8. Red blood cell experiment The trout experiment reported by Gutsell (Biometrics, 1951) was described in Exercise 15 of Chap. 3. As part of the same experiment, the red blood cell counts in the blood of brown trout were measured. Fish were put at random into eight troughs of water. Two troughs were assigned to each of the four levels of the treatment factor “sulfamerazine” (0, 5, 10, 15 grams per 100 pounds of fish added to the diet per day). After 42 days, five fish were selected at random from each trough and the red blood cell count from the blood of each fish was measured in two different counting chambers, giving two measurements per fish. The observations reported in Table 18.12, when multiplied by 5000, give the number of red blood cells per cubic millimeter of blood. A possible model for these data is Yijkt = μ + αi + Bj(i) + Ck(i j) + ijkt , ijkt ∼ N(0, σ2) , Bj(i) ∼ N(0, σ2 B(A)) , Ck(i j) ∼ N(0, σ2 C(AB)) , i = 1, 2, 3, 4; j = 1, 2; k = 1,..., 5; t = 1, 2;

where αi is the effect of the ith level of sulfamerazine in the diet, Bj(i) is the effect of the jth randomly selected trough assigned to the ith level of sulfamerazine, and Ck is the effect of the kth randomly selected fish from the (i j)th trough, and random variables on the right hand-side of the model are assumed to be mutually independent.

(a) What are the experimental units and observational units in this experiment?

(b) Since the data are counts, examine the assumptions of normally distributed errors and equal error variances by treatment. If the assumptions are not approximately satisfied, is there a transformation that can be used to correct the problem?

(c) Write out the degrees of freedom and the expected mean squares for each term in the model.

(d) Test the hypothesis that sulfamerazine has no effect on the red blood cell counts. Examine the linear and quadratic trends.

(e) If the test in part

(d) is rejected, calculate simultaneous 95% confidence intervals for pairwise comparisons in the effects of the sulfamerazine levels.