(a) Dropping a predictor that is orthogonal to the others doesn't change the

coefficient estimates. More specifically, suppose that the n × p design matrix X, assumed

to be of full rank, is partitioned into [XA XB] with XA being n× PA and XB being n X PB,

with p = PA + PB. Let 3 be the least squares estimate using X and BA that using XA.

Suppose that XB is orthogonal to X₁: XÂXA = 0. Show that

Bi = BA,i

for i=1,..., PA.

(b) Consider a one way ANOVA model

Yij = μl + αį + €įj, i = 1, ..., I, j = 1,..., nį.

Suppose that the design is balanced, n₁ = n₁ for all i. Consider the design matrices corre-

sponding to "treatment" and "sum" contrasts. In each of the two cases, is the intercept

column orthogonal to the factor columns? What if the design is unbalanced? Explain.

(c) Now consider the coefficient differences a; - aj in the balanced one way ANOVA

model. Do their estimates âi - â; depend on whether treatment or sum contrasts are

chosen? Explain.