The data on the yield of F3 families of wheat presented in Exercise 3 in Chapter 3 are a subset from a larger investigation, in which two crosses were studied, and the F3 families were grown with and without competition from ryegrass. The experiment had a split-split plot design, with the following relationship between the block and treatment structures: • ‘cross’ was the treatment factor that varied only between main plots; • ‘family’ was the treatment factor that varied between sub-plots within each main plot, but not between sub-sub-plots within the same sub-plot; • presence or absence of ryegrass was the treatment factor that varied between sub-sub-plots within the same sub-plot. The mean grain yield per plant was determined in each sub-sub-plot. (When the subset of the data is considered in isolation, it has a randomised block design, as described in the earlier exercise.) The first and last few rows of the spreadsheet holding the data are presented in Table 7.14: the full data set is held in the file ‘wheat with ryegrass.xls’ (www.wiley.com/go/mixed modelling). (Data reproduced by kind permission of Soheila Mokhtari.) 248 Two case studies Table 7.14 Yield per plant (g) of F3 families from two crosses between inbred lines of wheat, grown with and without competition from ryegrass in a split-split plot design. 0 = ryegrass absent; 1 = ryegrass present. A B C D E F G H 1 rep mainplot subplot subsubplot cross family ryegrass yield 2 1 1 1 1 1 29 0 15.483 3 1 1 1 2 1 29 1 5.333 4 1 1 2 1 1 26 0 13.4 5 1 1 2 2 1 26 1 3.483 6 1 1 3 1 1 40 0 11.817 7 1 1 3 2 1 40 1 3.583 . . . . . . 382 2 2 47 1 2 4 0 7.55 383 2 2 47 2 2 4 1 4.02 384 2 2 48 1 2 15 0 6.867 385 2 2 48 2 2 15 1 3.183

(a) Analyse the data according to the experimental design, both by analysis of variance and by mixed modelling.

(b) Modify your mixed model so that family-within-cross is regarded as a randomeffect term.

(c) Interpret the results of your analysis. In particular, consider the following points: (i) Does the presence of ryegrass affect the yield of the wheat plants? If so, in which direction? (ii) Is there a difference in mean yield between the crosses? (iii) Do the families within each cross vary in yield? (iv) Do the effects of crosses and families interact with the effect of ryegrass?

(d) Compare the results with those that you obtained from the subset of the data in Chapter 3.