STAT 510
Homework 6
Due Date: 11:00 A.M., Wednesday, March 2
1. Consider an experiment conducted at two research labs. Within each lab, four mice were assigned
two treatments using a completely randomized design with two mice per treatment. Let yijk be the
response variable measurement for the kth mouse that received treatment j in research lab i (i =
1, 2; j = 1, 2; k = 1, 2). Suppose
yijk = µ + λi + τj + ijk ,
(1)
where µ, λ1 , λ2 , τ1 , and τ2 are unknown parameters and the ijk terms are independent normal random
variables with mean 0 and some unknown variance σ 2 . Let
y = (y111 , y112 , y121 , y122 , y211 , y212 , y221 , y222 )0 ,
= (111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 )0 ,
and
β = (µ, λ1 , λ2 , τ1 , τ2 )0 .
(a) Give the entries in a matrix X so that the model defined in equation (1) above can be written as
y = Xβ + .
(b) Prove that τ1 − τ2 is estimable.
(c) Give the entries in a matrix X ∗ that has all of the following properties:
• X ∗ has the same column space as X.
• X ∗ has full-column rank.
• The columns of X ∗ are orthogonal; i.e., if xu and xv are any two columns of X ∗ , then
x0u xv = 0.
(d) Define the elements of a vector β ∗ in terms of µ, λ1 , λ2 , τ1 , and τ2 so that Xβ = X ∗ β ∗ .
(e) Without the help of a computer or calculator, derive the ordinary least squares estimator of
τ1 − τ2 . To get full credit, fully simplify your answer so that it contains no matrices or vectors.
(Hint: You may want to work with X ∗ rather than X to derive the ordinary least squares
estimator of τ1 − τ2 .)
2. If y = 1µ + , where Var() = σ12 I + σ22 110 for some σ12 , σ22 > 0, then the BLUE of µ is ȳ· , the
average of the entries in the vector y. Use this fact, along with whatever else you may know about
best linear unbiased estimation, to find a simplified expression (in terms of the elements of y) for the
BLUE of µ in the following situation.
Suppose



y=


y1
y2
y3
y4
y5






 ∼ N 




µ
µ
µ
µ
µ
 
 
 
,
 
 
5
1
1
1
0
1
5
1
1
0
1
1
5
1
0
1
1
1
5
0
0
0
0
0
4



 .


Note that finding the BLUE of µ in this situation does not require inverting any matrices. All the
computations needed to obtain the BLUE can be carried out by hand in a minute or two.
3. The following questions refer to the slide set 12 entitled The ANOVA Approach to the Analysis of
Linear Mixed-Effects Models.
(a) Derive the expected mean square for ou(xu, trt) for the ANOVA table on slide 9 using the
technique illustrated on slides 15 through 17.
(b) Repeat part (a) using the technique alluded to on slide 19. You may assume t = 2, n = 2, and
m = 2 to make things easier to visualize.
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