9.3 A chemist wants to compare three treatments. The experimental material he plans to use comes from four different manufacturers. He expects systematic differences among the material from the different manufacturers. Moreover, he is interested in finding out whether differences between treatment effects depend on the manufacturer. There is sufficient experimental material from each manu- facturers for 12 experimental units. (i) What experimental design should he use for his experiment? (ii) Suppose he comes to you with the following partial ANOVA table:

Fill in the ANOVA table. (iii) Give the SAS statements (classes, model) for the ANOVA in (ii). (iv) The chemist claims that he has used four replications per treatment, per manufacturer. After some questioning you find out that the four "repli- cations" are actually two replications and two duplicate measurements for each experimental unit. In this case what should the ANOVA really look like? Give sources of variation and d.f. (v) Give the SAS statements (classes, model) for the ANOVA in (iv). (vi) Unfortunately, the computer file with his original data was destroyed. In order to correct the situation and come up with a reasonable ANOVA table, you devise a small experiment which will allow you to obtain an estimate of the sampling (observational) error variance component. Suppose this estimate equals 1. Assuming that this is a reasonable estimate for the actual experiment, that is, assuming that this value was obtained from the actual experiment, complete now the ANOVA table given in (iv) by filling in d.f., SSS and MSS. (vii) Based on the ANOVA in (vi), give the test statistic for testing the null hy- pothesis that there is no M x T interaction. (viii) Based on the ANOVA in (vi), give the standard error for a simple treatment comparison (that is, the differences between the estimates of two treatment effects).

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Source Manufacturers (M) Treatments(T) MxT Error Total d.f. SS MS 20 25 5 2.3