Definition
: If there are observations at all combinations of all factors,
the design is
complete
, otherwise it is
incomplete
.
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Osborne, ST512
124
Exercise
1. Estimate the mean difference in cholesterol between young men
and young women.
2. Estimate the mean difference between old men and old women.
3. Estimate the mean difference between men and women.
4. Estimate the mean difference between older and younger folks.
5. Estimate the mean difference between the differences estimated
in 1. and 2.
6. Provide standard errors for all of these estimated contrasts
7. Specify the vectors defining these contrasts.
For example, the
first contrast of cohort means can be written
θ
1
= (

1
,
1
,
0
,
0)
μ
1
μ
2
μ
3
μ
4
=
μ
2

μ
1
Osborne, ST512
125
Consider the following contrasts of the cohort cholesterol means in
the population:
θ
3
= (

1
,
1
,

1
,
1)
μ
θ
4
= (

1
,

1
,
1
,
1)
μ
θ
5
= (

1
,
1
,
1
,

1)
μ
Q: Are these contrasts orthogonal?
Q: True/False:
SS
(
ˆ
θ
3
) +
SS
(
ˆ
θ
4
) +
SS
(
ˆ
θ
5
) =
SS
[
Trt
]
Another exercise:
1. Compute the sums of squares for the estimated contrasts in 3.,
4. and 5. using the exercise just completed and the fact that if
ˆ
θ
=
∑
c
i
¯
y
i
+
then
SS
[
ˆ
θ
] =
ˆ
θ
2
∑
c
2
i
n
i
.
2. Formulate a test of
H
0
:
θ
i
= 0 for each of these three contrasts.
Obtain the
F
ratio for each of these tests.
3. Obtain the
α
= 0
.
05 critical region for each test. Compare the
observed
F
ratios to critical value and draw conclusions about
(a) an age effect
(b) a gender effect
(c) an age
×
gender interaction
Osborne, ST512
126
Types of effects
Twoway ANOVA model for the cholesterol measurements:
Y
ijk
=
μ
+
α
i
+
β
j
+ (
αβ
)
ij
+
E
ijk
i
= 1
,
2 =
a
and
j
= 1
,
2 =
b
and
k
= 1
,
2
, . . . ,
7 =
n.
E
ijk
iid
∼
N
(0
, σ
2
). Parameter constraints:
∑
i
α
i
=
∑
j
β
j
= 0 and
∑
i
(
αβ
)
ij
≡
0 for each
j
and
∑
j
(
αβ
)
ij
≡
0 for each
i
.
Factor A: AGE has
a
= 2 levels 
A
1
: younger and
A
2
: older
Factor B: GENDER has
b
= 2 levels 
B
1
: female and
B
2
: male
Three kinds of effects in 2
×
2 designs:
1.
Simple
effects are simple contrasts.
•
μ
(
A
1
B
) =
μ
II

μ
I
 simple effect of gender for young folks.
•
μ
(
AB
1
) =
μ
III

μ
I
 simple effect of age for women
2.
Interaction
effects are differences of simple effects:
μ
(
AB
) =
μ
(
AB
2
)

μ
(
AB
1
) = (
μ
IV

μ
II

(
μ
III

μ
I
))
 difference between simple age effects for men and women
 difference between simple gender effects for old and young folks
 interaction effect of AGE and GENDER.
3.
Main
effects are averages or sums of simple effects
μ
(
A
) =
1
2
(
μ
(
AB
1
) +
μ
(
AB
2
))
μ
(
B
) =
1
2
(
μ
(
A
1
B
) +
μ
(
A
2
B
))
Exercise: Classify the contrasts in the last exercise as simple, inter
action or main effects.
Osborne, ST512
127
Partitioning the treatment
SS
into
t

1 orthogonal components
12281 =
SS
[
Trt
] =
SS
[
ˆ
θ
3
] +
SS
[
ˆ
θ
4
] +
SS
[
ˆ
θ
5
] = 5103 + 6121 + 1056
•
(
a

1)(
b

1)
df
for
AB
interaction
•
(
a

1)
df
for main effect of
A
•
(
b

1)
df
for main effect of
B
F
test for interaction effect
To test for interaction,
H
0
: (
αβ
)
11
= (
αβ
)
12
= (
αβ
)
21
= (
αβ
)
22
= 0
vs.
H
1
: (
αβ
)
ij
= 0 for some
i, j
use
θ
5
=
μ
(
AB
) and
F
=
SS
(
ˆ
θ
)
/
((
a

1)(
b

1))
MS
[
E
]
on 1 and 28

4 = 24 numerator, denominator
df
. For cholesterol
datathe estimated interaction effect is
ˆ
θ
5
= ˆ
μ
(
AB
) = (251
.
3

209
.
4)

(212

194
.
7) = 41
.
9

17
.
3 = 24
.
6
the associated sum of squares is
SS
(
ˆ
θ
5
) =
(24
.
6)
2
1
7
+
(

1)
2
7
+
(

1)
2
7
+
1
7
=
(24
.
6)
2
4
7
= 1056
and
F
= 1056
/
1185 = 0
.
9
which isn’t significant at
α
= 0
.
05 on 1,24
df
.