11.22. 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 % A&2, where A $ . 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).