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.

- Write down the orthogonal decomposition corresponding to the mixed model with associated dimensions of the orthogonal subspaces (the $V$ and $\tilde V$ spaces).
- Write down the decomposition of mean vector and the covariance matrix. Write down the factorization of the likelihood.
- Derive the estimates of fixed effects parameters and variances. Also derive F-tests for the fixed effects and F-tests for the variances.
- 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).
- 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.
- 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 ?
- Check residuals from the fitted model.
- 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 ?