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* Instituto de Ciências Biomédicas Abel Salazar and
Centro de Estudos de Ciência Animal, Universidade do Porto, Rua do Monte-Crasto, 4485-661 Vairão, Portugal
Department of Animal Science, Cornell University, Ithaca, NY 14853
Corresponding author:
J. Carvalheira; e-mail:
jgc3{at}mail.icav.up.pt.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Abbreviation key: CL = confidence limit, HTD = herd-test-date, LTE = long-term environmental effect, STE = short-term environmental effect, TD = test day
Key Words: test-day animal model • dairy cattle • autoregression • genetic evaluation
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Other approaches (Meyer et al., 1989; Ptak and Schaeffer, 1993; Schaeffer and Sullivan, 1994; Swalve, 1995; Carvalheira et al., 1998) directly analyzed daily yields instead of TD-adjusted cumulative 305-d lactation records to genetically evaluate sires and cows for milk production. Date of test effects may explain more of the systematic environmental variations (Ptak and Schaeffer, 1993; Swalve, 1995). More TD records per cow also promote more accurate prediction of genetic merit. Especially following advice in Quaas (1984) and Wade and Quaas (1993), Carvalheira and co-workers (1998) relaxed the usual assumption of unitary correlation between TD within cow by implementing first-order autoregressive structures within and between lactations to fit short- and long-term environmental covariances among repeated TD records. Results from this study indicated that an autoregressive TD animal model would provide larger additive genetic variance and heritability, more accurate estimates of individual genetic merit, and nearly double the theoretical annual genetic gain (200 kg) in milk for US Holstein. Consequently, our objectives were to define and evaluate a multiple lactation autoregressive repeatability model, challenging its capacity to accurately estimate the variance components and correlations underlying simulated data, which either included or ignored long-term environmental effects.
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MATERIAL AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Records of daily milk yield may be depicted as repeated observations. For example, a simple repeatability TD model may be defined as:
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where
yijkmnis the observed daily yield,
HTDiis the fixed effect due to cows tested in the same herd (H) and test date,
Age(H)jis the fixed effect due to the jth age class at calving within herd,
DIM(H)kis the fixed effect due to cows tested in the kth days in milk (DIM) class within herd,
amis the random effect of animal,
pmis the random effect accounting for environmental covariances among TDs within cow, and
eijkmnis the random residual term.
When nonadditive genetic components are assumed to be negligible, a simple repeatability model is an extension of the usual breeding value model, with the further assumption of an additional nongenetic covariance from repeated observations on the same animal. This covariance among residuals of repeated records on the same animal results in a structure typically assumed to be:
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where 
ijis the Kronecker delta, or, in matrix notation for three repeated records
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and having the phenotypic (co)variance structure
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where
represents the variance unique to each observation,
represents the environmental covariance between pairs of records on the same animal (assumed equally correlated for all pairs), and
amm
2a represents the genetic covariance assuming unitary genetic correlation between records on the same animal, which implies that the same genes are responsible for milk production throughout the lactation.
Quaas (1984) called this the "simplistic repeatability model" because it is unlikely that all records are equally correlated regardless of adjacency. A more realistic general structure was suggested (Quaas, 1984), which would impose an autoregressive covariance structure for residuals. The simplest of these structures is a stationary first-order autoregressive process, which is applicable with equal intervals (i.e., TD records taken 1 mo apart), as shown here for three repeated observations
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where
2e represents the environmental variance, and
represents the autocorrelation with || <1, and which involves no more parameters (two in this case) than a structure ignoring the autoregressive process. Contiguous records in an autoregressive model are equally correlated if the interval between them is constant, which yields a decaying correlation between noncontiguous records with increasingly greater separation in time.Fcan be easily factored intoLDL', whereL(L') is a lower (upper) triangular matrix andDis a diagonal matrix. This factorization is especially useful for computing the determinant ofF, e.g., for evaluating the likelihood function.
This structure is also computationally tractable because its inverse is easily obtained without formal inversion, resulting in a tridiagonal matrix, if all contiguous observations are present and equidistantly spaced:
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or
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However, it may create problems for interpreting environmental influences because its structure is assumed entirely autocorrelated, which precludes independent effects peculiar to each TD record. Therefore, a potentially more realistic portrayal of random environmental effects on daily milk yield may be (for three repeated observations):
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where
tandrare two environmental components resulting from partitioning the environmental variance (e) withtfollowing a first-order autoregressive process in repeated TD yields andris an independent effect,
2t represents the environmental covariance among repeated daily yields,
Frepresents the first-order autocorrelation structure associating records from each cow as defined above, and
2r represents the residual variance common to all observations.
Thus portrayed, the two environmental components represent separate influences on each TD record. One component (2t) comprises short-term effects (STE) within a lactation (e.g., dietary quality and intake, weather, minor injury, estrus). These positive or negative fluctuations may be canceled or reversed with time, thus imparting a pattern or structure (i.e., a stationary, first-order autoregressive process between equally spaced, monthly TD records), where the correlations between TD records decay in time. The other component(2r) comprises all other sources of unaccounted temporary variation that independently affect TD records.
Environmental effects that permanently influence subsequent lactations include body size, disease, and other health events, and physical injury. Conceptually, this is the classical definition of a permanent environmental effect, where close events in an autoregressive structure are assumed more highly correlated than distant events. These correlated long-term environmental effects (LTE) may not be separately partitioned from the STE effect (which accounts only for correlations between TD within a lactation). A repeatability TD model involving multiple lactations needs to anticipate potential covariances, including those occurring between repeated lactations. The unequal time lag between consecutive lactations (the dry period) precludes using the STE effect to also portray potential covariances between lactations. The report by Harville (1979) may have contained the earliest suggestion to model the cow’s permanent environmental effect as a first-order autoregressive process to more accurately reflect temporal effects due to cow from one lactation to the next. Factors that may cause correlation between lactations have long-term influence in the sense that they should similarly influence repeated lactations (but possibly in different degrees) with noncanceling, carryover effects. Consequently, an autoregressive structure may be a realistic approach to portray these LTE effects, which relaxes the restriction (or assumption) that covariances across lactations are equal and invariant (Harville, 1979; Quaas, 1984).
Consequently, the conceptual multiple-lactation TD animal model was defined:
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where
yijkLmnis the TD record,
HTDiis the fixed effect due to measuring the milk yields of cows on the same test date in the same herd,
Agejis the fixed effect due to the jth age at calving,
DIM(H)k(L)is the fixed effect due to cows tested in the same DIM class within herd and lactation,
am is the random effect of animal,
pm(L)isthe random effect of LTE following a first-order autoregressive process across lactations,
tn(mL) is the random effect of STE nested within cow and lactation, assumed independent between lactations, and following a first-order autoregressive process within cow and between TD, and
rijkLmn is the random residual term.
The model in matrix notation is:
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wherey
N(Xß,V),ßis the unknown vector of fixed effects that, with knownX, defines the mean;a,p, and t are vectors representing the random effects due to animal, LTE, and STE, which are, respectively, associated with records inybyZ,M, andQ;ris the vector of residuals; andVis the (co)variance matrix. The expectations and (co)variance for this model are (for the case ofLequal to three lactations per cow and invoking previous definitions):
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and then
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where q1equals the number of animals being evaluated,q2equals the number of animals having records,nequals the number of TD of a particular female, andNequals the total number of records in the analysis; 
represents the Kronecker tensor product;Ais the numerator relationship matrix,Iis the identity matrix, andFis the autocorrelated block diagonal corresponding to themth cow within theLth lactation. This parameterization also permits different variances of the STE effect for multiple lactations (2tL). Note thatQ=Iwith this design. Therefore, the corresponding mixed model equations are:
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Simulated Data
Two datasets were simulated according to model assumptions to evaluate the multiple lactation autoregressive repeatability methodology, and to assess the efficiency of achieving the desired convergence. One dataset contained an LTE effect; the other one did not. The TD animal model was challenged to accurately retrieve the variance components and correlations underlying the data simulated with these different effects. Although plausible parameter estimates are not a guarantee that procedure is correct, the reverse implies incorrect or inaccurate methodology.
A pedigree file containing 50 sires and 500 cows was generated and used to simulate datasets of 15,000 records (10 TD records per cow in each of three lactations). The data were organized in groupings according to fixed effects in the model: 44 test dates, 20 DIM subclasses per lactation, and 4 age-at-calving classes. Fifty replicates of each dataset were analyzed: the one including the LTE effect had 10 parameters to estimate (six variance components and four autocorrelations), and the one ignoring it had eight parameters (five variance components and three autocorrelations). Starting values for variance components and correlations were deliberately 10% larger than the true values for every analysis. Convergence was achieved when the variance of the –2 log likelihood functions from all points defining the simplex polytope was <10–6. For each replicate, the same log likelihood (up to the fourth decimal place; Boldman et al., 1995) in each of the two last runs was obtained with three or four cold starts.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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. Each model performed acceptably well for the data possessing identical effects (emboldened values, in Table 1). For these scenarios, the resulting parameter estimates were close to the true values within the limits of confidence (CL) at 
= 0.05. Furthermore, the model including an LTE effect was sensitive to the absence of this effect in the data that did not contain an LTE component. The small values obtained for
and 
p were plausibly from a sampling effect from imposing nonzero requirements on the parameter space. However, the analytical model omitting LTE overestimated most parameters from data containing an LTE effect.
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The number of parameters to estimate (variance components and correlations) may restrict the applicability of a linear mixed model for TD genetic analysis. The relatively small number of parameters considered by an autoregressive model is advantageous in this sense. The additive genetic (co)variance structure in this study and analysis of field data (Carvalheira et al., 1998; Carvalheira, 2001) was fitted with unitary correlation between TD and lactations under assumption that the same genes similarly affect milk yield expression throughout a cow’s productive life. Study is warranted to further evaluate models that would represent genetic (co)variances with autocorrelation structure.
As expected, substantial iteration was required to achieve convergence due to the number of parameters to estimate, which, as pointed out by Boldman et al. (1995), is worth considering when using DFREML with the simplex in variance component estimation. When the analytical model mismatched effects contained in the data, convergence sometimes required over 400 iterations. Therefore, besides "poor" starting values, an incorrect model may also affect convergence requirements. Table 2 shows the average numbers of iterations required to achieve convergence for the replicates requiring fewer than 400 iterations.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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ão para a Ciência e a Tecnologia (FCT) and FEDER (EU). Received for publication October 19, 2001. Accepted for publication February 26, 2002.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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This article has been cited by other articles:
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  J. Vasconcelos, F. Santos, A. Bagnato, and J. Carvalheira Effects of Clustering Herds with Small-Sized Contemporary Groups in Dairy Cattle Genetic Evaluations J Dairy Sci, January 1, 2008; 91(1): 377 - 384. [Abstract] [Full Text] [PDF]
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R. M. Sawalha, J. F. Keown, S. D. Kachman, and L. D. Van Vleck Genetic Evaluation of Dairy Cattle with Test-Day Models with Autoregressive Covariance Structures and with a 305-d Model J Dairy Sci, September 1, 2005; 88(9): 3346 - 3353. [Abstract] [Full Text] [PDF]
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R. M. Sawalha, J. F. Keown, S. D. Kachman, and L. D. Van Vleck Evaluation of Autoregressive Covariance Structures for Test-Day Records of Holstein Cows: Estimates of Parameters J Dairy Sci, July 1, 2005; 88(7): 2632 - 2642. [Abstract] [Full Text] [PDF]
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J. Vasconcelos, A. Martins, M. F. Petim-Batista, J. Colaco, R. W. Blake, and J. Carvalheira Prediction of Daily and Lactation Yields of Milk, Fat, and Protein Using an Autoregressive Repeatability Test Day Model J Dairy Sci, August 1, 2004; 87(8): 2591 - 2598. [Abstract] [Full Text] [PDF]
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