ANOVA for split-plot experiments - theory and practice

Consider the split-plot experiment described in Section 3.1 in this paper. Formulate a mixed model for this experiment with main effects of harvest and sowing time, an interaction between harvest and sowing time and random block and whole-plot effects.

  1. Write down the orthogonal decomposition corresponding to the mixed model with associated dimensions of the orthogonal subspaces (the $V$ and $\tilde V$ spaces).
  2. Write down the decomposition of mean vector and the covariance matrix. Write down the factorization of the likelihood.
  3. Derive the estimates of fixed effects parameters and variances. Also derive F-tests for the fixed effects and F-tests for the variances.
  4. Fit the mixed model to the beets data using both anova/lm/aov and lmer (data set is available in doBy package so install and load this package).
  5. Investigate the effects of sowing time and harvest time using F-tests (try also to use KRmodcomp together with lmer). Test also whether variances are zero.
  6. Check residuals from the fitted model.
  7. Try to remove a few randomly selected observations (so that the data set is no longer balanced). How does this change the results ? What results do you get with KRmodcomp ?