Suppose that we approximate a response surface with a model of order d1, such as y = X1????1 + ????, when the true surface is described by a model of order d2 > d1; that is, E(y) =

X1????1 + X1????2.

(a) Show that the regression coefficients are biased, that is, E(̂

????

1) = ????1 + A????2, where A = (X′

1X1)−1X′

1X2. A is usually called the alias matrix.

(b) If d1 = 1 and d2 = 2, and a full 2k is used to fit the model, use the result in part

(a) to determine the alias structure.

(c) If d1 = 1, d2 = 2, and k = 3, find the alias structure assuming that a 23−1 design is used to fit the model.

(d) If d1 = 1, d2 = 2, and k = 3, and the simplex design in Problem 11.3 is used to fit the model, determine the alias structure and compare the results with part (c).