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Genetic analysis of longevity in the Canadian and New Zealand dairy herds Mwansa, Pius Bwalya 1997

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GENETIC ANALYSIS OF LONGEVITY IN THE CANADIAN AND NEW ZEALAND DAIRY HERDS by PIUS BWALYA MWANSA B.Sc. (Agric), The University of Zambia, 1986 M.Sc, The University of Alberta, 1991  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in FACULTY OF GRADUATE STUDIES (Department of Animal Science)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1997 © Pius Bwalya Mwansa, 1997  In  presenting this  degree at the  thesis in  partial  University of  fulfilment  of  the  requirements  for  an advanced  British Columbia, I agree that the. Library: shall make it  freely available for reference and study. I further agree that permission for extensive copying  of this thesis for  department  or  by  his  scholarly purposes may be granted  or  her  representatives.  It  is  by the  understood  that  head of copying  my or  publication of this thesis for financial gain shall not b e , allowed without my written permission.  Department of  A-r-UrViA<-  S^tr^O?  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  ABSTRACT Two longevity measures, duration of productive life (DP) and duration of total life (DT) were investigated. Within each of these measures, two traits were defined based on the definition of failure of the cows. The first definition considered cows to have failed only if they were disposed of due to involuntary reasons giving functional duration of productive life  (/DP). The second considered cows to have failed  regardless of whether disposal was voluntary or otherwise yielding true duration of productive life  (rDP). The  same scheme was followed when defining functional duration of total life (/DT) and true duration of total life  (/DT). These four traits were studied by modeling the hazard rate of daughters of Canadian and New Zealand Holstein sires. The data used was derived from a field trial involving the mating of 20 proven Holstein sires from each country to Holstein cows in 20 New Zealand and 10 Canadian herds. The New Zealand data set consisted of 898 daughters of Canadian and New Zealand sires. In Canada, 239 daughters of Canadian and New Zealand sires were available. In order to measure the cow's true and functional duration of productive and total life, Cox's proportional hazard models were constructed In both Canada and New Zealand, daughters of Native sires had lower hazard rates. The results suggested that daughters of imported sires were culled at higher rates relative to daughters of native sires. Heritabilities for the traits studied were estimated and ranged from 0.03 to 0.29. The results from analysis of genetic correlations between the four traits in the same environment suggested that it is not the measure of longevity that is important but the definition of failure in the Canadian environment. Genetic correlations between the same traits in different environments suggested the existence of genotype by environmental interactions at both macro and micro levels suggesting that sires must be evaluated for superiority of survival traits in the environment within which their progeny are expected to perform.  ii  TABLE OF CONTENTS  ABSTRACT  ii  TABLE OF CONTENTS  iii  LIST OF TABLES  vi  LIST OF FIGURES  vii  ACKNOWLEDGMENT DEDICATION  viii «  .  ix  1. CHAPTER ONE: LONGEVITY ANALYSIS IN DAIRY HERDS: THEORETICAL AND PRACTICAL ASPECTS  1  1.1 INTRODUCTION  1  1.2 MEASURES OF LONGEVITY AND THEIR DEFINITIONS  2  1.2.1 Measures ofLongevity  2  1.2.2 Definitions OfLongevity Measures  4  1.2.3 Definition Of Failure  4  1.2.4 Voluntary And Involuntary Disposal  6  1.2.5 Censoring In Longevity Data  7  1.2.6 Survival Analysis  9  1.3 CONCLUDING REMARKS  19  1.4 REFERENCES  21  1.5 APPENDIX 1. CATEGORIES FOR DISPOSAL REASONS IN THE CANADIAN AND NEW ZEALAND RECORDING SCHEMES  23  2. CHAPTER TWO: COMPARISON OF THE FREQUENCY OF REASONS FOR DISPOSAL AND HAZARD RATES OF DAUGHTERS OF CANADIAN AND NEW ZEALAND SIRES IN THE HOME AND FOREIGN ENVHtONMENTS  25  2.1 ABSTRACT  25  2.2 INTRODUCTION  26  2.3 MATERIALS AND METHODS  27  2.3.1 Definition OfLongevity Traits Studied 2.3.2 Source Of Data 2.3.3 Statistical Analysis 2.4 RESULTS AND DISCUSSION  27 28 29 31  2.4.1 Frequency Distribution Of Reasons For Disposal  31  2.4.2 Hazard Curves Within Production Environments And Sire Groups  34  2.5 CONCLUDING REMARKS  42  2.6 REFERENCES  44  2.7 APPENDIX 2. THE DISTRIBUTION OF DAUGHTERS PER SIRE IN EACH ENVIRONMENT  46  3. CHAPTER 3. HERITABILITY ESTIMATES OF MEASURES OF LONGEVITY USING FAILURE TIME ANALYSIS IN CANADIAN AND NEW ZEALAND EXPERIMENTAL DAIRY HERDS  47  3.1 ABSTRACT  47  3.2 INTRODUCTION  48  3.3 REVIEW OF VARIANCE COMPONENT ESTIMATION METHODS  51  3.4 MATERIALS AND METHODS  54  3.4.1 Source Of Data  54  3.4.2 Statistical Analysis  55  3.5 RESULTS AND DISCUSSION  57  3.6 CONCLUDING REMARKS  60  3.7 REFERENCES  63  iv  4. CHAPTER FOUR: GENETIC CORRELATIONS AMONG MEASURES OF LONGEVITY WITHIN AND ACROSS PRODUCTION ENVIRONMENTS  66  4.1 A B S T R A C T  66  4.2 I N T R O D U C T I O N  67  4.3 M A T E R I A L S A N D M E T H O D S  68  4.3.1 Definition OfLongevity Traits Studied 4.3.2 Source Of Data  68 68  4.3.3 Statistical Analysis 4.4 R E S U L T S A N D D I S C U S S I O N  69 71  4.4.1 Genetic Correlations Among Measures Of Longevity Within And Between Production Environm 4.4.2 Genetic Correlations Estimated Within Environment And Sire Group  72  4.5 C O N C L U D I N G R E M A R K S  75  4.6 R E F E R E N C E S  76  5. CHAPTER FIVE: GENERAL DISCUSSION  78  5.1 CONCLUSIONS  78  5.2 THEORETICAL A N D PRACTICAL CONSIDERATIONS IN LONGEVITY ANALYSIS  78  5.3 DISPOSAL REASONS A N D HAZARD RATES  80  5.4 HERITABILITY ESTIMATES  80  5.5 GENETIC CORRELATIONS  82  5.6 GENERAL REMARKS  83  5.7 REFERENCES  86  v  LIST OF TABLES T A B L E l-l.  ACCOUNTING FOR CENSORING IN SURVIVAL DATA  10  T A B L E 1-2. IGNORING (EDITING) CENSORED OBSERVATIONS  11  T A B L E 1-3. CONSIDERING CENSORED AND UNCENSORED OBSERVATIONS ALIKE  11  T A B L E 2-1. FREQUENCIES OF REASONS FOR DISPOSAL IN THE CANADIAN ENVIRONMENT  32  T A B L E 2-2. FREQUENCIES OF REASONS FOR DISPOSAL IN THE N E W Z E A L A N D PRODUCTION ENVIRONMENT  33  T A B L E 2-3. SUMMARY STATISTICS OF FAILURE TIMES FOR THE MEASURES OF LONGEVITY IN THE CANADIAN PRODUCTION ENVIRONMENT WITHIN SIRE GROUPS  34  T A B L E 2-4. SUMMARY STATISTICS OF FAILURE TIMES FOR THE MEASURES OF LONGEVITY IN THE N E W Z E A L A N D PRODUCTION ENVIRONMENT WITHIN SIRE GROUP  35  T A B L E 2-5 T H E KRUSKAL-WALLIS H TEST OF THE EQUALITY OF THE HAZARD CURVES BETWEEN SIRE GROUPS IN THE CANADIAN AND N E W Z E A L A N D PRODUCTION ENVIRONMENTS FOR A L L MEASURES OF LONGEVITY  41  T A B L E 3-1 SIRE GROUP SOLUTIONS FOR MEASURES OF LONGEVITY WITHIN PRODUCTION ENVIRONMENTS  57  T A B L E 3-1. LIKELIHOOD RATIO TESTS FOR THE EFFECT OF SIRE  59  T A B L E 3-2 HERITABILITY ESTIMATES FOR THE MEASURES OF LONGEVITY  59  T A B L E 4-3 GENETIC CORRELATIONS AMONG MEASURES OF LONGEVITY USING BOTH SIRES GROUPS WITHIN ENVIRONMENT  72  T A B L E 4-4 GENETIC CORRELATIONS AMONG MEASURES OF LONGEVITY FOR CANADIAN SIRES T A B L E 4-5 GENETIC CORRELATIONS FOR N E W Z E A L A N D SIRES (CALCULATED WITHIN ENVIRONMENT AND STRAIN OF SIRE)  1  73  PRODUCTION 74  vi  LIST OF FIGURES FIGURE 1-1. T H E EFFECT OF THE PRESENCE OR ABSENCE OF CENSORED OBSERVATIONS ON THE ESTIMATED SURVIVAL CURVE  12  FIGURE 2-1. CUMULATIVE HAZARDS OF DAUGHTERS OF CANADIAN AND N E W Z E A L A N D BULLS FOR DURATION OF PRODUCTIVE LIFE (DP)  IN THE CANADIAN ENVIRONMENT  37  FIGURE 2-2. CUMULATIVE HAZARDS OF DAUGHTERS OF CANADIAN AND N E W Z E A L A N D BULLS FOR DURATION OF TOTAL LIFE (DT)  IN THE CANADIAN ENVIRONMENT  38  FIGURE 2-3. CUMULATIVE HAZARDS OF DAUGHTERS OF CANADIAN AND N E W Z E A L A N D BULLS FOR DURATION OF PRODUCTIVE LIFE (DP)  IN THE N E W Z E A L A N D ENVIRONMENT  39  FIGURE 2-4. CUMULATIVE HAZARDS FOR DAUGHTERS OF CANADIAN AND N E W Z E A L A N D BULLS FOR DURATION OF TOTAL LIFE (DT)  IN THE N E W Z E A L A N D ENVIRONMENT  40  vii  ACKNOWLEDGMENTS I would like to first express my gratitude to my supervisor, Dr. R. G. Peterson for his guidance throughout the period of this study. I also wish to thank my other supervisory committee members, Drs. K. M . Cheng, J. Shelford, and G. Namkoong. My sincere thanks to the Head of the Department of Animal Science for putting all the facilities of the department at my disposal. Thanks to the Canadian International Development Agency (CIDA) for funding the entire study. I would also like to sincerely thank Dr. V. P. Ducrocq for his unfailing help through numerous email communications as well as providing me with the software needed for the data analysis. Finally my deepest thanks to my wife (Eugenia) and my two daughters (Chanda and Mwansa) for their patience and understanding during the course of this study.  viii  In memory of one person I deeply loved:  ANDERSON MAKANI MWANSA  1. CHAPTER ONE: LONGEVITY ANALYSIS IN DAIRY HERDS: THEORETICAL AND PRACTICAL A S P E C T S  1.1 INTRODUCTION  Longevity in dairy cattle may broadly be defined as length of life of animals in a given herd. The length of an animal's life is a function of the disposal process on the farm. The decision to cull an animal from a herd is usually based upon comparative evaluations of all the animals in the herd given a disposal criterion. Reasons for disposal of animals are many and varied (Allaire et al., 1977). It is generally accepted that when the disposal decision is entirely under the control of the farmer, disposal is termed voluntary and involuntary otherwise. Voluntary disposal is practiced when a cow is considered less profitable to the enterprise than her potential replacement. Both voluntary (e.g. low milk production) and involuntary disposals (e.g. reproductive problems) may occur much earlier than senescence. It is not, however, easy to distinguish between voluntary and involuntary reasons for disposal (Ducrocq, 1987). There is clearly, a need for more precise definitions of reasons for disposal and classifications of the same. From a genetics stand point, interest has mostly been in the reduction of involuntary disposal of otherwise profitable cows. This is because, animal breeders have been interested in improving the ability of the cows to delay involuntary disposal thereby making room for voluntary disposal. There is interest in determining the genetic component of variation in the ability of the cows to delay involuntary disposal to assess potential for direct selection for this trait. Although milk, fat and protein yield are the most important dairy cattle selection criteria, routine genetic evaluation schemes have tended to include secondary traits.  These so called  secondary traits include type traits such as dairyness, feet and legs, stature etc. It is hoped that improvement of these type traits would reduce the probability of early disposal of, otherwise, highly productive cows. The rewards from such efforts are questionable since genetic correlations between type traits and direct measures  1  of longevity are usually low (Schaeffer and Burnside, 1974). Benefits of a correlated response in longevity by selection based on first lactation milk yield is doubtful since an expected attendant reduction in involuntary disposals has not been observed (Ducrocq, 1987). Currently, there seems to be much more emphasis put on direct selection for longevity. This has necessitated renewed research in defining the various measures of longevity as well as improvements in the statistical analysis procedures of the same.  This  chapter outlines theoretical and practical considerations in the analysis of survival data as well as problems commonly encountered in the analysis of longevity. It focuses on problems encountered in defining reasons for disposal, deciding on whether disposal is voluntary or not, defining various measures of longevity and the meaning and causes of censoring in survival data in dairy herds. Appropriate methodology for longevity 1  analysis is discussed. Censoring as a unique characteristic of survival data is especially highlighted.  1.2 MEASURES OF LONGEVITY AND THEIR DEFINITIONS 1.2.1 Measures of Longevity In this manuscript, longevity is defined very broadly, as the animals' ability to delay disposal. The term 'animal' is preferred to 'cow' in this broad definition since 'cow' infers attainment of reproductive age in animal breeding. Longevity measures may be grouped into two categories: Measures related to time and those related to life time milk production. Within each of the categories, some measures may be categorical and others continuous. The following gives a description of some of the numerous measures of longevity related traits encountered in the field of animal breeding. Ducrocq (1987) gives an excellent and exhaustive review of most, if not, all of these.  Censored data are observations on animals whose culling dates are not unequivocally known, e.g. animals still alive at the end of an experiment or at time of performing a survival analysis for field data. 1  2  1.2.1.1  Measures  Related To Time  Most continuous measures of longevity characterize the difference in time between some origin point (birth, date of first freshening, etc.) and disposal date. Examples are age at disposal or duration of total life (DT) and length of productive life or duration of productive life (DP). DT is the difference in time between birth date and disposal date on any desirable scale. DP is the difference between age at disposal and age at first freshening. To carry out a genetic analysis of these two longevity measures, disposal information is necessary. Note that DP and DT are one and the same trait in the event that birth dates and pre-first calving disposal information is missing. This is because in the estimation of the survival function, the same (post-first calving) information would be utilized for both measures. Categorical longevity traits are described by defining opportunity groups. This is an effort to circumvent the problem introduced by censored records. An opportunity group includes all animals which have been given the same opportunity to freshen a fixed number of times or to be still alive at a fixed age (Ducrocq, 1987). Threshold points used to define the opportunity groups were chosen to fall between date of first freshening and disposal. Examples are survival to 48 months and survival to 72 months (Everett et al., 1976). The coding of the binary trait then involves the choice of 'awarding' a one to those animals that survived beyond the threshold and a zero otherwise. The opposite coding scheme introduces no new concept.  1.2.1.2 Measures  Related To Life Time Milk  Production  Longevity measures related to milk production may be closely related to DP and/or DT. Lifetime milk yield or production (Gill and Allaire, 1976), production (milk, fat, protein yield) per day of total life and production (milk, fat, protein yield) per day of productive life are continuous traits related DT and DP. Number of completed lactations is a categorical trait closely related to DP.  3  1.2.2 Definitions Of Longevity Measures For any measure of longevity to make sense, it is not only necessary that the unit of measure be clearly defined, but that the definition of failure be clearly defined as well. For instance, when DP is the longevity measure of choice, it is necessary to describe what constituted failure of the cows in the scheme. In animal breeding the description and determination of failure is a source of much discussion (Ducrocq, 1987) leading to a fair number of longevity measures prevalent in the literature.  1.2.3 Definition Of Failure In this manuscript, the condition that ends the life (total or productive) of a cow will be referred to as a reason for disposal. Many authors (Van Arendonk, 1986; Ducrocq, 1987; Ruiz, 1991) have tried to group reasons for disposal into two categories: involuntary (when the reason for disposal is beyond the control of the farmer) or voluntary (when the decision to cull depends only on the farmer's choice). This classification is largely subjective. The problem is not only due to lack of a clear distinction between voluntary and involuntary disposal, but also due to the lack of exclusivity in the definitions of reasons for disposals. Indeed, cows may leave the herd for multiple reasons. Some of the recorded individual reasons for disposal of North American Holsteins are that the animal has been sold for dairy or sold for beef, or had poor conception, cystic ovaries, abortion, calving problems, mastitis, bad temperament, slow milker, bloat, udder size, poor legs, injury, poison, and old age. For a better understanding of the reasons for disposal, Allaire et al., (1977) partitioned them into the following categories: REPRODUCTION (poor conception, cystic ovaries, abortion), LOW PRODUCTION, MILKING CHARACTERISTICS (temperament, hard milker), MASTITIS, DISEASE (pneumonia, diarrhea), ACCIDENT (injury), TYPE (udder, legs, size), SURPLUS (sold for dairy purpose, sold for beef) and GENERAL HEALTH (small size, premature calf).  4  In Canada, the reasons for disposal are defined as 12 different categories, thus: 1. Dairy purposes 2. Export 3. Low production 4. Slow milker 5. Mastitis 6. Udder breakdown or teat injury 7. Feet and leg problems 8. Reproductive problems 9. Other sickness or disease 10. Other injury 11. Old age 12. Died In New Zealand, the data system defines 50 different categories! (See appendix 1). The higher the number of categories the more specific the description one would expect. This may not necessarily be true. In New Zealand, there is a category (among the 50) called culled/died injury or accident and another called injured, which causes obvious overlap or the appearance of such in the grouping of reasons for disposal. The same may be said for the situation in Canada for category 5 and 6 since mastitis frequently leads to udder breakdown and mastitis often follows injury. The purposes of recording individual reasons for disposal may be considered to be two fold: herd management purposes and genetic evaluation for longevity. The frequency of occurrence as well as the genetic variance for individual reasons for disposal is generally too low for the estimation of sire breeding values for these traits. Category 12 in the Canadian scheme is not very informative. The cause(s) of death may be the relevant piece(s) of information in the management and disposal decision making process.  5  Depending on the intended use of the disposal information, certain categories may be more informative when combined. For example, categories 1 and 2 are useful singly for management purposes but make better sense when combined as a single category for survival analysis. Reasons for disposal are, especially, difficult to interpret when a cow is removed due to multiple reasons. This may indeed be the more realistic condition in practice. This indicates that the various longevity measures in the literature are only as good as the recording schemes from which the data came and the ability of the researchers in clearly defining failure. This also emphasizes the need to determine the relationships between these various measures and definitions of longevity both within and between environments.  1.2.4 Voluntary And Involuntary Disposal For the purpose of estimating the genetic merit of the ability of animals to delay involuntary culling, a more useful categorization of these reasons for disposal is, perhaps, that breaking them into only two groups; whether they indicate voluntary or involuntary disposal. As it was mentioned earlier, it is generally accepted that when the disposal decision is entirely under the control of the farmer, disposal is termed voluntary and involuntary otherwise The distinction between voluntary and involuntary disposal is then necessary for an informative description of the disposal process. This distinction is not easy in practice since the realities of most disposals are due to multiple and, sometimes, interconnected causes. Expressed in another way, it is hard to envision a situation in which any disposal is indeed under the total control of the farmer since the overall economic environment usually dictates the decision. Even voluntary disposal for heifer replacement is a long-term economic decision on the part of the dairyman. However, there is an implied freedom of choice in the notion of voluntary disposal when the farmer is not 'forced' to get rid of the animal. The distribution of reasons for disposal between these two categories is somewhat arbitrary. This underscores the importance of an accurate and unique definition as well as recording of any reason for disposal. Any ambiguity in the definition and recording of reasons for disposal in this initial and early stages of the record keeping system renders, more or less, difficult, any attempt to group them into voluntary and involuntary categories, in any  6  attempt to model and analyze the genetic merit of animals to delay involuntary disposal, the distinction and assignment of reasons for disposal to and between these two categories is pivotal. This partition is not only intuitively and conceptually important, the whole notion of modeling the genetic merit of the ability of an animal to delay involuntary disposal rests squarely on it.  1.2.5 Censoring In Longevity Data Since longevity suggests a duration, measures of longevity in animal breeding usually characterize a difference in time units between the date of disposal and some origin point. As pointed out by Robertson and Barker (1966), productive lifespan is an awkward measure. It takes years to observe the full lifespan of an animal. Consequently, longevity measures, like most failure time data are subject to censoring. In a direct fashion, censoring occurs when some animals are still alive at the termination of an experiment (or some are still alive at the time of analysis for herd life) or their records have currently disappeared from the file and therefore incomplete (e.g. missing a last lactation). Over and above that, differences in the definition of which cows were actually disposed of may yield differing amounts of censoring in the same data set (Smith, 1983; Smith and Quaas, 1984). It is the existence of these incomplete observations that has presented animal breeders with interesting statistical problems with regard to handling of such records (Smith and Allaire, 1986, Ducrocq, 1987, Ruiz, 1991) in the analysis of longevity. In biological studies, there are many causes for and types of censoring (Lagakos, 1979). Ruiz (1991) explained that, sometimes the starting point is unknown but preceded the recorded value, or the end point is known to be above the recorded value (animal is still alive at time of data analysis in animal breeding), or both start and end point of a time interval may not be known exactly. He referred to these as left, right and center censored observations, respectively. Right censored observations are the most commonly encountered in animal breeding applications. For a complete treatment of censoring in endurance measures, the interested reader is referred to an excellent discussion by Lagakos (1979).  7  1.2.5.1 Censoring  And Statistical  Analysis  Until recently, longevity and associated survival measures in dairy cattle have generally been analyzed using linear models that either do not accommodate censored data well or ignore such data altogether. In some cases, longevity has also been analyzed as an all-or-none trait (Van Doormaal et al., 1985) which leads to a general loss of information. Even if statistical methods dealing with censored data have been in the literature and used in some fields ( Kaplan and Meier, 1958; Cox, 1972), due to their complexity, they have not been used in animal breeding until recently (Smith, 1983). Most animal breeders have tried to accommodate censoring in survival data by either ignoring censored observations  or defining  thresholds/opportunity groups in the linear model construct (Van Doormaal et al., 1985). Consequently, longevity traits like survival to given ages or lactations have been analyzed in a binary framework. In the linear model methodologies, continuous longevity traits such as DT and DP were analyzed by ignoring censored observations on animals still alive or those whose records disappeared when they were still on test. Ducrocq (1987) explained that the definition of opportunity groups (in linear model methods) removes problems of dealing with censored records while at the same time suffers substantial drawbacks. He pointed out that the thresholds have not been uniquely defined while forcing observations to be binary leads to an obvious loss of information. For example consider stayability to 48 months: within this opportunity group, an animal's record is 1 if it is no longer in the herd at 48 months or 0 otherwise. It does not matter whether an animal was culled a day or 24 months before this threshold. This demonstrates a serious loss of information due to a scale problem. Gianola (1980) described and encouraged the use of logistic regression techniques as being more adequate statistical methods that account for the binary nature of survival data. De Lorenzo and Everett (1986) used logistic regression methods to predict sire effects for probability of survival. These methods, however, do not adequately account for censored observations. Gianola (1980) and De Lorenzo (1983) described the consequences of using linear models and the assumption of normality when analyzing binary data. It was noted that the necessary assumption of independence between the mean and variance no longer held true.  8  Smith (1983), Smith and Allaire (1986), Ducrocq (1987) and Ruiz (1991) have demonstrated a powerful statistical method that not only accounts for the presence of censored observations but also takes advantage of the continuous nature of longevity or endurance measures. The tool is generally referred to as Survival Analysis. Smith and Allaire (1986) referred to the statistical technique as Failure Time Analysis.  1.2.6 Survival Analysis Survival data, are subject to random variations and, therefore, form a distribution. This distribution is usually described or characterized by three functions: the survival function, the probability density function and the hazard function. The three functions are mathematically equivalent, if one is given the other two may be derived. Let T denote the survival time. The survival function (S(t)) may be defined as the probability that an animal survives longer than time t . That is: S(t) = P(T>t)  [1.6]  and its nonparametric estimate is A  n-T:  number of animals alive just after t:  [1-7]  S(t)= n —= — t(r)st n - r +1 number of animals alive just prior to t j f  where (n - r^ is the number of animals alive just after tj and (n - r +1) represents the number of animals still ;  alive just prior to tjwith n representing the total number of animals in the data set. With survival times arranged in ascending order, r represents the number of failures in the time interval. It runs through those positive integers for which t(r) <t and t(r) is uncensored. The values of r are consecutive integers l,2,...,n if there are no censored observations. If an observation is censored, the value of r used is that just prior to the censored observation, otherwise it is the rank of the observation. The tabular form of explanation may be more clear. Three of the most common scenarios are considered. (1) The researcher chooses to account for censored observations (Table 1-1) (2) The censored observed are ignored/edited from the data files (Table 1-2) (3) Censored and uncensored observations are considered the same (Table 1-3)  9  For the first scenario an example data set of Table 1-1 with ten hypothetical animals is given. The data are first arranged in ascending order with respect to survival times and ranked sequentially. Survival at t = 0 is ;  taken as 1 or 100%. The data is then coded for censoring status (column 4) with 0 indicating censored data and 1 otherwise. The reverse coding scheme introduces no new principle. If no animal had a censored record, r would run as 1,2,3..., 10 consecutively. The estimation of the survival function (column 6) accounts for the presence of animals with censored observations within the time interval under consideration.  Table 1-1. Accounting for censoring in survival data Cow number  Survival Time In  Rank(i)  Days (ti)  Censoring  r  indicator (8;)  s(t)-n "' n  t(r)<t n - r  +1  (Transformed observations) -  0  -  -  -  1.00  303  250  1  1  1  0.90  102  370  2  0  -  0.81  403  385  3  1  2  0.72  206  390  4  0  0.64  300  400  5  1  3  0.56  400  500  6  1  4  0.48  500  600  7  0  ~  0.41  600  700  8  1  5  0.34  700  800  9  0  ~  0.28  800  920  10  1  6  0.22  n=10  Table 1-2. depicts the second scenario in which the animals with censored observations have their records edited from the data files. Animals numbered 102, 206, 500 and 700 (Table 1-1, column 1) are not considered for analysis and thereby reduces the data set to six animals. The ordering, ranking and coding of the data set may then look like Table 1-2. The last column indicates an overestimation of the failure rate or underestimation of the survival rate relative to that of the first scenario. The third scenario is demonstrated in Table 1-3. In this case, all ten animals are retained for analysis. In this scheme, r runs consecutively through  10  10 (Table 1-3, column 5). Again, column 6 of Table 1-3 indicates an underestimation of the survival rate relative to thefirstscenario. It is obvious that ignoring censored observations or considering them as actual failure times will affect the shape of the survival curve.  Table 1-2. Ignoring (editing) censored observations Cow number  Survival Time In Days  Rank (i)  Censoring indicator (5;)  r  s(t>= n  n-r  t(r)st n - r  Oi)  +1  1.00  0 303  250  1  0.83  403  385  2  0.66  300  400  3  0.50  400  500  4  0.33  600  700  5  0.17  800  920  6  0.00  n=6  Table 1-3. Considering censored and uncensored observations alike Rank(i)  Cow  Survival Time In  number  Days (tj)  -  0  -  303  250  1  102  370  403  Censoring indicator (8  r  s(t)=n t(r)<t n - r  -  1.00  1  1  0.90  2  1  2  0.80  385  3  1  3  0.70  206  390  4  1  4  0.60  300  400  5  1  5  0.50  400  500  6  1  6  0.40  500  600  7  1  7  0.30  600  700  8  1  8  0.20  700  800  9  1  9  0.10  800  920  10  1  10  0.00  +1  n=10  11  The foregoing hypothetical example data set is simplistic but serves to demonstrate the effects of various treatments of survival data sets. The graphical representation of scenarios are depicted in Figure 1-1 demonstrating the effect of censoring on the estimated survivor function. Figure 1-1 shows that survival rate is underestimated by either considering censored observations as actual dates of disposals or ignoring (editing them out) in the calculation of any measure of longevity. When some observations are tied, the likelihood construction of the hazard is a little complicated (Smith, 1990) and is explained later in this section. Note that when censored records are accounted for, survival rate at the end of an experiment or time of survival analysis is not necessarily zero as is with the case when there is failure to account for censored data (Figure 11).  -4—Accounting for censored observations -a—Ignoring censored observations -±—Censored and uncensored observations treated alike o o  8.  0  200  400  600  800  1000  Time in days  Figure 1-1. The effect of the presence or absence of censored observations on the estimated survival curve  The survival function can also be seen as: S(t) = l-F(t)  [1.1]  where F(t) is the probability that an animal fails before time t or the cumulative distribution of T or Cumulative Density Function.  12  The density function /(?) is defined as the limit of the probability that an animal is culled in the short interval tj to tj +At divided by At as At tends to zero. It is the slope of the cumulative distribution function at time t. The graph of /(t) is the density curve with the area under the curve equal to 1. It is also a nonnegative function, since: /(0>0forallti >0 /(O = 0foralltj <0  Statistically, this density function may be defined as: lim  /(t)= t->o  P(an animal dying/being culled in the interval (tj, tj + At)) 77  A  [1-2]  It is also known as the unconditional failure rate. This function is estimated as: y  number of animals culled in the interval beginning at time t total number of animals in the interval  [13]  ;  if there are no censored observations. When some observations are censored, the probability density function A  A  is estimated as thefirstderivative of l-S(t) where S(t) has been estimated as described previously for data with censored observations. Fundamental to the understanding of survival analysis is the concept of a hazard function denoted by O(t). The hazard function <J>(t) of survival time T gives the conditional failure rate. It is defined as the probability of failure during a very small time interval, assuming that the animal has survived to the beginning of the interval. In other words, it is the limit of the probability that an animal fails in a very short interval, tj to tj +At, given that the animal has survived to time tj. It, therefore, equals the conditional probability function for failure at time t, (or age t j) given non failure to t . It can simply be thought of as a f  failure rate. Thus, the relative size of  <X>(t) compared to some other O(t') reflects the relative risk of  failure at time t compared to some other time t'j. Defined in terms of the cumulative distribution function ;  F(t) and the probability density function /(t) , the hazard function <I>(t) is as follows: 13  ®(t)= /(t)/[l-F(t)] or  [1.4]  <£(t) = /(t)/S(t)  [1.5]  since l-F(t) is S(t). Plainly, the hazard function is a ratio of the density function to the survival function. In a parametric sense, using the Cox (1972) regression model, the hazard function has a particular form. The hazard rates are affected by factors independent of time. The hazard for the i  t n  observation may be  modeled as: [1.6]  Oi(t) = O (t)exp(Xb) 0  where, <£j (t) is a positive function of t , Oo (t) is a positive function of t defined as the baseline hazard (It is the hazard when the effects of all the explanatory variables in the model are zero), X is a known incidence matrix, b is a vector of unknown estimates for both the fixed effects and random effects. As may already be evident, under the Cox model (Cox, 1972), each observation has a characteristic multiplier that alters a common baseline failure rate Oo 0) • Therefore, O, (t) may be described as product of a baseline hazard function O (t), representing the aging process, and a positive function of the explanatory 0  variables assumed to influence disposal rate (Ducrocq, 1991). Because of this, [1.6] is referred to as the proportional hazards model. Model [1.6] is popular since b can be estimated without ever knowing or estimating <J) (t). They are estimated using what is known as a partial likelihood, a part of the full likelihood 0  in which the baseline hazard function does not appear. This is because only the marginal information is used. The likelihood constructed from the marginal information is invariant to cl> (t). When there is no censoring, 0  the marginal information is equivalent to the rank information. For example, if we observe the uncensored observations t,=250, t=300 and t=400, the marginal information is Ti<T <T where T's are the underlying 2  3  2  3  failure times. With censoring, the marginal information corresponds to what is known about the rank information of underlying failure times. If t, is censored, the marginal information is T <T , since we lost the 2  3  first observation and the underlying failure time T, may have any rank order. If t is censored, the marginal 3  information is still T,<T <T . However, if t is censored, the marginal information is T,<T and T,<T . In a 2  3  2  2  3  continuous model, failure times will be distinct. However, because measurements are not perfect and the scale of measure may be crude, failure times will occasionally be tied in practice. When this happens, our  14  knowledge about the correct rank order is incomplete. In the construction of the likelihood, marginal information is used to represent ties. If ^=250, t=300 and t=300 are the observed failure times, the marginal 2  3  information is the union of T,<T <T and T,<T <T (Smith, 1990). This information is what is incorporated 2  3  3  2  in the likelihood. Generally, the marginal likelihood is Pr[M] where M represents the marginal information and Pr[M] is the probability of event M. When there are no ties, Pr[M] is calculated under [1.6] as: m  L(b) = nexp(x]b)/[ Zexp(x|b)], j=l  [1.7]  i R(tj) £  where t,,^,...,^ (m<n) are assumed to be the uncensored failure times, and R(tj) is the index set of all subjects at risk prior to tj. A subject is at risk prior to tj if it survived beyond tj or failed at tj. The marginal likelihood is difficult to evaluate when there are ties. Peto (1972) suggested the following approximation to handle ties in the data set. L(b) = flexp(w;b)/[ I i=l  exp(x]b)]- i D  [1.8]  J£R(VJ)  where, v,,v ,...,v are distinct uncensored failure times, Dj is the number of failures at Vj and 2  r  jrf-(vi)  where F(Vi) is the index set of all subjects that failed at Vj. Without censoring [1.8] is identical to [1.7]. Model [1.6] was suggested and used by Smith and Quaas (1984) in a failure time analysis of bull progeny groups for productive lifespan. In this investigation b consisted of either herd effects (nested in the year of birth) and sire effects (cross-classified with years of birth) or had a genetic group effect considered as well. Herd and sire effects were assumed to have null means. An extension of model [1.6] was proposed by Ducrocq (1987). He argued that the hazard function for a measure of longevity can be approximated by a Weibull distribution: <l> (t) = ® (®ty-  1  0  p  [1.9]  which is a generalization of the exponential model [1.6]. Then a fully parametric Weibull model was defined as:  15  cD(t) = <D (Ot)^ exp(Xb)  [1.8]  1  p  where &0(i) = cj) (0 t)''" is a Weibull baseline hazard function, with parameters <S> and p ; X and bare as 1  p  previously described in [1.6]. Covariates in b can be quite elaborate (Ducrocq, 1991; Ruiz, 1991).  1.2.6.1 Problems  Related To The Definition  Of Failure In Measures  Of  Longevity  In the analysis of measures of longevity, one major problem is encountered. It is almost always difficult to determine which animals actually failed due to inadequacies in the recording schemes with regard to reasons for disposal. To get a better understanding of this problem, Ducrocq (1987), identified two types of longevity as being of interest to animal breeders. He explained that true longevity is the ability of an animal to delay culling (involuntary or otherwise) and functional longevity is the ability of an animal to delay involuntary culling. It is then obvious that the two definitions yield different amounts of censoring. Even though Ducrocq (1987) only defined DP in terms of true and functional terms, other measures of longevity such as DT can also be defined similarly. Smith and Quaas (1984) attempted a similar grouping strategy within the functional definition of failure in an attempt to account for the peculiarities in the culling reasons. In one case, a cow was considered culled if the culling reason of her last lactation indicated "died or sold for beef. The other definition considered a cow to have failed if the culling reason indicated "died, sold for beef or if the herd remained on test but the cow's record disappeared from the datafiles".The second definition is similar to Ducrocq (1987) functional form. In both cases, animals "sold for dairy purposes" were taken to have censored records. Another level of complexity is added to the argument when researchers consider appropriate adjustment factors for the various measures of longevity with the aim of removing biases in the parameter estimates. The importance of these adjustment factors may depend on the longevity measure of choice. For instance, it has been suggested that DT should be adjusted for age at first freshening to avoid underestimating the breeding values of bulls with fast maturing daughters (Ducrocq, 1987). This adjustment is not considered necessary with regard to DP. However, there is a danger of overestimating the breeding value of bulls with poor survival to first freshening when culling occurring between birth andfirstfreshening is ignored in modeling  16  DP. Dekkers and Jairath (1994) concurred with Ducrocq (1987) that adjustment for milk production or herdmate deviation for milk is desirable since culling for low production is based on the cows' relative ranking for milk production within herds. This adjustment, however, is not necessarily relevant to the definition of functional longevity when it is defined as the ability of the animal to delay involuntary culling. Only animals that were culled due to involuntary reasons are considered to have failed in this definition. Culling due to reasons such as low milk production, bad temperament, milking speed and dairyness are voluntary and, therefore, not considered failures in the functional definition. In this study this definition was strictly adhered to. From the foregoing, it is easy to see that animal breeders are far from resolving the problems presented by genetic evaluations for longevity. In summary, we may conceive the problems to be: (1) Related to the scale of measure (binary vs. continuous) (2) Related to the relevant segment of the animal's life (DP vs. DT) (3) Related to definition of failure (voluntary vs. involuntary or functional vs. true) of the measure (4) Related to relevant adjustment factors for the longevity measure of choice (e.g. adjusting for age at first freshening vs. none; adjusting for total milk production vs. herdmate deviation for milk vs. none) (5) Related to model choices with regard to linear or non-linear analytical models Even if the above is not an exhaustive list, the many possible alternatives (combinations) that are possible in defining a measure of longevity becomes a mathematical permutations problem. In general, most animal breeders agree that since it is not overly difficulty to get a continuous measure of longevity within most current milk recording schemes, use of a categorical or binary measure may not be justified due to the attendant loss of information. With regard to problem number 2 (above), most researchers prefer DP over DT. They point out that this is the income generating portion of a dairy cow's life. The other reason that has been advanced in support of DP is the lack of the need to adjust for age atfirstfreshening. This is due to the fact that, in most North American milk recording schemes, the existence of a record is conditional upon initiation of afirstlactation record. Therefore, information on culling beforefirstcalving is consequently not available. In such situations DP and DT are essentially the same trait since information used in estimating the hazard function would be the same for both measures. Advocates for the use of DT argue that the profitability  17  of a cow may be known only after accounting for income and expense items over the entire cow's life time. Most of the studies on profitability of cows in the dairy industry ignore income and expense items realized before first calving. In thefield,the condition or form of the records may dictate the longevity measure chosen. Several scenarios may arise infielddata. Unlike data collected from planned experiments,fielddata may lack some information needed for analysis; 1. A larger proportion of animals may be missing the date of birth. In cases like that, and assuming date of first freshening as well as disposal are available, it is best to consider DP life for survival analysis. DP life is said to intrinsically account for age atfirstfreshening (Ducrocq, 1987) or more precisely ignores culling occurring beforefirstfreshening. One of the greatest short comings of DP as a measure of longevity is that it is conditional upon the cows' ability to start a first record. An explicit adjustment for age at first freshening (if desired) is not possible under this situation since its calculation is impossible without the date of birth. Since culling before first freshening is ignored, breeding values of bulls with poor daughter survival to first freshening would be overestimated. 2. A higher proportion of records may be missing entries on date offirstfreshening. Assuming both birth date and disposal date are available, DT would be the trait available for analysis. This is the most difficult trait to handle since, in the absence of date at first freshening, DT can not be adjusted for age atfirstfreshening. As has been pointed out earlier, breeding values for survival of bulls with early maturing daughters may be underestimated. The biggest problem with DT is thatfielddata comes from milk recording information. This means that records of animals that do not start a lactation are usually missing. Nonetheless, this condition may occur in otherfielddata, especially in countries where the development of record keeping systems is still in its infancy. 3. The best situation is when all the information mentioned above is available. Then DT may be modeled with an adjustment for age at first freshening. 4. Consequently, DP life may also be modeled with an adjustment for age atfirstfreshening if desired.  18  1.3 CONCLUDING REMARKS After the forgoing exposition, the interest of this study was in the estimation of hazard curves as well as genetic parameters and genetic relationships between the alternative measures of longevity under different definitions of failure within a given production environment. This may answer the question of whether or not the measures and definitions of failure reflect one and the same trait. The next question may then be to investigate these measures and definitions across environments (countries) in the framework of a genotype by environmental interaction at the micro and macro levels. The quantitative genetics model assumes that the 2  3  phenotype of an individual is given by P = G + E in the absence of genotype by environmental interaction, where G is the genotypic value and E is the environmental deviation. When the interaction exits, however, the equation becomes P = G + E + I  GE  where I  GE  is the added interaction term (Falconer, 1989). He explained  that there may be two forms the interaction may take. There may be a change in the order of merit of genotypes measured under different environments. In other cases, a specific difference of environment may have a greater effect on some genotypes than on others without necessity for change of order of merit. Chapter 2 was focused on non-parametric estimation of hazard curves among daughters of bulls from two countries (Canada and New Zealand). Hazard rates within country and sire group were estimated, plotted and compared (tested for homogeneity). Reasons for disposal among daughters of Canadian and New Zealand sires were also compared within production environments. Chapter 3 involved estimation of heritabilities for the various measures of longevity traits defined in the study using appropriate statistical methodologies (failure time analysis) which are extensions of the Cox regression methods. Interest was in demonstrating the existence or lack thereof, of genetic variability for the longevity traits studied. Heritabilities were estimated within production environments (countries) for each of the traits across strain of sire.  2  Differential ranking of animals in different environments  3  Differential ranking of breeds or strains in different environments  19  Chapter 4 was dedicated to the investigation of the existence or lack thereof, of genetic correlations between the various measures (traits) of longevity within production environments and the presence of genetic by environmental interactions for traits studied. Heritabilities estimated in Chapter 3 were used as priors in the estimation of breeding values for the traits studied.  20  1.4  REFERENCES.  1. Allaire, F. R., H. E. Sterwerf, and T. M. Ludwick. 1977. Variations in removal reasons and culling rates with age for dairy females. J. Dairy Sci., 62:254. 2. Cox, D. R. 1972. Regression models and life tables (with discussion). J. R. Statist. Soc, B, 34:187. 3. Gianola, D. 1980. A method of sire evaluation for dichotomies. J. Anim. Sci., 51:1266. 4. Gianola, D., and J. L. Foulley. 1982. Nonlinear prediction of latent genetic liability with binary expression: an empirical Bayes approach. Proc. 2nd World Cong. Genet. Appl. Livestock Prod., Madrid, Spain, 7:293. 5. Dekkers, J. C. M., and L. K. Jairath. 1994. Requirements and uses of genetic evaluations for conformation and herd life. Proc. World Cong. Genet. Appl. Livestock Prod., 17:61. 6. De Lorenzo, M. A. 1983. Non-linear estimation of dairy cow survival to fixed ages. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 7. De Lorenzo, M. A., and R. W. Everett. 1986. Prediction of sire effects for probability of survival to fixed ages with a logistic linear model. J. Dairy Sci., 69:501. 8. Ducrocq, V. 1987. An Analysis of Length of Productive Life in Dairy Cattle. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 9. Ducrocq, V. 1991. Statistical analysis of length of productive life of dairy cows in the Normande breed. 42nd Annual meeting of the European Association for animal production. Berlin. Germany. 10. Everett, R. W., I. F. Keown and E. E. Clapp. 1976. Production and stayability trends in dairy cattle. J. Dairy Sci., 59:1532. 11. Falconer, D. S. 1989. Introduction to Quantitative Genetics. 3rd ed. Longman Scientific and Technical, John Wiley and Sons, Inc., N. Y., 8:135. 13. Freund, J. E., and E. W. Ronald. 1987. Mathematical Statistics. 4th ed. Prentice-Hall, Inc. Englewood Cliffs. N.J. 07632., 1:2.  21  14. Gill, G. W. And F. R. Allaire. 1976. Genetic and phenotypic parameters for a profit function and selection method for optimizing profit in dairy cattle. J. Dairy Sci., 59:1325. 15. Kaplan, E. L., and P. Meier. 1958. Non-parametric estimation from incomplete observations. J. Am. Statist. Assoc., 53:457. 16. Lagakos, S. W. 1979. General right censoring and its impact on the analysis of survival data. Biometrics, 35:139. 17. Peto, R. 1972. Contribution to the discussion on the paper of DR Cox. J R Stat Soc. Ser B, 34:205 18. Robertson, A and J. S. F. Barker. 1966. The correlation betweenfirstlactation milk production and longevity in dairy cattle. Anim. Prod., 8:241. 19. Ruiz, F. 1991. Relationships among Length of Productive Life, Milk Yield, and Probability of United States, Canada, and Mexican Holstein sires in Mexico. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 20. Schaeffer, L. R., and E. B. Burnside. 1974. Survival rate of rested daughters of sires in an artificial insemination. J. Dairy Sci., 57:1394. 21. Smith, S. P. 1983. The extension of failure time analysis to problems of animal breeding. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 22. Smith, S. P., and F. R. Allaire. 1986. Analysis of failure times measured on dairy cows: Theoretical considerations in animal breeding. J. Dairy Sci., 69:217. 23. Smith, S. P., and R. L. Quaas. 1984. Productive lifespan of bull progeny group: Failure time analysis. J. Dairy Sci., 67:2999. 24. Van Arendonk, J. A. M. 1986. Economic importance and possibilities for improvement of dairy cow herd life. Third world congress on genetics applied to livestock production, Lincoln, Nebraska, July 16-22, 1986, IX:95. 25. Van Doormaal, B. J., L. R. Schaeffer and B. W. Kennedy. 1985. Estimation of genetic parameters for stayability in Canadian Holsteins. J. Dairy. Sci., 68:1763.  22  1.5  APPENDIX  CANADIAN  1. CATEGORIES  AND NEW ZEALAND  FOR DISPOSAL RECORDING  1. CANADIAN RECORDING SCHEME 1. Dairy purposes 2. Export 3. Low production 4. Slow milker 5. Mastitis 6. Udder breakdown or teat injury 7. Feet and legs problems 8. Reproductive problems 9. Other sickness or disease 10. Other injury 11. Old age 12. Died  REASONS SCHEMES  IN THE  2. NEW ZEALAND RECORDING SCHEME 1. Abortion  18. Johnes disease  35. Progeny test below standard  2. Bloat  19. Low fertility  36. Salmonella  3. Brucellosis  20. Lame  37. Culled/died sickness or disease  4. Cataract  21. Low production  38. Sucker  5. Calving trouble  22. Leptospirosis  39. Sires proof below standard  6. Daughter conformation  23. Mastitis  40. Surplus to requirements  7. Died - Cause unknown  24 Milk fever  41. Tuberculosis  8. Eczema  25 Empty  42. Unsuitable temperament  9. Feet or leg problems  26 Natural proof below standard  43. Temperament of progeny  10. Failed veterinary examination  27 Unsatisfactory or non-server  44. Traits other than production  11. Grass staggers  28 Old age  45. Unsuitable type  12. Heritable defect in bull  29 Other causes  46. Udder breakdown  13. Heritable defect in dam  30 Other diseases  47. Unknown  14. Heritable defect in progeny  31 Other metabolic diseases  15. Culled/died injury or accident  32 Physical defect in bull  49. Udder breakdown  16. Infertility or poor fertility  33 Physical defect in progeny  50. Feet or leg problems  17. Injured  34 AB proof not up to standard  ,48. Weight gain below standard  24  2. CHAPTER TWO: COMPARISON OF THE FREQUENCY OF REASONS FOR DISPOSAL AND HAZARD RATES OF DAUGHTERS OF CANADIAN AND NEW ZEALAND SIRES IN THE HOME AND FOREIGN ENVIRONMENTS 2.1  ABSTRACT  Frequencies of reasons for disposal were estimated for daughters of Canadian and New Zealand dairy sires within Canada and New Zealand. Hazards curves were estimated for functional duration of total life (/DT), true duration of total life (/DT), functional duration of productive life (/DP) and true duration of productive life (/DP) for daughters of Canadian and New Zealand sires separately, within the Canadian and New Zealand production environments. The data were collected from afieldtrial involving the mating of 20 proven sires from New Zealand and 20 proven sires from Canada to cows in 20 New Zealand and 10 Canadian dairy herds. In Canada, 13.7% of the daughters of New Zealand sires were disposed of due to mastitis while only 6.2% of daughters of Canadian sires were culled for the same reason. In New Zealand, 1.4% of daughters of Canadian sires were culled due to bloat compared to 0.2% for daughters of New Zealand sires. This difference was found to be significant at p<0.5. Results of the estimated hazard curves suggested that daughters of native sires were culled at a relatively lower frequency compared to daughters of imported sires during the course of their total or productive lifespan.  25  2.2 INTRODUCTION Longevity has long been acknowledged as a trait of economic importance in dairy cattle herds (Dekkers and Jairath, 1994). McDaniel (1994), pointed out that two main methods of improving efficiency in dairy cattle are (1) diluting maintenance and other fixed costs by increasing yield per cow and (2) by reducing costs through improvement of secondary traits, including measures of longevity. In addition, Thaller et al., (1994) noted that the current research interest in secondary traits may be attributed to the fact that very high milk production levels have already been achieved in the dairy populations coupled with the existence of quota systems that restrict milk production per farm. This reality has necessitated discussions on new breeding objectives to include functional or secondary traits in models aimed at improving the efficiency of dairy farm production. This assertion may be true in most developed countries but not necessarily so in developing countries. Although the economic importance of longevity is well demonstrated and understood, the measures of choice that may be candidates for the inclusion in national genetic evaluation schemes are still debatable (see Chapter 1, page 3). Other problems are related to the definition of failure as well as use of appropriate statistical procedures that account for censored data. There are a fair number of statistical procedures developed for the analysis of survival data in the field of animal breeding. Animal breeders like Smith (1983), Ducrocq (1987) and Ruiz (1991) have adapted and demonstrated statistical procedures (like failure time analysis) used extensively in biomedical and engineering professions to those suitable for the analysis of survival and endurance characteristics encountered in animal breeding. Smith's (1983) procedure adequately accounts for censored observations. This manuscript presents experimental data analyzed for frequencies of disposal reasons for daughters of Canadian and New Zealand sires within production environment and sire group. The data were also analyzed using a nonlinear Cox proportional hazards regression procedure (Ducrocq and Solkner, 1994) to estimate hazard curves of daughters of Canadian and New Zealand sires in their home and foreign environments. The study investigated the relationships between hazard rates for the various measures of longevity.  26  The objectives of this study were: (1) To compare the frequency of occurrence of reasons for disposal among daughters of Canadian and New Zealand sires within their home and foreign environments. (2) To compare the hazard rates of daughters of Canadian and New Zealand sires in their home and foreign environments for duration of productive life and duration of total life.  2.3 MATERIALS AND METHODS 2.3.1 Definition Of Longevity Traits Studied In this study, the measures (traits) of longevity studied were all related to time. Duration of total life (DT) and duration of productive life (DP) were the focus of this study. DT was defined as the difference in days from birth to disposal or censoring date while DP was defined as the difference in days between date of disposal/censoring and date of first freshening. Ducrocq (1987), defined two variants within DP related to the definition of failure. True duration of productive life (rDP) was defined as the ability of an animal to delay culling (involuntary or otherwise) during its productive life and functional duration of productive life (/DP) was defined as the ability of an animal to delay only involuntary culling during the same time period. In this study the same scheme was followed for DP but extended to DT as well, such that rDT and /DT, represented true DT and functional DT, respectively. Involuntary culling was defined as culling due to disposal reasons beyond the control of the farmer (Abortion, Bloat, Calving trouble, Empty, Feet and Leg problems, Injury, Lame, Mastitis, Sickness/disease) and voluntary culling as when the decision to cull depends only on the farmer's choice (Economics, Low production, Milking speed, Unsuitable temperament). In this study therefore, four traits (/DP, rDP,/DT, and rDT) were analyzed. Within production environments DT and DP were adjusted for age at first freshening before performing the main analysis in order to avoid underestimating the breeding values of bulls with fast maturing daughters. In the preliminary and exploratory analysis of the data, convergence problems were encountered when statistical models were fit to the data unadjusted for age at first freshening, particularly in the New Zealand environment. This problem disappeared when data were pre-adjusted for age atfirstfreshening most likely due to the fact that animals  27  had different ages at first calving (in days) even though they were culled at the same time as seasonal calving and culling were practiced in the New Zealand environment.  2.3.2 Source Of Data The data were collected from a field trial (Peterson, 1988) involving the random mating of 20 proven Holstein sires from New Zealand and 20 proven Holstein sires from Canada to Holstein cows in 20 New Zealand and 10 Canadian herds for a total of 40 sires and 30 herds. In New Zealand, the mating plan used 10 bulls from each country in each herd in a structured design to ensure ties between all sires and herds. Each bull was given the opportunity to sire daughters in at least 10 herds and no herd had the same set of sires. In Canada all sires were afforded the opportunity to sire daughters in all herds. Sufficient frozen semen was allocated to each herd to mate approximately 90% of the herd in equal proportions by Canadian and New Zealand bulls. The project sires were selected in 1984 based on progeny test information and represented the top 20 available New Zealand bulls for fat yield and top 20 available Canadian bulls selected for milk yield for a total of forty (40) sires. The relationship between sires from the two countries was assumed to be zero. This field trial was initially set up to determine whether there is a genotype x environmental interaction for production traits. Nineteen commercial herds and the number four (4) herd at Massey University were the original cooperators and participated in the planned mating program, for a total of twenty (20) herds in New Zealand. These herds were recruited by the New Zealand Dairy Board (Livestock Division) from the factory supply herds which had a long history of milk recording and a high level of record keeping. All New Zealand herds used forage as the exclusive feed during lactation and relied principally on intensive pasture management programs rather than conserved forages and concentrate feeds of their Canadian counterparts. No supplemental concentrates were available to the New Zealand herds. Seasonal calving was practiced in line with the availability of pasture. The end of lactation was determined primarily by lack of available feed on pasture which led to groups of cows being turned dry as well as culled in a batch without consideration of calving date or days milked. The 20 New Zealand herds included in the study were from the 5 major dairy  28  regions on the North Island. The ten (10) Canadian herds included in the experiment were the Agriculture Canada herd at Agassiz in British Columbia, University of Guelph, University of Manitoba, University of British Columbia (Oyster river and Campus), Macdonald College, Nova Scotia Agricultural College and Olds Agricultural College. This study included records of daughters of Canadian and New Zealand sires that had either completed their last lactations, failed before 1989 (in Canada) or 1991 (in New Zealand) or were still alive at this time. Information collected included reasons for disposal, date of birth, day of first freshening, day animal left the herd as well as pedigree information. As has been already mentioned, these records were an offshoot of a field trial aimed at evaluating the existence or lack thereof, of genotype by environmental interactions for milk and milk component traits. Records of 898 daughters of Canadian (435) and New Zealand (463) sires were available in the New Zealand data set. The Canadian data set from an extract of the Agriculture Canada database yielded records of only 239 daughters of the same sires (144 Canadian and 95 New Zealand). The distribution of daughters per sire was better in New Zealand than in Canada due to differences of sample size (Appendix 2.7). In New Zealand all the animals were born in 1985 while in the Canadian environment, animals were born between 1985 and 1989.  2.3.3 Statistical Analysis Frequency of occurrence of the various disposal reasons were calculated within production environments and sire group (Table 2-1 and 2-2). Significance tests were performed on the frequency of reasons for disposal (Steel and Torrie, 1980) between Canadian and New Zealand sire groups. The calculated chi-square was compared to the expected chi-square at <x=0.05 and 1 degree of freedom between the daughters of Canadian and New Zealand sires. It was assumed that, if the incidence of reasons for disposal were equal between daughters of Canadian and New Zealand sires in a given environment (country), they should occur in the ratio equal or close to that reflecting the proportion of each progeny group in the total experimental  29  population. In each category of disposal, therefore, the expected numbers and the chi-square were calculated thus: " (Observed-Expected) _ . , X= L • This was then compared with the table values at °c=0.05 and 1 degree of i Expected 2  freedom. Non-parametric Kaplan-Meier (Kaplan and Meier, 1958) estimates of the hazard rates for the daughters of Canadian and New Zealand sires were calculated and plotted for each of the measures of longevity within environment and sire group (Figures 2-1 to 2-4). The curves of the cumulative hazard were tested for homogeneity (equality) between sire groups within environment. The aim was to test whether the hazard curves of daughters of sire groups within a production environment came from the same population. The Kruskal-Wallis H statistic tests the hypothesis that the distributions in the populations are the same against the alternative that they are different. Within each production environment, the Kruskal-Wallis H test consisted of pooling the observations (hazards) in samples A (hazards of daughters of Canadian sires) and B (hazards of daughters of New Zealand bulls), ranking the values from smallest to largest and adding the ranks for each sample. Then the test criterion was computed as 2  H=  —  ("  bRa  N(N + l) where n  A  2  )-3(N  +r>ARB  +1), (Steel and Torrie, 1989).  nn A  B  = the number of daughters of Canadian sires and n = the number of daughters of New Zealand B  sires. N the total number of observations (daughters of all sires). R  A  and R  B  are the sums of the ranks for  samples A and B, respectively. When n and N are sufficiently large, H is taken to be distributed as % with 2  k -1 degrees of freedom where k is the number of sire groups. Ties are given the mean rank and when in different sire groups, a correction is made to H whose divisor is then XT  D= l  — (N-\)N(N  + l)  Where T ={t-l)t(t+l) for each group of ties and t is the number of tied observations in the sire group. D is used as a divisor of H to give a corrected H. The differences in hazard rates (curves) between sires groups partially represent genetic group effects in the same environment. 30  2.4 RESULTS AND DISCUSSION 2.4.1 Frequency Distribution Of Reasons For Disposal Table 2-1. depicts the frequency distributions of reasons for disposal in the Canadian production environment. About one half (50.6%) of the cows had no reason for disposal indicated on their records (censored). Disposal due to bad temperament was the highest (11.3%) followed by mastitis (9.2%). Slow milker accounted for 6.3% of the disposal reasons with disposal due to reproductive problems accounting for 5.0%. Sickness/disease, injury and feet and legs problems showed incidences of lower than 2.0%. When reasons for disposal were analyzed within environment for each sire group, 13.7% of the daughters of New Zealand bulls were disposed of due to mastitis compared to 6.2% in daughters of Canadian bulls in the Canadian production environment. The difference was not, however, statistically significant. Although not statistically significant (p>0.05), daughters of Canadian bulls tended to be disposed of due to bad temperament, slow milker, and reproductive problems at a lower frequency than the daughters of New Zealand bulls. In point of fact, daughters of New Zealand bulls were disposed of at a lower rate only due to sickness/disease, injury and economics. In the New Zealand production environment (Table 2-2) the majority (79.62%) of the animals' records had no indication of the reasons for disposal (censored). Except for the ill-defined reason for disposal (other causes), all the individual reasons for disposals accounted for less than 4.0% each. Analysis within sire groups showed that daughters of Canadian bulls exhibited an incidence and died of bloat at a significantly (p<0.05) higher rate (1.4%) compared to the daughters of New Zealand bulls at only 0.2%. Canadian cows are bred and selected for performance under a high concentrate feeding nutritional regime whereas their New Zealand counter-parts are bred and selected for performance under pasture feeding regimes. This might explain the higher incidence of bloat among daughters of Canadian sires in the New Zealand environment.  31  Table 2-1. Frequencies of reasons for disposal in the Canadian environment  Disposal reason  Total n Overall (%)  2  Daughters of Canadian bulls (%)  Daughters of New Zealand bulls (%)  X  Disposal Class  Censored  121  50.6  56.2  42.1  2.26  unknown  Bad temperament  27  11.3  10.4  12.6  0.25  voluntary  Slow milker  15  6.3  6.2  6.3  0.00  voluntary  Mastitis  22  9.2  6.2  13.7  3.41  involuntary  Feet and Leg problems  1  0.4  0.0  1.0  1.52  involuntary  Reproductive problems  12  5.0  4.9  5.3  0.02  involuntary  Sickness or Disease  3  1.3  1.4  1.0  0.05  involuntary  Injury  4  1.7  2.1  1.0  0.36  involuntary  Economics  4  1.7  2.1  1.0  0.36  voluntary  30 239  12.6  10.4 144  15.8 95  0.00  voluntary  Other causes Total n  1  assumed to be voluntary disposal A consideration of the results depicted in Table 2-1. and Table 2-2. indicates that although the incidence of bloat was low overall (0.78%), daughters of Canadian bulls (6) in the New Zealand production environment contributed a higher proportion to the incidence in New Zealand compared to daughters of New Zealand bulls (1). The 6 animals were sired by 6 different bulls and the incidence occurred in 3 different herds (3 in one, 2 in another and 1 in yet a different herd).  32  Table 2-2. Frequencies of reasons for disposal in the New Zealand production environment Disposal reason  Total n  Overall (%)  Daughters of Canadian bulls (%)  Daughters of New Zealand bulls (%)  Censored  715  79.6  77.5  Abortion  2  0.2  Bloat  7  Calving trouble  2  1  Disposal Class  81.6  0.49  unknown  0.0  0.4  1.88  involuntary  0.7  1.4  0.2  3.89*  involuntary  4  0.5  0.5  0.4  0.00  involuntary  Injured  2  0.2  0.0  0.4  1.88  involuntary  Lame  1  0.1  0.2  0.0  1.06  involuntary  Low production  26  2.9  2.3  3.5  1.04  voluntary  Mastitis  5  0.6  0.7  0.4  0.27  involuntary  Empty  30  3.3  3.4  3.2  0.03  involuntary  Milking speed  1  0.1  0.2  0.0  1.06  voluntary  Unsuitable temperament Other Causes  2  0.2  0.2  0.2  0.10  voluntary  103  11.5  13.6  9.5  3.31  voluntary  Total n  898  435  463  1  •Significant at p<0.05 'assumed to be voluntary  In general, daughters of Canadian bulls tended to be culled at a lower rate for most individual reasons for disposal compared to daughters of New Zealand bulls in the Canadian production environment. The two tables, also, show the problem encountered in the analysis of individual reasons for disposals in small data sets. Most commonly, not enough records were found for individual reasons for disposal and therefore a more complicated contingency chi-square accounting for important effects such as herd-year-season of calving could not be fit (especially in Canada where animals were not born in the same year). Such an attempt would have resulted in many empty cells contributing nothing to the overall analysis. As can be seen from Tables 2-  33  1 and 2-2, the frequencies of occurrence are also generally low. Real differences may not be discernible simply because of the smaller numbers of cows in each disposal class. The most important use of the disposal information in the formulation of the hazard rates as per definition of failure. The formulation of the hazard function takes into account the fact that these disposals not only occurred in different herds but also serially across time.  2.4.2 Hazard Curves Within Production Environments And Sire Groups Tables 2-3 and 2-4 show the summary statistics of failure time for measures of longevity considered in this study. There were 144 and 95 daughters of Canadian and New Zealand sires, respectively, in the Canadian environment. Table 2-3. Summary statistics of failure times for the measures of longevity in the Canadian production environment within sire groups  Duration of productive life Involuntary n  Total  Duration of total life Involuntary  Total  (/DP)  (/DP)  (/DT)  (tDT)  21-1542  1-1542  804-2392  747-2392  Time to 50% failure (days)  758  826  1552  1620  Time to 80% failure (days)  1150  1064  1942  1885  Time to 90% failure (days)  1176  1177  1963  1991  Censored observations (%)  81.9  43.5  81.9  43.5  15-1298  15-1678  956-2086  893-2374  Time to 50% failure (days)  853  782  1665  1642  Time to 80% failure (days)  1109  1054  1915  1915  Time to 90% failure (days)  1121  1121  1927  1940  Censored observations (%)  71.6  31.6  71.6  31.6  Daughters of Canadian bulls  144  Minimum-Maximum failure time (days)  Daughters of New Zealand bulls Minimum-Maximum failure time (days)  95  Total represents failure defined as disposal due to both involuntary and voluntary reasons 34  In the New Zealand environment there were 435 daughters of Canadian sires and 463 for New Zealand sires. The tables revealed that the functional (involuntary) definition of failure for a measure of longevity entailed more censored observations relative to the true form of the definition. Recall that the functional definition considered an animal as culled (failed) only if the reason(s) for disposal were deemed involuntary. The true form considered an animal to have failed regardless of reason(s) for disposal. It is, therefore, expected that the functional definition yields more censored observations relative to the true form. Table 2-4. Summary statistics of failure times for the measures of longevity in the New Zealand production environment within sire group Duration of productive life Involuntary n  Daughters of Canadian bulls  (/DP)  Total*  Duration of total life Involuntary  (rDP)  (/DT)  Total (tDT)  435  Minimum-Maximum failure times (days)  1-1046  1-1280  555-1714  555-2000  Time to 50% failure (days)  244  563  931  1207  Time to 80% failure (days)  659  772  1302  1483  Time to 90% failure (days)  713  967  1415  1679  Censored observations (%)  79.26  25.80  79.26  25.80  1-1303  1-1614  562-2015  640-2268  Time to 50% failure (days)  206  358  916  1088  Time to 80% failure (days)  653  784  1364  1489  Time to 90% failure (days)  1002  981  1713  1687  Censored observations (%)  84.05  37.50  84.05  37.50  Daughters of New Zealand bulls Minimum-Maximum failure time (days)  463  Total represents failure defined as disposal due to both involuntary and voluntary reasons  35  For example, when DP was defined as /DP for daughters of Canadian bulls, 81.9 % of observations were censored whereas only 43.5% of the observations were censored when DP was defined as /DP (Table 2-3). The same was true for /DT and /DT. For daughters of New Zealand bulls, 71.6 and 31.6% of observations were censored for the functional and true definitions of DP and DT, respectively (Table 2-4). In the New Zealand production environment (Table 2-4), 79.3% of observations were censored while only 25.8% were for the functional and true definitions of failure, receptively, for daughters of Canadian bulls. Daughters of native bulls had 37.5% observations censored for the true definitions of failure. Tables 2-3 and 2-4 also show the time to 50, 80 and 90% failure for /DP, /DP, /DT and /DT within production environment and sire group. In Canada (Table 2-3), time to 80 and 90% failure for daughters of Canadian bulls were higher than those for daughters of New Zealand bulls for all the traits studied. This shows the trend that daughters of Canadian bulls experienced lower hazards relative to daughters of New Zealand bulls for all the traits studied in the Canadian environment. In New Zealand (Table 2-4), except for /DP, the results were opposite to the results obtained in Canada. Time to 80 and 90% failure were higher for the daughters of New Zealand bulls compared to daughters of Canadian bulls. The daughters of New Zealand bulls experienced lower hazards relative to daughters of Canadian bulls in the New Zealand production environment. The cumulative hazard rates of animals for the two definitions of DP (/DP and /DP) within sire groups in the Canadian environment is depicted in Figure 2-1. The graph indicates the proneness of the animals to failure at time t given that they have survived to time tj relative to the origin for the same trait. The graph ;  indicates that animals were more prone to failure at any time afterfirstfreshening when DP was defined as /DP compared to /DP definition. This was as expected, since by definition, more animals will be culled per unit time when /DP was defined as the ability of the animal to delay involuntary as well as voluntary culling, were as, /DP was defined as the ability of the animal to delay only involuntary culling. Although not very pronounced, small 'jumps' were observed around 365-d (12 month) intervals indicating as well as lending credibility to the notion that perhaps heavier culling was practiced around the ends of the lactation periods compared to any other period of productive life across sire groups.  36  It is worth noting that the difference in the hazard rate between /DP and /DP generally tended to increase with time within sire groups. The increase in the difference is likely a function of an increase in voluntary disposals with time relative to that in involuntary disposals. Note that in the functional definition animals without an indication of reasons for disposal were not considered as failures whereas as in the true definition they were considered failures. This grouping may underestimate the hazard curves for the /DP. The grouping has the opposite effect on the hazard curves for /DP. Needless to say, this observation has the same effect as it relates to /DT and /DT.  0  365  730  1095  1460  Duration of productive life (days)  Figure 2-1. Cumulative hazards of daughters of Canadian and New Zealand bulls for duration of productive life (DP) in the Canadian environment  Even though the differences in the hazard curves (Table 2-5, page 41) were not statistically significant (p>0.05), the cumulative hazard rate for daughters of New Zealand bulls tended to be slightly higher for all survival traits (/DP, /DP, /DT, /DT) in the Canadian production environment (Figures 2-1 and 2-2). Differences in the hazard rates between sire groups also tended to increase with time for all the traits considered in this study. In other words, with time relatively more daughters of imported bulls were culled compared to daughters of native bulls.  37  Daughters of New Zealand bulls (tDT) Daughters of Canadian bulls (tDT) Q _ Daughters of New Zealand bulls (fDT) ^  365  730  1095  1460  1825  Daughters of Canadian bulls (fDT)  2190  Duration of total life (days)  Figure 2-2. Cumulative hazards of daughters of Canadian and New Zealand bulls for duration of total life (DT) in the Canadian environment  The results for the cumulative hazard in the New Zealand production environment are shown in Figures 23 and 2-4. The curves show steps and jumps due to the seasonality of heavy disposals that coincide with the non-availability of pasture. It is a reflection of a higher number of tied observations discussed earlier compared to less in the Canadian environment. The small 'jumps' in the hazard curves were again observed towards the end of the 365-d interval indicating that, still, heavier culling occurred towards the end of each lactation in the New Zealand environment. Due to pasture feeding practiced in New Zealand, animals would be turned dry without completing a lactation depending on the availability of feed. Therefore, culling decisions would be made earlier than the 365-d period. Figure 2-3 is a depiction of the results of the cumulative hazard for /DP and rDP within sire groups. The observations are similar to those observed in the Canadian production environment with regard to animals showing higher cumulative hazard rates relative to the origin when the trait was /DP compared to /DP at any time, tj of productive life.  38  Daughters o f Canadian bulls (tDP) Daughters of New Zealand bulls (tDP) Daughters o f Canadian bulls (fDP) >  1.5  Daughters of New Zealand bulls (fDP)  3  £ 3  u  0  365  730  1095  1460  Duration of productive life (days)  Figure 2-3. Cumulative hazards of daughters of Canadian and New Zealand bulls for duration of productive life (DP) in the New Zealand environment  Daughters o f N e w Zealand bulls exhibited a lower hazard compared to daughters o f Canadian bulls for all the survival traits studied  (/DP, <DP,/DT, rDT)  at any time, tj, relative to the base (Figures 2-3 and 2-4). These  results indicate a lower proneness to failure for daughters o f native sires compared to daughters o f foreign sires since the hazard for daughters o f native sires is lower than that o f daughters o f foreign sires at any time, tj for each o f the traits studied. It could be speculated that this is indicative o f the superior adaptability o f the native sires to the local conditions compared to their foreign counter-parts. It is important to note that these sires were a selected group and, therefore, their relative functional fitness in their o w n countries w o u l d be relatively higher than the average o f their respective populations. The difference i n hazard rates between sire strains observed in this study may not be found i f random samples o f sires from the two countries were compared. Nonetheless, farmers are more likely to import semen o f superior bulls for use i n their herds.  39  365  730  1095  1460  1825  2190  Duration of total life (days)  Figure 2-4. Cumulative hazards for daughters of Canadian and New Zealand bulls for duration of total life (DT) in the New Zealand environment  It was not possible to attribute specific differences i n the two environments to differences i n survival rates between sire groups for the traits studied. Notable differences that might have contributed to the observed higher failure rate for daughters o f Canadian bulls i n the N e w Zealand environment are: smaller handling facilities designed for daughters o f N e w Zealand bulls, extensive pasture grazing systems, pathogens and climatic factors. Climatic factors (severe winters), conserved forage and high concentrate supplements may have contributed to the relatively poor performance o f daughters o f N e w Zealand bulls i n Canada.  2.4.2.1  Tests Of The Equality Of The Hazard Curves Over Sire  Groups  The equality o f the hazard curves o f daughters o f Canadian and N e w Zealand sires were tested in each production H: A  O  z l  environment.  ^ O  z  2  The  where <t>  zX  null  hypothesis  H: 0  <S  zl  = <D  w z2  a  s  tested  against  the  alternative  was the hazard curve for daughters o f Canadian sires and <J> for daughters o f z2  N e w Zealand sires by the K r u s k a l - W a l l i s H test (Steel and Torrie, 1989) as explained earlier.  40  Table 2-5 is a depiction of tests for the equality of the hazard curves between strain of sire within environment. Results depicted in Table 2-5 indicated that in the Canadian production environment, the hazard curves for sire groups were not significantly different (p>0.05) for any measures of longevity studied (/DP, /DP,/DT and rDT). This indicates that either the conditional failure rates were similar between daughters of Canadian and New Zealand sires for all of the traits considered in this study or that the data were not sufficient enough to detect the small differences between sire groups in their conditional failure rates. Although not statistically significant (p>0.05), daughters of Canadian sires were less prone to failure than their New Zealand counter-parts in the Canadian environment as can be seen in the graphic representation of the cumulative hazard curves (Figure 2-1 and 2-2) with /DP and /DT approaching significance.  Table 2-5 The Kruskal-Wallis H test of the equality of the hazard curves between sire groups in the Canadian and New Zealand production environments for all measures of longevity CanadaTrait  Duration of productive life: functional (/DP) true (/DP)  %  2  3.5444 2.0757  New Zealand P>Xuf  0.0597 0.1497  X  2  0.3363 7.4637  P>X\df  0.5620 0.0063  Duration of total life: functional (/DT) 3.3009 0.0697 4.3288 0.0375 true (rDT) 1.7350 0.1878 12.1236 0.0005 The first two rows are the results of the test for equality of curves of duration of productive life and the last two for curves of duration of total life between sire groups.  In the New Zealand production environment, significant differences (p<0.05) for the hazard curves between sire groups were observed for /DP,/DT and rDT (Table 2-5). When either duration of productive or total life were defined in their true form (i.e. /DP and /DT), the difference in the hazard curves between sire groups were highly significant (p<0.01). This is an indication that in the New Zealand environment, daughters of Canadian bulls were culled at a significantly (p<0.05) higher rate when the traits considered were /DP,/DT and /DT. In summary, significant differences in the hazard curves for some of the traits studied were  41  observed and a change in rank of the sire groups hazard curves between environments was evident in this study. This suggested a GxE interaction at the macro level.  2.5 CONCLUDING REMARKS This study showed that the frequency of reasons for disposal may not differ substantially between daughters of imported bulls relative to the daughters of native bulls. The overall frequency of disposal reasons were very low in both environments. Even though the frequency of bloat was very low overall, daughters of Canadian bulls experienced more problems due to bloat in New Zealand perhaps due to the fact that they are daughters of bulls that are selected for performance under intensive conserved forage and concentrate feeding regimes. On the other hand, although statistically insignificant, daughters of New Zealand bulls experienced more mastitis relative to daughters of Canadian bulls in the Canadian environment. The tests for equality of the hazard curves between sire groups within production environments suggested that daughters of imported bulls exhibited a higher proneness to failure compared to daughters of native bulls under the same rearing conditions. Bar-Anan et al. (1987), found a lower mean culling rate of daughters of Israeli bulls compared to the mean culling rate of daughters of bulls imported from Canada, New Zealand, Sweden and the United States of America. This may result from true genetic superiority of local sires due to the fact that they are better adapted to the local conditions relative to their foreign counterparts. As mentioned earlier, it was not possible in this study to attribute specific differences in selection pressures between the two environments that may have contributed to the differences in the hazard rates between sire strains. However, differences in the nutritional regimes (since animals in New Zealand are raised on pasture drying off is based on the availability of the pasture whereas in Canada animals are fed conserved forages with supplemental concentrates and days milked are an important consideration before cows are turned dry) and differences in handling facilities. The existence of milk quotas may have an effect on culling decisions in Canada but not in New Zealand. In general it is concluded that the results indicated "fitness" of the daughters of native sires to the general local conditions compared to daughters of their foreign counter-parts. Since the sires used in this study were a selected group representing the top 20 for production traits, the observed differences in hazard  42  rates between sire strains in the two environments may be larger than if comparisons were between random sires from each country. This is because the sires used in this study may have higher relative functional fitness than the average of their respective populations. These traits should be considered as different traits in the two environments.  43  2.6  REFERENCES  1 Bar-Anan, R., M. Heiman, M. Ron and J. I. Weller. 1987. Comparison of proven sires from five HolsteinFriesian strains in high-yield Israeli dairy herds. Livest. Prod. Sci. 17:305. 2. Dekkers, J. C. M., and L. K. Jairath. 1994. Requirements and uses of genetic evaluations for conformation and herd life. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:61. 3. Ducrocq, V. 1987. An Analysis of Length of Productive Life in Dairy Cattle. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 4 Ducrocq, V. P., and J. Solkner. 1994. "The survival kit" - A FORTRAN package for the analysis of survival  data. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada.  22:51. 5. Kaplan, E. L. and P. Meier. 1958. Non-parametric estimation from incomplete observations. J. Am. Statist. Assoc., 26:457. 6. Lagakos, S. W. 1979. General right censoring and its impact on the analysis of survival data. Biometrics, 35:139. 7. McDaniel, B. T. 1994. Genetic evaluation of secondary traits in dairy cattle. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:59. 8. Peterson, R. 1988. Res. Bull. No. 5 Livest. Improvement Corp., Hamilton NZ. 9. Robertson, A and J. S. F. Barker. 1966. The correlation betweenfirstlactation milk production and longevity in dairy cattle. Anim. Prod. 8:241. 10. Ruiz, F. 1991. Relationships among Length of Productive Life, Milk Yield, and Profitability of United States, Canada, and Mexican Holstein sires in Mexico. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 11. Smith, S. P. 1983. The extension of failure time analysis to problems of animal breeding. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y.  44  12. Smith, S. P., and F. R. Allaire. 1986. Analysis of failure times measured on dairy cows: Theoretical considerations in animal breeding. J. Dairy Sci., 69:217. 13. Smith, S. P., and R. L. Quaas. 1984. Productive lifespan of bull progeny group: Failure Time Analysis. J . Dairy Sci. 67:2999. 14. Steel R. G. D., and J. H. Torrie. 1980. Principles And Procedures Of Statistics: A Biometrical Approach. Second Edition. McGraw-Hill Book Company, p 495. 15. Thaller, G., M. Gierdziewicz, G. Averdunk and J. Auman. 1994. Animal model for the genetic evaluation of functional traits in German Fleckvieh. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:105. 16. Veekamp, R. F., S. Brotherstone, A. W. Stott, W. G. Hill and G. Simm. 1994. Combining transmitting abilities and linear type in an index for selection on production and longevity. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:69. 17. Weigel D.J. and B. G. Cassell. 1994. Differences in multiple trait prediction of transmitting abilities for erdlife due to data source. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:73.  45  2.7 APPENDIX 2. THE DISTRIBUTION OF DAUGHTERS PER SIRE IN EACH ENVIRONMENT Minimum number  Average number  Maximum number  of daughters per sire  of daughters per  of daughters per  sire  sire  Country  Total n  Canada  239  2  5798  13  New Zealand  898  10  22.45  48  46  3. CHAPTER 3. HERITABILITY ESTIMATES OF MEASURES OF LONGEVITY USING FAILURE TIME ANALYSIS IN CANADIAN AND NEW ZEALAND EXPERIMENTAL DAIRY HERDS  3.1  ABSTRACT  Heritabilities were estimated for functional duration of total life (fDT), true duration of total life  (/DT),  functional duration of productive life ( / D P ) and true duration of productive life (/DP) using non linear Cox regression models via the derivative-free restricted maximum likelihood (REML) approach in the Canadian and New Zealand production environments. The data were collected from a field trial involving the mating of 20 proven sires selected from New Zealand and 20 proven sires selected from Canada to cows in 20 New Zealand and 10 Canadian herds. A total of 1,137 daughters of Canadian and New Zealand sires were used. 898 daughters born of Canadian (435) sires and New Zealand (463) sires were used in New Zealand. In Canada 144 daughters of Canadian sires and 95 daughters of New Zealand sires were used. Heritabilities were estimated within each production environment (country) across strain of sire for each of the traits. In Canada, heritability estimates were 0.19, 0.03, 0.29 and 0.06 for / D P , In New Zealand estimates were 0.05, 0.05, 0.09 and 0.08 for / D P ,  /DP,/DT  /DP,/DT  and  /DT,  and  /DT,  respectively.  respectively. Since the  heritabilities for all the survival traits studied were not statistically different from zero, more studies with larger data sets should be conducted to arrive at more reliable estimates.  47  3.2 INTRODUCTION Longevity in dairy cattle may simply be defined as length of life of cows in a given herd. The length of cows' lives is a function of the culling process(es) on any given dairy farm. Culling decisions of a dairy producer are dependent upon comparative evaluations of all cows in a herd. Producers consider all factors they deem important at the time of the decision. Reflecting the sensitivity of dairy producers to economic constraints, dairy cows are presently kept milking for only 3 to 3.5 lactations in the herd (Ducrocq, 1987). A cow is usually voluntarily culled when she is considered less profitable than her potential replacement. This usually occurs much earlier than senescence. Animal breeders are interested in manipulating the cows' genetic potential to have a longer lifespan. Major reasons for disposal of dairy cattle afterfirstfreshening are often cited as low production, reproductive problems, udder problems, sold for dairy purposes and death (Allaire et al., 1977). Sold for dairy purposes, is usually considered voluntary while disposal due to reproductive problems is likely involuntary. Animal breeders' interest in this regard has been aimed at reducing involuntary culling in dairy populations. Although milk, fat and protein yield are indisputably the most important dairy cattle selection criteria, dairy producers desire to include other traits such as productive lifespan in the indices. Selection for length of productive life appeared to be profitable even when discounted for increased generation interval (De Lorenzo and Everett, 1985). Increasing the number of lactations from 3.3 to 5.3 increased income by 20% (Renkema and Stelwagen, 1976) whereas Congleton (1984) found income increased by 8.9% over the same range after discounting for increased generation interval. Disposal is considered voluntary when the culling decision depends on the dairyman's choice. It is involuntary when the culling decision is beyond the control of the farmer. The distinction between voluntary and involuntary culling is not always clear cut. The removal of a sick but otherwise profitable cow is involuntary. The sale of a healthy but poor producing cow for beef may be considered voluntary. However, culling for low milk production is almost always imposed by economic factors far beyond the control of the farmer. Until recently, statistical methods designed for censored (survival) data, like failure time analysis (Smith and Quass, 1984) have not been employed in survival studies of dairy cattle. In an attempt to analyze the  48  extent of variation in cow survival that is attributable to genetic causes, animal breeders have tended to utilize the same analytical procedures designed to seek answers to such questions as "Is milk production inherited?" or "Is growth rate inherited?". In other cases, animal breeders have tried to tackle the problem of censoring prevalent in survival data, by performing evaluations within defined opportunity groups. Such a group may include all animals which have been given the opportunity to freshen a given number of times or to still be alive at a given age. In some cases, censored observations are ignored altogether. Since longevity is a duration, death is usually viewed as a hazard, and data on lifespan is usually censored, methods (like failure time analysis) designed for survival analysis would seem appropriate (Vaupel et al., 1991). Measures of certain kinds of traits, like survival measures or failure times, may be described poorly by linear models. More especially, linear model methods do not accommodate censored data well. In addition most heritability estimates for survival of dairy cattle found in the literature have been estimated under the constraint that survival is an all-or-none trait (e.g. stayability). Generally, these estimates are low and are found to differ between breeds (strains) of dairy cattle. Robertson (1966) studied heritabilities of survival to different lactations in Ayrshires and British Freisians. In Ayrshires, heritabilities were 0.072, 0.126, 0.132, 0.152 and .203 for survival to 2 , 3 , 4 , 5 nd  rd  m  tn  and 6  tn  lactations, respectively. The estimates were 0.036,  0.052, 0.058 and 0.056 for survival to 2 , 3 , 4 and 5 lactations, respectively for British Freisians. nd  rd  th  th  When considered as a categorical trait, longevity is seen to vary in a discontinuous manner on the visible scale. Its inheritance, however, is not a simple Mendelian one. The clue to the understanding of the inheritance of such characters lay in the idea that the trait (character) has an underlying continuity with a threshold which imposes a discontinuity on the visible scale (Falconer, 1981). For instance, when the underlying culling variable is below this threshold level, the animal has one form of phenotypic expression (not culled); when it is above the threshold level the animal has the other phenotypic expression (culled). The threshold model is intuitively appealing for binomial traits. It should be noted that under the threshold model the continuous variation of the underlying culling variate is both genetic and environmental in origin. It may be visualized as a compound of several different physiological processes (traits) without necessity for the knowledge of how these are combined to give the underlying culling variate. In fact, we may not completely know what they really are. It is now generally acknowledged that the genetic determination of categorical  49  measures of longevity, although showing a discontinuous phenotypic distribution, cannot be readily analyzed as a Mendelian character but that they can be explained by a polygenic inheritance model (Foulley, et al., 1987). It, therefore, suffices to postulate that if longevity is measured as a duration, the measure would be continuous and its analysis may not necessitate the direct invocation of a threshold model. Smith and Quass (1984) observed that heritabilities for productive lifespan seemed to depend on how the trait was measured. For categorical observations like stayability, estimates were between 0.02 and 0.08 (Hudson and Van Vleck, 1981; Schaeffer and Burnside, 1974). Survival traits measured based on number of lactations have heritability estimates between 0.05 and 0.14 (Hogue and Hodges, 1980; Miller et al., 1967). When survival was measured in days of productive life, heritabilities were between 0.1 and 0.26 (Hogue and Hodges, 1980; Gill and Allaire, 1976). Smith and Quass (1984) speculated that heritabilities for stayability are generally not statistically different from zero, perhaps, due to scale problems as well as the fact that stayability is directly related to cow disappearance (censoring), which in their view, dilutes genetic variation associated with the process that governs death or sale for beef. They concluded that, for survival, heritabilities are bound to be higher when scales of measure are allowed to be quantitative and better still with models that adequately accounts for censored records. Smith (1983) reviewed the main objections against direct selection for longevity as being due to the following observations: (1) Longevity measures have low heritabilities. Direct selection would, therefore, yield little or no genetic response. (2) Longevity measures have shown a high positive correlation with milk yield and, therefore, longevity may be improved indirectly as a correlated response from selection on milk yield which is moderately heritable. (3) On average, measures of longevity are obtained later in life compared to first lactation milk yield. The practice of including measures of longevity in selection programs would see an attendant increase in the present generation interval thereby reducing the current genetic progress attainable from direct selection on milk yield. Thefirsttwo observations may not be valid arguments. Most heritability estimates available in the literature were obtained under the constraint that survival is an all-ornone trait. It has been observed that heritability estimates are generally higher for quantitative measures of longevity than qualitative ones (Smith and Quass, 1984). Ducrocq (1987) argued that the positive correlation (genetic and phenotypic) between milk yield and measures of longevity is misleading. He explained that  50  involuntary culling does not seem to be reduced by selection on milk yield whereas voluntary culling is determined more by the relative, rather than absolute levels of production of a particular cow within a given herd. With regard to argument number 3, advances in statistical methods designed to handle censored data, sires may be evaluated for their daughters' superiority in longevity much earlier in life without waiting to observe the daughters full lifespan. Other measures of longevity such as duration of total life (DT) have not been adequately studied in the literature. This is because the data used comes from milk recording schemes which only allows the modeling of duration of productive life (DP). DP was defined as the difference in days between first freshening and disposal. DT was defined as the difference in days between disposal date and birth date. In this study DP and DT are defined in their two variant functional and true forms. Functional duration of productive life (/DP) is defined as the ability of the animal to delay involuntary culling whereas true duration of productive life (/DP) is defined as the ability to delay culling, voluntary or otherwise. Functional and true duration of total herd life (fDT and tDT) are defined similarly. Relationships between the different definitions of failure of the cows in the herd (Involuntary vs. Involuntary + voluntary) for the measures of longevity (DP vs. DT) have not been adequately studied for animal breeding applications. This study was aimed at estimating heritabilities for measures of longevity, /DP, /DP,/DT and tDT in the Canadian and New Zealand production environment using failure time models.  3.3 REVIEW OF VARIANCE COMPONENT ESTIMATION METHODS 3.3.1.1  Estimation  Of Variance Components  In Linear  Models  A brief mention of methods used in estimating variance components in linear models is necessary for the understanding of some methodology employed in variance component estimation for non-linear models. Variance components in linear models have been estimated in many ways. In 1949 C. R. Henderson derived Henderson's method 1 as a result of his Ph.D. thesis work at Iowa State University. His method was not published until 1953 together with Methods 2 and 3. Method 1 extended the analysis of variance for balanced data to unbalanced data and applied to models in which all the factors were  51  assumed random except for the common mean. Method 2 was derived to accommodate mixed models provided that interactions betweenfixedand random factors, and/or nesting of random factors within fixed factors, did not exist in the model. Method 3 which is also referred to as 'the method offittingconstants' is capable of handling general mixed models, even though it is computationally more demanding. The method uses reductions in sums of squares due tofittingsub models of the full model. Some unresolved problems with this method of variance component estimation are: (1) It may yield more equations than there are components to be estimated; (2) It provides no guidance as to which equations are to be used (Searle, 1971). These methods of analysis assume random sampling and, therefore, estimates derived from them may be biased by selection (Meyer, 1989). In addition, estimates may be outside the parameter space (i.e. estimates of heritabilities may be less than 0 or greater than 1 and correlations may be less than -1 or greater than 1). Henderson's Method 4 (1984) was simply aimed at simplifying computations while maintaining a method as good as Method 3. This method has also been referred to as the Diagonal Minimum Variance Quadratic Unbiased Estimators method.  3.3.1.2 Maximum  Likelihood  (ML)  Hartley and Rao (1967) described the Maximum Likelihood (ML) approach to the estimation of variance components. The procedure maximizes the log likelihood function. These estimates are consistent, asymptotically normal and efficient (Harville, 1977) and account for selection (Kennedy, 1981). The major drawback of the ML approach is that, in a mixed model, thefixedeffects are treated as known since the loss in the degrees of freedom due tofittingthese effects is completely ignored. A modification of ML described by Patterson and Thompson (1971) designed to overcome this problem is known as Restricted Maximum Likelihood (REML). This procedure assumes that the data vector y has a normal distribution. With REML, the idea is to maximize that part of the likelihood which is invariant to thefixedeffects in the mixed model. The method also restricts the estimators within the allowable parameter space to avoid 'embarrassing' negative estimates that may arise from ML procedures.  52  3.3.1.3 Estimation  Of variance  Components  in A Non-Linear  Model  Methods for estimating variance components in non-linear models (e.g. Failure Time Models / proportional hazards models) are generally, extensions of methodology for linear models. They are derived by linearizing a non-linear model in accordance with large sample theory (Smith, 1990). Procedures for estimating variance components in Cox (1972) non-linear regression models involves procedures that implement a linearization of the model to allow the utilization of methodologies designed for linear models whose estimator properties are well known and tested. Variance component estimation methods designed for linear models may then be employed. The general representation of a linear model is: y = Xb + Zu + e  and  G 0"  u  Var  [3.01]  0 R  e  where b and u are vectors representingfixedand random effects, respectively. X and Z are known design matrices describing precise relationships offixedand random effects to the data vector y, while e is the random error term. G is the matrix of additive genetic variance and covariance between the elements of u and R is a matrix of error variance and covariance. If G and R are known and are nonsingular, then the best linear unbiased predictor of u is u where u is obtained from solution of the following mixed model equations (Kennedy, 1981): X'R X  X'R" »z  Z'R X  Z'R" 'Z + G  -1  _1  - 1  b  X'R"  u  Z'R~  V  [3.02]  with the description of the symbols given above. If a sire model is used, a simpler form of the mixed model equations is XX  X'Z  Z'X  Z'Z + IA.  * b ii  "X'y"  [3.03]  which is obtained after multiplying both sides of [3.02] by (l-l/4h jo 2  where ^ = ( 4 - h ) / h 2  2 y  2  = a /a 2  (Kennedy, 1981). Then applying REML, estimates of error and sire variances are obtained as:  53  2  a =(yy-b'X'y-u'Z'y)/[«-r(X)]  [3.04]  and G = u'u + a tr(C ) Iq  [3.05]  2  2  2  s  where C  7 7  is from  c, " 2  _ 12 C  -1  22  C  22.  X X X'Z Z'X Z'Z + IA,  [3.06]  The sire model assumes that the sires and the dams they were mated to are all unrelated and that they are random samples from the population (i.e. they are not a selected sample). To account for all of the relationships, an animal model is used by employing the additive relationship matrix of all animals in the data set. Non-linear Cox regression models (proportional hazards models) designed to handle endurance or survival data are generally represented as follows: 4>i(t) = O (t)exp(Xb + Zu) 0  with the X, b, Z and u as described previously. G>j(t) is the hazard function of an individual animal depending on time (t). Oo (t) is considered a positive function of t defined as the baseline hazard related to the aging process (Ducrocq, 1987).  3.4 MATERIALS AND METHODS. 3.4.1 Source Of Data The data were collected from afieldtrial involving the mating of 20 proven sires selected from New Zealand and 20 proven sires selected from Canada to cows in 20 New Zealand and 10 Canadian herds. In New Zealand, the mating plan used 10 bulls from each country in each herd in a structured design to ensure ties between all 40 sires as explained in Chapter 2, page 28. Survival records of 898 cows were used in New Zealand while records of 239 cows were used in Canada. The traits studied were as described in Chapter 2, section 2.3.1.  54  3.4.2 Statistical Analysis In this study the maximum a posteriori approach (MAP) technique (Smith, 1990) was used to linearize the non-linear Cox regression model for the estimation of variance components in order to satisfy the numerical requirements for the implementation of the REML procedure. The end result of linearization is a so-called 'constructed linear model', Smith (1990), in which the constructed coefficient matrix is T'WT and the constructed right-hand side is T'Wy. In this scheme, T is the known global design matrix, W is the matrix of second derivatives of the log-likelihood function (Smith, 1990) and y is a vector of hazard rates. Preliminary absorption of fixed effects was accomplished by setting up the following array with the assumption that G = A a where A is the additive relationship matrix: 2  X'WX X'WZ X'Wy Z'WX Z'WZ Z'Wy  [3.07]  and absorbing the rows representingfixedeffects as: [Z'MZ Z'My ]  [3.08]  where M = W-WX(X'WX)~X'W . Absorption is accomplished by using the Gaussian elimination strategy (Smith and Graser, 1986). Z'MZ is the coefficient matrix (C) and Z'My is the right-hand side after absorption offixedeffects. To accommodate relationships a matrix L is defined such that L 'L = A . Then equation [3.08] is reparameterized as ZL replaces Z in the following manner: [L'Z'MZL] = [L'Z'My]  [3.09]  The estimation of a via the expectation-maximization algorithm involved an iterative scheme such that: 2  2[k l] ,[k] [k] [k]  CT  +  =(u  u  +trC  )/q  [  3  1  Q  ]  where the k in brackets defines the iteration number, q is the order of uand u are the solutions to the [k]  equations {L'Z'MZL + I/(a  2[kI s  )}u  [kl  =L'Z'My  [3.11]  55  and C is the inverse of the coefficient matrix in [3.11]. The system was taken to have converged when the [ J  differences in the two successive solutions were less than 10". Smith's (1983) procedure which is used in this 6  study employs a model in which the likelihood is constructed from rank information. Consequently the error variance could not be estimated. Instead, a standardized residual was used (Smith, 1983). The method 2 /  standardizes the residual variance to / K  6  , which is the variance of a unit exponential distribution (Ducrocq,  1987), such that in a strictly paternal half-sib analysis, and since the likelihood is constructed from rank information, the heritability on the underlying scale was estimated as 4a . /(4a + (n 16)), (Smith, 1990). 2  2  2  This implies an assumption of normality of the distribution of the trait values on the underlying scale. In this study, the following model was used: Oi(t) = <J)(t) exp (Xb + Zu) in a paternal half-sib analysis, where Oi(f) is the hazard function of an 0  individual animal depending on t. <X>o(t) is considered a positive function of t defined as the baseline hazard related to the aging process (Ducrocq, 1987). band uare vectors representing the fixed effects of herd and sire group, and random effects of sire, respectively. X and Z are known design matrices describing precise relationships of fixed and random effects to the data. Heritability estimates for measures of longevity were obtained separately within production environments (Canada and New Zealand) using Smith's (1983) procedure. Since there is no accepted method for estimating standard errors for heritability estimates they were approximated using a formula suggested by Falconer (1989), thus: G  2  ~ (32 *h )/N 2  2  for a strictly paternal halfsib analysis, such that  o *^2*h )lN 2  hl  where N is the total number of observations in the sample. Note, however, that this is a very rough approximation of standard errors for heritability estimates in this study since the method is best suited for experiments with optimal design. The herd, sire group and sire effects in the model were tested via the likelihood ration tests (Ducrocq and Solkner, 1996). These tests were effected by comparing the log likelihood of the full model with log likelihood of the models excluding one effect at a time for each effect. The likelihood ratio tests compares the 56  difference between the log likelihood of the full model and the model excluding the effect of interest. In the Canadian environment the full model had 51 degrees of freedom and in the New Zealand environment the full model had 61 degrees of freedom.  3.5 RESULTS AND DISCUSSION The herd effect was found to be statistically insignificant in both environments. Table 3-1 shows the sire group solutions (relative culling rate) for the traits studied in both environments. Larger values indicate relatively higher culling rates and lower values are, therefore, desirable (Ducrocq, 1996, personal communication). In the Canadian environment, the effect of sire group was not significant (p>0.05) for any of the traits studied. These results agree with those obtained in Chapter 2 where the differences in the hazard curves were found to be non-significant (p>0.05) in the Canadian environment. Although statistically nonsignificant (p>0.05), estimates of the Canadian sire group solutions tended to be lower than those of the New Zealand sire group. Table 3-1 Sire group solutions for measures of longevity within production environments 1  sire group Canadian environment: Canadian sires  New Zealand sires  New Zealand environment: Canadian sires  New Zealand sires  /DP  /DP  /DT  /DT  -0.19 (0.62)  -1.05 (1.07)  -0.15 (0.63)  -0.58 (1.08)  0.10 (0.65)  0.85 (1.10)  0.11 (0.66)  0.40 (1.11)  0.15 (0.34)  0.39 (0.19)  0.17 (0.33)  0.37 (0.18)  -0.16 (0.32)  -0.37* (0.16)  -0.16 (0.32)  -0.36* (0.17)  'Standard errors of the estimated sire group solutions are in brackets and were estimated from the prediction error variances obtained from the SURVIVAL KIT program (Ducrocq and Solkner, 1996). * Indicates significant (p<0.05) differences between sire groups.  57  In the New Zealand environment, the sire group effect was significant (p<0.05) for /DP and /DT whereas no significant sire group effects were observed for /DP and /DT. This implies that daughters of New Zealand sires exhibited significantly (p<0.05) lower culling rates (lower sire group solutions) relative to daughters of Canadian sires for /DP and /DT. The results showed that voluntary losses were different between sire strains in New Zealand. Even though differences between sire group solutions are significant (p<0.05) only in a few cases, the change in the rankings between environment for all the traits studied are striking. The change in rank between environments, revealed that sire groups ranked higher (lower culling rate) in their native environments relative to the imported sires as a group. This observation considered together with the results of the study of hazard curves in Chapter 2 suggested the existence of a genotype by environmental interaction at the macro level (strains change ranks in different environments). If there was no bias against daughters of imported bulls, the apparent superiority of native sires as a group would be attributed to the fact that their daughters are more adapted to the local environment compared to the daughters of foreign counter-parts. The differences in culling rates for /DP and /DT in the Canadian environment were, in fact, larger than in the New Zealand environment. However, the lack of statistical significance in the Canadian environment is likely due to the small number of observations in the data set. Table 3-2 shows the results of the likelihood test for the sire effect. The effect of sire was found to be significant (p<0.05) in both environments for the longevity traits investigated. This suggested significant variation due to differences between sires for the longevity traits studied. Table 3-3 depicts the estimates of heritabilities, their standard errors, the number of iterations at convergence and the log likelihood at convergence for various longevity traits. Since the standard errors are a function of sample size and heritability, they were rather large in this study. The number of iterations at convergence ranged from 57 to 203 for the traits investigated. In the Canadian environment, the heritability estimates ranged from 0.03 to 0.29 for the traits studied. The approximate standard errors were larger than heritability estimates for /DP and /DT.  58  Table 3-1. Likelihood ratio tests for the effect of sire Trait  n  Canadian Environment:  239  -2 log likelihood (herd + sire group)  -2 log likelihood (herd + sire group + sire)  Duration of productive life: functional (/DP) true (tDP) Duration of total life:  461.55 1311.69  433.68 1276.47  functional (/DT) true (rDT)  461.41 1311.79  433.34 1277.07  2067.17 7639.83  2014.67 7567.71  2067.22 7641.05  2014.55 7569.09  New Zealand Environment:  Sig  1  *  *  898  Duration of productive life: functional (/DP) true(rDP) Duration of total life: functional (/DT) true (rDT)  *  * Significant at p<0.05  Table 3-2 Heritability estimates for the measures of longevity  Trait  n  Canadian Environment: Duration of productive life: functional (/DP) true(rDP) Duration of total life: functional (/DT) true (rDT)  239  New Zealand Environment: Duration of productive life: functional (/DP) true(rDP) Duration of total life: functional (/DT) true (rDT)  898  h ±stderr  no. of iterations at convergence  0.19±0.16 0.03±0.06  68 203  433.68 1276.48  0.29±0.20 0.06±0.09  57 134  433.34 1277.08  0.05±0.04 0.05+0.04  92 79  2014.67 7567.71  0.09±0.06 0.08±0.06  96 72  2014.55 7569.09  2  -2 log likelihood at convergence  59  Estimates in the New Zealand environment were not as variable as in the Canadian environment. They ranged from 0.05 to 0.09. These results are consistent with the range of heritability estimates reported in the literature for /DP. Ducrocq (1987) reviewed the heritability estimates of longevity traits in dairy cattle and showed that they varied anywhere from 0.008 to 0.39. Heritability estimates as high as 0.12, 0.13, 0.148, 0.18, and 0.37 for various measures of longevity have been reported by De Lorenzo (1983), Smith and Quaas (1984), Plowman and Gaalaas (1960), Hargrove et al., (1969) and Wilcox et al., (1957), respectively. Perhaps heritability estimates for measures of longevity are expected to be low. since these measures of survival are components of 'functional' fitness (fitness in an artificial environment), and therefore, much of the additive genetic variance may be near exhaustion within stable environments. Falconer (1989), however, mentioned that, if the environment to which an equilibrium population is adapted changes, it alters the relative weighting of the components of fitness such that some additive genetic variance is introduced. Even though genetic improvement may be achieved by direct selection for the longevity traits studied, the progress is expected to be low and likely uneconomical. Further research in genetic improvement through correlated response is recommended.  3.6 CONCLUDING REMARKS The results indicate that there exists some genetic variation associated with measures of longevity. The functional definition of failure of a measure of longevity shows the importance of involuntary disposal whereas the true definition deals with the actual culling rate in the herd representing an indirect measure of the overall economic excellence of a cow as viewed by the dairyman. Genetic improvement in measures of longevity would allow better and more profitable cows to live longer and thereby increase voluntary disposals (higher replacement rates). Since selection based on milk yield alone has not brought about an attendant reduction in involuntary disposals (Ducrocq, 1987), direct selection for longevity traits is still desirable. In a sense overall economic excellence of a cow can be known only after discounting for income and expense items over its entire life, the longevity measure of choice would be duration of total life. However, this appealing characteristic of duration of total life is eroded in the true form of the failure definition. Note  60  that all animals (except those still alive) are considered culled regardless of reason for disposal in the true definition of failure. Since voluntary culling is more discretionary compared to involuntary disposal, variations (due to lack of uniformity) in thresholds used for voluntary culling between sire groups, herds and countries makes the comparisons of failure rates difficult to interpret. Comparisons of differences between functional measures of longevity (failure due to involuntary reasons) between strains, herds and countries does not have this inherent difficulty. This is precisely because involuntary losses are beyond the control of the farmer and therefore less likely to contribute to differences attributable to preferential treatment. In this regard therefore, functional duration of total life may be preferred over other longevity traits studied. However, since this trait would require new record keeping systems different from the existing ones (Dairy Herd Improvement and others), this approach would not be cost effective. Since functional duration of productive life does not have either of the two problems discussed above, it may be the final choice for use in selective breeding programs. Note, however, that since disposal occurring before first calving is ignored when modeling functional duration of productive life, there is an inherent danger of overestimating the breeding values for the bulls with poor daughter survival tofirstfreshening. The reverse is true for bulls with superior daughter survival tofirstfreshening. It is likely that this loss of information on disposals occurring before first calving is economically unimportant at least when compared to the cost of implementing new data collection schemes that is required to measure functional duration of total life. It is important to note that even though the heritability estimates in this study fell within the ranges reported in the literature, they were estimated using very small data sets and therefore had large errors. More research needs to be done using larger data sets to archive higher accuracies. Even though direct selection for the longevity traits studied is theoretically feasible, the progress may be slow and uneconomical. Since the rate of genetic improvement by direct selection for a quantitative character is determined by the intensity of selection, the accuracy of evaluation, the genetic variance as well as the generation interval it is suggested that use of analysis procedures (like failure time analysis) which allows bull daughters to be evaluated earlier in life would reduce the generation interval and thereby hasten genetic progress. Analysis of longevity traits by use of large data sets would go a long way in improving the accuracy of evaluation for the genetic parameters (heritability estimates). Use of large data sets would also allow the fitting of complicated models  61  to test the importance of such effects as herd-year-season of calving in influencing longevity traits. Technologies using multi-trait analysis already in use in linear models would be a welcome addition in nonlinear models used in this study. When these technologies become available, more reliable genetic correlations between longevity traits and other production traits should be investigated in the failure time construct. This would allow investigations into the economic feasibility of improving longevity related traits by correlated selection response.  62  3.7 REFERENCES 1. Allaire, F. R., H. E. Sterwerf, and T. M. Ludwick. 1977. Variations in removal reasons and culling rates with age for dairy females. J. Dairy Sci. 62:254. 2. Congleton, W. R. 1984. Profitability of dairy cow herd life. J. Dairy Sci. 67:661. 3. De Lorenzo, M. A. 1983. Non-linear estimation of dairy cow survival tofixedages. Ph.D. thesis, Cornell Univ., Ithaca, N.Y.; Microfilms, Intl., Ann Arbor, M.I., pl08. 4. De Lorenzo, M. A., and R. W. Everett. 1985. Prediction of sire effects for probability of survival to fixed ages with a logistic linear model. J: Dairy Sci. 69:501. 5. Ducrocq, V. 1987. An Analysis of Length of Productive Life in Dairy Cattle. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 6. Falconer, D. S. 1989. Introduction to Quantitative Genetics. 2nd Edition. Longman. London, p 270. 7. Foulley, J. L., D. Gianola, and S. Im. 1987. Genetic Evaluation for Discrete Polygenic Traits in Animal Breeding. In Advances in Statistical Methods for Genetic Improvement of Livestock. D. Gianola and K. Hammond (Eds). Springer-Verlag. p 361. 8. Gianola, D. And J. L. Foulley. 1983. Sire evaluation of ordered categorical data with a threshold model. Genetic Sel. Evol, 15:201. 9. Gill, G. S., and F. R. Allaire. 1976. Genetic and phenotypic parameters for a profit function and selection methods for optimization of profit in dairy cattle. J. Dairy Sci. 59:1325. 10. Hargrove, G. L., J.J. Salazar and E. Legates. 1969. Relationships amongfirstlactation and lifetime measurements in a dairy cattle population. J. Dairy Sci., 52:651. 11. Harville, D. A. 1977. Maximum likelihood approaches to variance component estimation and related problems. J. Am. Stat. Assoc. 72:3320. 12. Hogue, M., and J. Hodges. 1980. Genetic and phenotypic parameters of lifetime production traits in Holstein cows. J. Dairy Sci. 63:1900.  63  13. Hudson, G. F. S., and L. D. Van Vleck. 1981. Relationship between production and stayability in Holstein cattle. J. Dairy Sci. 64:2246. 14. Kennedy, B. W. 1981. Variance Component Estimation and Prediction of Breeding values. Can. J. Genet. Cytol. 23:565. 15. Meyer, K. 1989. Estimation of genetic parameters. Pages 161-167 in: Evolution and Animal Breeding. W. G. Hill and T. F. C. Mackay, eds. CAB. International, Oxon, UK. 16. Miller, P., L. D. Van Vleck, and C. R. Henderson. 1967. Relationships among herd life, milk production, and calving interval. J. Dairy Sci. 50:1283. 17. Plowman, R. D. and R. F. Gaalaas. 1960. Heritability estimates of longevity in Holstein Freisian cattle. J. Dairy. Sci., 43:877(Abstract). 18. Renkema, J. D., and J. Stelwagen. 1976. Economic evaluation of replacement rates in dairy herds. I. Reduction of replacement rates through improved health. Livest. Prod. Sci., 6:13. 19. Robertson, A. 1966. A mathematical model of the culling process in dairy cattle. Anim. Prod. 8:95. 20. Robertson, A and J. S. F. Barker. 1966. The correlation between first lactation milk production and longevity in dairy cattle. Anim. Prod. 8:241. 21. Ruiz, F. 1991. Relationships among Length of Productive Life, Milk Yield, and Profitability of United States, Canada, and Mexican Holstein sires in Mexico. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 22. Searle, S. R. 1971. Topics in variance component estimation. Biometrics, 27:1 23. Schaeffer, L. R. 1993. Variance Component Estimation Methods. University of Guelph, Guelph , Ontario. 21. Schaeffer, L. R., and E. B. Burnside. 1974. Survival rate of rested daughters of sires in artificial insemination. J. Dairy Sci. 57:1394. 24. Smith, S. P. 1983. The extension of failure time analysis to problems of animal breeding. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 25. Smith, S. P. 1990. Estimation of genetic parameters in non-linear models. Advances in Statistical Methods For Genetic Improvement of Livestock. D. Gianola and K. Hammond (Eds). Springer-Verlag. Germany.  64  26. Smith, S. P., and K. Hammond. 1988. Rank regression with log-gamma residuals. Biometrika, 75:741. 27. Smith, S. P., and R. L. Quaas. 1984. Productive lifespan of bull progeny group: failure time analysis. J. Dairy Sci. 67:2999. 28. Wilcox, C. J., K. O. Pfau and J. W. Bartlett. 1957. An investigation of the inheritance of female reproductive performance and longevity and their relationships within a Holstein-Friesian herd. J Dairy Sci, 40:942. 29. Van Doormaal, B. J , L. R. Schaeffer and B. W. Kennedy. 1985. Estimation of genetic parameters for stayability in Canadian Holsteins. J. Dairy Sci, 68:1763. 30. Vaupel, J. W. Harvald, B.Holm, N. V. Yashin, A. I, and Xiu, L. 1991. Survival analysis in genetics: Danish twin data applied to a gerontological question. In Survival Analysis: State of The Art. J. P. Klein and P. K. Goel (eds.). Kluwer Academic Publishers. Netherlands. 121-138.  65  4. CHAPTER FOUR: GENETIC CORRELATIONS AMONG MEASURES OF LONGEVITY WITHIN AND A C R O S S PRODUCTION ENVIRONMENTS. 4.1 ABSTRACT Data collected from afieldtrial involving the mating of 20 proven sires selected from New Zealand and 20 proven sires selected from Canada to cows in 20 New Zealand and 10 Canadian herds yielding 1,137 survival records used in this study. Breeding values for functional duration of total life (/DT), true duration of total life (/DT), functional duration of productive life (/DP) and true duration of total life (/DT) were estimated using nonlinear Cox Regression models both within each environment (Canada and New Zealand) and within environment and sire group. Solutions (relative culling rates) for sire groups within each environment indicated that Canadian sires ranked higher, as a group, in the Canadian environment for all the measures of longevity whereas New Zealand sires ranked higher in New Zealand. Significant differences between sire groups were observed in the New Zealand environment for /DP and /DT. This result, together with the general trend suggested the existence of GxE interaction at the macro level (strain changed ranks between environments) for /DP and /DT. Estimated genetic correlations (correlation of breeding values weighted by the accuracies of evaluation) between the same measure (e.g., DP) but different definition of failure (/DP and /DP) were lower compared to correlations between different measures (DP and DT) but same definition of failure (say /DP and /DT) in the Canadian environment. In New Zealand, estimated genetic correlations between all the traits studied were high. Estimated genetic correlations between the same traits in different production environments for each sire group were low and not significantly (p<0.05) different from zero. This suggested a GxE interaction at the micro level (individual sires changed ranks in different environments) for all the traits studied. It is  66  suggested that sires should be evaluated for superiority of survival traits in the environment within which their progeny are expected to perform  4.2 INTRODUCTION With the multitude of measures of longevity prevalent in the animal breeding literature, the question of genetic relationships between these measures becomes relevant. The concept of genetic correlation can be used to resolve problems related to the interaction of genotype with environment (Banos, 1994; Falconer, 1989) or simply whether the measures represent the same or different traits. The knowledge of genetic correlations is also used in selection decisions. When traits are highly correlated genetically, the trait with the higher accuracy of evaluation (r ) and ease of measurement would be chosen and selected for directly TI  expecting a correlated selection response in the other correlated trait(s). When traits are measured in different environments (countries), the knowledge of genetic correlations between them may help in the pair-wise comparison of bull proofs from different countries (Banos, 1994). Genetic correlations of less than unity shows the existence of genotype by environmental interactions. This may mean that the best genotype in one environment is not necessarily the best in another or simply an animal considered to have genetic superiority in one environment may fail to exhibit it in another. There are a number of measures of longevity available in the literature. For animal breeding (dairy) applications most researchers use the time scale in defining longevity. In this case, longevity may be defined as the age at disposal (DT) or survival from birth until disposal. Longevity is sometimes measured as length of productive life (DP) or survival from first freshening to disposal. Ducrocq (1987) showed some advantages of defining longevity as length of productive life compared to age at disposal thus: 1) Calves and heifers are culled for different reasons from those for lactating cows. Inclusion of calves, heifers and cows in the same evaluation would render the results difficult to interpret since these age groups are not necessarily culled for the same reasons. 67  2) The recording of culling (and reasons for) occurring before first freshening requires a new and different recording system than for older cows. (3) If age at disposal is not corrected for age at first calving, the evaluations of sires of early maturing daughters would be underestimated. Ducrocq (1987) explained that if you have two sires (A and B) whose daughters are culled at the same rate after entering the herd, and assuming daughters of sire A are, on average, freshening for the first time, two months later than those of sire B. The average age at disposal of the daughters of sire A will be higher than for that of sire B. Such an evaluation would result in selection against sire B for his daughters' superiority in maturity! Very little research work has been done on the relationships between various measures of longevity. The same is true for longevity studies of genotype by environmental interactions between different production environments. The aims of this study were, therefore: (1) To estimate the genetic correlations among measures of longevity within production environments and sire groups. (2) To test for G x E for the same traits measured in different environments.  4.3 MATERIALS AND METHODS 4.3.1 Definition Of Longevity Traits Studied In this study, the measures (traits) of longevity studied were all related to time and defined in Chapter 2, section 2.3.1.  4.3.2 Source Of Data The data were collected from a field trial as detailed in Chapter 2, page 28.  68  4.3.3 Statistical Analysis The general representation of the univariate proportional hazards model for the estimation of the breeding values within each environment for each of the four traits studied was: 0>(0 = Oo(0-exp {x(r)'b + z(/)'u}  [4.1]  where, O(r) is the hazard function of an individual depending on time / . Oo(0 is considered a positive function of t defined as the baseline hazard related to the aging process (Ducrocq, 1987). When the estimation of breeding values was done within production environments, the fixed effects in b were herd and sire group. Sire and error were the only random effects included in z. When the analysis was done within production environment and sire group, b included only herd and z included sire. \(t)' is a matrix of fixed effects with the corresponding parameter vector b, z(r)' is a matrix of random effects with corresponding parameter vector u . Estimates of b and u are obtained using a partial likelihood, a part of the full likelihood in which the baseline hazard function does not appear. The random effects in vector z(/)'were defined to follow a multivariate normal distribution where the covariance structure between individuals modeled by the matrix of genetic relationships as ~( 0 , Aa ) • &l is the sire variance and A is a matrix of additive genetic 2  relationships between sires. It has diagonal elements equal to inbreeding of the k  tn  l+Fk  , where  Fk  is the coefficient of  sire and off-diagonal elements equal to the numerator of Wright's coefficient of  relationship which is the additive relationship between two animals in question. If the two animals are not inbred, the coefficient of relationship is the same as the additive relationship. Therefore, A a describes the 2  matrix of additive genetic variances and covariances. The model, therefore, implies that the hazard O(r) is a product of a time dependent term (<X>0(0) which is considered a positive function of t defined as the baseline hazard related to the aging process (Ducrocq, 1987), and a term (exp(x(r)'b + z(/)'u}) representing how the explanatory variables influence failure independent of time. As it has been pointed out earlier, the hazard is defined as the conditional failure rate. That is, the probability of failure at time t given that the animal has survived to time t. The baseline hazard 69  (<Do(0) is taken to be the hazard when all the effects of the factors in the model are zero. Under the Cox model (Cox, 1972), each observation has a characteristic multiplier (i.e. exp()) that alters a common baseline hazard rate <po(0 • $(0 is described as the product of a baseline hazard function and a positive function of the explanatory variables supposed to influence the rate of disposal (Ducrocq, 1991).  4.3.3.1 Genetic Correlations Sire variances corresponding to heritabilities estimated previously (Chapter 3) within each production environment were used as priors in the procedure for estimating sire solutions (relative culling rates). After the estimation of breeding values for each measure of longevity, product-moment correlations of breeding values were calculated between and within production environments for all the traits studied. However, since simple product-moment correlations do not reflect fully the genetic relationship between any two traits, adjustments were made to estimate the genetic correlations using the Calo's method ( Calo et. al., 1973) as: r  cy  =riy/(RPf/*RPT -)°-  S  where r  7  Gij  is the estimated genetic correlation between traits / and j, ty is the simple product-moment  correlation between breeding values of sires for traits i and j and RPT is the average of the repeatability values of the breeding values for traits / and j within each production environment. The repeatability of a breeding value is the square of the accuracy of evaluation defined as the correlation between the predicted and true breeding value (r ). The r ' s in this study were calculated from the prediction error variances TI  T1  obtained from the survival analysis program written by S. P. Smith (1983) for the estimation of genetic parameters. Genetic correlations between the same trait in different environments were also estimated.  70  4.4 RESULTS AND DISCUSSION 4.4.1 Genetic Correlations Among Measures Of Longevity Within And Between Production Environments. Within environments, breeding values were obtained across strain of sire with the herd and sire group effects removed. The results in Table 4-2 show genetic correlations between the traits within environment (off-diagonal entries). Within production environment, the results generally showed that genetic correlations between different measures ( duration of productive life and duration of total life) but same definition of failure (functional and true) for the trait are close to one. For example, in the Canadian environment (lower off-diagonal entries), the estimated genetic correlation between /DP and /DP was only 0.40 where as the genetic correlation between /DP and /DT was 0.99. In the New Zealand production environment (upper offdiagonal entries), the genetic correlation between /DP and /DP was 0.54 and yet that between /DP and /DT was 0.99. Generally, the results suggested that it is the definition of failure (voluntary or voluntary + involuntary) and not the measure of longevity chosen that is important, when considering duration of productive life or duration of total life for genetic improvement programs. If data on culling before first freshening is absent , DP and DT would essentially represent the same measure due to the part-whole relationship contribution to the correlation. The contribution by the part-whole relationship to the genetic correlations between different measures but the same definition of failure may be high in the Canadian environment since only 2.4% of data (5% of data with disposal information) represented culling occurring before first freshening. In New Zealand this contribution was less due to the fact that a slightly higher proportion (4.2% of total data set) of animals had information on culled before first freshening. This represented about 21% of animals with disposal information occurring beforefirstfreshening. Therefore, the New Zealand data set provided more information on culling that occurred before first freshening whereas fewer herds contained information on culling beforefirstfreshening in the Canadian environment. However,  71  the fact that the genetic correlations between traits in the New Zealand environment mirrored those in the Canadian environment lends credence to the above conclusions Estimated genetic correlations between the same traits in different environments were low and negative (diagonal entries) for all the traits investigated in this study. This indicated the existence of a GxE for the traits and, therefore, the traits should be considered as separate traits in different environments.  Table 4-3 Genetic correlations among measures of longevity using both sires groups within environment. 1  Duration of productive life /DP  /DP  Duration of total life /DT  /DT  Duration of productive life: /DP  -0.21  0.54  0.99  0.59  tDP  0.40  -0.09  0.60  0.99  /DT  0.99  0.58  -0.19  0.53  /DT  0.48  0.99  0.44  -0.06  Duration of total life:  Entries in the lower off-diagonal represent correlations in the Canadian production environment. The upper off-diagonal entries are correlations in the New Zealand production environment. Entries in the diagonal (bold) are the correlations between the same traits in different environments.  4.4.2 Genetic Correlations Estimated Within Environment And Sire Group Breeding values for the measures of longevity studied were calculated for sires in each production environment expressed relative to their own strain base for this aspect of the study (within environment and strain of sire). Previously, breeding values were calculated and expressed within environment but across strain of sire. As Goddard and Beard (1994) as well as Banos (1994) pointed out, international bull evaluations are complicated not only by the potential presence of genuine genotype-by-environmental  72  interactions, but also due to the fact that records associated with imported animals may be biased due to preferential treatment. Harbers, Lohuis and Dekkers (1994) reported the impact of preferential treatment on the accuracy of evaluations. They found that accuracies were reduced by as much as 25% due to preferential treatment in a simulated data set. Table 4-3 shows the estimated genetic correlations between traits within environments. The diagonal entries indicate correlations between same measures and definitions of failure between production environments.  Table 4-4 Genetic correlations among measures of longevity for Canadian sires Duration of productive life /DP  /DP  Duration of total life /DT  /DT  Duration of productive life: /DP  -0.23  0.75  0.99  0.82  /DP  0.29  0.13  0.83  0.99  /DT  0.99  0.40  -0.19  0.74  /DT  0.35  0.99  0.32  -0.07  Duration of total life:  Entries in the lower off-diagonal represent correlations in the Canadian production environment. The upper off-diagonal entries are correlations in the New Zealand production environment. Entries in the diagonal (bold) are the correlations between the same traits in different environments.  The results are the same as those for genetic correlations calculated within environment across strain of sire. That is, correlations between different measures (DP and DT) but same definition (functional or true) of failure have higher correlation coefficients in the Canadian environment. Genetic correlations between traits in the New Zealand environment were all high (lowest was 0.74). It appeared that, in the New Zealand environment, all the measures of longevity studied could be considered as representing the same trait. Genetic correlations between the same trait in different environments were low and not significantly (p>0.05) different from zero. This suggest the existence of a GxE interaction at the micro level (re-ranking of sires) for all traits studied.  73  Table 4-4 shows the genetic correlations for New Zealand sires calculated within and between production environments. Diagonal entries are correlations between the same trait in different environments. The lower off-diagonal entries represent genetic correlations in the Canadian production environment while the upper off-diagonal entries those in the New Zealand production environment.  Table 4-5 Genetic correlations for New Zealand sires (calculated within production environment and strain of sire). 1  Duration of productive life /DP Duration of productive life: /DP  tDP  Duration of total life /DT  tDT  -0.22  0.87  0.99  0.97  0.09  -0.34  0.96  0.99  /DT  0.99  0.22  -0.21  0.88  tDT  0.12  0.99  0.10  -0.21  tDP Duration of total life:  Entries in the lower off-diagonal represent correlations in the Canadian production environment. The upper off-diagonal entries are correlations in the New Zealand production environment. Entries in the diagonal (bold) are the correlations between the same traits in different environments.  The results in the lower off-diagonal entries are very similar to those obtained for Canadian bulls (Table 4-3) in that the correlations within Canadian environment show a greater genetic relationship (higher correlation coefficient) between different measures but same definition of failure. In the New Zealand environment, the results again suggested that all measures could be taken to represent the same trait since the estimated genetic correlations between the traits were high (lowest was 0.87). Genetic correlations between the same trait in different environments (diagonal entries) mirrored those obtained using Canadian sires. They were low suggesting, again, that the traits studied should be considered as different traits in different environments.  74  4.5 CONCLUDING REMARKS The results of this study suggested that not much weight should be attached to differences between duration of productive life and duration of total life (DP and DT). It was found that the definition of failure was the critical element. In general, genetic correlations between different measures but the same definition of failure were higher than between same measure but different definitions of failure. Suggestions of genotype by environmental interactions (at the macro level) for /DP and tDT were also found. GxE interactions at the micro level were found for all traits studied. Banos (1994) reported that animals may rank differently under varying environments due to reasons such as: genuine genotype by environmental interaction (where animals perform differently under various production systems); differences in trait definitions; and heterogeneity of genetic parameters associated with distinct cattle populations. It is important to also note that selection for the same trait can cause divergence in two stocks if the environmental constraints in the two environments are different. Preferential treatment and heterotic effects may also contribute to the observed lack of unit genetic correlations between different production systems. Note that daughters of imported bulls in each environment were 'cross-breds' in this study. However, no superiority in culling rates was observed for daughters of imported bulls in this study. As pointed out by Dempfle and Grundl (1988), the phenomenon of genotype by environment interaction makes the definition of a superior animal very difficult. In this regard, Preston and Willis (1970) recommended that prospective herd sires should be evaluated under the same environmental conditions under which their progeny are expected to perform. Due to apparent superiority of the daughters of native bulls relative to daughters of imported bulls in relative culling rate observed in the results of Chapter 2 and the GxE interactions for the traits studied in this Chapter, it is suggested that the longevity traits studied be considered as different traits under the control of different cascades of genes in different environments. It is also suggested that a multi-trait analysis of the traits studied may yield more reliable genetic correlations for use in genetic improvement programs. In across country genetic evaluations for longevity, it would not be prudent for an importing country to put any emphasis on information on culling collected from the bull's native country.  75  4.6 REFERENCES 1. Banos, G., 1994. International genetic evaluation of dairy cattle. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:3. 2. Bar-Anan, R., M. Heiman, M. Ron and J. I. Weller. 1987. Comparison of proven sires from five Holstein-Friesian strains in high-yield Israeli dairy herds. Livest. Prod. Sci. 17:305. 3. Calo, L. L, R. E. McDowell, L. D. Van Vleck and P. D. Miller. 1973. Genetic Aspects of beef production among Holstein-Friesians pedigree selected for milk production. J. Anim. Sci. 37:676. 4. Cox , D. R. 1972. Regression models and life tables (with discussion). J. R. Statist. Soc, B, 34:187 5. Dempfle, L., and E. Grundl. 1988. Identification of superior animals and their use in improvement programs. Advances in Animal Breeding. Proc. Wld. Sympos.Agricultural University, Wageningen, Netherlands. P 56. 6. Ducrocq, V. 1991. Statistical analysis of length of productive life of dairy cows in the Normande breed. 42nd Annual meeting of the European Association for animal production. Berlin. Germany. 7. Ducrocq, V. 1987. An Analysis of Length of Productive Life in Dairy Cattle. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 8. Ducrocq, V. P, and J. Solkner. 1994. "The survival kit" - A FORTRAN package for the analysis of survival data. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 22:51. 9. Ducrocq, V. P., and Solkner. 1996. The Survival Kit, VI.0. User's manual. p26. 10. Falconer, D. S. 1989. Introduction to Quantitative Genetics. 2nd Edition. Longman. London, p 270. 11. Goddard, M. E., and K. T. Beard. 1994. The use of international genetic material in an importing country. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:23.  76  12. Harbers, A. G. F, M. M. Lohuis and J. C. M. Dekkers. 1994. Correlation of preferential treatment of moet families by including an environmental correlation in genetic evaluations. Proc. 5th World Cong. Genet. Appl. Livestock Prod. Guelph, Ontario, Canada. 17:11. 13. Preston, T. R., and M. B. Willis. 1970. Intensive Beef Production. 2nd Edition. Pergamon Press. Ontario. P166. 14. Smith, S. P., and R. L. Quaas. 1984. Productive lifespan of bull progeny group: Failure Time Analysis. J. Dairy Sci. 67:2999. 15. Steel R. G. D., and J. H. Torrie. 1980. Principles And Procedures of Statistics: A Biometrical Approach. Second Edition. McGraw-Hill Book Company, p 495.  77  5. CHAPTER FIVE: GENERAL DISCUSSION  5.1 Conclusions A summary of the general conclusions with respect to objectives outlined in each of the thesis Chapters is presented below:  5.2 Theoretical And Practical Considerations In Longevity Analysis A review of problems encountered in the analysis of longevity first emphasized a peculiar characteristic of endurance or survival data; the problem popularly known as the presence of censored observations. It has been pointed out that an attempt by animal breeders to account for censoring in survival data by defining opportunity groups has led to the definition of survival measures such as stayability to a given age or month. As pointed out by Ducrocq (1987), this forces longevity measures to be analyzed as categorical traits using linear model threshold approaches leading to scale problems and an obvious loss of information. In addition, many thresholds (e.g, 24 months, 36 months, 48 months, 78 months etc.) for dairy cattle have been used in research circles without an apparent agreement on a single threshold acceptable to all. Another problem is that of determining the portion of an animal's life that should be considered for longevity analysis. In the estimation of sire proofs, animal breeders commonly prefer the use of duration of productive life rather than total life because this information is regularly collected from milk recording schemes. The existence of an animal's record in a milk recording scheme usually starts with the calving date for the first lactation. This means that records of culling before first freshening are non existent in such schemes and, therefore, duration of total life is impossible to model. Advocates for the use of duration of productive life argue that this period is the most productive period of a dairy cows life and, therefore, more important to farmers. Critics argue that the profitability of a cow may only be known after discounting for income and expense items over the entire life span of the cows in the herd.  78  When measures of longevity are taken to be continuous (e.g. duration of productive life), animal breeders seem to be in agreement that, these measures should be adjusted for age at first freshening to avoid biased estimates of genetic parameters. Ruiz (1991) and Ducrocq (1987) have further argued that the measures should be adjusted for levels of milk production...or better still, herdmate deviations for milk since cows are culled based on their milk production level relative to the herd average. It is the opinion of this author that this may be particularly true when interest is in modeling the ability of the cows to delay involuntary culling (functional definition of failure) as most voluntary disposals are likely due to low milk production. However, when interest is in modeling the ability of the cows to delay disposal, involuntary or otherwise, it is unlikely that voluntary disposal or more particularly disposals due to low production would always account for the highest percentage of disposals. The best approach is to survey each data set for this characteristic to determine if these adjustments are necessary. It should be pointed out that in defining longevity traits the duration and failure scheme need to be explicitly defined. Adjustment factors and factors included in our models need not affect how we define these longevity traits. The definitions should simply be based on duration and failure scheme for simplicity and clarity. Until recently, animal breeders have not used statistical procedures appropriate for longevity measures (Smith, 1983). They have largely applied linear methods to survival data even if these methods do not account for censoring prevalent in endurance measures. The reason was that linear methods were numerically tractable and computationally feasible. With the advent of high speed computers and availability of software designed specifically for analysis of endurance measures (Smith, 1983; Ducrocq, 1987; Ruiz, 1991), animal breeders now can research longevity measures using the appropriate tools available to them. The sections that follow discuss, in general, results of longevity analysis using statistical methodologies employed on data collected from a field trial.  79  5.3 Disposal Reasons And Hazard Rates It has long been known that dairy cattle strains exhibit differential responses to extreme environmental constraints. This stems from the fact that animals selected and bred for performance in a given environment tend to acquire some local adaptations to that environment. Daughters of imported sires were culled at higher rates for most disposal reasons relative to daughters of native sires in both environments. This observation was further supported by results from the analysis of hazard curves of daughters of Canadian and New Zealand sires within each production environment. In New Zealand, daughters of New Zealand sires had significantly (p<0.05) lower hazard rates than those of daughters of Canadian sires for all traits except /DP. A similar trend in favor of daughters of native sires was observed in the Canadian environment even though differences in hazard rates between strains of sire were not significant (p>0.05) for all the traits studied. Similar results were observed in a study comparing mean culling rates of daughters of sires from several countries in Israel (Bar-Anan et al, 1987). This could be genuine genotype-by-environment interactions in which case animals exhibit different culling rates under different environmental conditions.  5.4 Heritability Estimates It is generally accepted that heritability estimates for measures of longevity are low. These traits may be seen as components of 'functional' fitness. Fitness in this case is defined under the production environment's constraints whereby selection is not directly related to reproduction (as in the definition of natural fitness) but rather to production in dollars. If selection and environmental constraints have been constant over time then much of the additive genetic variance associated with these traits may be near exhaustion. Falconer (1989), however, mentioned that if the environment to which an equilibrium population is adapted changes, it alters the relative weighting of the components of fitness such that some additive genetic variance is introduced. Most heritability estimates available in the animal breeding literature are for categorical measures of  80  longevity (Robertson, 1966; Hudson and Van Vleck, 1981) using linear methods. In their review of available literature, Smith and Quaas (1984), observed that heritabilities for productive lifespan seemed to depend on how the trait was measured. They reported that heritability estimates seemed to be higher for continuous compared to categorical measures of longevity . They went on to speculate that heritabilities for categorical measures are generally near zero due to scale problems as well as the fact that stayability is directly related to cow disappearance (censoring) which they viewed as diluting the genetic variability associated with the process governing disposal. Chapter 3 presented results of heritability estimated using an extension of Smith's (1983) nonlinear Cox regression model and the derivative-free REML approach. In Canada, heritability estimates ranging from 0.03 to 0.29 were found for measures of longevity. In New Zealand, estimates ranged from 0.05 to 0.09. However, due to the small size of the data set in Canada, the approximated standard errors were rather large. Even though these heritability estimates are from a relatively small data set, they are in general agreement with those reported by others using larger data sets and similar statistical methods (Smith, 1990). More studies with larger data sets need to be conducted utilizing the Cox (1972) non-linear proportional hazards approach to obtain reliable estimates for use selective breeding programs. These methodologies can handle large data sets but have not yet been extended to the multi-trait case. The use of large data sets would also facilitate examination of more complicated models allowing the estimation of the magnitude of herd-year-season of calving effects which were not studied in this project. No specific differences in environments considered here were attributed to the lack of unit genetic correlations between the same traits in different environments. However, in New Zealand, environmental conditions such as extensive pasture feeding , lack of housing facilities, smaller (inadequate for larger Canadian cows) handling facilities, pathogens, and climatic factors may have favored the daughters of adapted New Zealand bulls. Similarly climatic factors (much more severe winters), intensive rearing condition (conserved forages and supplemental concentrates) and pathogens may have favored the daughters of adapted Canadian bulls. Investigations of the genetic correlations between these longevity traits and specific environmental factors is recommended to shade more light on the specific causes of the apparent GxE observed in this study.  81  5.5 Genetic Correlations There was interest in investigating genetic relationships between the various measures of longevity studied, as well as, investigating the existence of GxE interactions for the measures of longevity in different environments. As has been mentioned in the previous chapters, knowledge of genetic relationships among traits is important in multi-trait selection strategies, in designing across country genetic evaluation systems as well as in estimating correlated selection responses. Genetic correlations of less than unity between the same traits in different environments suggest the existence of a GxE. In this study, genetic correlations were estimated between traits in the same production environment as well as between different production environments. Note that due to low heritability estimates obtained in Chapter 3 and relatively small data set, these genetic correlations estimated in Chapter 4 were estimated with large error variances. In the Canadian environment, genetic correlations between traits indicated that differences in the definition of failure (total Vs functional) were more important than differences in the measurement itself (productive Vs total life). Some proportion of the correlation could be due to the part-whole relationship that exists between duration of productive (part) and total life (whole). If there is no prepartum disposal information available, then they are the same trait. Data collected from milk recording schemes have this problem inherent in the system. However, when disposal information prior to first freshening is available, the contribution of the part-whole relationship to the observed genetic correlation is minimized since culling before first freshening is likely based on different reasons compared to disposal after first freshening. The New Zealand data set contained slightly more animals with pre-first freshening disposal information than the data from the Canadian herds (4.2% Vs 2.5% of the data sets). This represented about 5% and 21% of data with disposal information in the Canadian and New Zealand environments, respectively. In addition there is the problem of the presence of censored observations and its effect on the observed hazard curves. The discussion on this is deferred to the next section (section 5.6).  82  In the New Zealand environment, genetic correlations between traits were all relatively high (0.74-0.99) when estimated within environment and sire group. It was concluded that the measures of survival studied essentially represented one and the same trait. Neither the segment of time nor the definition of failure would affect selection decisions greatly. The genetic correlations between the same traits in different environments were near zero for all the traits studied. This result suggested the existence of GxE for survival in these two environments. In international bull evaluations, if the genetic correlation is substantially less than one a straight forward pooling of data for genetic evaluation purposes is inappropriate. If the correlation is greater than zero the information can be pooled by weighting the information by the genetic correlation (Banos, 1994). If the correlations are zero the information collected in one environment gives no information on performance in the other production system and therefore, genetic evaluations should be done only within production environments (countries). In other words, animals must be evaluated for superiority of survival traits in the environment within which their progeny are expected to perform.  5.6 General Remarks The preceding chapters have demonstrated the use of statistical applications suitable for the analysis of survival data in the dairy industry. The methods are also suitable for survival and endurance analyses in other fields of animal breeding including beef cattle breeding. These methods account for censoring and allow measures of survival to be analyzed as continuous variables. Some work is still needed to develop these methods for multi-trait analyses. In this study the proportion of animals found without a reason for disposal ranged from 51% (Canada) to 80% (New Zealand). In the study of 12,705 survival records of Holstein dairy cows in Australia from field data, Smith (1990) reported that 74% of the cows did not have a disposal code. This leads to a higher percentage of data having censored records as well as underestimation of failures. In the functional definition of failure for the traits studied, a large number of censored observations (especially those lacking disposal codes) may bias the estimated hazard rates downwards since these animals are assumed to have survived. The opposite effect is expected in the true definition of failure for the traits studied. This is so  83  because, in the functional definition, all the censored observation as well as those that were voluntarily culled were not considered culled where as in the true definition, all except those still alive were considered culled. The desired scenario is to have every animal with disposal information. Just as there is an elaborate system of milk recording, perhaps, animal breeders should rededicate their efforts in finding ways of encouraging milk producers to consistently indicate disposal information in their records. More efforts should be directed towards resolving problems of ill defined and, sometimes, overlapping disposal reasons introducing more subjectivity when it comes to partitioning culling into voluntary and involuntary categories. Perhaps farmers should be encouraged to record simply whether they felt the disposal was voluntary or involuntary either with individual reasons for disposal or without! If the disposal reason or reasons cannot be distinctly categorized into voluntary or involuntary classes, the farmer is in a better position to subjectively favor one or the other of the two classes. In summary, this study showed that daughters of native bulls tended to exhibited lower culling rates relative to daughters of imported bulls perhaps due to superior adaptability to local conditions. It also generally showed that the segment in an animals life may not be as significant in modeling survival as is the definition of failure during whatever period in an animals life the researcher chooses to use. However, this observation may not hold for all environments as was the case in New Zealand where the definition of failure had little impact (genetic correlations were high between all the traits studied). Genetic correlations between the same traits in different environments were near zero indicating the existence of a GxE interaction at the micro level for all traits studied. Further research in the relative economic importance of survival as well as in genetic correlations between these measures, between them and other production traits and between them and other environmental factors (e.g. nutrition) all in a frame work of multi-trait failure time construct is necessary to shade more light on the complex nature of the relationships. A scientific demonstration of the economic importance of survival relative to milk and type traits already included in lifetime production index may go a long way in its acceptability by producers who currently rely on type traits as predictors of longevity. Since the heritabilities for all the survival traits studied were estimated with low accuracies, more research with larger data sets should be encouraged to arrive at more reliable estimates for use in genetic improvement programs. Studies in  84  indirect selection for improvement of these longevity traits studied by use of direct selection of better recorded traits are also encouraged. From a purely statistical consideration, it can be pointed out that the small size of the data set used in this study presented a few problems. Over-stratification by use of more complicated models resulted in the failure of the failure time procedure to run to completion due to numerical problems. Therefore, very simple models were fit resulting in some factors such as year-season of calving being excluded from the analysis. A balance has to be reached between the number of strata and the amount of information contained in them. Too many strata with very little information in them present numerical problems in these procedures (Smith and Quass, 1984).  85  5.7 References 1. Ducrocq, V. 1987. An Analysis of Length of Productive Life in Dairy Cattle. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 2. Falconer, D. S. 1989. Introduction to Quantitative Genetics. 2nd Edition. Longman. London, p 270 3. Hudson, G. F. S., and L. D. Van Vleck. 1981. Relationship between production and stayability in Holstein cattle. J. Dairy Sci. 64:2246. 4. Robertson, A. 1966. A mathematical model of the culling process in dairy cattle. Anim. Prod. 8:95. 5. Robertson, A and J. S. F. Barker. 1966. The correlation between first lactation milk production and longevity in dairy cattle. Anim. Prod. 8:241. 6. Ruiz, F. 1991. Relationships among Length of Productive Life, Milk Yield, and Profitability of United States, Canada, and Mexican Holstein sires in Mexico. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 7. Smith, S. P. 1983. The extension of failure time analysis to problems of animal breeding. Ph.D. Thesis, Cornell Univ., Ithaca, N.Y. 8. Smith, S. P., and R. L. Quaas. 1984. Productive lifespan of bull progeny groups: Failure Time Analysis. J. Dairy Sci. 67:2999.  86  

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