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* Department of Animal Science, Agricultural University of Norway, P. O. Box 5025, N-1432 Ås, Norway
Department of Animal Sciences, University of Wisconsin, Madison 53706
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Key Words: Bayesian methods • clinical mastitis • genetic evaluation • longitudinal threshold model
Abbreviation key: CM = clinical mastitis, EFD = expected fraction of days without mastitis, NRF = Norwegian Cattle
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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There has been an increased interest in using test-day (longitudinal) models for genetic analysis of dairy production traits, e.g., see reviews of Jensen (2001) and Swalve (2000). Applications have focused mainly on continuous traits, such as milk yield and SCC, and several random regression models have been discussed in the literature. A different treatment is needed when the longitudinal series consists of discrete responses. Rekaya et al. (1998) suggested a Bayesian approach for analyzing longitudinal binary traits, and presented an application to clinical mastitis in Holsteins, with data consisting of series of binary responses. The advantages of using a longitudinal model over a cross-sectional model for binary data include the ability of taking multiple treatments and time aspects into account, as well as of accounting for environmental effects peculiar to each test-interval. A longitudinal model can give a dynamic description of genetic variation in the course of lactation, although this is highly dependent on the adequacy of the definition of "genotype at time t". Also, records in progress and "incomplete records" due to culling can be handled in a longitudinal model. See Heringstad et al. (2001) for a description of difficulties involved when handling such records in a cross-sectional analysis. A disadvantage, however, is that the longitudinal model is more computationally demanding. This is because more records per animal results in larger datasets and because more parameters usually are needed than in a cross-sectional specification.
In this study, sequences of binary CM records on first-lactation NRF cows were analyzed with a longitudinal threshold model with the following objectives: 1) to examine alternative criteria for ranking and selection purposes, and 2) to compare sire evaluations obtained from the longitudinal model with those from cross-sectional models.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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. The pedigree file represented 437 males (via relationships through sires and maternal grandsires of the 245 sires above). All cases of veterinary treated CM in first-lactation, from 30 d before calving to culling, second calving or 300 d after calving, whichever occurred earlier, were included. About 77% of the cows did not have CM during first lactation, while 16%, 5%, and 2% had 1, 2, or 3 cases, respectively. Only 315 cows had more than four recorded CM treatments. Within cows with mastitis, the mean number of CM treatments during first lactation was 1.5. For each cow, the period going from 30 d before calving to 300 d after first calving was divided into 11 intervals of 30 d length each. Within each such interval, absence or presence of mastitis was scored as "0" or "1", respectively, based on whether or not the cow had at least one veterinary treatment of CM recorded. Cows culled before 300 d after calving, were scored also in the interval in which they were culled, irrespectively of the number of completed days in the last interval. Cows that were culled and did not have mastitis in the last interval, were scored as "0" even though the period at risk was shorter than 30 d. This would overstate the resistance to mastitis of that cow in that particular period.
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). If a cow had mastitis within a given interval, the day of treatment was used in the time dependent function (described later); otherwise, the midpoint of the interval was used as time. About 26% of the cows were culled before 300 d after calving, and the cumulative culling rate is shown in Table 2. About 6%, 12%, and 20% of the cows were culled prior to 60, 150, and 240 d of lactation, respectively.
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) to CM was expressed as:
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where
| ijklmt | = | liability to CM at time t of daughter k of sire m calving in year i, herd j, and age x season class l;
| yi | = | effect of year of calving i (i = 1990, 1991, 1992);
| hj | = | effect of herd j (j = 1, 2, ..., 5286);
| pk | = | "cow-specific" effect common to all intervals (k = 1, 2, ..., 36,178);
| al | = | 5 x 1 vector of "fixed" regressions by age x season of calving class l. There were 12 age x season classes, resulting from combining 3 age classes (<24, 24 to 27, and >27 mo) with 4 season levels (March-May, June-August, September-November, and December-February);
| sm | = | 5 x 1 vector of random regressions peculiar to sire m (m = 1, 2, ..., 437);
|  4(t) | = | incidence vector containing Legendre polynomials on time up to order 4 relating time t to the liability of a cow at a specific time; and
| eijklmt | = | residual effect, which follows a normal process N(0,1).
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The common "cow-specific" effect, pk, accounts for covariances between liabilities in different periods, assuming a constant correlation between periods. Note that pk includes permanent environmental effects and genetic effects other than those accounted for in the "transmitting ability at time t" of the sire of the cow, which is 
. The variance of the residual distribution was assumed constant from period to period, and set equal to 1.
Bayesian Analysis
Prior distributions.
Independent proper uniform priors were assumed for each of the year effects and for the "fixed" regressions by age x season classes:
 | ([1]) |
 | ([2]) |
where l = 1, 2, ..., 12 and o = 0, 1, ..., 4. For example a3,0 denotes the random intercept of the regression function for the third age x season class. Hyperparameter values were ymin = amin = -99 and ymax = amax = 99. The herd (hj) and "cow-specific" (pk) effects were assigned independent normal priors with means zero and unknown variances. The prior distributions were:
 | ([3]) |
 | ([4]) |
where 
is the variance between herds, and 
is the variance of "cow-specific" effects. Independent scaled inverse Chi-square prior distributions were assumed for the variances of herd and "cow-specific" effects:
 | ([5]) |
 | ([6]) |
where 
h = p = 2 are the degrees of freedom parameters, and 
h = p = 0.1 are the scale parameters. A multivariate-normal prior distribution was used for the sire regression effects, as follows:
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where sm is the 5 x 1 vector of regression coefficients for sire m, and G = {gij}, i,j=0,1,...,4, is the 5 x 5 covariance matrix of the sire regression coefficients, assumed common to all sires. Letting 
, where 437 is the number of sires in the pedigree, the joint prior distribution of all regression coefficients for all sires was:
 | ([7]) |
where A is a 437 x 437 matrix of additive relationships between sires. The matrix G was assigned an inverse Wishart prior distribution:
 | ([8]) |
where g is the degrees of freedom parameter, and Vg is the scale matrix.
Posterior distributions.
The joint posterior density of all the unknowns is proportional to the product of the densities in [1
] through [8], times the conditional distribution of the observations. The latter was product Bernoulli over intervals and cows, and with a probability of response (mastitis) peculiar to each cow and each interval. Draws from the posterior distributions of the parameters were obtained using a Gibbs sampler with data augmentation (Sorensen et al., 1995). After augmentation with the cow-interval liabilities, all the fully conditional posterior distributions of the parameters can be derived in closed form as described by Sorensen et al. (1995) and Sorensen and Gianola (2002). Chang (2002) gives details of the Gibbs sampling scheme applied.
Convergence diagnostics.
Visual inspection of trace plots and the convergence diagnostic method of Raftery and Lewis (1992) were used to decide total chain length and the length of burn-in. Inferences were based on 160,000 samples, with the previous 40,000 iterations discarded as burn-in.
Genetic Evaluation
Sire evaluations.
The probability of mastitis at time t for a specific daughters of a given sire is given by 
where 
(.) is the standard normal distribution function. Sire evaluations obtained with the longitudinal model may be presented as curves depicting the probability of mastitis along lactation. The curve for an individual cow depends on the specific value of the systematic effects associated with the records. In order to circumvent this dependence, the required sire-specific probability curve can be approximated as (µmti), where 
is the probit for time ti, and 
is the 5 x 1 vector of Legendre covariates evaluated at time ti. The information contained in the curve can be summarized into several single numbers that can be used for ranking and selection of sires. For example, consider the expected fraction of days without mastitis (EFD) in an arbitrary time interval (t1,t2). This may be calculated for each sire as
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Here t1 and t2 are the first and last days of the interval, respectively, (µmti) is the probability of mastitis at d ti for an infinite number of daughters of sire m, and 
gives the expected number of days with mastitis in the interval from t1 to t2. For EFD, (µmti) was computed as 
, where µti, as noted earlier, is the probit for interval ti; 
m is the 5 x 1 posterior mean vector of the sire-specific regression coefficients, and 4(ti) is as before. EFD were calculated for the following time intervals: 1) the total 330 d period, EFD(-30, 300); 2) from 30 d before to 30 d after calving, EFD(-30, 30); 3) from 30 d before to 120 d after calving, EFD(-30, 120); and 4) from 120 d to 300 d after calving, EFD(120, 300). Also, the sire rankings from EFD were compared with those obtained from two cross-sectional threshold model analyses where mastitis was treated as a single binary trait. One cross-sectional evaluation was for the interval from 30 d before to 300 d after calving (P300), and the second was from 15 d before to 120 d after calving (P120), which is the interval used for genetic evaluation of CM in Norway. These evaluations had the form (µ + 
m), where m is the posterior mean of transmitting ability of sire m and µ is the probit corresponding to the overall mean incidence of mastitis.
Heritability and genetic correlations.
The longitudinal model induces a covariance function, from which time-dependent genetic parameters can be contrived. Define the "transmitting ability at d ti", at the liability scale, as 
. Then, the between-sire variance of liability to CM at d ti can be defined 
, where 
is as before and G is the covariance matrix of the sire regression coefficients. The intra-herd heritability of liability to CM at any d ti of lactation can be defined as 
, and the genetic correlation between liability to CM at d ti and tj can be calculated as 
. The time-dependent heritabilities and genetic correlations can be inferred by replacing G by its posterior mean. Note that the only time-varying elements in the preceding functions are the Legendre covariates 
and 
. Hence, the values of the covariates drive the relationship between the resulting genetic parameters and time, as G is static. Caution should be exercised when interpreting these genetic parameters, i.e., the covariance function would have a biological meaning provided 
is an adequate representation of the genotype of sire m at time ti.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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and are given in Table 3. The posterior mean of g00 was 0.061, and a 90% credibility set is given by the interval (0.035 to 0.094). To the extent that the longitudinal model describes well the time-evolution of genotypes for liability to CM, this implies that the genetic variance at calving time in the underlying scale is about 0.24. The variances of the other regression coefficients were smaller. The evolution of posterior means of heritabilities of liability to CM along lactation is shown in Figure 1. Heritability ranged between 0.07 and 0.13 before calving, from 0.04 to 0.15 during the first 270 d of lactation, and increased sharply thereafter. Except for the last interval of lactation (270 to 300 d) the contrived heritability values are in accordance with estimates from cross-sectional threshold model analyses of clinical mastitis treated as a single binary trait, which range from 0.06 to 0.12 (Lund et al., 1999; Kadarmideen, et al., 2001; Heringstad et al., 2001; 2003). The trajectory of heritability along lactation, up to 270 d, was in agreement with Chang et al. (2001; 2002) who analyzed the same data set with an 11-variate threshold model, assuming that mastitis was a different trait in each period. It must be emphasized, however, that the steep increase in heritability observed after 270 d is driven primarily by the sequence of the Legendre covariates 
(270), (271), ..., (300). The disagreement with the multi-trait analyses of Chang et al. (2002) suggests that the definition of genotype in the longitudinal model approximates badly the genotype as defined in a multi-trait model, at least in this interval.
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. Most correlations were larger than 0.5. Genetic correlations between adjacent days were close to 1, but the values decreased as the interval between days increased, and the more so for early and late lactation. That is, the genetic correlation function is more sensitive with respect to time at the edge of the time domain. Negative correlations were found only when days in late lactation, i.e., subsequent to 270 d after calving, were involved. In a multivariate threshold model analysis, Chang et al. (2001; 2002) found that the posterior mean of the genetic correlations between eleven 30 d intervals ranged from 0.13 to 0.55. Lund et al. (1999) estimated genetic correlations of 0.89, 0.80, and 0.67 between clinical mastitis measured in early lactation (-10 to 50 d after first calving) and in the periods 50 to 180 d, 180–350 d, and 50 to 350 d after first calving, respectively. Genetic correlations between liability to CM at different stages of lactation much lower than 1 indicate that gene expression for CM varies in the course of lactation.
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illustrates that the trajectory of the expected probability of CM of daughters along first lactation varies between sires. For example, sire 10 had a peaked curve, i.e., high probability of CM in early lactation, whereas sire 139 had a relatively flat curve, with a low probability of CM throughout lactation. Sire 157 would be expected to sire daughters with increased probability of CM both in early and late lactation. Variation in the trajectory of the curve suggests that sire ranking depends on the part of lactation considered. Time-dependent re-ranking of sires is illustrated in Table 5, where sire rankings based on the average expected probability of CM of daughters in each of the eleven 30 d intervals were calculated for the top 10 sires in interval 2 (0 to 30 d after first calving). Some sires had a consistently high ranking throughout lactation, while others ranked highly in early lactation, but had much lower ranking later on. For example, the sire ranked as the second best in interval 1 was ranked among the top 3 sires except for the last 2 intervals, where his rank was 45 and 232, respectively.
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. The "best" sires had a much lower peak around calving, and a lower probability of CM throughout lactation than the lowest ranking sires.
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. In general, rank correlations between all sire evaluations that addressed the first part of lactation were high (
0.89). Rank correlations between EFD (120, 300) and other evaluations were lower (0.64 to 0.87). This is expected, as EFD (120, 300) makes use of the regression curve for the last part of lactation only. Although, rank correlations were strong, this does not imply that the same sires would be selected, especially if selection intensity is high. For example, if 10 of the 245 sires were to be selected, only 5, 4, or 7 of them would be in common if selection were based on EFD (-30, 300) versus EFD (-30, 30), P120 or P300, respectively. Table 7 shows the top 10 sires based on EFD (-30, 300) and their ranking for the other sire evaluations. For some sires there was good agreement between different selection criteria. For example, the top sire based on EFD (-30, 300) was also the top sire for four out of the five other selection criteria. On the other hand, the sire ranked as 9th in EFD (-30, 300) was ranked as 74th based on the cross sectional model P120.
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As opposed to, e.g., milk production traits that are measured at fixed test-days, mastitis can occur at any day of lactation. In order to make sequences of binary responses, the first lactation was divided into intervals, and mastitis was scored as 0 or 1 within each interval. Here, we used eleven intervals of 30 d length. Further studies are needed to examine effects of the number and length of such intervals.
For illustration purposes, we used the expected fraction of days without mastitis for sire ranking. However, other criteria may be of interest. For example, more weight could be placed on mastitis in early lactation (since these cases may result in higher costs) by computing the probability of no mastitis for given intervals for each sire, and then weighting the information in some manner.
A main reason for using clinical mastitis information only from the early part of lactation, as in the current genetic evaluation in Norway, is to reduce possible sampling biases caused by culling of cows. In a longitudinal or multi-trait model (Chang et al., 2002) this is not a problem because both "incomplete lactations" and records in progress can be accommodated. Other advantages of using a longitudinal model for CM are the ability to take multiple treatments and time aspects into account, and the possibility of accounting for environmental effects peculiar to each test-interval. However, the longitudinal model may not be effective in capturing differential gene expression in different parts of lactation. This is due to the fact that the genetic or sire variance-covariance structure (e.g., the matrix G) is static. The dynamics of the model are induced by the Legendre function, which, obviously, does not have a genetic component. A multiple-trait model may therefore be a more biologically sensible specification.
The results illustrate clearly that choice of the model and of the selection criteria can affect selection of sires markedly, even when rank correlations are high.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Corresponding author:
Bjørg Heringstad; e-mail:
bjorg.heringstad{at}ihf.nlh.no.
Received for publication November 28, 2002. Accepted for publication March 6, 2003.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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