OVERDUE FINES: 25¢ per 60 per item RETURNIM LIBRARY MATERIALS: Phce in book return to remove charge fm circulation records "K VO/WF mum Vb SELECTING FOR LACTATION CURVE SHAPE AND MILK YIELD IN DAIRY CATTLE By Theodore A. Ferris A Thesis Submitted to Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Dairy Science 1981 Q"//x5 7/‘ 1:4" ABSTRACT SELECTING FOR LACTATION CURVE SHAPE AND MILK.YIELD IN DAIRY-CATTLE by Theodore A. Ferris Efficiency of production has not been included in selection of dairy cattle. Feeding efficiency in lactating cows is greatest in the early stage of lactation followed by a gradual decline, but health care costs also follow a similar trend. Potentially, then, it may be desirable to select for cows which either produce more in the early part of lactation to take advantage of feeding efficiency, or to select for cows having lower lactation peaks to reduce stress and health care costs. This study determines whether the shape of the lactation curve can be changed, in what way, and to what extend the change would affect 305—day milk yield. An equation by Wood, yt = atbexp(-ct), was used to depict the shape of the lactation curve. Estimates of curve parameters, for initial yield (a), the ascent (b) and the decline after peak (c) were obtained for first lactations for each of 5,927 Michigan Holsteins on DHIA in 557 herds. The model for variance component estimation included effects for herds which were absorbed into seasons and sires. Mixed model equations with Best Linear Unbiased Predictor (BLUP) solutions were used with restricted maximum likelihood estimators to compute variance components in an iterative process. The heritabilities Theodore A. Ferris for these lactation curve characteristics were .06, .09 and .15 for a, b and c, respectively. The genetic correlations for 305-day milk yield with a, b, c and peak yield were -.367, .397, .004 and .911, respectively and the phenotypic correlations were .17, .071, -.107 and .849, respectively. To examine the potential of changing the shape of the lactation curve in conjunction with selecting for 305-day milk yield, selection indexes were set up for three strategies: 1) To increase the ascent to peak production and increase peak yield. This would shift more of the lactation production to the early stage where cows have higher feed efficiency and thereby, potentially increase overall efficiency of production. 2) To delay the time of peak and to decrease the slope to peak while either ignoring or considering persistency. This effort is directed toward reducing stress and health care costs in the early stage of lactation. 3) To flatten the lactation curve by decreasing the peak yield, then at the same time increase the initial yield and persistency which would make up for some of the loss in yield due to decreasing peak yield. Results from indexes in the first strategy suggested that selecting for both an increase in ascent and peak yield was successful and did not decrease 305-day milk greatly. Sire rankings on these indexes were very similar to their rankings on SOS-day milk alone. The second strategy was slightly successful in delaying time to peak and in decreasing the ascent to peak but it decreased the genetic gain in 305-day milk to between -38 and'76 lbs Theodore A. Ferris per generation. This is compared to a gain of 359 lbs when 305-day milk is selected alone. For indexes of the third strategy, selection resulted in flattening the lactation curve, but doing so at a great loss in genetic gain of BOB-day milk. Generation gains ranged from -282 to 6 lbs. The use of indexes in the first strategy were most desirable from the Standpoint of changing the shape of the curve in the desired direction without decreasing 305-day milk appreciably. Indexes in strategies two and three could possibly be useful if more weight were applied to 305-day yield. However, the desired change in the curve shape would be much slower. DEDICATION The thesis is dedicated to my parents, Joseph W. and Wilda S. Ferris, remembering their guidance and advice during my boyhood. ii ACKNOWLEDGEMENTS The author wishes to thank Dr. Ivan L. Mao for his advice, encouragement and support during this graduate program. I wish to express my appreciation to the members of my graduate guidance committee, Dr. J. L. Gill, Dr. W. T. Magee, Dr. C. E. Meadows and Dr. J. A. Speicher for their support and suggestions and in particular Dr. C. R. Anderson for the many discussions on computer analyses during my research program. My appreciation goes to the other faculty members in the Department of Dairy Science who gave me encouragement during my graduate program. A special thanks to a number of friends who were very supportive and giving of their time. A special thanks to Cindy Curtis-Coombs for her effort and assistance in editing and typing of this dissertation and Julie Drake for her help in preparations. And finally, thanks goes to my entire family for their support during my endeavor. iii TABLE OF CONTENTS Page List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . x I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1 II LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . 4 11.1 Merit . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.2 Mathematical Expressions of Lactation Curves . . . . . . 7 11.2.1 Work by Wood . . . . . . . . . . . . . . . . . . . 8 11.2.2 Comparison of equations used to fit lactation cuwes O O O O O O O I C O O O O O O O O O O O O O 12 11.2.2.1 Weighted vs. unweighted log-linear form of Wood's equation . . . . . . . . . 12 11.2.2.2 Linear vs. nonlinear form of Wood's equation . . . . . . . . . . . . . . . . 12 11.2.2.3 Comparisons of other equations . . . . . 15 11.2.3 Problems fitting lactation curves . . . . . . . . 19 11.3 Environmental Effects on Lactation Curve Character- iStiCSo O O O C O O O O O O O I O O I O O O O O O O O O O 20 11.4 Genetic Parameters for Lactation Curve Characteristics . 22 11.5 Selection Index Method . . . . . . . . . . . . . . . . . 27 11.6 Genetic Progress of Using Selection Index . . . . . . . . 32 111 MATERIALS AND METHODS . . . . . . . . . . . . . . . . . . . . 37 111.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . 37 iv IV 111.1.1 Source-defining the population . . . . . 111.1.2 Calculation of 305-day production from test day information . . . . . . . . . . . 111.1.3 Data screening procedure . . . . . . . . . 111.2 Selecting the Method to Fit Individual Lactation Curves . . . . . . . . . . . . . . . . . . . . . 111.3 Model . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Adjusting data for age at freshening . . . 111.3.2 Equations and assumptions of model . . . . 111.3.3 Absorption of herds . . . . . . . . . . . 111.4 Variance Component Estimation . . . . . . . . 111.5 Heritability, Genetic and Phenotypic Correlations 111.6 Select Indexes . . . . . . . . . . . . . . . . . . 111.6.1 Justification and strategies . . . . . . . 111.6.2 Computation of selection index criteria 111.6.3 Computing genetic change and correlated genetic response . . . . . . . . . . . . . 111.6.4 Computing new curves after selection . . 111.6.5 Computing and ranking sires on indexes . RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . IV.1 Test for Normality . . . . . . . . . . . . . . . . IV.2 Marquardt's Method for Least-Squares Estimation on Nonlinear Parameters . . . . . . . . . . . . IV.3 Data . . . . . . . . . . . . . . . . . . . . . . . IV.3.1 Variance Components . . . . . . . . . . . . IV. 3. 2 Heritabilities, genetic and phenotypic correlations . . . . . . . . . . . Page 37 38 4O 42 47 47 48 60 62 66 69 69 72 74 75 76 78 78 80 83 92 98 IV.4 Genetic and Correlated Genetic Change . . . . . . . . . 102 IV.4.1 Changes in lactation production . . . . . . . . 119 IV.4.2 Changes in the shape of lactation curves . . . . 142 IV.4.3 Summary of changes caused by selection indexes . , . . . . . . . . . . . . . . . . . . 161 IV.5 BLUP Solutions for Sires . . . . . . . . . . . . . . . 163 IV.5.1 Ranking sires by indexes . . . . . . . . . . . . 164 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . 167 List of References . . . . . . . . . . . . . . . . . . . . . 173 vi Table 11.4.1 111.1.1 111.1.2 111.1.3 111.4.1 III.6.l IV.3.1 IV.3.2 IV.3.3 IV.3.4 IV.3.5 IV.3.6 LIST OF TABLES Genetic and phenotypic correlations among lactation curve parameters for first lactation by Shanks et al. ’ 1980 O O I O O O O O O O O O O O O O O O 0 Frequency distribution (percent range) of first lactation records by season and age . . . . . Frequency distribution of first lactation records by age 0 O O 0 O O O O O O O O O O O O O O O O O 0 Frequency distribution of first lactation records by sires . . . . . . . . . . . . . . . . . . . . . Average of heritabilities reported in the literature and initial ratios used for 305-day milk and lact- ation parameters . . . . . . . . . . . . . . . . . Indexes for the three strategies and their weights . Amount of data after each step of screening Total and adjusted sums of squares . . . . . . . . Means, standard deviations and ranges of 305-day milk yield and lactation curve parameters before adjusting for age and age squared.‘ . . . . . . . Comparison of lactation curve parameter means in the present study and other studies reported in the literature for first lactation cows . . . . . . REML estimates of error and sire variance com- ponents for 305-day milk yield and lactation curve parameters . . . . . . . . . . . . . . . . Genetic variances (diagonals) and covariances (off-diagonals) for 305-day milk and curve parameters . . . . . . . . . . . . . . . . . . . . vii 24 40 41 41 65 85 88 9O 91 93 94 Table IV.3.7 IV.3.8 IV.3.9 IV.3.10 IV.3.11 IV.4.1 IV.4.2 IV.4.3 IV.4.4 IV.4.5 IV.4.6 IV.4.7 IV.4.8 IV.4.9 Page Phenotypic variances (diagonals) and covariances (off-diagonals) for 305-day milk and curve parameters . 95 Standardized genetic variances (diagonals) and covariances (off-diagonals) . . . . . . . . . . . . . . 96 Standardized phenotypic variances (diagonals) and covariances (off-diagonals) . . . . . . . . . . . - 97 Heritabilities, genetic and phenotypic correlations for 305-day milk yield and lactation curve parameters . . . . . . . . . . . . . . . . . . . . . . 99 Heritability values reported for lactation curve parameters and 305-day milk . . . . . . . . . . . . . .100 Change in lactation curve parameters after one generation of selection for 305-day milk yield alone . . . . . . . . . . . . . . . . . . . . . . . . .103 Change in each lactation curve parameter when selecting alone for itself . . . . . . . . . . . . . .104 Change in 305-day milk yield when selecting for lactation curve parameters alone . . . . . . . . . . .104 Change in lactation curve parameters when selecting for a, b or c alone . . . . . . . . . . . . . . . . . .105 Genetic change in 305-day milk and curve parameters for various indexes after 1 generation of selection .106 Percent genetic change for 305-day milk and curve parameters in various indexes relative to change when selecting for milk alone for 1 generation . . . .109 Percent genetic change in 305-day milk and curve parameters relative to change when milk or parameter is selected alone for 1 generation . . . . . . . . . .112 Generations 1,5 and 10 genetic values for 305—day milk, peak and time of peak for some group 1 indexes and milk selected alone . . . . . . . . . . . . . . .121 Generation 1, 5 and 10 genetic values for 305-day milk, peak and time of peak for group 2 indexes . . . .125 viii Table IV.4.10 IV.4.11 IV.4.12 IV.4.13 IV.4.14 IV.5.1 IV.5.2 Generation 1, 5 and 10 genetic values for 305-day milk, peak and time of peak for group 3 indexes . . 127 Genetic values for curve parameters a, b and c for generations 1, S and 10 using group 1 indexes . . . 129 Genetic values for curve parameters a, b and c for generations 1, 5 and 10 using group 2 indexes . 131 Genetic values for curve parameters a, b and c for generations 1, 5 and 10 using group 3 indexes . 132 Discrepancies between expected and estimated 305-day milk yield and peak yield for various indexes . . . . . . . . . . . . . . . . . . . . . . 134 Standard deviations of BLUPs for 305-day milk yield and lactation curve parameters for 150 Sires O O I O O O I O O O O O O O O O O O O O O O O 163 Rank correlations between sires ranked for milk only and other indexes . . . . . . . . . . . . . . 165 ix Figure 10 ll 12 13 LIST OF FIGURES A hypothetical curvilinear relationship between two traits ’ x and Y. O O O O O O C C O C C O O C 0 Change in the shape of the lactation curve when i is changed from 25 to 43 in Wood's equation . . Change in the shape of the lactation curve when b is changed from .24 to .33 in Wood's equation . Change in the shape of the lactation curve when c is changed from .032 to .041 in Wood's equation . . . . . . . . . . . . . . . . . . . . Change of the shape of the lactation curve one generation of selection on a_alone . . Change of the shape of the lactation curve five generations of selection onIQ alone . Change of the shape of the lactation curve after after after ten generations of selection onIE alone . . . . . Change of the shape of the lactation curve after one generation of selection on b alone . . . . . . Change of the shape of the lactation curve after five generations of selection on b alone . . . . . Change in the shape of the lactation curve after ten generations of selection on b alone . . . . . Change in the shape of the lactation curve after one generation of negative selection on c alone . Change in the shape of the lactation curve after five generations of negative selection on c alone . Change in the shape of the lactation curve after ten generations of negative selection on c alone . 143 144 145 147 148 149 151 152 153 154 155 156 Figure 14 15 16 17 Page Change in the shape of the lactation curve after one generation of selection on index 1 milk, 103, 1b, —10c and 1 peak yield . . . . . . . . . . . . . 157 Change in shape of the lactation curves after five generations of selection on index 1 milk, 10a, lb, -10c and 1 peak yield . . . . . . . . . . . . . 158 Change in the shape of the lactation curve after ten generations of selection on index 1 milk, 103, 1b, -10c aid 1 peak Yield 0 o o o o o o o o o 159 Change in shape of the lactation curve after ten generations of selection in index 1 milk and 6 peak yield . . . . . . . . . . . . . . . . . 160 xi I INTRODUCTION Selection of dairy sires has primarily been based on single trait evaluation of the total lactation milk production of daughters and/or butterfat yield, and in some cases, type traits. These traits are also considered in cow selection. More recently, milk, fat and overall type have been considered in an index as an alter- native or supplemental method of ranking sires. Total merit of an individual, in the strict sense, refers to the genotype for a particular trait or group of traits weighted according to their economic value. Selection index is referred to when the phenotypes of a number of traits, usually of economic importance, are considered jointly. The index of a particular individual may be defined in as many ways as there are indexes combining a number of traits by various weights. In the broad sense, total merit of an individual represents its overall economic value genetically plus the return over costs generated by the individual. This would include total value of milk, meat and offspring minus any costs associated with the out- puts. One may consider such variables as feed costs, costs of reproduction, health care costs, and loss of production due to disease and physical characteristics. Presently, it is difficult to get information on many of these traits in order to determine genetic parameters and include them in a total merit scheme. Efficiency of production is defined as dollars of output divided by dollars of input. Cows with greater efficiency of production would produce a larger net return. Selection on 305—day milk and fat yield is essentially selecting for gross milk income per lactation. Efficiency at which lactations are produced has been ignored,partly due to the difficulty of obtaining data on inputs. However, it is known that cows utilize feed more efficiently in early lactation. Granting, that part of this efficiency is due to catabolism of body fat. Cows producing more in the early stage of lactation may be more efficient, i.e., produce the same amount of milk for less cost. On the other hand,hea1th care costs are typically greater in early lactation. These costs may be related to the stress associated with high production and the cow's inability to consume enough feed to meet her requirements. Then,it may be advantageous to select individuals which do not peak as high and are more persistent. To better define merit,then, it would be more desirable to consider the efficiency of production. This efficiency is likely to be related to the manner in which a single or several lactations of a cow are produced. That is, the shape of a cow's lactation curve is probably strongly related to the magnitude of total out- puts minus total inputs or overall efficiency. Also the shape of one lactation may influence the following lactations or life- time productivity. Without addressing the question of what would be the optimum shape of the lactation curve, one first needs to know if the shape is heritable. Then second, how can the shape be altered. A number of studies have dealt with fitting mathematical equations to lactation curves. Several researchers have estimated heritabilities of parameters within the equations used. To change the shape of the lactation curve, one would select for curve characteristics (parameters) of the equation along with total yield in a selection index. An added gain would occur if measur- able lactation curve characteristics are more heritable and are highly correlated genetically to 305-day milk yield. Then, they can be used in a selection index to increase genetic gain in 305-day milk, as well as change the curve shape. The objectives of this study are: 1) Compute the genetic parameters of the lactation curve characteristics and 305-day milk yield, their heritabilities, and genetic and phenotypic correlations; 2) Devise selection index criteria for lactation productivity using the curve parameters and 305-day milk yield; 3) Compare the genetic changes in 305-day milk yield and the curve characteristics achieved by these indexes with progress when selecting for milk only; 4) Compare sire rankings by these indexes and their ranking considering 305-day milk yield alone. II LITERATURE REVIEW The merit of a sire or a cow can be defined many ways, depending upon the traits under consideration. For the context of this study, merit will be a function of 305-day milk yield and desired change in the shape of the lactation curve. However, the optimum shape of the curve will not be defined. Productivity will be defined as the amount of milk yield achieved in a 305-day lactation by any selection index used to change milk yield and the shape of the production or lactation curve. The shape of the lactation curve can be defined by an appropriate mathematical expression. Change in the shape will be a function of the change, due to selection, in the constants of the mathematical equation used. These concepts will be used to determine the flex- ibility of the shape of the lactation curve and the influence of change in shape on 305-day milk.production. The review background covered,wi11 then include discussions on efficiency of production,which may suggest how the shape of the curve should be altered, merit, mathematical descriptions of the shape of lactation curves, selection index and genetic progress through selection. 11.1 MErit Everett (1975) developed equations to predict differences between sires in return over investment for milk sold and percent return on investment for heifer and milking cow sales. Pearson (1976) discussed including sire's conception rate with a sire's predicted difference dollar value (PD$) to estimate the profitability of an ampule of semen. This was an attempt to express the joint effects of these two traits in deviations between sires. McGilliard (1978) computed net returns for genetically superior sires when considering income of milk and fat for daughters. Semen cost per ampule was included for each sire, while a number of other variables were simulated, sudh as conception rate, probability of female calves and age at freshening. These simulated variables were included to map out the income function, for all lactations over a number of generations, derived from the initial ampule of semen of a sire. Everett (1975) and McGilliard (1978) were computing by various methods, a more precise value of semen for a particular sire by considering the sireksgenetic merit (Predicted Difference) and semen cost. This is reflected by income over semen cost. By doing so,they suggested a number of variables that influence the profitability of a sire's daughters. They did not address the sires' genetic merit for these traits. Bakker at al. (1980) derived a profitability index for sires which included milk, fat yield and stayability, i.e., how long daughters remain in the milking herd. This is an attempt to expand the genetic merit of sires to traits other than milk, fat and type as was Pearson's (1976) work. Shanks et al. (1978), Hansen et a1. (1979) and Shanks et a1. (1981) investigated the effect of selection for milk production on reproduction, health and health care costs of daughters. These studies suggest there is a positive correlation between milk 6 production and health care costs. However, the higher production more than paid for the cost of health problems. Shanks (1979) further computed heritabilities and genetic correlations for some health problems. Total health costs and total health disorders had heritabilities of .03, .12, .11 and .02, .ll, .05 for lactations l, 2 and 3,respective1y. Mammary cost and mammary disorders both had heritabilities of .11 for first lactation cows. Heritabilities, in general, were low for health problems. The genetic correlations between first lactation mature equivalent (ME) milk and total health costs was .07, ME milk and total health disorders -.22 and ME milk and mammary costs was -.47. The highest genetic corre— lations with ME milk, outside of those computed to be greater than 1, were .76 with locomotion disorders and .69 with locomotion costs. Five variables associated with reproduction had genetic correlations of greater than 1 with ME milk. However, these traits had heritabilities of less than .02. Work by Shanks et a1. (1978), Shanks (1979), Hansen et al. (1979) and Shanks et a1. (1981) suggest merit of sires can be expanded to encompass other traits which reflect losses or gains in economic value of their offspring. This would better indicate the productivity of daughters in terms of total output and the net income of daughters,i.e., outputs minus inputs. Efficiency of daughters can also be determined by dividing output by units of input and then put into terms of merit. 7 Hooven et a1. (1968) found the genetic correlation between feed efficiency and milk production for their data was .92, with heritabilities of .46 for feed efficiency and .62 for milk production. Miller and Hooven (1969) further found that feed efficiency is greatest during early lactation and decreases throughout lactation. Part of this efficiency in early lactation is attributed to catabolism of body fat. With feed efficiency the greatest in early lactation, it may be desirable to select for individuals which pro- duce more of their milk during this period of lactation. Hansen et a1. (1979) and Shanks et a1. (1981) on the other hand, found health costs were greatest during higher production in early lactation. Therefore, one may want to select cows with lower peaks to reduce stress and possibly lower health costs. The above two cases would consider merit either in terms of net income or efficiency of production within a lactation. Then, the manner in which a cow produces a lactation may influence the efficiency ‘for the overall lactation. This would lead to the importance of the shape of the lactation curve, which reflects the distribution of milk production during a lactation. 11.2 Mathematical Expressions of Lactation Curves The curve of a typical lactation by a dairy cow can be de- scribed as having three stages. The first stage is an incline in production after freshening, followed by the second stage, peak production, which occurs 4 to 8 weeks after calving. The third stage is a steady decline after the peak. Numerious studies have dealt with describing the shape of lactation curves for milk production in dairy cattle, and several are reviewed in the following sections. 11.2.1 Work by Wood Wbod (1967) stated that a number of factors may influence the total yield for a single lactation, but the general shape of the curve remains substantially unaltered. He believes that the shape of the curve is economically important and suggests that cows which produce at a moderate level throughout a lactation are to be preferred to those which produce much at their peak and little thereafter. But no reasons for these arguements were given. Wood (1967) mentioned Gaines' (1927) formula as one of the first attempts to describe lactation curves by a mathematical function. Gaines' (1927) formula was: Kt (11.2.1) Y - ae- where y is yield to week t; e is the base of natural the logarithm and a and K are constants. This equation was an attempt to describe the decline in production after peak. The log-linear form of the equation was fit using a hand drawn approximation of the regression line. WOod also mentioned Nelder (1966) who described an inverse polynomial: 1x + bzxz) (11.2.2) where yx is the yield at week x; and b0, b1 and b2 are constants. yx - x/(b0 + b Expected maximum yield occurs when x equals the square root of (bO/bz) and this yield is: -1 (2/b0b2) + b1) . Wood (1967) believed that because the lactation curve initially rises to a peak following calving and then declines gradually, that the shape is essentially a gamma curve: yt - atbexp(-ct) (11.2.3) where exp(-ct) represents the base of the natural logarithm and can be written as e-Ct, and yt is the average daily yield in week t and g, b, and c are constants. Wood defined two other curve characteristics, each as a function of these constants: Peak yield occurs when t I b/c and Peak yield is ymax = a(b/c)bexp(-b). Wood (1972) later mentioned that g is a constant and a general scaling factor indicating the average daily yield at the start of lactation; that b is a parameter representing the rate of increase to peak yield; and that c represents the rate of decline after peak. The parameter §_will be underlined when it appears in a sentence. Wood (1967) also took the integral of average daily yield to estimate total yield to the t-th week: y - aOfT tbexp(-ct) dt, (11.2.4) t which can be evaluated using tables of the incomplete gamma function. Total yield is then: y . a/cb+l F(b +.1) (11.2.5) where T is the gamma function. 10 Because yt = g_when tbexp(-ct) I 1, then for lactations starting at the same level, Wood (1967) has suggested that the total yield, y, becomes a function of c‘-(b + 1). He defined this function as "persistency", and referred to it as the extent to which peak yield was maintained. WOod symbolized his term of persistency as S which will be used in this text to indicate c—(b + 1). To estimate the parameters, Wood (1967) used a log-linear form of equation (11.2.3) which was solved by multiple linear regression: 1n yt I 1n a + b 1n t - ct (11.2.6) where 1n symbolizes the natural logarithm. Multiple linear re- gression establishes the regression line with minimum residual error or sum of the squared deviations between the data points and the regression line. For regression, the equation takes the form: In yt = 1n a + b lnt - ct + e (11.2.7) 1: where et refers to the descrepency between the observed and estimated yield at week t. Equation (11.2.6) is deterministic, i.e., having no error, and (11.2.7) is probablistic which accounts for error of measurement. Hereafter, equations which are deterministic will be referred to as equations and those that are probabilistic will be referred to as models. WOod's 1969 study investigated further the characteristic of (11.2.7). Noting it compared favorably to Nelder's inverse ll polynomial curve. He showed that it accounted for 95.4% of the variation in monthly yield against 84.4% for the inverse polynomial. This comparison was made, however, with a small set of data. Wood (1967) used weekly samples of 859 Friesian lactations classified by parity and month of calving. The parameters 3, b and c_of (11.2.6) are evaluated for each cow's curve. Goodness-of-fit for the natural logarithm form of the equation was determined by the amount of variation of weekly yield accounted for by the function, i.e., the square of the multiple correlation coefficient (R2). Nelder (1966), however, argued the inability of Rz's to indicate the best model. Nelder pointed out, that for the known model, y I X2, that using values of X I l, ..., n, a straight line fit yielding an R2 greater than .93 could be obtained. Using Fisher's Z—transformation of the multiple correlation coefficient, Wood (1969) showed that an analysis of variance of Z for individual cows indicated more variation between months of calving than within months. The equation accounted for 73.8 to 91.22 of the variation in log weekly yield with 82.32 as an average for these cows. Wbod (1976) suggests it is possible to estimate b and c only by reference to the whole population when monthly weights are used, because there would be too few points to provide any precision on individual cows. However, Wood remarked that the deviations of individual cows from a general equation with population values for the parameters (a, b and c) can provide an estimate for 12 goodness-of-fit. It would be necessary, however, to find estimates for individual cows to determine genetic parameters for g, b and c. 11.2.2 Comparison of equations used to fit lactation curves 11.2.2.1 ngghted vs. unweighted log:linear form of Wood's equation Comparing the log-linear form of Wood's (1967) equation using weighted and unweighted regression, Shimizu and Umrod (1976) indicated the weighted regression equation provided a better fit. Equation (11.2.7) is the unweighted form. The weighted regression equation was: 1n yt I In a + b 1n t - ct + et/yt (11.2.8) where the inverse of the observed variable, yt, is the weight. Results from Shimizu and Umrod (1976) suggest the weighted equation provided better fit in the early stage of lactation and the unweighted equation produced better fit in late lactation. This was determined by calculating mean deviations from the computed regression line for each cow. The weighted equation had slightly fewer abnormal curves. Abnormal curves being those with either a negative b or c. II.2.2.2 Linearvs. nonlinear form of Wood's equation Kellogg et al. (1977) investigated the assumption Wood (1967) made in using the log-linear equation. Wood (1967), by using the logarithm transformation of (11.2.3), made the assumption that as daily milk yield increased (peaked), the variance increased. Therefore, it was assumed that a logarithm transformation was needed to achieve homogeneous variance. Kellogg et al. (1977) used a nonlinear method of Marquardt (1963) to obtain deviation estimates in the 13 untransformed form of Wood's equation (11.2.3). With 36 cows having four lactations, Kellogg and coworkers then found, with certain considerations, the scatter of data around the estimated lactation curves appeared uniform. This supports the use of a non- linear equation and indicates logarithm transformation may not be appropriate. They also found the variance in the first month was smaller than for some later months but otherwise no differences among variances were observed. They suggest then,this supports the assumption of homogeneous variance for months 2 to 10 using un- transformed data. Kellogg and coworkers' data consisted of monthly weights except for weekly averages used in the first two months of lactation. They were able to compare variances from month to month because all cows were tested close to the same times post- partum. Kellogg and coworkers suggested that besides random variation contributing to comparison of variances over lactation curves, two other factors are involved. "Cows have different lactation curves so individuals following different curves will differ much more at the second than the eight month. Secondly, the actual days postpartum for the second record of monthly production can range from about 35 to 70." The authors concluded from this that there is more diversity in stages of lactation represented in early months than in later months. They also suggested that the nonlinear form of Wood's equation (11.2.3), using intrinsic nonlinear regression, accounted for both these factors. 14 Cobby and LeDu (1978) also fitted data to the untransformed equation (11.2.3) and compared results to fitting by unweighted leasts-squares of the logarithm form (11.2.7). With their data, they found unweighted leasts-squares accounted for 94.2% of the variation. Plotted residuals showed a positive to negative trend from week 2 to 18, and the estimated curve overestimated the data between the 2nd and 10th week of lactation. The curve estimated by the nonlinear regression fit the data better and produced residuals that were more uniformly distributed. Cobby and LeDu (1978) indicated there was an average reduction in residual mean square of 14% when using nonlinear techniques as opposed to linear regression on the logarithm transformed equation. It is noted that the reduction was due to minimizing squared deviations from y,instead of natural logarithm of y,which is the case with linear regression on the log-transformed equation. To compare MSE's,Cobby and LeDu first untransformed the residuals of the log-linear model and then computed a new MSE. Anderson (1981) alluded to the fact that this comparison is meaningless because untransformed residuals after a log-linear fit should not produce a smaller MSE when the nonlinear fit is expected to produce the minimum MSE for the untransformed data set. Guest (1961) pointed out that for nonlinear equations which are transformed by logarithms, the appropriate weight for weighted least-squares is proportional to the square of the dependent variable. This gives an approximation of the nonlinear model. Cobby and LeDu (1978) used such a model: yt - ln a + b 1n t - ct + et/yi (11.2.9) 15 and found the weighted log-linear equation produced a curve similar to that of the nonlinear equation. 11.2.2.3 Comparisons of other equations Further comparisons between equations were done by Yadav et a1. (1977), using 745 lactation records from 249 cows in two breeds (Hariana and Friesian-Hariana crosses). Four equations were examined: the exponential function yt I A exp(—Kt) (11.2.10) the inverse quadratic polynomial 2 yt t/(b0 + blt + bzt ) (II.2.11) the gamma-type equation yt I Atb exp(-ct) (11.2.12) the parabolic exponential function yt - A exp(bt + ct2) (11.2.13) Using the R-square value as a measure of fit, they found that the inverse quadratic polynomial and the gamma-type equations gave better descriptions of the lactation curves. The transformed versions of these four functions were explored by Basant and Bhat (1978) who used weekly milk production records of 1,202 Hariana cows to compare the relative efficiencies of the functions. After transforming the observed milk (yield, yt) to allow for linear multiple regression methods the equations become respectively: 1n yt I 1n A - Kt (11.2.14) 2 ln yt - b0 + blt + b2t (11.2.15) 16 1n yt I ln A + b 1n t I-ct (11.2.16) ln yt I 1n A.+ bt + ct2 (11.2.17) Using the R-square obtained as a measure for relative efficiencies of these functions,the authors concluded that for those first lactations that were 44 weeks in length, the gamma-type equation (11.2.16) fit the best, while shorter length lactations were best fitted by the parabolic exponential function (11.2.17). For lactations two through six the average weekly yield was best fit by the inverse polynomial (11.2.15). Schneeberger (1981) used two models to estimate lactation curves for Swiss Brown cows: 1n (yi) I ln (a) + b 1n (ti) — cti + ei (11.2.18) ln (yi) I 1n (a) + b 1n (ti - to) - c(ti - to) + ei (11.2.19) where tO indicates the time of initiation of lactation which occurs prior to calving. Equation (11.2.19) gave smaller mean squared errors than (11.2.18). Schaeffer et al. (1977) compared a nonlinear technique for predicting 305-day lactation production with methods using multi- plication or extension factors and regression coefficients. The authors describe their equation as a one-compartment open equation which is: yij . A exp(-B (1 — t0)) [1 - exp (-6 (1 - t0))]/ B exp(Eij) (11.2.20) 17 where yi is the amount of milk given on the i-th day of the lactation of the j-th cow; to is a lag time parameter and may indicate when a cow's udder begins to lactate prior to calving; B is the slope of the lactation curve during the increasing pro- duction stage. A is associated with peak production; 8 is the slope during the decline in production after the peak; 811 is a residual effect which subsequently was split into: exp(ei ) I exp(Yi.sin (1p)) exp(ei ) where i sin .04 then (p < .05). B - Values in parenthesis are standard errors. If the absolute value of the phenotypic correlations 25 hundred days of lactation as a percentage of yield in the 1-th hundred days. The heritability estimates for these measures of persistency ranged from .19 to .29 with the largest for P3:l. Genetic correlations were .05 to .16 for persistency measures and 305-day yield, and -.23 to -.35 for persistency measures and lOO-day yield. The c was not genetically correlated with 305-day yield but positively correlated with lOO-day yield. The author reported that a genetic relationship between b and c, and 305-day yield was non-exsistent. However, the genetic correlations among 305-day yield and measures of persistency ranged from .05 (305-day fat yield with P2:l) to .16 (305-day milk yield with P2:l). ,There were positive genetic correlations for b and c with lOO-day milk yield, .24 and .29, respectively. Schneeberger (1981) concluded that the genetic correlations between lOO-day yield and c, lOO-day yield and measures of persistency, 305-day yield and c, and 305-day yield and measures of persistency suggest that breeding for high yield at the beginning of the lactation would lower persistency as he measured it, while genetic improvement of the standard (BOB-day) lactation would not affect it negatively. These conclusions agree with those by Gravert and Baptist (1976) who found a negative genetic correlation between initial yield and persistency measured by the slope of the lactation curve. Shanks and coworkers (1980) also found a low negative genetic relationship between initial yield and S. Shanks (1979) adjusted the early part of the lactations by modifing Shook's (1975) factors to compute yield on day six. 26 Using Shook's factors the yield on day six is always going to be less than the first monthly test after day six. This was done to reduce the number of atypical lactations shapes, i.e., negative b's. Shanks reported less than 1% atypical curves. Schneeberger (1981) on the other hand used a parameter to, (11.2.19), for the same purpose. Schaeffer and coworkers (1977), also used to, where to indicates the time of initial lactation but used it in a different equation (11.2.20). This is assumed to be some point prior to freshening,where the lactation process starts. These adjustments insured a curve which increases from day one to a peak. Therefore, a negative b is not possible, i.e., an ever decreasing curve. Shimizu and Umrod (1976) noted 34% while Schneeberger (1981) noted 22% atypical lactation curves. Schneeberger (1981) noted that for both of his models the percentage of atypical curves decreased as lactation number increased. Atypical shapes were greater for flat curves (42%). On the other hand the smallest MSE was for first lactation and greatest for second lactations. Both percentage of atypical shapes and MSE was lower in the second model (11.2.19) which included t For Schneeberger's data, the majority 0' of the atypical curves would probably be negative c's since esitmating t0 should eliminate most negative b's. Schneeberger (1981) remarked that when estimates for 305-day and lOO-day yield were computed by integrating the estimated lactation curve,the heritabilities were high (.4). The heritabilities 27 for b was .15 for milk and .12 for fat, and for c, .20 and .18 for milk and fat,respectively. A summary table of mean values for curve parameters is included in.the Results and Discussions section (Table IV.3.4). 11.5 Selection Index Method Smith (1936) first applied selection index theory to plant breeding while Hazel (1943) applied the theory to animal selection. The principle mathematical results and many of the mathematical and statistical difficulties involving the construction and use of selection indexes are discussed by Cochran (1951). Henderson (1963) provided proofs for a number of the properties of selection index criteria and also expressed the selection index procedure with matrix notation for practical computation. Selection index refers to selecting individuals from a popu- lation based on a criterion for the purpose of making genetic gain in a single trait or a number of traits. The phenotypic observations of the particular traits of interest are combined by computed weights (b's) which will be noted as a vector by the underscore character, ~ i.e., b. This is also to differentiate it from the parameter b of Wood's equation. All vectors will be denoted as underscored lower case letters, while upper case under- scored letters will represent matrices. The goal is to predict the total merit, or aggregate genotype of an individual using the selection index. For total merit, the aggregate genotype is T :- a'g (II.5.1) ~ 28 where a is an k x 1 vector of relative economic weights, g is a ~ k x 1 vector of additive genetic values expressed as deviations from their means of the k economically important traits. Equation (11.5.2) demonstrates the form of an index. This is the estimator of total merit, the selection index: 1 a 9'2 - g'g'g‘lg (11.5.2) where b' is a m x l coefficient vector which is equated to P-1§a from the equation Pb I Ga, where 2-1 is the inverse of the m x m phenotypic variance-covariance matrix P, G is a k x k matrix of genetic variances and covariances and p is an m x l observation vector of phenotypes expressed as deviations from the mean of the estimated fixed effects (y — §b). The selection index equation for unrelated animals can be written as g1 I 913:121’ where for the i-th animal, g1 is the vector of additive genetic values of the traits considered, and Bi is the vector of corresponding phenotypic deviations where E 2 therefore bT I G’TI/oI I l, I then AT I A1. Mac (1971), using matrix notation, describes the computing formula for AT, the true genetic progress when using the optimum index is AT - .15-'21; D - /§'§g’lg§ D (11.6.12) when constructed with known parameter values and with phenotypic observations on all the traits in the total index. 35 When pOpulation parameters are not known, as in practice, and the optimum index is not available, then one uses estimates of the optimum weights, b, obtained from the equation: Pb I Ga or «9 AA Cb I t. Mao (1971) notes that when truncation selection is performed ~~ utilizing such an index involving b, i.e., 1 I b'p the improvement in T will be: A r—A -1A -1 _1A AT' - rTfnoT . (a'Gbl/b'Pb)D = (a'Gp GaI/a'Gp pP Ga)D. (11.6.13) The selection intensity for upper truncation selection in a normal distribution would be D I z/q and for lower truncation selection D I -z/q. Therefore, AT is the maximum attainable progress and -AT is the minimum. Harris (1963) stated that a population of AT' values exists with upper and lower limits of +AT and -AT. He further remarked that with repeated estimations, different AT' values giving a "population" of AT' values will be distributed closer to the AT or true values. This occurs as the accuracy of estimation improves. In the practical situations, the progress from selection is estimated by subsituting estimates for the true values in AT of (11.6.12) to obtain the estimated gain: AT I rTEDST I /é'§§-1§a D. (11.6.14) One of the practical uses of selection index occurs when selection is desired on an unobservable or lowly heritable trait which has a high genetic correlation with a trait of higher 36 heritability. By selecting for the trait with a higher heritability, progress in the trait of interest will be greater than selecting for italone. Following this notion, when selection index is used, the genetic response of a single trait within an index frequently is of interest. Van Vleck (1979) demonstrated the genetic response of an individual trait included in an index by: AGl = C°V(G1’ I) (11.6.15) 01 A where 01 comes from (11.6.8) and Gov (61’ 1) represents the genetic correlation between trait l and the index: A A “ 2 Cov(G,I)=bo +bo +...+bo 1 1 cl 2 6162 N GlGN (11.6.16) The genetic response for a trait not included in an index can be computed by substituting Cov (G1, I) with Cov (GN+l I) where: C°V (GN+1, 1' ’ b1 OchN+1 + ... + bNoG G (11.6.17) N N+l and N+l refers to the first trait not included in the index. III MATERIALS AND METHODS 111.1 Data 111.1.1 Source — defining thegpopulation Monthly records from the Michigan Dairy Herd Improvement (DHI) papulation of 168,193 cows and 2,390 herds were taken for the period between August 1978 through August 1980. Records used were monthly records on first lactations, with the first test day prior to 35 days into the lactation and the last test occuring after 280 days, with the requisite that these cows be Holstein and identified by sire. Test refers to official monthly test day recording of milk and butterfat produced on that day. Any cows with a reported abortion during this record were discarded as well as cows on unofficial test. After the editorial process for the above criteria, the total useable records were 10,107 lactations. One must note that this population is a subpopulation of all DHI cows in Michigan (1144 of the 2,390 DHI herds) and is not necessarily a true random sample of the DHI population since those animals with sire identification may constitute a superior population. It would be logical for one to suggest this if those cows sired by superior artificial insemination (AI) sires are identified more frequently than those by poorer AI sires or unidenti- fied home bred bulls. It is also generally noted that the DHI population itself is superior to the overall population of dairy COWS o 37 38 111.1.2 Calculation of SOS—day production from test day information DHI 305-day records currently are estimated by the test day interval method using daily milk weights recorded monthly. This method takes the average of the test day weights for two consecutive months and multiplies it by the number of days between these test dates. This then is the amount of milk estimated to be produced during this interval between tests. The daily milk for all days between the calving data and the first test is estimated to be the same as produced on the first test day. Like- wise, if the last test occurs prior to 305—days, the daily yield estimates between that test and 305-days is computed to be the same as the amount produced at the last test day. These estimates produce a positive bias because cows are usually increasing in production in the early stage of lactation and decreasing when they are approaching 305—days. Shook (1975) presented adjustments to the test interval method for the first, second and last tests.- The adjustment for the first test accounts for the usual incline to a peak around 45 days into lactation. Because a cow is normally increasing in production prior to 30 days, less milk is actually produced than is credited by the test interval method. Therefore, a Shook factor is used to adjust this part of the cows estimated production. For the second test, an adjustment is made if the typical peak time occurs between the first and second test date, which would cut off the tap of the peak. Therefore, a Shook factor 39 here adds to a cow's production estimate. Finally, when the last test occurs prior to 305-day and since a cow is normally declining at this point, the test interval estimate for this period would be biased upward. Therefore, a Shook factor is used to make the adjusted estimate for the last interval. Examples of the computations of these adjustments used on this data set (Shook factors are in parenthesis) are given below: For the first lactation record with the first test of 46 lbs at 30 days in lactation, one would have 46 lbs x 30 days x (.84), giving 1159.2 lbs where .84 is the appropriate Shook factor. Then for a second test of 50 lbs at 62 days in lactation, one would have [46 + 50]/2 x 32 days x (1.01) giving 2294.72 lbs. Then for a last test of 32 lbs at 280 days and dry at 305 days, one would have 32 x 25 days x (.96) giving 768 lbs. If a test after 305 days was reported, the interval between 305 days and the previous test was computed by interpolation. For example: with a yield of 31 lbs at 290 days and 25 lbs at 320 days, one would have 320 minus 290 giving 30 days and 305 minus 290 giving 15 days so that: 15/30*[3l - 25] I 3 lbs then 25 + 3 (28 lbs) is the estimate on day 305 then: [31 + 28]/2 x 15 days I 442.5 lbs . 4O 111.1.3 Data screening procedure In addition to the pre-requisites for records to be included [111.l.l], more records were deleted for: l) sires having fewer than 8 daughters,2) herds having only one sire, and 3) herds having fewer than 3 cows. This was done simultaneously. The restriction on the number of daughters per sire was arbitrary. Herds having only one sire were deleted because sire would be confounded with herds and would not contribute to the estimation anyways. Herds with fewer than 3 cows also would not have enough degrees of freedom to contribute to the estimation of the factors in the model. The total usable records was then reduced from 10,107 to 5,927 cows after 3 rounds of deletions. Tables 111.1.1, 111.1.2, and 111.1.3 show the distribution of records by seasons, ages and sires. Table 111.1.1. Frequency distribution (percent range) of first lactation records by season and age. age in months of freshening Season 22-30 31-36 All 1 Jan-Feb 6.8-12.8 5.0-7.7 9.38 2 Mar-Apr 7.4-15.0 2.5-14.5 11.15 3 May-June 9.9-15.0 7.4-17.3 11.87 4 July-Aug 19.3-28.9 29.0-34.6 26.87 5 Sept-Oct 24.7—34.3 25.6-34.2 28.95 6 Nov-Dec ‘ 9.3—15.7 8.5-19.5 11.85 100.00 41 Table 111.1.2. Frequency distribution of first lactation records by. age . Age Freq. (%) Age Freq. (%) _<_18 .24 29 7.16 19 .16 30 6.61 20 .20 31 4.64 21 .79 32 4.11 22 1.85 33 3.79 23 4.98 34 2.76 24 10.77 35 2.32 25 12.82 36 1.61 26 13.10 37 1.35 27 10.20 38 .89 28 9.64 100.00 Table 111.1.3. Frequency distribution of first lactation records by sires. daughters per sire freq. of sires(N) l-7 .18 1-10 48 11-20 50 21-50 26 51-100 12 >100 15 range (4-339) total 151 Crosstabulation of age by season indicated a similar distribution within ages across the six seasons. Table 111.1.1 indicates the seasonal distribution within the two ranges of ages are very similar. For example, cows freshening in November and December make up a similar percentage within each of the two age ranges, 9.3 to 15.7% vs. 8.5 to 19.5% for age ranges of 22 to 30 and 31 to 36 months, respectively. Table 111.1.2 indicates that the majority of first 42 lactation cows freshened between 24 and 28 months of age. By crosstabulation, it was noted that this age distribution was similar for most sires. Table 111.1.3 shows the number of daughters per sire, which range from 4 to 339 with only 18 sires having fewer than 8 daughters. 111.2 Selecting_the Method to Fit Individual Lactation Curves The criteria for selecting the appropriate method and model to fit individual lactation curves should be based on their compliance with the assumptions of regression analysis. Therefore, the method, be it linear regression, weighted linear regression, or nonlinear regression, plus the model used, should produce homogeneous variance with normally distributed and independent residuals. Homogeneous error variance requires equal variance regardless of magnitude of the dependent variable, y. Therefore, there is no correlation between the magnitude of y and the amount of error in estimating daily milk production by the regression line. Normality refers to a normal distribution of the residuals. Independence of residuals refers to having no correlation in magnitude or sign between residuals (autocorrelation). When homogeneous variance does not exist among residuals but residuals are independently and normally distributed.the parameter estimates curve characteristics 3, b, and c obtained by least-squares still are unbiased and consistent (i.e., as sample size goes to infinity the variance of the estimator goes to zero), but they are no longer minimum variance unbiased estimates 43 (Neter and Wasserman, 1974). Several test statistics for detecting heterogeneous variance were compared by Layard (1973) and Brown and Forsythe (1974), using MOnte Carlo methods. For a population with a normal distribution, Bartlett's test had more power. Those tests which were found to be more robust than Bartlett's under certain non-normal distributions were not robust to all non-normal distributions. Layard (1973) suggests a minimum of 25 points to achieve good power to determine homogeneous error variance. This means 25 cows tested on or near the same days over the entire lactation would be needed. Kellogg et a1. (1977) used 36 cows having 4 lact- ations and having weekly milk weights for the first two months and monthly weights thereafter, to look at variance over the entire curve after a nonlinear fit had been used. Since time of sample days after parturition were similar for the 36 cows, comparing variance between cows at the same days postpartum was possible. They suggested that the variances were equal after the first month. Intuitively then, a linear fit of the same data could not also produce equal variances and, therefore, Kellogg and co- workers concluded nonlinear fit was more appropriate. However, they included the cow by lactation interaction in the error term which may have influenced the results if the interaction exists. On the other hand, it has been generally implicitly assumed, by those who have used Wood's (1967) equation(Wood 1967; Congleton and Everett, 1980a, b; Shanks et al. 1980) that as daily milk yield increased, so did variance. Therefore, a logarithm trans- formation of the data was thought necessary to achieve homogeneous variances across the entire lactation curve. 44 In this study, however, it was not possible to test for homo- geneous error variance because: 1) Grouping cows by similar test days over the lactation, as Kellogg and coworkers did, would not be practical because days into lactation at test dates would be the same only for cows freshening at the same time in the same herd and therefore, few cows could be grouped. 2) Individual cows have only 10-12 tests, where a number of consecutive daily tests would be needed at different times postpartum to test for homogeneous variance within a lactation for a single cow. Independence of errors refers to the assumption that there is no autocorrelation. Further, Kendall and Buckland (1971) defined autocorrelation as "correlation between members of series of observations ordered in time or space!‘ The occurence of autocorrelation in a least-squares model may produce a number of important consequences (Neter and Wasserman, 1974): First, though the parameter estimates are unbiased, they no longer have the prOperty of minimum variance and may be inefficient thus making the reliability of the estimates dubious. Second, the use of mean square error may seriously underestimate or overestimate the variance of the error term. Third, the least-squares procedure may greatly underestimated the true standard deviation of the estimated regression coefficient. Fourth, confidence intervals may not be valid. 45 The correlation between residuals for monthly test measurements of each cow after fitting a 10 to 12 month lactation is likely to trivial. There is little reason to suggest the residuals, after fitting each cow, would follow some repetitive sequence over a lactation. For this study, it is assumed that the 30 days between tests breaks up any autocorrelation between residuals. If data points were more closely related in time, then autocorrelation may be more likely to occur. In this study, it was not deemed necessary to test for autocorrelation of residuals. The assumption that errors are normally distributed is not essential to derive point estimates of parameters but is required when making probability statements about the reliability of estimates in the form of confidence limits. Normality of the residuals has not been tested for either the linear or nonlinear methods of fitting Wood's equation to lactations of dairy cows. If non-normality exists, tests for homogeneous variance may be in error (Brown and Forsythe, 1974). For :1 lactation curve of 290 to 360 days there are 8 to 12 monthly sample points. For testing normality it is suggested by Gill (1978) and noted by Shapiro and Wilk (1965) that the W statistic developed by Shapiro and Wilk (1965) is well suited for samples of less than 50. They also noted that the W-test is sensitive to a wide range of non-normality. Because testing for homogeneous variance was impossible for these data and autocorrelation is likely to be trivial for 46 monthly tests, the decision of which model to use for fitting lactation curves will be made based on results from testing for normality of residuals. Testing for normality will be performed on two models: Yt ' 1n (a) + b 1n (t) +Lct + e y I In (a) + b 1n (t) + ct«+ e/y2 t where yt is the daily yield at time to. The first equation is the log-transformation of Wood's (1967) equation, and the second is the weighted form of the first using l/y2 as the weight. It is noted that the Taylor series is one method of estimating nonlinear parameters (Marquardt, 1963). For these types of equations above, the second equation is the first order approximation of the nonlinear function (Guest, 1961), which is the first degree of the Taylor series, i.e., the function plus the first derivative in the series. Therefore, without fitting the data by nonlinear regression, which would be costly, one can compare results of weighted regression, which is one step closer to nonlinear regression, to those of the simple log-linear model. The General Linear Model (GLM) procedure of the Statistical Analysis System (SAS), (Barr et a1. 1979), using weighted regression, will be used for the fitting of the two equations and testing of normality. The Shapiro-Wilk (1965) W statistic will be used on each of 500 randomly chosen cows to test the residuals for normality. Individual cows will be tabulated by probability levels (P) of having non-normally distributed residuals. A probability level of P < .25 will be used. Levels lower than 47 P < .25 allow for larger type II error, i.e., accepting a set of residuals as normal when they are not. One would expect at a P level of .25 that 25% of the cows, in a population with a normal distribution, would fall outside the acceptable range of normally i.e., P < .25. A binomial test will be used to determine if the observed ratio of normal to non-normal is equal to the expected ratio. The model producing the highest probability will then be the one most likely to produce normally distributed residuals. III.3 Model III.3.l Adiusting data for age at freshening The 305-day milk lactation records in DHI data files are typically adjusted for age at fresehning when used for comparisons (McDaniel et a1. 1967, Mao et a1. 1974). These age adjusted records are called mature equivalent records. It is possible that age would also influence the lactation curve parameters within the first lactation.‘ Records in this data file were adjusted for age of freshening, as well as for its quadratic term, by regression analysis. The GLM procedure of SAS (Barr et al., 1969) was used for the model: Age2 + e (III.3.1) 3'13 2 1 ij where yij is the j-th observation of the i-th age for any of the I u + b1 Agei + b dependent variables, i.e., 305-day milk yield or the lactation curve parameters, a, b, c, time of peak yield, peak yield or S. 48 The residuals from this regression procedure become the new y values of 305—day milk and the curve parameters adjusted for age at freshening. This adjustment is valid only when there is not a significant interaction between age and the factors in the subsequent model for variance component analysis, i.e., herd, season and sire effects or when the correlations are simply to be removed and no bilogical interpretation of age and age2 is desired. A crosstabulation of data indicated that the ages of daughters within sires appeared to be distributed similarly for most sires. Also, ages within seasons were distributed similarly CTable III.1.l) and it was assumed that ages within the 557 herds would be similar for most of the herds. III.3.2 Equations and assumptions of model For a model describing each of the variables of interest. The equation used will be: yijkm = u + bi + fj + 3k + eijkm where: yijkm is the residual after the corresponding observation was adjusted for age of freshening and age of freshening squared, for the k-th sire in the j-th season in the i-th herd from a population of first lactation cows on Michigan DHI, having their sires identified, and lactating between July 1978 and August 1980, of either the a, b or c of Wood's (1967) equation (yt) - atb exp(-ct), time of peak yield (b/c), peak yield (a(b/c)b exp(—b)), S, 305-day milk yield or any of the pairwise combinations of these variables; u is the mean of the named fixed effects; hi is the effect of the i-th herd, 1 a 1, ..., 557; fj is the effect of the j-th season in which a cow freshened; j - l, ..., 6 which represents six seasons combining the months of January and February, March and April, May and June, July and August, September and October, and November and December; 49 3k is the effect of the k-th sire; k I l, ..., 150; and eijkm is the residual effect associated with yijkm' Factors hi and fj are assumed to be fixed, while yijkm’ 3k and eijkm are assumed to be random. Further assumptions include: 1) E(y) I 3b and the variance of j I y a g g g' + 3; 2) Var (g) I g I log (n is the number of observations) which implies that the e's associated with each observation of y are not correlated with other e's and that each e is independently drawn from the same population with mean zero, variance 0% and independence, i.e., no correlation between residuals; 3) Normal distribution of residuals; 4) Cov(s, e) I 0, which implies no correlation between e and the random factor, 3; 5) Var(s) I G I 115002 which implies that there is no covariance between s 's i. e., no additive genetic relationship and inde- pendent sampling between s 's, and that each s is drawn from the same pepulation with mean zero and varianCe 02; 6) The sire effect, 8, is normally distributed; 7) No correlation between the ranking of 3k and the number of observations for s and 8) Two and three—way interactions i.e., hiby f, h by s, f by s and h by f by s are trivial and negligible. Sire and season effects are of primary interest while herds are considered a nuisance factor. Converting to matrix form one obtains: z'§2+§2+s where: y is the observation vector on either a, b, c, 305 day yield, ~ peak yield, time of peak yield and S or any of the pairwise combinations of these values after adjustment for age. is an n x p incidence matrix, where n I 4818 cows and P is the sum of 557 herds, 6 seasons and one column for u. It contains 1's and 0's corresponding to the presence or absence of the observations in the herd and season classes, and for each observation a l in the column for u. is a vector of length 564 containing the unknown constants of the fixed effects. b' I [u hl ... hSS7 f1 ... f6]. 1:: IO‘ 50 (N is a 4,818 x 150 incidence matrix containing 1's and 0's correspond- ing to the presence curabsence of observations within each sire. is a 150 by 1 vector of non—observable random effects for s, g I [s ... s1 0]. is a 4,818 by 1 vector of non-observable random residuals correspond- ing to y. (C 20 Noting then: 9 ~ NID (0, I 02) 13(2) '9 ” e “2) ' X2 E<§> . Q Covcg. 2) =9 Var(y) I V I Z G 2' + R, where ~ G ~ var(§) I E(§§') EIg - E(§)] [§ - E(§)]' rt. ‘ E 01 [S1 s2 ... 3150] S2 15150; r '1 U: 0'S 0'S 1 1, 2 1, 150 0' 0': . . 0's S1, 2 2 2, 150 0': 0' . . . 0': L 1, 150 S2, 150 150 J and B I Var(s) I E(ss') = E[e I ECS)][§ - E(e)]' 0' L [el 0 0 e1, 4818 e2, 4818 €4,818] 1, 4818 2, 4818 2 0' e4818 It is assumed for s and e all covariances are zero and that both 2 2 have homogeneous variance i.e., 108 and Ice then: r 02 S 0 G: O L and (02 0 e o 0’2 e R: 0 0 L O 2 0' s e J O 0 ‘ 2 £150 03 J 150 x 150 E4818Oe 4818 x 4818 52 Then the variance-covariance matrix for all random factors can be written Var ry‘ [ V 202 I 02 ‘ ~ ~n ~ 3 ~n e . 2 2 E E 0's EISOCS 0 e I o2 0 I o L~J L~n e ~n e J To illustrate the model, a hypothetical example of 10 cows was used: Data: 305-day Sire Herd Season milk 14266 15984 18067 15332 13367 16691 17605 16525 16001 15785 Lphouahikahu~curvrd wHHNwNHNHH NNNHNwNHHH Then the data layout of a 10 cow example cross-classified for herds, seasons and sires is: Sire Sire 1 2 3 no. ’ 1 2 3 no. 1 2 2 1 ' 5 1 2 2 o 4 Herd 2 2 l 0 3 Season 2 2 1 2 5 3 l 0 1 2 3 1 0 0 1 no. 5 3 2 10 no. 5 3 2 10 53 Herd l 2 3 no. 1 2 2 0 4 Season 2 3 0 2 5 3 0 1 0 1 no 5 3 2 10 Then: Y = X F14,266‘ r1 1 o o 1 o 0‘ 15,984 1 1 0 0 1 0 0 18,067 1 0 1 .0. l 0 0 15,332 1 1 0 0 0 1 0 13,369 I 1 0 1 0 0 0 1 16,691 1 0 0 1 0 1 0 17,605 1 O 1 0 1 0 0 16,525 1 1 0 0 0 1 0 16,001 1 1 0 0 0 1 0 15,985 L1 0 o 1 o 1 o L J J 2 g + e r ‘ 1 1 o 0 s1 felll 0 l 0 82 e112 0 l 0 S3 + e212 1 0 0 e121 1 0 0 e231 1 0 0 e321 1 0 0 e211 0 0 1 e123 0 1 0 e122 L0 0 1} e313J The normal equations are then: L NH] tN (N (N (N (N (N IN IN 20) (0‘ (N (>4 (‘4 2‘4 54 and become for the example: L10 5 3 2 4 5 ‘ 1 5 3 2 5 5 o o 2 3 0 2 2 1 3 0 3 0 2 0 1 2 1 o 2 0 0 2 o 2 0 1 o 1 4 2 2 o 4 0 o 2 2 o 5 3 o 2 0 5 0 2 1 2 1 o 1 o o o 1 1 0 0 5 2 2 1 2 2 1 5 o 0 3 2 1 0 2 1 0 o 3 o L 2 1 0 1 o 2 o 0 o 2 1 r w L u 159,625 hl 78,108 h2 49,041 h3 32,476 f 65,922 X f: a 80,334 f 13,369 ...2.. 31 77,263 92 50,052 3 32 310 L 3 J L ’ J The variance of y I V I Z G 2' + R: r1 0 0‘ 2 0 0 1 o S 2 0 1 0 o 0 o 3 2 o 1 0 0 Us 0 1 0 ° 3 x 3 1 o o 1 o 0 1 o o o 0 1 o 1 0 o o 1 L J 10 x 3 O 1 O l 3 x 10 55 0 O O O 0 0 000002 000028 00028 O O 02 80 U 0280 O 0 0028000 0 0280000 0 02800000 0 028000000 0 280000000 (0 + C 0 0 J U 0 o 8 000 0 0 0 O O 0 28 0 2e OZSZSO 0 0 0 0 0 O O 28 0 2e 00 0 0 0 0 0 w 028 28 O O 28 28 282828M 00 0 O 0 02080 0 0 2M... 28 2828 280 0 0 GO 0 O 0280 0 2e . 28 28%28280 0 0 00 0 O O O 28 .0 2e 28 00 0 ”2828280 0 O 280 O O 0 2e 28 00 0 0 0 0 0280 28 U G 28 own/.80 0 O 0 0280 280 C O 28 WO 0282828280 0 0 208 O 0 O O 56 1 Formulating the mixed model equations then, X'R- X developes from: ninunu1inu1imw 1i1invnvnv1ZO «i1inunvnv120 1inv1inv1.nZU ainunu1inv120 1inu1inunvnvl 1i1inunvnv1IU 1inv1inu1.nZU 1100100 1100100 r L 7 x 10 d R~ 0 1/0 1/0 2 1/oeJ mwAununv 0001... 1110 0000 0010 1i1.nfl 1111 r 1inunvnvnvmw 010111 001000 010001 101000 000110 llllldk 57 x'R'lx r10 5 3 2 4 5 1‘ 5 5 o o 2 3 o 3 o 3 o 2 o 1 2 o o 2 o 2 o 2 4 2 2 o 4 o 0 1/0e 5 3 o 2 o 5 o L 1 o 1 o o o 1) Similarly, -1 5 2 2 1 2 2 1 Z'R x-- 3 2 1 o 2 1 o 12 1 o 1 .o 2 o 3 x 7 then X'R-lz having the dimensions of 7 x 10, 10 x 10 and 10 x 3 becomes: is 3 2‘ 2 2 1 2 1 0 2 1 0 1 Use 2 2 0 2 1 2 1 0 0 L J 7 x 3 ' -1 -1 and finally Z R Z + G is: r5 + 1/02 0 0 ‘ o 3 + Us2 0 Us2 3 e 0 O 2 + 1/02 L SJ -1 where G is: Flfisz o o ‘ s 0 1/0 0 0 0 1/02 S 58 the right hand side is: ”159,625‘ KISS 49,041 32,476 65,925 80,334 .légééfl 77,263 50,052 L 32,310J The partitioned mixed model equations are now: X'§-1§ §§-lz E §'R- Y z'Rflx z'R'iz + 6'1 s z'R'ly and they are multiplied by R, and 0'1 is multiplied by ofi/o:, the variance ratio for error and sire components. 59 4 "\ i am 0 m D\ND+N o a 1. can an N .0 0+ N \N m Nao.om. me~.- (D CD 0 o\ o+m NU) asm.ma «mm.om «Rho ohq.mm qu.me mofi.ma , ,mmo.mmai a HNMHNF‘IHNM SSS‘HM'H MMNQWHU‘MN mm a: 6: r4 a: a: para m<3 <3 p4 c: F1 c: c: c> F4 F4 c> c> q- a: on c> <- c> c: o: o: c: «1 c: «a c: a: c: F4 6: F4 :3 vs vs c> C) a: «a c1 0: 04 F1 NHOHONOO MNHONHOO MMONOV‘ONHN NOONCNOI-IOH 0 F4 g um>ms cmnu m3 60 where 3‘1 was cancelled from both sides leaving a ratio of 0§fl3§ in the diagonal of the 3'? portion. These equations will now yield Best Linear Unbiased Prediction solutions (Henderson, 1975) for sires only. It is noted then that the mixed model equations are equivalent to the normal equations of Generalized Least—squares for the fixed effects. In this example the herds were not absorbed as will be done for the large data set used for the variance component analysis. III.3.3 Abosorption of herds To solve the mixed model equations, the nuisance factor, herds, will be absorbed into the effects for season and sire. This will be accomplished by using a FORTRAN program which absorbs one herd at a time using a row by colum technique while setting up equations pertaining to seasons and sires. In setting up the normal equations: : X'Z ~~ iN (>4 25': ”N M cc >zo‘z E'E 5'2 9 herds will be absorbed into six seasons and 150 sires reducing the §'§ and §'§ and g'g portions to 6 by 6, and 6 by 156 and 156 by 6 respectively, while leaving the E'E portion 150 by 150 for sires. The non-unique solutions of the fixed season effects are E and the unique estimates for sires are g. The algorithm is as follows. First, data is sorted by herds, then sires are sorted within herd. Then, the following computations are done within each herd and summed across herds. 61 For absorbing right hand side (RHS) terms in X'y of herds into those of seasons and Z'y of sires:. ~ Absorbing herds into season RHS I season sums - (number of cows in season * herd sums/number of cows in herd). Absorbing herds into sire RES - sire sums - (number of daughters of sire * herd sums/number of cows in herd). Noting that sire, season and herd sums refer to the sums of obser- vations on a trait of a sire, sums of observations for cows within a season and sums of observations for cows within a herd. For absorbing portions of §'§ for herds into portions of §'§ for seasons, §'§ for seasons by sires and E'E for sires: Absorbing herds into seasons (X'X, diagonals) 8 Number of cows in season - (number of cows in season)2/number of cows in herd. Absorbing herds into season (X'X, off diagonals) - eNumber of cows in season i * number of cows in season i'/number of cows in herd, for i f i'. Absorbing herds into season by sire‘X'Z - Number of daughters of a sire in a particular season - (number of cows in the season * number of daughters of the sire/number of cows in herd). Absorbing herds into sire (Z'Z, diagonals) = number of daughters of sire - (number of daughtErE of sire)2/number of cows in herd. Absorbing herds into sire by sire (Z'Z, off diagonals) a - Number of daughters of sire i * number of daughters of sire i'lnumber of cows in herd, for i # 1'. After absorption of one herd, the column and row for that herd is zeroed out for the next herd. The herd, sire and season sums are set to zero. With this procedure, only one pass of the data is required to complete the abSorption and set up the normal equations. The resulting normal equations will have only six seasons and 150 sires, leaving a 156 by 156 coefficient matrix and a 156 by 1 vector for each trait and trait pair. Setting up the mixed model 62 - 2 2 Equations, then, requires only the addition of G 1Oe/GS to the random portion (E'E) for sires prior to solving for E and ,3. 111.4 Variance Component Estimation An iterative restricted maximum likelihood (REML) procedure using solutions from mixed model equations (MME) will be used to compute variance components (Mao, 1981). Some desirable characteristics of REML are: 1) when MME solutions are used in maximum likelihood equations, non-negative estimates of variance components result. 2) the restricted maximum likelihood procedure maximizes the random portion of the likelihood which is invariant to the fixed effects in a mixed model. It does not assume that the fixed effects are known, as in maximum likelihood (ML), and therefore the estimates computed are unbiased. A reduction in degrees of freedom must accompany the estimation of the fixed effects. 3) REML can be used in iterative computations. From the MME (III.4.1) b and u will be computed for each trait b' x'x x'z " x'y Z a " " " ” _1 “‘ “ (111.4.1) u z'x z'z + c I, Z'y where i is a diagonal matrice of the variance ratio of 02/02, and G.1 is the inverse of the relationship matrix for sires. Let E be the generalized inverse of the coefficient matrix x'x x'z _ c c Z'X Z'Z + G K C C 63 The REML estimators are c: . (y'y - b'X'y - u'Zy)/[n - r(X)] (111.4.3) where b and u are MME solutions. When only one random factor is involved as in the present model A 2 V(us) E US (111.4.4) and 82 - [3'3 + 32(crc )]/q (III 4 5) s s s e ~ss s ' ° where qS is the number of classes in the random sire effect. The estimators in III.4.3 and III.4.S will be non-negative. A REML lends itself to iterative computaion because u and b ~ A A A A A ~ depend on K; o: relies on u, 0e and Z; a: relies on u, b, and K; and 3: and 3: are needed to compute new estimates of K. To begin the iterative process, initial values for the variance ratios, K, will be based on the heritability estimates found in the literature for the traits of interest (Table III.4.l). The first computation involves solving (111.4.1) for E and Q, then computing 3: in (III.4.3) and 3: in (111.4.5). The trgss in (111.4.5) is the trace of the sire portion of the generalized inverse in (111.4.2) and q8 is 150 for the total number of sires. This process is then continued by replacing i with the new ratio of 82/8: and recomputing the REML estimators until the current and new ratio are not very different, i.e., converge. To speed up the convergence of the iterative process, three times the difference between the current and newly computed ratio will be added to the current ratio instead of replacing the current 64 ratio with the new ratio. The iteration process will stop when the difference between the current and new ratio is less than .2. For the equation (III.4.3), z'y must be adjusted not only for the mean but for herds because herds have been absorbed for other terms in (III.4.3). Therefore, z'y becomes the total sums of squares minus sums of squares for the mean minus sums of squares due to herds. The denominator is n - r(§) or the total number of cows - number of herds - number of seasons + 1, i.e., 4818 - 557 - 6 + 1 = 4256. The covariances between traits will be computed from the variance of the sum of each pair of traits and the variances of the two traits using the equation Cov(i, i') = 8[V(i + i') - V(i) - V(i')] for i # i'. (III.4.6) The initial variance ratios for these new paired traits will be the average of the variance ratios of the two traits making up the paired traits. Table III.4.l contains these initial variance ratios. An example of computing the variance ratio from the heritability of a trait is: A 2 A h2 = .25 - 4(Us) , set 0: = l , A2 A2 0+0 3 s then 4 (l; = .25, l + oz e and 4 _‘AZ - .25 1“0e 15’ so 02/02 = 15/1 8 15. e s 65 Using these initial ratios and the iterative process, more pre- cise estimates of the variance components can be computed for the data set used than without iteration. In order to keep the total sums of squares for traits and trait totals for sires within the significant digit computation range of the computer, scaling down of the magnitude of some traits was done. Those traits having large values were scaled down by division. Table III.4.1 Average of heritaRilities reported in the literature and initial ratios used for 305-day milk and lactation parameters Milk a b c b/c Peak Trait Heritability Ratios (Si/3:) MilkB .25 15 22 22 18 3s 16 35 In a .10 39 39 29 66 24 66 b .10 39 30 - 66 24 66 c .17 22 56 19 56 b/c time °f .02 200 39 100 peak Peak .23 16 39 s .02 ' 200 A - Values are from Schneeberger (1981) and Shanks et a1. (1980). B - Milk is 305-day milk yield. From (III.4.2) the variance of estimation for the fixed effects and the variance of prediction and variance of error of prediction for the random effects can be computed for the BLUPs. 66 These estimates are: V(F) - £11 3: variance of estimation for seasons; V(g) 8 (E - 922)3: variance of prediction for sires; V(g - S) a 922 3: variance of error of prediction for sires; where E - Egg/8:. A A Because F is a constant vector; V(F - F) = V(F), and the variande of error of estimation equals variance of estimation. III.5 Heritability, Genetic and Phenotypic Correlations Lush (1940) defined heritability in the "narrow sense" as the proportion of the total variance in a trait that is attributed to the average or additive effects of genes. He defined heritability in the "broad sense" as the fraction of total variance due to genetic variance, which contains variance due to additive effects plus variance due to dominance and epistatic effects. In the literature, heritability usually refers to that in the "narrow sense". Heritability will be estimated for each of the parameters in Wood's (1967) equation, a, b, and c,plus peak yield, time of peak yield, 8 and 305-day milk yield by: h afi—f . (IIIoSol) The denominator a: + o: is the phenotypic variance after the variance due to named fixed effects in the model, which were adjusted for age, have been removed. With this method there are two possible sources of bias: 67 1) Epistatic bias (Dickerson, 1969) and 2) Ratio bias (Kendall and Stuart, 1969). The expectation of the estimation of heritability is then, E(h2) = (h2 + epistatic bias)(1 + ratio bias) The general formula for the approximate standard error of the ratio of variance components will be used to compute the standard errors of heritability (Dickerson, 1969): 602/?) = 4/§2/ {Ezvoh + 322m?) - 23h? cov (22,1?) (111.5.2) where in this case X and Y are the additive genetic and phenotype variances, respectively. I The covariance may be estimated as a simple linear function of the variance of i, i', and (i + i') from (III.4.6). Then the genetic correlation between traits for sire becomes: 2 2 ' Cov sii.//bs o s 1 31' to where Cov s is the estimate of the sire component of covariance 11' between traits i and 1'. 0:1 is the estimate of the sire component of variance for trait i; and 0:1. is the estimate of the sire component of variance for trait i'. The estimate for phenotypic correlations will be computed similarly by adding the component for error to the sire component. Standard errors of the genetic and phenotypic correlation estimates were calculated by procedures outlined by Grossman (1970). The equation for estimating the variance of the correlation coefficient is: 68 2‘2 2 2 2 2 f Est Var (r ) = ‘9 a (U11U22 + U12 b (V11V22 + V12) “2 9 2 {[ + 1/912 (nd) u—l v-l azuil szil .2 azugz b2V§2 .2 + [ + ]/2911 + [ + ]/2922 u-l v-l u-l v—1 2 2 '2 U11U12 b v11V12 “ “ - zf———————- +-———-———J /9r 9 11 12 u-l v-1 2 2 a U12U22 b V12V12 - 2L———————- -+-—————-—:]/912922 u-l v—l azUi2 bZVi2 A' A + [ + ]/911922} (III.5.1) u-l v-l where f2 is l and 16 for the variances of the phenotypic and genetic correlations; respectively; r is the square of the correlation 9 between the two traits considered; U11 and U22 are the mean squares for sires for traits l and 2, respectively. U2 is the square for the 12 2 11, V22 and V12 represent the same mean squares for error; u and v are degrees of freedom for sires and error, respectively; Gil, 922’ 9:2 etc, represent the variances and covariances of traits 1 and 2; a2 and b2 are both 1 for the variance mean square for trait (l + 2); V of the genetic correlation, and for the variance of the phenotypic correlation, a2 is the square of the degrees of freedom for sires minus 1 and b2 is the square of the degrees of freedom for error minus 1. One is subtracted from u and v to give unbiased estimates. The standard errors for correlation coefficients can be computed by taking the square root of (III.5.1). III.6 Select Indexes III.6.1 Justification and strategies Several strategies will be considered in getting up selection indexes to select the lactation curve characteristics and 305-day milk jointly. 1) The first strategy is an attempt to increase the amount of production in the early stage of lactation. This may be done by increasing the rate of ascent to the peak or increase the ascent and the peak yield without regard to persistency in later lactation. This strategy considers that cows are typically more efficient in utilization of feed during the early stage of lactation (Miller and Hooven, 1969). Realizing that part of this efficiency is due to mobilizing body fat CMiller and Hooven, 1969). Potentially then, more net income could be derived if cows increase in pro- duction earlier, and peak higher. These indexes (1:1 through 1:19) and their weights are listed in Table III.6.1. 2) The second strategy is an attempt to decrease the ascent to the peak or increase the time to peak in conjunction with increasing persistency. Decreasing the stress of high peak production may be possible in both cases. Hansen and coworkers (1979) found higher costs for health care in the early stage of lactation, during which time production and stress are the highest. If cows reach their peak at a more gradual rate, this may reduce stress and allow body reserves to be used more slowly. This strategy will determine if this change in shape is genetically possible 69 70 and what influenceilzwould have on total milk yield. Obviously too, increasing persistency should have a positive effect on total yield. These indexes (2:1 through 2:9) and their weights are in Table III.6.1. 3) The third strategy is an attempt to increase initial yield (parameter a) and increase persistency while decreasing the peak, thus flattening the curve. This strategy considers decreasing the stress of peak production and possibly allowing body reserves to be used up more slowly while maintaining production in the later stage of lactation. This strategy is chosen, as is the second strategy, to decrease stress, but in this case by decreasing peak yield greatly as opposed to delaying it. Increasing persistency and increasing the initial yield as part of the strategy is an attempt to negate some of the loss in total production due to cutting off the peak. These indexes (3:1 through 3:6) and their weights are in Table III.6.1. These strategies have been chosen to determine the potential of changing the shape of the lactation curve and 305—day yield through selection, using Wood's (1967) equation. Indexes in Table III.6.1 with zeros for some of the weights are restricted indexes. These indexes attempt to restrict the genetic change in the traits with zero weights, while selecting for change in the other traits in the index. Kempthorne and Nordskog (1959) discuss the computations of restricted indexes. After indexes are formulated, two methods will be pursued: 71 Table III.6.1. Indexes for-the three strategies and their weights. Strategy 1 Milk b Peak Strategy 2 Milk b/c Peak SA 1=1 3 1 2 2:1 1 15 1 1 1:2 3 1 1 2:2 1 6 1 6 1:3 5 1 1 2:3 7 3 1 1 1:4 1 1 6 2:4 1 10 1 10 1:5 1 6 6 1:6 1 1 15 Milk c b/c Peak Milk a b c Peak :5 1 -10 15 1 B 2:6 1 -10 10 10 1:7 1 O 3 0 6 2:7 1 -6 6 1 1:8 1 O 6 0 6 1:9 5 0 1 O 1 Milk c Milk’ Peak 2:8 1 -1 1=10 1 6 2:9 1 —10 1:11 3 1 ' 1:12 6 1 Strategy 3 Milk a b Peak sA Milk _ b 3:1 1 10 1 1 10 1:13 1 6 3:2 1 10 -5 -5 10 1:14 1 15 3:3 1 10 -10 -10 10 1:15 3 1 1:16 6 1 Milk a b c Peak Milk a b c. 1:17 1 0 6 0 3:4 1 10 1 -10 1 1:18 3 0 1 0 3:5 1 10 -5 -10 -5 1:19 6 o, 1 _o 4 3:6 1 10 -10 -10 -10 A - S is c—(b+l), adjusted for a. B - Indexes containing weights of zero are restricted indexes, where traits with zero weights are those being restricted. 72 1) Genetic change will be determined for each of the traits in the indexes. The correlated responses of the curve parameters (a, b, and c) when not included in a particular index will also be computed. 2) Best Linear Unbiased Prediction (BLUP) solutions (3) for sires from MME will be linearily combined by the weights to give a Total Merit (Index) for each sire. Henderson (1963) noted the BLUP of k'B + m'u is k'B + m' Z'V-1 1 (y - X8) and u is equal to GZ'V- (y - X8). From MME, u is computed and therefore solving GZ'V"1 (y - X8) to get u is not necessary. ~ Also 8 is equal to the solutions for the fixed effects in MME, i.e., ~ b. Mao (1981) notes that T = a'g = m'u the aggregate merit. Thus total merit can be computed by the linear combination of weights (a or m) and BLUP solutions (u). III.6.2 Computation of selection index criteria The genetic and phenotypic variance-covariance matrices used in the selection index equation AA A Pb - Ea (III.6.1) will be standardized. This is done by dividing both sides of (III.6.1) by the phenotypic variance such that the diagonals (variances) in P and g are divided by the phenotypic variance for each trait and the off-diagonals (covariances) are divided by the product of the phenotypic standard deviations for the two traits making up the covariance. This treats both sides of (III.6.1) the same. So, P becomes: 73 1 r r P12 P13 r 1 r P21 P23 r r 1 L P31 P32 1 where 0% [UP a l 1 l andO’ loo sr P1P2 P1 P2 P12 which is the phenotypic correlation. G becomes: r ,2 A W A W ‘ h r (h h r /h h l Gle 1 2 G1G3 l 3 r /h h h r Vh h G261 2 1 2 G2G3 2 3 “ “2‘2 “ 72‘2" “2 r /h h r /h h G3G1 3 l G3G2 3 2 3 L 1 where a: /0: = h2 l l A A A W A A A T273 and o IO = r o o o = r 7h h 0102 p192 01027001 cz/ P1 p2 0102 1 2 With the standardized P and E matricies, the solutions for b in (III.6.1) will be computed. The values of b are standardized partial regression coefficients and will be denoted as d. To compute the partial regression coefficients, d is divided by the phenotypic standard deviation for its related trait. These standard values for E and P will only be used to compute the d's. It is pointed out that P and 9 are positive semi-definite matrices, such that all principal leasing minors have determinants 74 greater than or equal to zero. This is necessarily the case when all heritabilities are between zero and l and all correlations are between -1 and +1. III.6.3 Computing_genetic chaggEJand correlated genetic response Equation (II.6.3) expresses the genetic change of total merit as a result of the use of an index, but in this study this is not of interest. Instead, the genetic change of individual traits either within the index or not included in the index are of primary interest. In particular, 305-day milk yield and the curve parameters,_a, b, and c are of interest, and in some cases peak yield, time of peak and S will be of interest. Computation of genetic gain of an individual trait included in the index is as follows: A Cov(Gi, I) AGi = A X Z/q, (III.6.2) 0I where Cov(Gi, 1) represents the genetic correlation between trait i and the index, and o comes from: I oi a bio2 + 2b + ... + bfio2 , (III.6.3) y1 y1y2 yN where o2 , o , etc., come from the phenotypic variance-covariance y1 yiyz matrix. The b's in (III.6.3) are partial regression coefficients and the phenotypic variances and covariances are not standardized. The Cov(Gi, I) is computed using the genetic variance of the trait in question and its covariances with all traits in the index. If i a 1 then: 75 Cov(Gl, I) = EfGl[blpl + b2p2 + ... + prN]} = b C + b o + ... + b C . (III.6.4) 1 G1 2 Gle N GlGN For a trait not included in the index, only the covariances between the trait and all other traits in the index are used: Cov(G , I) = b O + ... + b C . N+1 l GIGN+1 N GNGN+1 (III.6.5) Correlated reaponses of an unselected trait, i', when a single trait, i, is selected is computed by: AG., a r ' (o /o )Au , ' (III.6.6) i 6161' 61' G1 1 where Aui is the change in trait i due to single trait selection of i, i.e., Aui h OP z/q. (III.6.7) i . Equations III.6.2, and 111.6.4 through III.6.7 are discussed in various forms by Van Vleck (1979). For the purpose of comparing genetic gain, z/q will remain constant and for simplicity. avalue of 1 is chosen. If 5% of the sires and 90% of the cows are selected as parents, then for q and 5%, z/q = 2.1 and for q of 90%, z/q - .2, then (.2 + 2.l)/2 = 1.15. Therefore, 1 is a reasonable choice. III.6.4 Computingvnew curves after selection For each index in the three strategies, estimates for genetic change in the curve parameters a, b and c in Wood's (1967) equation (II.2.3) will be computed for 1, 5 and 10 generations of selection. 76 Using these new estimates and the appropriate form of Wood's equation, be it log-linear or nonlinear, new lactation curves will be plotted for each index. The shapes of the curves generated by the indexes will be compared to the shape generated if only milk is selected. The integrals for the new curves will be computed for 305-days into lactation and compared to the expected change in 305—day milk yield determined by AGhilk. For those indexes relating to changing the peak yield, there will be a comparison between the expected change in peak (AGpeak), and that estimated from the new values for the curve parameters in the equation: Peak = a(b/c)bexp(-b). (III.6.10) Computing the change over 10 generations, as described above, assumes that the genetic response of individual traits as well as their correlations are linear. This may not be the case. III.6.5 Computing and ranking sires on indexes Computing total merit for sires for each index is done by standardizing the BLUP (8) solutions and combining them linearily with the apprOpriate weights. Standardization of the BLUPs is done by dividing the BLUP for each sire by the standard deviation of the BLUPs for that trait. Standardizing puts all traits in terms of their standard deviations so that traits of low numerical value are not over-shadowed in the total merit score by traits which have high numerical values. Then, total merit is computed by: T - a'g* .. m'u* (III.6.11) 77 A where a or m is a vector of weights and g* or 3* is a vector of standardized BLUPs. In'selection'index theory, g is replacing Side-1‘9) 1!" T = B'E'Y’”: - ESE) ' 2'3? The weights are the same as the weights listed for the indexes in Table III.6.1. These weights are relative only to changing the shape of the lactation curve and are therefore arbitrary, depending upon the direction and amount of change desired. The ranking of the sires by their total merit for each index will be compared to their ranking on 305-day milk yield. Spearman's ranked correlation analysis will be used (Gill, 1978) to determine if selection by various indexes have Significantly changed the ranking of the sires from their ranking on milk alone. IV RESULTS AND DISCUSSION IV.1 Test for Normality Monthly milk weights for first lactation records for 481 randomly chosen cows were fitted to two equations below, and the residuals were tested for normality using the Shapiro-Wilk W-test (1965). The two equations are: l) yt = 1n a + b 1n t + ct + e 2) yt - 1n a + b 1n t + ct + e/yi whereyt is daily milk yield at time t, a, b and c are constants, e is simple error and e/y: is the weighted regression form of error. Each cow had 8 to 11 monthly milk weights and received a probability (P) level indicating the probability of the 8 to 11 residuals being normally distributed when the hypothesis of normality is rejected. For example, P < .25 means the probability of type I error is less than .25. A binomial test was used to determine the probability that the observed number (N) of cows with P levels less than .25 was not different than the expected number. These results for model 1 and 2 are reported in Table IV.1.l for the random sample of 481 cows. Table IV.1.l Binomial probability of observed number of cows being not different from expected num- ber having probability levels below .25. Model P < .25- Binomial ObservedA ZB ExpectedC Probability ' A N N l .0080 98.0 20.3 120 2 .36 123 - . .25.5 120 A - Observed number of cows below P < .25. B - Percent of cows below P < .25 (total = 481). C - Expected number of cows below P < .25. 78 79 Foraisample of cows from a population with normally dis- tributed residuals, the test for normality should produce a percentage of cows having a probability level less than the chosen level of P, which is equal to the chosen P level. Therefore, if one tabulates all cows with P levels less then .25, this should include 25% of the cows. The binomial test compares the expected N with the observed N and yields the probability that they do not differ. At P < .25 one expects N - 120. In model (1), N is 98, and for model (2), N is 123. The binomial probabilities are .0080 and .36 for model (1) and (2), respectively. From the results, the model which provided the highest probability of having normally distributed residuals was model (2), an approximation of a nonlinear model. The real concern was the comparison between the log-linear and a nonlinear model. ~Because model (2), the weighted linear regression model, is an approximation of a nonlinear model, and is less expensive to compute, it was used in the test for normality of residuals in place of the nonlinear model. The results suggest that the nonlinear model would be more appropriate than the log-linear model, from the standpoint of normality. Based upon these results and findings by Kellogg et a1. (1977), Cobby and LeDu (1978) and Shimizu and Umrod (1976), the nonlinear model was used. Marquardt's (1963) technique of nonlinear regression was used to fit 5,927 lactations to the nonlinear form of Wood's (1967) equation. 8O IV.2 Marquardt's Method for Least-Squares Estimation on Nonlinear Parameters Marquardt (1963) develOped a maximum neighborhood method. This method utilizes the Taylor series and gradient (steepest-descent) methods of nonlinear estimation. Marquardt mentions that these two methods, when used separately, have difficulties in estimating nonlinear parameters. The maximum neighborhood method is stated to perform an optimum interpolation between the Taylor series and gradient method. The interpolation is based upon the maximum neighborhood in which the truncated Taylor series gives an adequate representation of the nonlinear model. Marquardt states the problem as follows. Choosing a model: E(y) =- 12 (x1, x2, xm: 81, 82, Bk) (IV.2.1) to be fitted to a set of data where X ..., Km are independent 1’ x29 variables and B ..., Bk are the population parameter values 1, 829 or a, b and c of Wood's (1967) equation for an individual cow or group (population) of cows. E(y) is the expected value of the dependent variable y. Data points are denoted by: (yi, yli’ X21, ..., Xmi) i = 1, 2, ..., n. (IV.2.2) The nonlinear form of Wood's (1967) equation is: yt = atb exp(-ct) + e, (IV.2.3) where X1 becomes t and X11 is any time, ti’ in lactation and y1 equals yt which is daily milk yield at some time, ti. Thus, t is the only independent value. 81 It is then desired to compute those estimates of the parameters which will minimize: n A2 (I? = Z'YisYi] 181 where Y1 is the value of y predicted by (IV.2.1) at the i-th data point. When the function for the expected value of Y1 is linear in the 8's, the contours of constant, ¢, are ellipsoids but for the nonlinear case, they are distorted, depending upon the degree of nonlinearity. But with nonlinear models, the contours are nearly elliptical in the immediate vicinity of the minumum of O. Marquardt also mentions that the contour surface of ¢ is very narrow in some directions and elongated in others such that the minimum lies at the bottom of a long curving trough. Using Marquardt's notations, the equations used for iteration to a point which the residual sum of squares (0) is minimized are as follows: a I A* = a311, -- ( 31 ), (IV.2.4) {“13 “a: '1' where A* is a scaling matrix to scale the b-space in units of the standard deviations of the derivatives 3fi/3bj taken over the sample points i . 1, 2, ..., n. This makes the A matrix one of simple correlation coefficients of the afi/Bbj's. ajj’ ajj‘ and a J 'j' represent various sums of squares and sums of cross-products. The algorithm used is: 82 (Mr + 2‘1) 5*: a g*r, (IV.2.5) representing the equation at the r-th iteration, where scaled vector g* 8 Afidt , (IV.2.6) and 81 ‘ 8* = (a?) = ( ) (IV.2.7) 311 and 5: is the Taylor series correction 5j 8 63/Vajj . (IV.2.8) * Equation (IV.2.5) is solved for 6 r and (IV.2.8) is used to obtain 6r. A new trial vector: b(r+1) - br + 6‘ (IV.2.9) will then produce a new residual sums of squares, ¢(r+l). Marquardt noted it is essential to select At such that 0(r+1)< 0‘ (IV.2.10) meaning the new residual sums of squares are less then the current. A form of trial and error is used to find a value Ar which will satisfy (IV.2.10) and produce rapid convergence of the algorithm to the least-squares values. Marquardt's strategy was: Let v be greater than 1 (usually use 10) and let 1(r-l) denote the value of A from the previous iteration, but initially 10 is equal to 10-2. Compute @(X'r-1)) and 0(1(r-l)/v). 1) if ¢(X(r-l)/v) §_¢r; let Ar 8 A'r-l)/v. 2) if ¢(A(r-1)/v) > 0‘, and 0(2'r‘1)) §_0r; 1et Ar = 1(5’1). 3) if ¢(X(r-1)/v) > ¢r’ and ¢(X(r'l)) > or; 83 increase 1 by successive multiplication by v until for some smallest W: (A(r'1)vw) 5 9"". Let 1‘ = A'r‘l)vw. By this algorithm, Marquardt suggests a feasible neighborhood is obtained. The iteration cdnverges when 16"! --J-;:- < e, for all j, T + ijl for suitably small e > O, i.e. 10..5 and a suitable T, i.e. 10-3.‘ For v, a value of 10 has been found to be a good choice. In the determination of the parameters a, b and c for Wood's (1967) nonlinear equation (IV.2.3),a grid search was performed for each cow to arrive at an initial best guess for the values of .a, b and c. Then Marquardt's method was used to refine the estimates of a, b and c by further minimizing the sums of squares of the residuals. The whole procedure was computed using SAS NLIN procedure (Barr et al., 1979). The partial derivatives of (IV.2.3) needed for Marquardt's procedure were: ny/Ba 8 tbexpC-ct) afy/ab 8 1n t(atb exp(—ct)) Sfy/Bc 8 (atbexp(-ct))—t. IV.3 Data Table IV.3.1 describes the transition of records used at various steps of the analysis. The initial selection criterion were : 84 1) let lactation records; 2) having sire identification; 3) having 8 to 12 monthly tests; 4) lactation not coded with an abortion; and 5) one test prior 35 days and one after 280 days. This yielded 10,107 records over the two year period, July 1978 to August 1980. Records were further dropped for sires having fewer than eight daughters and herds with fewer than three cows. This left 5,927 records. Upon fitting these 5,927'records with the nonlinear form of Wood's (1967) equation using nonlinear regression, 887 lactations (15%) yielded negative values for parameter c. There were 14 additional lactations with negative values for b. These lactation curves were estimated to have peaked before calving and therefore would have a continuous decline from freshening. They are, therefore, classified atypical lactations. Shimizu and Umrod (1976) reported 34 and 29% atypical lactation curves for an unweighted and weighted regression model of the logarithm form of Wood's (1967) equation. Schneeberger (1981) compared two models (11.2.18) and (11.2.19), which produced 26.6 and 25.9 atypical shapes, respectively. Schneeberger also noted the later lactation animals produced fewer atypical lactations (198222). If tests prior to peak are missing, then the curve would take on an ever decreasing shape and have a negative b. First lact- ation animals do not peak as high as later lactation animals, and they may peak earlier. Therefore, it would be likely that first 85 Table IV.3.1. Amount of data after each step of screening. Step Records . Herds Sires After selecting lst lactations with 8 to 12 tests and sire identification. 10,107 1,114 717 After deleting sires <7 daughters and herds <3 cows (used for non- linear regression). 5,927 678 152 Fit models to test normality. 481 (random records) after deleting records ' with negative values _ for c. 5,040 678 151 After deleting records with nega- tive values for b. 5,026 678 151 After last deletion of herds with one sire or less than 3 cows. 4,818 557 150 86 lactation animals would have more atypical curves i.e., negative b's. Likewise, it is more likely that a first lactation animals would be increasing in production near 305-days and therefore have a negative c. In the current study the number of atypical curves was minimized by deleting cows which did not have a test prior to 35 days into lactation. Also, first and second monthly tests were adjusted using Shook's (1975) factors, thereby accounting for the typical increase to peak. These procedures have been responsible for having fewer (15%) atypical curves in the present study than reported by Shimizu and Umrod (1976), and Schneeberger (1981).. Shanks (1979) also accounted for the typical increase from parturition by using Shook's (1975) factors to compute milk yield on day six. He reported less than 1% atypical curves for all parities. Almost all ofthe 15% atypical curves in the present study were due to negative c values, meaning the last part of the curve was increasing. If Shook (1975) factors are used to compute the last test, then a decline is forced and a negative c is less likely to occur. .In the present study this was done only if the last test occurred prior to 305 days. The atypical records were dropped from the data leaving 5,026 records. Before computing the variance components, a total of 121 herds each having only one sire or fewer than three cows were dropped. In these herds, sires would be confounded with herd, and in herds with only two cows, one degree of freedom would be lost for herd, leaving only one degree of freedom for estimation of sire and error. This left 4,818 records, 557 herds and 150 sires. 87 Equations for 557 herds were absorbed in setting up mixed model equations for seasons and sires. For the computations of variancecomponents,sequencialsums of squares were computed after removing sums of squares due to the mean and herds. These are in Table IV.3.2. These reduced sums of squares were used to compute the REML estimate for error variance (3:) in (III.4.3), i.e., “2 oe-( (‘4 I ~ 2-332-§¥me-49L where y y is the total sums ofsquares after removing the mean and herd effects, i.e., values in the 3rd column of Table IV.3.2. Noting that the sums of squares due to age were previously removed. . Table IV.3.3 shows the means, standard deviations and ranges for 305-day milk yield and the lactation curve parameters. These are the values before records were adjusted for age and age squared as mentioned in the method section. The average 305-day actual milk yield for the 5,927 records in this study was 14,801 lbs, which is for first lactation animals. The Michigan DHIA lactation' average is currently 15,463 lbs and the Holstein breed mature equi- valent (ME) is 15,416 lbs. 'Using the average 26 month age adjustment factors the first lactation 305-day records (ME) would be 18,192 lbs which is considerably greater then 15,463 lbs. One would expect the ME average of two yr olds to be greater than the pOpulation average if genetic progress exists. A portion of the difference may, also, be due to selecting a sub-population in which sires are identified, and requiring sires to have eight or more daughters. Table IV.3.2 Total and adjusted sums of squares. 88 2:232:38" ' ss'rA ssnB SST-SSHC 305-day milk 36,753,873,96l 14,409,519,760 22,344,354,201 a 886,152 176,247 704,405 b 99.4666 18.6552 80.8107 c .0112946 .00237596 .00891864 b/c time of peak 27,350,364 3,653,198 23,697,165 peak yield 530,856 225,713 305,143 S 13,970,035,907 2,587,541,657 ll,383,494,249 A - Sequencial sums of squares corrected for the mean. t” I O I Sequencial sums of squares for herds. Sequencial sums of squares after mean and herds. _ c-cbfl) , adjusted for a. 89 The mean value for a_was 31.6. This value compares closely to those reported in the literature and summarized in Table IV.3.4, except for Schneeberger (1981). Noting here, that values for a_are untransformed values of 1n a, except for the present study, which computed a using the nonlinear form of Wood's equation. Schneeberger's data represents a lower producing population. The mean value for b, .212, can only be compared to the value in Schneeberger's study (.409) lwere, as in the present study, time was computed in days. Values for b and c are not comparable between models which are computed using time in days as opposed to time in weeks. Values for c would necessarily be smaller when time is computed in days. Peak yield in the present study was higher than for other studies. This would be expected after comparing the 305—day pro- duction levels with those available from the other studies. The 305-day production was considerably higher than Wood (1967), 14,801 vs. 7,898 to 11,669 lbs and Schneeberger (1981) 7,132 lbs, and higher than Shanks (1979) when comparing mature equivalents, 18,192 vs. 16,465 lbs. Time to peak was greatest in those studies with higher peaks. This is expected if WOod's (1967) equation is used. Shanks (1979) reported a late peak time of 12 weeks, but when calculated from b and c values reported, b/c was equal to 10.1 weeks. Table IV.3.3 Means, standard deviations and ranges of 305-day milk yield and lactation curve parameters before adjusting for age and age squared.A Variables or Parameters mean standard deviation range 305-day milk yield 14800 2800 4,630-27,6OO a 31.6 13.7 .264-86.0 b .212 .145 .000119-l.330 c .00302 .00154 -.000479-.Ol6 b/c time of peak 69.1 133 -7340-3880 peak yield 58.0 10.6 20.4-118 s B -- 3880 -- A - 5,927 records were used before deletion of the atypical curves. B - S is c-(b+1> values which is zero. adjusted for a, therefore, the mean is of residual 91 any :a mo mmsflm> moshommamuucs mum mo=Hm> I m .am>aw mumumemuma Scum vousano I n .uamam>wsco musumz I u .mxmma H.0H u o\n I m .mmmv :« vmusano mm3 mafia I < NcH o.m5Ic.om w emH.5 moan.ea che.sauooo.m oom.ee Ameav eases assumes om.o mo.~H 955.5 ooc.m o5H.o o.oH smog «0 xmma o\n m.o~ m.mn n5.mm n~.5m o.mcIo.om o.mm Assay smog 5mqoo. 5mmo. Como. one. come. «once. u mow. mmN. omN. OmH. mad. NHN. n . . . . . . m ma ma ma 5m mm mm mm mm me On e an exsmaev Aeneas. .Aoeaev Assess a5easv <56=e6 muoseeeeee umwuonmocnom mxcmnm vooz moo: @003 usmmmua no moanmwum> .mzoo nowumuoma umufim How musumum Iowa mnu cw vmuuomou assuage nonuo was zesum ucmmmum mnu as memos amusemuma o>uso cofiumuoma mo somwumasoo «.m.>H manmh 92 IV.3.1 Variance Components The variance components estimates for sire and error from mixed model equations and the apprOpriate restricted maximum likelihood estimators (REML), are shown on Table IV.3.5. All estimates have positive values. The starting values of the variance ratios for iteration in REML are in Table III.4.1. The iterations required for convergence ranged from 2 to 10. Rounds of iterations required were greatly reduced by adding three times the difference between the current ratio, and the new variance ratio, to the current ratio, instead of replacing the current ratio with the new ratio. The genetic variances and covariances for 305-day milk yield and the lactation parameters a, b and c, plus time of peak, peak yield and S are in Table IV.3.6. Their phenotypic variances and covariances are in Table IV.3.7. The genetic covariance between 305-day milk and the curve parameters are all positive except for that with a and b/c (Table IV.3.6). The phenotypic covariances between 305-day milk and the curve parameters are all positive except for c (Table IV.3.7). Persistency as Wood (1967) defined it is c-(b+l), S, but this assumed that a_was constant for all cows. Since this is not the case, a new value for persistency was computed by adjusting for 3, using regression in the GLM procedure of SAS (Barr et al., 1979). Due to the drastic difference in the magnitude of the genetic and phenotypic variances and covariance for the traits, the variance-covariance matrice (P and G) were standardized. This was 93 done to minimize rounding error when P was inverted for the selection index equation: P. ' 2-199 - The standardized values for the genetic and phenotypic values are in Tables IV.3.8 and IV.3.9, respectively. Computations of these values are demonstrated in [III.6.2]. The standardized genetic variances are the heritabilities for the traits, while the standardized phenotypic variances are equal to 1. All standardized values are between +1 and -1. Table IV.3.5. REML estimates of error and sire variance components for 305éday milk yield and lactation curve parameters. Variables or parameters .Error variance Sire variance 305-day milk A 5,042,870 201,252 a 161.398 2.61620 b .0179950 4.30302 E-4 c .0191470 7.34878 E-8 b/c time of peak 5,484.95 99.84450 peak 66.9376 2.52968 8 . 2,641,490 25,559 A - S is, c-(b+1), adjusted for a. .m How umumsfipm . 0 m« m I 0 2+6: .mamwh moon «0 mafia I m .sees aeeImom I a 94 oo~.NCH om n.5o~ ~H.0H xmmm onH Nq.qu «.mam mo\n omamo.I. mlm mec.m.l 55Nmoo.I 5Im Hem.m o om~.m coO¢o. coq~.I NHoHoooo. H~5Hoo. a m.eom NHnm.~I Nw.om mocaoo.I cHNH.I o¢.oH m o.mmH ooo.~ mmm.HI 5¢waoo. c5.qa moo.HI ooo.mom hno mam xHHE 5vamom now Amamcommfin Imwov moosmwum>oo cam AmHmcowmwmv moocmfium> oaumcmo o.m.>H manme 95 .m How mmumsmpm . 2+3: 6 6e m I 6 .pfimfim good no mafia I m .&H«8 5273m I s coo.aee.~ 6m c.66H Ne.me s66e 6mm.mn .m6.een emm.m m6\e asa~.u meONoO. cuneoo.n one mma.a 6 ~5.m~ Haas. Nea.H cameooo. Nessa. e o.Hm~- No.mH 6.mm~- Haoeo.n NHA.HI o.eeH 6 oam.o~ oo~.eH omm.me mmem.: so.- eHo.m ooo.ae~.m «see: m game 6\e 6 e 6 see: nemfimamaommfiuImwovmoocmfium>oo vamAmHmsowmfinvmoocmaum> ofia5uoso£m .muMuoemuma m>uso was Mass zmcImOm 5.m.>H mHan 96 .m How pmumsnpm ma . Aa+nvlomfi m I U .vamah Jame mo mafia I m .361 6:16 barman I < mmo. om mamfio. ma. xmmn omNHo. cmmoo.I H50. m0\n enmoo.l auooo.l Namqo.l ma. 0 c5eao. moomc. onmmo.l 5Hmmo. mac. n Heemo. 5mm~o.I O5HNo. mommo.I camco.l coo. m cameo. ma5ma. Hmaaoo.l accoo. 5m5qo. m5emo.l 0H. H 3an 97 .6 How woumsnpm . 0 ma m I o HH+HVI .6H6H5 s66e «6 6eH6 I a .6H6Hs xHHa s66Imom I < H um 5mo. H xmma 5mq. «No.l H mo\n mNH.I «NH. Neo.I H . 6 eHH. eaH. NaH. ~65. H e NHo.I mNH. O5~.I eoe.I cam.I H 6 mmo. sew. sac. HoH.I Hao. HAH. H «xHHz m 3665. 6\e 6 a 6 _ xHHz .Amamcowmvauwov moocmwum>oo use Amamcowmfimv moosmwum> ofinmuocmzq vowwvummcmum .¢.m.>H magma 98 IV.3.2 Heritabilities, genetic and phenotypic correlations Computed heritabilities for 305-day milk yield and the lactation curve parameters are in Table IV.3.10. The heritabilities and their standard errors are on the diagonals with the genetic correlations on the upper off-diagonals and the phenotype correlations on the lower off-diagonals. The heritability for 305-day milk (.16) is lower than reported by Shanks (1979). Heritabilities are low for all curve parameters except for c and peak yield. The heritability for S adjusted for a, is .038 compared to .034 when not adjusted for a, Table IV.3.ll, contains the heritability values for the curve parameters for the present study and those reported in the literature. Keeping in mind the models are not the same. Then, the parameters in different models represent different traits. Schneeberger's (1981) heritability values for 305-day milk, a, b and c are all greater than found in the present study. Values in the present study are comparable to those of Shanks (1979) for a, b, c and 8, while Shanks values for peak (.15 vs. .23) and 305-day milk (.16 vs. .27) were higher. Genetically, 305-day milk was positively correlated to b, c, peak yield and S and negatively correlated to §_and time of peak yield. These correlations suggest that selecting for 305-day milk should decrease a, increase b, but little change in c would be expected because it has a low genetic correlation with 305-day 6111: (.004) .' 99 .m>wumwmc mumz AO5mHv :mEmmouo An wouammoua masshom mnu magma umusaaoo mmumEHumm mocmwnm> I m . m m: . o m I 6 66m 6 6 he6 HH+HVI H m a 6H6H5 x666 H6 meHe I o .6H6H5 sHHa s66ImOm I m .mwmmnucmuma a“ mum muouum mumpcmum mom mQOfiumHmuuoo ofia5uoCmsa mum mamaowmapImwo HmBoH mSu .mcofiumamuuoo ofiumcmw mum mamsowmamlwwo Home: mnu .mmaufiafinmufiuoc mum mascowmfiv 0:9 I < Aeoq.vmmo. Amma.vccmo. 505.Hv5m¢. Aoom.vaH.I AmeN.VoHH. Ammo.v5HHo.I Aooa.vammo. mm AwN.Hquu. Aqu.va. A5~o.v5mwo.l Awo.qu5a. Aaea.vc5a. Ammm.vm~H. mmww. xmmm Acmm.vwm~. 5~5H.vmme.l Acm~.va5o. Ammo.v5aqo.I Amwm.v~aa. AH¢¢.VO5N.I Awem.vww5o. Uo\n Am5.mv¢mH.I Am5m.va~oo.I Aqa.~v5wc.l AvoH.VmH. mmm5. AoNH.vqoo.I Aaaq.v5oH.I o ANm~.qu~. A<¢5.v5mo. Aoo.mvomN.I 55Hq.vo~5. AoMN.Vmao. mO5m.I Aoao.vma5o. n Amc.avmmq. 55o.vac~.I Amm.wvmmm. 555o.v5oc.l Amm¢.veom.I Aooq.v¢co. Ammm.VO5H. m Aqq.av~mm. Accu.vaaa. Aoam.vaowo.I A5Ne.vfiamoo. 5Hc5.v5mm. AmH.Hv5om.I 55oa.vma. mxHHz m zoom o\n o n m JHHZ .muoumamuma m>u=o cofiumuoma new mama» xawa amulmOM pom mooHumHmuuoo owahuoomna mam owumcmw .H manna 100 Schneeberger (1981) found negative correlations between 305-day milk and b (8.09) and milk and c (-.14) and a positive correlation (.37) with a, The measures of persistency Schneeberger used were positively correlated to 305—day yield (.07 to .16). The phenotypic correlations between 305-day milk and the curve parameters in the present study were all low except for peak yield (.849)(Table IV.3.lOL Phenotypic correlations between a and b (-.87), and §_and c (-.604) were negative. The phenotypic correlation between b and c (.792) was positive. Table IV.3.11 Heritability values reported for lactation curve parameters and 305-day milk. Variables or Present Shanks Schneeberger parameters Study (1980) (1981) 16 a .06A .10 .09 b .06 .06 .15 c .09 .14 .20 b/c time of peak .15 .02 peak .15 .23 S .04 .02 305-day milk .16 .27 ' .42 A - model used computed a, not 1n(a). The genetic correlation between a and b (-.906) and §_and c (-.607) were negative and between b and c (.726) was positive. The parameter c is itself a measure of persistency because it 101 represents the slope after peak yield and persistency may be defined as the ability to maintain peak production. To increase persistency, c needs to be decreased. Therefore, selecting negatively for c to increase persistency would tend to increase a and decrease b both genetically and phenotypically. Shanks (1979), using the logarithm form of Wood's (1967) equation, found very low positive genetic correlations for 1n(a) with b and 1n(a) with c. This differed with the preSent study and with results of Schneeberger (1981) who found high negative genetic correlations between a_and b (-.79) and a_and c (-.67). Based on the genetic correlations, one would expect that selecting to increase §_wi11 decrease 305-day milk and b, and decrease c, which would increase persistency. lhzwould'also increase S, persistency as WOOd (1967) defined it. Selecting to increase b would increase milk, decrease a_and increase c. Selecting neg- atively on c in order to increase persistency would increase a_and decrease b and increase 8. Since lactation persistency increases as c decreased.it would be expected to be negatively correlated to Wood's (1967) definition for persistency. Although this correlation (—.184) is negative, it is low. Also, those variables positively correlated to c would be expected to be negatively correlated to S and vice-versa. This is not true for b, because b has a positive correlation with both c (.726) and S (.248). 102 IV.4 Genetic and Correlated Genetic Change Genetic change was computed for 305-day milk yield selected alone. This and the correlated genetic responses in the lactation curve parameters after one round of selection are reported in Table IV.4.1. The genetic change in milk was 359 lbs. With this, an estimated change of —.475 in 3, .006589 in b, 8.678 10‘7 in c and an increase of 1.15 lbs in the peak yield would be expected. The change in the curve parameters when selecting for milk alone is also expressed relative to their means (Table IV.4.1). The change in the curve parameters when selecting for milk alone was compared to expected change when selecting for each parameters alone, Table IV.4.1. By selecting for milk alone the percent change in a, relative to selecting for 3 alone, was -l60, 52% for b, .4% for c, -113 for time of peak, 95% for peak yield and 113% for S. The genetic change when selecting alone for each curve parameter is reported in Table IV.4.2, along with the change relative to their means. The genetic change as a percent of the means when each trait was selected alone were 2.4% for 305-day milk 2.6% for a, 5.9% for b, 6.9% for c, 7.7% for time of peak and 2.1% for peak. Table IV.4.3 contains the correlated genetic change in 305-day milk when selecting for each curve parameter alone. The greatest lose in 305-day milk occurs when selecting for §_alone (-120%), while the greatest gain occurs when selecting for peak alone (87%). Table IV.4.4 contains the change in the other curve parameters when selecting for either a, b, or c alone. When selecting for a, the mean changes in b and c were -4.5% and -2.8%, respectively. 103 Selecting for b alone resulted in a mean change of -2.8% and 4% in a and c, respectively, and'selecting for c alone resulted in a mean change of 2.4% for a_and —5.5% for b. These indexes are compared to genetic gain when selecting for milk alone. Table IV.4.S lists the indexes with the genetic change for each parameter when that index is used. Table IV.4.6.lists. the indexes with the percent genetic change for each parameter relative to the change when selecting for milk alone. Table IV.4.6 lists the indexes with the percent genetic change for each parameter relative to the change when selecting for the parameter alone. Table IV.4.1 Change in lactation curve parameters after one generation of selection for 305-day milk yield alone. Variables or parameters unit change. % change lA % change 2B 305-day milk 359.0 ’ . 2. 4 100 a -.475 -l.5 -l60 b .006589 3.1 52.2 c 8.678 E—7 .021 .411 b/c time of peak -.6950 -1.0 —113 Peak yield 1.159 , 2.0 95.3 3" .70 . 62 , - 114 A — Change 1 is percent change relative to mean of parameters. B - Change 2 is percent change relative to change when selecting for parameter alone. C - c-(b+1), adjusted for a. Mean for S is zero because values are residuals of regression on a. 104 Table IV.4.2 Change in each lactation curve parameter when selecting alone for itself. V::::::::r:r Unit change % changeA a .8196 2.6 b .01262 5.9 c .0002087 6.9 b/c time of peak 5.306 7.7 Peak yield 1.217 2.1 sB 62.06 --C A - relative to mean of parameter. -(b+l) B - S is c , adjusted for a. C - mean for S is zero because values are residuals of regression on a. Table IV.4.3 Change in 305-day milk yield when selecting - for lactation curve parameters alone. Parameter being selected Change in milk % changeA a -83.4 -120 b 108 30 c 1.38 .38 b/c time of peak -20.7 -106 Peak yield 313 87 sB . 96.0 27 A - change in 305-day milk yield relatiVe to its mean. -(b+l), adjusted for a. 105 Table IV.4.4 Change in lactation curve parameters when selecting for a, b or c alone. Parameter Unit Change ;;.-Z Change 1A % Change 2B When selecting for a a .8196 2.6 -273 b -.OO9522 -4.5 -244 c -.00008339 —2.8 -9700 Peak yield -.l999 .' -.34 -l77 When selecting for b a -.8920 82.8 47.5 b .01262 5.9 191 c .0001198 4.0 13800 Peak yield .6651 1.1 57.4 When selecting for -c a .7557 2.4 —259 b -.01159 -5.5 -276 c -.0002087 —6.9 -24100 Peak yield .002448 0.0 0.211 A - Change relative to mean of parameter. B - Change relative to change when selecting for milk alone. 106 666.H 6-6 666.6 6H6666. 6666.- 666 H 6 6HuH H66.H 6-6 666.6 666H6. 6666.- 666 H 6 6HuH 66H.H 6-6 666.H 666H6. 6666.- 666 6H H 6H6H 66H.H 6-6 666.6 666H6. 6666.- H66 6 H 6HuH - 6 6H6: 66H.H 6-6 66H.6- HH6H6. 6666.I 666 H 6 6HuH 66H.H 6-6 666.6I H66666. 6666.- 666 H 6 HHHH 666.H . 6-6 666.6- 666H6. 6666.- 666 6 H 6H1H I 6666 6H6: 6666. 6-6 666.6I 666666. 666H66. 666 H 6 H 6 6 66H 6H6.H 6-6 66H.6I 666H6. 666666.I 66H 6 6 6 6 H 66H 666.H 6-6 666.6- 666666. 666666.- 66H 6 6 6 6 H 66H 6666 6 6 6 6H6: 666.H 6-6 666.6 666H6. 6666.- H66 6H H H 66H 666.H 6-6 666.6 666H6. 6666.- HH6 6 6 H 66H 666.H 6-6 H6H.6 666H6. 6H66.- 666 6 H. H 61H 666.H 6-6 666.6 .666H6. 6666.- 666 H H 6 66H 666.H 6-6 66H.6 66HH6. 6666.- 666 H H 6 66H 6H6.H 6-6 66H,6 666666. 6H66.- 666 6 H 6 HuH 6:36 666 .66666 6 6 . 6 6666 6666 6 6H6: moanmfipm> mo mwcmso oauonmu musw603\moaamwum> NovsH .co6uooaom_mo.cowumuoamw H umuwm.moxmu:6 650666> you mumuoamumn m>uso can xH6E zmvlmom :6 owsmnu ofiuosou m.q.>H manna 107 66.66 656H6. In 666.6- H6HH6.I 6665. H.6- 6H- H 666 66.H6 6666. 616 666.HI 6-6 566.6I 666H. 6.666 T H 666 6 6HH: 666.6- 66HH.I 666.H- 6-6 666.H 666H6. 6666.- 66 H 6 6- H 566 666.6 6666. 6H6.6I 6I6 666.H 666H6. 6666.- 65H 6H 6H 6H- H 666 66.66 6566.- 566.5 6I6 656.HI 666H6.I 6666. 6.56- H 6H 6H- H 666 6666 6\6 6 6HH: 6H.66 66666.I 666.6 6I6 6H6.6I 6H6666.I 666H.I 6HH 6H H 6H H 666 66.66 566.H 6665. 6-6 666.6 666666. 6566.I 666 H H 6 5 666 66.56 H6666. 666.5 6I6 656.6- 6H6666.- 6666.I 56H 6 H 6 H 666 66.66 6666.I 6H6.6 6Im 566.HI 6H6666.I 56666.I H.5- H H 6H H H66 , o .6 6 6666 656 6 6 6HH: 66 _6 6666 656 6HH: 666 H566. 5-6 H6H.6- 6-6 666.5- 666666. 666 6 H 6 6 6H6H 5666. 5-6 666.6- 6-6 6H6.H 666666. 666 6 H 6 6 6HuH 66H.H 5I6 H56.6 H55666. 666666.I 66H 6 6 6 H 5HuH 6666 n 6 6HH: 666 6 6 6 6HH: 666 mmenmaum> mo mwamnu oaumcou munw6m3\mmanmfium> vacH 56.6666 6.6.>H 6H666 108 .63H :6 666 @6666 xmma can wae 66666-mom I < cmMHO.I mHo.H. 66.H6- 6666.- 6-6 666.H- 666- 6H-_ 6H- 6H- 6H H 666 66.66- 6666.- 6-6 666.6- H6HH6.- 6HH.H H5H- 6- 6H- 6- 6H H 666 66.6H 6665. 6-6 666.6- 666666.- 6566. 6.6H- H 6H- H 6H H 666 6666 6 6 6 6HH: 66.6H 666.H- 6-6 655.6- 666H6.- 6666. 666- 6H 6H- 6H- 6H H 666 6H.66 6666.- 6-6 666.6- 6H6H6.- 5666. 666- 6H 6- 6- 6H H 666 6.56H 65H6. 6-6 666.6- 6H666.- 6666. 6 6H H H 6H H H66 6. 6666 6 6 6 6HH: 666 .6 6666 6 6 6HHz. mmapmaum> mo mwcmno UHumcwu mom munw663\mmHnmfium>. xmvsw Au.:ooV m.c.>H anmH 109 HOH moHI 66H 66 mm H 6 NHuH NOH NNHI 5HH mm mm H m HHuH ooH H661 H6~ Hm mm 6 H CHnH xmmm 3H6: no 6Hm| om ooH- 66 H o H o m 66H 6OH 65ml 66H H am 6 o o o H muH 60H 666: m6H. H cm 6 o m o H 66H xmmm u 6 m xHHz 6HH O6~m 66H 66H 66 mH H H 66H NHH cane CNN nnH 6m 6 6 H 66H 6HH 0666 66H qu mm o H H 66H NHH one 66H 66H 00H H H m 66H 6HH 0656 6mm 66H 00H H H m. NuH oHH Onwe m6H 66H ma N H m HnH 6H Hz .666 . x666 o a a mom xmmm a xHHz oGOHm xHHE now wcHuomHmm cmn3 mmamso o» 6>Humeu mwcmnu 6cmoumm mustmB\moHanum> xwvcH .:OHumuoamw H new wGOHm xHHB Mom waHuomHmm cuga mwsmno cu m>HumHmu mwxmvcH m:OHum> c6 mumumsmuma m>uso 6:6 xHHE haw-mom you mwcmno UHumaww uamuumm ©.c.>H mHan 110 moHI OHHI MHN OO55H omH omH HH H 6 on H 5HN m 56 moo oommH mNN moH . 66 OH OH CHI H cum 56 5MHI oHHHI OO5HNI omnl ceml OHHI H mH oHr H mum xwmm o\n u 6HH: 06H 5oH| 6wNHI COHHI meI .mm mm OH H OH H 666 MMH .m¢ Hom- ooHH 65. cm 66 H H .m 5 mum wMH mo omHHI wo5l m5HI a6 66 o H o H mum om m6H| O56HI ommwl oqml wH NoHI H H mH H Hum 6HH=.666 . m 666m o\n o n 6 non m xmmm , o\n 6HH: mm oMHI HoHt HoH- me o H o o mHuH mm OMHI N HOHI me o H o m mHuH QNH o5 5m H mm o e o H 5H6H o n m 6HH: ¢OH ammo 66H w6H 00H H o oHuH moH oon mmH 5mH 00H H m mHuH mm oomHH mew mew N5 mH H 6HuH OOH oooHH 5am 6am w5 o H MHHH 6666 6 6 . . 6 6HH: 666 6666 6HH: mom ocon xHH66uom waHuumHmm c623 mustms\mmHanum> xmwaH mwcmzu Cu m>HumHmH mwcmao unwoumm 66.6666 6.6.>H 6H666 111 66H- 65H- 6666H- HH6- 6H6- 66H- 6H- 6H- 6H- 6H H 666 66H- 66H- 66666- 656- 666- 66H- .6- 6H- 6-. 6H H 666 H6 66 66566- 66H- 666- 66H- H 6H- H 6H H 666 6666 , 6 6 6 . 6 V6HHz 66 66H- 6656- .666- 656- 65H- .6H 6H- 6H- 6H. .H 666 65 65H- 6566- 6H6- 556- 66H- .6H 6- 6- 6H H 666 66H 56 666- 56H- 66H- 6 . 6H H H 6H H H66 6HH: 66d . 6. 6666 6 6 6 - 666 6 6666 6 6 6HH: H6 656-.. 666- H6H- 6H- H 666 56 66H- 66H- 66 H- H 666 6 .6 6HH: 666 6- , 6HH: 666 maOHm xHHE you waHuumHmm :653 mwsmso ou m>HumHmu mwsmso uawoumm muan63566Hanum> xmvcH H6.6o66 6.6.>H 6H666 112 om mo. «OH quI mm H . o NHuH mm H. Ho ~MHI ma H m HHuH OOH H QNH 56H: m¢ o H OHuH xmmm xHHz No 5.H mm H. we H o H m auH am N mm HOHn mm o o o H mnH NoH o.H m5 HOHI on o o m H 56H xmmm o a xHHz 66H 66H: 66H 66H; 66 ,6H H .H 66H noH wNHI nHH meu mm c o H muH aoH OHHI mOH mmHI mm o H H 66H ooH mHHI em quI OOH H H m muH acH mHHI No omHI OOH H H m NuH moH omHI mm an| m¢ N H m HuH 6666 6 6 6 6HH: 666 6666 6 , 6HH: < men mcon vmuUMHmm 6H umumsmuma muanm3\mMHanum> xovcH no xHHS :mLB mwcmsu Cu m>HumHmu mwcmso uawouwm .60Humumamw H 606 acon kuomHmm mH umumamumm no xHHa :053 mmamso ou m>HumHmu mumumemumm m>u=o can xHHE hmvlmom cH mwcwnu UHumamw acmuumm .m.q.>H mHan 113 66H- 66H- 66H- 65H- H6 65H- HH H 6 6- H 566 6 66 56H- H6H- 6HH 66H- 66 6H 6H 6H- H 666 66 66H- 66H 66 56H- 66 6HH- H 6H 6H- H 666 6666 656 6 6HH: 66H 56H- 66H 6 66H- 66H- 66 6H H 6H H 666 H6H 66 6H 66H- 66 66H- 66 H H 6 5. 666 66H 5 66H 6.6 H6H- 66H- 66 6 H 6 H 666 66H 66H- 65H 6 65H- 6HH- 66H- H H 6H H H66 .6 6666 656 6 6 6 6HH: 666 6. 6666 656 6HH: 666 56 H. H6H- 6. 66 6 H 6 6 6HuH 66 H. 6.H 5.. 66 6 H 6. 6 6H6H 66 66H- 66 H6H- 66 6 6 6 H 5HuH 6 6 6 6HH: 66H 6HH- 65 66H- 66H H 6 6H6H 66H H6H- 66 H6H- 66H H 6 6HHH 66 66H- 66H 6H6- 65 6H H 6H6H 66 66H- 66H 6H6- 65 6 H 6HuH 6666 6 6 6 6H6: 666 6 6HH: 666 acon wmuumHmm 6H umumEmuma muanmz\mmHanum> xmvaH can: mwcmnu ou m>HumHmu mwamnu acmuumm 66.6666 5.6.>H 6H666 .u aH mwcmso m>Humwm= m mucmmmuamu mcon u now wcHuumHmm cu m>HumHmu mwsmso unmuuma m>HuHmoa 6 660666656 .me>Humwmc kuuonw 663 .mCOHm cmuumHmm cmga .o umumamumm I < 114 66H- 65H- 65 6H6- 66H 66H- 6H- 6H- 6H- 6H H 666 66H- 66H- 6HH 66H- 66H 66H- 6- 6H- 6- 6H H 666 66 66 66H 66H- 6HH 66H- H 6H- H 6H H 666 6666 6 6 6 6HH: 66 66H- 66 666- 66H 65H- 6H 6H- 6H- 6H H 666 66 H5H- 66 6H6- 66H 66H- 6H 6- 6- 6H H 666 666 66 H 66H- H6 6 6H H H 6H H H66 6 6666 6 6. 6 6HH: 666 6 6666 6 6 6HH: 666 66 66H 6 66H- 66 H6H- 6H- H 666 66H 66 66H- 66 66 H- H 666 6 6 6 6 6HH: 666 6 6HHz 666 «GOHm cmuomem muanmz\mmHanum> xmvsH 66:3 mnmuosmuma mo mwcmno unmoumm 66.6666 5.6.>H 6H666 115 Because none of the lactation curve parameters that have a high correlation with BOSoday milk, has a heritability higher than that of milk, it is expected that selecting for milk alone will increase 305—day milk the fastest. This is evident by the results in Table IV.4.5 where genetic gain in milk is less than 359 lbs for all but a few indexes. Indexes 1:1 through 1:19 attempt to increase the ascent to peak and/or increase the peak yield while selecting for 305-day milk. The notation 1:1 refers to the first strategy and the first index within that strategy. The greatest increases in milk yield occurred in indexes where milk had weight of 3 or greater, except for index 1:9 table IV.4.5 Index 1:9 is a restricted index, along with 1:7 and 1:8. These restricted indexes restricted the change in a and c while selecting for b. This was done to increase peak yield, a(b/c)bexp(-b). However, this also restricted the gain in any correlated traits, even when they were selected for, i.e., milk, b and peak yield. Kempthorne and Nordskog (1959) discussed restricted index method where 2 = B‘lss becomes 2 - [2-2’1§s<9'92'199>’19'§12'1§2 and 9'9 is the rows of the genetic variance-covariance matrix for the traits being restricted. The greatest increase in b occurred using indexes 1:5, 1:6, 1:10, 1:12, 1:13 and 1:14 (Table IV.4-.5). Index 1:10 increased b, the ascent to the peak, the greatest, although b was not included in the index. This index selected for milk and peak with weights of 1 and 6, respectively. 116 The largest increase in peak of this first group of indexes occurred in indexes 1:4 and 1:6. The least occurred in two restricted indexes, 1:18 and 1:19, where a and c were restricted. In 1:17, a and c were also restricted, but by putting more selection pressure on b, peak yield increased much more than in 1:18 and 1:19. The effect of this first group of indexes on c varied from an increase of 9.58 10"5 in 1:13 to a more desirable change of -4.13 10.6 in 1:8. However, 1:8 restricts c and c only changed 2% of what it would decrease if it were selected alone (Table IV.4.7). However, c changes -5772 of that when selecting for milk alone, (Table IV.4.6), but remembering the genetic correlation is only .004 between 305-day milk and c. In summary, the use of indexes for strategy 1, in which the attempt was to improve b and peak yield, resulted in: 1) The greatest change in b, relative to selecting for milk alone occurred in 1:3 and 1:5, and for peak occurred in 1:2, 1:4, 1:6 and 1:17. 2) Index 1:2 provided the greatest increase in b and peak while maintaining the same change in milk, as when selecting for milk alone. The second group of indexes, 2:1 through 2:9 attempted to delay the time of peak and increase persistency. Selecting for persistency was either done by selecting negatively for c or selecting positively for S, Wood's (1967) definition for -(b+l) persistency, c , but adjusted for the scaling parameter, a. 117 Only one of these indexes, 2:3, maintained genetic gain for milk production. This index used weights of 7, 3, 1 and l for milk, time of peak, peak and 8, respectively. Several indexes produced negative gains in 305—day milk; 2:1, 2:5 and 2:9. In Wood's (1967) equation, selecting for b increases the time to peak. However, for these indexes, time of peak (b/c), is selected for directly. Indexes 2:1 and 2:4 produced the greatest increases in the time to peak. Both of these indexes had negative changes in b and c. To get a larger value for b/c, c must decrease faster than b. S increased the most in 2:2, but nearly as much in 2:3, when milk and time of peak had higher weightings. Indexes 2:5, 2:8 and 2:9 decreased c the greatest, but this decrease, which indicates an increase in persistency was not consistent with WOod's (1967) measure of persistency, S, which only increased mildly due to use of these indexes (2:5, 2:8, 2:9). Peak yield did not increase in all indexes, in the second strategy, in which it was selected. It had mild increases compared to the increases in the first group of indexes (1:1 to 1:19). In fact, the greatest increase in peak in the second group occurred when milk was selected the heaviest (2:3). Most of the group 2 indexes produced negative gains in b. Selecting for b alone would increase time of peak, due to the relation of b/c. However, selecting positively for b was not included in these indexes to delay time of peak. The greatest 118 decreasejx-b occured in 2:5, when c was selected negatively. A large increase in b/c resulted (7 days). Thus, both b and c decreased but c is decreased faster. Comparing indexes 2:8 and 2:9 indicates that selecting negatively for c is detrimental to 305-day milk. This is true even when c is equally weighted with milk (2:8). When c recieves a weight 10 times milk (2.9), the genetic change in milk becomes negative. But the greatest change in c occurs in this index. Index 2:5 gives the greatest decrease in b and c relative to the genetic change expected when they are selected alone (Table IV.4.7), while increasing time to peak 132% of that when selected alone. The greatest change in b/c relative to selecting for it alone occurred in 2:1 (1792). Likewise the greatest change in b/c relative to when selecting for milk alone occurred in 2:1 (13692). The third group of indexes, 3:1 through 3:6, attempts to flatten the lactation curve by increasing a, decreasing the peak and increasing persistency or decreasing c. Indexes 3:1 to 3:6 (Table IV.4.5) indicate this is not possible without decreasing milk yield considerably. The greatest loss in milk occurred in 3:2, 3:3 and 3:6. The increase in a was greatest in 3:5 and 3:6. All indexes used caused a decrease in b, with the greatest decrease in 3:2 and 3:3, in which b was being selected negatively. 119 Selecting for S was most successful in 3:1 and the only index where the genetic change in 305-day rthg was not negative. This index produced a 220% increase in S, relative to selecting for it alone (Table IV.4.7). Indexes 3:4 and 3:5 where most successful in selecting negatively for c. In fact, the most successful in all three groups of indexes. Index 3:4 and 3:5 produced greater decreases in c than selecting negatively for it alone, -l35 and -1102, respectively. Indexes 3:4, 3:5 and 3:6 produced greater increases inla than selecting for it alone, 118, 135 and 123%,respectively. Concurrently, 3:4 and 3:5 produced the most desired results for g_and c based on the goals of this group of indexes. The percent change in.§ and c relative to selecting for them alone was 118 and -135 for 3:4 and 135 and -110 for 3.5, respectively. IV.4.1 Changes in lactation production The genetic change in 305-day milk and peak yield were computed in two ways. 1) From the expected genetic change in 305—day milk and peak for each index. These will be called the expected values. 2) From the expected genetic change in a, b and c, estimates for 305-day milk and peak yield were computed from: A _ 05 b y305-day milk - 30f3 t exp(—ct) dt and A Peak 8 a(b/c)bexp(-b). These values will be called estimated values. 120 For selected indexes, these expected and estimated values are reported in Tables IV.4.8 to IV.4.10 for 1, 5 and 10 generations of selection. In the same tables, change between generations and the accumulated change from generation to generation for expected and estimated 305-day milk are reported. The expected values for the curve parameters a, b and c are in Table IV.4.11 to IV.4.13 for generations 1, 5 and 10. The base values, which are those computed for the current population, are16,684lbs for 305-day milk, 62.9 lbs for peak, 70.2 days for time to peak, 31.6 for a, .212 for b and .00302 for c. These values are listed in Table IV.4.ll as generation zero in the milk only index. The value for 305-day milk, 16,684 lbs,is the estimated value from the integral produced by bbease values for a, b and c. These base values are the population means prior to adjustment for age and are reported in Table IV.3.3. When selecting for milk only (first index, Table IV.4.8 and IV.4.11), the expected 305-day milk yield was 17,043, 18,479 and 20,274 lbs for generations 1, 5 and 10, respectively. However, this was not equal to the estimated change from computing new curves when selecting for milk alone. They were 16,949, 18,011 and 19,329 lbs, respectively for 1, 5 and 10 generations. The expected values were about 100 lbs more per year than those values estimated by the integrals (Table IV.4.14). Comparison of the expected and estimated values for peak yield also shows the estimated values were less than the expected values (Table IV.4.14). 121 n.mOH N.om m.mm some oqqm mmmmm OO6m «moon oH m.mm m.mo n.mm Hon ommm e6me OONH 6wmmH n m.mm n.6o $.60 6cm coo omNNH 06m «NONH o H H H566H zoom n 6HH: o.mw o.on 6.06 mmou NNo wmme Ommm cowom oH n.mn m.¢o 0.50 HmHH mam onmH ocNH 666wH m «.mm N.6o m.me mmm mmm «NooH «mm enamH N H m H5HHH Hmom n xHHz N.Hm m.qn m.Hu seem meH mNmmH ommm «swam CH m.ow m.wo H.no QNMH HQOH HHowH mqu mmqu m m.~m c.6o “.mc 60m 60m mcooH mmm m¢QNH H H5xHHz 6HH: ~.c~ ¢.~o ¢.No I: II 6mcoH II «mocH :OHuoHoooo ommm O5 o5: xmom Moon 6HH: xHHz xHHz xHHz 6HH: nouoooxm wouoooxm .umm owcoso mowcmsu hon owcmsu %ma asoo< mom Eooo< mom 0 < woumaHumm oouoooxm mustozxmoHanmm> sou\xovsH .ocOHm omuooHom xHHE coo moxopoH H moouw oEom pom Homo mo oaHu mom Homo .xHHE zoo-mom How moon> oHuooow oH pom m .H mooHumHocou .m.6.>H oHnt 122 0.0NH 0.06 0.6HH 0606H H060 0~0H0 0060 60000 0H 0.00 0.00 0.60 NNHO 0000 60000 00~H 6000H 0 6.06 H.60 0.00 NO0H ~00H NONNH 060 «NomH 0 H H50HHH xmom 6HH: 0.00 6.06 0.60 00H0 0600 06000 0000 6000H 0H 0.H0 6.00 0.06 H600 000m 0~00H 06HH «NONH 0 6.06 0.00 6.60 H00 H00 0HNNH 0mm NHO0H H 0 H 0 0 H50HH 6660 o a m xHHz H.50H 0.0m 0.HOH H6HHH 0000 000mm 060H qman 0H 0.00 0.00 0.06 0606 0600 000H~ 000 «ommH 0 0.06 H.60 0.00 000 000 6006H 60H 00m0H 0 0 0 0 H H50HH 6660 o n m 6HH: 0.00H 6.06 6.60 0HO0H 0H00 00600 060H 60n~H 0H 0.60 H.00 H.0m 0066 0000 ~00H~ 000 0HNNH 0 0.06 H.60 0.00 N06 606 ~066H 50H H0n0H 0 0 0 0 H H5NHH o\n xmom xmom 6HH: 6HH2 3H6: 6HH: 6HH: vouooaxm 06600000 .660 owomfio owomsu >00 owsmzo 0mm 6660 o n m xHHz Esoo< 0 000 Eooo< 000 0 < woumaHumm wouooaxm muan635moH00Hum> :o0\x60oH 66.6666 6.6.>H 6H666 123 0.00 6.66 0.06 00H0 000H 0600H 000H 6066H 0H 0.06 0.00 H.00 000H 000H 0000H 0H0 60H6H 0 6.H6 0.60 0.00 000 000 0600H 00H 0060H 0 0 0 H H\6H"H u 0 m 0H0: 0.H0 0.06 0.06 0600 060H 0000H 0000 60000 0H 6.H0 0.00 0.00 000H 00HH 00HOH 000H 6060H 0 0.06 0.60 0.60 600 600 H000H 000 6606H H 0 H\0H"H 0 xHHz 0.60 0.66 0.06 H060 000 0HHOH 0H00 6060H 0H 0.00 6.00 0.00 006H 06HH 00HOH 006H 0000H 0 6.06 H.60 0.60 0H0 0H0 6006H H00 0000H 0 H H\0HuH 0 3H0: 0.00 0.66 0.H0 0060 0000 00000 0000 60000 0H 0.00 0.00 6.H6 6H00 H600 0000H 006H 6660H 0 6.06 H.60 0.60 066 066 HOH6H 000 0006H H 0 H\HH"H o\0 xmmm xmmm xHHz xHHz xHHz xHHz xHHz 0muomax0 0muomax0 .umm mwcmco mwcmso 0mm mwcmno 6mm xmmm 3H0: 563. m mom 883. mom 0 < 0mumEHumm 0muom0xm mustmB\moH0mHum> cmo\xm0:H 60.6000 0.6.>H anmH 124 .xHHE >m0I000 cmumEHumm Mom QOHumHmamw mmmn scum mmamnu 00aMH=ESUU< I 0 .xHHa 6m0I000 umumEHumo How 0:0Humumcmw :QQSumn mmamco I 0 .xHHE 6m0I000 0muuwaxm How :oHumnmamm mmmn Scum mwcmno 0mumH=Ebou< I < 0.06 0.00 0.00 0.6HI 6.6I 0600H 0000 6000H 0H H.06 0.60 0.00 0.6I 0I 6600H 00HH 0006H 0 0.06 0.00 0.00 0.HI 0.HI 0000H 000 0000H 0 H 0 0 H\0HuH o\0 xmmm xmmm 3H0: xHHz xHHz. xHHz xHHz 0muumaxm 0muomaxm .uxm «mango 0mwsmao 6mm mwcmnu 0mm 0 A m xHHz aboo< 000 aboo< 000 0 4 0mumEHumm 0muom0xm mu£0H03\moH0MHum> cm0\x00:H Au.:oov m.6.>H «Heme 125 0.06H 0.66 6.00 0.H0 060HI 000I 0600H 000I 6000H 0H 0.00H 0.06 0.00 6.60 066I H00I 0000H 00HI 6060H 0 0.66 6.06 0.00 0.H0 06HI 06HI 0000H 00I 0600H H 0H 0HI H H\0"0 xmmm exp 0 xHHz .00H 0.00 H.00 0.06 6006I 0000I 0H6HH 00HH 6006H 0H .HHH 0.00 0.00 0.00 0000I H0H0I 0000H 060 0006H 0 6.06 0.00 0.00 0.00 060I 060I 00HOH 0HH 0060H 0H H 0H H H\6"0 02 xmmm o\0 xHHz 6.00H 0.H6 6.60 H.06 6000I 0000I 0000H 06I 6HO0H 0H 6.6HH 0.00 0.00 0.H0 0000i 0000I 0000H 00I 0600H 0 6.06 6.60 6.00 0.00 606I 606I 0000H 6I 6600H H H 0H H H\H"0 0; xmom o\0 xHHz ~.o0 N.oN ¢.No a.~o I- .. «mega -I «mega aoflumumamu «mam o\H“~ o\0 u\0 swam xmwm xHHz xHHz xHHz xHHz xHHz cmuommxm .umm 0muumax0 .umm wwcmzo mwamso 6mm mwcmno 6mm Iueauo< 0 000 Aimasoo< 000 UmumEHumm 0muumaxm mustm3\meAMHum> am0\xmwcH .mmxmvaH 0 asouw How xmma mo mEHu 06m xmma .xHHE mm0I000 pom mmsHm> oHumamw 0H 0cm 0 .H :OHumumcoo .0.6.>H «Home 126 .xHHE 6m0I000 vmumawumm now =0Humumcm0 mmmn scum mwcmco vmumHaasuu< I 0 .xHHE 6m0I000 vaumEHumm you mGOHumumamw ammSumn mmcmno I 0 .xHHE 6m0I000 vmuummxw uom GOHumumcmw mmmn scum mwcmco wwumHsazoo< I < II .00H H.00 0.00 060I 600I 0000H 00I 6000H 0H II 6.66 0.00 0.00 0HHI 00HI 0000H 0HI 0000H 0 II 0.H6 0.00 0.00 6.0I 6.0I 0600H 0I H000H 0HI H H\0"0 o xHHz H.60 0.06 0.00 0.06 0000 600H H060H 006H 6660H 0H H.66 0.66 0.00 0.H6 060H 000H 6000H 000 6006H 0 0.00 0.H6 0.00 0.60 000 000 6006H 06H 0000H 0H 0H 0HI H\0u0 o\0 U\n 3mmm xmmm xHHz xHHS 0HH: xHHZ xHHz 0muumaxm .umm 0muoonxm .umm mmcmao mowcmso 6mm mwamso 6am xmmm u\0 o xHHz Enou< 000 sauo< 000 0 I< 0mumEHumm vmuommxm muanm3xmmH0mHum> am0\xw0:H Hu.aoov a.6.>H mHama 127 0.60 0.00 0.06 0060I 6H00I HO0HH 0000I 6000H 0H 0.06 0.60 H.00 0000I 0000I 0060H 0H6HI 6600H 0 6.00 0.H0 0.00 000I 000I 0600H 000I 0060H 0H 0HI 0HI 0H H H\0"0 0 3mmm 0 m xHHz 0.00 H.00 6.00 000I 06HI 0000H 00 6660H 0H 0.00 0.60 0.00 6HHI 60I 0600H 00 6H60H 0 H.00 0.00 0.00 6HI 6HI 6000H 0 0000H 0H H H 0H H H\H"0 0 xwmm 0 m JHHZ II 0.00 0.00 II II 6000H II 6000H :oHumuowu mmmm 0\ o\0 xmmm xmmm xHHz xHHz xHHz xHHZ xHHz vauomaxm 0muumaxm wwquHumm owcmso 00cm00 6mm 00cmso 0mm Enuo< 000 azuu< 000 0mumEHumm 0muomaxm mu£0H03\moHanum>. :m0\xm0aH mo mEHu cfim xmma .xHHE 0m0I000 ~00 .mmxmvcH 0 macaw H00 xmma mmnHm> UHumcmw 0H 06m 0 .H coHumumcmw .0H.6.>H mHnme 128 .xHHE 0227000 039503 .80 :OHumuwcmw 923 50.5 «0530 0mumHasnou< I 0 .xHHE 0m0I000 nauwaHumm you mCOHumumamw ammzumn mwcmno I 0 .xHHe >m0I000 0muomaxm How coHumumcmm mmmn Bonn mmcmno vmuaHse:oo< I 4 H.H0 6.60 0.H0 0H00I 0H0HI 0666H 0000I 6006H 0H H.60 6.00 0.60 000HI 000I 0000H 06HHI 0000H 0 0.00 H.00 0.H0 06HI 06HI 0000H 000I 0660H 0HI 0HI 0HI OH H H\0"0 xmmm o 0 m xHHz 060 0.06 0.00H 6000H 0600 00060 06HI 6HO0H 0H 0HH 0.00 0.66 0066 6000 0H6H0 00I 0000H 0 0.06 0.00 0.60 000 000 0006H 6HI 6000H H 0HI H 0H H H\6”0 U\0 xmmm xmom xHHz xHHz xHHz xHHz 0HH: cuuomaxm vmuomaxm voumaHumm mwcmnu 000cmso 6mm mwamnu 6mm xmmm o 0 m xHHz Esuo< 000 asuo< 000 0 < 0mumEHumm vmuomaxm muanm3\mmHanum> :00\xmucH Au.a000 oH.6.>H magma 129 Table IV.4.11 Genetic values for curve parameters a, b and c.for generations 1, 5 and 10 using group 1 indexes. Index/Gen Variables/Weights a b c /0 Base Generation 31.6 .21222 .0030243 Milk Milk/1 1 31.1 .21881 .0030251 5 29.2 .24516 .0030286 10 26.8 .27811 .0030329 Milk Peak 1:1/l 3 2 30.9 .22131 .0030662 5 28.1 .25766 .0032339 10 24.5 .30309 .0034437 Milk Peak 1:4/1 1 6 30.8 .22521 .0030455 5 28.1 .27719 .0031303 10 24.5 .34217 .0032360 Milk Peak 1:7/l 1 6 31.6 .22208 .0030210 5 31.5 .26156 .0030074 10 31.5 .31091 .0029905 Milk Peak 1:8/l l 6 31.6 .22295 .0030201 5 31.5 .26586 .0030036 10 31.5 .31951 .0029829 Milk . Peak 1:9/l 5 1 31.6 .21877 .0030207 5 31.6 .24496 .0030063 10 31.6 .27771 .0029883 Table IV.4.11 (con't) 130 Index/Gen Variables/Weights Milk Peak 3 b c 1:10/1 1 6 31.2 .22819 .0030222 5 29.6 .29172 .0030138 10 27.7 .37122 .0030034 Milk Peak 1:11/l 3 1 31.3 .21995 .00300240 5 30.3 .25088 .0030231 10 29.0 .28954 .0030219 Milk b 1:13/l l 6 30.6 .22585 .0031201 5 26.7 .28038 .0035035 10 21.9 .34854 .0039827 Milk b 1:16/l 6 1 30.9 .22204 .0030612 5 28.0 .26130 .0032088 10 24.5 .31038 .0033933 Milk b c 1:17/1 l 6 O 31.6 .21599 .0030249 5 31.5 .23107 .0030276 10 31.5 .24993 .0030309 Milk b c 1:19/l 6 l 0 31.6 .21215 .0030239 5 31.6 .21186 .0030227 10 31.6 .21150 .0030211 131 Table IV.4.12 Genetic values for curve parameters a, b and c for generations 1, S and 10 using group 2 indexes. Index/Gen Variables/Weights a b c /0 Base Generation 31.6 .21222 .0030243 Milk b/c Peak S 2:1/l l 15 l 1 31.5 .20260 .0030049 5 31.1 .l6414 .0029270 10 30.7 .11607 .0028310 Milk b/C Peak 3 2:4/l l 10 l 10 31.4 .20570 .0030158 5 30.6 .17964 .0029820 10 29.7 .14707 .0029416 Milk c b/c Peak 2:5/1 1 ~10 15 1 32.3 .19992 .0028374 5 35.0 .15070 .0020880 10 38.5 . .089190 .0011510 Milk c b/c Peak 2:6/l l -10 10 10 30.8 .22722 .0031932 5 27.5 .28723 .0038683 10 23.5 .36225 .0047131 Milk c 2:9/l l -10 32.3 .20061 .0028164 5 35.4 .15415 .0019833 10 39.1 .096090 .00094231 132 Table IV.4.13 Genetic values for curve parameters a, b and c for generations 1, 5 and 10 using group 3 indexes.‘ Index/Gen Variables/Weights a b c /0 Base Generation 31.6 .21222 .0030243 Milk 3 b Peak S 3:1/1 l 10 1 l 10 32.0 .20909 .0030216 5 33.7 .19656 .0030111 10 36.8 .18091 .0029979 Milk a b Peak 8 3:3/1 1 10 -10 -10 10 32.4 .19702 .0029665 5 35.7 .13624 .0027356 10 39.9 .060263 .0024469 Milk a b c Peak 3:4/l l 10 1 -10 1 32.5 .20777 .0027411 5 36.4 .18995 .0016081 10 41.2 .16769 .00019230 Milk a b c Peak 3:6/l 1 10 -10 -10 -10 32.6 .19826 .0028641 5 36.6 .14242 .0022232 10 41.7 .072621 .0014222 133 These discrepancies in 305-day yields when selecting for milk alone indicate that the expected genetic change in 305-day milk based on the equation 2 AGmilk hmilk x GP x z/q milk is not precisely reflected by the change in the shape of the curve generated by the expected values for a, b and c over a number of generations. Discrepancies of this nature occur to greater and lesser degrees in the indexes listed in the Table (111.6.1). Comparison between the expected and estimated accumulative genetic changes in 305-day milk are helpful in seeing the amount of discrepancy that occurs for each index (Tables IV.4.8 to IV.4.10). The discrepancies occur in both directions, i.e., the estimated values both overestimated and underestimated the expected values (Table IV.4.14). Indexes with the greatest overestimation of the expected values were 1:7, 1:8, 1:10 and 3:4. Index 1:10 overestimated 305-day milk by 10,130 lbs after 10 generations of selection (Table IV.4.14). This means the shape of the curve is much higher and the integral or the area under the curve (31,328 lbs,Table IV.4.8) is much greater than is likely to occur through selection. As mentioned earlier, none of these indexes should yield more 305-day milk than selecting for milk alone. These indexes also overestimated peak yield. For example 1:10 estimated a peak of 114.3 after 10 generations, and the expected value was 75.2 lbs. 134 0.60 6.06 0000 06000 6000H 0H 0uH 0.06 6.00 006H 0000H 0H06H 0 xmmmH 00 6.60 6.00 600 0H06H 0HO0H 000 H AH mo 80 0.HOH 0.06 00HOH 00060 6066H 0H 0HH 0.06 0.00 0006 000H0 6006H 0 xmmm 0 0.00 6.60 006 6006H 0060H 60H H 00 00 mo EH 6.60 6.06 0660 00600 6066H 0H 6uH H . 06 H . 00 0000 600H0 0H06H 0 3600 0 0.00 0.60 000 0066H H060H 60H H 00 00 mo EH 0.00 0.06 0000 00000 60000 0H 6HH 6.06 0.00 006H 0600H 6000H 0 xme0 0H EH 0 60 0 60 000 0006H 6006H 060 H 6.06 0.06 006HI 0060H 60000 0H HuH 0.60 0.00 000I 0H06H 6660H 0 a 98 ~.$ .62.. ~82 one: as H .18 N 3 am 0.H6 0.66 060I 0000H 66000 0H H.60 6.00 006I HHO0H 0660H 0 6.00 0.60 60I 0600H 0606H 000 H xHHz vmumEHumm vmuooaxm MNMIhmm vmumEHumm wmuumaxm cmm\=Hm0 GOHumuoamw xmvcH nmuowaxm 6H00» xmma eHmfis Mafia smeumom .mmxmvcH maOHum> you 0H0H0 xmma 0cm 0HoH6 xHHa >60I000 umumsHumm 0cm vmuowaxo smwSumn mmHoamamuomHn .6H.6.>H mHamH 135 H.06 6.60 6000I 0000H 6HO0H 0H H00 0.H0 0.00 H000I 0000H 0600H 0 0H xmmmH 0.00 6.00 606I 0000H 6600H. 6I H .000H EH 0.00 0.00 6000I 0600H 6000H 0H 0HuH 0.00 0.60 00HHI 6600H 0006H 0 00 0H mo E0 0.00 0.00 000I 0000H 0000H 000 H 0.06 6.66 00H0 0600H 6066H 0H 6HuH H.00 0.00 0HOH 0000H 60H6H 0 oo 00 m0 EH 0.00 H.60 00H 0600H 0060H 00H H 0.06 0.06 006I 0000H 60000 0H 0HHH 0.00 0.00 000I 00HOH 6060H 0 0H E0 0.60 0.60 00I H000H 6606H 000 H 0.06 0.66 060I 0HHOH 6060H 0H 0HuH 0.00 6.00 00: 60HOH 0000H 0 n0 EH 0.60 H.60 00 6006H 0000H H00 H 0.H0 0.66 HO0H 00000 60000 0H HHuH 6.H6 0.00 006 0000H 6660H 0 xmwmfin E0 0.60 H.60 00H HOH6H 0006H 000 H 0.6HH 0.06 660HH 000H0 60000 0H 0HuH 0.60 0.00 0666 60000 6000H 0 36000 EH 0.00 H.60 066 6066H 6006H 060 H vmumEHumm vmuooaxm ANNIamm nmumEHumm vmuomaxm cmw\:Hmw :oHumuwcmw xmvaH cmuomaxm 0HmHm xmma cams» xHHe smenmom Au.:000 6H.6.>H magma 136 o.moH ~.o6 moaoH ummkm 6HmeH oH 6nm m.ex 0.00 HHm6 oH6H~ aamoH m xmmmH_ ooH- m.6o “.mo new NnnNH NeooH a.eHI H 0H mOH EH c.m6 o.~n mmmmn HmNHH eommH oH mum H.mm m.am aaeHI makHH ekNmH m moH xmwmmHI m.oe w.He mmm- ascoH Noqu New- H noHI mOH EH “.mo H.60 mem- nameH eqaoH OH Hum m.~e n.60 66H- cameH 6HmoH m mOH xmmmH o.~e N.mo mm- HoeoH oaeoH o H 0H «CH EH m.mm H.mo menu momoH emeoH oH aum «.mm o.mo ooHI aemoH aeooH n oOH. EH ~.~o a.~e m- caeeH H000H m- H m.w~ H.00 NNNH HNNaH 666mH oH ou~ ~.Hx o.me mos NNHmH eomaH m xmmmoH 08H 0.60 m.mo 60H eucaH oomoH 00H H ooHI 5H ¢.Hm m.mn NGNHI ~6omH 6omeH OH mum 6.0m m.oe owns moamH 6¢60H n xmmmH oan a.Ho m.~o koHI ommeH 0600H mm: H UoH... EH e.m6 H.~o nHHoI aHNHH 6mmmH oH eu~ n.mm m.~0 6a~mu mmmHH amHnH m mOH 0mmmH m.oe m.~o moon coHeH oaNoH mHH H unoH sH umumEHumm wouomaxm meIhmm woumEHumm wouoonxm :00\:Hmw :oHumumcmw xovcH umuomaxm eHmHs anus eHmHs xHHa smeumoH 10....e «H.q.>H .Hn.e 137 0.H0 6.60 00H 0666H 6006H 0H 0u0 n.5m s.mn H6H NwomH mommH m xmmmoH- UoHI 0.H0 H.00 00 0000H 0660H 000I H 00HI H“.0H EH woumEHumm wouomaxm mxmlymm woumEHumm cmuomaxm :mw\aHmw :oHumumcow x00CH vmuomaxm 3.3:H xmma BHH.H xHHa smeumom Hu.:ou0 6H.6.>H mHnma 138 Indexes which underestimated expected genetic changes in 305-day milk the greatest were 1:19, 2:1, and 2:4 (Table IV.4.8 and IV.4.9). Peak yields were also underestimated in these three indexes. Indexes 1:19 and 2:4 estimated negative changes in 305-day milk, -14.9 and -4,964 lbs by 10 generations when the expected genetic changes were positive, 2,250 and 1,150 lbs, respectively (Table IV.4.8 and IV.4.9). For 2:4, the difference in 305-day milk by 10 generations was 6,115 lbs and the difference in peak yield was 62.1 - 45.6 = 16.5 lbs (Table IV.4.14). Indexes which had estimates close to the expected 305-day values were 1:13, 1:16 and 3:6. The difference at 10 generations for 3:6 was 760 lbs. Indexes which had estimated peaks close to the expected values where 1:13, 1:16, 1:17, 3:1 and 3:6. The expected 305-day and peak values were the expected genetic gain times the number of generations. Therefore, the increments between generations were equally spaced. This is not true for theaintervals between generations computed from the estimated values. Notable changes in the rates of change from generation to generation occurred in 1:7, 1:8, 1:10, 1:11, 2:9, 3:1 and 3:4 (Tables IV.4.8 to IV.4.10). All of these had an increas- ing rate of change in estimated 305-day milk from generation 1 to generation 10. The rate of change decreased in 1:13 (Table IV.4.8). Typically, the rate of genetic change is considered to be constant for a given population and a given selection intensity over a number of generations. This is because h2 and 0 do not P change greatly after a few generations of selection. Therefore, 139 £0 = h2 x OP x z/q would produce a relatively constant 00 from one generation to the next. It is therefore disturbing to note that for a constant rate of change of the curve parameters a, b, and c, the rate of change in the shape of the curve is not constant i.e., the rate of change in estimated 305-day yield. In fact, rather disproportion- ate genetic changes occurred when several indexes were used (1:7, 1:8, 1:10 and 3:4). The dispropotionate genetic change is estimated 305-day milk from one generation to the next or the change in the rate of change is due to estimation using a nonlinear equation. When a, b and c change linearly in the equation: 305th y305—day milk ’ aof ex9('°t) dt’ then y305-day milk’ the total area under the curve, changes non- linearly. This explains why the estimated change by 10 generations is not 10 times the estimated change in one generation for all indexes (Tables IV.4.8 to IV.4.10). This is also responsible for a small part of the discrepancies between expected and estimated genetic gain in 305-day milk. This is an inherent problem when nonlinear models are used, and one desires to estimate genetic progress with the model. The computation of genetic correlations assumes linear relation- ships between traits. Therefore, the relationships between a_and 305-day milk, a and b, a and c, b and c, etc., are assumed to be linear. It is possible that some of these relationships are 140 curvilinear as demonstrated in Figure l. A curvilinear relation- ship suggests that the correlations change notably as genetic change in the traits occur. This makes it difficult to estimate correlated genetic responses over time. In Figure 1, the correlation would be computed as the best estimate of a linear relationship between x and y. This is represented by the straight line. The linear correlation would only be appropriate within a certain range of x and y. If the relationship between any of the curve parameters and 305-day milk is nonlinear, then the true correlated genetic response between them would be nonlinear. That is, if 305-day milk and b are nonlinearly related, then as milk changes linearly, b changes curvilinearly or vice—versa. Therefore, when an index is used and curvilinear relationships exist, correlations used for the first generation would not be the same as those used in later generations to compute genetic responses. Therefore, 9 and P would become dynamic, i.e., contain different covariances over time. It then follows that the b, O and Cov (61’ I) become I dynamic. Then, in the example of expected 305-day milk, a curvilinear response could be computed. The correlations would be computed using a polynomial model. Between 305-day milk and a for example, the possibilities may be: 2 y305-day milk bla + b2a + e or 2 3 y305-day milk b1“ + bza + baa + e where the change in milk is a polynomial or curvilinear function of a. 141 .0 0cm x .muHmuu ozu Emmsumn chmsOHumHmu ummcHHH>u=u HmoHuonuo00: < .H musth 142 A more complicated situation may exist where the curve parameters a, b and c have curvilinear relationships among them- selves. Then a polynomial model is needed for each curvilinear relationship to define the correlations at any level of the para- meters. IV.4.2 Change in the shape of lactation curves Figures 2 through 4 are plots of the change in the shape of the curves when a, b and c are changed when using the nonlinear form of Wood's (1967) equation. These changes do not consider correlated change in the other curve parameters. Therefore, these curves demonstrate the change due to changing one parameter while holding the other two constant. Figure 2 shows that as §_increased from 25 to 43, the curve maintains its shape, but starts at a higher point. For this reason, a_is referred to as the scaling parameter. Figure 3 shows the change in shape as b is increased. The ascent to the peak becomes steeper as b is increased from .24 to .33. Also, the peak and the later stage of the curve increase in height, with the decline after peak, c,remaining constant.Therefore, the area under the curve increases. Figure 4 shows the change in shape as c is increased. The largest value for c (.041) yields the bottom curve with the greatest slope after peak. The most persistent curve is the highest curve which represents the lowest value for c (.032). As c decreases the peak also rises. This is a function of a(b/c)bexp(—b), which also increases as b increases. 143 :OHumuoma :H mxmmB 0+ 0+ 00 60 0H 0 0 «I v a» H “II «I ,4 w. (IwI u «I u 0 :00 L. 106 L. 1r00 Lr 0 .EOHumscm m.0ooa :H 06 ou 00 E000 cowcmno mH 0 sons m>u=o coHumuomH mnu mo mamsm 050 :H mwsmno .0 muame P191; aIIN KIIea 144 EOHuMuUMH cH mxmmz 06 00 *0 0H 0 06 o P pl P r pl p Pl r P P r .1 p «I 1 u q q 1 d u q 1 a u 3 it :00 4.. w. .3 1.! Lazy Lu 0 .:0Humavm m.0ooz :H 00. 00 60. E000 000:m:o mH A :05: m>h=o :oHumuomH mam mo mamnm 050 :H mwamno .0 muame pIaIA HIIH AIIPG 145 06 coHumuomH 0H 00003 06 «n 60 on 0 1r- d- d P b F 3 3 d 1‘ J G db .00 .06 .00 .00H00000 0.0003 0H H60. 00 000. E000 0000000 0H 0 00:3 0>000 00H00000H 0:0 00 00000 0:0 0H 000020 .6 0H00Hm ptau Hm £1qu 146 These curves are a function of t in weeks as Wood (1967) defined time. The values for a, b and c were deviates from those published by Wood (1970), 30, .28 and .036, respectively. The curves in Figure 5 to Figure 17 are produced using the nonlinear equation y . atbexptwfi$, where t goes from 0 to 305 days. The values for a, b and c are the expected values computed from their correlated genetic change (Table IV.4.5). The discrep- ancies between the integrals of these curves and the expected 305-day milk need to be kept in mind. Comparisons can be made between the integrals in Tables IV.4.8 through IV.4.lO and the shape of the curves at l, 5 and 10 generations plotted in these figures. Figures 5, 6 and 7 show the change in shape at l, 5 and 10 generations when a_is selected alone. The three curves in each figure represent the base (B) or zero generation, the shape when selecting on milk alone (M) and the shape when selecting on the index (I). Again, the base generation was computed using the mean values for a, b and c for the 5,927 first lactation records. As.a increases, the index curve has an increase in initial production, but now the peak drops and the slope after peak increases. This is because the correlations between §_and b (—.906) and a and c (-.607) were negative. Therefore, as‘a increases, both b and 0 decrease, but b decreases faster causing the peak (a(b/c)bexp(-b)) and the time of peak (b/c) to decrease. The 305-day milk yield is also negatively correlated with a and therefore some decrease in 147 000 coHumuomH 0H 0600 000 060 000 000 00: 00 06 0 T .r .r l T L« 0 .p 0 0 l + 0 4r 0 on 0r :06 ,, .00 0 .H .00 i. .r :06 000H0.M 00 00H000H00 n H 000H0 0HHE now 00H000H00 u z 06 0 00H0000000 0000 .000H0 0.00 00H000H00 00 00H0000c00 000 00000 0>uso 00H00000H 0:0 00 000:0 0:0 mo 00:0:0 .0 0000Hm man. Hm KIIE’CI 148 00H000004 0H 0000 000 000 060 000 00— F 00W r 00. F oer ? 000 LU L06 , .8 H :00 m l‘ 2 .60 000H0.M =0 00H000H00 n H 000H0 xHHE 000 00H000H00 u z 00H0000000 0000 n 0 .I 0 .000H0 0 :0 00H000H00 0o 000H0000000 0>H0 H0000 0>uso 00H00000H 0:0 00 000:0 0:0 00 0000:U .0 0u=0Hm ptau mm fined 149 _Hwn :oHu0u00A :H 0>0Q cmw ovw cow om— ou— on :0 o b! P F r P D r F r P r r Pi F r 1‘ d d d 1 id 1 nd a d d fl 1 1 11 an r r: r lam H row m ' lap 000H0 m.co :OHu00H0m u H 000H0 xHHE you cOHu00H0m n z z aoHu0u0c0w 000m u m a ham .000H0.m :o coauu0H00 mo 0:0Hu0n0a0w c0u u0um0 0>uao :OHu0uo0H 0nu Mo 00050 050 no 0wa0no .H 0u=th pIaIA XIIN fitted 150 the integral would be expected. However, the expected change in 305-day milk was -83, -415 and -830 compared to +116, -666 and -501 for estimated 305-day milk for l, 5 and 10 generations, respectively (Table IV.4.11). Figures 8, 9 and 10 graph the change in shape when b is selected alone (I). A slight increase in peak occurs at first and a decrease in time to peak occurs while an increase in c causes an increase in the slope after peak. The greatest change compared to selecting for milk alone is in c, which increases much greater when selecting for b (13,805Z) (Table IV.4.S). The correlation between b and c was estimated at .726 while that between 305-day milk and c was insignificant (.004). The expected change in milk is 108, 540 and 1,080 lbs compared to 223, 986 and 1,597 lbs computed by the integral for generations 1, S and 10, respectively. Finally, selecting alone for negative c is represented in Figures 11, 12 and 13. This causes a flattening of the curve (I) and a large loss in production from the base generation (B) even though the expected change in 305-day yield is +1.38 lbs per generation. Figures 14, 15 and 16 show the curve changes for index 3:4. This is an exmaple of an index in which estimated 305-day milk greatly overestimates expected 305-day milk. The intent of this index was to flatten the shape of the curve by increasing a and decreasing c. The weights were 1, 10, 1, ~10 and 1 for milk, a, b, c and peak yield, respectively. Keeping in mind that none 151 cum" 00Hu0u00H :H 0>0n cow new cam om— ou— co ov o TVT+vnvuuvvuvarmom .1 10¢ // .1 tom 5 , .60 :ow 000H0 n :o coHuo0H0m n H 000H0 xHHE pom couuo0H0m n z .. :OHu0H0c0w 000m n m m .0coH0 a co GOHuo0H00 mo . COHu0u0o0m 0:0 u0um0 0>uso coHu0uo0H 05u no 00050 0:0 mo 0wc0£o .w 0ustm PIBTA XIIN XIIea :oHu0uo0H CH 0%09 ecu emu own Do 6* o own new owm, ‘P P d 1" 4- ‘F 1h- 1- q:- q‘ d- q 1 152 .0m .50 +05. ptau Hm fined ll mZH 0coH0 A no ooHuo0H0m 000H0 xHHE new cowuo0H0w :oHu0u0c0w 0000 u .000H0 n so coauo0H00 mo 0:0Hu0u0S0w 0>Hm H0uw0 0>Mso cowu0uu0H 0£u mo 00050 050 mo 0ws0zo .m 0u=me 153 COHu0u004 CH 0>0Q DGN O+N DON 00H ONH Dc 6* G LI ‘91 a w v u v kw. v V. $r u» v. 9 u “um LT toe \ .Lum \ L. m _ LAxw LU \_ :2. 000H0 n so :OHuo0H0m u H 2 H 000H0 xHHE How cOHuo0H0m u z L. :oHu0H0C0m 000m n m _Um .0COH0 A no aOHuo0H00 mo 0coHu0u0c0w :0u u0uw0 0>u=o :oHu0uu0H 0£u mo 000:0 may CH 0wa0£o .OH 002000 P1311 HIFN 51190 154 00Hu0uo0a GH 0000 cum omw Q¢N cow :00 o- om ov 0.0.100010T0Tin0t1T m .Ov .cm H 8 m u z for 000H0 on so soHuo0H0m u H 000H0 xHHE co 00H000H0m u z . 00Hu0u0a0w 0000 u m m .000H0 o co cOHuo0H00 0>Hu0m0c mo coHu0H0G0w 0:0 u0uw0 0>p=o :oHu0uo0H 0£u mo 000:0 0£u 0H 0w00co .HH 0usz0 PIBIL HIIN fitted coHu0uo0A CH 0000 cum omN D¢N DON om— 0N0 an Dc 0 b 1 db 4. 4+- . D b bl 1 d d. a d d1 J. 1. q L 9. pram mm KUPCI 155 000H0 on new coauo0H0m 000H0 xHHE How :OHuo0H0m :oHu0u0a0w 0000 u 0 ll" 2H .000H0 o GOHuo0H00 0>Hu0m00 mo 0coHu0u0a0w 0>Hm u0um0 0>p=o cowu0uo0H 0zu Mo 00050 0:0 0H 0w00£o .NH 0uawH0 156 _me 00Hu0uo00 GH 0000 DQN O+N cow owH ONH co 0* o m .D¢ .om H Aum Auh 000H0 on you coHuo0H0m u H z 000H0 xHHE co coHuo0H0m u x L. coau0u0c0w 0000 u 0 o 000Hu0u000w 00a H0uw0 .0coH0 o co soHuo0H00 0>Hu0m0c mo 0>u=o coHu0uo0H 0:0 mo 00000 000 0H 0w00co .MH 0u:wH0 pram Hm AUBG 157 _me coHu0uo0H CH 0000 DON OvN DON Own ON— 69 0* O 0 V o, 9‘ “w 9. 0» w. v n, W, n» at V "I “um 0 .Lum Lt H 2 tap. L. _Lcm K000H 0o 00Hu00H0m u H 000H0 xHfia 00 00Hu00H0m u z COHu0u000w 0000 u 0 fan— .0H0H0 0000 H 000 00HI .nH .00H .xHHE H x000H co 0oH000H00 mo cowu0u000w 00o u0u00 0>H:o 0OHu0uo0H 00a 00 00000 000 0H 0w00su .qH 0ust0 ptau Hm Aueq 158 own coHu0uo00 0H 0000 cow 0+N cow OOH own on at o 'F P .1 q d T F r "[ 1| 1 d 0“ P 0 1 I 4 q d pIaIA.xIIN.AIqu r00 x000H 0o :oHuo0H0m n H 0030 03.2: .80 00H000H0m u z .éuu 00Hu0u000w 0000 n 0 .0000» xmmq 0,0cm ooa- .00 .moa .3005 H xmeafi so cofiuomamm 0o 000Hu0u0c0w 0>Hu u0um0 00>u=o coHu0uo0H 0su mo 000:0 0H 0w00£u .mH 000000 159 0N0 00Hu0u000 CH 0000 omw 0+N 00w 02 0mm r 00. F 0.? . 0 m L. L60 0 1r 2 :2. ..r L.00 H x000H coHuo0H0m u H 000H0 00:... How coHuo0H0m u : .10—H coHu0u000w 0000 u 0 .0H0H0 0000 H 000 00HI .0H .00H .xHHS H x000H co 00Hu00H00 mo 000Hu0u000m :0u u0u00 0>uso coHu0uo0H 0nu 0o 00000 000 0H 000000 .0H 00:0H0 Ptau Hm Rueq 160 ONE" coHu0uo0H CH 0000 cow cvu 000 000 0N0 00 0. o .r u v 0 .p J; 0 v 0 0 + 4. w .p 0 m .;om 0 it :00. 2 .:0m H x035 co 00H000H0m léaH 00oH0 xHHE 0o 00H000H0m z coHu0u000w 0000 u 0 .0H0H0 0000 o 000 0HHE H xmwcH co 00Hu00H00 mo 0coHu0u000w 000 00000 0>u=0 COHu0uo0H 000 00 00000 0H 000000 .0H 0ust0 9181A HIIN AIqu 161 of the indexes increased expected 3050day milk more than selecting for milk alone. Therefore, none of the index curves (I) should have a greater area (integral) than the curves representing selection for milk alone (M). Noting here that the curves for selection on milk alone (M) represent the 305-day estimates noted in Table IV.4.8 and they underestimate expected 305-day milk slightly. The integral for (I) at 10 generations is 27,282 lbs while that for (M) is 19,329. The expected values for (I) and (M) are 16,514 and 20,274 lbs, respectively. The curve for (I) should therefore be lower than that for CM) (Figure 16). These curves give an idea of the overestimation that occurs when the expected values for a, b and c are used to compute a new index curve (I) for generations 1, 5 and 10 (Figure 14, 15 and 16). Figure 17 represents another index (1:10) which greatly overestimates expected 305-day milk. This index includes milk and peak yield with weights of 1 and 6, respectively. By 10 generations the (I) curve represents 31,328 lbs and a peak yield of 114 lbs, while the expected values are 20,084 and 76 lbs, respectively. This exceeds the estimated curve for selection on milk only (M) by a staggering amount. IV.4.3 Summary of changes caused by_selection indexes Without knowing the optimum shape of the lactation curve with regard to efficiency of milk production, one can draw some conclusions about the indexes investigated. If we consider expected values for 305-day milk, we can exclude those indexes 162 that decrease or slow greatly the genetic progress in 305-day milk yield. It would be unlikely that any change in the shape of the lactation curve which reduces 305¥day milk extremely, would produce greater net profit due to reductions in stress and/or inputs. Therefore, of the indexes listed in Tables IV.4.8 through IV.4.10, 1:7, 1:8, 1:17, 2:1, 2:4, 2:5, 2:9, 3:1, 3:3, 3:4 and 3:6 can be excluded. This excludes all but one index (2:6) in group 2 which are attempting to delay peak and/or increase persistency. It also excludes all indexes which attempted to increase §_and decrease c or increases (third strategy). It may be that these indexes would be more desirable if more weight were applied to milk. It was intentional that milk was not selected strongly so that extremes could be compared to selecting for milk alone. From the values for peak and 305—day production, index 1:13 appears to do a reasonable job of increasing the portion of milk produced in the early part of lactation (Tables IV.4.8). This would be desirable if cows have higher daily net profit in early (peak period) lactation and if this higher production in early lactation is not detrimental to production in subsequent lactations. Index 2:6 which attempted to delay time to peak, had weights of l, -10, 10, 10 for milk, c, time of peak and peak, respectively. The expected values for b/c indicate it did not delay time to peak. They were 66, 47 and 24 days for generation 1, 5 and 10, respectively. 163 Indexes 2:1, 2:4 and 2:5 successfully delayed expected time to peak but greatly decreased 305-day milk. The estimated time to peak was actually reduced in 2:1 and 2:5 (Table IV.4.9). Perhaps with more weight on milk,these indexes would produce the desired changes in the shape of the curve without great loss in production. It appears that the indexes of the third strategy could be feasible only if more weight were put on milk. These indexes in general do flatten the curve by decreasing b (the ascent), and by increasing §_plus decreasing c. IV.5. BLUP Solutions for Sires The standard deviations for the BLUPs are in Table IV.5.1 for 305-day milk and the lactation curve parameters. The standard devia- tion of the BLUPs for 305-day milk was 267. The means of the BLUPs by definition, are zero. The range of the BLUPs for 305-day milk was 645 to -611, and 3.47 to —2.81 for peak yield. The top and bottom ranking sires for 305-day milk and peak yield were the same two sires. Table IV.5.1 Standard deviations of BLUPs for 305-day milk yield and lactation curve parameters for 150 sires. Variables or Parameters Standard Deviation 305-day milk a b c b/c time of peak Peak yield .SA 267.345 .736637 .0107281 1.59818 E-4 4.82325 .934166 062.4614 -(b+1) A - S is c , adjusted for a. 164 IV.5.1 Ranking sires by indexes Using the weights of an index, an index for each sire was computed. These indexes were computed by a linear combination of the BLUPs for each trait and the weights of the index, i.e., IS - mluls + m2u23 + m3u33. The subscript 3 refers to a specific sire, s - 1, ..., 150. This was done for a number of the indexes in the three strategies. The rank of the 150 sires for several indexes was compared to their rank on milk alone. This comparison was done using Spearman's correlation of ranks. These correlations are in Table IV.5.2. The sires' rankings by indexes 1:1, 1:10 and 1:16 are not greatly diff- erent from those for milk alone. This is consistent with the change in milk expected when these indexes are used (Table IV.4.8). These three indexes represent large weighting on milk (1:1 and 1:16) or a heavy weight on peak (1:10) which is highly correlated to yield. In general, indexes of the first strategy had the highest correlations with rankings on milk alone. Indexes of the third strategy had the lowest, two of which were negative, 3:3 and 3:6. While those of the second strategy fell in the middle. This is consistent with the amount of genetic change in 305-day milk expected for the indexes when correlated responses, via the covariances, are considered [IV.4]. That is, for those indexes with genetic change in 305-day milk near that change expected when selecting for milk alone, the Spearman's correlation of ranks were high. Conversely, for indexes 3:3 and 3:6 the genetic change in milk was 165 Table IV.5.2 Rank correlations between sires ranked for milk only and other indexes. Index COrrelationA 1:1 30 1b 2peak .9503 1: 1m 6b 6peak .6435 1:10 1m 6 peak .9023 1:13 10 6b .2783 1:16 6m lb .9863 2:1 1m lec 1pgak. 1S .4379 2:5 1m -10c 15bc lpeak. .3078 2:6 1m -ch 10bc lOpeak .7757 2:9 1m -10c .1963 3:3 1m 10a -10b -10p§ak 10S -.2526 3:4 1m 10a 1b -10c lpeak .2323 3:6 1m 10a -10b -10c -10peak -.2178 A - Spearman's correlation of ranks. 166 considerably negative, -282 and -235 lbs and the ranked correlations were negative -.25 and -.22 for 3:3 and 3:6, respectively. In order to consider the covariance between traits when computing I = m'u for each sire, the covariances must be incorporated in the BLUPs i.e., g. This can be done by expanding the random (sire) portion of the mixed model equations to include a variance- covariance matrix for each sire for the traits considered in the index. Multiple right hand sides i.e., one for each trait in the index are needed. This produces multiple BLUP solutions for each sire which are then combined by the weights, a to yield an index, I, for each sire. Computationally, this increases the random portion of the MME by a factor equal to the number of traits in the index. However, the method which was used in this study to combine the BLUPs for each.trait into an index value, I, will yield the same value for I as the method just mentioned. The advantage in the procedure used in this study is the individual BLUPs can be computed ignoring the covariances and later combined into an index value, I. V SUMMARY AND CONCLUSIONS Currently, selection for milk production is based on total 305—day lactation yield. Although it is known that feed efficiency is the greatest and health costs are the highest in early lactation, these efficiency factors of a lactation have not been considered in selection. 4Considering the efficiency in early lactation, one may want to select cows that produce more in early lactation. 0n the other hand, if health costs are extensive during the high production, high stress period, then it may be economical to select cows which peak lower and later and are more persistent. The purpose of this study is to fit first lactation records to Wood's equation and compute genetic estimates for the parameters a, b and c in the equation. Then, using selection indexes, change in the shape of the lactation curve along with 305-day milk yield will be selected jointly. This is an attempt to determine the flexability of the lactation curve shape and how it will affect total lactation yield. Lactations of two year old cows in the Michigan DHI population were fit to the nonlinear form of Wood's equation resulting in parameters estimates for a, b and c for each cow; Using Shook's factors to adjust the first and/or second monthly tests,reduced the number of cows having curves with negative b values. This insured an ascent to the peak as opposed to lactations with estimated first day production greater than all subsequent test days. Also, two year olds are more likely to be increasing in production 167 168 at 305-days than later lactation cows. Therefore, using Shook's factors to compute production on the 305th day, based on the previous test, may underestimate it for some two year olds. It is likely, then, that adjusting the end of all two year old records will eliminate negative c values by causing a downward lepe, but may do so in error. In the present study, this was done only for cows when their last test date was between 280 and 305 days. Upon using Wood's equation, it is noted that b and c are not entirely independent. As c decreases, b increases and therefore peak yield and time of peak increase. It is also noted that as b increases, peak yield and yield after peak are greater. Cows that increase in yield faster (larger b values) and maintain or decrease c (are more persistent) are expected to have a higher peak due to the relation in the computation for peak of the curve (a(b/c)bexp(-b)). For two cows with the same a_and b, peak yield dictates their persistency due to the relationship in the equation for peak. The cow with the higher peak will necessarily have a lower c and therefore, be more persistent. These conditions may not be true biologically. A more flexible equation would allow the ascent, peak, time of peak and persistency to be independent. This flexibility would improve the fit of lactation curves. 169 Computation of variance components using Best Linear Unbiased Prediction solutions from mixed model equations, and restricted maximum likelihood estimators in an iteration process was successful. Convergence occurred in ten or less iterations by using a relaxation step between iterations. The heritability for milk production was less than is usually reported (.16). Heritabilities computed for a (.06), b (.09), c (.15),time of peak yield (.07), peak yield (.15) and S (.04) were all less than that for 305-day milk. Therefore, selection on milk yield alone produced greater genetic gain in 305-day milk yield than selecting for milk jointly with the lactation curve parameters. Indexes including 305-day milk, the lactation curve parameters a, b and c, time of peak yield, peak yield and S, were set up for three strategies. The first strategy was to increase the amount of milk produced in the early part of lactation by increasing b and peak yield. The second strategy was an attempt to delay time of peak or decrease b, the ascent to the peak with or without con- sidering persistency. The third strategy attempted to flatten the lactation curve by increasing a, decreasing peak yield and increasing persistency. Indexes including milk, b and peak which are of the first strategy, resulted in nearly as much gain in 305-day milk as select- ing for milk alone. These indexes have potential if it becomes desireable to increase yield in the peak part of lactation. In the first strategy, several restricted indexes were used to f170 restrict the genetic change in a and c so that b could be increased without decreasing a_or increasing c. This was done in an attempt to increase peak yield (a(b/c)bexp(-b)) by increasing b only. The progress made in 305—day milk by these restricted indexes was reduced considerably. Therefore, the restricted indexes were not useful. Selecting for a delay in time of peak,the second strategy, in general, resulted in much lower gain in 305-day milk. When weights were 7, 3, l, l for milk, time of peak, peak and 8, respectively, the index, decreased the gain in milk somewhat less. However, the gain in time to peak was only .7 days per generation. In the third strategy, selecting negatively for c, with equal weights for milk, greatly reduced the genetic change in 305—day milk (244 lbs) compared to selecting for milk alone (359 lbs). Conversely, selecting for milk alone had little influence on c. The correlation between 305—day milk and c suggest that high pro- ducing ability is not genetically related to persistency as measured by c. Selection for persistency is feasible, but if milk is to be maintained, it must have greater weighting than c. Indexes which attempt to flatten the lactation curve, the third strategy, do so at the expense of 305-day milk, and with extreme weights, cause negative genetic gains in milk. These indexes selected positively for milk, S, and a, and negatively for b, c and peak. Therefore, the decrease in milk is to be expected. If these indexes are to be beneficial, the weights would have to be more in favor of milk. If flattening the curve results in decreasing 171 stress and inputs substantially, then these indexes could be help- ful. This is not likely, because increasing a and decreasing peak quickly decreases 3050day milk. Of the three strategies, that first seems to be more in line with maintaining a reasonable gain in milk production while chang- ing the shape of the curve. This is due to the positive relationship between peak yield and 305-day milk. Indexes for each sire were computed by combining the BLUP estimates for each trait for each sire by the weights used in the selection indexes. This yielded an index for each sire. The sires were then ranked according to their indexes. Then, rank correlations for sires were computed between the ranking on each index and the ranking for milk alone. Ranking the sires using their BLUPs and the index weights suggest: (l) Rankings by indexes of the first strategy were very similar to rankings by milk alone, except when a_and c were restricted. This suggests that most of the sires ranking high for milk alone also rank high for increasing peak yield and b. (2) In general, indexes which had genetic gain in milk close to that of selecting for milk alone, had high rank correlations. (3) Sires' rankings for indexes selecting to flatten the curve were poorly correlated to their ranks on milk alone and for some of these indexes negatively correlated. The final step was to plot the shape of the lactation curve after 1, 5 and 10 generations of selection on each index. This 172 was done by putting the new genetic values for a, b and c, after selection by each index, into the equation (y = atbexp(-ct)) and changing t from 1 to 305 days. Two problems occur when attempting to plot the shape of the lactation curve after selection by indexes. The first is related A y305—day milk a a0f305tbexp(-ct) dt. When a, b and c are changed linearly from to the nonlinear form of the equations used: generation to generation, the integral computed, i.e., estimated 305-day milk (y3OS-day milk)’ changes nonlinearly. Therefore, to a small degree, the increments between generations are not equal A for y305-day milk' This is due to the nonlinear relationship of the equation. Second, the estimated values for 305-day milk computed by the integral of the new curves were not equal to the expected genetic change in 305—day milk when selecting on the indexes. These differences for some indexes were great. Both positive and negative differences occurred. One possible cause of this discrepancy is that the relationship between 305-day milk and somecnrall of the curve parameters is curvilinear. This means as genetic change in 305-day milk occurs in a linear fashion, the curve parameters change curvilinearly or vice-versa. Therefore, the correlations between 305-day milk and the parameters may change considerably when the selection process continues over 10 generations. Also, the relationship among some of the curve parameters may also be curvilinear. LIST OF REFERENCES LIST OF REFERENCES Appleman, R. D., S. D. Musgrave, and R. D. Morrison. 1969. Extending incomplete lactation records of Holstein cows with varying levels of production. J. Dairy Sci. 52:360. Anderson, C. R. 1981. A biometrical and genetic study of Tribolium egg production curves as a model for lactation curves. Un- published Ph.D. Thesis, University of Illinois. Bakker, J. J., R. W. Everett, and L. D. Van Vleck. 1980. Profitability index for sires. J. Dairy Sci. 63:1334. Barr, A. J., J. H. Goodnight, J. P. Sall, W. H. Blair and D. M. Chilko. 1979. Statistical Analysis System (SAS) Users Guide. SAS Institute Inc. Raleigh, N.C. Basant, S. and P. N. Bhat. 1978. Models of lactation curves for Hariana cattle. Indian J. Anim. Sci. 48:791. Brown, M. B., and A. B. Forsythe. 1974. Robust tests for the equality of variances. J. Amer. Stat. Assoc. 69:364. Cobby, J. M. and Y. L. P. Le Du. 1978. On fitting curves to lactation data. Anim. Prod. 26:127. Cochran, W. G. 1951. Improvement by means of selection. Proc. Second Berkely Sym. Math. Stat. and Prob. 449. Congleton, W. R., Jr. and R. W. Everett. 1980a. 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. 1980b. Application of the incomplete gamma function to predict cumulative milk production. J. Dairy Sci. 63:109. Dickerson, G. E. 1969. Techniques for research in quantitative animal genetics. Techniques and Procedures in Animal Science Research. In Mbnograph of Amer. Soc. Anim. Sci. Champaign, ILL. pp. 36. 173 174 Everett, R. W. 1975. Income over investment in semen. J. Dairy Sci. 58:1717. Everett, R. W., H. W. Carter, and J. D. Burke. 1968. Evaluation of the Dairy Herd Improvement Association record system. J. Dairy Sci. 51:153. Gaines, W. L. 1927. Persistence of lactation in dairy cows. Ill. Agric. Exp. Sta. Bul. 288. pp. 353. Gill, J. L. 1978. Design and Analysis of Experiments in the Animal and Medical Sciences. Vol. 1, Iowa State University Press, Ames, IA. Gooch, M. 1935. An analysis of the time change in milk production in individual lactations. J. Agr. Sci. 25:71. Gravert, H. 0., and R. Baptist. 1976. Breeding for persistency of milk yield. Livest. Prod. Sci. 3:27. Grossman, M. 1970. Sampling variance of the correlation coefficients estimated from analyses of variance and covariance. Theoret. Applied Genetics. 40:357. Guest, P. G. 1961. Numerical Methods of Curve Fitting. Cambridge University Press, London, England. pp. 334. Hansen, L. B., R. W. Touchberry, C. W. Young, and K. P. Miller. 1979. Health care requirements of dairy cattle. II Nongenetic effects. J. Dairy Sci. 62:1932. Harris, D. L. 1963. The influences of error or parameter estimation upon index selection. Statistical Genetics and Plant Breeding. National Academy of Sciences National Research Council Publication 982. pp. 491. Harvey, W. R. 1972. General outline of computing procedures for six types of mixed models. Ohio State Univ. memeo. Hazel, L. N. 1943. The genetic basis for constructing selection indexes. Genetics. 28:426. Henderson, C. R. 1953. Estimation of variance and covariance components. Biometrics. 9:26. Henderson, C. R. 1963. Selection index and expected genetic advance. In Statistical genetics and plant breeding. NAS-NRC 982. pp. 141. 17S Henderson, C. R. 1973. Sire evaluation and genetic trends. Proc. Anim. Breeding and Genetics Sympos. in Honor of Dr. Jay L. Lush, ASAS, Champaign, IL. Henderson, C. R. 1975. Best linear unbiased estimation and prediction under a selection model. Biometrics. 31:423. Johansson, I., and A. Hannson. 1940. Causes of variation in milk and butterfat yield of dairy cows. Kungl. Landbr. tidskr. 72 (6 1/2). Kellogg, D. W., S. Urquhart and A. J. Ortega. 1977. Estimating Holstein lactation curves with a gamma curve. J. Dairy Sci. 60:1308. Kempthorne, O. 1957. An introduction to genetic statistics. John Wiley and Sons, Inc., New York. Kempthorne, O. and A. W. Nordskog. 1959. Restricted selection indices. Biometrics. 15:10. Kendall, M. G., and W. R. Buckland. 1971. A Dictionary of Statistical Terms. Hafner Publishing Company, Inc. New York. Kendall, M. G. and A. Stuart. 1969. Advanced Theory of Statistics. Vol 1: Distribution theory. 3rd ed. Griffin Co. London. Keown, J. F., and L. D. Van Vleck. 1973. Extending lactation records in progress to 305-day equivalent. J. Dairy Sci. 56:1070. Lamb, R. C., and L. D. McGilliard. 1960. Variables affecting ratio factors for estimating 305-day production from part lactations. J. Dairy Sci. 43:519. Layard, M. W. J. 1973. Robust large-sample tests for homogeneity of variance. J. Amer. Stat. Assoc. 68:195. Lush, J. L. 1940. Intra-sire correlations or regressions of off- spring on dam as a method of estimating heritability of char- acteristics. Thirty-third Annual Proceedings of the Amer. Soc. of-Anim. Prod. 293. Madalena, F. E., M. L. Martinez and A. F. Freitas. 1979. Lactation curves of Holstein-Friesian and Holstein-Friesian x Gir Cows. Anim. Prod. 29:101. Madden, D. E., J. L. Lush, and L. D. McGilliard. 1955. Relations between parts of lactations and producing ability of Holstein cows. J. Dairy Sci. 38:1264. 176 Mahadevan, P. 1951. The effects of environment and heredity on lactation. II Persistency of lactation. J. Agr. Sci. 41:89. Mao, I. L. 1971. The effect of parameter estimation errors on the efficiency of index selection and on the accuracy of genetic gain prediction. Unpublished Ph.D. Thesis. Cornell university. Mac, I. L. 1981. Lecture notes on multi—factor unbalanced data. Mao, I. L., J. W. Wilton and E. B. Burnside. 1974. Parity in age adjustment for milk and fat yield. J. Dairy Sci. 57:100. Marquardt, D. W. 1963. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Indust. Appl. Math. 1:431. McDaniel, B. T., R. H. Miller, E. L. Corley and R. D. Plowman.v 1967. DHIA age adjustment factors for standardizing lactations to a mature basis. ARS-44-l88. Dairy-Herd-Improvement Letter. Vol. 43, No l. McGilliard, M. L. 1978. Net returns from using genetically superior sires. J. Dairy Sci. 61:250. Miller, R. H., and N. W. Hooven, Jr. 1969; Variation in part-lactation and whole-lactation feed efficiency of Holstein cows. J. Dairy Sci. 52:1025. Nelder, J. A. 1966. Inverse polynomials, a useful group of multi- factor response functions. Biometrics. 22:128. Neter, J. and W. Wasserman. 1974. ‘Applied Linear Statistical Models. Richard D. Irwin, Inc. Homewood, Illinois. Norman, H. D., P. D. Miller, B. T. McDaniel, F. N. Dickinson, and C. R. Henderson. 1974. USDA-DHIA factors for standardizing 305-day lactation records for age and month of calving. ARS-NE—40. O'Connor, L. K., S. Lipton. 1960. The effect of various sampling intervals on the estimation of lactation milk yield and composition. J. Dairy Res. 27:389. Pearson, R. E. 1976. Managing herd breeding: Economic considerations. Nat. Workshop on Genetic Improvement of Dairy Cattle. pp. 137. Schaeffer, L. R., C. E. Minder, I. McMillan and E. B. Burnside. 1977. Nonlinear techniques for predicting 305-day lactation production of Holsteins and Jerseys. J. Dairy Sci. 60:1636. 177 Schneeberger, M. 1981. Inheritance of lactation curve in Swiss Brown Cattle. J. Dairy Sci. 64:475. Shanks, R. D. 1979. Relations among health cost, persistency, postpartum length, and milk production in Holstein cows. Unpublished Ph.D. Thesis, Iowa State University. Shanks, R. D., A. E. Freeman, P. J. Berger and D. H. Kelley. 1978. Effect of selection for milk production on reproductive and general health of the dairy cow. J. Dairy Sci. 61:1765. Shanks, R. D., P. J. Berger, A. E. Freeman, and F. N. Dickinson. 1980. Genetic aspects of lactation-curve parameters. Paper 126, presented at 75th Annual meeting of the American Dairy Science Association, Blacksburg, Virginia. Shanks, R. D., A. E. Freeman and F. N. Dickinson. 1981. Postpartum distribution of costs and disorders of health. J. Dairy Sci. 64:683. Shapiro, S. S. and M. B. Wilk. 1965. An analysis of variance test for normality (complete samples). Biometrika. 52:591. Shimizu, H. and S. Umrod. 1976. An application of the weighted regression procedure for constructing the lactation curve in dairy cattle. Japanese J. Zootech. Sci. 47:733. Shook, G. E. 1975. Outline of the proposed factors for improving accuracy of DHI estimates of lactation yield. Presented at National DHI Computing Workshop, Seattle, Washington, October. Smith, H. F. 1936. A discriminant function for plant selection. Ann. Eugenics. 7:240. Tallis, G. M. 1962. A selection index for optimum genotype. Biometrics. 18:120. Van Vleck, L. D. 1979. Notes on the theory and application of selectiongprinciples for the genetic improvement of animals. Cornell University. Third printing. Williams, J. S. 1962. The evaluation of a selection index. Biometrics. 18:375. Wood, P. D. P. 1967. Algebraic model of the lactation curve in cattle. Nature, Lond. 216:164. Wood, P. D. P. 1969. Factors affecting the shape of the lactation curve in cattle. Anim. Prod. 11:307. 178 Wood, P. D. P. 1970. A note on the repeatability of parameters of the lactation curve in cattle. Anim. Prod. 12:535. Wood, P. D. P. 1972. A note on seasonal fluctuations in milk production. Anim. Prod. 15:89. Wood, P. D. P. 1976. Algebraic models of the lactation curves for milk, fat and protein production, with estimates of seasonal variation. Anim. Prod. 22:35. Yadav, M. C., B. G. Katpatal and S. N. Kaushik. 1977. Study of lactation curve in Hariana and its Friesian crosses. Indian J. Anim. Sci. 47:607.