n d 8 a n d those with V2 are given in T a b l e s 7 a n d 8 of V a n V l e c k a n d H e n d e r s o n (17).
EXTENDING PART LACTATION RECORDS
1527
there is little difference between the errors of prediction for the two procedures. Prediction of total yield ignoring herd effects from cumulative bimonthly tests appears to merit consideration as a replacement for testing every month. Testing costs for the herd owner could be approximately halved without any appreciable loss of accuracy in estimating total yield. This conclusion has previously been advanced by Gifford (7), McDowell (11), McKellip and Seath (12), and Alexander and Yapp (1). Bayley et al. (2) reported that bimonthly tests are about 96% as accurate as monthly tests, but cautioned "the frequency of the large errors indicates that bimonthly and quarterly records should be satisfactory for sire provings and population studies, but they may be unsatisfactory when used as individual lactation records." CONCLUSIONS
The slight additional inaccuracy of predicting total milk yield from regression ignoring herd effects as compared with intra-herd regression does not appear as important as the extra computational difficulties encountered with intra-herd prediction for most practical situations. The practical situations considered include predicting total yield from five or more cumulative monthly test-day records, predicting total yield from cumulative bimonthly test-day records, and predicting total yield from a single test-day record obtained in the fourth, fifth, or sixth month of lactation. The high correlations which exist between the total yield and cumulative monthly records 5 or more mo. in length and between total yield and cumulative bimonthly records strongly indicate the possibility of sire proving based on short records or on bimonthly records. The former could reduce the costs of proving young sires, increase genetic progress through a shorter generation interval or additional sampling, or provide a combination of these results, while the latter could reduce by about half the testing cost to herd owners. REFERENCES (1) AL~XANI)F31%M. It., AND YAPP, W. W. Comparison of Methods of Estimating !Ylilk and F a t Production in Dairy Cows. J. Dairy Sci., 32: 621. 19~9. (2) BAYL~Y, N. D., LIss, R. M., AN]) S~ALL~D, J . E. A Comparison of Bimonthly and Quarterly Testing with Monthly Testing for E s t i m a t i n g Dairy Catt]e Production. J. Dairy Sci., 35: 350. 1952. (3) CA~N0~r, C. Y., ~ V E , J. B., AND SI~S, 5. A. Predicting 305-Day Yields from Short Time Records. J. Dairy Sei., 25: 991. 1942. (4) DASSA~T, P. ~ possible anticiparee rendere pid accurate il progeny test nel bestiame da latte? Atti IV. Riunione Biometric Society e I Riunione Assoclazone Genetica Italiana, Roma, 27-28 Matzo. 1954. (5) FRITZ, G. R., ~]~C~ILLIARD, L. D., AND MADDEN, D. E.
E n v i r o n m e n t a l I n f l u e n c e on
Regression Factors for Estimating 305-Day Production from P a r t Lactation. J. Dairy Sci., ¢3~: 1108. 1960. (6) GAINES, W. L. Deferred Short-Time Test as a ~VIeasure of the Performance of Dairy Cows. J. Agr. Research, 35: 237. 1927. (7) GIPFORa), W. The Reliability of Bimonthly Tests. J. Dairy Sei., 13: 81. 1930. (8) JSsT, 1K. Die Beurteilung verschledener Ffirsenkurzleistungsabschnitte im IIinbliek auf eine vorl~iufige Erbwertsch~itzung bei Bulien. Ziichtungskunde, 31: 201. 1959.
1528
L. D. VAN VLECK AND C. R. HENDERSON
(9) KITSCH, J., .~-~D H0~F~I~, J. Untersuchungen fiber die MSglichkeit eincr friihzeitigen Erbwertermittlung auf Grund der F~irsenkurzleistung yon 200 und 305 Tagen. Z. Tierziicht. Zfichtungsbiol., 64: 153. 1954. (10) M ~ D ~ , D. E., MC~ILLIAP~D, L. D., AND RALSTON, 1~. P. Relations Between Test-Day Milk Production of Holstein Dairy Cows. J. Dairy Sei., 42: 319. 1959. (11) McDow~ur., J. C. Testing Cows for Production Every Other Month. Holstein-Friesian World, 24: 1970. 1927. (12) MoKEhL~, I., AND S F ~ H , D. A Comparison of the Different Methods of Calculating Yearly Milk and B u t t e r f a t Records. J. Dairy Sci., 24: 181. 19@1. (13) O'CONNOR, L. K., A~-D ST$WAItT, A. The Use of 180-Day Records in Contemporary Comparisons. Rep. Prod. Div. Milk Mktg. Bd., 8: 93. 1958. (1~) R~rDE~, J. M., ROBEKTSOIT, A., ASKE~, A. A., KHISHIN, 8. S., AND RKGAB, M. W. The Inheritance of Milk Production Characteristics. J. Agr. Sci., 48: ~26. 1957. (15) T u ~ , P. Some Factors Influencing Milk Yield. Rept. Proc., W o r l d ' s Dairy Congr., 9: 151. 1931. (16) VA~rVL~oK, L. D., ~ H E a D , SON, C.. R. Ratio Factors for A d j u s t i n g Monthly TestDay Data f o r Age a n d Season of Calving a n d Ratio Factors for Extending P a r t Lactation Records. J. Dairy Sci., 44.: 1093. 1961. (17) V ~ V I ~ C K , L. D., AND H~NDE~SON, C. R. Regression Factors for Extending P a r t Lactation Milk Records. J. Dairy Sci., 44: 1085. 1961.
M a n y reports (3, 4, 6, 8, 9, 13, 14, and 15) have shown relatively high correlations between milk yields of part lactations of varying lengths and complete lactation yield. Most of these reports have been of total correlations which were obtained without considering herd or other effects. Recently, Madden et al. (10), also ignoring herd effects, have reported regression coefficients for extending monthly test day records and cumulative monthly test-day records to a complete yield basis. These regressions ignored herd effects but were calculated separately for less than 3 yr. of age at calving and more than 3 yr. of age at calving. Fritz et al. (5), also using Michigan D.H.I.A. records, concluded that herd effects on regression factors for extending part records were not important. This conclusion was based on an analysis of variance of regression coefficients. VanVleck and Henderson (17) reported within herd regressions of complete lactation milk yield on various combinations of age and season adjusted monthly test-day records. All such reports either considering herd effects or ignoring herd effects have emphasized the relatively large correlations between part lactation yields for 5 or more too. and 10-too. lactation yield. The accuracy of prediction on a within-herd basis (intra-herd regression) is greater than of prediction ignoring herd effects (total regression), but the difficulty of using within-herd regression coefficients is considerable for commercial or cooperative dairy records processing centers. In order to use intra-herd regression coefficients in prediction equations herd averages for total yield and for part lactation yield are necessary. These averages can not be obtained economically in m a n y cases. The herd means are also likely to be estimated with considerable error. Therefore, regression equations ignoring herd effects would be desirable in order to facilitate economical extension of part records. Estimates of population averages would be used in the prediction equations rather than herd averages. T h e former Received for publication ~Vfarch 3, 1961. 1519
1520
L. D. V A N V L E C K
AND
C. R.
HENDERSON
would be easy to obtain and would be subject to less error than would estimates of herd means. Such prediction equations, however, must be reasonably accurate. The purpose of this paper is to present regression coefficients which m a y be used to estimate lactation milk yield from various functions of monthly test-day records ignoring herd effects and to compare the efficiency of these predictors with comparable predictors obtained from regression on a within-herd basis. DATA
The records used in this study were the same as those used by VanVleck and Henderson (17) for the estimation of within-herd regression coefficients. The D.H.I.A. milk records were made b y 9,036 Holstein cows in five New York counties. Ten monthly test-day records were available for each cow's lactation beginning with the first month of lactation and ending with the tenth. Zero production was defined to be a valid test-day yield. The first of the test-day records had to be taken less than 50 days after calving. Otherwise, the lactation record was discarded as were lactations having missing test-day records. The sum of the first ten test-day records was defined to be a complete yield. Multiplication by 30.5 estimates the 305-day record. The monthly records were adjusted for age at calving and season of calving with ratio factors described by VanVleck and Henderson (16). ANALYSIS
PROCEDURE
The usual normal equations were solved to estimate the desired regression coefficients. The right- and left-hand sides of the normal equations were made up of total sums of squares and cross-products corrected for means. The reduction in total sum of squares of total yield due to regression (the multiple or simple correlation coefficient squared, R 2) was computed for each regression function. Functions were estimated for the same combinations of test-day records as those treated by Van Vleek and Henderson (17) on a within-herd basis. Clearly, a method is needed to determine the relative effectiveness of the two regression procedures (within-herd versus ignoring herds). Comparison of the multiple or simple R's does not adequately show the relative efficiency of the two procedures. The R 2 associated with within-herd regression describes the amount of variation in within-herd variance which is accounted for by within-herd regression, while the R 2 associated with regression ignoring herds is the amount of the total variance (corrected for the mean) which is accounted for by regression ignoring herd effects. Certainly, these R's can not be compared. What is of concern is the variance of the difference between actual and predicted yield. The residual variance not accounted for by total regression is ( 1 - RI)Z~,~ ~2 where R[ is the square of the multiple or simple correlation coefficient when herd N N ( Z Yi) ~ effects are ignored and d~ is the estimated variance of total yield, ~=1
N-1
N
On the other hand, the variance not accounted for by within-herd regression is
EXTENDING
PART
1521
LACTATION RECORDS
~2 where R2w is the square of the correlation coefficient on a within-herd ( 1 - - R ~ ) ze, basis and a2eis the estimated error variance of total yield after eliminating herd varini
( ~ Yjk) 2
ance, j=l k=~
y2 jk N--s
.
j=l
n~-
•
Note that~ ~ ~
k~l
N
y2
ik = ~
i=1
y2.
,,
j=l
n~. = N =
the total number of observations, and s = the number of herds• These two values are the residual variances of prediction for the two procedures. The prediction procedure which provides the smaller error variance is defined to be the better of the two procedures. Experience leads to the belief that (1 --Rw)a~ ~ 2 is usually smaller than (1 - RI)a~. 2 2 Then the relative efficiency of the two procedures as a per cent can be defined as E = 100
= 100
-
R~ )a;,- I
R~)~,-]2 ~- . The absolute
errors, of course, are x / ~ and v ' ~ , which correspond to the standard deviations of the difference between actual and predicted records for the two procedures. Even with a measure of relative efficiency defined there remains another difficulty. Suppose Method 2 is 20% more efficient than Method 1. Should Method 1 be used rather than Method 2? This problem, naturally enough, is not amenable to an easy solution. The experimenter or the user of the procedures must make the final decision. He will be guided by the importance of the additional error and by the importance of the relative difficulties of computation of the two procedures. Perhaps, in the present situation, the ease of computation of Procedure 1 would more than counterbalance an increased error variance of 20% associated with Procedure 1. Then the using agency would choose Procedure 1. Or, if the user decided that the ease of Procedure 1 is not as important as the increased inaccuracy, then Procedure 2 would be used. RESULTS AND DISCUSSION Regression coefficients. The means of the monthly test-day milk records and the means of cumulative monthly test records are shown in Table 1. These values are comparable to those reported by Madden et al. (10) for Michigan data for cows 3 yr. and older at calving, except that the New York means drop off more rapidly toward the end of the lactation period. Estimates of average monthly milk production can be obtained by multiplying the test-day means by 30.5. Regression factors ignoring herd levels for predicting total yield from a single monthly test-day record are given in Table 2. These values may be compared with the comparable within-herd regression coefficients given in Table 2 of VanVleck and Henderson (17). For each monthly test-day period the regression coefficient, b, is larger when herd effects are ignored. The b's ignoring heMs are larger than the within-herd b's. A not very precise test of significance of the difference can be accomplished. The standard error of a b is about 0.06 which makes the variance about 0.0036. The variance of (b~ - b j) is less than 2(0.0036) since any covariance between bi, total regression, and bj, intra-herd regression, in our problem is ex-
1522
L.D.
VANVLECK AND C. R. I-IENDERSON
TABLE 1 M e a n s of a g e a n d season a d j u s t e d m o n t h l y t e s t - d a y r e c o r d s a n d m e a n s of c u m u l a t i v e m o n t h l y t e s t - d a y rec ords ~¢[onth
Single months Cumulative months
1
2
3
4
5
6
57.1 57.1
59.8 116.9
56.9 173.8
51.7 22'5.5
46.7 272..2
41.5 313.7
7
8
3 ~ 5 . 2 28.3 348.9 377.2
9
10
18.9 396.1
11.8 407.9
TABLE 2 Regression factors for e s t i m a t i n g total lactation yield from a single monthly test record ~ Month of l a c t a t i o n
b sb R
1
2
3
4
5
6
7
8
9
10
5.52 0.06 0.67
5.59 0.04 0.82
5..86 0.04 0.86
6.58 0.04 0.89
7.14 0.04 0.90
7.51 0.04 0.89
7.83 0.05 0.87
7.33. 0.06 0.81
6.08 0.07 0.68
4.33 0.07 0.5.2
" b is the r e g r e s s i o n coefficient, sb is t h e s t a n d a r d e r r o r of t he r e g r e s s i o n coefficient, a n d R is the c o r r e l a t i o n coefficient b e t w e e n the p r e d i c t e d a n d a c t u a l value.
TABLE
3
Regression factors for estimating total lactation yield from sequential test-day data Sequential months 1
2 3 4 5 6 7 8 9 10
M o n t h l y r e g r e s s i o n coefficients 1 5.52 1.68 1.32 1.04 0.77 0.74 0.83 0.88 0.95 1.00
10
Ra
...................................................... 4.57 ................................................ 1.46 3.86 .......................................... 0.95 1.35 3.86 .................................... 0.90 1.09 1.5.2 3.51 ...... .................. ...... 0.89 1.07 1.11 1.36 3.11 ........................ 0.9'8 0.97 0.97 1.06 1.14 2.89 .................. 1.07 1.01 0.91 0.90 0.98 0.89 2.68 ............ 1.03 1.04 0.96 0.97 0.98 1.00 0.95 1.90 ...... 1.00 1.0'0 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2
3
4
5
6
7
8
9
0.6,7 0.84 0.89 0.92 0.95 0.96 0.97 0.99 1.00 1.00
" E is the m u l t i p l e c o r r e l a t i o n coefficient b e t w e e n t he p r e d i c t e d a n d a c t u a l value.
TABLE 4 Regression factors for estimating total lactation yield from sequential monthly test day records when first m o n t h s a r e m i s s i n g Sequential months
M o n t h l y r e g r e s s i o n coefficients 1
2
3
4
5
6
7
8
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 3 4 5 6 7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
9 10
.......................................... .................................... .............................. ........................ 3.68 .................. 3.00 1.47 ............ 2.03 1.30 1.22 ...... 1.47 1.03 1.03 1.13 1.00 1.00 1.00 1.00 1.00
4.60 1.72 1.22 1.19 1.08 1.00
5.80 1.79 1.26 0.99 0.89 0.95 1.00
6.87 1.61 1.14 0.SG 0.83 0.92 1.02 1.00
9
10
4.33 6.0~8 --0.11 0.15 0.55 0.59 0.76 0.6,8 0,.91 0.81 0.99 0.92 1.00 1.01 0.94 1.00 0.9'4 1.00 1.00
0.52 0.68 0.82 0.89 0.93 0'.96 0.98 0.99 1.0:0' 1.00
EXTENDING
PART
LACTATION
RECORDS
1523
peered to be positive. Tabulated t-values can be compared with our sample t statistic,
b~ - bj The degrees of freedom associated with the test can only x/2 (0.0036)"
be approximated but are probably not very critical in the present situation because of the large number of degrees of freedom associated with the variances of b~ and bi. Even though the total and intra-herd b's are significantly different, this result does not necessarily preclude the use of total regressions over all herds, as will be shown later. The total correlations between single monthly test-day records and total yield are very similar to those presented by Madden et al. (10). An example of the use of regression factors to estimate total yield from a single monthly test-day record follows: suppose the fourth test-day record is 61.7 lb. of milk. Then the predicted sum of ten test-day records is ?) = 407.9 -4- 6.58(61.7 51.7) = 473.7. Multiplication by 30.5 gives the predicted 305-day yield, 14,448 lbs. of milk. The predicted sum is given in general by ~ = p~ + bl (Xi - hl) where by is the estimate of the population average sum of test-day records, b~ is the regression coefficient associated with the i th month, and p~ is the estimate of the population average test-day record in the i th month. Multiple regression factors ignoring herd effects are given in Table 3 for the situation when the last monthly test-day records are lacking and in Table 4 for the situation when the first monthly records are missing. Relatively great accuracy of prediction can be obtained from a linear function of the first three or more monthly test-day records. The expected correlation with total yield when using the first three records is 0.89. The correlation increases with the addition of more monthly test records. The total multiple regression coefficients appear to be different from the intra-herd coefficients reported in" Tables 3 and 4 of VanVleck and Henderson (17) unless five or more monthly records are included in the function. The total regression factors for extending cumulative monthly test-day records are of more interest than the multiple factors, since records are maintained in a cumulative manner by most records-processing laboratories. These coefficients are reported in Table 5 for the situation when the last monthly records are missing, and in Table 6 when the first monthly records are not available. It is interesting to notice that the accuracy of prediction using cumulative records is only slightly less than using a linear function of the same monthly records. The corresponding intra-herd regression factors for cumulative months are given in Tables 5 and 6 of VanVleek and Henderson (17). The intra-herd and total regression coefficients are very similar for the cumulative records, especially when the last months are missing and the first four or more monthly test records are included in the cumulative sum. Bimonthly or trimonthly testing is a possibility for reducing the cost of testing for herd owners. The high correlation between bimonthly and monthly testing reported by several researchers, (1, 2, 7, 11, and 12), advances this possibility. The within-herd prediction equations based on bimonthly and trimonthly testing presented in Tables 7 and 8 of VanVleck and Henderson (17) further suggest the practicality of such testing procedures. Total regression coefficients for linear rune-
1524
L.D.
VAN VLECK AND C. R. HENDERSON
TABLE 5 R e g r e s s i o n f a c t o r s for e s t i m a t i n g t o t a l l a c t a t i o n y i e l d f r o m c u m u l a t i v e t e s t - d a y records C u m u l a t i v e m o n t h of l a c t a t i o n
b R
1
2
3
4
5
6
7
8
9
5.52 0.67
3.26 0.82
2.28 0.87
1.79 0.91
1.5.2 0.93
1.34 0.95
1.21 0.97
1.12 0.98
1.05 0,.99
TABLE 6 R e g r e s s i o n f a c t o r s f o r e s t i m a t i n g t o t a l l a c t a t i o n y i e l d f r o m c u m u l a t i v e t e s t - d a y re c ords when f i r s t - m o n t h records a r e m i s s i n g C u m u l a t i v e m o n t h of l a c t a t i o n
b R
10
9
8
7
6
5
4
3
2
4.83 0.52
2.91 0.63
2.40 0.74
2.11 0.83
1.83 0.89
1.59 0.93
1.40 0.96
1.22 0.98
1.08 1.00
TABLE 7 Regression factors for estimating total lactation yield from sequential bimonthly and t r i m o n t h l y t e s t - d a y records M o n t h of l a c t a t i o n
Monthly set
1
2
3
4
5
6
7
8
9
1 2
1.32 ......
...... 1.98
2.17 ......
..... 2.03
1.94 ......
...... 2.15
2.00 .....
...... 2.0'5
2.36 ......
1
1.58
......
4.15
2
......
3
........
........... 2.57
3.17
............
3.20
e:87
...........
............
3.34
10
R
..... 1.32
0.99 0.99
.................. 3.51 ............ .......... 2.59 ......
0.96 0.98 0.98
TABLE 8 Regression factors for estimating total lactation yield from cumulative bimonthly and t r i m o n t h l y t e s t - d a y records B i m o n t h l y set
b R
T r i m o n t h l y set
1
2
1
2
3
1.96 0.99
1.95 0.99
2.94 0.95
2.92 0.97
3.09 0.98
TABLE 9 R e l a t i v e efficiency of t o t a l a n d i n t r a - h e r d r e g r e s s i o n for p r e d i c t i n g t o t a l l a c t a t i o n y i e l d f r o m s i n g l e m o n t h l y t e s t - d a y m i l k re c ords Month
V1a V~b /~
1
2
3
4
5
6
7
8
9
10
4,585 3,774 121
2,828 2,4.25 117
2,252 1,916 118
1,802 1,599 113
1,736 1,56.0 111
1,82'8 1,612 113
2,14.5 1,775 121
2,998 2,209 13.6
4,78¢ 3,176 15.1
6,446 4,0'82 158
a V1 is the r e s i d u a l v a r i a n c e of t h e sum of t h e first t e n t e s t re c ords ( t o t a l y i e l d ) n o t a c c o u n t e d f o r b y t o t a l r e g r e s s i o n . The r e g r e s s i o n coefficients a r e shown i n T a b l e 2. b V~ is the r e s i d u a l v a r i a n c e of t o t a l y i e l d n o t a c c o u n t e d for b y i n t r a - h e r d r e g r e s s i o n . The i n t r a - h e r d r e g r e s s i o n coefficients a r e shown i n T a b l e 2 of V a n V l e c k a n d H e n d e r s o n ( 1 7 ) .
C E i s the ratio 100 ( ~ ) , w h i e h
is the relative efficiency of the two procedures.
EXTENDING PART
LACTATION RECORDS
1525
tions of bimonthly and trimonthly test-day records are shown in Table 7 and coefficients for cumulative bimonthly and trimonthly test records in Table 8. The total regression coefficients do not appear different from the intraherd coefficients. Some are almost identical. Again, it is apparent that the linear functions of bimonthly or trimonthly test-day records are only slightly better predictors of total yield than the corresponding cumulative sums of bimonthly and trimonthly test records.
Comparison of the e~ciency of the total regression and intra-herd regression equations. As stated previously, the residual variances not accounted for by regression can be used to compare the two procedures which are used to predict total yield. Procedure 1 is prediction based on regression ignoring herd effects and Procedure 2 is prediction based on within-herd regression. The relative efficiencies of the two procedures when using single test-day records are given in Table 9, together with the residual variances. The relative efficiencies are smallest for the fourth, fifth, and sixth m o n t h s - - t h e same months which provide the best estimates of total yield. The difference in variances is about 200 for these residuals, yet the differences in the average errors (the square roots of the residual variances) are only about 2 or 3 lb. of milk. For example, the square roots of the residual variances for the fourth month are: (V1)t = (1802)t = 42.5 and (V2) t = (1599) i = 40.0. This, then, shows an average error of measurement of 2.5 lb. more for Procedure 1 than Procedure 2. The average total yield from Table 1 is 407.9 lb. The percentage of additional error is thus less than 1% relative to total yield. It would appear that if prediction were based on the fourth, fifth, or sixth monthly test day record, the total regression equation could be used without compromising accuracy very much and with more computational ease. On the contrary, if the least desirable monthly test (the tenth) were used then the decision would probably be to use Procedure 2. Since the cumulative test-day records are nearly as accurate as linear functions of the same test-day records for predicting total yield, the relative efficiencies of the sequential functions will be discussed only briefly, but are shown in Table 10. For the conventional situation when the last monthly records are lacking the regression procedure ignoring herd effects can apparently be used rather than the intra-herd procedure, because the difference in accuracy is relatively small. On the other hand, when the first months are missing the differences in accuracy of the two procedures are quite large unless four or five monthly tests are available. The relative effieieneies of Procedures 1 and 2 when prediction is based on cumulative monthly test-day records are given in Table 11. These effieiencies follow the same pattern as those shown in Table 10. The effieieneies of the cumulative procedures when the last months are missing are similar, but when the first months are missing Procedure 2 is much more efficient. As was seen in Table 5, the average correlation between the yield predicted from a 5-mo. cumulative record and total yield is O.93. The increase in average error of prediction over within-herd prediction is (1176) t - (1045) '~ = 2.0 lb. Therefore, it seems that prediction of total yield ignoring herd effects based on 5-mo. records is a possibility which could be considered for earlier proving of young sires. Table 12 presents the relative efficieneies between Procedures 1 and 2 for linear functions of bimonthly and trimonthly test records. It is readily apparent that
1526
L.D.
VAN YLECK AND C. R. HENDERSON T A B L E 10
Relative efficiency of total a n d i n t r a - h e r d r e g r e s s i o n f o r p r e d i c t i n g t o t a l l a c t a t i o n yield f r o m linear f u n c t i o n s o f s e q u e n t i a l m o n t h l y t e s t - d a y milk records '~ S e q u e n t i a l m o n t h s of l a c t a t i o n - - l a s t m o n t h s m i s s i n g
V1 172 E
1
9
3
4
5
6
7
8
9
4,585 3,774 121
2,638 2,293 115
1,914 1,664 115
1,378 1,226 112
93'0 839 111
654 598 109
~¢1 4,10' 108
190 182 104
51 50 10'2
Sequential m o n t h s of l a c t a t i o n - - f i r s t m o n t h s m i s s i n g
171 17~ E
10
9
8
7
6
5
4
3
2
6,4~6 4,082 158
~,784 3,164 151
2,965 2,130 139
1,789 1,384 129
1,159 9~9 122
727 622 117
348 306 113
166 149 112
65 60 108
" T h e linear f u n c t i o n s c o r r e s p o n d i n g to V~ a r e g i v e n in T a b l e s 3 a n d 4, a n d t h o s e corres p o n d i n g to V., a r e s h o w n in T a b l e s 3 a n d 4 of Y a n V l e c k a n d H e n d e r s o n ( 1 7 ) .
T A B L E II Relative efficiency of t o t a l a n d i n t r a - h e r d r e g r e s s i o n f o r p r e d i c t i n g t o t a l l a c t a t i o n yield f r o m c u m u l a t i v e m o n t h l y t e s t - d a y milk records " C u m u l a t i v e m o n t h s of l a c t a t i o n
V1 172 E
1
2
3
4
5
4,585 3,774 121
2,82.9 2,462 115
2,080 1,820 114
1,594 1,407 113
1,176 1,045 113
last months missing 6 854 763 112
7
8
9
594 534 111
325 295 110
105 96 109
C u m u l a t i v e m o n t h s of l a c t a t i o n - - f i r s t m o n t h s m i s s i n g
171 172 E
10
9
8
7
6
5
4
3
2
6,446 ~,082 158
5,292 3,396 156
3,995 2,644 151
2,794 1,927 145
1,871 1,354 138
1,191 906 131
645 512 126
281 2'32 121
86 75 115
" T h e r e g r e s s i o n coefficients c o r r e s p o n d i n g to V1 a r e g i v e n in T a b l e s 5 a n d 6 a n d those c o r r e s p o n d i n g to 172 a r e shown in Tables 5 a n d 6 of V a n V l e c k a n d H e n d e r s o n (17).
T A B L E 12 Relative efficiency of total a n d i n t r a - h e r d r e g r e s s i o n f o r p r e d i c t i n g t o t a l l a c t a t i o n yield f r o m b i m o n t h l y a n d t r i m o n t h l y t e s t - d a y milk records ~ Sequential function B i m o n t h l y set 1 2 V1 V~ E
166 163 102
171 16,3 105
1
Cumulative
T r i m o n t h l y set 2 3
678 644 105
4,26 414 103
398 370 108
B i m o n t h l y set ] 2 214 207 104
213 194 110
1
T r i m o n t h l y set 2 3
904 851 106
457 444 103
421 384 110
T h e r e g r e s s i o n coefficients a s s o c i a t e d w i t h V1 a r e given in T a b l e s 7 a n d 8 a n d those with V2 are given in T a b l e s 7 a n d 8 of V a n V l e c k a n d H e n d e r s o n (17).
EXTENDING PART LACTATION RECORDS
1527
there is little difference between the errors of prediction for the two procedures. Prediction of total yield ignoring herd effects from cumulative bimonthly tests appears to merit consideration as a replacement for testing every month. Testing costs for the herd owner could be approximately halved without any appreciable loss of accuracy in estimating total yield. This conclusion has previously been advanced by Gifford (7), McDowell (11), McKellip and Seath (12), and Alexander and Yapp (1). Bayley et al. (2) reported that bimonthly tests are about 96% as accurate as monthly tests, but cautioned "the frequency of the large errors indicates that bimonthly and quarterly records should be satisfactory for sire provings and population studies, but they may be unsatisfactory when used as individual lactation records." CONCLUSIONS
The slight additional inaccuracy of predicting total milk yield from regression ignoring herd effects as compared with intra-herd regression does not appear as important as the extra computational difficulties encountered with intra-herd prediction for most practical situations. The practical situations considered include predicting total yield from five or more cumulative monthly test-day records, predicting total yield from cumulative bimonthly test-day records, and predicting total yield from a single test-day record obtained in the fourth, fifth, or sixth month of lactation. The high correlations which exist between the total yield and cumulative monthly records 5 or more mo. in length and between total yield and cumulative bimonthly records strongly indicate the possibility of sire proving based on short records or on bimonthly records. The former could reduce the costs of proving young sires, increase genetic progress through a shorter generation interval or additional sampling, or provide a combination of these results, while the latter could reduce by about half the testing cost to herd owners. REFERENCES (1) AL~XANI)F31%M. It., AND YAPP, W. W. Comparison of Methods of Estimating !Ylilk and F a t Production in Dairy Cows. J. Dairy Sci., 32: 621. 19~9. (2) BAYL~Y, N. D., LIss, R. M., AN]) S~ALL~D, J . E. A Comparison of Bimonthly and Quarterly Testing with Monthly Testing for E s t i m a t i n g Dairy Catt]e Production. J. Dairy Sci., 35: 350. 1952. (3) CA~N0~r, C. Y., ~ V E , J. B., AND SI~S, 5. A. Predicting 305-Day Yields from Short Time Records. J. Dairy Sei., 25: 991. 1942. (4) DASSA~T, P. ~ possible anticiparee rendere pid accurate il progeny test nel bestiame da latte? Atti IV. Riunione Biometric Society e I Riunione Assoclazone Genetica Italiana, Roma, 27-28 Matzo. 1954. (5) FRITZ, G. R., ~]~C~ILLIARD, L. D., AND MADDEN, D. E.
E n v i r o n m e n t a l I n f l u e n c e on
Regression Factors for Estimating 305-Day Production from P a r t Lactation. J. Dairy Sci., ¢3~: 1108. 1960. (6) GAINES, W. L. Deferred Short-Time Test as a ~VIeasure of the Performance of Dairy Cows. J. Agr. Research, 35: 237. 1927. (7) GIPFORa), W. The Reliability of Bimonthly Tests. J. Dairy Sei., 13: 81. 1930. (8) JSsT, 1K. Die Beurteilung verschledener Ffirsenkurzleistungsabschnitte im IIinbliek auf eine vorl~iufige Erbwertsch~itzung bei Bulien. Ziichtungskunde, 31: 201. 1959.
1528
L. D. VAN VLECK AND C. R. HENDERSON
(9) KITSCH, J., .~-~D H0~F~I~, J. Untersuchungen fiber die MSglichkeit eincr friihzeitigen Erbwertermittlung auf Grund der F~irsenkurzleistung yon 200 und 305 Tagen. Z. Tierziicht. Zfichtungsbiol., 64: 153. 1954. (10) M ~ D ~ , D. E., MC~ILLIAP~D, L. D., AND RALSTON, 1~. P. Relations Between Test-Day Milk Production of Holstein Dairy Cows. J. Dairy Sei., 42: 319. 1959. (11) McDow~ur., J. C. Testing Cows for Production Every Other Month. Holstein-Friesian World, 24: 1970. 1927. (12) MoKEhL~, I., AND S F ~ H , D. A Comparison of the Different Methods of Calculating Yearly Milk and B u t t e r f a t Records. J. Dairy Sci., 24: 181. 19@1. (13) O'CONNOR, L. K., A~-D ST$WAItT, A. The Use of 180-Day Records in Contemporary Comparisons. Rep. Prod. Div. Milk Mktg. Bd., 8: 93. 1958. (1~) R~rDE~, J. M., ROBEKTSOIT, A., ASKE~, A. A., KHISHIN, 8. S., AND RKGAB, M. W. The Inheritance of Milk Production Characteristics. J. Agr. Sci., 48: ~26. 1957. (15) T u ~ , P. Some Factors Influencing Milk Yield. Rept. Proc., W o r l d ' s Dairy Congr., 9: 151. 1931. (16) VA~rVL~oK, L. D., ~ H E a D , SON, C.. R. Ratio Factors for A d j u s t i n g Monthly TestDay Data f o r Age a n d Season of Calving a n d Ratio Factors for Extending P a r t Lactation Records. J. Dairy Sci., 44.: 1093. 1961. (17) V ~ V I ~ C K , L. D., AND H~NDE~SON, C. R. Regression Factors for Extending P a r t Lactation Milk Records. J. Dairy Sci., 44: 1085. 1961.