Section 2.4.5 considered the “main effects” model for a balanced 2×2 layout, showing there is orthogonality between each pair of parameters when we constrain ∑

i ????i = ∑

j ????j = 0.

a. If you instead constrain ????1 = ????1 = 0, show that pairs of columns of X are uncorrelated but not orthogonal.

b. Explain why ????2 for the coding ????1 = 0 in

(a) is identical to 2????2 for the coding ????1 + ????2 = 0.

c. Explain how the results about constraints and orthogonality generalize if the model also contains a term ????ij to permit interaction between A and B in their effects on y.