Chapter 3:
K
Sample Methods
0. Comparing more than two groups (treatments)
•
overall comparison: whether or not differences exist among groups
•
multiple comparison: which groups differ significantly from the others
•
Assumption: the experimental units are assigned to the
k
treatments in a
completely random design or the observations have been randomly selected
from
k
populations
•
Hypothesis:
H
0
:
F
1
(
x
) =
F
2
(
x
) =
· · ·
=
F
k
(
x
)
H
a
:
F
i
(
x
)
≤
F
j
(
x
) or
F
i
(
x
)
≥
F
j
(
x
)
for at least one pair (
i, j
) with strict inequality holding for at least one
x
•
Shift Alternative Hypothesis:
H
0
:
F
1
(
x
) =
F
2
(
x
) =
· · ·
=
F
k
(
x
) =
F
(
x
)
H
a
:
F
i
(
x
) =
F
(
x
−
μ
i
)
for at least one
i
of
i
= 1
, . . . , k
1.
K
Sample Permutation
F
Tests
•
OneWay Data Layout
X
ij
=
μ
i
+
ǫ
ij
where
ǫ
ij
∼
iid
F
(
ǫ
)
N
=
n
1
+
n
2
+
· · ·
+
n
k
Treatments
Observations
Sample Sizes
Means
Variances
1
X
11
, X
12
, . . . , X
1
n
1
n
1
¯
X
1
S
2
1
2
X
21
, X
22
, . . . , X
2
n
2
n
2
¯
X
2
S
2
2
. . .
. . .
. . .
. . .
. . .
k
X
k
1
, X
k
2
, . . . , X
kn
k
n
k
¯
X
k
S
2
k
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
•
OneWay ANOVA test
–
Assumptions:
∗
for
i
= 1
, . . . , k j
= 1
, . . . , n
i
X
ij
∼
iid
N
(
μ
i
, σ
2
)
∗
X
i
’s are independent
–
Hypothesis:
H
0
:
μ
1
=
μ
2
=
· · ·
=
μ
k
H
a
:
μ
i
negationslash
=
μ
j
for at least one pair (
i, j
)
–
F
test:
¯
X
=
∑
k
i
=1
∑
n
i
j
=1
X
ij
/N
Source
DF
SS
MS
F
Treatment
k
−
1
SST
r
=
∑
k
i
=1
n
i
(
¯
X
i
−
¯
X
)
2
MST
=
SST
k

1
F
=
MST
MSE
Error
N
−
k
SSE
=
∑
k
i
=1
(
n
i
−
1)
S
2
i
MSE
=
SSE
N

k
Total
N
−
1
SS
total
=
∑
k
i
=1
∑
n
i
j
=1
(
X
ij
−
¯
X
)
2
–
Under
H
0
,
F
∼
F
k

1
,N

k
–
reject
H
0
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 SST, ni Ri, strict inequality holding

Click to edit the document details