An experiment was performed to determine the effect of the enzyme lactase (which hydrolyses the sugar lactose) on the composition of the milk in a sheep’s udder. In each side of the udder in each of eight lactating sheep, the levels of fat and of lactose in the milk (%) were measured during a pre-experimental period. This was followed by a treatment period, at the beginning of which lactase was injected into one side of the udder, chosen at random. It was assumed that this injection did not affect the composition of the milk in the other side, which therefore served as a control. A day later, the levels of fat and lactose were measured again. This was followed a week later by a second treatment period, at the beginning of which lactase was injected into the other side of the udder. Again, a day later the fat and lactose levels were measured. The results are presented in Table 7.9. (Data reproduced by kind permission of Roberta Bencini, Faculty of Natural and Agricultural Sciences, The University of Western Australia.) The final levels of fat and lactose are the response variables in this experiment.

(a) Identify the block and treatment terms in this experiment. (‘Period’ might be placed in either category: classify it as a treatment term.) Specify the block structure and treatment structure models.

(b) Determine the effects of the block and treatment terms on each response variable, by analysis of variance and by mixed modelling. For each significant relationship observed, state the nature of the effect, e.g. does lactase raise or lower the level of lactose? In each mixed-model analysis, the mean initial value of the response variable for each sheep can be added to the model. The difference between this mean and the corresponding individual initial value on each side of the udder in each sheep can also be added. 240 Two case studies Table 7.9 Fat and lactose content of sheep’s milk, with and without injection of lactase into the udder Sheep Side Period Treatment Fat (%) Lactose (%) initial post- initial posttreatment treatment indiv. value sheep mean indiv. value sheep mean Surprise Right 1 lactase 6.06 5.990 4.81 3.17 3.600 3.57 Surprise Left 1 control 5.92 5.990 5.70 4.03 3.600 4.06 159 Right 1 lactase 5.92 6.095 3.58 3.93 3.930 2.00 159 Left 1 control 6.27 6.095 5.06 3.93 3.930 2.98 338 Right 1 control 4.35 5.130 3.46 3.06 3.705 3.19 338 Left 1 lactase 5.91 5.130 5.82 4.35 3.705 3.22 356 Right 1 lactase 6.39 6.410 4.99 4.50 4.540 3.54 356 Left 1 control 6.43 6.410 4.86 4.58 4.540 4.01 369 Right 1 lactase 6.27 6.485 4.73 4.28 4.325 2.16 369 Left 1 control 6.70 6.485 5.21 4.37 4.325 4.03 389 Right 1 lactase 5.80 6.045 5.04 4.44 4.540 3.53 389 Left 1 control 6.29 6.045 5.11 4.64 4.540 4.56 477 Right 1 control 6.15 6.085 5.52 3.89 4.085 4.44 477 Left 1 lactase 6.02 6.085 5.43 4.28 4.085 3.62 486 Right 1 control 4.48 4.480 3.50 3.66 3.660 3.26 486 Left 1 lactase 4.48 4.480 2.95 3.66 3.660 0.37 Surprise Right 2 control 6.06 5.990 5.79 3.17 3.600 3.93 Surprise Left 2 lactase 5.92 5.990 7.00 4.03 3.600 0.66 159 Right 2 control 5.92 6.095 5.46 3.93 3.930 3.73 159 Left 2 lactase 6.27 6.095 5.48 3.93 3.930 4.60 338 Right 2 lactase 4.35 5.130 3.88 3.06 3.705 2.35 338 Left 2 control 5.91 5.130 4.37 4.35 3.705 3.73 356 Right 2 control 6.39 6.410 4.76 4.50 4.540 3.97 356 Left 2 lactase 6.43 6.410 6.12 4.58 4.540 3.71 369 Right 2 control 6.27 6.485 4.75 4.28 4.325 3.75 369 Left 2 lactase 6.70 6.485 5.52 4.37 4.325 3.79 389 Right 2 control 5.80 6.045 3.41 4.44 4.540 4.13 389 Left 2 lactase 6.29 6.045 4.11 4.64 4.540 4.39 477 Right 2 lactase 6.15 6.085 3.67 3.89 4.085 0.59 477 Left 2 control 6.02 6.085 3.66 4.28 4.085 3.73 486 Right 2 lactase 4.48 4.480 4.02 3.66 3.660 0.68 486 Left 2 control 4.48 4.480 4.01 3.66 3.660 3.76

(c) Should these terms be placed in the fixed-effect or the random-effect model?

(d) Make this change to each mixed model, and repeat the analysis. Is the final value of each response variate related to its initial value? Does the adjustment for the initial value give any improvement in the precision with which the effects of the treatments are estimated?