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A bivariate analysis is described for one continuous and one discrete trait to estimate sire effects in a progeny test. Unequal design matrices, in terms of both missing data as well as different fixed effects are allowed for. Three independent sub data sets are formed, for which the corresponding log-likelihoods are expressed. The parameters are estimated as their maxima a posteriori by maximizing the log-posterior density using a Newton Raphson algorithm; equations are given in a mixed model form. A simulation study is presented showing the benefits of this bivariate analysis for the evaluation of beef bulls for calving difficulty, using birth weight as a correlated trait. Standard errors obtained from the information matrix were close to empirical standard errors, and it was shown that bivariate analysis with unequal design matrices can correct for a bias which may occur when birth weight records are not missing at random.

Starting from the approach of Wright (1931), some methods were developed for predicting the selection limit for a quantitative trait. These methods, however, do not take into account the effects of selection on the genetic variance, which are mainly (i) the induction of linkage disequilibrium and (ii) the change in family structure. Some algorithms intended to predict the evolution of the genetic variance and mean under the infinitesimal model are presented and discussed. Simulation results are compared with the expected values. They clearly show the irrelevance of equations taking into account drift or selection separately. On the other hand, taking into account both effects leads to prediction of responses to selection close to those observed. However, no algorithm provides a strictly exact prediction of the genetic variance, and the most critical point lies in the prediction of the effect of selection on drift. More complex situations are also discussed: (1) the consequences of selection when an animal model and BLUP techniques are used; (2) the case of a finite number of loci, with more or less linkage, (3) the impact of non-random mating or mutation; (4) trying to make predictions under practical situations such as overlapping generations. Practical applications in breeding programmes are outlined.