Verify the expected mean squares in Table 13.8. What changes occur if we include the term (ed) N(0,

a) in the model?

#!# 13.18. Brownlee (1960) describes an experiment in bacteriological testing of milk. Twelve milk samples were examined in all six combinations of two types of bottles and three types of tubes. Ten tests were run on each combination and the data shown below give the number of positive tests in each set of ten. Assume the three-factor mixed model in which tube, bottle, and their interaction are fixed effects, and sample is a random effect. Noting that we have only one observation per cell, use the convention that o, -0.

a. Fit the appropriate linear model, complete the analysis of variance table, including the expected mean squares, and test the usual hypotheses on the fixed effects.

b. Estimate the appropriate function of the variance components to construct confidence intervals on all pairwise differences in marginal means for the tube effect, and comment on the source of significance of that factor.

c. Estimate the individual variance components, and test the hypotheses that da, and d are zero where samples is the third factor.

d. Examine the data to see if you can explain the negative estimate.

e. Generalize the results in Table 13.17 to obtain the expected mean squares obtained by Brownlee. In particular, note that he gives the following expression for the sample expected mean square: A = $123 +603 Section 13.6