‘ saw:- 4 I in .z‘iiwnc iii-LL»; ‘ f—""c~m'- IHIHHlIllHIIIMWU|l||||ltl|lU|Hl||H|llIhHISHHHI 3 1293 01399 46 THESIS L LIBRARY Michigan $tate Unlverslty This is to certify that the dissertation entitled EVALUATION OF METHODS FOR ESTIMATING 305-DAY LACTATION YIELD IN DAIRY CATTLE presented by PETER MALACHI SAAMA has been accepted towards fulfillment of the requirements for Ph.D. degree in Animal Science W Major professor 0mm 61/??? MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE ll RETURN BOXtomnavothh chockomfrom your mood. TOAVOIDFINESMumonorbdmddoduo. ' DATE DUE DATE DUE DATE DUE MSU loAnAfflnndivoAction/Equd Oppommlylmwon mm: EVALUATION OF METHODS FOR ESTIMATING 305-DAY IACTATION YIELD IN DAIRY CATTLE BY PETER MALACHI SAAMA A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Animal Science 1995 ABSTRACT EVALUATION OF METHODS FOR ESTIMATING 305-DAY IACTATION YIELD IN DAIRY CATTLE BY PETER MAIACHI SAAMA Sampling variations in mean of daily milk yield within parity groups were approximated using bootstrap resampling. The variance of mean milk yield was widest at the peak to mid lactation. The optimal sample size for bootstrap resampling was found to be 80% of the original sample. Differences in the mean and variance of daily milk yield at morning and evening during a lactation in different parity and season ofcalving groups were investigated. Across regions, mean morning yield was higher than evening yield. The variance in daily total yield was mostly determined by variance in daily evening yield. Second lactation cows had the largest variance in daily morning and evening yield. The greatest variance occurred at peak lactation during December to February. Mathematical functions for various shapes of mean and variance of mean yield curves were established. These functions were representative of all breeds, regions, parities, ages at calving, and seasons of calving. A data set of sample lactation curves (SLAC) was generated from these functions. Fixed effects of missing test-day, and starting days of recording were included in SIAC. The SIAC was used to compare the relative accuracy ofsix methods for estimating 305d milk yield. When no yields were missing, the overall prediction bias for all methods was generally very small. Within shape oflaetation curve, variability in the accuracy of the methods was evident. With missing test-day records and varying starting days of recording, some methods had smaller bias than others. The establishment ofSLAC was commissioned by the International Committee ofAnimal Recording to examine the accuracy of current as well as future methods of estimating 305d milk yield. DEDICATION To the loving memory ofSarah Kemoli ACKNOWLEDGMENTS The author expresses sincere thanks to Dr. I. L. Mao for his companionship, guidance, and advice during the course of this study. Dr. R. LePage is cited for his resourcefulness and patience in understanding the body of this work. Dr. Gill is cited for his counsel and friendship. Utmost gratitude is extended to Dr. Ferris and Dr. Banks for their advice and guidance. The love and support of my parents, Elizabeth Okwenje, William Kelly, Judith Saurman, Sylvia Knight, Robert Wright, Alfred Stephanik, Terri Swezey, and Lee Coryell is fondly remembered. The friendship of Dr. M. C. Dong, Dr. J. Jensen, Dr. G. Jeon, Dr. G. Maria-Levrino, Ms. T. Moore, Dr. F. Ngwerume, Dr. D. Krogmeier, Ms. G. Ferreira, Mr. F. Grignola, Ms. S. Chung, Dr. Zhiwu Zhang, Mr. M. Sillanpaa, and Mr. C. Chang is greatly appreciated. Dr. H. E. Koenig, Dr. B. E. Koenig, and Dr. J. Anderson are cited for their support and collaboration in a Systems Sciences project. The author is grateful to Rick Halbert, Dr. M. Delorenzo, and Dr. L Schaeffer for their assistance in providing some of the research data. The author acknowledges Riccardo Alleandri, Todd Meinert, Anne Kjeldsen, Hans Wilmink, Nicole Bouloc, and Felipe Ruiz for their assistance in the computation of lactation totals. This research was supported by a grant from The International Committee of Animal Recording. Computing resources were provided by the Center for Genetic Improvement of Livestock ,University of Guelph, Canada. iv TABLE CONTENTS List ofTabIes ................................................................................................................................ vii List of Figures ............................................................................................................................... viii 1. Introduction ............................................................................................................................. 1 2. Objectives ................................................................................................................................ 4 3. Review of literature .................................................................................................................. 5 3.1 Sources of variation in daily milk yield ............................................................................ 5 3.1.1 Variation on testday ............................................................................................... 5 3.1.2 Variation between testdays .................................................................................... 8 3.1.3 Variation between lactations .................................................................................. 9 3.1.1 Variation between cows .......................................................................................... 10 3.2 Sampling frequency during lactation ............................................................................... 11 3.3 Fitting lactation curves ..................................................................................................... I I 3.4 Methods for computing lactation totals ........................................................................... 14 3.4.1 Test interval method (MSU) .................................................................................. 14 3.4.2 Test interval method (France) ................................................................................ 15 3.4.3 Test interval method with adjustment factors (USA) ............................................. 16 3.4.4 Linear interpolation with standard curves (Netherlands) ...................................... 17 3.4.5 Centering date method (Denmark) ........................................................................ 19 3.4.6 Multiple trait projection method (Italy) .................................................................. 20 3.5 Bootstrap resampling ....................................................................................................... 21 3.6 Smoothing of continuous functions ................................................................................ 23 3.7 Principal factor analysis .................................................................................................... 25 3.7.1 Orthogonal varimax factor rotation ........................................................................ 26 3.7.2 Oblique procrustean factor rotation ....................................................................... 26 3.7.3 Oblique Harris-Kaiser factor rotation ..................................................................... 27 3.7.3.1 Harris-Kaiser rotation with Cureton-Mulaik weights .................................... 28 4. Bootstrap assessment of the sampling variations in mean daily milk yield .............................. 30 4.1 Abstract ........................................................................................................................... 30 4.2 Introduction .................................................................................................................... 30 4.3 Materials and methods .................................................................................................... 31 4.4 Results and discussion ..................................................................................................... 33 4.5 Conclusions ..................................................................................................................... 36 5. Variation in morning and evening yield during lactation of Holstein cows ............................. 38 5.1 Abstract ........................................................................................................................... 38 5.2 Introduction .................................................................................................................... 38 5.3 Materials and methods .................................................................................................... 39 5.4 Results and discussion ..................................................................................................... 39 5.4.1 Overall variance ofdaily milk yield ......................................................................... 42 5.4.2 Within parity variance ofdaily milk yield ............................................................... 43 5.4.3 Within season variance ofdaily milk yield ............................................................. 43 5.4.3.1 Alberta .......................................................................................................... 43 V 5.4.3.2 Ontario ......................................................................................................... 43 5.4.3.3 Florida ........................................................................................................... 44 5.5 Conclusions ..................................................................................................................... 44 6. Mathematical description of lactation curves in dairy cattle ..................................................... 52 6.1 Abstract ........................................................................................................................... 52 6.2 Introduction ..................................................................................................................... 52 6.3 Materials and methods .................................................................................................... 54 6.4 Results and discussion ..................................................................................................... 57 6.5 Conclusions ..................................................................................................................... 60 7. Description of the standard lactation curves project ................................................................ 62 7.1 Summary ......................................................................................................................... 62 7.2 Design of lactation curves for daily total yield .................................................................. 63 7.3 Design of lactations for AM and PM yield ...................................................................... 66 7.4 Designation of trait/breed/ parity/ calving season categories ............................................ 67 7.5 Data files of standard lactation curves (SIAC) ................................................................. 69 7.6 Data files of calculated lactation totals to returned to MSU ............................................. 70 7.7 Participating data records processing center .................................................................... 71 7.8 Data files for computed lactation totals returned to MSU ............................................... 71 8. Comparisons of methods of predicting 305-d milk yield ......................................................... 72 8.1 Abstract ........................................................................................................................... 72 8.2 Introduction .................................................................................................................... 72 8.3 Materials and methods .................................................................................................... 74 8.3.1 Design ofSIAC ..................................................................................................... 74 8.3.2 The methods for calculating 305-d lactation yield .................................................. 75 8.3.3 Statistical analyses ................................................................................................... 76 8.4 Results and discussion ..................................................................................................... 78 8.4.1 Biases for recording schemes for daily yield ........................................................... 78 8.4.1.1 All test-d records available ............................................................................. 81 8.4.1.2 Not all test-d records available ....................................................................... 83 8.4.1.3 Starting day of recording ............................................................................... 86 8.4.2 Biases for the AP/4 scheme for morning and evening yields ................................. 89 8.4.2.1 All test-d records available ............................................................................. 89 8.4.2.2 Not all testd records available ....................................................................... 91 8.4.2.3 Starting day of recording ............................................................................... 93 8.4.3 Validation of sampling design ................................................................................ 93 8.5 Conclusions ..................................................................................................................... 93 9. Summary ............................................................................................................................... 95 10. Appendices ............................................................................................................................ 96 1 1. References .............................................................................................................................. 114 vi Table 1. Table 2. Table 3. Table 4. Table 5. Table 6. Table 7. Table 8. Table 9. Table 10. Table 11. LIST OF TABLES Average and standard deviations of daily milk yield ................................................. 7 Test-day data for Holstein cows utilized for analyses by location ............................... 4O Product-moment correlations between mean morning, evening, and total milk yields ......................................................................................................................... 42 Number of records in a complete lactation by scheme and starting day for milk recording ................................................................................................................... 64 Weeks in which data were missing by scheme and starting day of milk recording 65 Number of curves by scheme, starting day, and missing pattern .............................. 66 Number of curves for AP/4 by starting day and missing pattern .............................. 67 Combinations of fixed effects for calculating lactation totals ..................................... 69 Mean, standard deviation, standard error, and coefficient of variation of bias by method of estimating lactation totals and schemes A1 and A4 ................................. 76 Mean, standard deviation, standard error, and coefficient of variation of bias by method of estimating lactation totals and schemes A6 and A8 ................................. 80 Mean, standard deviation, standard error, and coefficient of variation of bias by method of estimating lactation totals and scheme AP/ 4 ........................................... 89 vii Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9. Figure 10. Figure 11. Figure 12. Figure 13. Figure 14. Figure 15. LIST OF FIGURES Mean milk yield of 305-day milk yield for first and second lactation cows (A). Effect of seasonality on curve of lactation and total milk yield (B). ........................... 6 Bootstrap confidence intervals for daily milk yield. Data were for first lactation cows following sampling a light proportion (p=.2) of the original data. The pseudo-population mean is also shown on each plot ................................................ 34 Bootstrap confidence intervals for daily milk yield. Data were for first lactation cows following sampling a light proportion (p=.8) of the original data. The pseudo-population mean is also shown on each plot ................................................ 35 A sample lactation curve simulated by random sampling from 99% bootstrap confidence intervals ................................................................................................... 36 Number of cows with lactation records by location: A = total milk yield, B = morning milk yield, and C = evening milk yield ................................................ 41 Estimated 95% confidence interval of mean daily milk yield for Alberta, Ontario, and Florida: A = total yield, B = morning yield, and C = evening yield ..... 45 Estimated 95% confidence interval of mean daily milk yield within parity for Alberta, Canada: A = total yield, B = morning yield, and C = evening yield ............. 46 Estimated 95% confidence interval of mean daily milk yield within parity for Ontario, Canada: A— “ total yield, B= morning yield, and C — evening yield ............ 47 Estimated 95% confidence interval of mean daily milk yield within parity for Florida, USA: A = total yield, B = morning yield, and C = evening yield ................. 48 Estimated 95% confidence interval of mean daily milk yield within calving season for Alberta, Canada: A-— " total yield, I3: morning yield, and C * evening yield ........................................................................................................................... 49 Estimated 95% confidence interval of mean daily milk yield within calving season for Ontario, Canada: A = total yield, B = morning yield, and C = evening yield ........................................................................................................................... 50 Estimated 95% confidence interval of mean daily milk yield within calving season for Florida, USA: A = total yield, B = moming yield, and C = evening yield ........................................................................................................................... 51 Lactation curves for daily yield ................................................................................... 55 Lactation curves for morning and evening yield ........................................................ 56 Variance curves for mean daily yield ......................................................................... 57 viii Figure 16. Figure 17. Figure 18. Figure 19. Figure 20. Figure 21. Figure 22. Figure 23 Figure 24. Figure 25. Mean :t standard errors for lactation yield (A), 5 sample lactation curves (B), and 1 sample lactation curve (C) ............................................................................... 61 Least 5 uares mean bias b ’ method, sha e oflactation curve and schemes C1 ) P Al , A4, A6, and A8 .................................................................................................. 82 Reference structure for weighted Harris-Kaiser rotation of factors for schemes A1, A4, and A6. Procrustean rotation of factors for scheme A8. .............................. 83 Least squares mean bias by method, pattern of missing test-d records and schemes A1, A4, A6, and A8 .................................................................................... 85 Reference structure for weighted Harris-Kaiser rotation of factors for scheme A1, A4, A6, and A8. ................................................................................................. 86 Least squares mean bias by method, starting day of recording and schemes A4, A6, and A8 ................................................................................................................ 87 Reference structure for weighted Harris-Kaiser rotation of factors for schemes A4, A6, and A8. ........................................................................................................ 88 . Least squares mean bias by scheme AP/4, method, and shape oflactation curve (A), pattern of missing test-d records (B), and starting day of recording (C). ................... 90 Reference structure for weighted Harris-Kaiser rotation of factors for scheme AP/4 and method for lactations with no missing data(A), lactations with missing data (B), and lactations with varying starting day of recording (C) ............... 91 Least squares mean bias by method, data set, and shape of mean curve for scheme Al ................................................................................................................. 92 1. INTRODUCTION Total yield in a lactation is commonly estimated from production on a few sample test days during a lactation. Milk production in a lactation has been measured as the production during the first 305 days following parturition. This standard length allows records to be compared without concern for the varying length of the production period. The term “lacta- tion curve” refers to the graphical representation of the relationship between milk yield and length of time since calving. A frequent application of the lactation curve is in the extension of records in progress to predict yield up to a standard length. Lactation curves also can be used to monitor the cow during a lactation for health and other managerial purposes. Numerous methods for estimating lactation totals from test-day yields have been devel- oped. Dairy records processing centers in different regions and countries use different methods. Different methods give results of a varying degree of accuracy. For a given method, accuracy of estimation may vary due to 1) length of test interval, 2) shape of lactation curve, 3) Missing test-d yields, and 4) starting day of recording. Furthermore, programs for genetic improvement of dairy cattle have developed rapidly over the last 40 years. This progress has been due to the availability of artificial insemination, biotechnological advances such as embryo transfer, better information processing technolo- gies, and more accurate methods for assessing the genetic merit of individual animals. Conse- quently, there is an ever increasing exchange of germplasm and information between coun- tries. Meaningful international genetic evaluations require accurate estimates of lactation totals. However, differences in methods for calculating lactation totals present an obstacle to intema- tional genetic evaluations. The International Committee ofAnimal Recording (ICAR) was established 34 years ago. It is an organization which is concerned with the coordination of livestock recording world- wide. Presently, ICAR has adopted only two methods of lactation yield calculation as “official ICAR methods”, namely: the centering date method (CDM) and the test interval method (TIM). Some countries have adopted CDM and TIM. Other countries, such as USA and Italy, use different methods. In 1990, the ICAR board commissioned a working group on “lactation calculation methods” to set up guidelines and standards for lactation calculation methods and related matters. The group decided to examine the efficiency of current methods and from this information methOt s would be evaluated for possible use as official methods by ICAR. This study was undertaken to examine the relative accuracy of methods for calculating lactation totals. The data from this study would then be the fundamental basis for the recom- mendations made by the “lactation calculation methods” working group of ICAR. In §2 the objectives of this project are outlined. In §3, a review of concepts important to the understanding of the body of this work are discussed. Namely, sources of variation in milk yield, sampling frequency during lactation, algebraic models for the lactation curve, methods for computing lactation totals, bootstrap resampling, data smoothing, principal factor analysis, and factor rotations. The body of the research is summarized in §4, §5, §6, §7, and §8. A necessary aspect of this project was the generation of a data set of standard lactation curves (SIAC). Different methods for estimating lactation totals would then be applied to SIAC to compare their relative accuracy. In order to create this data set, we needed to know the shapes of mean of yield from morning milking and evening milking and daily total yield and the variance associated with each of these measures, during lactation.We had to under- stand how the mean and variance of daily morning and evening yield affect the mean and variance of total daily yield. From a series of preliminary analyses and after incorporating information from the literature and disucssions with the working group of ICAR, a set of curves for mean and variance of daily yield was created. Mathematical functions for these curves were determined. From these functions, a Monte-Carlo method was used to generate the SIAC. The total yield for lactations in SIAC was estimated using various methods. The methods were thereafter compared. In §4, the bootstrap resampling method is used to approximate sampling variations in mean daily yield. In §5, data sets for three regions in North America were used to investigate daily variation in total, morning, and evening yield. From §4 and §5, we had preliminary understanding of typical shapes of curves for mean and variance of daily total, morning, and evening yields. In $6, a set of curves for mean and variance of daily, morning, and evening yields was formally presented. Mathematical functions for these curves were stipulated. A Monte-Carlo method for generating sample lactations were discussed. In §7, a complete description of how the SIAC was created and utilized was given. In §8 methods for estimating lactation totals were outlined and compared. Concepts presented in the earlier chapters were summarized and the results of the project were discussed. sperm] ll It) .I TO etaniu I“ SCI‘A accurat 2. OBJECTIVES The overall goal of this study was to compare methods of calculating total lactation yield. The specific aims were: 1) 2) 3) 4) 5) 6) to examine sampling variations in mean daily milk yield; to examine differences in mean and variation in morning and evening yields; to develop mathematical functions for the mean daily yield in a lactation; to develop mathematical functions for the variance of mean daily yield in a lactation; to develop a data set of standard lactation curves; to establish the relative accuracy of methods for computing lactation yields. The results from aim l and 2 were used to establish the shape of some of the curves examined in aims 3 and 4. The mathematical functions developed in aims 3 and 4 were used to generate the standard lactation curves data set. Six methods were evaluated for their accuracy in estimating the total yield for each of the standard lactation curves. 3. REVIEW OF LITERATURE 3.1 Sources of variation in daily milk yield While different types of management have a marked influence on variations in yields, it is not so easily understood why this variation should exist for those herds following all the currently recommended practices. Age at calving and season of calving are main factors affecting milk production in dairy cattle (Everett and Wadell, 1970; Schultz, 1974). Milk yield increases with age at a decreasing rate and reaches a maximum at maturity (Auran, 1973; Mao et al., 1974) as is shown in Figure IA. The effect of seasonal variation was anlalysed into its two components, seasonality of production (‘spring hump seasonality’) and calving month seasonality (Wood, 1969) as is depicted in Figure 1 B. Daily yield was depressed during the winter months and stimulated during the spring to an extent which was independent of stage of lactation. Cows calving in winter months tended to produce more in total lactation than spring calvers (Wood, 1969). The standard deviation of daily production among cows varied from 6 to 4.5 kg through the lactation, the magnitude being closely related to the mean (Table l , Anderson et al., 1989). 3.1.1 Variation on testday Much of the variation in test-day milk yield has been attributed to the interval between milkings (Everett and Wadell, 1970; Putnam and Gilmore, 1970; Shook et al., 1980), com- pleteness of milking (Dodd and Foot, 1948), dry matter intake (Polan et al., 1986), estrus (Humik, et a1. 1975; Palmer, 1982), and water intake (Murphy, 1992). Age effects which are frequently confounded with production group effects can affect test-day variation. Stanton et a1. (1992) used a test-day model to study the effects of age on test- day production and concluded that age at calving would account for more of the variation in test-day production that age on test-day. The reproductive status of a cow may contribute to test-day variation. Modest (1 lumik et al., 1975) to significant (Palmer, 1982) increases in milk yield have associated with the onset of estrus. A I 7 Parity: I L+ First lactation + Second lactation f 5700 E *N/t—‘t’I 5500 E why? I 5100 E I ,-.—H“"‘ ’ +111 I Milk yield (kg) 5 \l 8 LJJALLLLA 1'1141 I lLlLll 1 l . i i .- i 18 20 22 2 26 29 31 33 35 37 39 41 43 45 47 Age in months B I Seasonality of: I :Spring -+-Calvingl. 20 I T i 15 g p ‘A‘ g 10 E ! k-fl _ + I “5 5 F ‘4 7L * g “f“ + \ I o — — - . ——— — a —+ LL___L__,-_ — \s “a 'O + ’ _ g -5 F 1,, + + ‘\ I ‘10 E . I \“’ P -15£/,' ; I 1 I I I I ___l - Jan. Feb. March April May June July Aug. Sep. Oct. Nov. Dec. Month of lactation Figure 1. Mean milk yield of 305-day milk yield for first and second lactation in A (Source: Mao et al., 1974). Effect of seasonality on curve of lactation and total milk yield in B (Source: Wood, 1969). TABLE I. Averages and standard deviations of daily milk yield Days in lactation Number of cows Average milk Standard Coefficient of yield deviation variation (kg) 10 255 24.5 6 0.24 15 255 28.7 5.4 0.19 20 255 29.7 5.1 0.17 30 255 30.4 5.1 0.17 40 255 30.5 5.5 0.18 50 255 30.7 5.5 0.18 60 255 30.4 5.2 0.17 70 255 29.8 5.1 0.17 80 255 29.5 5.4 0.18 90 255 28.4 5.1 0.18 100 255 27.5 5 0.18 120 255 26.7 5 0.19 140 255 25.6 4.7 0.18 160 255 24.6 4.6 0.19 180 255 23.5 4.8 0.2 200 255 22 4.6 0.21 220 254 21 4.5 0.21 240 25.3 19.8 4.5 0.22 260 248 17.9 4.7 0.26 280 232 16.5 4.7 0.28 300 193 15.1 4.5 0.3 Source: Anderson et al. (1989) Shook et a1 (1980) and Everett and Wadell (1970) concluded that differences between morning and evening milk yield were primarily a function of milking interval and the number of days in lactation. 3.1.2 Variation between test-days The relationship between milk yield and month of calving is caused in part by the seasonal variations in feeding and care. The effect of month of calving on persistency has been observed by Sanders (1923 and 1930), Gaines (1927a), Gooch (1935), Johansson and Hansson (1940), Woodward (1945), Sikka (1950), Mahadevan (1951 ), Pickard (1952), Appleman (1969), Wood (1972), and Schultz (1974). These studies indicated that those cows calving in fall and winter are more persistent than those calving in spring and summer. Johansson and Hansson (1940) found that 5% of the total variation in persistency was due season of calving while Sikka (1950) reported this figure to approach 9.57%. Interactions between month of calving and stage of lactation were observed (Dannell, 1981; Miller et al., 1967). This source of variability in day to day milk yield suggested that shape of the lactation curve is dependent on month of calving. However, Auran (1973) showed that month of calving was not as important as age at calving. Month of calving accounted for about 1.800 of the total variation in the first test-day and about 7.8% in the seventh and eight test-days. The influence of age at calving on monthly test-day yields decreased with advancing lactation, accounting for about 41% to 50% of total variation for first monthly test to about 2% to 5% for the last three days (Auran, 1973; Dannell, 1981). Thus, contrary to the age effects, the effect of month of calving is largest towards the end of lactation. In addition, the relationship between morning and evening yields influences the shape of the lactation curve. Daily morning to evening ratios increased during later stages of the lactation (Palmer et al., 1994). The effect of month of test-day on test-day production has been investigated (Syrstad, 1965; Everett and Wadell, 1970; Shook et al., 1980; Dannell, 1981; Ng-Kwai-Hang et al., 1984). Lindgren et al. (1980) concluded that a cow's production is less affected by month of testing immediately after calving than later in the lactation. Gavin (1912), Brody et al. (1923), Hammond et al. (1923), Gaines et al. (1926), Johansson and Hansson 1940), Turner (1943), Louca and Legates (1968), and Chazal and Chilliard (1986) reported that the effect of pregnancy on daily milk yield is not noticeable until five months into gestation. Smith and Legates (1962) showed that production seemed to decline more rapidly 16-20 weeks following conception. Corley (1956) summarizes, ".. results agree in that neither persistency nor total yield is appreciably influenced by pregnancy during the first five months of gestation. However, if cows conceive early or carry a calf over 200 days of any lactation, a slight decline in total yield and a definite drop in persistency will likely occur." The calving interval may impact upon variation in yield during lactation. Sanders (l 92 3), Gaines (1927a), Bonnier (1935), Johansson and Hansson (1940), Klein and Woodward (1945), Smith and Legates (1962) have shown that persistency increases with increased length of calving interval. The effects of days open on milk yields have been studied by Wilton et al. (1967), Smith and Legates (1962), Ripley et a1. (1970), Schaeffer and Henderson (1971), and Schultz (1974). Production losses due to long open periods have been reported by Louca and Legates (1968). The effect of bovine somatotropin (bST) administration on milk yield during the lactation of cows maintained in cold environmental conditions was studied (Becker et al., 1990). Under farm conditions, bST treated cows produced 11% more milk than control-treated cows and in environmentally controlled chambers produced 17.4% more milk. 3.1.3 Variation between lactations Ng-Kwai-Hang et al. (1984) indicated that milk production increased markedly between two and five years of age and then increased at a slower rate between five and six years of age. 10 Wood (1967) investigated the effect of parity on the constant a for average daily production. This constant on a log scale was 3.53, 3.72, 3.97, 3.86 for first, second, third, and forth or later parities respectively. Thus, first lactation cows were shown to be more persistent than later lactations. Results of Gowen (1920), Gaines (1927a), Turner (1927), Sanders (1923 and 1930), Gooch (1935), Dickerson et al. (1939), Johansson and Hansson (1940), Ludwick (1942), Sikka (1950), Ripley (1970), and Keown (1986) show similar findings. Sikka (1950) noted that differences in lactation number accounted for 17.7% of the variation in persistency and 31 .2% of the variation in maximum yield. Ludwick (1942) studied the records of 130 Guernsey, Holstein, and Jersey cows and reported an 810% drop in persistency from first to second lactation. Corley (1956) showed that cows in first lactation were 8% more persistent than cows in their second lactation. He found little difference however among subsequent lactations. Appleman (1969) showed a significant interaction between lactation number and season of calving as did Miller et al. (1970). They found that older cows were more severely affected by summer calving than their younger counterparts. 3.1.4 Variation between cows Cows that produce moderately with high persistency throughout lactation usually will be under less stress than cows that are less persistent have a large differential between produc- tion at peak and end of lactation. Age effects on lactation yield were demonstrated by Mao et al. (1974). Between cow differences have been attributed to herd (Auran, 1973; Everett and Wadell, 1970; Goodger et al., 1988; Mao et al., 1974), region (Mainland, 198 5), additive and nonadditive genetic variance (Grossman et al., 1986), disease (Bartlett, 1991; Simerl, 1992), temperature (Becker et al., 1990; Elvinger et al., 1992), stocking rate (Baker and Leaver, 1986). Ambient temperatures affect the performance of cows. Heat stress cows increased rectal temperatures, respiration rates, and decreased milk yield. (Elvinger et al.). Differences between morning and evening yield may contribute towards variation be- 11 tween cows. Daily morning and evening ratios were plotted for selected lactations (Palmer et al., 1994). All showed large daily variations and evidence of cow differences. 3.2 Sampling frequency during lactation Accepted intervals between recording yields of cows have gradually changed over the years. The 7-day interval was shown by Yapp (191 5) to be a poor indicator, while Gaines (1927b) later showed-that 7-day tests conducted after 60 days in lactation were more indicative as an estimate of lacmtion yields. M'Candlish and M’Vicar (1925) found that a 1-day test per month yielded results within 2% of actual yield, and Dick (1950) observed an average error of 2.32% from actual when cows were tested at 28d intervals. Houston (1932) found that weekly test intervals gave estimated yields approximating actual, and that to keep errors within a range of 10%, the testing interval should not exceed 30d. McDowell (1927) found that monthly and bimonthly tests varied from actual an average of 2.91 and 3.80 per cent, respectively. Clearly, there is no unanimous agreement on length of test intervals. However, a four week interval between tests is most common globally. Anderson et al. (1989) found that the four week equal interval sampling procedure gave acceptable estimates of total lactation milk yield. The crux of the matter remains that the total yield in a lactation has to be determined from periodic test- day yields. Both accuracy and precision in estimating total lactation yield increase with fre quency of sampling. 3.3 Fitting lactation curves Since the 1920’s there has been considerable interest in the mathematical description and analysis of the lactation curve in dairy cattle. When a functional form is used to describe a lactation curve then: 1) the milk yield at any given stage of lactation can be predicted. Such predictions, if accurate, can be used as a basis to cull or to retain for breeding stock; 2) an individual animal’s lactation and thus the average curves of groups of animals may be compared in terms of parameters of 12 the functional form; 3) it provides a mathematical description of average milk yield needed in any simulation model of a dairy enterprise; 4) concise summaries of patterns can be generated from 'which cumulative curves can be generated. The general approach has been to exploit parameters of the function in order to fit different lactation curves. Mathematical functions for the shape (Wood, 1967; Kumar and Bhat, 1979), peak time (Sikka, 1950; Cobby and Le Du, 1978; Molina and Boschini, 1979), and declining phase (Brody, Ragsdale and Turner, 192.3; Gaines, 1927a; ) of the lactation curve have been pro- posed. Wood’s equation, yn = a nbexp(—cn)' has been the most frequently applied. The variable n represents the length of time since calving. Coefficients a, b, c are constants determining the characteristics of curves. The equation can be estimated by ordinary least squares (01.3) in the form: loge (ya) = logc a + b log. n — cn . The curve reaches a turning point at n. = — (b / c)- demonstrating that b is the parameter for pre-peak curvature, and c the parameter of post-peak curvature. Congleton and Everett (1980a, 1980b) examined the prediction error of Wood’s equation and concluded that the function provided predictions of 305d cumulative milk that were comparable with the estimates obtained by other DHIA techniques. Ramirez et al. (1994) found this equation performed better than the functions proposed by Brody (1 92 3), Sikka (1950), Nelder (1966), and Colby and Le Du (1978). However, several authors (Cobby and Le Du, 1978; Danoa, 1981; Rowlands et al., 1982) have reported systematic lack of fit to lacta- tion milk yields recorded weekly using this model. Goodall and Sprevak (1984) retained Wood’s formulation and introduced an autocorrelation function to account for systematic lack of fit, logc8(t) zalogceh— l)+e(t) where e(t) is a random error term and or is a parameter such that Ial < 1. Values ofa greater 13 than zero were interpreted to be an improvement over Wood (1967). In order to account seasonal variation, Grossman et al. (1986) extended Wood’s equation by adding sine and cosine terms: yn : a nbexpfn (1 + usin(x) + v cos( x)) where a, b, c, u, and v are coefficients to be estimated, n 2 day of lactation; X:d'dy of year, computed as radians. The log transformation of this model can be fitted by OLS. Batra (1986) compared this extended function to the inverse polynomial (Nelder, 1966) and found that the latter function gave a better fit than the former, based on R2. Grossman and Koops (1988) proposed a multiphasic function to describe the lactation curve, based on the sum of logistic functions: Yr 2: (a.b,[1‘— tanliz(l~‘l(t _ C'))l) where Yt is milk yield at t days in milk, n is the number of lactation phases, tanh is the hyper- bolic tangent. Functions of parameters for each phase included initial yield computed as aibii contribution from each phase to 305d yield, computed by integrating each phase from t=0 to F305; time of peak yield for each phase, defined as c;; and duration of each phase as days required to attain 75% of total yield and computed as 2bi'1. DeBoer et al. (1989) fitted the multiphasic function to first through third-parity curves for milk yields for Israeli Holsteins and concluded that functions of parameters for each phase differed by parity, yield, and days open. Lactation curve estimates also have been obtained by solving for OLS estimates of the fixed effects of days in milk (DIM) on test-d yield for which the lactation curve is partitioned into numerous DIM classes (Ngwerume, 1994; Schaeffer and Dekkers, 1994; Schaeffer et. al., 1994; Stanton et al., 1992; Trus and Buttazoni, 1990). The primary advantages of the test- d model are that it can account for 1) information from different lactations; 2) permits esti- mates of fixed effects to vary across herds and stages of lactation, and 3) adjust for effects of sampling date. 14 3.4 Methods for computing lactation totals The traditional record of cow lactation yields is based on recording at weekly, monthly, or longer intervals. However, actual lactation yield can only be calculated by accumulating daily yields. Many procedures have been developed for computing total yield in a 305d period. The following is a review of the methods that were compared in this study. 3.4.1 Test interval method (MSU) The test interval method uses the time from one test-d until the following test-d as the test period (Appendix A of ICAR agreement). In general, a lactation record is calculated using three steps: 1. Estimate the sample-day yield for milk; 2. Estimate the yield from the previous sample day through the current sample day (credit for the test interval; and 3. Add the test interval credits for the lactation to determine the total lactation yield. For most intervals, the interval yield (or credit) is calculated by multiplying the average yield between sample days by the number of days in the interval. Average daily yield in an interval is estimated as the average of the yields for the preceding and current sample days. The credit for the first interval is calculated as yield on sample day x day oflactation. The test interval credit for an interval with a sample day after the first interval but prior to the last interval is calculated as (yield on preceding sample day + yield on sample day)/ 2 x days in interval. If a cow terminates her lactation before 305 d in milk, the credit for the last interval is calculated as 15 (yield on preceding sample day + yield on sample day)/ 2 x days in interval + (yield on sample day) x (days to 305 d). The first part of the formula gives credits for the interval immediately preceding the last one while the second part calculates credit for the days to 305 d. If the interval before 305 d is greater than 7 d, this situation is treated as resulting from an incomplete lactation and days to 305 d is set to zero. We can expect overestimation at the last interval and underestimation in the intervals spanning peak lactation. If the yield for the current sample day is missing but the yield for the immediately preced- ing and subsequent sample days was recorded, an estimate of the sample day yield can be calculated as, (yield on preceding sample day + yield on next sample day)/ 2. If the yield is missing for consecutive sample days, there is no estimate for the missing yield. For the moming/ evening yield schemes, yield on a sample day is estimated as (yield on sample day x 2) Calculation of test interval credits is then performed in a tnanner similar to the one for daily yield. The TIM method described above is illustrated in Appendix El 3.4.2 Test interval method (France) The test interval credits from the first to the last interval are similar to those performed at MSU (Letter from Nicole Bouloc, Institut de l’elevage, Paris, France, 11/ 8/ 94). However, credits for the last interval are calculated as (yield on preceding sample day + yield on sample day)/ 2 x days in interval + (yield on sample day) x c where c is 14 for schemes Al , A4, A6, and AP/4 or c is 28 for scheme A8. Lactation yield is 16 calculates as the sum of the test interval credits. There are no corrections for missing test-d yields. This method is illustrated in Appendix F.2. 3.4.3 Test interval method with adjustment factors (USA) In the USA, most common sampling plans require weighing the milk at all milkings and collecting a composite sample during the approximate 24-h period of the sample day. Varia- tions include AM-PM (AP) plans, for which only one milking is weighed and sampled each sample day for herds milked two times a day (2X) and only one or two milkings are weighed and one milking sampled each sample day for herds milked three times a day (3X). A com- plete description of this method for different sampling plans is given by Wiggans (1989) and adjustment factors. are shown therein. Interval credits are computed in order to obtain 305 d yield. The credit for the first test interval is calculated as: test interval creditfirst = factor x yield on sample day x days in milk. Factors are based on breed, region, season, trait, lactation number, and stage of lactation. The test interval credit for an interval with a preceding sample day before 40 d in milk is calculated with factors as: test interval creditpeak = factor x (yield on preceding sample day + yield on sample day)/ 2. The test interval credit for an interval with preceding sample day after 40 d is calculated as test interval creditpost-peak = factor x (yield on preceding sample day + yield on sample day)/2. The credit for the last interval is calculated as, test interval crediqast = factor x yield on last sample day x days to termination. The procedures for projecting lactation records of less than 305 d to 305 d is based on the number of days the cow actually milked, plus an estimate for the remainder of the 305-d lactation derived from the last available sample-day yield (Wiggans and Dickinson, 198 5). For records less than 1 55 d in milk, the average mature‘equivalent (ME) yield for cows freshening in the same herd 1 to 2 years prior to the records last sample day also is required. 17 Separate factors have been developed by trait, calving season, lactation number, region of the country, and breed (W iggans and Powell, 1980). Records less than 305 d can be projected by Tscs : YDIM + (TDX 305‘ DIM) where 730' = projected 305d yield, YDIM : yield for the partial record, and To = estimated average daily yield for the remainder of the lactation. For records with more than 155 d in milk, average daily yield for the remainder of the lacta- tion can be estimated as To = [as + Bs(DIM)] (Y5) + as + BF(DIM) where or = intercept, S 2 sample day, [3 = slope, YS = sample-day yield, and F 2 factor. For records with 155 d in milk or less, the ME herd average is included in estimating average daily yield: aH+BH(DIM)](YH) v.=[a.+s.(mw](v.)+[ 1000 where H = herd ME average and YH : herd-average yield. 3.4.4 Linear interpolation with standard curves (Netherlands) This method was developed by Wilmink and Ouweltjes (1991) with the following require- ments: 1 - The calculations should be independent of the recording scheme. The method should use all known test day yields as observations Cumulative yields should be estimated from these test day yields. 2 - Cumulative yield should be estimated by using corrections for the first part of lactation. The expected shape of the lactation curve is estimated from lactations of contemporaries and is used in the calculation of cumulative yields in order to improve accuracy. Standard 18 lactations were estimated for separate classes of herd production level, age at calving, and season of calving (W ilmink, 1987 and 1990). These standard curves are used in the prediction of 24-h yield by interpolation and the prediction of 24-h yield before the first test day or a future yield. Interpolation using standard lactations is performed using the equation: vi:g,+((y2-y,)-(g3-gl))x(xitxi)/(xz-X1)+(yl-g,) f [NRS'H where y; is an estimate for the uknown yield on the ith day of lactation for a cow that has a recorded yield V1 at day x1 and yield yz at day x2. Corresponding data froma pertinent standard lactation curve are yield g1 at day x1, yield g2 at day x2, and yield g; at day xi. Unknown test day yields are estimated by the following prediction equation: Vi : H, + b1 X (xp - up) + b; X (y305 - p305) [NRS-2] where y; — predicted yield at day i, it; expected yield at day i, up = expected yield at day p, [1305 2 expected yield over 305d in prior lactation, xp = realised yield at day p, y305 realised 305-d yield in prior lactation, b1, b2 = regression factors. All expected yields are taken from standard lactations. To allow for varying interval lengths, die cumulative yield is computed as follows: 1 - The 24-h yield at day 0 is predicted by [NRS - 2]. 2 - The 24-h yields at 30, 50, and 70 d are estimated by [NRS - l], as long as these days are surrounded by measured test day yields. If the first test day yield is measured after 30,50, or 70d, the 24-h yield at 30, 50, and 70 d is predicted using [NRS - 2]. 3 - If the lactation is completed, the 24h yield for the last day in lactation is predicted by [NRS ~21. 19 4 - Using all known test, day yields and calculated 24h yields (in 1, 2, and 3) the total yield is calculated by: 2;. [(INT..1+1)> 0 there exists a continous function g E L1(—00, 00) such that U) llf—gllr : II“ X)-g(x)ldx < 8- -ao Proof: Let a be a positive real number. Define fa, the moving average of f, by the formula X+ _L a : F(x+a)-F(x-a) [1] f.(x)- 23 [mar 23 where F(x) = j mar . 1 . . . . . It 18 easy to check that f. E L (‘00. 09), and that it is continuous. Hence, from integration theory, limof,(x) = f(x) a.e. Thus, f is the limit a.e. of a sequence of continuous functions. A detailed proof that 24 f, E Ll(—oo, 00) is given by Shapiro (1969). Generalizing, we rewrite [I] in the form, t‘.(x)= jf(x-t)o.(r)dt where OJX): Z fOI‘IXI S a 0 for [X] > a Whence, we define the convolution f* g of f and g by-the formula (f* gXX) = THX - t) e(th‘lt The following properties apply to the convolution product (i) {*g e L‘(-oc, w)-Infact ”its“. S Hill Halli; (ii) fag = gtf; (iii) f*(g*h) = (ftg)*h for f, g, h e L1(—oo, 00). Hence, notice that f is the convolution product of f and Ga. Moreover, if we set then and writing for h > 0, [(A(x) = hK(hx) we have I GJX) : KMX), 7» I "" a So, the ‘moving average’ method ofsmoothing is characterized by a certain function known as the kernel, 1((x). Writing f(x; 2‘.) for fa(x), we have 25 f(x; i) = (emu) : If(x-t)hl((ht)dt : £f(x T) K(t)dt. 3.7 Principal factor analysis Factor analysis was used to examine differences between methods for computing total lactation yield. Factor analysis-entails the statistical analysis of multivariate data from a mixture of finitely many populations. The task, at hand, is to find fundamental and meaningful dimensions of a multivariate domain by examining the intercorrelations between a set of traits of interest. In the initial step, a composite score measuring what these traits have in common is generated. This score must explain the maximum variance among the variables. The principal axis, or component, defines the factor or basic dimension the variables are measuring in common. This procedure is called principal components analysis (Morrison, 1976). The resulting principal factors are used as a set of reference axes for determining the most easily interpretable set of factors for the domain in question. This whole process, which Harman (1960) calls multiple-factor analysis, is reviewed within the context of the sample space model by Cooley and Lohnes (1971). Subjective procedures are proposed (Harman, 1960) for developing the transformation from some initial solution to the multiple-factor form of the solution. The methods consist of the build up of a series of rotations in a plane using simplification of the rows or columns of the factor matrix. After the initial factor extraction, the common factors are uncorrelated with each other. If the factors are rotated by an orthogonal transformation, the rotated factors are also uncorrelated. If the factors are rotated by an oblique transformation, the rotated factors become more correlated. However, oblique rotations often produce more useful patterns than 26 do orthogonal rotations. 3.7.1 Orthogonal varimax factor rotation The varimax criterion involves simplification ofthe columns of the factor matrix and has become the most widely accepted and employed standard for orthogonal rotation of factors since its development by Kaiser (1958). He defines the simplicity of the factor as the variance of its squared loadings (A factor loading is a correlation between the underlying factor and the observed trait in question): {PZfabfklz- (Z; 8.1)} V1,: 2 P where azk is the new factor loading for variablej on factor k; j=1, 2, ..., p, and k = l, 2, ..., n. ., - Then for the entire factor matrix the varimax criterion is: “ " P2,; (Silk) _(Zf=i Silk). 3 kzl k=l p" max To eliminate some slight bias associated with the column sums 29 152k, Kaiser redefined the l: l criterion by ‘normalizing’ the loadings, “ P2;(Sit/le—(Zfflka/hly ‘P kZI 13" V: max where hz is the communality of the jrh trait (A cornmunality is the proportion ofthe variance 1 of the jfh trait that is explained by all n factors). Kaiser (1958) delineates the method fully. The criterion V is maximized by the iterative application of trigonometric functions. 3.7.2 Oblique Procrustean [actor rotation First, the matrix of factor loadings is rotated to orthogonal simple structure using the 27 varimax criterion. Then these orthogonal results are rotated to a least squares fit to give the ideal oblique solution. Hendrickson and White (1964) define a matrix P:(Pk) such that: . . J ‘u+1 st, / Sky with u > 1. Thus, each element of this matrix is the ud‘ power of the corresponding element Pr, = in the row-column normalized orthogonal matrix. Then find the OLS fit of the orthogonal matrix of factor loadings to the pattern matrix, P: , L: (1315)1 E'P where L is the unnormalized transformationmatrix of the reference vector and E is the orthogonal rotated matrix. This is the ‘Procrustes‘ equation described by Hurley and Cattell (1962). The columns of L are normalized such that their sums of squares are equal to unity. This provides the transformation matrix from the orthogonal factors to the oblique reference V CCI’O IS. 3.7.3 Oblique Harris-Kaiser factor rotation A derived oblique solution which employs only positive definite diagonal matrices (D matrices) and orthonormal matrices (T matrices) is presented by Harris and Kaiser (1964). This feature then permits translating the problem of the developing an oblique solution into the problem of orthogonal rotation of a matrix that differs in certain ways from the initial orthogonal solution, F. The preliminaries are as follows: 1) R”, the correlation or covariance matrix of the traits of interest. 2) R'=QM2Q', Q'Q= 1. QQ'¢1 where M2 is positive definite and diagonal and Q consists of m columns of normalized eigenvectors corresponding to the nonzero eigenvalues of R'. The tautological expression is: R.:QM2Q':(QMT2D2T1 D1) (DI1 T'i DQ‘T'zM“ MZM“ T; Dé‘ T1 Dil)(Dr T'i DzT'z MQ') in which all T matrices are orthonormal (T’T= TT' : I) . An oblique solution is obtained by setting T» :1 and D3 = I (withT, i I, D2 i I, L $1) . Then, for an independent cluster solution, 28 define 4\*::(:lTFl[)li L = D,"1 T'r M2 Tr Dj', (Case II; Harris and Kaiser, 1964) B=Q M2 T; Dr‘- Here A is regarded as a pattern matrix and B as a structure matrix in the sense of Harman (l 964), and the matrix L designates the intercorrelations of the factors. llliilamsKarsem' - ' mammmLumtonMulflkweighm Kaiser’s iterative algorithm for the varimax retation fails when a) there is a substantial cluster of variables near the middle of each bounding hyperplane, and/ or b) there are appre- ciably more than m traits whose loadings on one of initial F-matrix, usually the first, are near zero. Cureton and Mulaik (1975) proposed an approach for overcoming these difficulties by weighting the factors, giving maximum weights to those likely to be near the primary axes, intermediate weights to those likely to be near hyperplanes but not near primary axes, and near-zero weights to those almost collinear with or almost orthogonal to the first initial F-axis. For a solution, normalize the rows of the initial F-matrix and call the result G, with elements fa / 111’ where h is the square root of the communality of trait j. Isolate all rows of G gar = whose first-factor loadings are negative, and call the result A. The desirable weighting function y... ’1: no ‘—1 " ‘ Q—l' o wrzcos" C°°—V--(~1/l“)—“’>—-fl‘lxgo +.001 if alkl2,/(1/m cos '1 (l/m) -—l / —1 1 ‘ ‘ 1 _ ‘ ‘ o w, =Cos~ 91,—5— —/“‘,) "wt—3K1 x 90 + .001 if 3,, < ,/(1/m 9O - cos «(I/m) Let W be a diagonal matrix of the m weights for the m traits. Then the weighted varimax orthogonal approximation to simple structure is: Vv=FAw’ where A is the transformation matrix of V =WA- Applications to the Procmstean rotation w w 29 are discussed by Cureton and Mulaik (1975). An extension to case II of the Harris-Kaiser oblique procedure exists if their T1 is replaced with the weighted varimax transformation matrix Aw. 4. BOOTSTRAP ASSESSMENT OF THE SAMPLING VARIATIONS IN MEAN DAILY MILK YIELD 4.1 Abstract The notion behind bootstrap is that by sampling repeatedly from data, one can approxi- mate the sampling variations which produced that data. The objective was to estimate sam- pling variations in mean of daily milk yield (DMY), throughout a lactation, within subclasses ' of parity and season of calving. We used 4 yridaily milk records ‘of 340 Holstein cows from a Michigan herd. Parity groups were 1st, 2nd, 3rd, 4th or higher. Season of calving groups were April through October and November through March. Bootstrap resampling was done within each of the eight parity-season groups: A random sample comprising of a fixed percentage of the total number of lactations was obtained to form a hypothetical random sample from the population. This sample was duplicated, or cloned, to form a proxy for the population. A random sample of the fixed percentage, the bootstrap sample, was drawn without replacement from the cloned population. Different percentages were studied to determine optimum size for resampling. This resampling was repeated 5000 times. The mean of DMY for each of the bootstrap samples was deviated from those calculated from the hypothetical sample to give an approximation to the sampling variations in the mean of DMY. 4.2 Introduction The availability of electronic identification and decreased cost of electronic data acquisi- tion have made feasible the daily monitoring of milk yield for individual cows. The physi- ological state of cows can be associated with abnormal fluctuations in daily milk yield. The sources of variation in milk yield have been examined (Everett and Wadell, 1970) and reviewed (Palmer et al., 1994). The gross standard deviation and coefficient of variation of total daily was 4.5 to 6 kg and 17 to 30% within 300-d lactations (Anderson et al., 1989). 30 31 Because variance estimates differ based on the sample, there is a need to have a better under- standing of the sampling variations around daily milk yields. However, the data needed to compute the true variance of daily milk yields are too expen- sive to acquire. The bootstrap (BS) method, introduced by Efron (1989), can provide a good approximation to the true variance given a relatively small sample that is representative of the population. It is a computer-intensive method that achieves this approximation by repeated sampling from the original sample. In simple terms, resampling does with a computer what an experimeter would do in practice, if it were possible: he or she would repeat the experi- ment. In resampling, the observed variable’s values are randomly reassigned to treatment groups, and the test-statistics are recomputed. These reassignments and recomputations are done thousands of times. The objectives of this study were: 1) to approximate sampling variations in mean daily milk yield; 2) to determine the optimum proportion of the sample data for bootstrap resampling; 3) to use bootstrap confidence intervals for generating lactation records. 4.3 Materials and methods The data were daily milk yield for 340 primiparous Holstein cows from a low somatic cell count herd in Michigan. There were 89 cows in the first lactation and 251 in the second and later lactations. 67% of the first lactation cows and 88% of the second and later lactation cows had > 305 days in milk. Parity subclasses were defined as 1) first parity and 2) second and later parities. To be consistent with the literature, all records were truncated at 305d. Sampling variations in mean DMY were approximated by bootstrap resampling. To determine the optimum proportion of the original data for resampling, two sampling propor- tions (p) were defined; p = .2 for light and p = .8 for heavy sampling. Within parity the BS method was implemented by the following steps: 3. Generate a proxy for the population. Note that each cow generates 305 data points. 32 i) Obtain random sample ofn cows, where n : p*N (e.g. N:89 for 1st parity). ii) Compute sample mean. iii) To reduce noise, smooth the sample mean by convolution: Define a suitable kernel estimator, k(y— x)=exp_2m(_ici‘s) ; if: 1,---, 305 Then let f(y) = a(ii). Then f(x) jf(y) k(y- x>dy and convolution = InverseFourierlFourier[f(x)] Fourierlk(y-x)l ] iv) Clone the sample (N / n) times. b. Generate the bootstrap sample. of i) Obtain a random sample of size n without replacement from the clone. This is known as the BS sample. ii) Compute mean of the BS sample. iii) To reduce noise, smooth the mean of the BS sample by convolution. iv) Compute difference a(iii) and b(iii). Repeat b 5000 times to give 5000 x 305 matrix of smoothed BS differences. To obtain naive 100(1- 00% confidence interval (CI), i) Sort smoothed BS differences in c ii) The half width for mean DMY is the irh, jrh element of e where, i = 5000(1- or) iii) CI = b(iii) i d(ii). Due differences between a(iii) and b(iii), we can expect some degree under or over-estimation by the BS approach. A lactation curve can be generated by random sampling of a real value between the upper and lower bounds described by d(iii) on each day in lactation. 33 4.4 Results and discussion The BS resampling analyses were performed on an Intel 486/ 66MH2 IBM PC compat- ible computer with 20MB of RAM. The average CPI.) time was 58 sec per iteration. Figure 2 shows the 100(1a)% BS confidence intervals of mean DMY for first lactation cows after light sampling (p:.2) of the original data. The mean of the original sample, which represents the population mean, is shown on each plot. These data were consistent with theory; the 99% C1 > 95% CI > 80% CI. Ilowever, the 80% CI for mean DMY did not cover the pseudo- population mean. The half-width was widest in the interval spanning peak lactation and narrowest at the beginning and end of 305-d lactation. The figure shows that the “sample mean“ lead to an overestimation during the middle part of the lactation. This overestimation was due to “sampling error”. The BS estimates of confidence intervals were considered to be unbiased and were the primary focus of these results. Contrary to the data reported by Anderson (1989), the variance of daily yield varied during lactation. However, our data are in agreement with those of Palmer et al. (1994). The Cl’s after heavy sampling of the original data are shown in Figure 3. The half-width following heavy sampling was narrower than that for light sampling. Furthermore, the 80% CI covered the pseudo-population mean. Because the BS samples contained more information about the population, the coverage was much better. Light sampling led to wider BS estimates of the half-width. Results for the second and later lactation cows (not shown) were similar. 34' 36‘ W ' 80% CI for mom DMY (n=18) Milk yield (lg/d) ji. “1141““ ‘l‘ll, ‘1 (kg/d) Milk yleld ..a O l . 12% . . r , . . r 4———. . . T , . . . O 34 68 102 136 170 204 238 272 3060 34 68 102 136 170 204 238 272 306 Days in milk Days in milk Figure 2. Bootstrap confidence intervals for daily milk yield. Data are for first lactation cows following sampling a light proportion (p=.2) of the original data. The pseudo-population mean is also shown on each plot. 35 36 80% CI for mean DMY (n=7‘l) Milli yield (iq/d) 12:. - a a - . e l 95% CI for mean DMY (n=71) l j 99% CI for mean DMY (n=71) (lQ/(ll Tank—:4 FEETIIEIIE: "mull 'u l' l llllllli'l'il Illll llilil' l'lli 24 “Ill yield 0 34 68 102 136 170 204 238 272 0 34 68 102 136 170 204 238 272 306 Days In milk Days In milk Figure 3. Bootstrap confidence intervals for daily milk yield. Data are for first lactation cows following sampling a heavy proportion (p:.8) of the original data. The pseudo-population mean is also shown on each plot. A sample lactation curve generated using the 99% Cl is shown in Figure 4. The f luctua- tions in milk yield depicted by this curve are comparable to those in the observed data. 36 Simulated lactation curve i ' ' v v 30: M31: yielttiootgld) 1 0| 0 . .4. 0| k4+L A 50 1 00 1 50 200 250 300 Days in milk Figure 4. A sample lactation curve simulated by random sampling from 99% bootstrap confidence intervals. 4.5 Conclusions The bootstrap method was used to approximate the sampling variations of mean daily milk yield. The bootstrap 100(1- 00% CI of mean daily milk yield were accurate. However accuracy of the sampling variations was sensitive to the size of the bootstrap sample. The coverage of the confidence intervals was much better when the size of the bootstrap sample was equal to 80% of the original sample size. When the bootstrap sample size was only 20% of the original sample size, the confidence intervals appeared to be over estimated. The data show that confidence interval of mean daily milk yield is widest at peak to mid lactation. The confidence intervals obtained are useful in generating biologically consistent lactation records. The method is relatively easy to implement. The main dis-advantage of bootstrap resampling is the heavy computation involved and availability of computer software to perform analyses. 37‘ Most applications require individually crafted programs. In addition, once software is in place, the thousands of simulations required can be uncomfortably timeconsuming, particularly for those with limited computing facilities. The use of bootstrap resampling in this study was not only descriptive but well suited to the problem of approximated the distributional characteris- tics of mean daily yield. Thejapproach was suitable because the mean for the original sample was unusual compared to the resampling distribution. The resampling method was a conve- nient and asymptotically valid way of incorporating the unknown dependence structures inherent in the data. In the absence of the requisite large sample sizes, the method can be used to verify analytic results frotn standard analyses. 5. VARIATION IN MORNING AND EVENING YIELD DURING LACTATION OF HOLSTEIN COWS 5.1 Abstract Differences in mean and variance of daily milk yield at morning and evening during a lactation in different parity and season of calving were investigated. Data on a total of 956,680 lactations of 3295 cows distributed in 26 herds from Dairy Herd Improvement Centers of Alberta and Ontario, and Florida Agricultural Experiment Station were used. Size of herds and number of cows with records had an irupact on the magnitude of the variances within subclasses of parity and season of calving. Across regions, mean morning yield was higher than evening yield. The shape of the mean curve for total yield followed that for evening yield closely. Rank correlations between daily morning, evening, and total yield were as high as .998 (P < .0001 ). Variance in daily total yield was mostly determined by variance in daily evening yield. Second lactation cows had the largest variance in daily morning and evening yield. Parity differences were least in the Florida data. The highest variation occurred at 50 d postpartum during the lactation of cows calving in December to February. Fluctuations in daily yield were highest for cows freshening in june through August. 5.2 Introduction Recent technological advances in identification and automated data capture have made feasible the daily recording of milk yields for individual cows. The accumulated data can be utilized in monitoring the health status and in facilitating the management of lactating cows. Variation in milk yield due to different intervals between morning and evening milkings has been reported (Anderson et al., 1989, Everett and Wadell, 1970; Palmer et al., 1994; Schmidt, 1960). Gilbert et al. (1 97 3) investigated diurnal variations of tnilk yield and found that milk production at morning (AM) and evening (PM) were not equal even though the milking interval was 12h. Hyde et al. (1981) observed that variation in milk yield was not constant throughout the entire lactation and varied between first and later parities. 38 39 Adjustment factors for the difference between AM and PM milk yields have been devel- oped. These adjustment factors accommodate differences among herds, cows, seasons, month of lactation milking interval, month of lactation, breeds, and age at calving (Palmer et al., 1994; Putnam and Gilmore, 1970; Shook et al., 1980). Palmer et al. (1994) calculated a range of 2 to 6kg in gross standard deviation of individual milk yield within 305-d lactations. How- ever, the literature is lacking in studies that show the actual magnitude of variation in the AM and PM milk yields throughout lactation. The objective of this study was to examine differ- ences in mean “and variation in AM and PM yields and their relationship to daily total yield. The effects of parity, season, and region on AM and PM yield both in mean and variation were also examined. 5.3 Materials and methods Data of morning and evening milk yield for Holsteins cows from Florida, Alberta, and Ontario were used. Data from the two provinces were collected in an automated milking systems project conducted by the Canadian Dairy Herd Improvement Program. To be in- cluded in analyses, all lactation records had to have calving dates and parity number specified and were truncated at 305 d postpartum (pp). The resulting data sets included whole and part- lactation records with either a AM or PM milk weight on each day of lactation. The intervals between morning and evening milkings varied. However the exact length of the interval could not be determined because milking times for the sampled and previous milkings were not available. The extent of the data is indicated in Table 2. The Ontario data set had the most records, lactations, cows, and herds. Six of the Ontario herds had less than 90 cows while all 5 Alberta herds had more than 160 cows. In contrast to the Florida data, the number of records in- creased with parity for the Canadian herds. Alberta herds tended to have more late Fall and Winter calvings whereas the Ontario data consisted of higher number of calvings during the Spring and Summer. Most of the Florida lactation records were initiated in late Fall. 40 TABLE 2. Test day data for Holstein cows utilized for analyses by location. Data source Category ~ Alberta Ontario Florida No..of lactation days i . , 287325 451423 217932 No. of lactations ‘ 1288 271 5 587 No. of cows ' 960 I924 41 1 No. ofherd ’ 5 20 1 Minimum herd size (cows) 161 28 __ Maximum herd size (cows) 4.37 490 _ 1 st lactation records 409 787 272 2nd lactation records 300 691 146 3rd or later lactation records 579 1237 169 Lactations started Dec. - Feb. 325 657 129 Lactations started Mar. - May 295 700 141 Lactations started Jun. - Aug. 308 723 130 lactations started Sep. - Nov. 360 635 187 Figure 5 shows the number of cows with lactations records by location. All locations had few cows with records from 1 to 7 d pp. Thereafter, the Alberta data showed a drop in number of cows with records (NCR) after 280 d pp. A steady decline in NCR occurred in the Ontario data after 100 d pp. The NCR remained the same after 7 d pp for the Florida data. Whereas, NCR for Alberta and FTorida was similar at a.m. and p.m., Ontario had more NCR for AM after 100 d pp. Most of the cows with records at a.m. had a record at par. Within each location, parity, and season of calving, the mean, standard error of mean, 95% confidence interval of mean and variance of mean AM, PM, and daily total yield were computed. Product-moment correlations between mean AM, PM and daily total yield were calculated. A test for significant differences between correlations was conducted. All analyses 41 a were accomplished using analytical procedures of SAS (1992). g , v--—Y———-1— - V . w——v‘ 0 34 1’38 10.7" ‘38 '70 .3114 238 .272 33 ' fury"; {10:31:26.1'111'11 ' Figure 5. Number of cows with lactation records by location: A = total milk yield, B = morning milk yield, and C = evening yield. 42 5.4 Results and discussion Table 3 shows the within location product-moment correlations between mean total, AM, and PM yields. Across locations, correlation coefficients were significantly high (P < .0001) but they were not significantly different from each other (P > .1). The magnitude of these correlations was not regarded as ample evidence for predictive ability because the sources of variation in total ( Everett “and Wadell, 1970; Murphy, 1992), AM, and PM yields are numer- ous (Amos et al., 1985; Becker et al., 1990; Elvinger et al., 1992; Lewis and Newman, 1984). TABLE 3. Product moment correlations between mean morning, evening, and total yields. Alberta Ontario Florida Total AM1 PM2 Total AM PM Total AM PM Total .. .994 .996 .. .970 .992 .. .993 .993 AM .. .080 .. .99 .. .. .982 TAM = daily morning yield. 2PM =daily evening yield. All correlation coefficients significantly different from zero (P < .0001) but not significantly different from each other. 5.4.1 Overall variance of daily milk yield Figure 6 shows 95% confidence intervals for mean total, AM, and PM yield for Alberta, Ontario, and Florida. In all cases, mean AM yield was higher than mean PM yield. This relationship between AM and PM yields was similar to that observed by Putnam and Gilmore (1970). However, the general shape of mean curve for total yield appeared to follow that for PM. For both Alberta and Ontario, the variance of AM yield rose sharply, peaked at 20 d pp, declined rapidly then started to rise in late lactation. Ontario had much smaller herds. In small herds, time of AM milking tends to be consistent but PM milking time tends to be less so due to field work and social life. Thus, more variation in milk yield would be associated with cows in small herds. On the contrary, in the Florida data, the variance ofAM peaked at 45 d and then declined steadily throughout lactation. In all cases, the variance of total yield followed that for PM. In this and subsequent illustrations, the apparently higher variance at 43 the beginning and end of the lactation could be attributed to more missing observations caused by incomplete lactations. 5.4.2. Within parity variance of daily milk yield 95% confidence intervals and mean total, AM, and PM yields for Alberta, Ontario, and, Florida are shown in Figures 7, .8, and 9 respectively. Mean yield increased with parity. The variance of daily morning, evening, and total yield was highest for 2nd lactation cows. This difference in the variance peaks was associated with a possible region by parity interaction. This variance peaked at 7 d pp in the Alberta, and Ontario data but at 50 d in Florida. In addition, differences between 1st and 2nd lactation curves were much less for Florida. Be- cause the Florida data were from a single herd, this lack of differences between lst and 2nd parities was attributed to management within that herd. 5.4.3 Within season variance of daily milk yield 5.4m There were seasonal differences in shapes of the curves for mean milk yield (Figure 10). The largest peak for mean and variance of AM and PM was at 50 d pp, for cows freshening in December through February. The data for mean daily yield are in agreement with those of Keown and Van Vleck (1973) who reported that cows calving in January through April had the highest peak production. The least variance during lactation was for cows giving birth in March to May. The variance in AM and PM peaked at 7 d pp in June through August. 11.3.2an The magnitude of the variance for AM, PM, and total yield was highest at 45 d pp for cows calving in March to May (Figure l 1). As few as 10 cows with records at the beginning and end of the lactation caused variances to be high. Fall freshening cows were more persis- tent and had the highest variance during lactation. The persistency in the Fall provides biologi- 44 cal evidence for the Spring stimulus to production discussed by Wood (1969). W The largest variance in AM, PM, and total yield occurred at 50 d pp for cows calving in December through February (Figure 12) and agreed with Florida. Mean milk yield was more depressed and had greater fluctuations in june to August but the variance declined gradually following the 50 d pp peak. Seasonal effects on mean daily yield differ from effects reported by Keown and Van Vlecki(1973), who found cows freshening in May through August had the highest average production. 5.5 Conclusions Mean morning yield was higher than evening yield. The data indicated that the shape of the mean curve for total daily yield followed that for evening yield. In these data, the variance of evening yield appeared to determine the variance of total daily yield. Regional differences in the magnitude of the variance of daily total, morning, and evening yield were observed. These differences could be associated, in part, with stage of lactation, parity, seasonal effects. The data did not contain complete morning and evening yields for each cow during lactation. Hence, it was not possible to quantify the observed region by season, region by parity, and parity by season interactions. Future studies should investigate these interactions. (OF—OJ“. C.n.<...2C (...Kmumdd‘ me .23.» chQé n O paw .Eor» MEEOE u m .23.» ~69 u< £255 pea .2530 .8332 no» 22.» 4:8 358 E88 Co 1285 cone—occoo 8mm woquwmm .0 23mm (EJuLOOaOOO n>00 00m. Nb" In“ CON Oh. 0n .. Now 00 On 0 O V .0 . W. .OpIHI. . m: p .81.” (D w ( VON T U «N O in #0. WW 4 M. ?np m ) )I t ..D It" VON w- m an O .I .n r 40' .0. I". L. in 'A .2 .M .. D. van) M1 W Ru 3" W won. ( r 01 (Ewan—u. 000 Nb“ on" Entooaooo n>.00 vON or. 00' NO— 00 in O T. 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NO. 00 in O O 7 v“ I fin: vnv ( 1 Y O ’ fl (bi) 918141144 N u ZO_._.<._.0<1_ (61) mail 1141 (61) well an .33.» @6526 n O 3:“ .33.» 9350:. u m .33.» 3.9 n < 83230 6:860 new team 5323 30:» 82:: 35o :88 .Lo 1285 85338 Axumo woumEcmm .m 8sz EJVLOAuunOQ I>OO 000 Nb" UflN ‘0“ Oh. Ofl. NO, 00 in _. u. ZO_._.<._.0<1_ (hi) all 1111 (51071911 11111 won EJCOOunOO u>00 NBN an.“ GON 0hr finnr "Op 00 0n 0 70' vflr 'ON on N I {IV on It (51) wall 11111 rON ij’ fi N won 0 n Y 40' V n "A ZO_._.<._.U<1_ (51) 9191411111 (91) mail 11111 #0." Nb“ 0n“ we EDuLOOunOO n>.OO VON Oh. On . mop 0n 0 .0. '0' .On .nn. 70v N u ZO_._.<._.U<1_ no. on.” [[3:003n00 n>.00 00H 0%. OH. NO. no .33.» 93:36 N O 38 .33.» MEEOE n m .33.» 39 n < "~ >\ it as 20‘ "i "i ‘5': ., 5 O 25 Milt yield (kg) O C. 23 23 C C 1 4 204 '7‘ ‘ in ‘5 4 a ‘3‘ v~cfizo4 o'tufl" DO\‘ DO‘\DOV‘U'V‘ o: ”no 204323-‘0 272 302 o 34 on no: 0 3‘ a. i V 3 Days oosio 230 272 106 Figure 12. Estimated 95% confidence interval of mean daily milk yield within calving season for Florida, USA: A = total yield, B = morning yield, and C = evening yield. 51 6. MATHEMATICAL DESCRIPTION .01: [ACTATION CURVES IN DAIRY ‘ ‘ CATTLE. ,~ 6.1 Abstract Tools for fitting mathematical functions to various shapes of lactation curves of dairy cows were established. A method for generating sample lactation curves from the functions was described. This method transcends any method that approximates the shape and variance of the lactation curve from a single mathematical model.. The method was demonstrated by fitting ten curves for mean yield and four variance curves for mean yield during lactation. 6.2 Introduction Frequent applications of the lactation curve include extension of records in progress to predict yield up to a standard length, calculation of lactation totals, and generation of yields for simulation studies. Lactation curves also can be used on a routine basis to monitor the health and progress of individual animals. Therefore, a clear, concise, and accurate mathematical description of the lactation curve is imperative. Such a description must be representative of different breeds, regions, seasons of calving, age at calving, days open, days dry, stage of pregnancy, stage of lactation, BST treatment, and other environmental factors. An early attempt to develop a model which would describe the lactation curve was made by Gaines (1927). He proposed the formula Yr: A exp(—Kt) in which Yt is the yield in month t, A is the starting yield (when t = 0) and K is the rate of decline per month of lactation. This expression however makes no attempt to explain the initial rise in production, a portion of the curve which is of extreme importance. Vujicic and Bacic (1961) suggested a modification of the formula of Gaines. They pro- posed the expression Yt = rm” exp(—mt), where Yt is the yield in period t while a and m are Parameters. Unlike Gianes’ formula, the ex ression to ,osed b Vu'icic and Bacic varies both P P P V l 52 .litctr early 1 53 directly as exponentially with time. Thus, it accounts for the initial rise in production during early lactation. Nelder (1966) described a family of inverse polynomial curves of the form, Y.‘ =(bo/X + b1 +b2X)-l where Yx is the yield in week x and b0, b1 ‘ and b2 are constants which could be estimated by Ordinary least Squares (OLS). Under this model, maximum weld occurs when x =(bc / b3 ). The maxrmum yield is equal toe/(bola) + b1)—l . Using a similar approach, Wood (1967, 1969, 1972) described the lactation curve as the incomplete gamma (IG) function Y“ 2 a where Yn is the average daily yield in the . nb extol-.01) nth week of lactation and a, b, and c are constants. The equation reaches a maximum when n = b/c. The expected maximum yield is a(b/c)b exp(—b)' The exponential form accounted for 95 4% of the variation in month yield as opposed to 84.4% for the inverse polynomial. The Wood's 1G function has been the most widely used because of the flexibility of fitting curves of different shapes by estimating the three curve parameters. Grossman et al. (1986) extended the IG equation by including sine and cosine terms to account for seasonal variations other than season of calving, Y" :anb exp(— cn)(1 + u sin(x) + v cos(x)) where x is the day of the year in radians and all other terms are as defined previously. This extended function gave a better OLS fit than the inverse polynomial function (Batra, 1986). In order to account the different stages of lacta- tion, Grossman and Koops (1988) proposed a multiphasic function, Y ::a b {1_ tanhzlb (f—c ))}, where Y is milk yield at time t (t : days in milk); n is the 1:] number of lactation phases; tanh is the hyperbolic tangent; ‘ are parameters for the irh 81 1131' 9 Ci phase. A diphasic function was found to be sufficient to describe the lactation curve. This function accounts for smaller and more random residuals and provides easily interpretable parameters that have biological importance. In addition, linear (Schaeffer and Dekkers, 1994; Rowlands et al., 1982; Stanton et al., 1992), non-linear (Freeze and Richards, 1992; Schaeffer et al., 1977 ) models to include shape, genetic and environmental effects have been proposed. 111116 let-.11 54 All of the above mathematical expressions and models require raw data in order to esti- mate the parameters of the lactation curve and they diverge from typical lactation curves because shapes of lactation curves differ for over a wide range of ages, parities, regions, climates, and breeds. For empirical studies, the hitherto approach of relying on estimates of curve parameters in orderto accommodate factors which affect the shape of the lactation curve is. not only unsuitable but prohibitive. The objective of this study was to propose math- ematical functions which describe typical shapes of the lactation curve in dairy cattle. 6.3 Materials and methods Ten curves that were representative of typical lactation curves were identified. These curves as shown in Figure 13. Two curves for morning and evening yield also were identified (Figure 14). Observed from the literature were mean curve 1 (Grossman and Koops, 1988), curve 2 (Mainland, 1985), and curve 3 (Congleton and Everett, 1980). Mean curves 4, 5, 6, 7, 8, 9, 10, 11, and 12 were perceived from a preliminary analysis of data from a herd in Michi- gan. Four curves that were representative of the variation in mean daily yield during lactation were identified. These variance curves, shown in Figure 15, were observed from preliminary analysis of data from a herd in Michigan. All the mean and variance curves were endorsed by the "Lactation Computation Working Group" of the International Committee of Animal Recording as representative of all breeds, regions, parities, ages at calving, and seasons of calving, and other grouping effects. A collection of approximately 40 basic mathematical functions from the literature (Abramowitz and Stegun, 1965; Papajcsik and Bodero, 1988) were used to handcraft the functions for the curves in Figure 13, 14, and 1 5. To reduce random noise, the ‘moving average’ approach was used to smooth mean curves 1, 2, 3, 4, and 5; and variance curve 1. This was achieved by defining the kernel estimator, k( —— )- 700(85): 305 y x - exp , l, ,. 55 \ 251 \ 231 a! \‘x 201 \ 15‘ 151 Dow. 90.th *— 151 \\ 151 \‘\---__,—l D»! \ Days \\ Dav: \\ | :1 l \ ‘l l S. l 301 \ 1 , _ , .__a__.l 301 01 x 101 \ . \ .. \ a \ f ..s \ 1 n \3 A n. A n. 3 A _ . _ /// r/ u \ I . ‘ /,. /./ lir/ xx /.l b ./I .l ll+...ll . +I.V.. l+ II.1l.v ‘ . l 1+ Jill II. o 1 .1 l i . i + . o ‘ illll +lr+1 ...I, ,o 1 — u 1' o 1 w w m w o. m w m w o w w m m o w w m w o w m m as 2; a... 9.; a... no.» a... as» a: .2.» llll .ll J1 .1 .1 1 Ill. . fl 1 1:1 .. . -.lAlIlll1 1.14 . i -l l-.v.~i _ 1H 3 m 33 m \ w ,. i m x. \ _ _. . _ a 9 w. 1 _ 3 h 5 u 7 i . a u n . _ n . a a. .2. 3 _ a _ a. x _ \ a w \ as m a 4 w r. m 3 m \. \ n a m m m t . m . . . a t. x \\ is“. \ ..m .m. .mm \ w . w \ W \ m, m m \ \ r w \ m \ a. w. . m a a z x n. H 1. u h m 3 .. m / ,./- / ,/ l-Tl.HdVl:+1 .l .. .r/r . . . -1 . .ltmlllJr .l . . T lilfllli « .i r w, t w w m m o m w m m o w w m m o w x m m o w w m a... 91> 65 92> 3.. a: a: no» a... 9.; 301 251 201 l5! Day- mtnfl‘unn I0! 51 o .—_~—~—o-——~~-+ daily yield. 151 20! 25‘ Days poummn s for 10! 51 Figure l 3. Lactation curve 56 1 3D; 1 25 3 3 I 11:20 i .3 015 * : -..g , >* : 10 > « l 1 Sr 0. g . ‘ - A L l 0 50 100 150 200 250 300 Days PP 35 » otal 12 f 30 3:1 3 ”'25 j -o . v-l 4 .E’, . *201 j 15. 1 10 *- , . A - - U 50 100 150 200 250 300 Days PP Figure 14. Lactation curves for morning (AM) and evening (PM) yield. ..Mu w“ man u .i 5 I. . i .34. 3. $3.21) Aalv .i.;-‘:'> S7 g: E 8 8 E .5 s 5; 110 E fl 100 + I 9° ' 3 so} a 2 7° 1 f: 2 °° + E .L. 5 5° a“ > ‘0 .41— 30 + m —0— 10 ~~ A o . t +1 + fi‘ f 1 51 101 151 201 251 301 1 51 101 151 201 251 301 Days postpartum Days postpartum Figure 15. Variance curves for mean daily yield during lactation. The unifying integral for the convolution was f(x) = [f(y) k(y- x>dy and convolution : InverseFourierlFourierIKxH Fourierlldy-xfl ] where f(y) is the raw data for the curve. All analyses were performed using Mathematica (Wolfram, 1988). 6.4 Results and discussion. The mathematical functions for the mean and variance curves were as follows: 58 Mean curve 1: f(x) = 14.7819 (1 - Tanhl.008471 (x-150.656)]); x, 1, ..., 305; Mean curve 2: , f(x)=1ox=5= Sech[.009x]; x=1....,305% Mean curve 3: f(x)=lem Sech[.009x]; x=1,...,305; Mean curve 4: f( y=10+24im3 Secl1[.006i];i=1,...,250,ISxSZSO x : y=23—.O6i—.00112; i=6,...,60, 250 szo‘Llli'mSechl‘OOéils i=1....,250. 1ng250 X Z . v=26.8—.00113—.01u1g[11; i=6....,60, 2500905; Mean curve 6: f(x)=30+Exp[-.O7x]—.02x—l.5Log[x]; x=1,....305; Mean curve 7: f(x)=20+.3Exp[.Ol1x]+.OO7x‘”+ .9ng[.2x]; x=1,...,305i Mean curve 8: f(x)=22.1539+.500083x—.00620379x3+.0000258156x3+ 3.68262*1O‘3x‘; x=1,...,305° Mean curve 9: f(x):15+.00019x+.00731x3—.00009x3+3.724* 10‘7x4-5.1379* 10"10 x5; x: 1,...,305' 59 Mean curve 10: Y1=5+105X3 SCCl‘LOOg]; 1:7... ., 5Q x:1,...,43 f(x) yz=263+.0002+.00592-.000.1i3+3.62*1U7i4—5.228'1(Tmi5; i=85...,283 x=44,...,243 313:1008+1 31* Secli.009];‘ i=244..., 305 x=244...,305 Mean curve 11: Yam 2 2.5 + 9.30139 (1 - Tanh[0.007471 (150,656 + x)]); x : 1‘ ..., 305 me : 2 + 7.61007 (1 - Tanh[0.006162(-130.546 + x)l); x = 1, ..., 305 Mean curve 12: y1=8.8+1 11°05 Sech[.006i], 1:1,...,250; 193250 “N y2=ylm—.001i2-.01Log[i], i=6....,60; 250' M 10 «~ 0 1 1 1 1 1 1 1 51 101 151 201 251 301 Days postpartum Figure 16. Mean i standard error for lactation yield (A), 5 sample lactation curves (B), and l lactation curve (C). 7. DESCRIPTION or“ THE STANDARD LACTATION CURVES (SLAC) PROJECT 7.1 - Summary The data files of standard lactation curves (SIAC) were generated. The SlAC consists of test- day yields for a total of 1,126,080 lactation records representing five testing schemes, six patterns of missing test-day yields, varying starting days of recording postpartum from 3600 sample lacta- tions of various shapes of lactation curves. Collaborating Data Processing Centers (DPC) were asked-to calculate the lactation totals of these lactations using their operating procedures. The calculated lactation totals were sent back to Michigan State University (MSU) for analyses and summarization. A total of 40 parameter curves were designed by combining ten curves of different shapes for mean daily yield in a lactation with four curves for variance of daily yield. For each of the 40 parameter curves, 90 sample curves were simulated to generate a total of 3,600 sample curves. True lactation total yields of the sample curves were calculated and kept at MSU. Four test schemes of A1 , A4, A6, and A8 were applied to each of the sample curves. Only one starting test-day postpartum was considered for the A1 scheme, but four starting days were consid- ered for the other three schemes. For each of the simulated curves under a specific test scheme with a specified starting day postpartum, up to five patterns of missing test-day records were established. The patterns in which missing records occurred at the end of a lactation were intended to simulate incomplete lactations which will need to be “extended” in order to obtain 305~d yields. After application of different test schemes with different starting days and patterns of missing test-days on the 3600 sample curves, there are test records on a total of 136,800 lactations. Mean daily AM yield curves and the corresponding mean PM yield curves were designed for each of two mean daily total yield curves, and two variance curves. From a composite of simulated AM and PM data, 90 sample curves on daily total yields were generated for each of the four parameter curves to give a total of 360 sample curves. The AP/ 4 test scheme was applied to these curves by taking monthly test records altemately from the AM and PM test results. After the 62 63 application of the four starting test-days postpartum and the patterns of missing test day records, there were AP/ 4 records on a total of 3,960 lactations. The total number of lactations was now 140,760. Each simulated lactation was coded for one trait, one breed, one of two panties, and one of four seasons of calving. Each cooperating DPC had to designate the codes to fit categories of its choice before processing the SIAC data for total lactation yields. The DPC was asked to inform MSU of its designations. For SIAC, we assumed that all codes (1x1x2x4=8 combinations) would be designated, thus, each, of the 140,760 lactations were repeated 8 times to give a total of 1,126,080 lactations. The SIAC data file on 1,126,080 lactations was partitioned into five parts by test schemes and was stored in a fixed format. Column positions and codes used for the files are described and a printout of the first three records of each file are shown in Appendix C. The original 103 MB data file was packed into a size of 5 MB. Instructions for retrieving SIAC by DPC with FTP service were provided. Transfer with IBM PC compatible floppy disks was a popular alternative as most DPCs did not have FTP services. Collaborating DPCs were asked to send the yield totals calculated from lactations in SIAC back to MSU. The fomiat and a sample printout of the first three records of each of the five return files are shown in Appendix D. Alternative means for transfer/ retrieval were also sug- gested. 7.2 Design of lactation curves for daily total yield A total of ten curves of different shapes for mean daily yield in a lactation and four curves for variance of daily yield were designed. These curves are shown in Figure 13 and 1 5, respectively. a.) Combining each of ten mean curves with each of four variance curves gives a set of 10 x 4 = 40 curves (see illustration in Figure 16). b) Mathematical functions were established for each of the 10 mean curves. Then for each of the mean curves, 90 sample curves were simulated according to the assigned variance curve. A 64 total of 3,600 curves was generated. For each of the curves, true lactation yield was calcu- lated by accumulating daily yields. The true lactation yield data was kept at MSU for analysis purposes. Note that the optimal size required to compute a significant difference between mean biases at the 99% level was found to be 90. c) For each of the simulated curves, four test schemes with different lengths of interval (A1, ' A4, A6, and A8) were applied. A1 stands for a 1 week interval between milk recording, A4 istands for a 4 week interval recording, A6, stands for a 6 week interval between recording, and A8 stands for an 8 week interval between recording. Starting day of recording was 7d postpartum for Al scheme, while there were four (4) different starting days, 7d, 14d, 21d, and 28d postpartum, applied to A1, A4, A6 schemes. The number of test-day records for each of the combinations of test schemes by starting day is shown in Table 4. TABLE 4. Number of records in a complete lactation by scheme and starting day for milk recording. Starting day Scheme 7d pp 14d pp 21d pp 28d pp 3.7—— 43 _ _ _ A4 11 11 11 10 A6 8 7 7 7 A8 6 6 6 5 d) For each of the simulated curves under a specific test scheme and starting day, up to five (5) patterns of missing test-day records were established. They are shown in Table 5 below and a graphical representation is in Appendix B. 65 The number of lactation curves before (not in parentheses) and after (in parentheses) applying schemes, starting days, and missing patterns is shown in Table 6. TABLE 5. Weeks in which data were missing by scheme and starting day of milk recording. Starting day Scheme 7d pp 14d pp 21d pp 28d pp A1 1, ..., 3 __ __ _ 6, ..., 9 10, ..., 13 26, ..., 43 36 ..., 43 A4 9, ,13 6,. , 10 3 _ 7 ll, ..., 15 27, ..., 43 35, ..., 43 A6 13 _ 3 _ 9 15 27, ..., 39 33, ..., 39 A8 9 _ 3 _ ll 19 27, ..., 43 35, ..., 43 66 TABLE 6. Number of curves by scheme, starting day, and missing pattern Starting day Scheme ‘ 7d pp _ 14d pp 21 d pp 28d pp Total T— 3600+(3600x5)=21600 . 21600 A4 . 3600+(3600)=7200 3600+(3600):7200 3600+(3600x5)=21600 3600 39600 A6 3600+(3600)=7200 ‘4 3600 ‘ 3600+(3600x5)=21600 3600 36000 A8 3600+(3600)=7200 - 3600 3600+i3600x§)=21600 3600+3600=7200 39600 ‘ T6611 43200 _ 14400 64800 . 14400 136800 7.3 Design of lactation curves for AM and PM yield Mean daily AM yield curves and the corresponding mean daily PM yield curves were designed for each of the two mean daily total yield curves. Two shapes of variance curves were chosen. The two mean curves are shown in Figure 14 and the two variance curves are shown in Figure 15 as curves ,3 and 4. a) Combining each of the yield curves with each of the two variance curves gave a set of 2 x 2 = 4 curves. b) For each of 4 curves, a total of 90 sample curves were simulated for a total of 360 curves. For each of the simulated curves, true daily yield was calculated by summing the corresponding AM and PM daily yields, and true lactation yield was calculated by summing the true daily yields. c) For each of the simulated AM/PM curves, the AP/ 4 test scheme was applied by taking the yield on AM curve at week 1, 9, 17, 25, 33, and 41 and the yield on the PM curve at week 5, 13, 21, 29, and 37 (see illustration in Appendix A). Also applied were four starting days of recording 7d, 14d, 21d, and 28d postpartum. 67 For each simulated curve, patterns of missing records were established in a manner similar to that for A4. The total number of curves for AP/4 is shown in Table 7.: TABLE 7. Number of curves for AP/4 by starting day and missing pattern. ’ Starting day 7 Scheme 7d pp 14d pp 21d pp 28d pp Total AP/4 360+360=720 ' _ 360+360=720 ‘ 360+(360x5)=2160 360 3960 The total number of AP/ 4 lactation curves was 3,960 and the combined total number of lactation curves for both daily total yield and AP/ 4 was (136,800+3,960):140,760. 7.4 Designation of trait/breed/parity/calving season categories Each simulated lactation included a trait designation, a breed designation, and designa- tions for parity and calving season categories. Each of the Data Processing Centers needs to fit their designation codes to the codes described below and, in the process, to choose the specific categories in order to capture the most popular categories. In every case, the Data Processing Center was required to inform MSU of the category definition of the chosen designation codes. Designation codes were required for the following categories: (1) Trait: Only “1 " for milk yield; (2) Breed: Only “ 1 ” for one cattle breed. We suggested that the breed be Holstein/ Friesian, which was the case for Italy and Canada for example. For Norway, the breed could be Norwegian cattle. For Switzerland, the breed could be either Simmental, Brown or Holstein; (3) Parity: Up to two categories. Code “1 ” denoted either first parity or all parities, while 68 Code “2” denoted either second or second plus later parities. For U.S.A., for example, “1 ” was for the first parity and “2" for second plus later parities. If the algorithm of a Data Processing Center did-not distinguish between parities, the DPC used “1 ” to denote all parities. (4) Season: For season of calving, up to four categories. For U S, “1 ” denoted Decem- ber through February, “2” was for March through May, “3” June through August, and “4” represented September through November. If a Data Processing Center considered only two calving seasons, for example, the designation would be either “1” or “2” only. The purposes of such designations were: (1) To ensure that the record format contains basic information necessary for inputting the data into the algorithm/ program for calculating total yields at a Data Processing Center, and (2) To suggest specific categories and a specific number of categories in order for a Data Processing Center to choose specific algorithm/ programs for those specific categories. (3) To achieve a reasonable degree of standardization, because of the great variety of category designations among Data Processing Centers. The number of categories for each designation was kept at a minimum in order to keep the data size manageable. It was assumed that the maximum number of the above categories (1,2,3, and 4) would be chosen. Thus each of the 140,760 curves was repeated 8 (1x1 x2x4=8) times. The total number of curves was therefore 1,126,080. The combinations of categories are shown in Table 8. 69 TABLE 8. Combinations of fixed effects for calculating lactation totals. Combination code (1,...,8) lactation (1,2) Breed (l) Trait (1) Season (1,...,4) I 1 '1 1 l 2 I I ' I ' 2 3 I . - I ' l .3 I 4 1 1 1 4 5 2 I 1 l 6 2 1 I 2 7 2 1 l 3 8 2 l 1 4 7.5 Data files of standard lactation curves (SLAC) The 1,126,080 curves made up the SIAC. The data were stored in a fixed format. The record format in SIAC was: Mean curve No.; Variance curve No.; Replicate No.; Scheme; Starting day; Missing pattern; Combination code for fixed effects; Days PP; Testday yield. The SIAC was partitioned into five (5) parts by test schemes. An exact description of column positions and codes used for the files and a printout of the first 3 records of each file are shown in Appendix C. This SIAC was made available in one of the following two ways to the Data Processing Center (DPC) of each of the ICAR member which agreed to collaborate: Choice 1: DPC’s with FTP service could retrieve SIAC via Guest FTP to 35.8.12445 (Guest login password was: slac ). Files could be transferred from the sub-directory \puln\slac\outgoing. All files were stored in an archived format. Instructions for transfer/ 70 retrieval of the archived files were contained in the file \pub\slac\outgoing\readme.lst and are included herein as Appendix E.2. Problems with file transfer were addressed to saamaCal 1n su.edu. Choice 2: DPC‘s without FTP services could receive SLAC on IBM PC compatible 1.44MB 31/2” or 1.2MB 5%” disk media with installation instructions included herein as Appendix E.1 . 7.6 Data files of calculated lactation totals to be returned to MSU These calculated lactation totals by the DPC will be sent to Michigan State University in the corresponding five (5) parts by test scheme with the format below: Type Numeric Numeric Numeric Numeric Numeric PP) Numeric Numeric Numeric Len gth v—ir-‘wr—‘N 5 Position 12 3 4-6 7 8-9 10 11 1216 Description Mean curve No. (1, ..., 10) Variance curve No. (1, ...,4) Replicate No. (1, ...,120) Scheme (1=A1, 2=A4, 3=A6, 4=A8, 5: AP/4) Starting day (727d PP, l4=l4d PP, 21:21d PP, 28:28d Missing pattern (0‘—none, 1=early lactation, 2'-‘-ear1y peak, 3=late peak, 4=late, 5=tai1 end) Combination code for levels of fixed effects (1 => lactation=l, breed=1, trait=l, seasontl; 2 => lactationZI, breed=1, trait=l, season=2; 3 => lactation—‘1, breed=l, trait=1, season=3; 4 => 1actation=1, breed=1, trait=l, season=4; 5 => lactation=2, breed=1, trait=1, season=1; 6 => Lactation=2, breedtl, trait=l, seasonZZ; 7 => Lactation=2, breed=1, trait=1, season-'33; 8 => LactationIZ, breed=1, trait=l, season14) Total yield (kg) in Integer format. The data could be returned to MSI I using only one of the following methods: Choice 1: Guest FTP to 35.8.124.45 (Guest login password was: slac ). Files could be placed in the subdirectory \pub\slac\incoming. A description of the files transmitted was required. After transmission, DPC’s were asked to send an E-mail message to saama@msu.edu. The E- mail message had to include data source and a list of the files transmitted to MSU. 71 Choice 2: IBM PC compatible 1.44MB 31/2” or 1.2MB 5%” disk (Provide disk catalog). Choice “3: ASCII or EBCDIC tape (Provide tape catalog). . A sample printout of the first 3 records of the files for the corresponding 5 parts is shown in Appendix F. Note that the total yield shown for each record in the sample printouts is solely for the purpose of illustration. 7.7 Participating data records processing centers A request for participation in the SIAC project was solicited by the Secretary Generalof ICAR. A respose was received at MSU from Australia, Italy, the Netherlands, New Zealand, LISA, Denmark, Jersey (England & Wales), France, Switzerland, Germany, Austria, and Mexico. Subsequently, the SIAC was sent to each of these countries. Results for the test- interval method were computed at MSU. 7.8 Data files for computed lactation totals returned to MSU Data files of computed lactation totals were returned from USA, Netherlands, Denmark, Italy, and France. As a result of in-house changes in the method for computing lactation totals, Germany, New Zealand, and Australia decided against computing or returning data files. These radical changes were not atypical of the DPC's but some may have been motivated by preliminary SIAC results presented to the working group on “Lactation computation and related matters” at the 29th session of the ICAR General Assetnby meetings in Ottawa, 8. COMPARISONS OF DIFFERENT METHODS OF ESTIMATING 305-D MILK YIELD 8.1 Abstract In order to evaluate the relative accuracy of methods for calculating total yield in a lactation, a dataset of standard lactation curves was generated. Standard lactation-curves were designed. Replicates by simulation, test schemes, missing data patterns and starting day of recording were . empirically imposed on each sample standard curve. Separately, morning and evening lactation curves were designed for an alternating testing scheme. The total number of sample test-day records was 1,126,080. A total of six methods for calculating lactation totals from test-d yields were compared. They were methods of centering date ignoring missing testday data, test interval without adjusmient factors but correction for missing data, test interval without adjustment factors and no correction for missing data, test interval with adjustment factors, interpolation with standard curves, and multiple-trait projection. The differences between actual and calcu- lated lactation yields were analyzed within each method by fixed classification models. Factor analysis using the squared multiple correlations of the methods as priors was conducted. Main effects of method, scheme, patterns of missing yields, and starting day of recording postpartum and two way interactions between scheme and shape of lactation curve, interactions between scheme and starting day of recording and, interactions between scheme and pattern of missing test-day records were significant (P < .0001). When no yields were missing, the overall predic- tion bias for all methods was generally small. Within shape of lactation curve, variability in the accuracy of the methods was evident. With missing test-day records and varying starting days of recording, some methods had smaller bias than others. 8. 2 Introduction Test-day records in a lactation are the basic unit of information on a cow’s production for management decisions and genetic evaluation. Lactation totals are calculated from information collected on testdays. The test-day yields are most commonly recorded at monthly intervals, which are then used to estimate a cow’s lactation yield over a standard lactation period, and the 72 73 convention is to use 305 days. Some of the officially approved testing schemes by the Inter- national Committee of Animal Recording (ICAR) are A4, A6, and A8 where the A denotes testing schemes in which data collection, handling, and transferring is done by authorized technicians at 4, 6, and 8 wk intervals. To calculate 305-d yield, typically, a continuous lactation curve is simulated by linear interpolation between test-d records. This approach is weakened by non-linearity in the shape of the lactation’curve at the beginning, peak, and end of lactation. To account for this weak- ness, various methods have been developed (Wood, 1972; Schaeffer et al., 1977; Grossman et al., 1986; Shook et al., 1980; Wilmink, 1987). The projection of incomplete lactations to 305 d also provides useful information for both management and genetic evaluation. Factors have been estimated that can be used to extend partial lactation records (Batra and Lee, 1985; Keown et al., 1986; Wiggans and Van Vleck, 1979; Wiggans, 1981; Wiggans, 1986). Most of these factors were estimated from empirical relationships between cumulative and last test-d yields and lactations from animals with complete lactation records (Shook et al., 1980, Palmer et al., 1994), but some used factors calculated from lactation curves (Schaeffer et al., 1977; Wilmink, 1987). Different methods for computing lactation totals are being used in different regions and countries. These methods are at various levels of sophistication and produce estimates of lactation totals with different degrees of precision and accuracy. With the increased need for pooling data bases and the tremendous growth in the exchange of germplasm between countries, a set of standard methods is needed. Such methods must be flexible enough to suit different shapes of lactation curves due to breed and management differences, different intervals of sampling and different rules of screening records in different schemes, different patterns of missing test-day records, different starting days of milk recording, and different production traits. To evaluate, summarize, and compare the precision and accuracy of current methods, a “lactation Computation” working group, consisting of 12 member countries, was established by ICAR. This study was commissioned by ICAR to establish the relative preci- 74' sion and accuracy of methods for computing 305-d lactation yields. Twelve countries partici- pated in the computation of lactation totals. 8.3 Materials and methods A test-day data set on‘ standard lactation curves (SIAC) was generated. 8.3.1 Design of SLAC A total of 40 parameter curves were designed for 10 curves of different shapes for mean daily yield in a lactation by 4 curVes describing variance of daily yield throughout a lactation. For each of the parameter curves, 90 sample curves were simulated to generate a total of 3,600 sample curves. True lactation yield of the sample curves were calculated. Four test schemes of Al , A4, A6, and A8 were applied to each of the sample curves. Only one starting day postpar- tum (7 d) was considered for scheme A1, but four starting days (7, 14, 21, and 28 d) were considered for the other three schemes. For each of the simulated curves under a specific test scheme with a specified starting day postpartum, up to six patterns of missing test-d yields (none, early, early peak, late peak, late, and tail-end of lactation) were established. After applica- tion of different test schemes with different starting days and patterns of missing test-d yields on the 3600 sample curves, there were test-d records on a total of 136,800 lactations. Mean daily morning and the corresponding evening yield curves were designed for each of two mean daily total yield curves, for each of which two variance curves were designed. Again, 90 sample curves were generated for each of the four parameter curves to give a total of 360 sample curves. An AP/4 test scheme was then applied to these curves by taking alternate morning and evening test-day yields from one month to the next. After the application of the four starting test-days postpartum and patterns of missing test-day records, there were AP/ 4 records on a total of 3,960 lactations. The combined total number of lactations was now 1 40,760. Each simulated lactation was coded for one trait, one breed, one of two parities, and one of 75 four seasons of calving to accommodate the application of different methods for calculating lactation totals. Thus, each of 140,760 lactatiOns was duplicated 8 times to give a total of 1,126,080 lactations. The SIAC curves were simulated using Mathematica (Wolfram, 1988). 8.3.2 Methods for calculdting 305-d lactation yield Six methods were compared: 1)Test interval method (TlM-Al: Wiggans, 1985; Wiggans and Dickinson, 1985) with prediction of missing test-day yields and 305-d projections using adjustment factors for parity, and season effects. No lactation totals were computed for lactations in scheme A6 and A8, and missing pattern of ‘late lactation’; 2) Linear interpolation using standard curves (ISC: Wilmink, 1987; Mimeo from Wilmink and Ouweltjes, March 1991, NAS-report, 91 0355/WO/ CA, Netherlands Royal Syndicate, AL Arhem) with prediction of missing test-day yields and 305-d projections using standard yield curves; 3) Centering date method (CDM: Letter from O. K. Hansen, Danish Agric. Adv. Centre, Aarhus on 9/7/94) with no prediction of missing test-day yields but 305-d projections using adjustment factors for parity, and season effects. No lactation totals were computed for scheme AP/4; 4) Test interval method (TIM-U1: Appendix A of Int. Comm. of Anim. Recording Agree- ment) with an estimate for a single missing test-d yield and 305-d projections for lactations that were at least 300d but with no adjustments for parity or season effects. Interval credits for scheme AP/ 4 were calculated after doubling the sample test-d yields; 5) Test interval method (TIM-U2: Letter from Nicole Bouloc, Institut de l’elevage, Paris, France, on 1 1/ 8/ 94) with no adjustments for parity and season effects and no corrections for missing test-d yield. Interval credits for the last interval computed as, ((Test-d yield + previous yield)/2 x interval length) + (test-d yield x c) where c = 14 for schemes Al, A4, AP/ 4, and A6 or c I 28 for scheme A8. No lactation totals 76 were computed for incomplete lactations; 6) Multiple trait projection (MTP: Trus and Buttazzoni, 1991; Mimeo from D. Trus, Ottawa, Canada, on 8/ 1 5/ 94) method with prediction of missing test-day yields and 305-d projections using expectations of parity and season sub-class means, and error estimates from the residual covariances among intervals (15). Lactation totals were computed for only schemes A4 and A6. 8.3 .3 Statistical analyses Bias in calculated lactation total yield from each of the above methods was expressed as bias : (actual yield - estimated yield). Thus, a positive mean bias was synonymous with overestimation while negative bias was indicative of underestimation. For purposes of analysis, biases were stratified into three data sets in a manner that was consistent with the SIAC design: 1) Data set A - For general inferences about the methods, biases from lactation records with starting d of 7 d postpartum and no missing test-d yields were used. Using data set A, crude means for bias were computed within each test scheme. Within shape of mean curve, sources of variation in bias were analyzed by the following linear models: qulmn : H + S + M, + Vk + P1“) + Tm(j) + MVa + SM”- + SVlk + Spam + BTW) + 8.,k1mn [1 a] y is bias (kg); Ll is a constant common to all observations; g is the fixed effect of I] run I where Y scheme i with i:l, 2, ..., 4, with 1 3 Al, 2 I A4, 3 1 A6, 4 I A8; M, is fixed effect of method j with j I 1, 2, ..., 6; VI: is fixed effect ofvariance curve k with k = 1, 2, 3, 4; PI is the fixed effect of parity 1 with l t 1, ..., 4 with 1 ‘—' first, 2 : second, 3 I second and later, 4 Z not used; Tm is the fixed effect of season of calving m with m : 1, ..., 7 with 1 = December to Febru- ary, 2 1 December to January, 3 : March to May, 4 = April to May, 5 -‘-’ june to August, 6 : September to November, 7 :— not used; Fwand Tm“) are nested effects and 77 MVfl“ SM,,~. SV.k, Spam. STm j) are interaction effects between the respective factors; Bijklmn is the random residual error distributed as N(0, I oi) , where 0'3 was assumed to be homoge- neous across all groups and zero covariances between groups. All other interactions were assumed to be negligible. Vain... '5 N ‘*' M) + Vk + Pm) + Tmm + MVa + 8mm [28] where y] m" is bias (kg) for scheme AP/ 4; all terms are as defined in [la]. 2) Data set B - For inferences regarding missing test-d yields, biases from all lactation records for scheme A1, but only lactations with postpartum starting d of 21 d for schemes A4, A6, A8, and AP/ 4. Within each shape of mean curve, sources of variation in bias were analyzed by the following linear models: Yuumq, = H +51 + Mj+ Vk + Pup'l' Tmm+ E, + MVa + SM“- + Sl’nm‘t STmm lb + SFiq + aijklmqr [ l where y is bias (kg); E, is fixed effect of pattern of missing test-d yield, q = 1, 2, ..., 6, ijklmqr with 1 = None, 2 1 early lactation, 3 —‘ early peak, 4 : late peak, 5 = late lactation, 6 = tail end; SF.q is an interaction effect between the respective factors. All other terms are as defined in [1 a]. Yaqu, : l1 + M,‘ + Vk + Pm) +Tm(j) + E, + MV,i +8,;d,,,q, [2b] where meqr is bias (kg) for scheme AP/ 4. All other terms are as defined in [2a]. 3) Data set C - For inferences about starting day postpartum of recording, biases from lacta- tion records for schemes A4, A6, A8, and AP/ 4 with no missing test-d yields were used. Within each method, sources ofvariation in bias were analyzed by the following linear mod- els: Yljklmno : H + Si ‘1” M, + Vk + Fun + Tmm + Dn + MV,k + SMtJ + SP.1(,)+ STmm + SDm + 8.,ldmno 11c} where y H is bias (kg); D is fixed effect of starting day postpartum, n11, 2, 3, 4 with 1 t I] mno n 7d, 2 14d, 3 = 21d, 4 28d postpartum; SD is an interaction effect between the respective .78 factors. All other terms are as defined in [1a]. demm, = l1 + Mj + Vk + Pm) + Tm) + Du + Mvjk + Sjldmno lZCl where yjldmno is bias (kg) for scheme AP/ 4. All other terms as defined in [3a]. To further delineate between the methods, principal factor analysis (Harman, 1960) using squared multiple correlations for theprior communality estimates was conducted across and within the schemes. Kaiser’s measure of sampling adequacy (MSA) is a summary, for each variable and for all variables, of how much smaller the partial correlations are than the simple correlations (Cemy and Kaiser, 1977). Kaiser’s MSA was computed (Cemy and Kaiser, 1977); values of .8 and .9 were considered very good, while MSA below .5 suggested sampling inadequacy. Orthogonal varimax prerotation (Harman, 1960) followed by oblique Procrustean rotation (Hendrickson and White, 1964) of the initial factor matrix was carried out. Harris- Kaiser’s rotation (Harris and Kaiser, 1964) with Cureton-Mulaik weights (Cureton and Mulaik, 1975) was used to obtain an independent cluster oblique solution to the simple structure of the factors. All statistical analyses were conducted using SAS (SAS, 1992). 8.4 Results and discussion 8.4.1 Biases of different methods associated with test schemes on daily yield The crude mean, standard deviation, standard error, mean : standard deviation, and coefficient of variation of bias by method of estimating lactation totals and schemes A1 and A4 are shown in Table 9 while data for schemes A6 and A8 are summarized in Table 10. For all methods, the mean bias did not increase with decreased sampling frequency. However, the standard deviation and standard error was higher under scheme A8 when compared with scheme A1 (P < .0001). The magnitude of mean : standard deviation followed the sampling frequency and was a reliable measure for ranking methods within each test scheme. The mean for TIM-A1 under schemes A4 and A8 was similar but larger than that under scheme Al; the mean under scheme A6 was smallest (P < .0001). The mean bias for ISC and TIM-L72 79 increased with decreased sampling frequency. However, the mean bias for TIM-L72 was smaller under scheme A8. The MTP method also showed an increase in the mean under scheme A6 when compared with scheme A4 (P < .0001). The mean for TIM~U1 under schemes Al and A4 were close but increased under schemes A6 and A8. This reduced accuracy was attributed to the increase in the interval between recording. TABLE 9. Mean, standard deviau'on, standard error, coefficient of variation, and mean : standard deviation of bias by method of estimating lactation totals and schemes A1 and A4. Scheme A1 A4 Method 32 SD 2 /SD SE cv X so )"t /59 SE CV __ (kg) _ (kg) % __ (kg) __ (kg) % nMA1 53.4 29.2 1.8 .2 54.7 120.5 61.6 2.0 .4 51.1 15c -8.8 28.1 -.3 .1 319.5 11.5 61.9 .2 .3 535.8 CDM .101 28.2 -.4 .2 279.4 8.7 61.4 .1 .4 706.2 TIM-Ul -101 28.2 -.4 .2 279.4 8.7 61.4 .1 .4 706.2 MTP 43.6 120.8 .4 .7 277.2 T1M-u2 29.3 67.9 ..4 .2 231.4 30.9 140.9 .2 .3 455.3 CDM = Centering date method, ISC = interpolation with standard curves, MTP = multiple trait projection, TIM-A1 = test interval method, TIM-Ul = TIM without adjustments for fixed effects, and TIM-U1 = TIM without adjustments for fixed effects and no corrections for missing testd yield. 80 TABLE 10. Mean, standard deviation, standard error, coefficient of variation, and mean : standard deviation of bias by method of estimating lactation totals and schemes A6 and A8. Scheme A6 A8 Method X SD )’( /SD SE CV K . SD )-(/SD SE CV . __ (kg) __ ' (kg) - o/o __ (kg) __ (k8) % TIM 32.5 83.9 .4 .5 258.2 61.4 112.5 .5 .7 183.1. ISC 63.9 87.8 .7 .4 137.6 89.2 120.9 .7 .5 135.5 CDM - 70.9 87.2 .8 .5 123.1 106.4 120.6 .9 .7 113.4 TIM-U1 70.9 87.2 .8 .5 123.1 106.4 120.6 .9 .7 113.4 MTP 63.8 115.0 .6 .7 180.0 TIM-U), 259.0 206.4 1.3 .4 79.7 178.1 282.2 .6 .6 158.5 CDM = Centering date method, ISC = interpolation with standard carves, MTP = multiple trait projection, TIM- A1 = test interval method, TIM-U1 = TIM without adjustments for fixed effects, and TIM-U1 = TIM without adjustments for fixed effects and no corrections for missing testd yield. The CDM and TIM-U1 gave equivalent results because, within scheme, the interval between recording was fixed. Agreement between the centering date and test interval methods was also observed by Sargent et a1. (1968). The increase in standard error with length of testing interval is comparable with the results of Erb et al. (1952) and Anderson et al. (1989). The considerably high coefficient of variation signified that the prediction error was due to numer- ous factors and that further partitioning of the mean bias was necessary. MLlfljestdaLrecmdamilable Within method, from model [1 a], main effects and interactions between scheme and shape of mean curve were significant (P < .0001). In this and subsequent analyses, the main effect of variance curve was not significant (P > .1 ). The fixed effects of parity and season of calving were assigned to lactation curves in order to: l) accomodate computing algorithms for the 81 various methods; 2) facilitate computation of lactation totals. The bias introduced by these effects was removed by consideration of these effects in the linear models. The least squares mean bias by method, scheme and shape of mean curve is shown in Figure 17. From this and other models, standard errors of least squares means ranged from .1 to .85 kg. Hence, least'squares means that were not within 2 standard errors of each other were consideredito be significantly different (P < .0001). Observe that, in this and all charts (showing adjusted mean bias, scales for plots vary from scheme to scheme. Under scheme A1, all methods, with the exception of TIM-A1 , tended to overestimate lactation yields for all shapes of the lactation curve with the exception of shapes 6 and 7. Estimation of lactation yields by TIM-A1 had the lowest bias for shape 1. In comparison with other methods, TIM- Al had the largest absolute bias for all shapes except shape 1. The CDM, TIM-U1, and TIM- L72 performed best for shape 6 and 7 while the ISC approach gave the lowest bias for shape 7. The results for scheme A4 indicate a tendency towards underestimation by all methods for most of the shapes except the MTP method which overestimated yields for shapes 2, 8, and 9. The ISC, CDM, and TIM-U1 tended to overestimate yields for shape 9. The MTP, TIM-A1, and TIM-U2 gave equivalent results for shapes 3 and 10. The MTP approach did best for shapes 1 and 4 but was not accurate for shape 7. The ISC, CDM, and TIM-L31 were similar and gave consistently lower biases across all shapes. A remarkable improvement in TIM-A1, relative to other methods, was observed from scheme A4 to A6. The bias for TIM-A1 was relatively low as compared to the other methods. All methods tended to underestimate yields for all shapes except shapes 6 and 7. The ISC, CDM, TIM-L11, and TIM-L12 were similar. Similar relationships between the methods were observed for scheme A8 which was in agreement with the crude statistics in Table 10 where the biases were much larger for scheme A8. The TIM-A1 had relatively lower absolute bias than all other methods. 82 A I mod 1 A1 j ' l 4 [EZTIM A1 DISC .wzeL—JTIM Ul .MTP CITIMU I Method ‘ _ . _ so ZJTNA‘I WISC .CDM CITN'U1 .mrUZ‘ m ———r————:-- -—_;_—T—i—T._—:u ——-i T ,1 j . . . . . . . . . . , . . . . ; 1 LN LS moan (kg) 1.8 mean (kg) A 1 1 A 1 A A ‘ 1 _2m . 1 ; l g 1 1 L . . ‘1 2 3 4 5 6 7 a 9 10 1 2 3 4 5 s 7 a 9 10 ' ~ f\ N Shape at mean curve Shape 0‘ M curve A6 M I T A8 EDI-A also I 6311 m unlwu DISC ICON Emu—U1 IMTP :1rmvuzLl 300 ‘ 0°“ “‘9‘ - '02] LS moan (kg) 8 § 0 400 '1 : . 1 ' 1 2 3 4 5 6 7 a 9 10 (1\r2\ra\&ri~\:J/r<&r\11 f\r\’\’\’\\/’\/\M Shapoofnmncuwe Shapodmoancurve Figure 17. Least squares mean bias by method, shape of lactation curve and test scheme A1, A4, A6, and A8. The total variation in the TIM, ISC, CDM, TIM-U1, and TIM-U2 was explained by at least two factors. Plots of the reference structure of the predominant factors for the methods within test schemes A1, A4, A6, and A8 are shown in Figure 18. A distance of 5mm was used as a measure of close proximity between methods. Methods within 3mm of each other were considered to have formed a cluster. The ISC and TIM-A1 formed a cluster which was close to CDM, TIM-U1, and TIM-U2 under scheme A1. However, ISC, TIM-A1, CDM, and TIM-U1 formed a cluster which was close to TIM-U2 but distant from MTP under scheme A4. For schemes A6, CDM, ISC, TIM-U1 and TIM-U2 clustered together and were close to TIM-A1. The MTP and TIM-A1 were close. Therefore estimation of lactation yields by CDM, 83 ISC, TIM-U1, and TIM-U2 was expected to give results that were similar and close to those given by TIM-A1 for scheme A6. The data for scheme A8 showed a similar‘relationship between the methods. However, TIM-A1 was isolated from the other methods. The MSA’s ranged between 7.67 and .91 implying that the data were suitable for the factor analysis. Results from factor analysis were in agreement with those from the linear model analysis. Although, the data were inadequate for factor analysis within scheme and shape of mean curves, this analysis added strength to interpretations by considering the (co)variance structure of methods. 1 1 1 I F 0.8 » l F u » j x ' as » ' os » f c . c . . l N L 1 0.4 » ° 11.2 L . l 0 0.2 » r o 4 2 I ' o 1 -oz » j 1 -o.2 » .06 > j 41.; » -o.s , .05 > ' -o.a r 4).- r j -1 ‘ -1 -1 -0.8 ~06 -04 ~02 0 02 04 0.6 08 1 -1 08 416 -O,4 -0.2 O 02 0.4 06 0.8 1 Futon . _ FoctorZ . rum . u . Lu- van-7113:. L. van . E( . (w 'Dm ‘ an x 1.1; 1 1 0! * l 08 L 0.6 8 1 i :2: . '~ J :3: j” . j l l 1 1 ROI-0"“ .9 ... qo—OI'H ‘ 412 » 4.4 L -°.‘ r 4). L . 1‘» u 1 -1 - -1 -08 -05 -0.4 -0.2 O 0.2 04 0.6 0.8 1 -1 -08 ‘06 -0.4 -0.2 0.2 04 05 0.8 FIC‘U 2 Factor 2 l - 'UM . nit Q in mg: ' an x— W.:._J : - mm . .—‘.( - . [Pg—rum x:~_:r_ ' Figure 18. Reference structure for weighted Harris-Kaiser rotation of factors for test scheme A4 and test scheme A6. Procrustean rotation of factors for scheme A8. Within method, from model [1b], the main effects and interactions between schemes and patterns of missing test-d records were significant (P < .0001). The least squares mean bias by method, scheme, and pattern of missing test-d records is shown in Figure 19. When all records were available, all methods were accurate under the different recording schemes 84 except TIM-A1, TIM-L11 and TIM-L2 in scheme A6. This reduced accuracy was attributed to a possible starting day by scheme interaction. For scheme Al, the accuracy of the ISC method was consistently high for all patterns of missing. data. The data for TIM-A1 were comparable to ISC expect when records were missing towards the later part of lactation. Therefore, the ISC method was more accurate than TIM-Al for the incomplete lactations. Biases for CDM were large when records were missing in all but the late lactation and tail end stages of lacta- tion. The TIM-U1 was not accurate in all patterns and grossly underestimated lactation yields for incomplete lactations. The TIM-L72 was comparable to ISC when records were missing during early and peak lactation. I 1 As the sampling frequency decreased from A1 to A8, biases increased but the relative accuracy between methods remained essentially the same. Biases in CDM and TIM-U1 became very exaggerated. The MTP method was relative more accurate than TIM-A1 when test-d records were missing during lactation but was comparable to ISC for incomplete lacta- tions in schemes A4 and A6. The TIM-L12 had lower biases than TIM-U1. Biases for TIM- Ul were low when test-d records were missing during early peak under scheme A4 and A8, and during late peak lactation under scheme A8. This implied that estimating missing yields using the average of the previous and next test-day yield improved the accuracy of TIM-L11 in schemes A4 and A8. 85 A1 l .— W; A4 ' Matted: .‘BF'LAL 45‘? 992'! :JN'U‘ -1I'MU3; ITlM-A1 CISC Icon 121mm um crmwa r ...a r - 2"” f 2000’ ... 2000:» . 9 E a 1500, g 1000? ~ 2 1°00» 3 t l m 1 5a);- ' ' _. soo, -5°° a. m.‘-‘~‘_~ :- 1.x... mimics}; 'n...‘ ‘ * 4...‘ w w I... w "Mdmmdm Worn-bemoan ~ ‘nwu DISC ICON DM1 .m-uz i . tanum also .0004 3M1 _-_1_1m>_:_;r1u ___4 .5 aooo -— — 1 a; 1500; l 2 1000' S i 1 ’ o ; . E 1000:» B 500 3 : . m9:— ’ 01> o; L. E 3 : 1 l l : '5‘» 1 . _1_ a L x 1 . 1 A L ; 2 A __ None Enny MML‘OM we Talend Non. Em. E-~ you t. a“ Lab 1‘ Ln.) Wham of M data Pattern at "tossing data Figure 19. Least squares mean bias by method, pattern of missing test-d records and test scheme A1, scheme A4, scheme A6, and scheme A8. Plots of the reference structure of the important factors for methods by scheme, for lactations with missing test-d records, are shown in Figure 20. Under scheme A1, all the methods, except ISC and TIM-U 2 were apart from each other. The furthest distance was between CDM and TIM-A1. The plot for scheme A4 showed that TIM-A1 was close to both MTP and ISC; ISC was close to TIM-A1 and TIM-U2; TIM-U2 was distant from MTP. For scheme A6, the TIM-A1 , MTP, and TIM-U2 methods formed a cluster that was close in proximity to ISC. The CDM and TIM-U1 were distant from this cluster. Therefore, these methods were not expected to yield equivalent results under schemes A6 and A8. The MSA’s ranged between .6 for scheme A8 and .9 for schemes A4 and A6. Hence, the data for scheme A8 were suitable for the factor analysis. 86 A1 "A4 1 j 1 4 l - ‘1 F 03* j 08 T ; 2 06» I Z 06» l 1 04+ 1 1' . j c 04 r " l I . 0 0.2 » 1 X 3 t 0.21" x r 0 - T , ‘r’ o e l a 1 -02 4 , ' . A .02 . ‘l .1 -04 t 1 .04 . l j -06 ‘ ' 1 -06- l 1 438L I j ~08» J ' _, a l . - J _, - . ' -1 -O.8 -0.6 -0.4 -0.2 . O 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -O.4 -O.2 O 0.2 0.4 0.6 0.8 1 FactorZ Factor2 l | Tin—M 0 ISC ‘ c0111 9 mun x 7111:1102 1' “W“ 0 1‘4» A170“ 0 W91 X W" X "W? 1 T j 1 fi 08 » ' F 08+ '9: T F 06 - 0 '0 l ‘ a 06 » 1 a I C | c 04 ' ( 0.4 t t 0.2+ ° 02 » ° 0 . s a r o 4 f ‘ l ' 0.2» ' , 1 412 ~ ‘ 1 04» ' 04- ‘05 ’ . ‘ -06L 1 08» i 1 cal I _1 a 4 _ - l -1 -O.8 -0.6 -O.4 -0.2 O 0.2 0.4 0.6 0.8 1 -1 -0.8 «0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Factorz FW 2 L. hum . 1st ‘ COM O 'M--1 x MTP x tun/1;} l I TIM-Ai . ISC ‘ CDM . “”1 x Tin-1J3: Figure 20. Reference structure for weighted Harris-Kaiser rotation of factors for test scheme A1, scheme A4, scheme A6, and scheme A8. 8:14me dayofrecordingpostpartum Within method, from model [1 c], the main effects and interactions between schemes and starting day of recording were significant (P < .0001). The least squares mean bias by method, scheme and starting day of recording is shown in Figure 21. For scheme A4, all methods performed well. The ISC, CDM, and TIM-L72 showed an increase in overestimation as the 87 duration to the starting day for recording was increased from 7 d to 28 d postpartum. Biases for CDM and TIMeUl had opposite signs and also were largest when the starting day of recording was 28 d postpartum. Erb et al. (1952) also found that the day of the initial super- vised test influenced the accuracy of lactation records. The data for scheme A6 show similar trends for ISC and CDM. Biases for TIM-Al and TIM-Ullwere large for all Starting days except for 7; biases for TIM-A1 were much smaller. The accuracy of TIM-U2 improved as the duration to the initial tested increased from 14 d to '28 d. The bias for MTP was small and relatively stable but decreased when the starting day of recording was 28 d postpartum. For scheme A8, all methods performed well for all starting days except for 28 d postpar- tum where biases for TIM-A1, TIMeUl, and TIM-L12 increased with the latter grossly under- estimating lactation yields. A4 Method: A Method: ”TIM- A1 DISC .0”! ClTlM- U1 INT? CTIM- U21 LBTIM- -A1 DISC ICON CITIM U-1 IMTP CDTIM- U2 soo . f 1&0 . . ‘00 : aoo : 200 - 1 100 :- E I “IN L 0E ;‘ _2m : 1 4‘ L 1 l i L _mi L 1 l i 1 L 1 7d 14d 21d 28d 76 14d 21!! 28d Postpamam starting day 0! recording Postpartum starting day at recording A82“) Method: ' EZTlM-M DISC .00“ Dm-Ut ITN-UZ LSmoan (kg) 21d 28:! Postpatum starting day 01 record‘ng Figure 21 . Least squares mean bias by method, starting day of recording and scheme A4, scheme A6, and scheme A8. 88 A4 1 1 06 "04> 02> O ‘ at. -02t_ I -O4> ' -06 e —0.8 fiOHON'fl —. .1. ~———-— - ._ .1 . 5 -1 -08 -06 -04 -O2 0 0.2 0.4 0.6 08 Factor2 l'mr Orsc ‘cm .TMJ‘ flaw ij i 1 . 1 08 » I l 0,8 » F 06 » j 06 » a t 02 > 0.2 t . O 0 ‘ 0 e A. r x 02 » { oz » 1 0.4 . 0.4 * ~06 » -06 l -0.8 r ~08 . -1 - - - * -1 1 4 -1 -O.8 -06 -O.4 -O.2 O 02 0.4 06 08 1 -1 -08 -O.6 -04 -O.2 0 02 0.4 06 0.8 1 Factor2 Factor2 I rum . use ‘ tau Q tu X um X Tuj I rm“ . 1‘51: ‘ (30M 0 Tim.” x tMth Figure 22. Reference structure for weighted Harris-Kaiser rotation of factors for test scheme A4, scheme A6, scheme A8. Plots of the reference structure of the important factors for methods by scheme, when starting days of recording were allowed to vary, are shown in Figure 22. Under scheme A4, the CDM and ISC formed a cluster that was close to TIM-A1; TIM-A1 was close to TIM-L12. All other methods were isolated. The plot for scheme A6 showed a cluster between CDM, ISC, TIM-L11, and TIM-U2 which was distant from MTP and TIM-A1. For scheme A8, the cluster was between CDM, ISC, TIM-U1, and TIM-L12; TIM-A1 was close TIM-U1. The MSA’s ranged between .5 for scheme A8, and .8 for schemes A4 and A6. Therefore, the data for scheme A8 were marginally suitable for the factor analysis. 89 8.4.2 Biases of different methods associated with the AP/4 test scheme The crude mean, standard deviation, standard error, coefficient of variation, and mean : standard deviation of bias by method of estimating lactation totals for scheme AP/ 4 are shown in Table 1 1. The ISC method had the smallest bias while TIM-Al had the largest bias but all biases and mean : standard deviation were much larger than those for scheme A4 shown in Table 9‘. However, the coefficient of variation for TIM-A1 was relatively low. The variance for TIM-U1 was small as compared to other methods. The mean : standard deviation were smallest for ISC and TIM-L12. Standard errors 'were similar but larger than those for other testing schemes shown in Table 9 and 10. TABLE 11. Mean, standard deviation, standard error, coefficient of variation, mean : standard deviation of bias by method of estimating lactation totals and schemes AP/ 4. Method 2 SD X/so SE CV ____ (kg) L___- (kg) % TIM 222.1 87.1 2.5 1.6 39.2 ISC 25.0 86.1 .3 1.6 344.4 TIM-L11 34.9 9.5 3.7 1.7 256.3 TIM-L12 -42.4 196.3 .2 1.2 463.2 ISC = Interpolation with standard curves, TIM-Al = test interval method, TIM-U1 = TIM without adjustments for fixed effects, and TIM-U1 = TIM without adjustments for fixed effects and no corrections for missing tested yield. WW From model [2a], the effect of shape of mean curve was significant (P < .0001). Least Squares mean bias by method and shape of mean curves is shown in Figure 23A. The biases 90 for TIM-Al, ISC, and TIM-L72 were invariant to the shapes of the underlying lactation curves. However, TIM-U1 had a larger bias under shape 11. This difference was attributed to more variability around peak and postpeak yields for shape 11. A ' Method. 1 [Irv-A1 :Jlsc ITlM-Ut Dru-02J 250 1 200 a a, E a) .J Shape of mean curve B ~————......,+--- -- m... amt A1 2150 ITIM-Ut ::T1M 0 0.2 i 0 0.2 * r o m L ”A a . - 102, y 143.2 » .o.4i .04» ‘ .05, l .05» l y 418* I -o.a» l I - a - l A - n - J - . A . 1 . A a . l -1 -0.8 -0.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 —0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 FactorZ Factor2 Ll TIM-A1 I Isc o TIM-U1 'Xllyg [ I TIM-A1 0 {SC 0 TIM-Q1 x mfg Figure 24. Reference structure for weighted Harris-Kaiser rotation of factors for scheme AP/ 4 and method for lactations with no missing data(A), lactations with missing data (B), and lactations with varying starting day of recording (C). MLZMssing test-day records From model [2b], the effect of missing test-d records was significant (P < .0001). Least squares mean bias by method and pattem of missing test-d records is shown in Figure 238. 92 for SLAC. A1 r Method: i {also (SLAC 1)_ EISC (SLAC 2) . 10 , 3 ; g f s 4 A , o i 71/ 2 a I c '5 i as I ° . E -10 - 0') .1 - -15 -20 -25 gl 1 L i a I . A ; L4 ‘ l E i i i 1 2 3 4 5 6 7 8 9 A 10 (\mmmmxfi'xmm Shape of mean curve Figure 25. Least squares mean bias by method, data set, and shape of mean curve for scheme A1. Contrary to the data for scheme A4, the ISC method had considerably large biases for incomplete lactations. The bias for TIM-Al was largest when testd yields were missing at the tail end of lactations. The largest biases for TIM—U1 occurred when data were missing early and peak periods of a lactation. During those periods, the performance of TIM-U1 was inferior to TIM-A1 and ISC. For incomplete lactations, the performance of TIM-U1 was comparable to TIM. The accuracy of TIM-U2 was highest when testod yields were missing in early lactation but was lowest when data were missing during early peak lactation. A plot of the reference structure for the important factors is shown in Figure 24B. This plot depicted close proximity between ISC and TIM-UZ. All other methods were distant. The overall MSA was .65 which implied that the data were well-suited for factor analysis. 93 From model [2c], the effect of starting day of recording was significant (P < .0001). Least squares mean bias by method and starting day of recording is shown in Figure 23C. Biases for TIM-A1 were large when the starting day of recording was 7d but appeared to decline as the starting day occurred later in lactation. A plot of the reference structure for the important factors is shown in Figure 24C. The TIM-Al and TIM-C2 were close to each other. All other methods were distant from each other indicating a lack of equivalence between them. The overall MSA of .7 implied that these data were marginally favorable for the factor analysis. 8.4.3 Validation of sampling design To examine if the performance of methods was caused by the sampling design of SLAC, a different data set (SLAC 2) was created as defined in §8.3.1. However, values for the variance curves used in creating SIAC 2 were twice those shown in Figure 15. Thus, test-day yields for lactations had more day to day variation. The ISC method was used to compute lactation totals for SIAC 2. Scheme A1 results for SIAC 1 and SLAC 2 are shown in Figure 25. The biases for both data sets were very similar. This consistency established that the performance of methods was not due to sampling design 8.5 Conclusions Generation of SIAC enabled the examination of the relative accuracy of different meth- ods for calculating lactation totals. The shapes of mean curves were designed to cover factors that affect milk yield in a lactation. The patterns of missing test-d records, starting days of recording, and testing schemes were representative of typical situations. The relative accuracy was examined under different shapes of mean curve, patterns of missing test-d records, and starting days of recording. Methods for computing lactation yields are many and some are currently being modified. The methods compared in this study were considered to be fairly 94 representative of different approaches. The sampling frequency did not affect the mean bias but increased the standard errors. The accuracy of methods for scheme AP/ 4 was lower when compared with scheme A4. When all test-d records are available, and when the duration to the initial test was 7d, all the methods performed well. The accuracy of these methods dif- fered according to underlying shapes of the lactation curve. In general the interpolation with standard curves, centering date, multiple trait projection were more accurate than the other methods. While the adjustment factors used with the test interval method appeared to under- estimate lactation yields in scheme Al, the method produced small biases for lactations with missing test-d records. The centering date and test interval methods were considered suitable for recording schemes that exclude lactations with missing test-d yields. When the starting day of recording was delayed to 28 d postpartum, the accuracy of the test interval methods without adjustment for fixed factors was reduced. The shapes of the lactation curves in this data set were known apriori. Thus, implementation of the interpolation with standard curves methods was easy. In most situations, information about the underlying shapes is not readily available. Therefore, the benefit of using this method should be considered after a careful review of its merit and the availability of such data. In order for a data processing center to adopt a given method, shapes of lactation curves, patterns of missing test-d records, and starting days of recording postpartum inherent in the data need to be considered. The SIAC enabled the International Committee of Animal Recording to evaluate the accuracy of any method for computing lactation totals. From these evaluations, standards for recognizing methods as official were established. 9. SUMMARY Sampling variations in mean daily milk yield were investigated using bootstrap resampling. The bootstrap confidence intervals of mean daily yield were accurate and consistent with the theory. The variation in yields was highest at the peak and postpeak stages of lactation. It was established that for a given mean and variance curve, sample lactation curves could be gener- ated. Variation in moming and evening yields of I lolstein cows was studied. Mean moming yield was higher than evening yield. The variance of evening yield appeared to determine the variance of total daily yield. The variance was highest at peak lactation of morning and evening yields. Regional differences in the magnitude of variance of daily total, morning, and evening yield were observed. These results provided a preliminary understanding of the shapes of curves for mean milk yield and its variance. A set of curves for mean and variance of total, morning, and evening yields that was representative of the global cattle population was established. Mathematical functions were fitted to these curves. A data set of standard lactation curves (SIAC) was generated using these functions. Thus lactation records of SLAC were biologically consistent and accomodated different missing test-day records, starting days of recording, parities, breeds, seasons of calving, ages at calving, regions, and other grouping effects. The SIAC was sent to six participating data processing centers. Each center calculated and returned lactation totals for SIAC. A total of six methods for calculating lactation totals were compared. When all test-d records were available, all methods performed well. The sampling frequency did not af‘ect the mean bias but increased the standard errors. Within shape of lactation curves, the accuracy of the methods varied. With missing test-d yields, some methods gave a lower bias than others. The SIAC was commissioned by ICAR to identify the sensitiv— ity of a given method to the various shapes of the lactation curve, patterns of missing test-d yields, and starting days of recording. 95 96 10. APPENDICES APPENDIX A: Illustration of parameter lactation curves for daily AM and PM yield using mean daily total yield curve 11 and variance curve 4 AM/PM MEAN CURVE 11 - VARIANCE CURVE 4: 40 . . - . 35* 30’ 1 N U1 AAAAAA‘A‘A‘AA Yield (Kg) N D 15; i 10. «j 5 i 1 0 A . . 1 so 100 150 200 250 300 Days PP AP/4 CURVES (AM/PM MEAN CURVE 11 - VARIANCE CURVE 4): 40 . r . . - . 35' 30- A25 b Yield (Kg 10. 50 100 150 200 250 300 Days PP no a E. .33 5. .E. _.< ”miwxom $0.502 Evans 9:81: mo mEBumm Co 50932;: uEQEO um 5Dmem< wo in? van ¢< "mmEmIUw mo w< "ms—mlow w< "ms—mzom APPENDIX C: Format for SLAC data files and sample printout of data (1) FILE 1 (slacl td.dat): Contains testday data from test scheme Al Number of records: 172,800 Type Length Position Numeric 2 1-2 Numeric l 3 Numeric 3 4-6 Numeric 1 7 Numeric 1 8-9 Numeric 1 10 Numeric 1 1 1 Numeric 1 1 2 Numeric 2 1314 " " 31-38 Numeric 3 39-41 Numeric 3 153.125 Numeric 3 1 26-1 28 Numeric 3 129-131 Numeric 3 152254 Description Mean curve No. (1, ..., 10) Variance curve No. (l , ...,4) Replicate No. (1, ...,120) Scheme (1=A1) Starting day (7'7d PP) Missing pattern (O‘none, 1=early lactation, 2=early peak, 3=late peak, 4'late, S'tail end) Combination code for levels of fixed effects (1 '> Lactation-1, breed-1, trait-1, season-1; 2 ‘> Lactation-1, breed-'1, trait=l , season-2; 3 => Lactau'onel, breed=1, trait=1, season-=3; 4 '> Lactation-1, breed'l , trait-1, season'4; 5 -> Lactation-2, breed-l , trait-1, season-1; 6 => Lactation-2, breed-1, traitul , season-2; 7 ‘> Lactationsl, breed=1, trait=1, season'3; 8 => 1actadon=2, breed-=1, trait=1, swsone‘i) Days PP at lst recording Days PP at 2nd recording Days PP at 14th recording Days PP at 15th recording Days PP at 43rd recording Testday yield at lst recording (divide by 10 to get yield, O-missing) Testday yield at 2nd recording( " ) Testday yield at 43rd recording ( ” ) ILLUSTRATION OF SLAC FILE 1 (slacl td.dat) with lst and last two days PP and corresponding test day yields shown Brstlrecotdmniilefllnformamdl: 11 11 701 714...294301228289 11 11 702 714...294301228289 11 11 703 714...294301228289 .. 294 301 1 1 1 1 7 0 1 7 14 . 1 1 1 1 7 0 2 7 14 ... 294 301 1 1 1 1 7 O 3 7 14 . 294 301 ..135118 ..135118 ..135118 22.8 28.9 ... 13.5 11.8 22.8 28.9 ... 13.5 11.8 22.8 28.9 ... 13.5 11.8 101 (2) FILE 2 (slacltddat): WWW Number of records: 316,800 Type Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Length Position F‘NHUHN w ’WNNN 1.2 3 4.6 7 8.9 10 11 12.13 14.15 1617 18-20 $6.41 42.44 45.47 7274 Description Mean curve No. (l, ..., 10) Variance curve No. (1 , ..., 4) Replicate No. (1, .... 120) Scheme (2'A4) Starting date (7'7d PP, 14-14d PP, 21 -21d PP, 28'28d PP) Missing pattern (0-none, l-early lactation, Z'airly peak, 3=late peak, 4=late, 5'- tail end) Combination code for levels of fixed effects (1 -> lactation-1 , breed-1, trait-1, season‘l; Z '> lactation'l, breed-‘1, trait-1, season-2; 3 e) lactation-1, breedel , trait=1, season-3; 4 "> lactation=1, breed=l , trait==l , season=4; 5 ="> Iactation=2, breedel , trait=l, seasonel; 6 a*> Lactation'Z, breed'I , trair'l, season=2; 7 '> Lactation-2, breed-1, trait=1, season=3; 8 => 1actation=2, breed-'1, trait=1, season=4) Days PP at lst recording Days PP at 2nd recording Days PP at 3rd recording Days PP at 4th recording Days PP at 11th recording (999 = Not applicable for scheme/starting date of recording) Testday yield at lst recording (divide by 10 to get yield, 0=missing) Testday yield at 2nd recording ( " ) Test-day yield at 11th recording (divide by 10 to get yield, 0=missing, 999 - Not applicable for scheme/ starting date of recording) ILLUSTRATION OF SIAC FILE 2 (slac2td.dat) with lst and last two days PP and corresponding test day yields shown Eirstlmordsomfilefllnforma trod): 11 12 701 735...259287228380 ...153131 11 12 702 735...259287228380...153131 11 12 703 735...259287228380 ...153131 EirsLLrecordsafteueadingiEmmattedl: 1 1 1 2 7 0 1 7 ... 259 287 22.8 38.0 ... 15.3 13.1 1 1 1 2 7 0 2 7 . 259287 22.8 38.0 15.3 13.1 1 1 1 2 7 0 3 7 . 259 287 22.8 38.0 ... 15.3 13.1 102 (3) FILE 3 (slac3td.dat): WW6 Number of records: 288,000 Type Length Position Numeric 2 1-2 Numeric 1 3 Numeric 3 4-6 Numeric 1 7 Numeric 2 8-9 Numeric l 10 Numeric I 1 l Numeric 2 12-13 Numeric 2 14-15 Numeric 3 16-18 Numeric 3 31.33 Numeric 3 34-36 Numeric 3 37-39 Numeric 3 55-57 Description Mean curve No. (1, ..., 10) Variance curve No. (1, ..., 4) Replicate No. (1, ..., 120) Scheme (3'A6) Starting date (7‘7d PP. 14'14d PP, 21 =21d PP, 28‘28d PP) Missing pattern (0*none, 1=early lactation, Z'early peak, 3=late peak, 4'late, 5' tail end) Combination code for levels of fixed effects (1 -> Lactation-1, breed=l , trait-l , season=1; 2 -> Lactation-l, breed=1, trait-l, season'Z; 3 -> lactation-1, breed-1, trait=l , season-'3; 4 '> lactation-'1, breed-1, trait-l , season‘4; 5 -> lactation-2, breed-1, traitel , season"1; 6 -> Lactation-2, breed=1, trait=l , season=2; 7 '> Iactarionr'Z, breed-1, trait-'1, season-=3; 8 -> Lactation-2, breed-1, trait=l , season'4) Days PP at lst recording Days PP at 2nd recording Days PP at 3rd recording Days PP at 8th recording (999= Not applicable for scheme/starting date of recording) Testday yield at lst recording (divide by 10 to get yield, 0°missing) Testday yield at 2nd recording( " ) Testday yield at 8th recording (divide by 10 to get yield. O'missing, 999 - Not applicable for scheme/starting date of recording) ILLUSTRATION OF SIAC FILE 3 (slac3td.dat) with lst and last two days PP and corresponding test day yields shown Eirstirecerdmfilclflnformattedl: 11 13 701 749...259301228379 11 13 702 749...259301228379 11 13 703 749...259301228379. imrdLaItechadingTEennatted): 49 ... 259 301 ... 259 301 49 ... 259 301 22.8 37.9 ... 22.8 37.9 ... 22.8 37.9 ... ..153118 ..153118 ..153118 15.3 11.8 15.3 11.8 15.3 11.8 103 (4) FILE 4 (slac4td.dat): Containuestdaadatafiemschemefl Number of records: 316,800 Type Length Position Numeric 2 1-2 Numeric 1 3 Numeric 3 4-6 Numeric 1 7 Numeric 2 8-9 Numeric 1 10 Numeric I 1 1 Numeric 2 1 2-1 3 Numeric 2 14-1 5 Numeric 3 16-18 Numeric 3 25-27 Numeric 3 28-30 Numeric 3 31 ~33 Numeric 3 43-45 Description Mean curve No.(1,...,10) Variance curve No. (1, ..., 4) Replicate No. (1, ..., 120) Scheme (4=A8) Starting date (7'7d PP, 14=l4d PP, 21 =21d PP, 28=28d PP) Missing pattern (O-none, 1-early lactation, 2=early peak, 3=lare peak, 4‘1ate, 5' tail end) Combination code for levels of fixed effects (1 -> lactation-1, breed-1, trait-1, season-1; 2 ‘0 Lactation-l , breed-1, trait-1, season-'2; 3 '> Lactation-'1, breed=1 , trait-1, season'3; 4 -> Lactation-1, breed'I, trait-1, season=4; 5 -> lactation-2, breed=1, trait-=1, season-=1; 6 => Lactation‘Z, breed=l , trait-=1, season=2; 7 '> Iactarion=2, breed=1, trait=1, season=3; 8 => lactation=2, breed=1, trait=1, season=4) Days PP at lst recording Days PP at 2nd recording Days PP at 3rd recording Days PP at 6th recording (999 = Not applicable for scheme/ starting date of recording) Testday yield at lst recording (divide by 10 to get yield, 0=missing) Testday yield at 2nd recording( " ) Testday yield at 6th recording (divide by 10 to get yield, 0=missing, 999 = Not applicable for scheme/starting date of recording) ILLUSTRATION OF SIAC FILE 4 (slac4td.dat) with lst and last two days PP and corresponding test day yields shown 11 14 701 763...231287228368 11 14 702 763...231287228368. 11 14 703 763...231287228368 ..182131 ..182131 ..182131 I 7 0 1 7 63 ... 231 287 22.8 36.8 ... 18.2 13.1 7 0 2 7 63 ... 231 287 22.8 36.8 ... 18.2 13.1 7 0 3 7 63 ... 231 287 22.8 36.8 ... 18.2 13.1 104 (5) FILE 5 (slac5td.dat): mmmammmmmm Number of records: 31,680 Type Length Position Description Numeric 2 12 Mean curve No. (11 *3 AM/PM curve 1, 12 = AM/PM curve 2) Numeric 1 3 Variance curve No. (5 - AM/PM variance curve 1, 6 '= AM/PM variance curve 2) Numeric 3 4-6 Replicate No. (1, ..., 120) Numeric 1 7 Scheme (5-AP/4) Numeric 2 8-9 Starting date (7'7d PP, 14"] 4d PP, 21=21d PP, 28'28d PP) Numeric 1 10 Missing pattern (0=none, 1=early lactation, 2=esrly peak, 3=late peak, 4=late, 5= tail end) Numeric 1 11 Combination code for levels of fixed effects (I '> Lactation=1, breed=l , trait-l, season°1g 2 -> Lactation-l , breed-l , trait-1, season-2; 3 -> Lactation=1, breed=l , trait-1, season=3; 4 '> Lactation=1, breed‘I , trait-=1, season=4; 5 '> Iactation=2, breed=1, trait-=1, season=l; 6 => lactation=2, breed—=1 , tmit=1, season=2; 7 => Iactation=2, breed=1, trait=1, season==3g 8 => Iactation==2, breed=1, traiFl , season‘4) Numeric 2 12-13 Days PP at lst recording Numeric 2 14-15 Days PP at 2nd recording Numeric 2 16-17 Days PP at 3rd recording Numeric 3 18-20 Days PP at 4th recording Numeric 3 3941 Days PP at 1 1th recording (999 = Not applicable for scheme/starting date of recording) Numeric 3 42-44 Test~day yield at 1 st recording (divide by 10 to get yield, 0=missing) Numeric 3 45-47 Testday yield at 2nd recording (divide by 10 to get yield, 0=missing) Numeric 3 72-74 Testday yield at I 1th recording (divide by 10 to get yield, O=missing, 999 = Not applicable for scheme/starting date of recording) ILLUSTRATION OF SLAC FILE 5 (slac5td.dat) with lst and last two days PP and corresponding test day yields shown F i rstlrecerdso n,- f ilc _.(Unfor_m_a tted); 115 15 701 735...259287 96148... 46 50 115 25 701 735...259287106147... 55 49 115 35 701 735...259287104141... 51 45 EirsLlrccotdsafteuead Lug. (Iimnattedl: 11 5 1 5 7 0 1 7 35 ... 259 287 9.6 14.8 ... 4 11 5 2 5 7 0 1 7 35 ... 259 287 10.6 14.7 ... 5. 11 5 3 5 7 0 1 7 35 ... 259 287 10.4 14.1 .. 5 i-atnm A.ntn Unoo 105 APPENDIX D: Illustration of files containing computed lactation total yield FILE 1 (slacl sum.dat) for SCHEME A1 Eirstlrecordsonfile: 111170110111 11 1170210112 111170310113 FILE 2 (slac25um.dat) for SCHEME A4 11 1270110111 11 1270210112 11 1270310113 FILE 3 (slac3sum.dat) for SCHEME A6 Ei_tsL3_uecords_o n.- fi 1c: 111370110111 11 1370210112 11 1370310113 FILE 4 (slac4sum.dar) for SCHEME A8 Eirstitecotd spmfile; 111470110111 11 1470210112 11 1470310113 FILE 5 (slac5sum.dat) for SCHEME AP/4 11515 70110111 115 25 70110112 115 35 70110113 106 APPENDIX E.1 Installation instructions for the SIAC dataset A. Distribution disks INSTALI/DATA DISK #1 Filename Type Length Date Description autoinst.bat A 3452 02-16-94 DOS batch file (alternative to install.exe) 2495 02-01-94 Instructions for data retrieval via FTP slac.fmt 13814 02-01-94 File formats for SLAC install.exe 175132 02-16-94 SLAC installation software readme.1st A A B *.b A Internal files (used by install.exe) A B A *.t: Internal files (used by install.exe) *.x_ Internal files (used by install.exe) slac1.1 797559 02-14-94 Part I of archive for File 1 of SLAC INSTALI/DATA DISK #2 Filename Type Length Date Description slac1.2 A 797559 02-14-94 Part II of archive for File 1 of SLAC INSTALL/DATA DISK #3 Filename Type Length Date Description slac2.zoo B 1287447 01-15-94 Archive for File 2 of SLAC (slac2td.dat) INSTALL/DATA DISK #4 Filename Type Length Date Description slac3.zoo B 940150 01-15-94 Archive for File 3 of SLAC (slac3td.dat) INSTALL/DATA DISK #5 Filename Type Length Date Description slac4.zoo B 934519 01-15-94 Archive for File 4 of SLAC (slac4td.dat) slac5.zoo B 119312 01-31-94 Archive for File 5 of SLAC (slacStd.dat) The SIAC is partitioned into 5 parts according to milk recording scheme. A detailed description of each part is given in the file slacfmt on INSTALI/DATA DISK 1. The uncompressed SIAC files should occupy 103 MB disk space. B. Data Installation To install SIAC: Place the diskette labelled INSTALL/DATA DISK l in the 3.5" drive, e.g. A: 53:11-2 Type A:INSTALL and press (Executes SLAC installation program). You should see a screen that looks like this: 107 In Inul-I I I (Lula .h.lu~r " i Q :"’-" Tubal bptllA: nucduil 1:.- In.) ”L. m L1. 1:. -.§;:. In the event that this screen does not come up, we recommend that you stream-line your system start-up files. If problems persist, skip to step 4. The following are required as input- a) The data source drive: To choose from a list of all available drives on your system. use [Up] and [Down] arrow keys or mouse. This should be the 3.5" drive. The default is A: b) The Destination drive: This is the drive on which the data will be decompresseed. Use die [Tab] key move forward from input field to input field. [SHIFT-Tab] moves backwards between input fields. c) The destination sub-directory: This is the location on the destination drive where the data will be stored. d) Scheme: Selective installation of data for any desired scheme is made possible by pressing the spacebar. The default choice installs data for all schemes. The [Up] and [Down] arrow keys can be used to move between the check boxes. Pressing [Alt] plus the [highlighted letter] or [number] selects or invokes the respective feature. For example Alt-l selects scheme A1 while [Alt-H] invokes the on-Iine help. The disk space needed is shown at the bottom of the screen. Press [Alt-I] to start the data installation. The program will prompt you for the appropriate disks. If the installation program fails to work, perform a full installation using the batch file autoins.bat on INSTALI/DATA DISK I. 108 a) Place the diskette labelled INSTALI/DATA DISK l in the 3.5" drive, e.g. A: b) Type AzAUTOINS [source drive:] [destination drive:\path] and press This should install SIAC on the [destination drive:\path]. If the installation program fails to work and you do not have at least 103MB free disk space, perform a selective installation using the batch file selins.bat on INSTALI/DATA DISK 1. a) Place the diskette labelled INST ALI/DATA DISK 1 in the 3.5" drive, e.g. A: b) Make a subdirectory on a fixed disk drive where the SLAC will be stored, e.g. mkdir c:\slac c) Change the default directory to the sub-directory created in b), e.g. c: cd\slac d) Copy the file selins.bat from INSTALL/DATA DISK 1 to this subdirectroy. e) Type SELINS [SOURCE drive:] [DESTINATION drive:\pathname] [scheme #] and press then follow the instructions on the screen. Valid values for [scheme #] are: 1 for scheme A1 2 for scheme A4 3 for scheme A6 4 for scheme A8 5 for scheme AP/4 For example: SELINS A: C:\SIAC 1 ---»- Installs SCHEME A1 data from A: to C:\SIAC This should perform a selective installation of SIAC. 109 APPENDIX 132 Installation instructions for electronic retrieval of SLAC via FTP Files were stored in the directory pub\slac\incoming\ (guest ftp 35 .8.1 24.45 , pwd: slac) Filename Type Length Date Description readme.1st A 2495 02-01-94 This file (Instructions for data retreival) slac.fmt A 3814 0201-94 File formats for SLAC slac1.zoo B 1595118 01-15-94 Archive for File 1 of SLAC (slac1td.dat) slac2.zoo B 1287447 01-15-94 Archive for File 2 of SIAC (slac2td.dat) slac3.zoo B 940150 01-15-94 Archive for File 3 of SLAC (slac3td.dat) slac4.zoo B 934519 01-15-94 Archive for File 4 of SLAC (slac4td.dat) slac5.zoo B 119312 01-31-94 Archive for File 5 of SIAC (slac5td.dat) B 20021 0.exe archives 55721 01-1 794 Self extracting - makes/ extracts/ views ZOO The SIAC is partitioned into 5 parts according to milk recording scheme. A detailed description of each part is given in the file \pub\slac\incoming\slac.fmt. Due to the large size of the files each one was compressed to enable transfer from MSU. Five (5) MB disk space is required for transferring SIAC. The uncompressed SLAC files take up 103 MB disk space. The following sequence of commands is suggested (rootclir is used to represent the destination directory): Command: Purpose: 1. ch \rootdir (Changes your default directory ) 1. bi (Sets file transfer mode to binary) 2. get filename.ext (Transfers filename.ext to \rootdir, repeat as desired) or mget *.* (Transfers all files to \rootdir) 3. quit (Closes the FTP connection). Note: The file com pression\decompression utility is Dhesi's ZOO version 2.10 for MSDOS 3.x or higher and should now be in your \rootdir. The compressed files should be compatible with the UNIX version of ZOO. 4. 200210 Executes 200210.exe and extracts the files ZOO.EXE 8L ZOO.MAN 5. zoo x are *.* Extracts the contents of \rootdir\arc.zoo (where are is the filename for the archive (e.g. slacl for \rootdir\slac1.zoo). See ZOO.MAN for further instructions. 110 APPENDIX F.1: Illustration of the test interval method (MSU) Example 1: All testday yields available (Scheme A4) SLAC record (Mean curve/ Variance curve/ Replicate No./ Scheme/ Starting date/ Missing pattern/ Code for fixed effects) = 10/ 4/ 01 5/ 2/ 07/ 0/ 8 Day of lact. l Testday yield l Interval days I Interval yield I Cum. Yield 7 26.9 7 188.3 188.3 35 35.1 28 868.0 1056.3 63 35.1 28 982.8 2039.1 91 31.9 28 938.0 2977.1 119 27.3 28 828.8 3805.9 147 24.2 28 721.0 4526.9 175 23.2 28 663.6 5190.5 203 27.5 28 709.8 5900.3 231 29.2 28 793.8 6694.1 259 26.6 28 781.2 7475.3 287 22.9 46 1105.2 8580.5 Notes: - length oflast interval = (287 - 259) + (305 - 287) = 46 days credited for. Example 2: Not all testday yields available (SCHEME A4) SIAC record (Mean curve/ Variance cutve/ Scheme/ Replicate No./ Starting date/ Missing pattern/ Code for fixed effects) = 10/4/010/2/21/2/8 Day of lact. l Testday yield | Interval days I Interval yield l Cum. Yield 21 32.6 21 684.6 684.6 49 ____ 28 914.9 1599.5 77 32.9 28 919.1 2518.6 105 28.5 28 859.6 3378.2 133 24.7 28 744.8 4123.0 161 23.6 28 676.2 4799.2 189 26.6 28 702.8 5502.0 217 27.6 28 758.8 6260.8 245 28.1 28 779.8 7040.6 273 26.3 28 761.6 7802.2 301 21.7 32 758.8 8561.0 Notes: - length of last interval = (301 - 273) + (305 - 301) = 32 days credited for. - estimate for missing yield on day of lactation = 49 was calculated as (32.6 + 32.9) / 2 = 32.8 kg 111 APPENDIX F.2 Illustration of the test interval method (France). Example: All testday yields available (SCHEME A8) Mean curve/ Variance curve/ Replicate/ Scheme/ Starting day/ Missing pattern/ = 7/4/1/4/ 21/0 Day of lact. l Testday yield [Interval days I Interval yield I Cum. Yield 21 21.4 21 449.4 449.4 77 23.4 56 1254.4 1703.8 133 25.1 56 1358.0 3061.8 189 25.6 56 1419.6 4481.4 245 27.8 56 1495.2 5976.6 301 31.1 56 1649.2 7625.8 305 31.1 4 124.4 7750.2 112 APPENDIX E3: Illustration ofthe centering date method (Denmark) Example 1: All testday yields available (Scheme A4) SLAC record (Mean curve/ Variance curve/ Replicate No./ Scheme/ Starting date/ Missing pattern/ Code for fixed effects) = 10/ 4/ 01 5/ 2/ 07/ 0/ 8 Day of lact. I Testday yield I Interval days IInterval yield I Cum. Yield 7 26.9 7 188.3 188.3 35 35.1 28 868.0 1056.3 63 35.1 28 982.8 2039.1 91 31.9 28 938.0 2977.10 119 27.3 28 828.8 3805.9 147 24.2 28 721.0 4526.9 175 23.2 28 663.6 5190.5 203 27.5 28 709.8 5900.3 231 29.2 28 793.8 6694.1 259 26.6 28 781.2 7475.3 287 22.9 46 1105.2 8580.5 Example 2: Not all testday yields available (SCHEME A4) SIAC record (Mean curve/ Variance curve/ Scheme/ Replicate No./ Starting date/ Missing pattern/ Code for fixed effects) = 10/ 4/ 010/ 2/ 21 / 2/ 8 Day of lact. Testday yield Interval days Interval yield Cum. Yield 21 32.6 21 684.6 684.6 49 0.0 28 456.4 1141.0 77 32.9 28 460.6 1601.6 105 28.5 28 859.6 2461.2 133 24.7 28 744.8 3206.0 161 23.6 28 676.2 3882.2 189 26.6 28 702.8 4585.0 217 27.6 28 758.8 5343.8 245 28.1 28 779.8 6123.6 273 26.3 28 761.6 6885.2 301 21.7 32 758.8 7644.0 113 11. REFERENCES Abramowitz, M., and I. A. Stegun. 1972. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Eds. US. Govt. Print. 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Ragsdale, and C. W. Turner. 1923. The rate of decline of milk secretion with the advance of die period of lactation. J. Gen. Physiol. 5:441. Cemy, B. A., and H. E. Kaiser. 1977. A measure of sampling adequacy for factor-analytic correlation matrices. Multiv. Behav. Res. 12:43. Chazal, M. P. , and Y. Chilliard. 1986. Effect of stage of lactation, milk yield and herd man- agement on seasonal variation in spontaneous Iypolysis in bovine milk. J. Dairy Res. 53:529. Cobby, J. M., and Y. L P. Le Du. 1978. On fitting curves to lactation data. Anim. Prod. 26:127. ' Congleton, W. R, Jr. and R. W. Everett. 19803. Error and bias in using the incomplete gamma function to describe lactation curves. J. Dairy Sci. 63:101. Congleton, W. R, Jr. and R. W. Everett. l980b. Application of the incomplete gamma func- tion to predict cumulative milk production. J. Dairy Sci. 63:109. Cooley, W. W., and P. R. Lohnes. I971. Multivariate data analysis. John Wiley & Sons, New York, NY. Corley, E. L. 1956. 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