The general problem of estimating breeding values of animals in a population under selection to improve performance for growth and reproductive success is addressed. Genetic and environmental (co)variances for correlated traits must be estimated from the data if they are unknown or expected to have changed due to the type of selection. The data were a set of four lines, one selected for increased pupa weight, another for increased family size, the third based on an index combining pupa weight and family size, and a randombred control. The analysis of genetic responses over 16 generations in these four selection lines derived from a common base population present some interesting phenomena likely to be encountered in the analysis of other populations under selection. Changes in genetic and environmental (co)variances associated with selection for pupa weight were found to have a profound effect on estimates of (co)variance components within and between generations. New insight is provided on ways to interpret restricted maximum likelihood estimates of genetic parameters. Base populations and control lines with 16 generations of data from two replicated experiments were used to show how insufficient data, misidentification of major fixed effect when combining data across experiments, and confounding of random effects can lead to widely different estimates of parameters for the same data. Gibbs sampling techniques were used to implement a full Bayesian analysis of the data. All (co)variance components in the model were not estimated with equal information from the data. Extensive use was made of the 95% central interval of the posterior distribution to graphically show the effect of different assumptions about prior knowledge or belief in the realized values of random variables. Even a small amount of weight on prior knowledge about parameters can overcome problems associated with the belief that all information must come entirely from the data. A multiple trait heterogeneous mixed model is proposed to adjust for the effects of genotype by environment interaction. It is argued that this model overcomes several deficiencies of other models proposed to account for heterogeneous (co)variances.