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Contemporary Groups
L. R. Schaeffer
April 1, 2001

INTRODUCTION

Contemporary groups (CG) are defined and formed by researchers to identify a group of animals that are roughly of the same age and sex, and that have undergone the same management and environmental conditions during a particular phase of their life. In dairy cattle, CG have been formed on the basis of herd (H), year of calving (Y), season of calving (S), and age (in number of lactations or parities) (P) for genetic analyses of 305-d lactation production by an animal model. The CG effects have nearly always been treated as fixed effects. From an estimability perspective a model with fixed HYSP effects is identical to a model with fixed H, Y, S, P, HY, HS, HP, YS, YP, SP, HYS, HSP, HYP, YSP, and HYSP effects in it. The estimate of a particular HYSP effect, therefore, contains the true effects of each of the main effects, two-way and three-way interactions as well as the four-way interaction effects.

If the HYSP effect were made random by adding the ratio of residual to HYSP variances to the diagonals of the HYSP equations in the mixed model equations, then the evaluation model should logically be augmented by inclusion of some or all of the main, two-way, and three-way effects as fixed. Failure to augment the model means that the main, two-way, and three-way effects are assumed to be non-existent. Time trends (YS effects) are almost always significant as are herd effects, thereby violating the implied assumption of non-existence. The effects of these ignored factors go into the solutions of other factors in the model creating bias. If YS effects are significant, for example, then their effects can appear in the estimates of breeding values of animals if not included in a random HYSP effects model. Caution should be exercised in deciding if CG effects are to be fixed or random.

Why have CG effects usually been treated as fixed factors? In 1973, Henderson presented best linear unbiased prediction (BLUP) methodology for use in genetic evaluation of dairy sires. First lactation milk yields of dairy cows were pre-adjusted for age at calving, and the model was

$$y_{ijk} = \mu + (HYS)_{i} + s_{j} + e_{ijk},$$

where $\mu$ was an overall mean, (HYS)_i was a herd-year- season of calving (or contemporary group) effect, s_j was a sire transmitting ability, and e_ijk was a residual effect. Sires were assumed to be random and unrelated from a common population with variance $\sigma^{2}_{s}$. To be more complete, this model should have included H, Y, S, HY, HS, and YS effects too. Henderson (1973) said that logically the HYS effects were random variables, but because the average value of true HYS effects were most likely not equal for all sires (i.e. non-random association of sires with HYS effects), then HYS effects treated as fixed would remove the possible bias from the expectations of solutions for the random sire effects. If HYS are fixed, then to get a solution to the mixed model equations, the solutions for H, Y, S, HY, HS, and YS would need to be set to zero (unless some of these were random effects). Henderson (1973) economized his presentation by not showing H, Y, S, HY, HS, or YS effects in his model, knowing that the solutions for these would be set to zero when HYS effects were fixed. The consequence of Henderson (1973) was that all subsequent genetic evaluation models treated contemporary groups (CG) as fixed effects in the model, regardless of CG subclass size or the magnitude of the ratio of residual to CG variances, or whether from sire models with unrelated sires or animal models with nearly complete additive genetic relationships among animals.

The proof that treating HYS effects as fixed provide unbiased predictions of sire transmitting abilities in a sire model with unrelated sires was given by Henderson (1973, 1975) based upon Pearson's (1903) selection model. Thompson (1979) questioned the validity of conditioning expectations on the selection differentials, and was confused by the matrix ${\bf L}$, which Henderson (1975) used to describe the selection process, because it was a random variable and not fixed. Gianola et al. (1988) went further by arguing against the idea of conceptual repeated sampling which Henderson (1975) found necessary to invoke. Gianola et al. (1988) pointed out that when Henderson (1973) required predictors of sire transmitting abilities to be unbiased by treating HYS effects as fixed, that this did not necessarily maximize expected genetic response from selection on sire estimated transmitting abilities. Bayesian methods were recommended to perhaps get around these problems. The conclusion from these two papers was that the procedures given by Henderson (1975) may not be appropriate. Therefore, the basis for treating HYS effects as fixed should at least be re-examined.

Two simulation studies were conducted by Ugarte et al. (1992) and by Visscher and Goddard (1993) to compare fixed versus random CG effects on the bias and accuracy of sire estimated transmitting abilities. Both studies used a simple sire model with sires assumed to be unrelated, and neither study had an aspect of time or selection of animals in their simulations. Therefore, there were no year of calving, season of calving, or herd effects involved in the simulations, and therefore, no need to account for time trends in their data. The results of Ugarte et al. (1992) basically supported the contentions of Henderson (1973), but concluded that benefits could arise from using random CG effects in certain situations depending on CG subclass size and parameter values. Visscher and Goddard (1993) took an analytical approach to show that the results of Ugarte et al. (1992) were not general, and how sire estimated transmitting abilities could be negatively correlated to true transmitting abilities when CG effects were random depending on the nature of the sire by CG associations. To date, there has been no study of fixed or random CG in animal models in which time trends have been incorporated.

A topic never addressed throughout this debate has been the proof of the existence or the magnitude of non-random associations of sires with true HYS effects. Henderson (1988) presented a method for measuring bias, but required inverse elements of the mixed model equations, which are invariably impossible to obtain from animal models. No experiments or analyses have been proposed to study this issue. The traditional assumption is that the non-random association between sires and true HYS effects exists, and therefore, HYS effects must be fixed. Arguments point to imported foreign sires that are highly priced, and therefore, are used in the more wealthy (i.e. better managed) herds, and their daughters tend to receive preferential treatment. The sires are highly selected and perhaps should be treated as fixed in an animal model rather than treating contemporary groups as fixed. Not all of these biases can be removed statistically through the use of an appropriate model without having more information.

The objectives of this study were to 1) discuss appropriate modelling of contemporary group effects in an animal model, 2) evaluate fixed versus random contemporary group effects in the context of animal models with animals dispersed over time, 3) propose a method of assessing biases in field data, and 4) estimate the magnitude of association between sires and HYS effects.

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