Factorial Structures: Two and three way ANOVA
Factorial experiments involve the simultaneous
observation of a response over the levels of 2 or more
factors.
Example1:
A horticulturist wants to study the effects of four different pesticides
on the yield of fruit from three different varieties of a citrus tree. We
are interested in pesticides and the variety.
Two Factors: Pesticide and Variety (BOTH FIXED)
The factor pesticide has four levels and factor variety has three levels
Response variable: fruit yield
Example2:
A store manager wishes to study the effects of height of shelf display
(bottom, middle, top) and width of the shelf display (regular, wide) on
sales of the bakery's bread.
Two Factors: Height and Width
The factor height has three levels and the factor width has two levels.
(Both fixed)
Response variable: bread sales
1
Example 3: (with data) (2 factors both fixed)
The Castle Bakery Company supplies wrapped Italian Bread to a large
number of supermarkets in a metropolitan area. An experimental study
was made of the effects of height of the shelf display (bottom, middle,
top) and the width of the shelf display (regular, wide) on sales of this
bakery’s bread (measured in cases) during the experimental period.
Twelve supermarkets, similar in terms of sales volumes and clientele,
were utilized in the study. Two stores were assigned at random to each
of the factor level combinations and the sales of the bread were
recorded.
Display Width
Display Height
Regular
Wide
Bottom
47, 43
46, 40
Middle
62, 68
67, 71
Top
41, 39
42, 46
Here we have 2 factors both FIXED.
Questions of Interest
1. Are the mean sales different for the three heights?
2. Are the mean sales different for the two widths?
3. Does the effect of display width on sales depend on display height?
In other words, do the two factors interact?
2
What is meant by interaction?
Two Factors A and B are said to interact if the difference in mean
responses for two levels of one factor is not constant across levels of
the second factor.
When the effect of one factor on a response variable y depends on the
level of a second factor, we say that the two factors interact.
In other words: The effect of A PLUS the effect of B is larger/smaller
than the effect of (A+B).
When the combined effect is larger we call it SYNERGISTIC
INTERCATION.
When the combined effect is smaller we call it ANTIGONISTIC
INTERCATION.
Examine the following scenarios. In which of the scenarios does there
appear to be interaction between Factors A and B?
3
B1
A1
A2
x =84
n=10 x =92
n=10
B2
x =74
n=10 x =83
n=10
B1
A1
A2
x =84
n=10 x =74
n=10
B2
x =76
n=10 x =90
n=10
B1
A1
A2
4
x =74
n=10 x =76
n=10
B2
x =75
n=10 x =90
n=10
Plot of Table 1:
Interaction Plot for y
Data Means
92.5
factor 1
a1
a2
90.0
87.5
Mean
85.0
82.5
80.0
77.5
75.0
b1
b2
Factor 2
Interaction Plot for y2
Data Means
factor 1
a1
a2
90.0
87.5
Mean
85.0
82.5
80.0
77.5
75.0
b1
b2
Factor 2
Plot of Table 3:
Interaction Plot for y3
Data Means
factor 1
a1
a2
90.0
87.5
Mean
85.0
82.5
80.0
77.5
75.0
b1
b2
Factor 2
5
Two-Factor Anova Hypotheses
1. H0: There is no interaction between the two factors
Ha: There is interaction between the two factors
2. H0: The population average response is the same for each level of
Factor A
Ha: The population means for the different levels of factor A are
not all equal.
3. H0: The population average response is the same for each level of
Factor B
Ha: The population means for the different levels of factor B are
not all equal.
Assumptions required for F-tests:
The observations on any particular treatment are independently
selected from a normal distribution with variance 2 (the same variance
for each treatment), and samples from different treatments are
independent of one another.
6
The Advantage of Multi-factor Studies
Suppose an experiment is run in which each of the four cell
combinations (A1B1, A1B2, A2B1, A2B2) is replicated r times.
We could perform marginal comparison for the factor A level
means by using an independent two-sample t test:
t

0
ˆ
ˆ


1
.
.
2
.
.
ˆ
ˆ


2
r
2

M
S
E
1 1


M
S
E




2

r2

r


1
.
.
2
.
.
degrees of freedom = 2r + 2r – 2 = 2r + (r + r - 2)
For simple comparisons the t-test has the following form:
ˆ
ˆ

t
1
j
.
0
2
j
.
ˆ
ˆ



r

f
o
r
j
=
1
o
r
2
1
1


M
S
E


 
rr


degrees of freedom = r + r - 2
7
1
j
.
2
j
.
2

M
S
E
If the marginal plot shows parallel lines (e.g., no interaction),
then the test using the marginal means is more powerful then the
test based on the cell means, because of the larger sample sizes.
In addition, the test of marginal means for the 2 by 2 factorial
structure requires only a single test, whereas the simple tests
require two tests. This would mean the experimentwise type I
error is greater for the simple tests.
8
The General Two-way Treatment Structure
Factors B
1
2
.
a.
.
Factor
A
1
2
Y
111,
,Y
11r
Y
121,
Y
211,
,Y
21r
Y
221,
.
Y
a11,
.
.
,Y
12r
,Y
22r
.
,Y
a1r
Y
a21,
.
.
,Y
a2r
...
...
...
...
...
...
Model:
𝑌𝑖𝑗𝑘 = 𝜇𝑖𝑗 + 𝜀𝑖𝑗𝑘
9
i = 1, 2,
,a
j = 1, 2,
,b
(cell means model)
b
Y
1b1,
,Y
1br
Y
2b1,
,Y
2br
...
. . .,Y
Y
ab1,
...
abr
k = 1, 2,
,r
Yijk - response for the ith level of factor A (i = 1, 2, , a), jth level
of factor B (j = 1, 2, , b) and the kth replicate (k = 1, 2, , r) of
the ijth cell.
𝜇𝑖𝑗 - true mean for the ijth cell (or the mean of the treatment
combination of AiBj)
𝜀𝑖𝑗𝑘 - error for the kth observation within the ijth treatment
combination
10
Effect Size Model: (Fixed effects)
𝑌𝑖𝑗𝑘 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 + 𝛼𝛽𝑖𝑗 + 𝜀𝑖𝑗𝑘 (effects model)
i = 1, 2,
, a, j = 1, 2,
, b and k = 1, 2,
,r
Yijk - response for the ith level of factor A (i = 1, 2, , a), jth
level of factor B (j = 1, 2, , b) and the kth replicate (k = 1, 2,
r) of the ijth cell.
𝜇 - grand mean (or  )
...
-𝛼𝑖 treatment effect for the ith level of factor A










Y

Y


i
i
.
.
.
i
.
.
.
.
.
i
.
.
.
.
.
𝛽𝑗 - treatment effect for the jth level of factor B










Y

Y


j
.
j
.
.
.
j
.
.
.
.
.
j
.
.
.
.
𝛼𝛽𝑖𝑗 - interaction effect between the ith level of factor A and the
jth level of factor B








































j
i
.
.
j
i
j
.
.
i
.
.
.
.
j
.
.
i
j
i
.
.
.
j
.
.
.
.
i
j i




Y

Y

Y

Y
j
. i
.
. .
j
. .
.
.
 i

𝜀𝑖𝑗𝑘 - error for the kth observation within the ijth treatment
combination
11
,
Assumptions:
𝜀𝑖𝑗𝑘 ~ 𝑁(0, 𝜎𝜀2 )





2
E
Y




V
Y




i
j
k
i
j
i
j
k
i
j
For the above parameters:
ij. Yij. 
- true mean for the ijth cell

r
ij 
 or



k1
r
i.. Yi.. 
r
Y
k1
r
ijk






- marginal mean for the ith level of factor A
b

j 1
 ij .
b
. j .
- marginal mean for the jth level of factor B
a

j 1
 ij .
a
12
Note: The above formulas are for a balanced design and require
equal sample sizes (r).
Parameter Estimation
The best linear unbiased estimate for the true cell mean    is
ij .
r
Y

k1
ˆij. Y

ij. 
ijk
r
The best linear unbiased estimates of the marginal means
row) and  (jth column) are, respectively,
i..
. j.
b
r
Y


ˆi.. Y

i.. 
j
1k
1
a
r
Y


ijk
ˆ.j. Y

.j. 
i
1 k
1
br

ijk
ar

The best linear unbiased estimate of the true grand mean is
a b r
Y



i
1 j
1k
1
ˆ... Y

... 
ijk
abr
 
Model Parameter Estimates
𝜇 is estimated by ˆ...  Y...
ˆ
ˆ
ˆ
ˆ
ˆ
Y



i
i
.
. Y
.
.
.
i
.
i
.
.
.
.
.
𝛼𝑖 is estimated by 
𝛽𝑗 is estimated by
13
ˆ
ˆ ˆ
ˆ ˆ

YY




j
.
j
.
.
.
.
.
j
.
j
.
.
.
.
(ith
𝛼𝛽𝑖𝑗 is estimated by
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ














ˆ
ˆ

Y

Y





Y

Y

Y

Y


i
j
i
j
i
j
.
i
.
.
.
.
.
j
i
j
.
i
j
i
.
.
i
j
.
i
.
.
.
j
.
.
j
.
.
.
.
Decomposition of the Total Sum of Squares
ˆ
ˆ
ˆ
ˆ
ˆ
Y


b

r


a

r









abr
2
i
j
k
i

111
j k

.
.
.
a
i

1
2
i
.
.

.
.
.
b
2
j

1
.
j
.

ˆ
ˆ
ˆ
ˆ

r


Y

ˆ






i
j
.
i
.
.
.
j
.
.
.
.
i
j
k
i
j
.
ab
2
i

11
j
abr
2
.
.
.
or
i

111
j k

Yy


b

r
y
y

a

r
y
y








i
j
k
.
.
.
i
.
.
.
.
.
.
j
.
.
.
.
abr
i

111
j k

2
a
2
i

1
b
j

1

r

y
y
y

Yy








i
j
.
i
.
.
.
j
. y
.
.
.
i
j
k
i
j
.
ab
i

11
j
2
abr
2
i

111
j k

or SSTotal = SSA + SSB + SSAB + SSError
14
2
Mean Square Errors (Fixed effect Model)
S
S
S
S
S
S
S
S
E
A
B
A
B
M
S

,
M
S

,
M
S

,
M
S

E
A
B
A
B
a
b
r

1

1b

1 
a

1
b

1

 a



2
a
r

b
r


i

j

1
2
2
2
1
E
M
S

, E
M
S

i
, E
M
S


,






E
A
B
a

1
b

1
b
a
2
i
r






i
j
i

1j

1
2
E
M
S



 A

B
a

1
b

1




ab
2
ANOVA Table
Source
SS df
MS
Factor
A
SS A
SS
M
SA  A
a1
Factor B
SS B
a-1
b-1
SS
M
SB  B
b1
EMS
F
a
2 
MS A
MS E
bri2
i1
a1
b
2 
MS B
MS E
ari2
j1
b1
a
b
Interacti
on
SS AB
(a-1)(b-1)
S
S
A
B
M
S
A
B
a

1
b
1
  

r
ij
i1 j1
2

a1b1
Error
SS E
ab(r-1)
S
S
E
M
S
E
a
br
 1
2
Total
SST
abr-1
15
2
MS AB
MS E
Tests:
Main Effects for Factor A:


H
:




v
s
.
H
:
N
o
t
a
l
l

e
q
u
a
l
H
:



0
v
s
.
H
:
N
o
t
a
l
l

0
o
r
0
1
a
a
i
01
.
.
a
.
.
a
i
.
.
M
SA
F

0
M
SError
Reject Ho if Fo > F( , a - 1, ab(r - 1))
Main Effects for Factor B:


H
:




v
s
.
H
:
N
o
t
a
l
l

e
q
u
a
l
H
:



0
v
s
.
H
:
N
o
t
a
l
l

0
o
r
0
1
a
a
i
0.
1
.
.
b
.
a
M
SB
F
0 
M
SError
Reject Ho if Fo > F( , b - 1, ab(r - 1))
16
.
j
.
Interaction Effect:





H
:



0
v
s
.
H
:
N
o
t
a
l
l
=
0
o
r






0
a
1
1
a
b
i
j
H
:
N
o
i
n
t
e
r
a
c
t
i
o
n
v
s
.
H
:
I
n
t
e
r
a
c
t
i
o
n
0
a
𝐹=
𝑀𝑆𝐴𝐵
𝑀𝑆𝐸
Reject Ho if Fo > F( , (a - 1)(b - 1), ab(r - 1))
Approach to Analysis of a Two Factor Design:
Test for an interaction.
i) If an interaction is present, simple tests can be performed to
assess a difference among factor A level means at a fixed level
of factor B. The same procedure can be used to assess the means
of factor B.
ii) If no interaction is determined then the main effects can be
assessed directly. If a significant F-test is found for a factor, then
multiple comparison procedures can be used to separate the
marginal means.
17
Checking Model Adequacy
Residuals:
ˆ
ˆ
e
Y
YY

i
j
k
i
j
k
i
j
k 
i
j
.=
i
j
k
i
j
. 
Based on the residuals we can assess:
1) Normality
i) Wilks-Shapiro or Anderson-Darling Test
ii) Normal Probability Plot
2) Constant Variance
i) Levene's Test
ii) Plot of residuals vs. Factor Levels
iii) Likelihood Ratio Test
18
Example 1:
A rancher is interested in determining if the average daily gain in weight of the
claves depends on the bull which sired the calf. Consider the following situations:
1. Rancher has only 5 bulls and then 5 are randomly mated with randomly
selected cows and the average daily gain of the calves produced are
recorded.
2. Rancher has hundreds of bulls and 5 are selected at random and mated
with randomly selected cows and the average daily weight gain are
recorded.
Example 2:
To study the effect of pasteurization and the breweries used, six randomly
selected breweries were used (among the many breweries owned by the
manufacturer) and different pasteurization processes were used. Ten randomly
selected beers from each brewery-method combination were analyzed for microorganism count.
Example 3:
A study was designed to study the effect of 4 different chemicals in the control of
fire ants. The researcher selected 5 random locations from a large selection of
locations (locations representing the environment) and at each location assigned
the 4 chemicals. The number of fire-ants killed (in thousands) in a one-week
period was recorded.
19
One Way ANOVA
Model for fixed effect
Yij =  + i + ij
Here i = 1,…,k, j = 1,…,n
Here ij follows N(0, 2).
E(MSA) = 2 + ni2
Model for Random Effect :
Yij =  + ai + ij
Here i = 1,…,k, j = 1,…,n
Here ij follows N(0, 2).
And ai follows N(0, a2).
E(MSA)= 2 + na2
20
Two Factor Models:
BOTH factors fixed with interaction:
Yij =  + i + j + ijijk
Here i = 1,…,A, j = 1,…,B, k=1,…,K
Here ij follows N(0, 2).
E(MSA) = 2 + ni2/(A-1)
E(MSA) = 2 + nj2/(B-1)
E(MSA) = 2 + nij2/((A-1)(B-1))
The divisor of the F statistic for testing A, B and AB is MSE.
21
Model for a 2 Factor Model with BOTH Random:
Yij =  + ai + bj + abij + ijk
Here i = 1,…,A, j = 1,…,B, k=1,…,K
Here ij follows N(0, 2).
And ai follows N(0, a2).
And bj follows N(0, b2).
And abij follows N(0, ab2).
E(MSA) = 2 + nBa2+ nab2
E(MSB) = 2 + nAb2+ nab2
E(MSAB) = 2 + nab2
The divisor of the F statistic for testing A and B is MSAB
The divisor for testing AB is MSE.
22
Mixed Model for a 2 Factor Model with one FIXED and the other
Random Effect:
Yij =  + i + bj + bij + ijk
Here i = 1,…,A, j = 1,…,B, k=1,…,K
And bj follows N(0, b2).
And bij follows N(0, ab2).
E(MSA) = 2 + ni2/(A-1)+ nab2
E(MSB) = 2 + nAb2+ nab2
E(MSAB) = 2 + nab2
The divisor of the F statistic for testing A and B is MSAB
The divisor for testing AB is MSE.
23
Examples of Analysis performed in Two-Factor ANOVA
A company was interested in comparing three different display panels
for use by air traffic controllers. Each display panel was to be examined
under five different simulated emergency conditions out of a large
number of possible conditions. Thirty highly trained air traffic
controllers with similar work experience were enlisted for the study. A
random assignment of controllers to display-panel-emergency
conditions was made, with two controllers assigned to each factor-level
combination. The time (in seconds) required to stabilize the emergency
situation was recorded for each controller at a panel-emergency
condition. These data appear below.
Emergency Condition
Display
Panel
1
2
3
4
5
1
18
16
31
35
22
27
39
36
15
12
2
13
15
33
30
24
21
35
38
10
16
3
24
28
42
46
40
37
52
57
28
24
Here we have a 2-Factor Design in a CRD set up with one factor
fixed and one random
24
Proc Mixed Output follows:
Type 3 Analysis of Variance
Source
DF
Sum of
Squares
Mean Expected Mean
Square Square
Error Term
Error
F Pr > F
DF Value
panel
2 1227.800000 613.900000 Var(Residual) + 2
MS(panel*condition)
Var(panel*condition)
+ Q(panel)
8 109.46 <.0001
condition
4 2850.133333 712.533333 Var(Residual) + 2
MS(panel*condition)
Var(panel*condition)
+ 6 Var(condition)
8 127.05 <.0001
panel*condition
8
44.866667
15
106.000000
Residual
5.608333 Var(Residual) + 2
MS(Residual)
Var(panel*condition)
7.066667 Var(Residual)
.
15
0.79 0.6167
.
Differences of Least Squares Means
Effect panel _panel Estimate Standard Error DF t Value Pr > |t| Adjustment
Adj P
panel 1
2
1.6000
1.0591
8
1.51 0.1693 Tukey-Kramer 0.3363
panel 1
3
-12.7000
1.0591
8
-11.99 <.0001 Tukey-Kramer <.0001
panel 2
3
-14.3000
1.0591
8
-13.50 <.0001 Tukey-Kramer <.0001
25
.
.
Interaction Plot for time
Fitted Means
60
panel
1
2
3
50
Mean
40
30
20
10
1
2
3
condition
4
5
Main Effects Plot for time
Fitted Means
panel
45
condition
40
Mean
35
30
25
20
1
26
2
3
1
2
3
4
5
Residual Plots for time
Normal Probability Plot
Versus Fits
99
3.0
Residual
Percent
90
50
10
1.5
0.0
-1.5
-3.0
1
-5.0
-2.5
0.0
Residual
2.5
5.0
10
20
3.0
7.5
1.5
5.0
2.5
0.0
27
50
Versus Order
10.0
Residual
Frequency
Histogram
30
40
Fitted Value
0.0
-1.5
-3.0
-2.4
-1.2
0.0
Residual
1.2
2.4
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
Observation Order
EXAMPLE 2:
An experiment was conducted to examine the effects of different levels of
reinforcement and different levels of isolation on children’s ability to recall. A
single analyst was to work with a random sample of 36 children. Two levels of
reinforcement (none and verbal) and three levels of isolation (20, 40, and 60
minutes) were to be used. Students were randomly assigned to the six treatment
groups, with a total of 6 students being assigned to each group.
Each student was to spend a 30-minute session with the analyst. During this
time, the student was to memorize a specific passage, with reinforcement
provided as dictated by the group to which the student was assigned. Following
the 30-minute session, the student was isolated for the time specified for his or
her group and then tested for recall of the memorized passage. These data
appear in the accompanying table.
Time of Isolation (Minutes)
Level of
reinforcement
None
Verbal
20
40
60
20
19
23
30
36
25
6
10
11
18
28
25
28
27
24
14
17
19
15
16
24
24
26
29
31
38
29
22
25
21
27
23
21
34
35
30
Design: 2 factors (in a CRD set-up) both factors FIXED.
28
ANOVA TABLE:
Source
DF
SS
MS
F
P
C1
2
181.56
90.778
5.80
0.007
C2
1
225.00
225.000
14.38
0.001
Interaction
2
1016.67
508.333
32.49
0.000
Error
30
469.33
15.644
Total
35
1892.56
Interaction Plot - Means for C3
C2
'None'
'Verbal'
'None'
'Verbal'
Mean
32
22
12
'20'
'40'
'60'
C1
Normal Probability Plot for Assessing Normality
Normal Probability Plot of the Residuals
(response is C3)
99
95
90
Percent
80
70
60
50
40
30
20
10
5
1
29
-10
-5
0
Residual
5
10
Plot of Residuals vs. Predicted Values for Assessing Equal Variance Assumption
Residuals Versus the Fitted Values
(response is C3)
8
6
Residual
4
2
0
-2
-4
-6
-8
10
15
20
25
Fitted Value
30
30
35
Three Factor Model Structure:
We have done 2 factor models and should look at one more
extension to the 3 factor model. At this point writing out the
model shouldn’t be too difficult.
Y


i
j
k
l 
ij
k






jk
i
j
i
k




e
i
j
k
i
j
k
l
f
f
e
c
t
s
m
o
d
e
l
e

i
=
1
,
2
,,
a
,
j
=
1
,
2
,,
b
,
k
=
1
,
2
,,
c
a
n
d
l
=
1
,
2
,,
r
31
Yijkl - response for the ith level of factor A (i = 1, 2, , a), jth level of factor B (j = 1,
2, , b), kth level of factor C (k = 1, 2, , c) and the lth replicate (l = 1, 2, , r) of
the ijkth cell.
μ - grand mean ( ... or .... )
αi - treatment effect for the ith level of factor A


o
ri


i
i
.
.
i
.
.
.
.
.
β j - treatment effect for the jth level of factor B


o
r



j
γ k - treatment effect for the kth level of factor C
.
j
.
j
.
j
.
.
.
.



o
r




k
.
.
k
k
.
.
k
.
.
.
αβij - interaction effect between the ith level of factor A and the jth level of factor B







j
i
j
.
i
.
.
..
j
αγ ik - interaction effect between the ith level of factor A and the kth level of factor C







i
k
i
.
k
i
.
.
.
.
k
βγ jk - interaction effect between the jth level of factor B and the kth level of factor C







j
k
.j
k
..
j
.
.
k
 ijk - interaction effect between the ith level of factor A, the jth level of factor B
and the kth level of factor C




















i
j
k i
j
k i
j
.
i
.
k .
j
k i
.
.
.
j
.
.
.
k
eijkl - error for the lth observation within the ijkth treatment combination
32
ANOVA table for a CRD three-factor factorial experiment
Source
SS
df
MS
Main Effects
F
MSA/MSE
A
SSA
a-1
MSA=SSA/(a-1)
MSB/MSE
B
SSB
b-1
MSB=SSB/(b-1)
MSC/MSE
C
SSC
c-1
MSC=SSC/(c-1)
AB
SSAB
(a-1)(b-1)
MSAB=SSAB/(a1)(b-1)
MSAB/MSE
AC
SSAC
(a-1)(c-1)
MSAC=SSAC/(a1)(c-1)
MSAC/MSE
BC
SSBC
(b-1)(c-1)
MSBC=SSBC/(b1)(c-1)
MSBC/MSE
ABC
SSABC
(a-1)(b-1)(c1)
MSABC=SSABC/(a MSABC/MSE
-1)(b-1)(c-1)
Error
SSE
abc(r-1)
MSE=SSE/(abc(r1)
Total
SST (TSS)
abcr-1
Interactions
33
Example
A marketing research consultant evaluated the effects of fee schedule (high,
average and low), scope of work (all performed in house and some
subcontracted out), and type of supervisory control (local supervisors and
traveling supervisors only) on the quality of work performed (numerical index)
under contract by independent marketing research agencies. The factor levels in
the study were as follows:
---------------------------------------------------------------Factor
Factor Levels
---------------------------------------------------------------Fee level
i = 1: High
i = 2: Average
i = 3: Low
Scope
j = 1: All performed in house
j = 2: Some work subcontracted out
Supervisor
k = 1: Local supervisors
k = 2: Traveling supervisors only
----------------------------------------------------------------
34
Data from the study:
Supervision
Fee level
1
2
3
35
1
2
scope
scope
1
2
1
2
124.3
115.1
112.7
88.2
120.6
119.9
110.2
96.0
120.7
115.4
113.5
96.4
122.6
117.3
108.6
90.1
119.3
117.2
113.6
92.7
118.9
114.4
109.1
91.1
125.3
113.4
108.9
90.7
121.4
120.0
112.3
87.9
90.9
89.9
78.6
58.6
95.3
83.0
80.6
63.5
88.8
86.5
83.5
59.8
92.0
82.7
77.1
62.3
The GLM Procedure
Dependent Variable: quality
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
11
16291.75562
1481.06869
200.34
<.0001
Error
36
266.13750
7.39271
Corrected Total
47
16557.89313
R-Square
Coeff Var
Root MSE
quality Mean
0.983927
2.718444
2.718954
100.0188
Source
fee
scope
fee*scope
supervis
fee*supervis
scope*supervis
fee*scope*supervis
DF
Type III SS
Mean Square
F Value
Pr > F
2
1
2
1
2
1
2
10044.27125
1833.97687
1.60125
3832.40021
0.78792
574.77521
3.94292
5022.13562
1833.97687
0.80062
3832.40021
0.39396
574.77521
1.97146
679.34
248.08
0.11
518.40
0.05
77.75
0.27
<.0001
<.0001
0.8977
<.0001
0.9482
<.0001
0.7674
Although each main effect shows a significant difference among the treatment means (fee with Pvalue < 0.0001, scope with P-value < 0.0001 and supervisor with P-value < 0.0001), there is a
significant two-way interaction between scope and supervisor (P-value < 0.0001). Therefore, one
should at least look at the margin plot of the scope by supervisor means, if not the marginal plot
based on the set of 12 cell means (e.g., fee by scope by supervisor cell means).
Means with the same letter are not significantly different.
36
t Grouping
Mean
N
fee
A
110.7250
16
High
A
109.7625
16
Average
B
79.5688
16
Low
A
Since there was no interaction between fee and either scope or supervisor, the main effects for fee
can be interpreted. Therefore, the above LSDs for the levels of fee are valid and indicate that Low
fee produces a significantly lower quality than either the High fee or the Average fee. Further on in
the analysis this result will again be shown through comparison of the simple effects for the fee by
scope by supervisor means.
37
Means with the same letter are not significantly different.
t Grouping
Mean
N
scope
A
106.2000
24
In_house
B
93.8375
24
Subcontr
Because the interaction between scope and supervisor was significant, these above LSD comparison
between the In-House and Subcontractor levels of scope may not be valid. It is best to check the
simple tests and the marginal plot of the means.
Means with the same letter are not significantly different.
t Grouping
Mean
N
supervis
A
108.9542
24
Local
B
91.0833
24
Travel
Because the interaction between scope and supervisor was significant, these above LSD comparison
between the Local and Travel levels of supervisor may not be valid. It is best to check the simple
tests and the marginal plot of the means.
Least Squares Means
quality
38
Standard
LSMEAN
scope
supervis
LSMEAN
Error
Pr > |t|
Number
In_house
Local
111.675000
0.784894
<.0001
1
In_house
Travel
100.725000
0.784894
<.0001
2
Subcontr
Local
106.233333
0.784894
<.0001
3
Subcontr
Travel
81.441667
0.784894
<.0001
4
Least Squares Means for effect scope*supervis
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: quality
i/j
1
1
2
3
4
<.0001
<.0001
<.0001
<.0001
<.0001
2
<.0001
3
<.0001
<.0001
4
<.0001
<.0001
<.0001
<.0001
NOTE: To ensure overall protection level, only probabilities associated with pre-planned
comparisons should be used.
Note, the comparison of the scope by supervisor simple means indicates that each pair is
statistically different. However, since the interaction between scope and supervisor was significant,
one should be careful and note whether the factor level means for one factor differs in the same
pattern across the levels of the second factor. For example, the comparison between the Local and
Travel levels of supervisor indicates that the Local level has a larger mean than the Travel level,
and this difference is consistent across the levels of scope.
39
fee
scope
supervis
LSMEAN
Error
Pr > |t|
Number
Average
In_house
Local
121.225000
1.359477
<.0001
1
Average
In_house
Travel
110.975000
1.359477
<.0001
2
Average
Subcontr
Local
116.250000
1.359477
<.0001
3
Average
Subcontr
Travel
90.600000
1.359477
<.0001
4
High
In_house
Local
122.050000
1.359477
<.0001
5
High
In_house
Travel
111.250000
1.359477
<.0001
6
High
Subcontr
Local
116.925000
1.359477
<.0001
7
High
Subcontr
Travel
92.675000
1.359477
<.0001
8
Low
In_house
Local
91.750000
1.359477
<.0001
9
Low
In_house
Travel
79.950000
1.359477
<.0001
10
Low
Subcontr
Local
85.525000
1.359477
<.0001
11
Low
Subcontr
Travel
61.050000
1.359477
<.0001
12
Dependent Variable: quality
i/j
1
1
2
3
4
5
6
<.0001
0.0138
<.0001
0.6704
<.0001
0.0094
<.0001
<.0001
0.8871
<.0001
0.0047
0.0134
<.0001
<.0001
2
<.0001
3
0.0138
0.0094
4
<.0001
<.0001
<.0001
5
0.6704
<.0001
0.0047
<.0001
6
<.0001
0.8871
0.0134
<.0001
<.0001
7
0.0316
0.0038
0.7276
<.0001
0.0114
0.0055
8
<.0001
<.0001
<.0001
0.2876
<.0001
<.0001
9
<.0001
<.0001
<.0001
0.5535
<.0001
<.0001
10
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
11
<.0001
<.0001
<.0001
0.0122
<.0001
<.0001
12
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Least Squares Means for effect fee*scope*supervis
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: quality
i/j
40
7
8
9
10
11
12
1
0.0316
<.0001
<.0001
<.0001
<.0001
<.0001
2
0.0038
<.0001
<.0001
<.0001
<.0001
<.0001
3
0.7276
<.0001
<.0001
<.0001
<.0001
<.0001
4
<.0001
0.2876
0.5535
<.0001
0.0122
<.0001
5
0.0114
<.0001
<.0001
<.0001
<.0001
<.0001
6
0.0055
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.6333
<.0001
0.0007
<.0001
<.0001
0.0026
<.0001
0.0063
<.0001
7
8
<.0001
9
<.0001
0.6333
10
<.0001
<.0001
<.0001
11
<.0001
0.0007
0.0026
0.0063
12
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
NOTE: To ensure overall protection level, only probabilities associated with pre-planned
comparisons should be used.
Blocking Factor: A Factor that contributes to the variability of our response but
we are not inherently interested in it. We generally use RANDOM effects for
blocks as we are not interested in the levels of the blocks.
So now that we know the Analysis of 1 way, 2 way and 3 way ANOVAs we are
ready to look at analyzing the different designs.
41
We still need to look at Random and Mixed in a 3 Factor set-up and Nested
Designs. We will do that after CRD and RCBD.
We will look at CRD, RCBD, Latin Square, Repeated Measures, Split Plot in our
next set of notes.
42