An experiment using 36 samples of Portland cement is described by Davies and Goldsmith (1984, Chapter 6, Section 6.75, pp 154–158). The samples were ‘gauged’ (i.e. mixed with water and worked) by three gaugers, each one gauging 12 samples. After the samples had set, their compressive strength was tested by three breakers, each breaker testing four samples from each of the three gaugers. The results are presented in Table 3.13. (Data reproduced by kind permission of Pearson Education Ltd.)

(a) Arrange these data for analysis by GenStat or R. ‘Gauger’ and ‘Breaker’ are both to be regarded as random-effect terms.

(b) Justify these decisions.

(c) Analyse these data by analysis of variance and by mixed modelling.

(d) Perform F tests for the significance of the following model terms: (i) Gauger (ii) Breaker (iii) Gauger × Breaker interaction.

(e) Obtain estimates of the variance components for the same model terms, and also for the residual term, from the analysis of variance. Confirm your answers from the mixed-modelling results. Table 3.13 Compressive strength of samples of Portland cement (psi) mixed by three gaugers, measured by three breakers Gauger Breaker 123 1 5280 5520 4340 4400 4160 5180 4760 5800 5020 6200 5320 4600 2 4420 5280 5340 4880 4180 4800 5580 4900 4960 6200 4600 4480 3 5360 6160 5720 4760 4460 4930 5680 5500 5620 5560 4680 5600 120 Estimation of the variances of random-effect terms

(f) According to the evidence from this experiment, which of these sources of variation needs to be taken into account when obtaining an estimate of the mean strength of samples of Portland cement? The breakers in this experiment were human assistants who operated the testing machine, and the variation in the results that they produce is due to personal factors in their preliminary adjustment of the machine. (g) Suppose that the cost of employing an additional breaker is 10 times the cost of getting the current breaker to test an additional sample. Determine the number of samples that each breaker should test in order to obtain the most accurate estimate possible of the strength of the cement, at a fixed cost.