ESTIMATES OF GENETIC PROGRESS IN THE DEVELOPMENT OF THE AMERICAN RED DANISH CATTLE By Norman Ray Thompson AN ABSTRACT Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Dairy Year 1955 Approve d_ 1 NORMAN RAY THOMPSON ABSTRACT Milk and butterfat yields and related information were collected on the foundation generation and on successive back-crosses to Red Danish sires. The relation of certain environmental factors to butter- fat production was investigated by a least squares procedure. Butter- fat yield did not vary significantly with month of calving, but did vary significantly (P ^ .01) with age at calving, previous calving interval, and length of lactation period. The regression of butterfat yield on age at calving was curvilinear and of the form Y - Y + 10.012(X-X) - 0.067(X^-X^), where X is age in months and Y represents butterfat. Yield of butterfat increased at a rate of 1.96 pounds for each additional month of previous calving interval, and 0.99 pound for each additional day of lactation period. Also, butterfat yield exhibited an upward environmental trend of 6.49 pounds per year (P ^ *05). Additive correction factors were developed from the least squares estimates, and used to adjust the original butterfat records. The foun­ dation generation, using fully adjusted records, averaged 351 pounds butterfat; the first-cross generation, 379; the second, 377; and the third, 389. The increase of 28 pounds from foundation to first-cross generation was very highly significant (P <^.G01), the 2-pound decrease to the next generation was not significant, while the gain of 12 pounds by the third-cross generation was very highly significant (P < .001). Selection in the foundation groups, based on comparison of all cows with those having daughters in the first-cross generation gave estimated additively genetic superiorities (for those having daughters) of zero to 2 NORMAN RAY THOMPSON ABSTRACT +13.4 pounds butterfat per lactation period. Selection in the first- and second-cross generations gave estimated additively genetic superior­ ities of zero to +5*1 pounds butterfat. These latter values are based on the amount of selection practiced from lactation to lactation. The effect of age at calving on butterfat production was investi­ gated in some detail, and tentative age correction factors were devel­ oped for the American Red Danish breed. Factors based on fitting an intra-cow quadratic regression to the data were the most efficient of any developed here, both in making use of more records than in the paired method and in accounting for a larger proportion of the variance than did any other method. Based on comparison of sets of regression factors, the second-cross generation reached peak production at an earlier age than did the first cross. Preliminary evidence was found that the presently recommended Bureau of Dairy Industry factors are too low for the 2- and 3-year age brackets. Heritability of butterfat yield, using the method of intra-sire regression of daughter on dam, was estimated as 0.39 t 0.11 on a single record basis. 0,43. Repeatability of butterfat production was estimated as The intra-cow correlation of age at calving with previous calving interval was 0.07 (P < #05). Other intra-cow correlations among age at calving, month of calving, and previous calving interval were numerically small and not significant statistically. Repeatabilities of calving interval, month of calving, and length of lactation period were estimated as 0.04, 0.37, and 0.18, respectively. values suggest rather low genetic determination. The first and third ESTIMATES OF GENETIC PROGRESS IN THE DEVELOPMENT OF THE AMERICAN RED DANISH CATTLE By Norman Ray Thompson A THESIS Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PUILOSPHY Department of Dairy 1955 ProQuest Number: 10008682 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10008682 Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 06- 1346 ACKNOWLEDGMENTS Grateful appreciation is expressed to the following persons, groups, and institutions: 1. To the breeders of American Red Danish cattle for use of the records which they so kindly reported, and to the extension workers who gave valuable aid in collecting the records. 2. To the faculty and staff at Michigan State University for their great help and cooperation in this investigation, and particularly to Dr, Earl Weaver, Dr. N. P. Ralston, Dr. W. D. Baten, Miss Norma Tashner, and Mr. Robert P. Witte. 3. To Mr. A. F. Teske and Mrs, Lura E. Whiter at Virginia Polytechnic Institute for invaluable service in the IBM operations. TABLE OF CONTENTS Page INTRODUCTION .............................................. REVIEW OF LITERATURE.................... . A. Genetic Progress 1 * 4 . .................................. 4 G e n e r a l ........................................ . . Basis . , 4 4 Definitions ........................................ 5 Estimation and Estimates .......................... 6 ..................... 7 B. Repeatability and Heritability C. Selection in Dairy Herds ........................... S D. Environmental Factors ............................... B General ............................................ B Nutrition .......................................... 9 Time of C a l v i n g ............................... . . * 9 C l i m a t e ............................................ 10 Dry P e r i o d .............. 10 Calving I n t e r v a l ................ 11 Gestation . . . . . . . . . . 11 Age at First C a l v i n g .............................. 12 Miscellaneous ...................................... 12 P r o b l e m s .................. 12 Numerical Estimates ................................ 13 Correction Factors 14 ................................ iv E. Age Correction F a c t o r s ....................... 1$ Historical . . . . . . . . . . . . . . . . . . . . . Age and Production.......................... F. 15 16 Methods for C o m p u t i n g ........ * .................. 17 Efficiency . . . . . . . .................. , . . • 18 P r o b l e m s .................................... 19 Mathematical Bases and P r o c e d u r e s ........... 19 M e t h o d s .............. * .......................... 20 PROCEDlTttES.............................. . .............. 23 A. P r e l i m i n a r y .................................. 23 Collection of D a t a .......................... 23 Planning the Investigation ........................ Preliminary Operations on the D a t a .......... 26 B. Least Squares Solution for Non-genetic Effects . . . . Nature and Properties of Least Squares Estimates . . Calculation of Terms for the E q u ations...... Solution of the Equations . . . . . . . . Development of Correction Factors Estimated Producing Ability of Each C o w .... 26 32 32 ................ Procedure for Correcting the Butterfat Data 26 28 ........ C. Correcting the Data for Non-genetic E f f e c t s . 23 .... 32 39 39 D. Heritability and Repeatability of Butterfat Production 40 E. Comparison of Foundation Generations with Successive Crosses to Red Danish S i r e s ................. ... 43 ................ 43 Methods and Assumptions . . . . . V F. Calculation of Age Correction F a c t o r s .............. 46 Gross F a c t o r s .................................... 46 Paired Factors .................................. 46 .............................. 46 G. Repeatability of Environmental Factors; Correlations among F a c t o r s ................................ 50 H. Selection in the Various G e n e r a t i o n s ............ • 50 RESULTS AND DISCUSSION .................................. 58 Regression Factors A. Effects of Environmental Factors on Butterfat Production . ............ . .................... 58 Estimates from Least Squares Analysis . . . . . . . 58 Interrelations of Environmental Factors; Genetic A s p e c t s ...................................... 60 Relative Importance of Effects 60 .................. Correction Factors for Environmental Effects ... 6l B. Heritability and Repeatability of Butterfat Production.............. 61 C. Estimates of Genetic P r ogress ...................... 62 Assumptions...................................... 62 Differences Among Generations . 62 ................ D. Age Correction F a c t o r s .......... 66 E. G e n e r a l ............................................ 68 S U M M A R Y ................................................. 69 vi LIST OF TABLES Table Page 1. Coefficients in the original equations ................ 29 2. Coefficients in the reduced equations ................ 30 3. Estimates of parameters obtained from least squares .................................... analysis 33 A# Mean squares for parameters included in least squares a n a l y s i s .................................. 34 5. factorsfor year of c a l v i n g .............. 35 6 . Correction factorsfor age at c a l v i n g ......... 36 7. factorsfor of lactationperiod . . . 37 Correction Correction length 8 . Correction factorsfor length of previouscalving i n t e r v a l ...................................... .... 41 10. Estimation of heritability of butterfat yields ........ 42 11. Changes in butterfat yield with relation to age at c a l v i n g .................... 47 12. Multiplicative age correction factors . . . . . . . . . 48 13. Analysis of variance of changes in butterfat yield with relation to age at calving ........ 49 14. Repeatability of month of c a l v i n g ................... 51 . 9. Estimation of repeatability of butterfat yields 38 15. Repeatability of length of calving interval .......... 52 16. Repeatability of length of lactation . . . . . . . . . . 53 17. Simple intra-cow correlations among month of calving, age at calving, previous calving interval, and length of l a c t a t i o n .......................... 54 13. Overall selection in each generation .................. 55 19. 56 Selection from lactation to lactation ................ vii 20. Average butterfat production in each generation group. I, Foundation breed groups .............. 63 21. Average butterfat production in each generation group, II, Crosses to Red Danish b u l l s ........ 64 22. Comparison of average butterfat production among generation groups ................................ 65 23. Original layout of IBM detail or individual lactation cards .................................. ix 24. Revised layout of IBM detail or individual lactation cards ........ . . . . . ............. x 25. Layout of IBM cow summary cards used in least squares analysis . . . . . ....................... xi 26. Layout of IBM cow summary cards used in comparing generations and in estimating heritability and repeatability of butterfat production . . .... xii 27. Layout of IBM detail cards used in calculating age correction f a c t o r s ............ xiii INTRODUCTION The development of populations of domestic animals which have useful or desirable traits has been marked by several means and procedures. Among these are inbreeding, selection, and migration. Inbreeding accompanied by selection has been used in the development of relatively uniform populations termed breeds. Migration, accom­ plished by man, has introduced new genes to populations and, through crossing and up-grading, has led to shifts in gene frequencies within these populations. Selection, of course, has been an ever-present tool, ready for use in each generation so long as rates of reproduction remained normal. The development of the American Red Danish breed of dairy cattle (Cranek, 1952) is an excellent example of the effects of migration on gene frequency. Although the foundation stocks were highly diverse genetically, repeated back-crossing to Red Danish sires soon led to much less diversity as evidenced by phenotypic characters. In addition, the frequency of genes which affect economic traits favorably appears to have been increased. Cranek reported gains in milk and butterfat production from the foundation groups to the successive generations. In making his investigation, Cranek corrected milk and butterfat records only for age at calving and number of days milked. The cor­ rection values used were based on information from the literature, rather than from the experimental data. He investigated the relation of month of calving and previous calving interval to milk and butter­ fat yields, but did not make use of the findings. In view of these 2 facts, it appeared desirable to make further investigation of the data, particularly with regard to factors other than changes in gene fre­ quency which might have affected phenotypic performance* Accordingly, the writer set up several investigational objectives* These were: 1. Obtain joint estimates of the effects of month of calving, previous calving interval, age at calving, year of calving, and length of lactation period on butterfat production* 2* •'Correct*1 the raw data for these effects, and make further evaluation of the changes in butterfat production from repeated back-crossing to Red Danish sires. 3* Construct a set of age correction factors for the American Red Danish breed* A. Estimate the repeatability of month of calving, length of lactation period, and length of calving interval. 5* Estimate heritability and repeatability of butterfat production, based on records that had been corrected for the environmental effects listed in the first objective. 6. Investigate rates and effectiveness of selection within each generation. The fourth, fifth, and sixth objectives were not anticipated at the beginning. However, it soon became evident that estimates of heritability and repeatability would be needed in the estimation of genetic progress from generation to generation. Further, estimates of heritability and repeatability have some general application, in that they indicate the relative roles of heredity and environment in 3 determining the characters considered. Information on the status of selection can be used to help determine the possibilities for both further and more effective selection. objectives were included. Therefore the second three REVIEW OF LITERATURE This review will cover the most pertinent reports in the fol­ lowing subject areas: 1* Genetic progress. 2. Repeatability and heritability. 3* Selection in dairy herds. 4. Envi ronmental factors • 5. Age correction factors. 6. Mathematical bases and procedures. In some instances, most of the available information will be cited; in others, only a single report by way of example. A. Genetic Progress General The concept of genetic progress in the breeding of domestic animals probably appeared, in some crude form, at a very early time. Selection of breeding animals was practiced long before anything was known about the science of genetics. Thus at the beginning of the twentieth century a number of domesticated breeds of birds and mammals had been developed that far surpassed their more distant ancestors with respect to characters of economic value. Basis The basis for genetic progress is the shifting of gene frequencies so that the genes which favor expression of desirable attributes or characters are more numerous in the population than their alleles. 5 The demonstration of such shift is relatively simple when only a single pair of genes is concerned, since genotypic and phenotypic classes are discrete and in many cases can be identified. In the absence of domi­ nance, each of the three genotypes coincides with the corresponding phenotype, and each phenotype and genotype can be identified. With complete or partial dominance, identification of certain genotypes becomes difficult to impossible, however. The effects of a single pair of genes, in numerous cases, are not noticeably modified by environ­ mental forces. However, the demonstration becomes somewhat more difficult with characters of economic value because (l) such characters appear to be determined by many gene pairs, (2) the sum of gene effects includes both individual additive effects and those due to interactions among genes, and (3) non-genetic forces exert large and variable effects on the final expression of the character. Definitions Definitions of genetic progress can range from the rough estimates secured from early grading up experiments to the somewhat more precise estimates of recent years. Several early grading up experiments (Olson and Biggar, 1922; Cunningham, 1926; Fairchild, 1926; Weaver et al.. 1928) demonstrated the substantial improvement that could result when mediocre females were bred to superior males. The composition of this improvement was not separated into genetic and non-genetic portions, although the effects of environment were recognized in two experiments and demonstrated in one. Improved definitions of genetic progress became possible after methods for separating genetic and non-genetic effects were elucidated 6 (Fisher, 1918; Wright, 1935) and applied to particular problems (Lush, 1940)• Perhaps the most conservative definition of genetic progress which we can state today is f,that improvement in economic value or performance which is due to increases in frequencies of desirable genes and to their individual roles in producing a superior phenotype". The phenomena of heterosis, hybrid vigor, "nicking", etc., are not included in this strict definition of genetic progress, even though they may be of great economic importance. Estimation and Estimates A more exact estimation of genetic progress has been evolved only as new techniques have been developed to estimate additive genetic effects separately from other genetic effects, and to evaluate the non-genetic effects that commonly are grouped and termed environmental. Wright (1939) recognized that environmental effects were large and that they might interfere in the estimation of genetic effects, Dickerson and Hazel (1944) used the heritability ratio (additively genetic variance/total phenotypic variance) to estimate the additively genetic superiority of a selected group of parents, from which the additively genetic superiority of their unselected offspring in turn could be predicted. Their method has been used by many recent workers to estimate genetic gains. Nelson (1943) indicated a procedure somewhat different from that of Dickerson and Hazel, in that comparisons were to be made over entire herds or flocks from year to year, rather than from generation to generation. The use of this latter procedure requires preliminary estimation of non-genetic year-to-year deviations in animal performance. 7 (These deviations should be considered regardless of method, if per­ formances of any group are measured over more than a single year,) Numerical estimates of potential genetic progress in dairy cattle include those by Lush (1949), and Rendel and Robertson (1950), Estimates of actual genetic progress have been reported by Rendel and Robertson (1950), Laben and Herman (1950), Mahadevan (1951a,b), the Iowa Station (1952), Harvey (1953a), and Stonaker (1953). In general, neither the theoretical nor the actual increases have been large; an approximate figure would be one per cent a year. Similarly, neither have the increases been large under conditions of artificial insemination with the use of a few carefully selected sires (Robertson and Rendel, 1950, 1954). B. Repeatability and Heritability Cranek (1952) reviewed the literature on repeatability and heritability of milk and butterfat yield, and discussed methods of estimation. He obtained values for heritability, using data on various groups of American Red Danish cows, which ranged from 0.3# 0.66 ± 0.20 for milk yield, and 0.42 ± butterfat yield. £ 0.03 to 0.10 to 0.67 ± 0.19 for Corresponding values for repeatability were 0.49 to 0.53 for milk yield, and 0.61 to 0.74 for butterfat yield. The environmental factors discussed in preceding paragraphs are not all necessarily non-genetic in an absolute sense. may be determined in part genetically. Some of them Repeatability of length of calving interval has been estimated variously at 0.133 (Legates, 1954) and 0.134 (Rennie, 1954), although heritability appears to be 8 essentially zero for this trait, Tandon (1953) observed a repeat­ ability of 0,8 for length of dry period in Indian cattle. Although very little is actually known, it is likely that length of lactation is determined in part genetically, G, Selection in Dairy Herds Only a few reports are available on this subject, Seath (1940) found culling rates of 28,6 per cent and 30,9 per cent per year, respectively, for Iowa and Kansas herds. Asdell (1951) reported an average removal rate of 21.9 p©r cent annually among herds in 17 states. The rate varied with age of cow, being as low as 6.0 per cent for ages 2-3 years and up to 35.2 per cent for ages 7-8 years. Since much of the culling ordinarily is for dairy purposes, disease, etc., the opportunity for making selections for breeding purposes is reduced accordingly. D. Environmental Factors General The term environmental is used here in a broad sense, in that it pertains to all factors nearly or completely of a nongenetic origin and which may affect quantity or quality of milk produced. These non-genetic or environmental factors for the most part exert temporary effects, although a few (injuries, disease, malnutrition, etc.) may affect the subject individual permanently. Non-additively genetic effects, resulting from genic interactions and commonly termed epistasis and dominance, similarly affect the individuals throughout 9 life, but these effects are determined at fertilization rather than during the subject’s lifetime. The effects of environmental factors on production have been recognized for many years (Pearl and Miner, 1919; Norton, 1932; Wright, 1939), and numerous investigations of them have been made. Nutrition, time of calving, climate, dry period, calving interval, gestation, and age at first calving all have been shown to affect production. Nutrition Nutrition was shown at an early time to have major effects on yields of milk and butterfat (Roberts, 1892; Doane, 1900; Wing and Foord, 1904), although the butterfat percentage apparently could not be altered appreciably except in an indirect manner (Eckles, 1912). The effects of successive increments of feed are curvilinear over more than a limited range of nutritional levels (Jensen et al., 1942). The effects of nutrition are not limited to those of a direct and immediate nature. Optimal nutrition during growth may improve performance after reaching maturity (Weaver et al., 1928). Time of Calving The effect of month or season of calving has been investigated by a number of workers. In general, under temperate zone conditions, cows calving in the fall and early winter months yield more milk than those calving at other times of the year (McDowell, 1922; Hammond and Sanders, 1923, Turner, 1923; Wylie, 1925; Sanders, 1927-28; Headley, 1933; Frick et al., 1947; Cranek, 1952). However, Oloufa and Jones (1948) found no significant differences that could be attributed to month of calving under the mild climatic conditions prevalent in 10 western Oregon, Several workers (Hammond and Sanders, 1923; Sanders, 1927-28; Cannon, 1933; Cranek, 1952) have constructed correction factors for month of calving. Differences in quantity and quality of feed available from month to month appear to be a causative factor (Bettenay, 1949; Cullity, 1949; Scott and Wilson, 1952), Jordlfo and Assiz (1948- 49) observed both higher milk yield and lower rate of decline among cows calving in May-August (winter) than in November-February (summer). Their observations were made on cows kept in the Southern Hemisphere. Climate The effect of season (apart from time of calving) has been the subject of several reports, Butterfat percentage tends to be low in summer and high in winter (Ragsdale and Turner, 1922; Headley, 1933; Becker and Arnold, 1935). Conversely, milk yield tends to be higher in summer than in winter (Arnold and Becker, 1935; Erb and Shaw, 1953). The recent review by Hancock (1954) strongly suggests that extremes of climate (and more specifically of temperature) affect not only butterfat percentage and milk yield but also solids-not-fat content. Specifically, moderately high temperatures favor low fat percentage, high milk yield, and low SNF, and moderately low temperatures the opposite. Erb and Shaw (1953) have devised sets of correction factors for adjusting monthly milk and butterfat yields according to calendar month in which secured. Dry Period The reports of investigations on length of dry period preceding the lactation (Carroll, 1913; Hammond and Sanders, 1923; Sanders, 1927-28; Dickerson and Chapman, 1939; Dickerson, 1940; Klein and 11 Woodward, 1943; Erb and Shaw, 1953) indicate the desirability of 1 to 2 months duration. While extremely short periods are detrimental, those longer than 2 months seem to offer no advantage in terms of increased yields. In fact, Dickerson (1940) found that dry periods longer than 2 months were accompanied by relatively low production. He observed that such low production was related to low persistency and producing ability, hence probably was of genetic rather than environ­ mental origin. Correction factors for length of dry period have been proposed (Hammond and Sanders, 1923; Sanders, 1927-28; Klein and Woodward, 1943; Erb and Shaw, 1953)* Calving Interval Length of time interval between calvings can affect yields both in the current lactation and in that which follows (Tyler and Hyatt, 1950). Production appears to increase with longer calving interval, in a linear fashion up to about 12 months (Gaines and Palfrey, 1931; Erb and Shaw, 1953) but at a lesser rate for longer periods (Cranek, 1952; Erb and Shaw, 1953). Gestation The effects of advancing gestation on milk yields have been in­ vestigated both from the standpoint of days of gestation while milking (Erb and Shaw, 1953) and of service period (days from calving to next conception) (Hammond and Sanders, 1923; Sanders, 1927-28; Jordao and Assiz, 1948-49). Production tends to decrease as service period decreases and as days of gestation increase. days of gestation have been prepared by Correction factors for Erb and Shaw (1953), and for service period by Hammond and Sanders (1923) and by Sanders (1927-28). 12 It should be noted that dry period, days of gestation, and service period all are components of calving interval. Age at First Calving Age at first calving depends partly on management. Davis (1953) reported that age at first calving was significantly correlated (r = 0.409, P < 0 .01) with butterfat yield in first lactation but not with productive life. Adjustment for this variable ordinarily coincides with correction for age at calving. Miscellaneous The number of times milked per day usually is assumed to affect production (Norton, 1932; Lush and Shrode, 1950)• The effect appears not to be constant from lactation to lactation, at least not for the first 2 or 3 , and provision is made for such inconstancy in adjusting records of cows milked 3 or 4 times daily when such records are used in proving sires (Kendrick, 1953)* Pathological factors, including chronic disease, can take a heavy toll of milk production. Records made under these handicaps may deviate excessively from the normal, and perhaps should be excluded altogether when making comparisons. Psychological factors such as association with numerous other individuals (Schein et al., 1955) may affect cows adversely, although a certain amount of competition among animals has been thought beneficial (Maynard, 1947)* Problems The magnitude and nature of environmental effects give rise to numerous problems in connection with evaluation of animal performance on a genetic basis. The problems become especially critical in evaluating 13 dairy sires (Laben, 1954). Korkman (1953) has noted the problem of non-genetic differences between herds that are due partly to unequal levels of nutrition, and has made comparisons among daughters of A.I. (artificial insemination) sires within similar planes of nutrition. McGilliard (1954) and Henderson et al, (1954) have considered use of the contemporary herd average to circumvent differences of an environ­ mental nature between herds. Robertson and Rendel (1954) compared progenies of A.I. and non-A.I. sires on an intra-herd basis for the same reason. Year-to-year variations have been observed by Libizov (1933), Plum (1933), and Laben and Herman (1950). Their existence tends to reduce the value of daughter-dam comparisons (Laben, 1954) and has led to the use (Robertson and Rendel, 1954) of daughter averages alone. Methods to separate yearly environmental effects have been reported by Henderson (194&, 1949) and extended to IBM computation by Harvey (1953b). Numerical Estimates The magnitude of various environmental effects may be esqpressed either as plus or minus deviations (percentages or constants) from the mean or as proportions of the total variance among records. Examples of the former (Hammond and Sanders, 1923; Sanders, 1927-28; Cranek, 1952; Erb and Shaw, 1953) generally have been computed from simple one-way tabulations or at best with only partial adjustment for corre­ lations among the several effects. In general, recommended correction values for any single variable (except for age at calving) have not been greater than plus or minus 10 per cent at the most. Erb and Shaw (1953) indicated that previous dry period and days calf was carried, 14 however, can influence records more than 25 per cent when acting together. Year to year deviations and environmental trends have been esti­ mated by Laben and Herman (1950), workers at the Iowa Station (1952), Harvey (1953a), and Dillon et al. (1955)* Legates (1949) found the variance component for year to year changes in herd average to be only about 5 per cent of the total variance, and further that almost ninetenths of this component was due to changes in individual herd averages from year to year. Bayley (1950) found that 9 environmental factors accounted for approximately 50 per cent of the variation in yields of milk and butterfat. The total size of all temporary environmental or genetic effects may be expressed as the total variance minus that due to repeatability of individual production records. On such a basis, temporary environment accounts for more than half of all variance in milk and butterfat records. Correction Factors Although correction factors for environmental effects have been developed by a number of workers, their use in practice has been limited chiefly to correcting records for age at calving and number of times milked daily. Length of lactation commonly is standardized to 305 days, at which length the effect of calving interval appears to be minimized (Dickerson, 1940) but not entirely eliminated (Erb and Shaw, 1953). Bayley (1950) devised an index, based on a multiple regression analysis, to adjust records for 6 environmental factors (selection rating, age at calving, TDN feeding rate, days carried calf, herd size, and condition of cow at time of calving), but no wide or 15 general use appears to have been made of this index. The applicability to data of corrections for environmental effects depends on (l) the reliability of the initial estimates of such effects, i.e., whether based on simple one-way tabulations or on appropriate least squares or maximum likelihood estimates, (2) the standard errors of such initial estimates and the limits of error when used to ’’correct" small samples of data, (3) the actual reduction in total variance from making corrections to data, and (4} the fraction of the variance due to a given effect that is removed by making corrections. Individual estimates of environmental effects, unless the various effects are not correlated, may be biased; joint estimates should avoid this pitfall. Further, the use of correction factors may not be justified if the limits of error in application are very large and/or only a small amount of variance is removed. E. Age Correction Factors One of the major environmental factors, age at calving, will be treated separately. Much work has been done on this factor, and some of the findings will be presented. Historical The increase in milk yield of cows from lactation to lactation, up to maturity, is a readily recognized phenomenon and was noticed at an early date (Hills, 1908). Early age correction factors arose partly from the need for making comparisons among A.R. (Advanced Registry) records of the several dairy breeds. A number of the earlier reports were based on such records (Holdaway, 1916; Pearl and Patterson, 16 1917; Gowen, 1920a, b, c; Hooper, 1921; Gowen and Gowen, 1922; McCandlish, 1922; Ragsdale et al., 1924; Norton, 1932). With the growth of Cow Testing Associations (now Dairy Herd Improvement Associations) the need arose for factors appropriate for records made under other than A.R. conditions. Clark (1924), using data from 11 Land Grant College herds, prepared age correction factors for the Holstein, Jersey, Guernsey, and Ayrshire breeds. A decade later the Bureau of Daily Industry, U.S.D.A., developed a set of "all-breed” age conversion factors from D.H.I.A. records available at that time, and put them to use in the proved sire program which began in 1935 (Kendrick, 1953). Factors for the various breed groups were developed and released by the Bureau of Dairy Industry in 1941, and subsequently individual sets were made available for most dairy breeds (Kendrick, 1953). To date (June, 1955) no separate set of factors has been reported for the American Red Danish breed. Age and Production Pearl (1914) found the increase in milk and butterfat production with advance in age to be curvilinear, first rising rapidly, then more slowly to a peak, then declining gradually. He postulated a curve of the form Y = A + bX + cX2 + d log X to describe the variation. Pearl and Miner (1919), using Scottish Ayrshire records, obtained the curve Y = 12.4766 + 0.6146 X - 0.0366 X2 + 3.6641 log X, with the highest point occurring at 10J years of age. Dickerson and Chapman (1939) found the increase "essentially linear up to about five years of age, when maximum production was reached". Other workers have found, in general, that the increase is curvilinear, with the highest yearly milk production somewhere between the fifth and eighth years. 17 The increase in milk yield from lactation to lactation appears to be associated in part with gains in live weight up to maturity (Illinois Sta. Rpt., 1934-35), and in part with the rise and decline of physiolo­ gical processes that relate to milk secretion. Genetic differences in the rate of increase exist, not only between breeds of dairy cows but also within breeds (Libizov, 1933; Dickerson and Chapman, 1939). Non- genetic factors such as level of nutrition may affect the increase one way or another. Methods for Computing At least three methods for computing age correction factors are available. In the first, all records at each age of calving are averaged, a smooth curve is fitted through the means, the high point of the curve is determined, and the multiplicative factors are developed from this curve to correct records in the various age classes to the production level of the highest class. This first method is termed the gross method by the writer, and is identical with the "lumped" lacta­ tion method noted by Hammond and Sanders (1923). In the second method, first and second records of the same cows are compared and a segment of the curve is established, then second and third records of the same cows are compared, et seq. In the third method the form of the curve is anticipated in advance and appropriate intra-cow sums of squares and cross products and terms of higher orders are computed, a set of equa­ tions set up and solved, and the resulting values used to establish the curve. The first method is simple, easy to understand, and the calcula­ tions are straightforward. However, any appreciable and effective culling of low-producing cows between first and second lactations (or later) will tend to throw bias into the age curve, both in elevating the portions representing second and later lactations (Hammond and Sanders, 1923) and in transferring the high point of the curve to an unduly late age. The second method, while somewhat more tedious to compute, avoids the bias inherent in the first method, but may intro­ duce a bias in the opposite direction (Lush and Shrode, 1950). The third method is superior to the second in that it utilizes a maximum of information and the computations are not unduly involved, but the bias of the second method may be present here also. Stonaker (1953) tried to avoid the second bias (as well as the first) by regressing the first records of each group of pairs back to the mean for all first records (paired and unpaired together) according to a repeata­ bility value of 0.5. Efficiency The obvious purpose of age correction factors is to minimize an otherwise large source of non-genetic variability and thus increase the accuracy of comparisons among individuals whose records were made at different ages. Lush and Shrode (1950) estimated that age of calving accounted for only about 14 to 16 per cent of the total vari­ ance among records of dairy cows. They further estimated that the B.D.I. factors (Kendrick, 1941) took out 91 per cent of the age vari­ ance. It is obvious, however, that even the most efficient age correction factors cannot remove more than about one-sixth of the total variance among records. The remaining variance still may be expected to contain substantial components due to other non-genetic 19 effects such as level of nutrition. Further, the application of age correction factors to small groups may be hazardous (Anthony, 1932), since the limits of error in the use of such standard values tend to vary inversely with the number of individuals concerned* Problems The problems attendant to the development and application of age conversion factors to dairy records have continued to be investigated in recent years. Ward and Campbell (1936), from results with New Zealand Herd Test data, suggest that neither percentage addition (multiplicative) factors nor those in which constant amounts are added to the original records are correct, but that a regression formula is preferable, Dickerson and Chapman (1939) found evidence that the increase in yield with age was related to initial level of production. Lush and Shrode (1950) showed that a bias opposite to that discussed earlier under the 11firstn method could occur when age curves were developed by the use of paired records, i.e., the higher portions of the curve would be depressed. However, they did not make any estimate of such bias from their data. F. Mathematical Bases and Procedures Biological processes, in general, are concerned with many variables. The effects of these variables seldom follow any simple law or pattern. The measurement of biological variables may vary from highly objective to highly subjective, or be well nigh impossible to specify at all. The measurements may give anything from discrete classes to (for all practical purposes) continuous variation. Further, numerous interactions 20 may take place among the variables. Therefore, a simple situation seldom if ever exists, and adequate analyses tend to become complex. At best, much variability remains unattributed and unexplained; thus the "error” variance is large. Multivariate analyses generally are necessary and a priori knowledge of the subject matter is highly desirable, Methods The methods for analysis of biological data began perhaps with calculation of what we now consider phenotypic correlations among various classes of genetically related individuals. Rietz (1909) appears to have pioneered such correlations, and Gowen and associates at the Maine Station made numerous contributions between the years 1915 and 1925. Fisher (191&) showed that, under certain assumptions, the parent-offspring correlation would include one-half of the additive genetic variance, and the full-sib correlation one-half of the additive and one-fourth of the dominance variance. procedure to include epistatic variance. Wright (1935) extended the Bywaters (1937) and Jafar et al. (1950) have made estimates of both linear and non-linear genetic variances, and estimates of the linear or additive portion have been made for a number of traits by numerous workers. Estimates of the heritability ratio (additive genetic variance/total phenotypic variance) have become fairly common in the literature. Kempt h o m e and Tandon (1953) have investigated the problem of variable numbers of offspring per parent when heritability is estimated from regression of offspring on parent. Lush (1953) has discussed the hazards and pitfalls in estimating heritabilities. These hazards and 21 pitfalls include sampling errors, biases due to selection of data, discontinuous phenotypic variation, non-linear scales of measurement, highly correlated environments for classes of relatives compared, and non-randomness in the mating systems used* The early development of practical methods of statistical analysis was characterized, among other things, by the use of planned experi­ ments with a state of complete orthogonality throughout. The methods of analysis of data from such experiments are relatively simple. However, much of the data in animal science, particularly field data, lacks the orthogonality of a planned experiment, and such procedures as the conventional analysis of variance are not adequate. cases ’’missing plot” techniques may suffice. In some In others, it is neces­ sary to go back to the more general procedures involving least squares and maximum likelihood. Yates (1934) and Hazel (1946) attacked the problems in the analysis of data with different numbers in the sub­ classes. Henderson (1948, 1949) developed specific methods and computational procedures, and applied them to a particular problem. Harvey (1953b) extended the procedures to include IBM operations on the data wherever feasible. A number of workers (Dickerson, 1942; Baker et al., 1943; Hetzer et al., 1944; Knapp et al., 1951; Touchberry, 1951) have used variance components to derive genetic variances and covariances, and to estimate heritabilities and genetic correlations. Estimates derived from least squares analyses of data can be used to "correct1’ the raw data for the effects concerned (Price et al,, 1953). It should be recognized that such correction of data does not remove all of the variance for a 22 given effect. For example, variance due to linear regression does not account for all of the variability in a factor, and the remaining variance (deviations from regression) stays in the error term in the analysis. Transformations apparently have not been used widely on animal data, although such use might be appropriate in certain instances. Cummings et al, (1947) used an arc-sine transformation on swine data. In closing, two aspects of the present status of mathematical procedures should merit comment. First, the increasing availability of high speed computers has lessened the computational burden of multivariate analyses (though not the planning). Second, the status of variance components as a genetic tool is far from static. Lowry (1955) has reviewed the use of variance components rather carefully. PROCEDURES A. Preliminary Collection of Data The major portion of the data used (3,270 lactation records) was collected earlier by Cranek (1952). In addition, 981 more lactation records were obtained from herd owners early in 1953* Milk and butter­ fat production, sire, dam, age and date of calving, days milked, etc., for each lactation period were obtained from the herd owner* The data was key punched into standard 80-column IBM cards (one card for each lactation period of each cow). These IBM cards (appropriately desig­ nated as "detail" cards) were used in subsequent operations ’with the data. Although both milk and butterfat production were reported, only the butterfat data were used in analyses by the writer. Planning the Investigation Preliminary analyses of a small sample of data suggested that the effects of age, season of calving, and previous calving interval on butterfat production might be correlated. Cranek (1952)found both month of calving and previous calving interval to affectmilk and butterfat production. on simpleone-way His analyses were based classifications, which ignored the possibilites of correlations among these and other factors affecting production. He made no attemptto correct records for these two factors, Cranek had used mixed breed age conversion factors on the records of the Red Danish crosses, and there was a possibility that such factors were not entirely appropriate for the breed. Also, a need had arisen 24 for a set of age factors that (l) were based on actual performance of Red Danish cattle at successive ages and (2) could be used to adjust the rapidly accumulating production records on immature cows to "mature equivalents". There was a possibility that year to year deviations and trends in production had occurred, due to factors such as changes in feeding and management. These deviations and trends, if they existed, could have thrown both random errors and biases into the comparisons between foundation generations and the successive Grosses to Red Danish bulls. Further, there was a possibility that more precise comparisons between generations than those made by Cranek could be obtained. Cranek, after correcting the records for age at calving and length of lactation, used the entire remaining error variance to test significance of differences among generation groups. It was possible that the remaining error variance could be reduced still further by correcting the records for effects related to year of calving, month of calving, and previous calving interval. In addition, the error variance after such reduction still would include both additively genetic, non-additively genetic, and environmental components, and only the latter two should be included in the appropriate error term for testing additively genetic differences among generation groups. In view of the preceding observations (possible correlations among variables, need for age conversion factors, possible yearly trends, and potential increase in precision of the comparisons among the generations), it was decided finally to (a) derive least squares estimates of several non-genetic effects on butterfat production, and (b) make corrections 25 accordingly to the individual lactation records. The variables included in the least squares analysis were year of calving, month of calving, age at calving, previous calving interval, and length of lactation period. Real producing ability of cows was considered, also, but not included in the least squares equations as solved. Inbreeding was excluded, since the analysis was on an intra­ cow basis and the effect of inbreeding therefore should be the same (and hence variance zero) among successive records of the same cow. Year effects were not expected to follow any particular pattern, and so one constant was allowed for each year. The effect of month of calving was expected (based oh results in the literature) to be curvi­ linear, and a quadratic curve of the form Y * A - bX + cX2 was postulated. The effect of age at calving was shown by the earlier workers to be curvilinear. Preliminary investigations by the writer on the data used here indicated that a curve of the form I s A + bX - cX for most of the variability due to age. would account The effects of previous calving interval and of length of lactation both were assumed to be linear. The resulting mathematical model thus contained 1 2 constants for years (19A1 to 1952, inclusive), 2 quadratic regressions (four terms), and 2 linear regressions, and necessitated a set of IS equations in IS unknowns. The mathematical model assumed was: Yi j = yu.+ a± + b ^ + d-jM + d ^ l 2 + d^A + d^A2 + d^K + d^L + e, in which Y^j is the record of the ith cow calving in month M of the jth year, at age A (in months), vdth previous calving interval K (in months) and length of lactation period J* (in days). The ai stand for 26 deviations (from the population mean) of real producing ability of individual cows, the bj for deviations associated with year of calving (and presumably due to causative factors present and operating during these years), and the d*s symbolize regression of butterfat yield on the respective variables. jla > is the population mean, and e is the error or random deviation from this mean. Preliminary Operations on the Data Of the 9,572 IBM detail cards originally at hand, only 4,251 were suitable for the analysis. The rest were set aside because of no milk or butterfat data, records shorter than 200 or longer than 365 days, obviously incomplete or sub-normal lactations, and miscellaneous dis­ crepancies. Further, £81 single-record cards (cows with only single records available) among the 4,251 were not used in the least squares analysis, since they would drop out automatically in the process of obtaining the 18 intra-cow equations. Similarly, 1,069 of the 3,370 cards (4,251 minus 881) had no previous calving interval. Therefore only 2,301 detail (or individual lactation) cards were used in the least squares analysis. B. Least Squares Solution for Non-genetic Effects Nature and Properties of Least Squares Estimates The nature of least squares procedure i3 such that the error or residual sum of squares (that which remains after removing the sums of squares due to the specified parameters from the total sum of squares) is minimized. The formal procedure is to (1) develop the error equa­ tion from the mathematical model, (2) take a partial derivative of 27 the error equation with respect to each variable in turn, (3) set 1 partial derivatives equal to zero, and (4) solve the resulting set of equations simultaneously for the unknown parameters. The properties of least squares estimates are such that (Henderson, 1948): 1. Estimates of the parameters are unbiased. 2. Sampling errors for the several parameters are (in effect) minimized. 3. Estimates of the parameters are independent of the distribution of the errors. 4. If the errors are assumed to be distributed normally, tests of hypotheses can be made. 5. Computations are always possible (barring cases of inconsis­ tency and dependency, e.g., denominator of the determinant not equal to zero). 6. Maximum information is obtained from the data. This is a consequence of minimizing the sampling and other experimental errors, the amount of information obtained being inversely proportional to the size of these errors. If the several parameters in a least squares estimate are correlated, the components of variance due to their interactions should be estimated, since otherwise the error term may be improperly increased in size. However, the above-mentioned properties will be true, regardless, and failure to estimate interaction effects will only render tests of sig­ nificance less sensitive than they should have been. 28 Calculation of Terms for the Equations In practice, the formal derivation of the least squares equations is not actually done. Instead, the appropriate equations are set up directly, and the needed sums of squares and cross products are com­ puted from the raw data. The procedure used in this investigation follows the example by Harvey (1953b) and the methods developed by Henderson (1948). Table 1 shows coefficients for the original equa­ tions, i.e., the overall sums, sums of squares, and sums of cross products, uncorrected for the mean. Since the matrix is symmetrical, the coefficients below the diagonal will be correspondingly the same as those above. For instance, will be the term for the lower left hand corner as well as the upper right hand comer. The zeros in cells containing the diagonal terms denote that all coefficients off the principal diagonal within these cells are zero. bjbji = 0 when j ^ j 1. large for juu+ coefficient. Symbolically, Note that the numbers of equations are very- several for bj, and only one for each regression A dot in a subscript denotes summation over that factor. These original equations actually were never set up, since more than 1,100 equations would have been necessary and the resulting computational load would have been truly formidable. Instead, a reduced set of equations (Table 2) was computed by the use of reduction formulas. (See examples immediately below.) These formulas actually obtain an intra-cow matrix, and the second terms of the right-hand members will be recognized as correction factors for obtaining intra­ cow sums of squares and cross products* For example, 29 td •r-3 •H s u in g •o • •<“3 •O •o •H •H oT Usi;: CM •o •rt CM ? CM * •rl *=*J I •rt CA TO • -=c Kl;~> crva CM *r "3 • •H •H i-3 Kl-^ -p •rt CM G •o •H a> a CD r—t TO •H G •H i-M*^ © o «o -^•f“5 -H is : w.-o . •rl •o • cm o (0 G -o ■H CD M rH TO B o o c 0 •<“ 3 •H C •r-3 •H cd -*• O •o « •rt to o u si +> o •rl cd •o 42 CM TO •cT1 no u\ TO s£> TO + a> 3 , -p o s cell are zero CM TO CM • •H W'-T* CM *0 0 Coefficients 1. ■^5 bsjr^ § ■f~5 •H CM •o • •H Table CM * 0 • < w •rt n equations. in the original CM oF CM s *o -cT TO •rt •o •H •o •rt •1-3 •o •r| Ug;^) -P 3 Ki:: •O •H si 30 Table 2. Equa­ tions bj Ax d2 d3 d4 d5 d, Coefficients In the reduced equations (/t+ ax eliminated). b. CCbjbj) d^ d2 dj dj d^ Sums CCbjdi) C(bjd2 ) C(bjd3 ) C(b.,d4 ) C(bjd5) C(bjd6) S(b ) C(d1 ) C(d1d2 ) C(d1d3 ) C(dxd4) C(djd5) C(dxd6 ) S(d1 ) C(d2 ) C(d2d3 ) C(d2d4) C(d2d5) C(d2d6) S(d2) C(d3 ) C(d3d4 ) C(d3d5) C(d3d6) S(d3 ) c(d4) c(d4d5) c(d4d6) S(d4) C(d5) C(djd6) S(d5) C(d6) s(d6 ) 31 C(bjbj) 1 = c(bjbd .) J j CCbjdp C(b;jd2) S(b j) n -0 = M .J = “ .§ S(di) - r i - i - X J • Y . •J ctap C(did2 ) i j nu ' z A - V y - n. _M. /n. ij i. l. n. M 2/n. ij i. i. 1 V i zj i = > 2 / 7 n±. A . (Mi.)2A z • -• zI V i V In the first equation, C(bjbj) is the reduced term (or equivalent to an intra-cow sum of squares), n^j is the original term (analogous to an uncorrected sum of squares), and Zn-?/n. is the correction J J i• factor. Equations for the remaining coefficients of the reduced equa­ tions are similar to these examples. By the use of the reduced equations, not only were ju, and the eliminated, but also the number of equations was reduced to 18. Since the bj equations were not independent, it was necessary to assume J b i = 0 and subtract the coefficient of ^ « in each equation from the J coefficient of each of the other bj terms, after which both the b ^ row and column of the matrix were deleted. Thus only a 17 x 17 matrix was left to solve. It may be w e H to note carefully the broader significance and import of the method described above. While the primary objective was to reduce the number of equations so that a solution would be feasible, the procedure actually led to an intra-cow matrix of variances and covariances. It is possible, in like manner, to compute 32 other reduced matrices, «.g.> intra-sire, and from their solutions to secure estimates of genetic variances and covariances. Solution of the Equations Numerous procedures for solving equations are described in the literature. 1941) The abbreviated Doolittle method (Doolittle, 1878; Dwyer, was selected for this particular problem, since the inverse matrix which appears during the solution was needed to calculate appropriate error terms for testing significance of the estimated parameters, The error mean square (Table 4) was obtained by subtracting from the total sum of squares (corrected for the mean) the sums of squares due to fitting constants for years and regression coefficients for the other variables, and dividing the remainder by 2,283 degrees of freedom. The square root of the product of the error mean square and the appropriate diagonal elements of the inverse matrix gave standard errors for the various constants and regression coefficients, C. Correcting the Data for Non-genetic Effects Development of Correction Factors Estimates of the various parameters obtained in the least squares solution (Table 3) were used to develop additive correction factors for adjusting the original butterfat records. The general procedure was to (a) find the mean for a given variable, (b) calculate the expected average deviation in butterfat yield for each class or level of the variable, and (c) reverse the sign of the deviation. Correction factors developed and used on the butterfat data are shown in Tables 5, 6, 7, and 8. 33 Table 3* Estimates of parameters obtained from least squares analysis. Parameter Numerical value Standard error t-ratio Year of calving: Linear (all years) 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 6.49 2.35 2.76** 15.576 -5.041 18.434 30.577 -0.096 4.303 -3.927 10.545 41.024 75.881 85.323 14.971 13.044 11.146 10.210 8.889 7.403 5.752 4.266 3.385 3.302 3.440 1.040 0.386 1.654 2.995** 0.0108 0.581 0.683 2.472* 12.119** 22.980** 24.803** -0.770 0.009 2.917 0.220 10.012 -0.067 0.3a 0.0022 Month of calving: Linear Quadratic 0.264 0.0004 Age at calving: Linear Quadratic 29.361** 30.455* Previous calving interval: Linear 1.955 0.383 0.994 0.0374 5.104** Length of lactation: Linear -^Significant at 5 per cent probability level. ■^Significant at 1 per cent probability level. 26.578** 34 Table 4* Mean squares for parameters included in least squares analysis. Parameter Years Degrees of freedom Mean square 11 5,384 Month of calving (linear & quadratic) 2 1,217 Age at calving (linear & quadratic) 2 406,390 Previous calving interval (linear) 1 14,039 Length of lactation period (linear) 1 520,408 2,283 757 Residual or error term 35 Table 5* Correction factors for year of calving. Year of calving Correction to butterfat lactation record of individual cow (lb.) 1941 32 1942 26 1943 19 1944 13 1945 6 1946 0 1947 - 6 1943 -13 1949 -19 1950 -26 1951 -32 1952 -39 36 Table 6* Correction factors for butterfat yield according to age at calving, based on the equation Y + 10.012(X - 1 ) - 0.067(X2 - X2 ). Age Corr. Age ^orr. Age mo. lb. mo. lb. mo." 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 40 37 33 30 27 24 21 18 16 13 11 9 7 5 3 2 0 -2 -3 -4 -5 -5 -6 -7 -7 -7 -8 -8 -8 -7 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 47 208 200 193 186 178 171 165 158 151 145 139 132 126 120 114 109 103 98 93 88 82 76 73 68 64 59 55 51 48 44 Note: For ages past 96 months, add 3 pounds for each additional month. 46 Corr. lb. -7 -6 -6 -5 -4 -3 -2 0 1 2 4 6 8 10 12 15 17 20 23 37 Table 7. Correction factors for butterfat yield for length of lactation period (number of days milked)• Number of days milked 200 Correction (lb.) 92 • • • • 250 42 • • • • • 291 292 293 • 1 0 -1 • • * 300 • -8 • • 365 Note: ♦ -73 Intermediate values (indicated by dots) varied by 0.994 pounds of butterfat for each day of difference in length of lacta­ tion period. Actual values used in correcting records were taken to the nearest whole pound. 38 Table 8 . Previous calving interval mo* Correction factors for butterfat yield for length of previous calving interval. Predicted deviation in butterfat production lb. Correction to butter­ fat lactation record of individual cow lb. 9 -3-634 4 10 -2*640 3 11 -1•646 2 12 - 0*652 1 0 0 13 0.342 0 14 1.336 -1 15 2*330 -2 16 3.324 -3 17 4.318 -4 18 5*312 -5 12.7 Note: For calving intervals longer than 18 months, a correction factor of -5 pounds was used uniformly* 39 Only the linear trend for year of calving (Table 3) was used. The estimate of the effect of month of calving on butterfat yield was not significant (Table 3) and consequently no correction factors were developed for this variable. The corrections for age at calving, unlike those commonly used, were additive rather than multiplicative and further based on mean age at calving (64 months) rather than the age (74 months) at which production reached a peak. A convenient example of the computations is afforded by the fac­ tors for previous calving interval (Table &). 12.7 months. The mean interval was The regression of butterfat yield on previous calving interval was 1.955 pounds per month. Therefore the correction factor (to the nearest whole pound) for 12 months w a s +1 pound, for 13 months 0 , for 14 months -1 pound, etc. Procedure for Correcting the Butterfat Data The detail cards were sorted into classes for one variable at a time, and the appropriate correction factors for that variable were gang punched into the detail cards. Then the original butterfat values were corrected by adding algebraically the correction factors punched in each card. An intermediate value, corrected only for year of calving, previous calving interval, and length of lactation, was punched in each detail card. These intermediate values were used in calculating age correction factors. The final fully corrected value punched in each card was used variously, as will be seen later. Estimated Producing Ability of Each Cow The mean or average of all fully corrected butterfat records of each cow was taken as the best estimate of her producing ability. A 40 summary card was punched for each cow (Table 25, Appendix). This card showed number of lactation records, and both total and average butter­ fat yield. The average butterfat record of each cow was used in calculating heritability and in making comparisons among generation groups. Individual lactation records were used in calculating re­ peatability of butterfat production and in the development of age correction factors. D. Heritability and Repeatability of Butterfat Production An estimate of the heritability of butterfat production (Table 10) was needed in connection with comparisons among generations and esti­ mates of genetic progress. The method chosen was that of intra-sire regression of daughter on dam (Lush, 1940). This method was used because (a) no bias should result from selection among dams (Eisenhart, 1939), (b) the errors inherent in this method are probably smaller than in the half-sib correlation method, and (c) an appropriate stan­ dard error of the regression is not difficult to calculate. The average butterfat records of 413 daughter-darn pairs, in the firstcross and foundation generations respectively, were used to estimate heritability of butterfat production. The initial estimate of 0.56, based on cow averages, was reduced to a single-record basis and a value of 0.39 by use of the formula reported by Laben and Herman (1950) . An estimate of the repeatability of butterfat production was needed in adjusting the heritability value to a single-record basis. This was obtained by use of 744 records on 237 cows in the first-cross generation. The computational procedure should be evident from Table 9. 41 Table 9* Estimation of repeatability of butterfat yield. Source of variation Degrees of freedom Mean square Total 743 Between cows 236 1,026 Within cows 507 272 CTo = ^ > 026 Repeatability = - 272)/ 3.72 = Expectation of mean square 203 203 203 + 272 r 0. 4274 cr z crz + 3.72 42 Table 10* Estimation of heritability of butterfat yield* Item Daughters8, Dams Covariance Overall unconnected S* S. & S. C. P. 62,171,123 56,314,428 58,911,145 Correction factors 60,730,749 56,263,076 58,333,750 1,440,374 2,051,352 577,395 Intra-sire corrected S. S. & S* C. P. — H • 2 f 1 577,395 ^2,U>1,352 I / = 0.56294 = heritability based on average records. Applying the formula by Laben and Herman (1950): H - H (r + 1 - r | - 0.56294 ( 0.4274 + \ r 0.3938 ± 0.1129 H 3 single-record heritability. r = repeatability. d 3 average number of records for the dams, fit Daughters by 31 sires. 1 -0,4274 2.1036 43 E. Comparisons of Foundation Generations with Successive Crosses to Red Danish Sires Methods and Assumptions The method of Dickerson and Hazel (1944) for estimating genetic shifts from generation to generation (and hence genetic progress) requires that the selection differential, in terms of phenotypic su­ periority, be known for both sexes in each parental generation. Since this information for the Red Danish bulls was not known, the method could not be used here. Instead, direct comparisons were made between foundation groups and the first-, second-, and third-cross generations of Red Danish females, both between groups as a whole and between daughters and dams. In making these comparisons between generations, it was assumed that: 1. The phenotypic mean of each group is an unbiased estimate of the genetic mean. (This assumption implies the absence of bias due to selection.) 2. The errors e-j^ are normally distributed with mean zero and variance u e . 3. The variance CfJ is divisible into an additively genetic portion (j£ 9 a p o r t i o n c o n t a i n i n g non-additively genetic and perma­ nent environmental effects, and a temporary environmental portion 4. The variance G q is equivalent (for all practical purposes) to the phenotypic variance G p . 5. The major part of the variation in temporary environmental fac­ tors was removed by corrections to the data, and the remainder 44 affects the data randomly. If the phenotypic mean of several performances of the same indi­ vidual is considered to i)e the best estimate of its genetic merit, and if the error variance of a single performance is equal to 0^,minus (Robertson, 1955) > the error variance of the mean of several performances becomes some specific function of length of calving interval (Table 15), and length of lactation (Table 16), and to develop simple correlations among these variables and age at calving (Table 17). Repeatabilities were estimated by the same method used for butterfat production. The correlations were estimated on an intra-cow basis. H. Selection in the Various Generations The comparison of cows in succeeding generations made possible certain estimates of selection (Table 13), since not all dams in a given generation had daughters. The phenotypic difference between mean butterfat yields for all dams and those dams having daughters can be interpreted as a selection differential. Such a differential, when multiplied by the heritability ratio, gives an estimate of the addi­ tively genetic superiority of the selected dams. The calcination of paired age correction factors gave opportunity for estimates of selection, within generations, from lactation to lactation (Table 19). Not all the cows completing first lactations 51 Table 14. Repeatability of month of calving. Source of variation Degrees of freedom Total 2,300 Between cows 1,117 20.642 CT2 Within cows 1,183 7.740 (j 2