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. Conduct analyses both with sugpct and yield as dependent variable.

  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. Assume for now that there is no harvest:sow interaction. Compute estimates of treatment contrasts between the two harvesting times and between first and second sowing time. Also compute variances of the contrast estimates and perform t-tests for the hypotheses that each of the contrasts are zero. Which treatment contrast is estimated with highest precision ?
  7. Check residuals from the fitted model.
  8. 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 ?