The Gauss-Markov Linear Model
The Gauss-Markov Linear Model
y = Xβ + Note that the model is not completely specified because the
distribution of y is not completely specified.
y is an n × 1 random vector of responses.
y = Xβ + ,
X is an n × p matrix of constants with columns corresponding to
explanatory variables. X is sometimes referred to as the design
matrix.
=⇒
=⇒
β is an unknown parameter vector in IRp .
is an n × 1 random vector of errors.
E() = 0,
Var() = σ 2 I
E(y) = Xβ, Var(y) = σ 2 I
y ∼ (Xβ, σ 2 I)
“y has a distribution with mean Xβ and variance σ 2 I.”
E() = 0 and Var() = σ 2 I, where σ 2 is an unknown parameter in
IR+ .
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The Normal Theory Gauss-Markov Linear Model
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Example 1
Researchers harvested five randomly selected ears of corn from a
field. For i = 1, . . . , 5; let yi denote the weight in grams of the ith ear.
y = Xβ + y1 , . . . , y5 ∼ N(μ, σ 2 )
i.i.d.
We often add an assumption of multivariate normality to the
Gauss-Markov linear model: ∼ N(0, σ 2 I).
yi = μ + i ,
Of course, ∼ N(0, σ 2 I) =⇒ y ∼ N(Xβ, σ 2 I).
The goal of analysis often focuses on answering questions
about certain linear functions of β of the form Cβ for a
specified matrix C.
Statistics 511
i = 1, . . . , 5;
1 , . . . , 5 ∼ N(0, σ 2 )
i.i.d.
y1 = μ + 1
y2 = μ + 2
y3 = μ + 3
1 , . . . , 5 ∼ N(0, σ 2 )
i.i.d.
y4 = μ + 4
The normality assumption is useful for constructing
confidence intervals and performing tests concerning Cβ.
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Statistics 511
y5 = μ + 5
3 / 32
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Statistics 511
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Example 1 (continued)
Example 1 (continued)
⎡
y1 = μ + 1
⎢
⎢
⎢
⎢
⎣
y2 = μ + 2
1 , . . . , 5 ∼ N(0, σ 2 )
i.i.d.
y3 = μ + 3
y4 = μ + 4
y5 = μ + 5
⎡
⎢
⎢
⎢
⎢
⎣
y1
y2
y3
y4
y5
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎦ ⎣
μ
μ
μ
μ
μ
⎡
⎡
⎤
⎥ ⎢
⎥ ⎢
⎥+⎢
⎥ ⎢
⎦ ⎣
1
2
3
4
5
⎤
⎡
⎥
⎥
⎥,
⎥
⎦
⎢
⎢
⎢
⎢
⎣
y1
⎢ y2
⎢
⎢ y3
⎢
⎣ y4
y5
⎤
1
2
3
4
5
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎦
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Statistics 511
5 / 32
Example 1 (continued)
⎡
⎢
⎢
⎢
⎢
⎣
y1
y2
y3
y4
y5
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎦ ⎣
1
1
1
1
1
⎤ ⎡
⎤
1
μ
⎥ ⎢ μ ⎥ ⎢ 2 ⎥
⎥ ⎢
⎥
⎥ ⎢
⎥ = ⎢ μ ⎥ + ⎢ 3 ⎥ ,
⎥ ⎢
⎥
⎥ ⎢
⎦ ⎣ μ ⎦ ⎣ 4 ⎦
μ
5
⎡
⎤ ⎡ ⎤
⎤
1
1
⎢ 2 ⎥
⎥ ⎢ 1 ⎥
⎢
⎥ ⎢ ⎥
⎥
⎥ = ⎢ 1 ⎥ [μ] + ⎢ 3 ⎥ ,
⎢
⎥ ⎢ ⎥
⎥
⎣ 4 ⎦
⎦ ⎣ 1 ⎦
1
5
⎤
⎡
⎡
⎤
1
⎢ 2 ⎥
⎢
⎥
⎢ 3 ⎥ ∼ N(0, σ 2 I)
⎢
⎥
⎣ 4 ⎦
5
⎡
⎤
1
⎢ 2 ⎥
⎢
⎥
⎢ 3 ⎥ ∼ N(0, σ 2 I)
⎢
⎥
⎣ 4 ⎦
5
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Statistics 511
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Example 2
⎡
⎤
y1
y2
y3
y4
y5
⎢
⎥
⎢
⎥
⎥ [μ] + ⎢
⎢
⎥
⎣
⎦
1
2
3
4
5
⎤
⎡
⎥
⎥
⎥,
⎥
⎦
⎢
⎢
⎢
⎢
⎣
1
2
3
4
5
⎤
Researchers randomly assigned eight experimental units to two
treatments and measured a response of interest. For i = 1, 2; let
yi1 , yi2 , yi3 , yi4 denote the responses of the experimental units in the ith
treatment group.
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎦
y11 , y12 , y13 , y14 ∼ N(μ1 , σ 2 )
i.i.d.
independent of
y = Xβ + ,
y21 , y22 , y23 , y24 ∼ N(μ2 , σ 2 )
i.i.d.
2
∼ N(0, σ I)
yij = μi + ij ,
11 , 12 , 13 , 14 , 21 , 22 , 23 , 24 ∼ N(0, σ 2 )
i.i.d.
Cβ = [1][μ] = μ
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i = 1, 2; j = 1, . . . , 4
Statistics 511
7 / 32
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Statistics 511
8 / 32
Example 2 (continued)
Example 2 (continued)
y11 = μ1 + 11
⎡
y12 = μ1 + 12
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
y13 = μ1 + 13
y14 = μ1 + 14
y21 = μ2 + 21
y22 = μ2 + 22
y23 = μ2 + 23
y24 = μ2 + 24
y11
y12
y13
y14
y21
y22
y23
y24
⎡
⎤
μ1
μ1
μ1
μ1
μ2
μ2
μ2
μ2
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥+⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
11
12
13
14
21
22
23
24
⎤
⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11
12
13
14
21
22
23
24
⎤
⎥
⎥
⎥
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎥
⎥
⎥
⎦
11 , 12 , 13 , 14 , 21 , 22 , 23 , 24 ∼ N(0, σ 2 )
i.i.d.
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Example 2 (continued)
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⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
y11
y12
y13
y14
y21
y22
y23
y24
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
⎤
⎡
⎢
⎥
⎢
⎥
⎢
⎥
⎥
⎢
⎢
⎥ μ1
⎢
⎥
⎥ μ2 + ⎢
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
11
12
13
14
21
22
23
24
⎤
⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11
12
13
14
21
22
23
24
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎥
⎥
⎥
⎦
y11
y12
y13
y14
y21
y22
y23
y24
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
⎤
⎡
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥ μ1
⎢
⎥
⎢
⎥ μ2 + ⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎦
⎣
y = Xβ + ,
Cβ = [1, −1]
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Example 2 (continued)
⎡
⎡
Statistics 511
Statistics 511
11 / 32
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11
12
13
14
21
22
23
24
⎤
⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11
12
13
14
21
22
23
24
⎤
⎥
⎥
⎥
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎥
⎥
⎥
⎦
∼ N(0, σ 2 I)
μ1
μ2
= μ1 − μ2
Statistics 511
12 / 32
Example 3
Example 3 (continued)
Suppose eight fertilizer amounts denoted x1 , . . . , x8 were randomly
assigned to eight field plots. For i = 1, . . . , 8; let yi denote the yield of
the plot that received fertilizer amount xi .
yi = β0 + β1 xi + i ,
⎡
i = 1, . . . , 8
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 , . . . , 8 ∼ N(0, σ 2 )
i.i.d.
y1 = β0 + β1 x1 + 1
y2 = β0 + β1 x2 + 2
y3 = β0 + β1 x3 + 3
y4 = β0 + β1 x4 + 4
y1
y2
y3
y4
y5
y6
y7
y8
⎡
⎤
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
β0 + β1 x1
β0 + β1 x2
β0 + β1 x3
β0 + β1 x4
β0 + β1 x5
β0 + β1 x6
β0 + β1 x7
β0 + β1 x8
⎤
⎡
1
2
3
4
5
6
7
8
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥+⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
⎤
⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2
3
4
5
6
7
8
⎤
⎥
⎥
⎥
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎥
⎥
⎥
⎦
y5 = β0 + β1 x5 + 5
y6 = β0 + β1 x6 + 6
y7 = β0 + β1 x7 + 7
y8 = β0 + β1 x8 + 8
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Example 3 (continued)
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⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
y1
y2
y3
y4
y5
y6
y7
y8
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
1
1
1
1
1
1
1
1
x1
x2
x3
x4
x5
x6
x7
x8
⎤
⎡
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥ β0
⎢
⎥
⎢
⎥ β1 + ⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎦
⎣
1
2
3
4
5
6
7
8
⎤
⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2
3
4
5
6
7
8
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎥
⎥
⎥
⎦
y1
y2
y3
y4
y5
y6
y7
y8
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
1
1
1
1
1
1
1
1
x1
x2
x3
x4
x5
x6
x7
x8
⎤
⎡
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥ β0
⎢
⎥
⎢
⎥ β1 + ⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎦
⎣
y = Xβ + ,
Cβ = [0, 1]
Statistics 511
15 / 32
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1
2
3
4
5
6
7
8
⎤
⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2
3
4
5
6
7
8
⎤
⎥
⎥
⎥
⎥
⎥
⎥ ∼ N(0, σ 2 I)
⎥
⎥
⎥
⎥
⎦
∼ N(0, σ 2 I)
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Example 3 (continued)
⎡
⎡
Statistics 511
β0
β1
= β1
Statistics 511
16 / 32
Example 4
Example 4 (continued)
Eight hogs were randomly assigned to two diets and two inoculations
such that two hogs received each combination of diet and inoculation.
This experiment involves two factors: diet and inoculation.
In this case, each factor has two levels (denoted here generically
as 1 and 2).
For i = 1, 2; j = 1, 2; and k = 1, 2; let yijk denote the average daily gain
of the kth hog that received diet i and inoculation j.
yijk = μ + ijk
i = 1, 2; j = 1, 2; k = 1, 2;
A combination of one level from each factor forms a treatment.
111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 ∼ N(0, σ 2 )
i.i.d.
In this case, we have four treatments:
Treatment
1
2
3
4
Diet
1
1
2
2
Inoculation
1
2
1
2
Under this model, neither diet nor inoculation affects average daily
gain.
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Statistics 511
17 / 32
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Example 4 (continued)
Example 4 (continued)
For i = 1, 2; j = 1, 2; and k = 1, 2; let yijk denote the average daily gain
of the kth hog that received diet i and inoculation j.
For i = 1, 2; j = 1, 2; and k = 1, 2; let yijk denote the average daily gain
of the kth hog that received diet i and inoculation j.
yijk = μ + αi + ijk
i = 1, 2; j = 1, 2; k = 1, 2;
yijk = μ + βj + ijk
111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 ∼ N(0, σ 2 )
111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 ∼ N(0, σ 2 )
i.i.d.
i.i.d.
Under this model, only diet affects average daily gain.
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i = 1, 2; j = 1, 2; k = 1, 2;
Under this model, only inoculation affects average daily gain.
Statistics 511
19 / 32
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Statistics 511
20 / 32
Example 4 (continued)
Example 4 (continued)
yijk = μ + αi + βj + ijk
yijk = μ + αi + βj + γij + ijk
i = 1, 2; j = 1, 2; k = 1, 2;
111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 ∼ N(0, σ 2 )
i.i.d.
111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 ∼ N(0, σ 2 )
i.i.d.
Under this model, factors diet and inoculation affect the mean
average daily gain in an additive manner.
There is no interaction between the factors diet and inoculation.
diet
1
2
diet difference
inoculation
1
2
μ + α1 + β1 μ + α1 + β2
μ + α2 + β1 μ + α2 + β2
α1 − α2
α1 − α2
inoculation difference
β1 − β2
β1 − β2
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Example 4 (continued)
Under this model, there is one mean for each combination of diet
and inoculation.
Those four means are free to take any four values with no
restrictions.
diet
1
2
Δdiet
inoculation
1
2
μ + α1 + β2 + γ12
μ + α1 + β1 + γ11
μ + α2 + β1 + γ21
μ + α2 + β2 + γ22
α1 − α2 + γ11 − γ21 α1 − α2 + γ12 − γ22
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Δinoculation
β1 − β2 + γ11 − γ12
β1 − β2 + γ21 − γ22
Statistics 511
22 / 32
Example 4 (continued)
yijk = μ + αi + βj + γij + ijk
An equivalent model is the so called cell means model:
yijk = μij + ijk
i = 1, 2; j = 1, 2; k = 1, 2;
i = 1, 2; j = 1, 2; k = 1, 2;
i = 1, 2; j = 1, 2; k = 1, 2;
y111 = μ + α1 + β1 + γ11 + 111
y112 = μ + α1 + β1 + γ11 + 112
111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 ∼ N(0, σ 2 )
i.i.d.
y121 = μ + α1 + β2 + γ12 + 121
y122 = μ + α1 + β2 + γ12 + 122
diet
1
2
Δdiet
inoculation
1
2
μ11
μ12
μ21
μ22
μ11 − μ21 μ12 − μ22
y211 = μ + α2 + β1 + γ21 + 211
Δinoculation
μ11 − μ12
μ21 − μ22
y212 = μ + α2 + β1 + γ21 + 212
y221 = μ + α2 + β2 + γ22 + 221
y222 = μ + α2 + β2 + γ22 + 222
111 , 112 , 121 , 122 , 211 , 212 , 221 , 222 ∼ N(0, σ 2 )
i.i.d.
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23 / 32
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24 / 32
Example 4 (continued)
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
y111
y112
y121
y122
y211
y212
y221
y222
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
μ + α1 + β1 + γ11
μ + α1 + β1 + γ11
μ + α1 + β2 + γ12
μ + α1 + β2 + γ12
μ + α2 + β1 + γ21
μ + α2 + β1 + γ21
μ + α2 + β2 + γ22
μ + α2 + β2 + γ22
Example 4 (continued)
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥+⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
111
112
121
122
211
212
221
222
⎤ ⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
111
112
121
122
211
212
221
222
⎡
⎤
⎥
⎥
⎥
⎥
⎥
⎥ ∼ N(0, I)
⎥
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
y111
y112
y121
y122
y211
y212
y221
y222
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
0
1
1
0
0
0
0
1
1
0
0
1
1
y = Xβ + ,
c
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Statistics 511
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Example 4 (continued)
inoculation
1
2
μ + α1 + β2 + γ12
μ + α1 + β1 + γ11
μ + α2 + β1 + γ21
μ + α2 + β2 + γ22
α1 − α2 + γ11 − γ21 α1 − α2 + γ12 − γ22
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
⎡
⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎢
⎣
μ
α1
α2
β1
β2
γ11
γ12
γ21
γ22
⎤
⎡
⎥
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥+⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎣
⎦
111
112
121
122
211
212
221
222
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
∼ N(0, σ 2 I)
c
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Statistics 511
26 / 32
Example 4 (continued)
β = [μ, α1 , α2 , β1 , β2 , γ11 , γ12 , γ21 , γ22 ]
β = [μ, α1 , α2 , β1 , β2 , γ11 , γ12 , γ21 , γ22 ]
diet
1
2
Δdiet
1
1
0
0
0
0
0
0
⎤
Δinoculation
β1 − β2 + γ11 − γ12
β1 − β2 + γ21 − γ22
diet
1
2
Δdiet
inoculation
1
2
μ + α1 + β2 + γ12
μ + α1 + β1 + γ11
μ + α2 + β1 + γ21
μ + α2 + β2 + γ22
α1 − α2 + γ11 − γ21 α1 − α2 + γ12 − γ22
Δinoculation
β1 − β2 + γ11 − γ12
β1 − β2 + γ21 − γ22
Is the difference between diet means for inoculation 1 the same as the
difference between diet means for inoculation 2?
Is the difference between inoculation means for diet 1 the same as the
difference between inoculation means for diet 2?
Cβ = [0, 0, 0, 0, 0, 1, −1, −1, 1]β = γ11 − γ12 − γ21 + γ22 = 0?
Cβ = [0, 0, 0, 0, 0, 1, −1, −1, 1]β = γ11 − γ12 − γ21 + γ22 = 0?
This questions asks if there is interaction between the factors diet and
inoculation.
This questions also asks if there is interaction between the factors diet
and inoculation.
c
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Statistics 511
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c
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Statistics 511
28 / 32
Example 4 (continued)
Example 4 (continued)
β = [μ, α1 , α2 , β1 , β2 , γ11 , γ12 , γ21 , γ22 ]
diet
1
2
Δdiet
inoculation
1
2
μ + α1 + β2 + γ12
μ + α1 + β1 + γ11
μ + α2 + β1 + γ21
μ + α2 + β2 + γ22
α1 − α2 + γ11 − γ21 α1 − α2 + γ12 − γ22
β = [μ, α1 , α2 , β1 , β2 , γ11 , γ12 , γ21 , γ22 ]
Δinoculation
β1 − β2 + γ11 − γ12
β1 − β2 + γ21 − γ22
diet
1
2
Δdiet
Is the average over inoculation means for diet 1 different than the
average over inoculation means for diet 2?
Cβ = [0, 0, 0, 1, −1, .5, −.5, .5, −.5]β = β1 − β2 + γ̄·1 − γ̄·2 = 0?
This question asks about the main effect of the factor diet.
Statistics 511
Δinoculation
β1 − β2 + γ11 − γ12
β1 − β2 + γ21 − γ22
Is the average over diet means for inoculation 1 different than the
average over diet means for inoculation 2?
Cβ = [0, 1, −1, 0, 0, .5, .5, −.5, −.5]β = α1 − α2 + γ̄1· − γ̄2· = 0?
c
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Dan Nettleton (Iowa State University)
inoculation
1
2
μ + α1 + β2 + γ12
μ + α1 + β1 + γ11
μ + α2 + β1 + γ21
μ + α2 + β2 + γ22
α1 − α2 + γ11 − γ21 α1 − α2 + γ12 − γ22
This question asks about the main effect of the factor inoculation.
29 / 32
Example 4 (continued)
c
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Dan Nettleton (Iowa State University)
Statistics 511
30 / 32
Example 4 (continued)
β = [μ, α1 , α2 , β1 , β2 , γ11 , γ12 , γ21 , γ22 ]
β = [μ, α1 , α2 , β1 , β2 , γ11 , γ12 , γ21 , γ22 ]
diet
1
2
Δdiet
inoculation
diet
1
2
Δdiet
1
μ + α1 + β1 + γ11
μ + α2 + β1 + γ21
α1 − α2 + γ11 − γ21
2
μ + α1 + β2 + γ12
μ + α2 + β2 + γ22
α1 − α2 + γ12 − γ22
Δinoculation
β1 − β2 + γ11 − γ12
β1 − β2 + γ21 − γ22
Is there a difference between the diet means for inoculation 1?
Cβ = [0, 1, −1, 0, 0, 1, 0, −1, 0]β = α1 − α2 + γ11 − γ21 = 0?
This question asks about the simple effect of the factor diet for the first
level of the factor inoculation.
c
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Statistics 511
31 / 32
inoculation
1
2
μ + α1 + β2 + γ12
μ + α1 + β1 + γ11
μ + α2 + β1 + γ21
μ + α2 + β2 + γ22
α1 − α2 + γ11 − γ21 α1 − α2 + γ12 − γ22
Are all four treatment means identical?
⎡
0 0 0 1 −1
Cβ = ⎣ 0 0 0 1 −1
0 1 −1 0 0
⎡
β1 − β2 + γ11 − γ12
= ⎣ β1 − β2 + γ21 − γ22
α1 − α2 + γ11 − γ21
c
Copyright 2010
Dan Nettleton (Iowa State University)
Δinoculation
β1 − β2 + γ11 − γ12
β1 − β2 + γ21 − γ22
⎤
1 −1 0
0
0 0
1 −1 ⎦ β
1 0 −1 0
⎤ ⎡ ⎤
0
⎦ = ⎣ 0 ⎦?
0
Statistics 511
32 / 32