An experiment was performed to investigate the factors that influence predation of seeds lying on the ground in an area on which a crop has been grown. Seeds of three species were placed on the ground in cages of four designs that excluded different types of predators, in areas on which three types of crop residue were present, and at four distances from an area of bushland. The levels of these treatment factors are given in Table 7.10. This experimental area was divided into three blocks, and each block was further divided into three strips. Each strip had a boundary with the bushland at one end. Each strip received a crop residue treatment, each treatment being applied to one randomly chosen strip in each block. In each strip, one cage of each design was placed at each distance from the edge of the bushland. At each distance, the four cages in each strip were positioned at random. Each cage contained seeds of all three species. After 15 days, the number of seeds of each species that remained in each cage was recorded, and the percentage predation was calculated. The layout of the experiment is illustrated in Figure 7.7. Strip 1 Strip 2 Strip 3 Block 1 Block 2 Block 3 Key Bushland Residue 1 Residue 2 Residue 3 Cage 0 m 25 m 50 m 100 m Figure 7.7 Layout of an experiment to investigate influences on predation of seeds. Crop residues are shown only in Block 1. Cages are shown only at 25 m from the bush in Strip 1 of Block 1. The first and last few rows of the spreadsheet holding the data are presented in Table 7.11: the full data set is held in the file ‘seed predation.xls’ (www.wiley.com/ go/mixed modelling). (Data reproduced by kind permission of Robert Gallagher.) Distance, residue, cage type and species are to be classified as treatment factors.

(a) Identify any treatment factors for which randomisation cannot be performed. Exercises 243 Table 7.11 Percentage of predation of seeds of three species at different levels of three experimental variables A B C D E F 1 distance block residue cage type species %pred 2 0 1 2 1 wo 100.0 3 0 1 2 2 wo 95.0 4 0 1 2 3 wo 30.0 5 0 1 2 4 wo 10.0 6 0 1 1 1 wo 95.0 7 0 1 1 2 wo 100.0 8 0 1 1 3 wo 100.0 9 0 1 1 4 wo 10.0 10 0 1 3 1 wo 100.0 11 0 1 3 2 wo 30.0 12 0 1 3 3 wo 100.0 13 0 1 3 4 wo 10.0 14 0 2 3 1 wo 10.0 15 0 2 3 2 wo 0.0 16 0 2 3 3 wo 5.0 17 0 2 3 4 wo 0.0 . . . . . . 430 100 3 3 1 arg 100.0 431 100 3 3 2 arg 12.2 432 100 3 3 3 arg 23.3 433 100 3 3 4 arg 11.1

(b) Determine the block and treatment structures of this experiment.

(c) Analyse the data by analysis of variance.

(d) Perform an equivalent analysis on the data by mixed modelling, and obtain diagnostic plots of the residuals. Note any indications that the assumptions underlying the analysis may not be fulfilled, and consider why this may be the case.

(e) Create a variate holding the same values as the factor ‘distance’, and add this to the mixed model. Consider carefully whether the factor ‘distance’ should now be regarded as a fixed-effect term or a random-effect term. Consider also the consequences of your decision with regard to the interaction of ‘distance’ with other terms. Fit your new model to the data.

(f) Assuming that any departures from the assumptions underlying analysis of variance and mixed modelling are not so serious as to render these analyses 244 Two case studies invalid, interpret the results of the analyses performed in Part (e). In particular, consider the following points: (i) Is there a linear trend in the level of predation over distance? If so, does the level increase or decrease with increasing distance from the bushland? (ii) Does this trend vary among the species? If so, what is the nature of this variation? (iii) Does this trend vary according to the crop residue? If so, what is the nature of this variation? (iv) Does this trend vary according to the cage type? If so, what is the nature of this variation? (v) Is there evidence of variation among the levels of distance that cannot be accounted for by the linear trend? (vi) Is there evidence of three-way or four-way interactions among the treatment factors? (g) Obtain predicted values of predation at representative distances from the bushland for each species, crop residue and cage type. Make a graphical display of the relationship between these variables. A refinement of this analysis, taking account of the problems identified in the diagnostic plots, is introduced in Exercise 2, Chapter 9. These data have also been analysed and interpreted by Spafford-Jacob et al. (2005).