In research on artificial insemination of cows, a series of semen samples from each of six bulls was tested for the ability to produce conceptions (Snedecor and Cochran, 1989, Chapter 13, Section 13.7, pp 245–247). The results are presented in Table 3.12.

(a) Arrange these data for analysis by GenStat or R.

(b) Analyse the data by analysis of variance and by mixed modelling, making the assumption that the bulls studied have been chosen at random from a population of similar animals. Perform the best available test to determine whether the variation among bulls is significant. The data are percentages based on slightly different numbers of tests: the assumption, made in the analyses specified in Part (b), that the variance among samples within each bull is constant, is therefore not quite correct. Table 3.12 Conception rates obtained from semen samples from six bulls Bull Percentage of conceptions to services 1 46 31 37 62 30 2 70 59 3 52 44 57 40 67 64 70 4 47 21 70 46 14 5 42 64 50 69 77 81 87 6 35 68 59 38 57 76 57 29 60 Exercises 119

(c) Obtain diagnostic plots of the residuals, and investigate whether there is evidence of a serious breach of the assumptions on which your analyses are based.

(d) Estimate the following variance components: (i) among bulls (ii) among samples within each bull. Suppose that the results from this experiment are to be used to design an assay procedure to estimate the fertility of similar bulls in the future. Suppose also that the cost of including an additional bull in the assay is three times the cost of obtaining an additional sample from a bull already included.

(e) How many samples should be tested from each bull included in the assay? Note that the sums of squares and mean squares in the anova of these data do not agree with those presented by Snedecor and Cochran.