Bias in variance and covariance component estimators due to selection on a correlated trait

Statistical methods generally used in the analysis of animal breeding data assume that these data are a random sample from the population concerned. However, as livestock improvement programmes consist largely of continuous selection decisions this is often not true, particularly for field data. Analyses neglecting these selection decisions are not likely to give appropriate estimates unless selection were acting on some character independent of the traits under analysis. Removal of selection bias will normally depend on the correct identification of the selection criteria and is only achieved when all information contributing towards the selection decision is included in the model of analysis. Yet, in many situations this is not the case. For the analysis of dairy records for first and second lactation, for instance, this would require that the decision whether or not a cow was to have a second lactation were solely determined by the first lactation record. In practice, however, additional fact6rs like type, fertility, temperament or health status play an important r61e. In the simplest case selection might be thought to be on one trait correlated with yield. The objective of this paper is to investigate for this case the consequences of carrying out analyses on yield.