In many types of plant, exposure to low temperature at an early stage of development causes flowering to occur more rapidly: this phenomenon is called vernalisation. An inbred line of chickpea with a strong vernalisation response and a line with little or no vernalisation response were crossed, and the F1 hybrid progeny were self-fertilised to produce the F2 generation. Each F2 plant was self-fertilised to produce an F3 family. The seed of each F3 family was divided into two batches. Germinating seeds of one were vernalised by exposure to low temperature (4◦ C) for four weeks. The other batch provided a control. All F3 seeds were then sown, and allowed to grow. The plants were arranged in groups of four: within each group the plants were of the same family and had received the same low-temperature treatment. Generally there were 12 plants (i.e. three groups of four) in each family exposed to each low-temperature treatment, but in some families fewer or more plants were available. The number of days from sowing to flowering was recorded for each plant. The first and last few rows of the spreadsheet holding the data are presented in Table 5.2: the full data set is held in the file ‘chickpea vernalisation.xls’ (www.wiley.com/go/mixed modelling). (Data reproduced by kind permission of S. Abbo, Field Crops and Genetics, The Hebrew University of Jerusalem.)

(a) Divide the data between two spreadsheets, one holding only the results from the vernalised plants, the other, only those from the control plants. Table 5.2 Time from sowing to flowering of F3 chickpea plants with and without exposure to a vernalising stimulus. A B C D E 1 plant group plant family low T days to flower 2 60 1 88 vernalised 63 3 60 2 88 vernalised 64 4 60 3 88 vernalised 61 5 60 4 88 vernalised 70 6 1 1 14 vernalised 75 7 1 2 14 vernalised 8 1 3 14 vernalised 9 1 4 14 vernalised 10 103 1 29 control 78 11 103 2 29 control 82 12 103 3 29 control 82 13 103 4 29 control 88 . . . . . . 1092 41 3 44 vernalised 70 1093 41 4 44 vernalised 64

(b) Analyse the results from the vernalised plants by mixed modelling, regarding ‘family’, ‘group’ and ‘plant’ as random-effect terms.

(c) Obtain an estimate of the component of variance for each of the following terms: (i) family (ii) group within family (iii) plant within group. Which term in your mixed model represents residual variation?

(d) Estimate the heritability of time to flowering in vernalised plants from this population of families. (Note that the estimate obtained using the methods described in Chapter 3 is slightly biased downwards, as some of the residual variance is due to genetic differences among plants of the same family.)

(e) Obtain the unadjusted mean and the shrunk mean, and the BLUP, for the number of days from sowing to flowering in each family.

(f) Extend Equation 5.6 to the present situation, in which two components of variance contribute to the shrinkage of the BLUPs. Use the values obtained above to check your equation. (g) Repeat the steps indicated in Parts (b)–

(d) of this exercise for the control plants. Comment on the difference between the estimates of variance components and heritability obtained from the vernalised plants and the control plants. (h) Plot the shrunk mean for each family obtained from the control plants against the corresponding value obtained from the vernalised plants. Comment on the relationship between the two sets of means.