The use of repeated measures analysis of variance (ANOVA) options for the analysis of in vitro ruminal fermentation gas production profiles is illustrated. Because of the different variances and covariance structures among profile observations, ordinary ANOVA for more than two-time points is not recommended. To mitigate this problem, the Greenhouse–Geisser epsilon correction can be applied to reduce the degrees of freedom, inflated by violation of the sphericity assumption, for F ratio probability calculations. After this correction, the Box–Greenhouse–Geisser ANOVA (modified ANOVA) layout appears similar to the layout of a split-plot design ANOVA with whole plots divided into subplots (incubation time). Any F tests in the main plot part are valid but F tests involving the time factor from the subplot part need modification because time factor, by its very nature, cannot be allocated at random. Application of multivariate ANOVA, distance multivariate ANOVA, ante-dependence and mixed model analysis are also considered. All these options lend themselves to wide application in the applied biological sciences.

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.