The seeds of some species of clover (Trifolium spp.) will not germinate immediately after ripening, but must undergo a period of ‘softening’, by exposure to fluctuating 328 Beyond mixed modelling high and low temperatures, usually on the soil surface. In an investigation of this phenomenon, seeds of eight clover species were sown into strips of fine mesh cotton bags, 100 seeds of a single species in each bag. Each strip comprised one bag of each species. At the start of summer, the bags were laid on the soil surface in full sun at a site cleared of vegetation. At intervals during the summer and autumn, two or four strips were removed from the field, and the seeds in each bag were tested to determine how many were still ‘hard’ (i.e. would not germinate when moistened in favourable conditions). The first and last few rows of the spreadsheet holding the data are presented in Table 9.9: the full data set is held in the file ‘clover seed softening.xls’ (www.wiley.com/go/mixed modelling). (Data reproduced by kind permission of Hayley Norman.) Table 9.9 Numbers of ‘hard’ and ‘soft’ seeds of several species of clover after exposure on the soil surface for varying lengths of time. Observations on the same replication at the same time come from a single strip of bags: time = interval in days from the start of exposure on the soil surface; nhard = number of hard seeds in a single bag; nsoft = number of soft seeds in a single bag. A B C D E 1 species! replication! time nhard nsoft 2 T. glomeratum 1 0 98 2 3 T. glomeratum 2 0 99 1 4 T. glomeratum 3 0 99 1 5 T. glomeratum 4 0 99 1 6 T. spumosum 1 0 94 6 7 T. spumosum 2 0 93 7 8 T. spumosum 3 0 95 5 9 T. spumosum 4 0 96 4 . . . . . . 234 T. scutatum 1 174 4 96 235 T. scutatum 2 174 11 89 236 T. scutatum 3 174 5 95 237 T. scutatum 4 174 5 95 238 T. pilulare 1 174 17 83 239 T. pilulare 2 174 28 72 240 T. pilulare 3 174 241 T. pilulare 4 174 Assume that the species studied are a representative sample from a large population of species potentially available for study. Exercises 329

(a) Specify a regression model for this experiment. Which term(s) in the model should be regarded as fixed, and which as random? What is the response variable?

(b) Fit your mixed model to the data, specifying an appropriate error distribution for the response variable, and an appropriate link function to relate the response variable to the linear model.

(c) Consider whether there is evidence that any terms can be omitted from the model. If so, fit the modified model to the data.

(d) Make a graphical display, showing the fitted relationship between the proportion of hard seed remaining for each species and the time elapsed since the start of exposure, and showing the scatter of the observed values around this relationship.