Scheffé’s Method
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Statistics 611
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Scheffé’s Method:
Suppose
y = Xβ + ε,
where
ε ∼ N(0, σ 2 I).
Let
c01 β, . . . , c0q β
be q estimable functions, where
c1 , . . . , cq
are linearly independent.
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Statistics 611
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Let
 
c01
.
.
C=
.
c0q
and define
  

θ1
c01 β
.  . 
.  . 
θ=
 .  =  .  = Cβ.
θq
c0q β
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Let W = C(X0 X)− C0 with diagonal elements denoted
w11 , . . . , wqq .
Then
Var(Cβ̂) = σ 2 W
and
Var(θ̂j ) = Var(c0j β̂) = σ 2 wjj
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j = 1, . . . , q.
Statistics 611
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For any q × 1 vector u and any k ∈ R, let L(u, k) denote the interval
√
√
[u0 θ̂ − k σ̂ 2 u0 Wu, u0 θ̂ + k σ̂ 2 u0 Wu].
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We want to find k 3
P(u0 θ ∈ L(u, k) ∀ u ∈ Rq ) = 1 − α.
Thus, we seek simultaneously coverage probability 1 − α for an infinite
set of intervals.
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Statistics 611
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Show that
u0 θ ∈ L(u, k) ∀ u ∈ Rq
⇐⇒
k2
(Cβ̂ − Cβ)0 (C(X0 X)− C0 )−1 (Cβ̂ − Cβ)
≤
.
qσ̂ 2
q
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Statistics 611
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Thus, we have
P(u0 θ ∈ L(u, k) ∀ u ∈ Rq )
(Cβ̂ − Cβ)0 (C(X0 X)− C0 )−1 (Cβ̂ − Cβ)
k2
=P
≤
.
qσ̂ 2
q
What shall we choose for k to make this probability equal to 1 − α?
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Dan Nettleton (Iowa State University)
Statistics 611
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Example:
Suppose an experiment was conducted using a completely
randomized design with 10 subjects in each of 4 treatment groups.
The treatment groups were defined by the combinations of levels from
2 factors: diet (1 or 2) and exercise program (1 or 2).
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Statistics 611
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The model
yijk = µij + εijk
was fit the response data
yijk =
measure of overall health
for diet i, exercise program j,
subject k (i = 1, 2; j = 1, 2; k = 1, . . . , 10).
The parameter µij represents the mean response for diet i, exercise
program j (i = 1, 2; j = 1, 2).
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The εijk terms are assumed to be iid N(0, σ 2 ).
A summary of the data is as follows:
ȳ11· = 9
ȳ12· = 7
ȳ21· = 8
ȳ22· = 3
σ̂ 2 = 5.
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Suppose we want to construct a set of confidence intervals using a
method that gives simultaneous coverage probability at least 95%.
Suppose the confidence intervals will be used to address the following
questions:
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1. Diet main effect?
2. Exercise program main effect?
3. Diet-by-exercise program interaction?
4. Difference between diet 1 and diet 2 under exercise program 1?
5. Difference between exercise program 1 and 2 under diet 1?
6. Diet 1, exercise program 1 vs. mean of other treatments?
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What estimable function of
 
µ
 11 
µ 
 12 
β= 
µ21 
 
µ22
is of interest in each of the questions 1 through 6, respectively?
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Compute an estimate and standard error for each estimable function of
interest.
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Determine appropriate intervals to address questions 1 through 6.
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Each interval is of the form
c0j β̂ ± ksej .
How shall we choose k?
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If we use the Bonferroni approach, then
k = t40−4,
0.05
(2)(6)
≈ 2.79.
This approach would be legitimate if we were interested in these 6
intervals, and only these 6 intervals, prior to observing the data.
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Alternatively, we can consider Scheffé intervals, k =
p
qFq,40−4,0.05 .
What is the value of q in our situation?
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