On the use of animal models in the analysis of selection experiments

The use of an animal model in the analysis of selection experiments offers the theoretical advantage of accounting for changes occurring in the genetic parame- ters in the course of the experiments. Explicit estimators of realized heritability (h2) are derived in this paper for balanced one-generation selection designs. Expressions are given for the expectations and variances of the estimators in relation to the true heritability and for the sensitivity of the estimators to the prior value of heritability. Sensitivity is generally high, except for high values of the true heritability and/or extremely large family sizes. The uncertainty on heritability may, however, be taken into account in a context of Bayesian inference, which allows a simultaneous esti- mation of the initial heritability and of the response. On the other hand, animal model estimators, being dependent on the genetic model assumed, may not provide adequate measures of the actual responses. They also tend to overestimate the ac- curacy of genetic trend evaluations, since genetic drift is not properly accounted for. Animal models, however, provide a way of evaluating the effects of selection and lim- ited population size in long-term selection experiments, and thus permit a check on the validity of the underlying infinitesimal additive genetic model. Some examples based on published results of long-term selection experiments on mice are discussed. © Inra/Elsevier, Paris


INTRODUCTION
One of the objectives of a selection experiment is to compare reality to theory, by checking whether the selection responses predicted are actually achieved. Use is made of the concept of realized heritability !', defined as the ratio of response (R) to selection differential (S) such that h! = R/S (see Falconer [5]). This parameter (h') is in fact one element of a set of realized genetic parameters, which can be derived from a properly planned multitrait selection experiment. In farm animals with long generation intervals, selection experiments are generally carried out for a limited number of generations, and the main interest is to evaluate the effect of selection on the means of the selected populations. On the other hand, long-term selection experiments with laboratory animals have somewhat different purposes, which are essentially to assess the limits to selection and to evaluate the effect of selection on the genetic parameters. Most often, individual selection is applied, allowing an easy calculation of S and a direct measurement of R/S. Family and combined selection, however, may also be applied and h' is then a more complex function of R/S involving the corresponding index coefficients (e.g. see Perez-Enciso and Toro [11]). With BLUP selection, no exact calculation of S for the selection criterion is possible, except in balanced designs, and realized heritability cannot be measured. Responses (R) are classically based on generation/line leastsquare estimators obtained in an experimental design properly controlling environmental differences between generations. However, responses may also be derived from individual breeding values. In the late 1970s it appeared that the BLUP method of evaluation could be taken to its 'logical conclusion', as noted by Thompson [16] in his review of sire evaluation, since genetic trends in dairy cattle using BLUP estimators began being presented at that time. In addition to the standard methods of analysis of selection experiments, essentially based on least-square estimators (see [8]), new methods based on mixed models were developed from then. Moving from sire models to animal models offered additional advantages. As shown by Sorensen and Kennedy !13!, the animal model has advantages in the estimation of selection response as well as in the study of the evolution of genetic variance. These two aspects will be considered in succession in this paper. A distinction will be made between inferences based on assumed prior values of the variances in the model and a more general approach integrating the uncertainty on those variances.  Thompson had pointed out that the responses derived from mixed models include information components based on the selection pressure applied. The estimator of R is then a function including S, which is not the case in the standard methods of analysis of selection experiments. The question explicitly put by Thompson [16] was whether &dquo;BLUP estimates of trend are just multiples of the selection differentials&dquo;. By considering simple one-generation designs, analytical expressions of the weight of S in the estimation of R can be obtained, as will be shown below. Simple designs have previously been investigated by Thompson !17!, who considered selection in one sex over several generations, and also by Sorensen and Johansson !12!, who considered selection operating in both sexes.

Design 1: no control line
Though this situation has been fully addressed by Thompson [17], it is again summarized here for the sake of completeness, and the derivation of the estimator of !2 is detailed in the Appendix. Using Thompson's notation, n unrelated males (n > 2) are measured for the trait of interest in generation 1, out of which one is selected and leaves n progeny measured in generation 2. A pool of dams unrelated to the sires is assumed, in which pedigree information is ignored. In such a situation, the individual (animal) mixed model applied is: where y2! is the value of the trait measured in generation i (i = 1, 2) on the individual j ( j = 1, ... n), m i the generation mean, a2! the individual additive genetic value with variance aa, eZ! a random environmental effect with variance Q e, and letting h 2 = a2/(a2 + (2) In this design, fixed effects are confounded with generation, and it was shown by Thompson !17! that the estimator of realized heritability is the prior value of h 2 assumed. The derivation presented in the Appendix may be extended to any balanced scheme implying selection of s sires leaving n offspring each. It has also been shown by Sorensen and Johansson [12] to hold when selection is in both sexes.

Design 2: control line
The situation considered here is the same as in design 1, with the addition of a very large pool of unrelated individuals of constant genetic merit, measured in both generations. This allows all measures to be expressed as deviations from a fixed control level. Environmental differences between generations are thus eliminated, and a common mean m may be taken in the model, which becomes: When selection operates in both sexes, assuming the extremes to be selected out of n candidates in each sex, and assortatively mated to produce two full-sib families of size n, it can be shown that the weighting factor of D/ 5 ' in equation (6) becomes k f/ (1 + lc f ), with: As with design 2, the situations can be extended to the case of 2sn unrelated candidates in generation 1 and 2s half-sib families of size n in generation 2, or sn candidates of each sex and 2s full-sib families of size n, since the expressions (6), (7) and (8) (1), and the response measured is strictly speaking a prediction, independent of the measures in generation 2 and of family size n. In designs 2 and 3, it can be seen that R combines an a priori information (0.5 h 2 S), in fact a multiple of the selection differential, and an a posteriori information (D), which is the observed response. The prior information dominates roughly in inverse proportion of h 2 , as shown by the k values (4), (5), (7) and (8), which are increasing functions of h 2 , as also noted by Sorensen and Johansson [12] for design 2. The statistical properties of the random variable !2 in designs 2 and 3 will now be examined in order to evaluate more precisely the efficiencies of those designs.
The estimators (3) and (6) of !2 have the following expectation, since E(D/S) = 0.5 5 h) , ho being the true heritability, as opposed to the prior value h 2 : As shown in figure !, this function varies from 0 to h) when h 2 increases from 0 to 1, and goes through a maximum which can be obtained by setting the derivative of equation (9)  one-line selection with control (design 2). One sees also that the advantage of design 3 over design 2 increases with increasing heritability and/or larger family size. When selection operates in both sexes, similar patterns can be shown to hold.
The variance of the estimators (3) and (6) for given fixed values of S is: Given the assumptions underlying model (1) and further assuming o-a + af = 1 in both generations, it can be shown that in the general case of s or 2s sires selected in generation 1 and half-sib family size of n: in designs 2 and 3, respectively. Equation (10) shows that the accuracy of estimation of h2, in terms of the inverse of its standard error, is inversely proportional to the relative weight k/(1 + k) given to the posterior information in this estimation. In designs yielding estimators very sensitive to prior heritability, i.e. with low heritability and small family size, animal model estimators of !2 will be extremely accurate.
It can also be seen that equation (11) does not include the drift variance associated with the limited effective size of the selected lines, and thus shows that the genetic drift variance is not properly accounted for in the animal model estimators. For instance, in the simple case of design 3 with s = n = 1, V(D) = 2 and does not include the drift variance due to an effective population size of N = 4 in each line, corresponding to one male and an infinite pool of unrelated females. Quite similarly, a strict application of least squares does not account for genetic drift either, but this effect may be incorporated into the variance of the estimators of realized heritability, through the procedures described by Hill !8!.

INFERENCES FROM SELECTION EXPERIMENTS WHEN THE VARIANCES IN THE MODEL ARE UNKNOWN
The sensitivity of the estimators considered so far to prior values of h 2 is clearly the consequence of the uncertainty as to the real value of this parameter. The problem, however, has a conceptually simple solution when framed in a Bayesian setting, as shown by Sorensen et al. !15!. Inferences about selection responses can be made using the marginal posterior distribution of selection response, and the uncertainties about variance components are then taken into account by viewing those components as nuisance parameters.
The marginal posterior distributions can be obtained by Gibbs sampling, and probabilities that the response R lies between specified values can be computed. The same reasoning applies to variance components and h 2 . In the simple designs considered in section 2, where S can be calculated, the posterior distribution of R/S could be obtained and compared to that of h 2 . Inferences are influenced by the amount of data available and the assumed type of a priori distribution of the variance components, as shown in the example in Sorensen et al. [15]. In this example h' cannot be obtained, since S cannot be easily calculated. But one can expect its properties to closely follow those of R, according to the amount of data and type of prior, i.e. the more data are available the less are the estimates of responses influenced by the choice of priors. And similarly for the variances of the estimate, they would be expected to be highly dependent on the type of prior, in addition to being larger than those obtained in the section 2 setting, since more uncertainty is taken into account.

EVOLUTION OF GENETIC VARIANCE IN SELECTION EXPERIMENTS OVER SEVERAL GENERATIONS
Moving from one cycle of selection, as considered above, to several successive cycles requires accounting for the effects of selection on the genetic variance. It is well known that selection induces linkage disequilibria tending to reduce the genetic variance, and leading to an asymptotic response lower than the response expected in the first generation !3!. In selected lines of limited size, an additional factor reducing the response is the decrease in genetic variance due to genetic drift, a decrease which itself depends on the selection criterion applied !18!. Consequently, the ratio R/S evaluated over several generations is not relevant, as it is expected to be systematically below the initial heritability. The animal model takes into account the two phenomena of variance reduction due to drift [13] and to the Bulmer effect [14]. This model, when applied to long-term selection experiments, thus yields unbiased estimates of selection responses over successive generations on the one hand, and provides an estimate of the initial genetic variance on the other, using the restricted maximum likelihood approach (REML: e.g. see !16!). A basic assumption of this approach is of course the additive genetic infinitesimal model. Selection experiments have been analysed increasingly according to the animal model methodology, since Blair and Pollak [2] evaluated selection response in a seven-generation experiment on sheep, and suggested that mixed models could be used to estimate genetic trends when no control is available. One of the first applications to long-term selection experiments has been presented by Meyer and Hill !10!, on 23 generations of selection for food intake in mice. In order to show the evolution of genetic variance, a two-step procedure of data splitting was implemented, first cumulating increasingly larger numbers of generations from the beginning of the experiment (analysis I), and then having separate groups of consecutive generations analysed independently (analysis II). As shown in table 11, analysis I indicates that, as expected, standard realized heritability (R/S) decreases when the number of generations included increases, whereas the animal model heritability also decreases, which is contrary to expectation, since in theory the animal model estimates the initial genetic variance. Analysis II indeed reveals a marked reduction of genetic variance already at generation 8, and the effect is enhanced at generation 14. The authors could then safely conclude that 'selection for appetite in mice has reduced the genetic variance over and above the effects of inbreeding and selection', and that the infinitesimal model does not apply. Another conclusion to be drawn is that the animal model underestimates the initial heritability and, consequently, responses are also underestimated initially, owing to the sensitivity of the estimator to prior heritability. A close examination of the graph of predicted values and phenotypic means over generations (in figure 2 of [10]) indeed seems to indicate a slightly larger observed divergence compared to the animal model prediction. In contrast, in another mouse selection experiment of similar duration, the animal model estimate of heritability over the whole experiment was found to be very close to the estimate obtained in the first seven generations, and, accordingly, the divergence predicted from the animal model was in good agreement with the actual phenotypic divergence observed [1].

DISCUSSION AND CONCLUSIONS
The theoretical advantages of the mixed animal model in the analysis of selection experiments have been frequently emphasized. Compared to a simpler least-square analysis, the method allows one to better account for environmental effects and avoids the need for an experimental design with controls [2,12,14]. It is also well known that the estimates of selection response obtained via the animal model are dependent on the prior values of the genetic parameters [2, 12, 17!. As shown here, this dependency can be precisely evaluated in simple one-generation selection designs and the usual designs yield estimates of !2 highly sensitive to the prior heritability in most cases (see table !. Such a conclusion can safely be extended to designs covering more generations, such as the repeat sire design investigated by Thompson [17] over three generations. The sensitivity of a design may also be evaluated a posteriori, by estimating responses with increasing values of the prior heritability, and in most cases responses have been shown to actually increase markedly when h 2 2 increases (see, for instance, [2] or [11]). A posteriori evaluations of responses with varying values of prior heritability should also be recommended in the more general case of field data. The sensitivity of the estimator to prior h 2 2 may be expected to be a decreasing function of the degree of overlap between generations, or of the degree of connectedness of the data. Obviously, when generations do not overlap the situation is that of design 1, with no control, and sensitivity is maximum. by Gianola et al. [6] that breeding values should be predicted using its REML estimate in the data. It was later shown that the problem of inferences about genetic change when heritability is unknown can be solved in a Bayesian setting !15!. It should be noted that the classical approach suggested by Gianola et al. [6] offers a good approximation to the full Bayesian method of Sorensen et al. [15] when the information about heritability in the experiment is large enough. The accuracy of BLUP evaluation has also been sometimes presented as an argument in favour of the method for the estimation of genetic trends.
However, the prediction error variance of BLUP estimates is highly dependent on the weight given to the prior information, as equation (11) shows. A false impression of high accuracy will then be obtained in designs highly sensitive to prior genetic parameters. In addition, drift variance as a source of error between replicates is partially ignored, since the incidence matrix Z of individual genetic values and the relationship matrix A are considered as fixed. A common feature of the graphs showing genetic trends based on animal model evaluations of breeding values is the smoothing out of the between-generation fluctuations, in contrast with the highly irregular evolution of the phenotypic means (e.g. figure 1 of !2!, or figure 2 of !10!). If a Bayesian approach is implemented, the choice of an appropriate prior distribution of heritability is an important issue to consider. As shown in the example simulated by Sorensen et al. (15!, the variance of the posterior distribution of the selection response is considerably reduced when an informative prior is used. Another issue, quite distinct from the problems of statistical inference previously discussed, is the genetic model assumed. The additive infinitesimal model is implicit in models (1) and (2) and it is also the most generally used model in the analysis of long-term selection experiments. The responses estimated are clearly model dependent.
In particular, ignoring dominance is known to lead to an overestimation of the responses. A simulation [9] has shown that for a trait showing 40 % additive genetic and 20 % dominance variance, the use of an additive animal model yielded a bias in the estimate of response over six generations which was 1.21 times the real response. Chevalet [4] has derived an expression for the bias expected in breeding value prediction when an additive model is applied in a dominance situation. In addition, the infinitesimal model cannot account for changes in gene frequency due to selection or mutational variance, which are likely to contribute substantially to changes in additive genetic variance. Heath et al. [7] have suggested an extension of the REML procedure to the estimation of changes in variance components over generations and they have shown that significant changes had occurred in their selected mouse lines.
In conclusion, the usefulness of the animal model approach for studying the evolution of genetic parameters in long-term selection experiments is now well documented. The model indeed provides a way of testing the adequacy of the genetic assumptions underlying the analysis of selection responses. As to genetic trends, the animal model, strictly speaking, only provides trends in breeding value predictions based on a specific genetic model. This dependency on the genetic model leads to questioning the adequacy of the animal model applied to evaluate genetic progress. It should be noted that the consequences of using a wrong genetic model for evaluating responses over several generations are expected to be different from the consequences on breeding value predictions and selection efficiency. In breeding value predictions precision is more important than bias, as pointed out by Johansson et al. (9!. When responses are evaluated, the errors may be cumulative over generations, and create a sizeable bias. In other words, one may doubt that a proper evaluation of past events (such as genetic progress over a long period of time) can be safely based on a method whose aim essentially is to predict the future (such as breeding values needed to carry out selection decisions).