Crop scientists occasionally compute sample correlations between traits based on observed data from designed experiments, and this is often accompanied by significance tests of the null hypothesis that traits are uncorrelated. This simple approach does not account for effects due to the randomization layout and treatment structure of the experiments and hence statistical inference based on standard procedures is not appropriate. The present paper describes how valid inferences accounting for all relevant effects can be obtained using bivariate mixed linear model procedures. A salient feature of the approach is that the bivariate model is commensurate with the model used for univariate analysis of individual traits and allows bivariate correlations to be computed at the level of effects. Heterogeneity of correlations between effects can be assessed by likelihood ratio tests or by graphical inspection of bivariate scatter plots of effect estimates. if heterogeneity is found to be substantial, it is crucial to focus on the correlation of effects, and usually, the main interest will be in the treatment effects. If heterogeneity is judged to be negligible, the marginal correlation can be estimated from the bivariate model for an overall assessment of association. The proposed methods are illustrated using four examples. Hints are given to alternative routes of analysis accounting for all treatment and design effects such as regression with groups and analysis of covariance.
