GENETIC EVALUATION OF MULTI-BREED BEEF CATTLE. A Thesis. Presented to. The Faculty of Graduate Studies. The University of Guelph

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1 GENETIC EVALUATION OF MULTI-BREED BEEF CATTLE A Thesis Presented to The Faculty of Graduate Studies of The University of Guelph by VANERLEI MOZAQUATRO ROSO In partial fulfilment of requirements for the degree of Doctor of Philosophy November, 2004 Vanerlei Mozaquatro Roso, 2004

2 Advisory Committee: Dr. Stephen P. Miller (Advisor) Dr. Flávio S. Schenkel Dr. Gary J. Umphrey Dr. James W. Wilton Dr. Lawrence R. Schaeffer

3 ABSTRACT GENETIC EVALUATION OF MULTI-BREED BEEF CATTLE Vanerlei Mozaquatro Roso University of Guelph, 2004 Advisor: Professor Stephen Paul Miller Three alternative methods for measuring the degree of connectedness among test groups (TG), including variance of estimated differences between TG effects (VED), connectedness rating (CR), and total number of direct genetic links between TG due to common sires and dams (GLT), which could be routinely used in genetic evaluation programs, were evaluated. Data were consecutive weights of bulls tested in central evaluation stations in Ontario, Canada. The Prediction error variance of differences in estimated breeding values of bulls from different TG (PEVD) was assumed the most adequate measure of connectedness and results from VED, CR, and GLT were compared relative to PEVD. Average PEVD of pairs of TG can be more accurately predicted on the basis of GLT than on the basis of either VED or CR. Average PEVD of each TG with all other test groups can be more accurately predicted on the basis of either CR or GLT. The GLT, which is not excessively computing demanding, was used to identify a set of connected contemporary groups including both purebred and crossbred animals from beef herds in Ontario. Estimates of variance components, breed additive genetic changes, direct and maternal breed, dominance, and epistatic loss genetic effects on pre-weaning weight gain (PWG) were obtained. Both direct and maternal dominance effects were assumed proportional to breed heterozygosity and showed favourable effects on PWG. Direct epistatic loss reduced the performance of the animals, whereas maternal epistatic

4 loss did not significantly affect the PWG. Breeds ranked similarly to what was expected, but estimates were highly unstable, with high standard errors, possibly due to multicollinearity, which can result in inaccurate across-breed estimated breeding values. A framework using ridge regression methods was developed to obtain more stable estimates of direct and maternal breed, dominance, and epistatic loss effects on PWG when multicollinearity is of concern. Two generalized methods were applied in the choice of the ridge parameter. Once the choice of the ridge parameter was made, its reliability and validity were evaluated through bootstrap resampling procedures. Mean squared error of prediction (MSEP) of both ridge regression methods were 3% lower than the MSEP from ordinary least squares. Ridge regression methods were effective in reducing the multicollinearity involving predictor variables of breed effects.

5 ACKNOWLEDGEMENTS I am particularly grateful to my advisor Dr. Stephen P. Miller for giving me the opportunity to develop my graduate studies at University of Guelph. His enthusiasm, encouragement, guidance, and friendship during my graduate program were appreciated. I would like to extend sincere acknowledgements to the other members of my advisory committee, Dr. Flávio. S. Schenkel, Dr. Gary J. Umphrey, Dr. James W. Wilton, and Dr. Lawrence R. Schaeffer for their time, advice, and contributions to the manuscript. Thanks to Dr. Peter G. Sullivan, Dr. Luiz A. Fries, and Dr. Roberto Carvalheiro for their suggestions. I would like to acknowledge the faculty, staff, students, and visiting scientists at the Department of Animal and Poultry Science for their help, kindness, and support, making my graduate program a pleasant experience. A special thanks to my friends Flávio, Sandra, Mariana, and Daniel, who made me feel at home during my stay in Guelph, and to my family, for their continuous support and love. I am thankful to my partners at GenSys Consultores Associados in Brazil, Fernanda V. Brito, Jorge L. P. Severo, Luiz A. Fries, and Mario L. Piccoli for their extra effort to cover my temporary leave of absence, which allowed me to pursue a Ph.D. at the University of Guelph. I would like to thank Beef Improvement Ontario (BIO) for providing data and financial support, Natural Sciences and Engineering Research Council of Canada, and Ontario Ministry of Agriculture and Food for financial support, and the Canadian i

6 Foundation for Innovation, Ontario Innovation Trust, and Compaq for supporting the required computing infrastructure. ii

7 TABLE OF CONTENTS 1. General Introduction 1 2. Degree of connectedness among groups of centrally tested beef bulls. 6 Abstract.. 6 Introduction 7 Material and Methods 9 Data 9 Statistical model. 9 Measures of the degree of connectedness.. 10 Prediction error variance of differences in EBV of bulls ( â ) from different test groups (PEVD).. 10 Variance of estimated differences between test groups effects ( ĝ ) (VED). 11 Connectedness rating (CR).. 11 Total number of direct genetic links between test groups (GLT) 11 Results 13 Connectedness 13 Prediction of PEVD on the basis of VED, CR, and GLT.. 14 Average PEVD of pairs of TG 16 On the basis of VED On the basis of CR.. 17 iii

8 On the basis of GLT. 17 Average PEVD of each TG with all other TG On the basis of VED On the basis of CR.. 19 On the basis of GLT 19 Simulation of disconnected test groups. 21 Discussion.. 22 Conclusions Additive, dominance, and epistatic loss effects on pre-weaning gain in crossing of different Bos taurus breeds.. 36 Abstract.. 36 Introduction 38 Material and Methods 39 Data.. 39 Connectedness analysis Predictor variables of fixed genetic effects. 40 Breed additive effects.. 40 Dominance effects Epistatic loss effects Genetic analysis Multi-breed additive genetic changes.. 44 Results 45 iv

9 (Co)variance components 45 Multi-breed additive genetic changes Dominance and epistatic loss effects Breed additive effects.. 47 Sampling correlations.. 48 Discussion.. 48 Conclusions Estimation of genetic effects in the presence of multicollinearity 69 Abstract.. 69 Introduction 71 Material and Methods 72 Data.. 72 Predictor variables of fixed genetic effects. 73 Breed additive effects.. 73 Dominance effects Epistatic loss effects Multicollinearity diagnostics Variance inflation factor. 74 Condition index Variance-decomposition proportions associated with the eigenvalues Genetic analysis v

10 Ridge regression.. 79 Objective methods for selecting the ridge parameter K.. 80 Generalized Ridge Estimator of Hoerl and Kennard (R1).. 80 Bootstrap in combination with cross-validation (R2). 82 Mean squared error of prediction and variance inflation factor.. 83 Bias measurement 84 Comparison of across-breed estimated breeding values.. 84 Additive-dominance models 84 Additive-dominance-epistatic models. 85 Results 86 Multicollinearity diagnostics.. 86 Ridge parameter K. 87 Convergence of estimates of fixed genetic effects. 88 Mean squared error of prediction and variance inflation factor. 88 Bias measurement.. 89 Dominance and epistatic loss effects Breed additive effects. 90 Sampling correlations 92 Comparison of across-breed estimated breeding values 93 Use of the same ridge parameter in subsequent genetic evaluations 95 Discussion.. 96 Conclusions 102 vi

11 5. General Discussion 124 Degree of connectedness among test groups of centrally tested beef bulls. 125 Practical implications Limitations and suggestions for further investigations Additive, dominance, and epistatic loss effects on pre-weaning gain in crossing of different Bos taurus breeds 129 Practical implications Limitations and suggestions for further investigations Estimation of genetic effects in the presence of multicollinearity. 134 Practical implications 136 Limitations and suggestions for further investigations References vii

12 LIST OF TABLES Table 2.1. Summary of the bull test data Table 2.2. Correlations among PEV of the difference between EBV of bulls from different test groups (PEVD), variance of estimated differences between test group effects (VED), connectedness rating (CR) and total number of direct genetic links between test groups (GLT) for pairs of test groups (above diagonal) and for averages of each test group with all other test groups (bellow diagonal) Table 2.3. Estimates of intercept, regression coefficients, and coefficient of determination (R 2 ) of the models to predict average PEVD of pairs of test 29 groups.. Table 2.4. Estimates of intercept, regression coefficients, and coefficient of determination (R 2 ) of the models to predict average PEVD of each test group with all other test groups Table 3.1. Coefficients of direct (H D ) and maternal (H M ) dominance and direct (E D ) and maternal (E M ) epistatic loss genetic effects for different mating systems involving two breeds, A and B.. 56 viii

13 Table 3.2. Distribution of observations among coefficients of direct (H D ) and maternal (H M ) dominance and direct (E D ) and maternal (E M ) epistatic loss genetic effects. 57 Table 3.3. Mean and standard deviation (SD) of pre-weaning gain (Gain), weaning age (Age), coefficients of direct and maternal breed additive, dominance (H D and H M ), and epistatic loss (E D and E M ) genetic effects.. 58 Table 3.4. Estimates of (co)variance components and genetic parameters of preweaning gain (kg) 59 Table 3.5. Multi-breed additive genetic changes in pre-weaning gain per year obtained through regression of average estimated breeding values of purebred calves on birth year (Average) and through regression of estimated breeding values on contribution of each breed to the breed composition of the calves (Regression).. 60 Table 3.6. Estimates and standard errors of direct and maternal dominance (H) and epistatic loss (E) effects on pre-weaning gain (kg). 61 Table 3.7. Estimates (as deviations from Angus) and standard errors of direct and maternal breed additive effects for pre-weaning gain (kg). 62 ix

14 Table 3.8. Sampling correlations among estimates of direct (D) and maternal (M) fixed genetic effects Table 4.1. Correlation coefficients among predictor variables of direct (D) and maternal (M) fixed genetic effects (n = 478,466) Table 4.2. Eigenvalues of the correlation matrix among predictor variables of fixed genetic effects and corresponding condition indices Table 4.3. Decomposition of the variance structure of the parameter estimates associated with the two largest condition indices Table 4.4. Values of the ridge parameter (K) obtained by ridge regression methods R1 and R2, for direct and maternal genetic effects. 107 Table 4.5. Summary of results obtained over one hundred bootstrap samples for ordinary least squares (LS) and ridge regression methods R1 and R Table 4.6. Estimates of direct and maternal dominance (H) and epistatic loss (E) effects on pre-weaning gain (kg), obtained by ordinary least squares (LS) and ridge regression methods R1 and R2 109 Table 4.7. Estimates of direct and maternal breed additive effects on pre-weaning x

15 gain (kg), as deviations from Angus, obtained by ordinary least squares (LS) and ridge regression methods R1 and R Table 4.8. Number of calves including records from 1986 to the indicated year, expressed as equivalent purebred calves. 111 Table 4.9. Values of the ridge parameter (K), obtained by ridge regression methods R1 and R2, using records from 1986 to xi

16 LIST OF FIGURES Figure 2.1. Average degree of connectedness for pairs of test groups (top) and for each test group with all other test groups (bottom) on the basis of PEVD, VED, CR, and GLT.. 31 Figure 2.2. Observed relationship of average PEVD per test group with number of bulls per test group, average PEVD per test group with number of sires per test group, CR with number of bulls per test group, and GLT with number of bulls per test group Figure 2.3. Observed relationship of average PEVD of pairs of test groups with VED, CR, and GLT Figure 2.4. Observed relationship of average PEVD of each test group with VED, CR, and GLT. 34 Figure 2.5. Observed relationship of average PEVD of each test group with number of bulls per test group, VED, CR, and GLT for connected and disconnected test groups 35 Figure 3.1. Percentage of calves, sires, and dams with 1, 2, 3, or 4 breeds in the genetic composition in the dataset containing 478,466 calves, 19,908 xii

17 sires, and 234,608 dams 65 Figure 3.2. Number of purebred and crossbred calves, sires, and dams containing some portion of the indicated breed in dataset including 478,466 calves, 19,908 sires, and 234,608 dams 66 Figure 3.3. Numbers of purebred and crossbred (expressed as equivalent to purebred) calves per breed 67 Figure 3.4. Multi-breed additive genetic changes in pre-weaning gain obtained through average breeding values of purebred calves per birth year (Average) and through regression of yearly breeding values on contribution of each breed to the breed composition of the calves (Regression) Figure 4.1. Variance inflation factor (VIF) associated with predictor variables of direct and maternal dominance (H), epistatic loss (E), and breed additive effects Figure 4.2. Convergence of the estimates of direct and maternal dominance (H), epistatic loss (E), and breed additive effects under ridge regression method R xiii

18 Figure 4.3. Convergence of the estimates of direct and maternal dominance (H), epistatic loss (E), and breed additive effects under ridge regression method R Figure 4.4. Variance inflation factor (VIF) associated with predictor variables of direct and maternal dominance (H), epistatic loss (E), and breed additive effects under ordinary least squares (LS) and ridge regressions methods R1 and R Figure 4.5. Estimates (as deviations from AN) and standard errors of direct dominance (H), epistatic loss (E), and breed additive effects under ordinary least squares (LS) and ridge regression methods R1 and R Figure 4.6. Estimates (as deviations from AN) and standard errors of maternal dominance (H), epistatic loss (E), and breed additive effects under ordinary least squares (LS) and ridge regression methods R1 and R Figure 4.7. Sampling correlations (multiplied by 1.0) between estimates of maternal dominance (H M ) and direct epistatic loss (E D ) effects and between estimates of direct and maternal breed additive effects given by ordinary least squares (LS) and ridge regression methods R1 and R Figure 4.8. Pearson and Spearman correlations, and percentages of coincidence for xiv

19 different proportions of selected (top 1%, 10%, 20%, and 40%) sires, dams, and calves on the basis of ABC yielded by different models compared to model ADE-R Figure 4.9. Estimates of direct and maternal dominance (H), epistatic loss (E), and breed additive effects (as deviations from AN), under ordinary least squares, using records from 1986 to the indicated year 121 Figure Estimates of direct and maternal dominance (H), epistatic loss (E), and breed additive effects (as deviations from AN), under ridge regression method R1, using records from 1986 to the indicated year (ridge parameter K was obtained using records from 1986 to 1996) Figure Estimates of direct and maternal dominance (H), epistatic loss (E), and breed additive effects (as deviations from AN), under ridge regression method R2, using records from 1986 to the indicated year (ridge parameter K was obtained using records from 1986 to 1996) xv

20 ABREVIATIONS KEY ABC = Across-breed estimated breeding value AN = Angus BD = Blond D Aquitane BEG = Bull estimated weight gain BLUP = Best linear unbiased predictor CH = Charolais CI = Condition index CR = Connectedness rating D = Dominance effect E = Epistatic loss effect EBV = Estimated breeding value E D = Coefficient of direct epistatic loss effect E M = Coefficient of maternal epistatic loss effect GLT = Total number of direct genetic links between test groups GV = Gelbvieh H D = Coefficient of direct dominance effect HE = Hereford H M = Coefficient of maternal dominance effect LM = Limousin LS = Ordinary least squares MA = Maine-Anjou MSE = Mean square error xvi

21 MSEP = Mean squared error of prediction PEV = Prediction error variance PEVD = Average prediction error variance of the difference between EBVs R1 = Generalized ridge estimator of Hoerl and Kennard (ridge regression method R1) R2 = Bootstrap in combination with cross-validation (ridge regression method R2) SA = Salers SH = Shorthorn SM = Simmental TG = Test group VED = Variance of estimated differences between test group effects VIF = Variance inflation factor xvii

22 Chapter 1 General Introduction Genetic selection and planned crossbreeding systems are two complementary strategies that have been applied in the beef cattle industry to generate animals with high levels of production and efficiency under varying management conditions and market preferences. Programs of genetic improvement taking advantage of between-breed additive and non-additive genetic effects are now common worldwide. A genetic goal is effectively accomplished by selection based on modern genetic evaluation. Considering the importance of crossbreeding in beef cattle production, genetic evaluation must consider animals of multiple breeds. Mixed model procedures, employing an animal model, are generally used in the genetic evaluations of multi-breed populations. For having highly accurate genetic evaluations and consequently high response to selection, breed additive and non-additive genetic effects must be properly accounted for. Moreover, estimated breeding values of animals should be comparable regardless of the breed composition and management units from which they come. The present research focuses on some problems related to statistical methods applied to the estimation of breeding values of animals in a multi-breed population of beef cattle, more specifically: (1) Estimation of the degree of connectedness among groups of centrally tested beef bulls; 1

23 (2) Estimation of additive, dominance, and epistatic loss effects on pre-weaning gain in crossing of different Bos taurus breeds; and (3) Estimation of genetic effects in the presence of multicollinearity. Central testing of beef bulls is an important component of genetic improvement programs for beef cattle in many countries. Because selection is carried out across test groups, evaluation of the degree of connectedness among test groups is of great concern. With few genetic links between test groups, comparison of bulls EBV from different groups is less accurate, even if the accuracy of the EBV are high within the groups (Kennedy and Trus, 1993). Different criteria for measuring connectedness have been proposed in the literature (e.g., Wood et al., 1991; Folley et al., 1992; Laloë, 1993; Kennedy and Trus, 1993; Fries, 1998; Hanocq and Boichard, 1999; Mathur et al., 2002). Ideally, PEV of comparisons between animals or average PEV of comparisons between groups of animals (PEVD), which is influenced by the average genetic relationship between and within management units, should be the basis for measuring connectedness (Kennedy and Trus, 1993). However, computing the PEV matrix is very difficult or impossible for large datasets. If obtaining a measure of connectedness through PEVD is impossible, alternative methods could be used to predict PEVD. In Chapter 2, three alternative methods are assessed and compared with respect to prediction of PEVD. Models to predict PEVD, which could be routinely used in genetic evaluation, are suggested. An indication of the degree of connectedness among test groups of beef bulls in Ontario, Canada, is obtained. Results from this investigation will be the basis for developing recommendations to increase the accuracy of comparisons of bulls across test groups. 2

24 Across herd genetic evaluations for growth traits is another significant component of genetic improvement programs for beef cattle in many countries. Similar to genetic evaluations of centrally tested beef bulls, across herd genetic evaluations for growth traits are based on additive-dominance genetic models. These models are justified based on the assumption that heterosis is mainly due to dominance effects, in agreement with results obtained by Gregory et al. (1997) in a large beef cattle crossbreeding experiment. Heterosis is modeled as being proportional to the probability that genes at a locus come from different breeds, which corresponds to the breed heterozygosity. Deviations from the linear association of heterosis with degree of heterozygosity are due to recombination loss (Dickerson, 1969, 1973). Recombination loss (epistatic loss) is attributed to the loss of favourable epistatic combinations present in the gametes from purebreds as a result of long-term selection. This loss is proportional to the probability that two non-allelic genes randomly chosen in the individual are from different breeds. Because it is difficult to estimate dominance and epistatic loss effects separately, research studies to estimate both dominance and epistatic loss effects in beef cattle are not abundant, particularly with field data. However, results obtained by Arthur et al. (1999) suggest that, when data structure allows, the inclusion of epistatic effects in the genetic evaluation model can significantly improve the accuracy of predictions. Estimates of (co)variance components, heterosis, breed effects, and additive genetic changes have been obtained in Ontario (Miller, 1996; Sullivan et al., 1999), but there were no available studies which separated direct and maternal dominance and epistatic loss effects associated with breed heterozygosities. An objective reported in Chapter 3 was to obtain estimates of direct and maternal breed additive, dominance, and epistatic loss effects for pre-weaning gain weight. (Co)variance components were also obtained 3

25 and breed additive genetic changes between 1986 and 1999 were examined. Estimates obtained in this study can be used to update the parameters currently used in the genetic evaluations to improve accuracy. For fitting breed additive, dominance, and epistatic loss effects, a multiple regression equation including predictor variables such as breed compositions and breed heterozygosities, and functions of the heterozygosities can be used. This has been generally done by ordinary least squares methods. The interpretation of the estimates given by ordinary least squares depends on the assumption that predictor variables are not strongly interrelated. If the vectors of predictor variables are multicollinear, the least square estimates typically have large standard errors, may have signs that are opposite to what would be expected, and are sensitive to changes in the data file and to addition or deletion of variables in the model, making modeling very confusing. Moreover, when taken in combination, the estimated coefficients often cancel out, indicating confounding. In the presence of multicollinearity, the least squares estimator is not adequate because it will be very unstable. Multicollinearity has been indicated as one of the main causes of unexpected signs and high degree of confounding involving estimates of direct and maternal breed additive and/or non-additive genetic effects (e.g., Kinghorn and Vercoe, 1989; Rodríguez-Almeida et al., 1997; Fries et al., 2000; Cassady et al., 2002), which can lead to the incorrect ranking of animals based on across breed comparisons. For overcoming difficulties caused by multicollinearity, Hoerl and Kennard (1970a, 1970b) suggested the use of the ridge regression estimator. With a suitable choice of the ridge parameter, the ridge regression estimator gives a more precise estimate of regression coefficients because its variance and mean squared error are smaller than those of the least squares estimator. The fact that ridge regression estimators have been 4

26 successfully applied in dealing with multicollinearity in diverse fields, including Chemistry, Econometrics, and Engineering (Gruber, 1998) suggests avenues for research and application in the context of animal breeding, particularly in the analysis of multibreed populations of beef cattle. Chapter 4 presents the development of a framework, using ridge regression methods, for obtaining stable estimates of direct and maternal breed additive, dominance, and epistatic loss effects on pre-weaning gain when multicollinearity is of concern, which could contribute to more accurate multi-breed genetic evaluation of beef cattle. After identifying the causes of dependencies among predictor variables, two generalized ridge regression methods were applied in the choice of the ridge parameter. Once the choice of the ridge parameter was made, its reliability and validity were evaluated through bootstrap resampling procedures in combination with cross-validation. Finally, some results obtained with ridge regression methods were examined to further illustrate application of ridge regression in routine large-scale genetic evaluations. The final chapter is a general discussion of results obtained in the previous chapters. Some practical implications of the results of this study, limitations, and suggestions for future research are presented. 5

27 Chapter 2 Degree of connectedness among groups of centrally tested beef bulls V. M. Roso, F. S. Schenkel, and S. P. Miller Published in Canadian Journal of Animal Science : Reproduced by permission of the Agricultural Institute of Canada ABSTRACT - The degree of connectedness among test groups (TG) of bulls tested in central evaluation stations from 1988 to 2000 in Ontario, Canada, was evaluated using the methods PEVD, VED, CR, and GLT. The model used in the analysis included the effects of breed and TG (fixed) and animal (random). PEVD was assumed the most adequate measure of connectedness and results from the alternative methods VED, CR, and GLT were compared relative to PEVD. Models to predict the average PEVD of pairs of TG and the average PEVD of each TG with all other TG on the basis of VED, CR, and GLT were developed. Results from all measures of connectedness indicated an unfavourable trend in the degree of connectedness after The average PEVD of pairs of TG can be 6

28 better predicted on the basis of the model that includes GLT. The average PEVD of each TG with all other TG can be better predicted on the basis of models that include either CR or GLT. Connectedness among TG of centrally tested beef bulls can be adequately assessed for specific pairs of TG or overall for each TG with all other TG using GLT. Key words: accuracy, central test, genetic evaluation, harmonic mean Abbreviations: BEG, bull estimated weight gain; CR, connectedness rating; EBV, estimated breeding value; VED, variance of estimated differences between test group effects; GLT, total number of direct genetic links between test groups; PEV, prediction error variance; PEVD, average prediction error variance of the difference between estimated breeding values; TG, test group. INTRODUCTION Connectedness among test groups (TG) is of interest in genetic evaluation of stationtested beef bulls because comparisons of estimated breeding values (EBV) of bulls tested in different groups are made. The EBV of bulls from different TG are comparable due to use of appropriate methodology (Best Linear Unbiased Predictor, BLUP) and genetic connectedness among groups. However, the accuracy of the comparisons depends upon the degree of connectedness among TG. With lower connectedness between TG, comparison of bulls EBV from different TG is less accurate, even if the accuracy of EBV is high within the groups (Kennedy and Trus, 1993). When genetic evaluation is under an animal model, connections occur through additive genetic relationships. Hence, two TG could be connected by direct and/or 7

29 indirect genetic links. Kennedy and Trus (1993) argued that the most appropriate measure of connectedness is the average prediction error variance of differences (PEVD) in EBV between animals in different management units (e.g., TG), which is influenced by the average genetic relationship between and within management units. However, computing this statistic is extremely time consuming and not feasible for routine application. When PEVD cannot be computed, Kennedy and Trus (1993) proposed to use the variance of estimated differences between management unit effects (VED), which was highly correlated with PEVD in their simulation study. Mathur et al. (1999) also suggested that VED could be used as a measure of connectedness between two management units and proposed to calculate the connectedness rating (CR), defined as the correlation between estimated effects of two management units. Following Mathur et al. (1999), CR is less dependent on the size and structure of management units than VED. For calculating CR, the authors proposed an iterative method, which captures the inverse elements for some rows and columns (corresponding to TG in the mixed model equations, for example) of any large matrix for which a direct inverse is not possible. Fries (1998) proposed the use of number of direct genetic links between TG (GLT) due to common sires and dams as a method for measuring degree of connectedness among TG. The objectives of this study were: (1) To obtain an indication of the degree of connectedness of test groups of beef bulls in Ontario, (2) To assess and compare the methods VED, CR and GLT for measuring the degree of connectedness among groups of station-tested beef bulls, and 8

30 (3) To define a model to predict the PEVD of pairs of test groups and the average PEVD of each TG with all other TG, which could be routinely used in genetic evaluation programs. MATERIAL AND METHODS Data Data were consecutive weights of bulls tested in central evaluation stations in Ontario, Canada, from 1988 to Bulls from multiple breeds and crossbreds, from different herds, were delivered to test stations and submitted to an adjustment period of 28 days before start of test. Bulls were weighed every 28 days during a period of 112 or 140 days on test. A summary of the data is presented in Table 2.1. Statistical model Consecutive weights of bulls were used to obtain the estimated weight gain (BEG). A fixed univariate linear regression of the weight (w ij ) on days on test (d ij ) for each bull i was estimated, using the model w ij = α i + β i d ij + e ij, where α i and β i are the intercept and linear regression coefficient of the i th bull, respectively, and e ij is the random residual term. The BEG was calculated multiplying β i by the number of days on test (140 days) and adjusted for heterosis on the basis of individual bull s heterozygosity. An ad hoc heterosis of 3% was assumed for an animal with heterozygosity of 100%, regardless of the breeds involved (Sullivan et al., 1999). Then, BEG was used as an observation in the follow genetic evaluation model: 9

31 ij 14 BEG = b B + g + a + e, k k=1 ik j ij ij where BEG ij is the estimated weight gain of the i th bull in the j th TG; b k is the linear regression coefficient on the breed composition for the k th breed; B ik is the contribution of the k th breed to the breed composition of the i th bull; g j is the fixed effect of the j th TG; a ij is the random additive genetic effect of the i th bull in the j th TG; e ij is the random residual effect. Random effects a and e were assumed independent with covariance matrices equal to Aσ 2 a and Iσ 2 e, respectively. All available pedigree information was incorporated into the additive numerator relationship matrix A. The required elements for calculating VED, CR and PEVD were obtained using PEST (Groeneveld, 1990), assuming ad hoc heritability of 0.43 (Sullivan et al., 1999), which was previously estimated for the same data set. Measures of the degree of connectedness The degree of connectedness among TG was measured using the following methods: (1) Prediction error variance of differences in EBV of bulls ( â ) from different test groups (PEVD). The PEVD of two animals, one from the i th and other from the j th TG was given by PEVDij = var(âi ai ) + var(â j a j) 2 cov(âi ai,â j a j). 10

32 (2) Variance of estimated differences between test group effects ( ĝ ) (VED). The VED between the i th and the j th TG was given by VEDij = var(ĝi ) + var(ĝ j ) 2cov(ĝi,ĝ j ). (3) Connectedness rating (CR), defined as the correlation between estimated effects of TG (Mathur et al. 2002). The CR between the i th and the j th TG was given by cov(ĝi,ĝ j ) CRij = 100. var(ĝ ) var(ĝ ) i j (4) Total number of direct genetic links between test groups (GLT), defined as the links between TG due to common sires and dams (Fries, 1998). The basic steps of the algorithm and the criteria used for computing GLT are: 1. Calculate the number of direct genetic links between pairs of TG due to common sires and dams. Then, for each TG, calculate the overall number of genetic links due to sires (GLs) and dams (GLd) with all other TG. 2. Calculate the total number of genetic links (GLT) as the sum of GLs and GLd. 3. Identify the TG with the largest GLT ( main TG ). 4. Identify all TG direct and/or indirectly connected to main TG. These groups constitute the principal mass. TG with less than 10 GLT and/or less than three different parents (sires + dams) were considered disconnected to principal mass and have their GLT zeroed. Other criteria could be used. 5. Repeat step 4 until the connected TG remain the same as at previous run. 11

33 6. Save records that were considered as connected to the principal mass. TG disconnected to the principal mass have GLT equal to zero and should be rerun through the program. This procedure allows identification of isolated subsets of connected TG. The average PEVD was assumed as the basic measure of connectedness of a TG, following Kennedy and Trus (1993). This statistic was considered the most appropriate measure of connectedness and the alternative methods VED, CR, and GLT were compared relative to PEVD. The degree of connectedness was calculated for pairs of TG and for each TG with all other TG. Connectedness between pairs of TG indicates accuracy in comparing EBV of animals from two TG. Average connectedness of each TG with all others indicates the average accuracy in comparing EBV of an animal with animals in all other TG. This measure is of greater importance in the evaluation of beef bulls in a station test because selection generally considers all TG instead of a few very well connected TG. High average connectedness of each TG with all other TG allows effective selection across all TG. As previously indicated, the GLT of a TG is the number of direct genetic links of the TG with all other TG. Obviously many pairs of TG that do not have any direct genetic links are indirectly connected and, consequently, can have high accuracy of comparisons of EBV between them. For this reason, the number of direct genetic links between pairs of TG is inadequate to indicate the degree of connectedness between pairs of TG. The arithmetic mean of GLT of each pair of TG is also inadequate because pairs of TG with equal arithmetic mean can have very different degrees of connectedness. A potentially adequate measure of connectedness between pairs of TG could be obtained through the harmonic mean of the GLT. This measure has the property of discriminating among pairs 12

34 of TG with different GLT, penalizing those expected to be more poorly connected. As a consequence, better relationship between PEVD with harmonic means than with arithmetic means of GLT may be expected. The harmonic mean of GLT was used in the prediction of average PEVD of pairs of TG. The harmonic mean of GLT of TG i and TG j (GLT ij ) was given by 2 GLT ij =, GLT GLT i j where GLT i and GLT j are the GLT of the i th and j th TG with all other TG, respectively. The harmonic mean is always smaller than the arithmetic mean unless the GLT of the two TG are identical. When the GLT of a TG was equal to zero, which means the TG is not connected to the principal mass, a harmonic mean equal to zero was assumed. The statistical analyses to define the models for predicting PEVD were performed using the general linear models procedure (GLM) of the SAS statistical software (SAS Institute Inc., 1990). The R 2 of the models and the level of significance (P < 0.05) of each effect considered were the criteria used to determine the final models. When segmented polynomial regressions were used, the knots (junction points between segments) were determined based on maximization of R 2 of the model. RESULTS Connectedness The average value of degree of connectedness among TG using PEVD and the alternative measures VED, CR, and GLT were 1599 ± 58, 286 ± 132, 1.23 ± 1.28, and 707 ± 503 for pairs of TG and 1726 ± 41, 286 ± 93, 1.21 ± 0.51 and 709 ± 690 for each 13

35 TG with all other TG, respectively. The overall results over the years are depicted in Figure 2.1. Small values of PEVD and VED, and large values of CR and GLT are desirable, because they indicate higher levels of connectedness among TG. All measures of connectedness showed the same trend, that is, an increase in the degree of connectedness from 1988 to 1994 and a substantial decrease after The highest PEVD and VED, and the smallest CR and GLT were observed in 2000 (last year with available information at the time of this research). Prediction of PEVD on the basis of VED, CR, and GLT Correlations among PEVD, VED, CR and GLT for pairs of TG and for averages of TG with all other TG are presented in Table 2.2. In general the correlations had moderate to high magnitude. The correlation between PEVD and VED was 0.71 both for pairs of TG and average per TG, in contrast with the almost perfect correlation obtained by Kennedy and Trus (1993) in their simulation study. The coefficient of correlation measures only the strength of the linear relationship between two variables. Because a better indication of the true relationship of PEVD with the alternative methods was needed for defining the models to predict PEVD, the observed relationship between PEVD and the other variables were graphically analyzed. The relationship of PEVD with both number of bulls and number of sires per TG was also analyzed. As shown in Figure 2.2, the observed relationship of average PEVD per TG with both number of bulls and number of sires per TG had the same pattern. By observation, TG with more than approximately 40 bulls or 20 sires were associated with values of PEVD smaller than 1750, otherwise TG showed large variation in PEVD. The variation depends on the genetic relationship between groups, which is not a direct function of number of 14

36 bulls or number of sires per TG. Because TG with a small number of bulls or a small number of sires showed large variation in PEVD, which indicate large variation in the degree of connectedness of these groups, neither number of bulls (size of the group) nor number of sires per TG are good predictors of the degree of connectedness between TG. Figure 2.2 shows also the relationship of both CR and GLT with number of bulls per TG. Although a large variation in the degree of connectedness was indicated by PEVD when the size of TG was small, CR was strongly associated with number of bulls per TG over the whole range of TG size. CR decreased linearly when the size of TG became smaller than approximately 40 bulls. Mathur et al. (2002) reported a similar trend in the application of CR for measuring connectedness in the Canadian Centre for Swine Improvement. VED seemed to be even more dependent on the size of TG than CR, where TG with less than 25 bulls were associated with increasingly higher VED (data not shown). On the contrary, GLT showed large variation across the range of TG sizes (Figure 2.2). Even small TG had large GLT, which could result in these TG having high accuracy of comparisons. The observed relationships of PEVD with VED, CR and GLT are depicted in Figure 2.3 for pairs of TG and in Figure 2.4 for the average of each TG with all other TG. In both cases, PEVD and VED were linearly, but not strongly, associated. On the other hand, the relationships of both CR and GLT with PEVD were curvilinear. When GLT of pairs of TG were represented by their arithmetic mean, large variation in PEVD was observed at the same level of GLT (Figure 2.3). However, when GLT of pairs of TG were represented by their harmonic mean, a stronger relationship with PEVD was observed. Therefore, in the prediction of PEVD of pairs of TG, superior results can be expected using harmonic mean instead of arithmetic mean of GLT. Figures 2.3 and 2.4 also 15

37 indicate that averages of CR smaller than approximately one and GLT smaller than approximately 250 per TG were associated with increasingly higher PEVD. The information provided by the correlations and graphical analyses were explored to define the models for predicting PEVD. Initially, VED, CR, GLT, number of bulls per TG, number of sires per TG, and the ratio of number of bulls per sire per TG were considered. In the final models, however, only those with significant effect (P < 0.05) were kept. The final models to predict the average PEVD of pairs of TG and the average PEVD of each TG with all other TG based on VED, CR, and GLT were the following: (1) Average PEVD of pairs of TG (1a) On the basis of VED The observed average PEVD of pairs of TG was modeled by a linear regression on VED and a quadratic regression on the ratio of harmonic means of number of bulls and number of sires of pairs of TG. PEVD ij = α + β 1 VED ij + β 2 (NB/S) ij + β 3 (NB/S) 2 ij + e ij, where PEVD ij is the observation of the average PEV of the difference between EBV of bulls in the i th TG with EBV of bulls in the j th TG; α is the intercept; VED ij is the variance of estimated differences between the i th and the j th TG; (NB/S) ij is the ratio of harmonic means of number of bulls and number of sires in the i th and j th TG; 16

38 β 1, β 2 and β 3 are the regression coefficients; e ij is the residual associated with PEVD of the i th and j th TG. (1b) On the basis of CR The observed average PEVD of pairs of TG was modeled using a quadratic-quadratic polynomial regression on CR and a quadratic regression on the ratio of harmonic means of number of bulls and number of sires of pairs of TG. PEVD ij = α + β 1 CR ij + β 2 CR 2 ij + β 3 Z + β 4 (NB/S) ij + β 5 (NB/S) 2 ij + e ij, where α is the intercept; CR ij is the connectedness rating between the i th and the j th TG; Z = 0 if CR < 1.9 or Z = (CR 1.9) 2 otherwise; (NB/S) ij is the ratio of harmonic means of number of bulls and number of sires in the i th and j th TG; β 1, β 2, β 3, β 4, β 4, and β 5 are the regression coefficients; e ij is the residual associated with PEVD of the i th and j th TG. (1c) On the basis of GLT The observed average PEVD of pairs of TG was modeled using a quadratic-quadratic polynomial regression on the harmonic mean of GLT of pairs of TG and a quadratic regression on the ratio of harmonic means of number of bulls and number of sires of pairs of TG. PEVD ij = α + β 1 GLT ij + β 2 GLT ij 2 + β 3 Z + β 4 (NB/S) ij + β 5 (NB/S) 2 ij + e ij, 17

39 where α is the intercept; GLT ij is the harmonic mean of the GLT of the i th and the j th TG; Z = 0 if GLT < 550 or Z = (GLT 550) 2 otherwise; (NB/S) ij is the ratio of harmonic means of number of bulls and number of sires in the i th and j th TG; β 1, β 2, β 3, β 4, β 4, and β 5 are the regression coefficients; e ij is the residual associated with PEVD of the i th and j th TG. (2) Average PEVD of each TG with all other TG (2a) On the basis of VED The observed average PEVD of each TG with all other TG was modeled by a linear regression on VED and a quadratic regression on number of sires per TG. PEVd i = α + β 1 VED i + β 2 S + β 3 S 2 + e i, where α is the intercept; PEVD i is the observation of the average PEV of the difference between EBV of bulls in the i th TG with EBV of bulls in all other TG; VED i is the average variance of estimated differences between the i th TG and all other TG; S is the number of sires represented in the i th TG; β 1, β 2 and β 3 are the regression coefficients; e i is the residual associated with PEVD of the i th TG. 18

40 (2b) On the basis of CR The observed average PEVD of each TG with all other TG was modeled using a quadratic-quadratic polynomial regression on CR, a quadratic regression on number of sires, and a quadratic regression on the ratio of number of bulls per sire. PEVD i = α + β 1 CR i + β 2 CR 2 i + β 3 Z + β 4 S i + β 5 S 2 i + β 6 (NB/S) i + β 7 (NB/S) 2 i + e i, where α is the intercept; CR i is the average connectedness rating of the i th TG with all other TG; Z = 0 if CR < 1.15 or Z = (CR 1.15) 2 otherwise; S is the number of sires represented in the i th TG; (NB/S) i is the average ratio of number of bulls per sire represented in the i th TG; β 1, β 2,β 3, β 4, β 5, β 6 and β 7 are the regression coefficients; e i is the residual associated with PEVD of the i th TG. (2c) On the basis of GLT The observed average PEVD of each TG with all other TG was modeled using a quadratic-quadratic-quadratic polynomial regression on GLT, a linear regression on number of sires and a quadratic regression on the ratio of number of bulls per sire. PEVD i = α + β 1 GLT i + β 2 GLT i 2 + β 3 Z 1 + β 4 Z 2 + β 5 S i + β 6 (NB/S) i + β 7 (NB/S) 2 i + e i, where α is the intercept; GLT i is the total number of direct genetic links between the i th TG and all other TG; 19

41 Z1 = 0 if GLT < 200 or Z1 = (GLT 200) 2 otherwise; Z2 = 0 if GLT < 800 or Z2 = (GLT 800) 2 otherwise. S is the number of sires represented in the i th TG; (NB/S) i is the average ratio of number of bulls per sire represented in the i th TG; β 1, β 2,β 3, β 4, β 5, β 6 and β 7 are the regression coefficients; e i is the residual associated with PEVD of the i th TG. Estimates of parameters and coefficient of determination (R 2 ) of the models are presented in Table 2.3 for prediction of average PEVD of pairs of TG and in Table 2.4 for prediction of average PEVD of each TG with all other TG. The R 2 of the models to predict average PEVD of each TG were higher than the R 2 of the models to predict PEVD of pairs of TG on the basis of VED, CR and GLT. These results were expected because extreme values observed in the pairwise comparisons were averaged out, reducing the variation on PEVD. The R 2 of the model to predict average PEVD of pairs of TG on the basis of VED was equal to 0.53 and VED accounted for 51% (partial R 2 ) of the total variation in PEVD. In the model to predict average PEVD of pairs of TG on the basis of CR, the R 2 was equal to 0.50 and CR accounted for 49% of total variation in PEVD. R 2 of 0.72 was obtained in the model that considered GLT, which accounted for 71% of the total variation in average PEVD (Table 2.3). In the models to predict average PEVD of each TG with all other TG, the R 2 of the model based on VED was equal to 0.55 and VED accounted for 50% of the total variation in PEVD. In the model to predict PEVD on the basis of CR, the R 2 was equal to 0.82 and 20

42 CR accounted for 73% of total variation in PEVD. R 2 of 0.79 was obtained in the model that considered GLT, which accounted for 76% of the total variation in PEVD (Table 2.4). The R 2 increased to 0.82 when GLT also included the genetic links due to grandparents (data not shown). Simulation of disconnected test groups In the data set, on the basis of GLT, there was only one completely disconnected TG. Thus, to evaluate the effect of complete disconnectedness, 36 TG had sire and dam identifications modified to generate completely disconnected TG, covering a range of TG sizes from very small to large (6 to 183 bulls). Because there were no relationships among bulls within the created disconnected TG, accuracy of bull EBV from disconnected TG would increase only with the size of the group. Figure 2.5 shows that increasing the size of disconnected groups reduced the average PEVD of each TG with all other TG from 1950, in a group with only 6 bulls, to an asymptotical minimum value around 1850, when 120 bulls were in the TG. Kennedy and Trus (1993) showed that relationships among bulls within disconnected TG would increase the PEV of comparisons of EBV across TG. Therefore, connected TG with average PEVD greater than or equal to 1850 would behave similarly to large disconnected TG of unrelated bulls with respect to PEVD. Disconnected TG were easily identified through GLT because it was equal to zero. However, the VED and CR of those disconnected TG varied between 164 and 739 and between 0.27 and 1.10, respectively (Figure 2.5). Therefore, completely disconnected TG presented a large range of VED and CR values and cannot be distinguished from connected TG. 21

43 DISCUSSION The genetic evaluation of bulls tested in central evaluation stations in Ontario, Canada, is currently performed using an individual animal model. With such a model, connections among TG occur through additive genetic relationships. Accurate comparison of estimated breeding values between animals in different groups is necessary to provide reliable ranking of animals across TG. The accuracy of comparison between animals in different TG is higher if groups are well connected. For a bull test station to operate in Ontario some requirements based on minimal number of bulls (12) and minimal number of sires (4) per TG are observed. Nevertheless, results of the current study have shown that these requirements were not sufficient to maintain a high level of connectedness among TG. Kennedy and Trus (1993) stated that PEV of comparisons between animals or average PEV of comparisons between groups of animals (PEVD) should be the basis of the measurement of connectedness. However, computing the PEV matrix is very difficult or impossible for large data sets. Approximate methods for obtaining diagonal elements of the PEV matrix of large data sets have been developed (Misztal and Wiggans, 1988; Meyer, 1989), but they generally do not provide the required off-diagonal elements to obtain PEVD. If obtaining a measure of connectedness through PEVD is not possible, alternative methods could be used to predict PEVD and, consequently, provide a measure of degree of connectedness among management units. Different criteria for measuring connectedness have been proposed in the literature. Wood et al. (1991) compared the effectiveness of different breeding programs for evaluation of pigs in test stations, using only the diagonal elements of the PEV matrix to measure connectedness. Foulley et al. (1992) proposed calculating the ratio of the 22