Consider using common random numbers (CRN) across all four design points of the full 22 fa...

Consider using common random numbers (CRN) across all four design points of the full 2² factorial design on the inventory model in Examples 12.2 and 12.3. Let C₁₂ = Cov(R₁, R₂), C₁₃ = Cov(R₁, R₃), etc., and assume that CRN “works,” i.e., all the covariances between the R_i’s are positive.

(a) Find expressions for the variances of both main effects as well as the interaction effect in terms of these covariances and the variances of the R_i’s. What can you conclude about whether CRN reduces the variances of the estimators for the expected effects?

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(b) Suppose that we are interested primarily in getting precise estimates of the expected main effects and care less about the expected interaction effect. Suggest an alternative random-number-assignment strategy that would do this. What happens to the precision of the estimate for the expected interaction effect?

Example 12.2

TABLE 12.4 Design matrix and simulation results for the 2² factorial design on s and d for the inventory model

Example 12.3

TABLE 12.5 Sample means and variances of the responses for the inventory model

TABLE 12.6 96.667 percent confidence intervals for the expected effects, inventory model