Suppose that we approximate a response surface with a model of order d₁, such as y = x₁β₁ + ∈, when the true surface is described by a model of order d₂ > d₁; that is, E(y) = X₁β₁ + X₁β₂.

(a) Show that the regression coefficients are biased, that is, that E(β̂₁)) = β₁ + Aβ₂, where A = (X'₁X₁)^-1X'₁X₂. A is usually called the alias matrix.

(b) If d₁ = 1 and d₂ = 2, and a full 2^k 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 2^3-1 design is used to fit the model.

(d) If d₁ = 1, d₂ = 2, 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).