Several times we have used the hierarchy principle in selecting a model; that is, we have included nonsignificant lower-order terms in a model because they were factors involved in significant higher-order terms. Hierarchy is certainly not an absolute principle that must be followed in all cases. To illustrate, consider the model resulting from Problem 6-1, which required that a nonsignificant main effect be included to achieve hierarchy. Using the data from Problem 6-1,
(a) Fit both the hierarchial and the nonhierarchial model.
(b) Calculate the PRESS statistic, the adjusted R2, and the mean square error for both models.
(c) Find a 95 percent confidence interval on the estimate of the mean response at a cube corner (x1 = x2 = x3 = ±1). Hint: Use the results of Problem 6-33.
(d) Based on the analyses you have conducted, which model do you prefer?
Often the fitted regression model from a 2k factorial design is used to make predictions at points of interest in the design space.
(a) Find the variance of the predicted response ŷ at a point x1, x2, . xk in the design space. Remember that the x’s are coded variables, and assume a 2k design with an equal number of replicates n at each design point so that the variance of a regression coefficient β̂ is σ2/(n2k) and that the covariance between any pair of regression coefficients is zero.
(b) Use the result in part (a) to find an equation for a 100(1 – a) percent confidence interval on the true mean response at the point x1, x2 xk in design space.
Problem 6-1.
An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results follow:
