The University of Sydney
School of Mathematics and Statistics
Homework 5
STAT3922: Applied Linear Model (Advanced) Semester 1, 2024
Consider the linear model y = Xβ + ε, where the n × p design matrix X has rank
r < p. In the advanced lecture week 12, we consider the general linear hypothesis test
H0 : Lβ = 0 where Lβ is a set of q estimable linear combinations. This general linear
hypothesis test is carried out via a F -statistic,
F =
SSH/q
SSE/(n− r)
where SSH = (Lβˆ)⊤
[
L(X⊤X)−L⊤
]−1
(Cβˆ) is called the “sum of squares due to hypoth-
esis,” with βˆ = (X⊤X)−X⊤y. The purpose of this homework is to show that this SSH is
simplified to many common sum of squares in regular ANOVA models.
1. Consider the one-way ANOVA model with t = 3 groups, and each group has ni = 2
replications, i.e
yij = µ+ αi + εij, i = 1, 2, 3, j = 1, 2.
Consider the test H0 : α1 = α2 = α3. Show that the SSH is the same as the sum
of squares between groups, i.e SSH =
∑t
i=1 ni(y¯i•− y¯••)2. Your proof need to show
the expression for (Cβˆ) and
[
L(X⊤X)−L⊤
]−1
explicitly.
2. Consider the two-way ANOVAmodel with interactions between two factors A (with
a = 2 levels) and B (with b = 2 levels). Each factor-level combination has r = 2
replications. That is, we assume the model
yijk = µ+ αi + βj + γij + εijk, i = 1, 2, j = 1, 2, k = 1, 2
Consider the test for interaction H0 : γ11 − γ12 − (γ21 − γ22) = 0. Show that the
SSH for this test is the same as the sum of squares due to interaction as written in
the mainstream lecture, i.e SSH = r
∑2
i=1
∑2
j=1(y¯ij• − y¯i•• − y¯•j• + y¯••)2.
You can use R to obtain the inverse or generalized inverse of a matrix.
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