Factorial Experiments
Analysis of Variance
Experimental Design
• Dependent variable Y
• k Categorical independent variables A, B,
C, … (the Factors)
• Let
–
–
–
–
a = the number of categories of A
b = the number of categories of B
c = the number of categories of C
etc.
The Completely Randomized Design
• We form the set of all treatment combinations
– the set of all combinations of the k factors
• Total number of treatment combinations
– t = abc….
• In the completely randomized design n
experimental units (test animals , test plots,
etc. are randomly assigned to each treatment
combination.
– Total number of experimental units N = nt=nabc..
The treatment combinations can thought to be
arranged in a k-dimensional rectangular block
B
1
1
2
A
a
2
b
C
B
A
Another way of representing the treatment combinations
in a factorial experiment
C
B
...
A
...
D
Example
In this example we are examining the effect of
The level of protein A (High or Low) and
The source of protein B (Beef, Cereal, or Pork)
on weight gains Y (grams) in rats.
We have n = 10 test animals randomly
assigned to k = 6 diets
The k = 6 diets are the 6 = 3×2 Level-Source
combinations
1. High - Beef
2. High - Cereal
3. High - Pork
4. Low - Beef
5. Low - Cereal
6. Low - Pork
Table
Gains in weight (grams) for rats under six diets
differing in level of protein (High or Low) and s
ource of protein (Beef, Cereal, or Pork)
Level
of Protein High Protein
Low protein
Source
of Protein Beef Cereal Pork Beef Cereal Pork
Diet
1
2
3
4
5
6
73
98
94
90 107
49
102
74
79
76
95
82
118
56
96
90
97
73
104 111
98
64
80
86
81
95 102
86
98
81
107
88 102
51
74
97
100
82 108
72
74 106
87
77
91
90
67
70
117
86 120
95
89
61
111
92 105
78
58
82
Mean
100.0 85.9 99.5 79.2 83.9 78.7
Std. Dev. 15.14 15.02 10.92 13.89 15.71 16.55
Example – Four factor experiment
Four factors are studied for their effect on Y (luster
of paint film). The four factors are:
1) Film Thickness - (1 or 2 mils)
2) Drying conditions (Regular or Special)
3) Length of wash (10,30,40 or 60 Minutes), and
4) Temperature of wash (92 ˚C or 100 ˚C)
Two observations of film luster (Y) are taken
for each treatment combination
The data is tabulated below:
Regular
Dry
Minutes 92 C
1-mil Thickness
20
3.4 3.4
30
4.1 4.1
40
4.9 4.2
60
5.0 4.9
2-mil Thickness
20
5.5 3.7
30
5.7 6.1
40
5.5 5.6
60
7.2 6.0
100 C
92C
Special Dry
100 C
19.6
17.5
17.6
20.9
14.5
17.0
15.2
17.1
2.1
4.0
5.1
8.3
3.8
4.6
3.3
4.3
17.2
13.5
16.0
17.5
13.4
14.3
17.8
13.9
26.6
31.6
30.5
29.5
30.2
30.2
4.5 4.5
5.9 5.9
5.5 5.8
25.6
29.2
32.6
22.5
29.8
27.4
31.4
29.6
8.0 9.9
33.5
29.5
Notation
Let the single observations be denoted by a
single letter and a number of subscripts
yijk…..l
The number of subscripts is equal to:
(the number of factors) + 1
1st subscript = level of first factor
2nd subscript = level of 2nd factor
…
Last subsrcript denotes different observations
on the same treatment combination
Notation for Means
When averaging over one or several
subscripts we put a “bar” above the letter and
replace the subscripts by •
Example:
y241 • •
Profile of a Factor
Plot of observations means vs. levels of the
factor.
The levels of the other factors may be held
constant or we may average over the other
levels
Definition:
A factor is said to not affect the response if
the profile of the factor is horizontal for all
combinations of levels of the other factors:
No change in the response when you change
the levels of the factor (true for all
combinations of levels of the other factors)
Otherwise the factor is said to affect the
response:
Definition:
• Two (or more) factors are said to interact if
changes in the response when you change
the level of one factor depend on the
level(s) of the other factor(s).
• Profiles of the factor for different levels of
the other factor(s) are not parallel
• Otherwise the factors are said to be
additive .
• Profiles of the factor for different levels of
the other factor(s) are parallel.
• If two (or more) factors interact each factor
effects the response.
• If two (or more) factors are additive it still
remains to be determined if the factors
affect the response
• In factorial experiments we are interested in
determining
– which factors effect the response and
– which groups of factors interact .
Factor A has no effect
70
60
50
40
B
30
20
10
0
0
20
A
40
60
Additive Factors
70
60
50
40
30
20
B
10
0
0
20
A
40
60
Interacting Factors
70
60
50
40
30
B
20
10
0
0
20
A
40
60
The testing in factorial experiments
1. Test first the higher order interactions.
2. If an interaction is present there is no need
to test lower order interactions or main
effects involving those factors. All factors
in the interaction affect the response and
they interact
3. The testing continues with for lower order
interactions and main effects for factors
which have not yet been determined to
affect the response.
Example: Diet Example
Summary Table of Cell means
Source of Protein
Level of Protein Beef
High
100.00
Low
79.20
Overall
89.60
Cereal
85.90
83.90
84.90
Pork Overall
99.50 95.13
78.70 80.60
89.10 87.87
Profiles of Weight Gain for
Source and Level of Protein
110
High Protein
Low Protein
Overall
Weight Gain
100
90
80
70
Beef
Cereal
Pork
Profiles of Weight Gain for
Source and Level of Protein
110
Beef
Cereal
Pork
Weight Gain
100
Overall
90
80
70
High Protein
Low Protein
Models for factorial Experiments
Single Factor: A – a levels
Random error – Normal, mean 0, std-dev. s
yij = m + ai + eij
i = 1,2, ... ,a; j = 1,2, ... ,n
mi
mi  themean of y when A  i
 m  ai
Overall mean
Effect on y of factor A when A = i
a
a
i 1
i
0
1
observations
Levels of A
2
3
a
y11
y12
y13
y21
y22
y23
y31
y32
y33
ya1
ya2
ya3
y1n
y2n
y3n
yan
m1
m2
Normal dist’n
Mean of
observations
Definitions
m + a1
m + a2
m3
m + a3
1 a
m  overallmean  m   mi
a i 1
ma
m + aa
1 a
a i  (Effect when A  i)  mi  m  mi   mi
a i 1
Two Factor: A (a levels), B (b levels
yijk = m + ai + bj+ (ab)ij + eijk
mij
i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,n
mij  themean of y when A  i and B  j
 m  a i  b j  ab ij
Overall mean
Main effect of A
a
a
i 1
i
Main effect of B
Interaction effect
of A and B
b
a
b
j 1
i 1
j 1
 0,  b j  0,  ab ij  0,  ab ij  0
Table of Means
Table of Effects – Overall mean, Main effects, Interaction
Effects
Three Factor: A (a levels), B (b levels), C (c levels)
yijkl = m + ai + bj+ abij + gk + (ag)ik + (bg)jk+
abgijk + eijkl
= m + ai + bj+ gk + abij + (agik + (bgjk
+ abgijk + eijkl
Main effects
Two factor
Three factor Interaction
Interactions
Random error
i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,n
a
a
i 1
i
b
c
a
c
j 1
k 1
i 1
k 1
 0,  b j  0,  g k  0,  ab ij  0,,  abg ijk  0
mijk = the mean of y when A = i, B = j, C = k
= m + ai + bj+ gk + abij + (agik + (bgjk
+ abgijk
Two factor
Overall mean
Main effects
Three factor Interaction
Interactions
i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,n
a
a
i 1
i
b
c
a
c
j 1
k 1
i 1
k 1
 0,  b j  0,  g k  0,  ab ij  0,,  abg ijk  0
No interaction
Levels of C
Levels
of B
Levels
of B
Levels of A
Levels of A
A, B interact, No interaction with C
Levels of C
Levels
of B
Levels
of B
Levels of A
Levels of A
A, B, C interact
Levels of C
Levels
of B
Levels
of B
Levels of A
Levels of A
Four Factor:
yijklm = m + ai + bj+ (ab)ij + gk + (ag)ik + (bg)jk+
(abg)ijk + dl+ (ad)il + (bd)jl+ (abd)ijl + (gd)kl +
(agd)ikl + (bgd)jkl+ (abgd)ijkl + eijklm
Overall mean
=m
Two factor
Main
effects
+ai + bj+ gk + dl
Interactions
+ (ab)ij + (ag)ik + (bg)jk + (ad)il + (bd)jl+ (gd)kl
+(abg)ijk+ (abd)ijl + (agd)ikl + (bgd)jkl Three factor
Interactions
+ (abgd)ijkl + eijklm
Four factor Interaction
Random error
i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,d; m = 1,2, ... ,n
where 0 = S ai = S bj= S (ab)ij  S gk = S (ag)ik = S(bg)jk= S (abg)ijk = S dl= S
(ad)il = S (bd)jl = S (abd)ijl = S (gd)kl = S (agd)ikl = S (bgd)jkl =
S (abgd)ijkl
and S denotes the summation over any of the subscripts.
Estimation of Main Effects and Interactions
• Estimator of Main effect of a Factor
= Mean at level i of the factor - Overall Mean
• Estimator of k-factor interaction effect at a
combination of levels of the k factors
= Mean at the combination of levels of the k factors
- sum of all means at k-1 combinations of levels
of the k factors +sum of all means at k-2
combinations of levels of the k factors - etc.
Example:
• The main effect of factor B at level j in a four
factor (A,B,C and D) experiment is estimated by:
bˆ j  y j  y
• The two-factor interaction effect between factors B
and C when B is at level j and C is at level k is
estimated by:
bg  jk  y jk  y j  yk  y
• The three-factor interaction effect between factors
B, C and D when B is at level j, C is at level k and
D is at level l is estimated by:
bgd jkl  y jkl  y jk  y jl  ykl  y j  yk  yl  y
• Finally the four-factor interaction effect between
factors A,B, C and when A is at level i, B is at level
j, C is at level k and D is at level l is estimated by:
abgdijkl  yijkl  yijk  yijl  yikl  y jkl  yij  yik  yil  y jk
 y jl  ykl  yi  y j  yk   yl   y
Anova Table entries
• Sum of squares interaction (or main) effects
being tested = (product of sample size and
levels of factors not included in the
interaction) × (Sum of squares of effects being
tested)
• Degrees of freedom = df = product of (number
of levels - 1) of factors included in the
interaction.
Analysis of Variance (ANOVA) Table
Entries (Two factors – A and B)
a
SSA  nb aˆ i2
i 1
b
SSB  na bˆ j2
j 1
a
b
SSAB  n ab ij
2
i 1 j 1
SSError   yijk  yij 
a
b
n
i 1 j 1 k 1
2
The ANOVA Table
Analysis of Variance (ANOVA) Table
Entries (Three factors – A, B and C)
a
SSA  nbc aˆ
i 1
a
b
SSB  nac bˆ
2
i
j 1
SSAB  nc ab 
i 1 j 1
c
SSC  nab gˆk2
b
a
2
j
c
k 1
b
SSAC  nb ag 
2
ij
i 1 k 1
a
b
c
j 1 k 1
SSABC  n abg ijk
2
i 1 j 1 k 1
SSError   yijkl  yijk 
a
b
c
n
i 1 j 1 k 1 l 1
c
SSBC  na bg  jk
2
ik
2
2
The ANOVA Table
Source
SS
df
A
B
C
AB
AC
BC
SS A
SS B
SS C
SS AB
SS AC
SSBC
SSABC
SS Error
a-1
b-1
c-1
(a-1)(b-1)
(a-1)(c-1)
(b-1)(c-1)
ABC
Error
(a-1)(b-1)(c-1)
abc(n-1)
• The Completely Randomized Design is called
balanced
• If the number of observations per treatment
combination is unequal the design is called
unbalanced. (resulting mathematically more
complex analysis and computations)
• If for some of the treatment combinations
there are no observations the design is called
incomplete. (some of the parameters - main
effects and interactions - cannot be estimated.)
Example: Diet example
Mean
mˆ  y
= 87.867
Main Effects for Factor A (Source of
Protein)
aˆ i  yi  y
Beef
1.733
Cereal
-2.967
Pork
1.233
Main Effects for Factor B (Level of Protein)
bˆ j  y j  y
High
7.267
Low
-7.267
AB Interaction Effects
ab ij  yij-yi-y j  y
Source of Protein
Beef Cereal Pork
Level
High 3.133 -6.267 3.133
of Protein Low -3.133 6.267 -3.133
Example 2
Paint Luster Experiment
Table: Means and Cell Frequencies
Means and Frequencies for the AB
Interaction (Temp - Drying)
Profiles showing Temp-Dry
Interaction
25
Regular Dry
20
Special Dry
Luster
Overall
15
10
5
0
92
100
Te mpe rature
Means and Frequencies for the AD
Interaction (Temp- Thickness)
Profiles showing Temp-Thickness
Interaction
30
1-mil
25
2-mil
Overall
Luster
20
15
10
5
0
92
100
Te mpe rature
The Main Effect of C (Length)
Profile of Effect of Length on Luster
16
Luster
15
14
13
12
10
20
30
40
Le ngth
50
60
70
Factorial Experiments
Analysis of Variance
Experimental Design
• Dependent variable Y
• k Categorical independent variables A, B,
C, … (the Factors)
• Let
–
–
–
–
a = the number of categories of A
b = the number of categories of B
c = the number of categories of C
etc.
Objectives
• Determine which factors have some effect on
the response
• Which groups of factors interact
The Completely Randomized Design
• We form the set of all treatment combinations
– the set of all combinations of the k factors
• Total number of treatment combinations
– t = abc….
• In the completely randomized design n
experimental units (test animals , test plots,
etc. are randomly assigned to each treatment
combination.
– Total number of experimental units N = nt=nabc..
Factor A has no effect
70
60
50
40
B
30
20
10
0
0
20
A
40
60
Additive Factors
70
60
50
40
30
20
B
10
0
0
20
A
40
60
Interacting Factors
70
60
50
40
30
B
20
10
0
0
20
A
40
60
The testing in factorial experiments
1. Test first the higher order interactions.
2. If an interaction is present there is no need
to test lower order interactions or main
effects involving those factors. All factors
in the interaction affect the response and
they interact
3. The testing continues with for lower order
interactions and main effects for factors
which have not yet been determined to
affect the response.
Anova table for the 3 factor Experiment
Source
SS
df
MS
F
A
SSA
a-1
MSA
MSA/MSError
B
SSB
b-1
MSB
MSB/MSError
C
SSC
c-1
MSC
MSC/MSError
AB
SSAB
(a - 1)(b - 1)
MSAB
MSAB/MSError
AC
SSAC
(a - 1)(c - 1)
MSAC
MSAC/MSError
BC
SSBC
(b - 1)(c - 1)
MSBC
MSBC/MSError
ABC
SSABC
(a - 1)(b - 1)(c - 1)
MSABC
MSABC/MSError
Error
SSError
abc(n - 1)
MSError
p -value
Sum of squares entries
a
a
SSA  nbcaˆ  nbc  yi  y 
i 1
2
i
2
i 1
Similar expressions for SSB , and SSC.
SSAB  ncab  nc yij  yi  y j   y 
a
i 1
a
2
ij
b
i 1 j 1
Similar expressions for SSBC , and SSAC.
2
Sum of squares entries
a
2
SSABC  nabgikj
i 1
a
b
 n yijk  yij  yik   y jk  yi
c
 y j   yk   yi 
i 1 j 1 k 1
2
Finally
SSError   yijkl  yijk 
a
b
c
n
i 1 j 1 k 1 l 1
2
The statistical model for the 3 factor Experiment
yijk/ 
m
 ai  b j  g k
mean effect
main effects
 ab ij  ag ik  bg  jk 
abg ijk
2 factor interactions
3 factor interaction
 e ijk/
random error
Anova table for the 3 factor Experiment
Source
SS
df
MS
F
A
SSA
a-1
MSA
MSA/MSError
B
SSB
b-1
MSB
MSB/MSError
C
SSC
c-1
MSC
MSC/MSError
AB
SSAB
(a - 1)(b - 1)
MSAB
MSAB/MSError
AC
SSAC
(a - 1)(c - 1)
MSAC
MSAC/MSError
BC
SSBC
(b - 1)(c - 1)
MSBC
MSBC/MSError
ABC
SSABC
(a - 1)(b - 1)(c - 1)
MSABC
MSABC/MSError
Error
SSError
abc(n - 1)
MSError
p -value
The testing in factorial experiments
1. Test first the higher order interactions.
2. If an interaction is present there is no need
to test lower order interactions or main
effects involving those factors. All factors
in the interaction affect the response and
they interact
3. The testing continues with lower order
interactions and main effects for factors
which have not yet been determined to
affect the response.
Examples
Using SPSS
Example
In this example we are examining the effect of
• the level of protein A (High or Low) and
• the source of protein B (Beef, Cereal, or
Pork) on weight gains (grams) in rats.
We have n = 10 test animals randomly
assigned to k = 6 diets
The k = 6 diets are the 6 = 3×2 Level-Source
combinations
1. High - Beef
2. High - Cereal
3. High - Pork
4. Low - Beef
5. Low - Cereal
6. Low - Pork
Table
Gains in weight (grams) for rats under six diets
differing in level of protein (High or Low) and s
ource of protein (Beef, Cereal, or Pork)
Level
of Protein High Protein
Low protein
Source
of Protein Beef Cereal Pork Beef Cereal Pork
Diet
1
2
3
4
5
6
73
98
94
90 107
49
102
74
79
76
95
82
118
56
96
90
97
73
104 111
98
64
80
86
81
95 102
86
98
81
107
88 102
51
74
97
100
82 108
72
74 106
87
77
91
90
67
70
117
86 120
95
89
61
111
92 105
78
58
82
Mean
100.0 85.9 99.5 79.2 83.9 78.7
Std. Dev. 15.14 15.02 10.92 13.89 15.71 16.55
The data as it appears in SPSS
To perform ANOVA select Analyze->General Linear
Model-> Univariate
The following dialog box appears
Select the dependent variable and the fixed factors
Press OK to perform the Analysis
The Output
Tests of Between-Subjects Effects
Dependent Variable: WTGN
Type III
Sum of
Source
Squares
Corrected Model
4612.933 a
Intercept
463233.1
SOURCE
266.533
LEVEL
3168.267
SOURCE * LEVEL 1178.133
Error
11586.000
Total
479432.0
Corrected Total
16198.933
df
5
1
2
1
2
54
60
59
Mean
Square
922.587
463233.1
133.267
3168.267
589.067
214.556
a. R Squared = .285 (Adjusted R Squared = .219)
F
4.300
2159.036
.621
14.767
2.746
Sig.
.002
.000
.541
.000
.073
Example – Four factor experiment
Four factors are studied for their effect on Y (luster
of paint film). The four factors are:
1) Film Thickness - (1 or 2 mils)
2) Drying conditions (Regular or Special)
3) Length of wash (10,30,40 or 60 Minutes), and
4) Temperature of wash (92 ˚C or 100 ˚C)
Two observations of film luster (Y) are taken
for each treatment combination
The data is tabulated below:
Regular
Dry
Minutes 92 C
1-mil Thickness
20
3.4 3.4
30
4.1 4.1
40
4.9 4.2
60
5.0 4.9
2-mil Thickness
20
5.5 3.7
30
5.7 6.1
40
5.5 5.6
60
7.2 6.0
100 C
92C
Special Dry
100 C
19.6
17.5
17.6
20.9
14.5
17.0
15.2
17.1
2.1
4.0
5.1
8.3
3.8
4.6
3.3
4.3
17.2
13.5
16.0
17.5
13.4
14.3
17.8
13.9
26.6
31.6
30.5
31.4
29.5
30.2
30.2
29.6
4.5
5.9
5.5
8.0
4.5
5.9
5.8
9.9
25.6
29.2
32.6
33.5
22.5
29.8
27.4
29.5
The Data as it appears in SPSS
The dialog box for performing ANOVA
The output
Tests of Between-Subjects Effects
Dependent Variable: LUSTRE
Type III
Sum of
Source
Squares
Corrected Model
6548.020 a
Intercept
12586.035
TEMP
5039.225
COND
5.700
LENGTH
70.285
THICK
844.629
TEMP * COND
15.504
TEMP * LENGTH
3.155
COND * LENGTH
9.890
TEMP * COND * LENGTH
6.422
TEMP * THICK
511.325
COND * THICK
1.410
TEMP * COND * THICK
.150
LENGTH * THICK
15.642
TEMP * LENGTH * THICK
11.520
COND * LENGTH *
7.320
THICK
TEMP * COND * LENGTH
5.840
* THICK
Error
87.995
Total
19222.050
Corrected Total
6636.015
31
1
1
1
3
1
1
3
3
3
1
1
1
3
3
Mean
Square
211.226
12586.035
5039.225
5.700
23.428
844.629
15.504
1.052
3.297
2.141
511.325
1.410
.150
5.214
3.840
F
76.814
4577.000
1832.550
2.073
8.520
307.155
5.638
.383
1.199
.778
185.947
.513
.055
1.896
1.396
Sig.
.000
.000
.000
.160
.000
.000
.024
.766
.326
.515
.000
.479
.817
.150
.262
3
2.440
.887
.458
3
1.947
.708
.554
32
64
63
2.750
df
a. R Squared = .987 (Adjusted R Squared = .974)
Random Effects and Fixed
Effects Factors
• So far the factors that we have considered are
fixed effects factors
• This is the case if the levels of the factor are a
fixed set of levels and the conclusions of any
analysis is in relationship to these levels.
• If the levels have been selected at random from
a population of levels the factor is called a
random effects factor
• The conclusions of the analysis will be
directed at the population of levels and not
only the levels selected for the experiment
Example - Fixed Effects
Source of Protein, Level of Protein, Weight Gain
Dependent
– Weight Gain
Independent
– Source of Protein,
• Beef
• Cereal
• Pork
– Level of Protein,
• High
• Low
Example - Random Effects
In this Example a Taxi company is interested in
comparing the effects of three brands of tires (A, B and
C) on mileage (mpg). Mileage will also be effected by
driver. The company selects b = 4 drivers at random
from its collection of drivers. Each driver has n = 3
opportunities to use each brand of tire in which mileage
is measured.
Dependent
– Mileage
Independent
– Tire brand (A, B, C),
• Fixed Effect Factor
– Driver (1, 2, 3, 4),
• Random Effects factor
The Model for the fixed effects experiment
yijk  m  ai  b j  ab ij  e ijk
where m, a1, a2, a3, b1, b2, (ab)11 , (ab)21 , (ab)31 ,
(ab)12 , (ab)22 , (ab)32 , are fixed unknown constants
And eijk is random, normally distributed with mean 0
and variance s2.
Note:
a
n
a
b
a   b   ab    ab 
i 1
i
j 1
j
i 1
ij
j 1
ij
0
The Model for the case when factor B is a random
effects factor
yijk  m  ai  b j  ab ij  e ijk
where m, a1, a2, a3, are fixed unknown constants
And eijk is random, normally distributed with mean 0 and
variance s2.
bj is normal with mean 0 and variance s B2
and
2
(ab)ij is normal with mean 0 and variance s AB
a
Note:
a
i 1
i
0
This model is called a variance components model
The Anova table for the two factor model
yijk  m  ai  b j  ab ij  e ijk
Source
SS
df
a -1
A
SSA
b-1
B
SSA
AB
SSAB (a -1)(b -1)
Error SSError ab(n – 1)
MS
SSA/(a – 1)
SSB/(a – 1)
SSAB/(a – 1) (a – 1)
SSError/ab(n – 1)
The Anova table for the two factor model (A, B – fixed)
yijk  m  ai  b j  ab ij  e ijk
Source
SS
df
MS
EMS
F
nb a 2
s 
ai
a  1 
i 1
MSA/MSError
A
SSA
a -1
MSA
B
SSA
b-1
MSB
AB
SSAB
(a -1)(b -1)
MSAB
Error
SSError
ab(n – 1)
MSError
2
s2 
na b 2
bj
b  1 
j 1
a
b
n
ab ij2
s 

a  1b  1 i 1 j 1
2
EMS = Expected Mean Square
s2
MSB/MSError
MSAB/MSError
The Anova table for the two factor model
(A – fixed, B - random)
yijk  m  ai  b j  ab ij  e ijk
Source
SS
df
MS
A
SSA
a -1
MSA
B
SSA
b-1
MSB
AB
SSAB
(a -1)(b -1)
MSAB
Error
SSError
ab(n – 1)
MSError
EMS
s  ns
2
2
AB
nb a 2

ai
a  1 
i 1
s 2  nas B2
2
s 2  ns AB
s2
Note: The divisor for testing the main effects
of A is no longer MSError but MSAB.
F
MSA/MSAB
MSB/MSError
MSAB/MSError
Rules for determining Expected
Mean Squares (EMS) in an Anova
Table
Both fixed and random effects
Formulated by Schultz[1]
1.
Schultz E. F., Jr. “Rules of Thumb for Determining
Expectations of Mean Squares in Analysis of
Variance,”Biometrics, Vol 11, 1955, 123-48.
1. The EMS for Error is s2.
2. The EMS for each ANOVA term contains
two or more terms the first of which is s2.
3. All other terms in each EMS contain both
coefficients and subscripts (the total number
of letters being one more than the number of
factors) (if number of factors is k = 3, then
the number of letters is 4)
4. The subscript of s2 in the last term of each
EMS is the same as the treatment
designation.
5. The subscripts of all s2 other than the first contain
the treatment designation. These are written with
the combination involving the most letters written
first and ending with the treatment designation.
6. When a capital letter is omitted from a subscript ,
the corresponding small letter appears in the
coefficient.
7. For each EMS in the table ignore the letter or letters
that designate the effect. If any of the remaining
letters designate a fixed effect, delete that term from
the EMS.
8. Replace s2 whose subscripts are composed
entirely of fixed effects by the appropriate sum.
a
s
2
A
a
by
i 1
a 1
a
s
2
AB
2
i
 ab 
by
i 1
2
ij
 a  1 b  1
Example: 3 factors A, B, C – all are random effects
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
F
2
2
2
s 2  ns ABC
 ncs AB
 nbs AC
 nbcs A2
2
2
2
s 2  ns ABC
 ncs AB
 nas BC
 nacs B2
2
2
2
s 2  ns ABC
 nas BC
 nbs AC
 nabs C2
2
2
s 2  ns ABC
 ncs AB
MS AB MS ABC
2
2
s 2  ns ABC
 nbs AC
MS AC MS ABC
2
2
s 2  ns ABC
 nas BC
MSBC MS ABC
2
s 2  ns ABC
s2
MS ABC MSError
Example: 3 factors A fixed, B, C random
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
s  ns
2
2
ABC
 ncs
2
AB
 nbs
s  nas
2
2
BC
F
a
2
AC
 nbc ai2
 a  1
i 1
 nacs B2
MSB MSBC
2
s 2  nas BC
 nabs C2
MSC MSBC
2
2
s 2  ns ABC
 ncs AB
MS AB MS ABC
2
2
s 2  ns ABC
 nbs AC
MS AC MS ABC
2
s 2  nas BC
MSBC MSError
2
s 2  ns ABC
s2
MS ABC MSError
Example: 3 factors A , B fixed, C random
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
F
a
s  nbs
2
AC
 nbc a i2
 a  1
MS A MS AC
s  nas
2
BC
 nac  b j2
 b  1
MSB MSBC
2
2
i 1
a
i 1
s 2  nabs C2
s  ns
2
a
2
ABC
b
 nc  ab ij
2
i 1 j 1
MSC MSError
 a  1b  1
MS AB MS ABC
2
s 2  nbs AC
MS AC MSError
2
s 2  nas BC
MSBC MSError
2
s 2  ns ABC
s2
MS ABC MSError
Example: 3 factors A , B and C fixed
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
a
F
s  nbc  a i2
 a  1
MS A MSError
s  nac  b j2
 b  1
MSB MSError
s 2  nbc  g k2
 c  1
MSC MSError
2
i 1
a
2
i 1
c
k 1
a
b
s  nc  ab ij
 a  1b  1
2
2
i 1 j 1
a
c
s 2  nb  ag ij
 a  1 c  1
2
i 1 k 1
b
c
s 2  na   bg ij
a
b
2
j 1 k 1
c
s 2  n  abg ijk
i 1 j 1 k 1
s2
2
MS AB MSError
MS AC MSError
 b  1 c  1
MSBC MSError
 a 1b 1 c 1
MS ABC MSError
Example - Random Effects
In this Example a Taxi company is interested in
comparing the effects of three brands of tires (A, B and
C) on mileage (mpg). Mileage will also be effected by
driver. The company selects at random b = 4 drivers at
random from its collection of drivers. Each driver has n
= 3 opportunities to use each brand of tire in which
mileage is measured.
Dependent
– Mileage
Independent
– Tire brand (A, B, C),
• Fixed Effect Factor
– Driver (1, 2, 3, 4),
• Random Effects factor
The Data
Driver
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
Tire
A
A
A
B
B
B
C
C
C
A
A
A
B
B
B
C
C
C
Mileage
39.6
38.6
41.9
18.1
20.4
19
31.1
29.8
26.6
38.1
35.4
38.8
18.2
14
15.6
30.2
27.9
27.2
Driver
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
Tire
A
A
A
B
B
B
C
C
C
A
A
A
B
B
B
C
C
C
Mileage
33.9
43.2
41.3
17.8
21.3
22.3
31.3
28.7
29.7
36.9
30.3
35
17.8
21.2
24.3
27.4
26.6
21
Asking SPSS to perform Univariate ANOVA
Select the dependent variable, fixed factors, random factors
The Output
Tests of Between-Subj ects Effects
Dependent Variable: MILEAGE
Source
Intercept
TIRE
DRIVER
TIRE * DRIVER
Hypothesis
Error
Hypothesis
Error
Hypothesis
Error
Hypothesis
Error
Type III
Sum of
Squares
28928.340
68.290
2072.931
87.129
68.290
87.129
87.129
170.940
df
1
3
2
6
3
6
6
24
Mean
Square
28928.340
22.763a
1036.465
14.522b
22.763
14.522b
14.522
7.123c
F
1270.836
Sig .
.000
71.374
.000
1.568
.292
2.039
.099
a. MS(DRIVER)
b. MS(TIRE * DRIVER)
c. MS(Error)
The divisor for both the fixed and the random main effect is MSAB
This is contrary to the advice of some texts
The Anova table for the two factor model
(A – fixed, B - random)
yijk  m  ai  b j  ab ij  e ijk
Source
SS
df
MS
EMS
A
SSA
a -1
MSA
B
SSA
b-1
MSB
s 2  nas B2
MSB/MSError
AB
SSAB
(a -1)(b -1)
MSAB
2
s 2  ns AB
MSAB/MSError
Error
SSError
ab(n – 1)
MSError
s2
2
s 2  ns AB

nb a 2
ai
a  1 
i 1
F
MSA/MSAB
Note: The divisor for testing the main effects of A is no longer
MSError but MSAB.
References Guenther, W. C. “Analysis of Variance” Prentice Hall, 1964
The Anova table for the two factor model
(A – fixed, B - random)
yijk  m  ai  b j  ab ij  e ijk
Source
SS
df
MS
EMS
A
SSA
a -1
MSA
B
SSA
b-1
MSB
2
s 2  ns AB
 nas B2
MSB/MSAB
AB
SSAB
(a -1)(b -1)
MSAB
2
s 2  ns AB
MSAB/MSError
Error
SSError
ab(n – 1)
MSError
s2
2
s 2  ns AB

nb a 2
ai
a  1 
i 1
F
MSA/MSAB
Note: In this case the divisor for testing the main effects of A is
MSAB . This is the approach used by SPSS.
References Searle “Linear Models” John Wiley, 1964
Crossed and Nested Factors
The factors A, B are called crossed if every level
of A appears with every level of B in the
treatment combinations.
Levels of B
Levels
of A
Factor B is said to be nested within factor A if the
levels of B differ for each level of A.
Levels of A
Levels of B
Example: A company has a = 4 plants for
producing paper. Each plant has 6 machines for
producing the paper. The company is interested
in how paper strength (Y) differs from plant to
plant and from machine to machine within plant
Plants
Machines
Machines (B) are nested within plants (A)
The model for a two factor experiment with B
nested within A.
yijk 
m

overall mean
ai
effect of factor A

b a  j i   e ijk
effect of B within A
random error
The ANOVA table
Source
SS
df
MS
F
A
SSA
a-1
MSA
MSA/MSError
B(A)
SSB(A)
a(b – 1)
MSB(A)
MSB(A) /MSError
Error
SSError
ab(n – 1) MSError
p - value
Note: SSB(A ) = SSB + SSAB and a(b – 1) = (b – 1) + (a - 1)(b – 1)
Example: A company has a = 4 plants for
producing paper. Each plant has 6 machines for
producing the paper. The company is interested
in how paper strength (Y) differs from plant to
plant and from machine to machine within plant.
Also we have n = 5 measurements of paper
strength for each of the 24 machines
The Data
Plant
machine
Plant
machine
1
1
2
3
4
5
98.7 59.2 84.1 72.3 83.5
93.1 87.8 86.3 110.3 89.3
100.0 84.1 83.4 81.6 86.1
3
13
14
15
16
17
83.6 76.1 64.2 69.2 77.4
84.6 55.4 58.4 86.7 63.3
90.6 92.3 75.4 60.8 76.6
2
6
7
60.6 33.6
84.8 48.2
83.6 68.9
8
44.8
57.3
66.5
9
58.9
51.6
45.2
10
63.9
62.3
61.1
11
63.7
54.6
55.3
12
48.1
50.6
39.9
22
37.0
47.8
41.0
23
43.8
62.4
60.8
24
30.0
43.0
56.9
4
18
19
61.0 64.2
81.3 50.3
73.8 32.1
20
35.5
30.8
36.3
21
46.9
43.1
40.8
Anova Table Treating Factors (Plant, Machine) as
crossed
Tests of Between-Subjects Effects
Dependent Variable: STRENGTH
Type III
Sum of
Source
Squares
Corrected Model
21031.065 a
Intercept
298531.4
PLANT
18174.761
MACHINE
1238.379
PLANT * MACHINE 1617.925
Error
5505.469
Total
325067.9
Corrected Total
26536.534
df
23
1
3
5
15
48
72
71
Mean
Square
914.394
298531.4
6058.254
247.676
107.862
114.697
a. R Squared = .793 (Adjusted R Squared = .693)
F
7.972
2602.776
52.820
2.159
.940
Sig.
.000
.000
.000
.074
.528
Anova Table: Two factor experiment B(machine)
nested in A (plant)
Source
Plant
Machine(Plant)
Error
Sum of Squares
18174.76119
2856.303672
5505.469467
df
Mean Square
F
3
6058.253731 52.819506
20
142.8151836 1.2451488
48
114.6972806
p - value
0.00000
0.26171
Analysis of Variance
Factorial Experiments
• Dependent variable Y
• k Categorical independent variables A, B,
C, … (the Factors)
• Let
–
–
–
–
a = the number of categories of A
b = the number of categories of B
c = the number of categories of C
etc.
The Completely Randomized Design
• We form the set of all treatment combinations
– the set of all combinations of the k factors
• Total number of treatment combinations
– t = abc….
• In the completely randomized design n
experimental units (test animals , test plots,
etc. are randomly assigned to each treatment
combination.
– Total number of experimental units N = nt=nabc..
Random Effects and Fixed
Effects Factors
fixed effects factors
• he levels of the factor are a fixed set of levels
and the conclusions of any analysis is in
relationship to these levels.
random effects factor
• If the levels have been selected at random from
a population of levels.
• The conclusions of the analysis will be
directed at the population of levels and not
only the levels selected for the experiment
Example: 3 factors A, B, C – all are random effects
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
F
2
2
2
s 2  ns ABC
 ncs AB
 nbs AC
 nbcs A2
2
2
2
s 2  ns ABC
 ncs AB
 nas BC
 nacs B2
2
2
2
s 2  ns ABC
 nas BC
 nbs AC
 nabs C2
2
2
s 2  ns ABC
 ncs AB
MS AB MS ABC
2
2
s 2  ns ABC
 nbs AC
MS AC MS ABC
2
2
s 2  ns ABC
 nas BC
MSBC MS ABC
2
s 2  ns ABC
s2
MS ABC MSError
Example: 3 factors A fixed, B, C random
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
s  ns
2
2
ABC
 ncs
2
AB
 nbs
s  nas
2
2
BC
F
a
2
AC
 nbc ai2
 a  1
i 1
 nacs B2
MSB MSBC
2
s 2  nas BC
 nabs C2
MSC MSBC
2
2
s 2  ns ABC
 ncs AB
MS AB MS ABC
2
2
s 2  ns ABC
 nbs AC
MS AC MS ABC
2
s 2  nas BC
MSBC MSError
2
s 2  ns ABC
s2
MS ABC MSError
Example: 3 factors A , B fixed, C random
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
F
a
s  nbs
2
AC
 nbc a i2
 a  1
MS A MS AC
s  nas
2
BC
 nac  b j2
 b  1
MSB MSBC
2
2
i 1
a
i 1
s 2  nabs C2
s  ns
2
a
2
ABC
b
 nc  ab ij
2
i 1 j 1
MSC MSError
 a  1b  1
MS AB MS ABC
2
s 2  nbs AC
MS AC MSError
2
s 2  nas BC
MSBC MSError
2
s 2  ns ABC
s2
MS ABC MSError
Example: 3 factors A , B and C fixed
Source
A
B
C
AB
AC
BC
ABC
Error
EMS
a
F
s  nbc  a i2
 a  1
MS A MSError
s  nac  b j2
 b  1
MSB MSError
s 2  nbc  g k2
 c  1
MSC MSError
2
i 1
a
2
i 1
c
k 1
a
b
s  nc  ab ij
 a  1b  1
2
2
i 1 j 1
a
c
s 2  nb  ag ij
 a  1 c  1
2
i 1 k 1
b
c
s 2  na   bg ij
a
b
2
j 1 k 1
c
s 2  n  abg ijk
i 1 j 1 k 1
s2
2
MS AB MSError
MS AC MSError
 b  1 c  1
MSBC MSError
 a 1b 1 c 1
MS ABC MSError
Crossed and Nested Factors
Factor B is said to be nested within factor A if the
levels of B differ for each level of A.
Levels of A
Levels of B
The Analysis of Covariance
ANACOVA
Multiple Regression
1. Dependent variable Y (continuous)
2. Continuous independent variables X1, X2, …,
Xp
The continuous independent variables X1, X2, …,
Xp are quite often measured and observed (not set
at specific values or levels)
Analysis of Variance
1. Dependent variable Y (continuous)
2. Categorical independent variables (Factors) A,
B, C,…
The categorical independent variables A, B, C,…
are set at specific values or levels.
Analysis of Covariance
1. Dependent variable Y (continuous)
2. Categorical independent variables (Factors) A,
B, C,…
3. Continuous independent variables (covariates)
X1, X2, …, Xp
Example
1. Dependent variable Y – weight gain
2. Categorical independent variables (Factors)
i. A = level of protein in the diet (High, Low)
ii. B = source of protein (Beef, Cereal, Pork)
3. Continuous independent variables
(covariates)
i.
X1= initial wt. of animal.
Dependent variable is continuous
Statistical Technique
Multiple Regression
ANOVA
ANACOVA
Independent variables
continuous
categorical
×
×
×
×
It is possible to treat categorical independent
variables in Multiple Regression using
Dummy variables.
The Multiple Regression Model
Y  b0  b1 X1 
 bp X p  e
The ANOVA Model
Y  m  ai  b j 
Main Effects
 ab ij 
Interactions
e
The ANACOVA Model
Y  m  ai  b j 
Main Effects
 ab ij 
Interactions
 g 1 X1  g 1 X1 
Covariate Effects
e
ANOVA Tables
The Multiple Regression Model
Source
S.S.
d.f.
Regression
SSReg
p
Error
SSError
n–p-1
Total
SSTotal
n-1
The ANOVA Model
Source
S.S.
d.f.
A
SSA
a-1
B
SSB
b-1
SSAB
(a – 1)(b – 1)
Main Effects
Interactions
AB
⁞
Error
SSError
n–p-1
Total
SSTotal
n-1
The ANACOVA Model
Source
S.S.
d.f.
Covariates
SSCovaraites
p
A
SSA
a-1
B
SSB
b-1
SSAB
(a – 1)(b – 1)
Main Effects
Interactions
AB
⁞
Error
SSError
n–p-1
Total
SSTotal
n-1
Example
1. Dependent variable Y – weight gain
2. Categorical independent variables (Factors)
i. A = level of protein in the diet (High, Low)
ii. B = source of protein (Beef, Cereal, Pork)
3. Continuous independent variables
(covariates)
X = initial wt. of animal.
The
data
wtgn
112
126
88
97
91
78
86
83
108
104
42
93
102
77
85
88
82
41
63
88
104
114
78
111
109
115
47
124
80
97
initial wt
1031
1087
890
1089
894
917
972
899
821
846
1041
1108
1132
1023
1090
921
909
1091
838
935
1098
888
1000
993
1043
992
834
1005
905
1059
Level
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
Source
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Pork
Pork
Pork
Pork
Pork
Pork
Pork
Pork
Pork
Pork
wtgn
56
86
78
69
76
65
60
80
78
41
68
67
71
76
85
37
119
91
51
57
96
67
85
17
67
54
105
64
92
62
initial wt
1044
1025
878
1193
1024
1078
965
958
1135
847
986
1003
968
1035
1018
882
1053
978
1057
1035
965
1025
970
836
961
931
1017
845
1092
932
Level
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Source
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Beef
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Cereal
Pork
Pork
Pork
Pork
Pork
Pork
Pork
Pork
Pork
Pork
The ANOVA Table
Source
Initial (Covariate)
LEVEL
SOURCE
LEVEL * SOURCE
Error
Total
Sum of Squares
3357.8165
6523.4815
2013.6469
2528.0163
19609.4835
31966.8500
df
1
1
2
2
53
59
Mean Square
3357.82
6523.48
1006.82
1264.01
369.99
F
9.075
17.631
2.721
3.416
Sig.
0.00397
0.0001
0.07499
0.04022
Using SPSS to perform
ANACOVA
The data file
Select Analyze->General Linear Model -> Univariate
Choose the Dependent Variable, the Fixed Factor(s) and the
Covaraites
The following ANOVA table appears
Tests of Between-Subjects Effects
Dependent Variable: WTGN
Source
Corrected Model
Intercept
INITIAL
LEVEL
SOURCE
LEVEL * SOURCE
Error
Total
Corrected Total
Type III
Sum of
Squares
12357.366a
24.883
3357.816
6523.482
2013.647
2528.016
19609.484
421265.0
31966.850
df
6
1
1
1
2
2
53
60
59
Mean
Square
2059.561
24.883
3357.816
6523.482
1006.823
1264.008
369.990
a. R Squared = .387 (Adjusted R Sq uared = .317)
F
5.567
.067
9.075
17.631
2.721
3.416
Sig .
.000
.796
.004
.000
.075
.040
The Process of Analysis of Covariance
140
Dependent variable
120
100
80
60
40
700
800
900
1000
1100
Covariate
1200
1300
1400
The Process of Analysis of Covariance
140
Adjusted Dependent variable
120
100
80
60
40
700
800
900
1000
1100
Covariate
1200
1300
1400
• The dependent variable (Y) is adjusted so that
the covariate takes on its average value for
each case
• The effect of the factors ( A, B, etc) are
determined using the adjusted value of the
dependent variable.
• ANOVA and ANACOVA can be handled by
Multiple Regression Package by the use of
Dummy variables to handle the categorical
independent variables.
• The results would be the same.
Analysis of unbalanced Factorial
Designs
Type I, Type II, Type III
Sum of Squares
Sum of squares for testing an effect
SS Effect  RSS  modelReduced   RSS  modelComplete 
modelComplete ≡ model with the effect in.
modelReduced ≡ model with the effect out.
Type I SS
• Type I estimates of the sum of squares
associated with an effect in a model are
calculated when sums of squares for a model
are calculated sequentially
Example
• Consider the three factor factorial experiment
with factors A, B and C.
The Complete model
• Y = m + A + B + C + AB + AC + BC + ABC
A sequence of increasingly simpler models
1. Y = m + A + B + C + AB + AC + BC + ABC
2. Y = m + A+ B + C + AB + AC + BC
3. Y = m + A + B+ C + AB + AC
4. Y = m + A + B + C+ AB
5. Y = m + A + B + C
6. Y = m + A + B
7. Y = m + A
8. Y = m
Type I S.S.
SS
I
ABC
 RSS  model2   RSS  model1 
I
SSBC
 RSS  model3   RSS  model2 
SS
I
AC
 RSS  model4   RSS  model3 
I
SSAB
 RSS  model5   RSS  model4 
SSCI  RSS  model6   RSS  model5 
SS  RSS  model7   RSS  model6 
I
B
SS  RSS  model8   RSS  model7 
I
A
Type II SS
• Type two sum of squares are calculated for an
effect assuming that the Complete model
contains every effect of equal or lesser order.
The reduced model has the effect removed ,
The Complete models
1. Y = m + A + B + C + AB + AC + BC + ABC
(the three factor model)
2. Y = m + A+ B + C + AB + AC + BC (the all
two factor model)
3. Y = m + A + B + C (the all main effects
model)
The Reduced models
For a k-factor effect the reduced model is the all
k-factor model with the effect removed
II
SSABC
 RSS  model2   RSS  model1 
II
SSAB
 RSS Y  m  A  B  C  AC  BC   RSS  model2 
II
SSAC
 RSS Y  m  A  B  C  AB  BC   RSS  model2 
II
SSBC
 RSS Y  m  A  B  C  AB  AC   RSS  model2 
SSAII  RSS Y  m  B  C   RSS  model3 
SSBII  RSS Y  m  A  C   RSS  model3 
SSCII  RSS Y  m  A  B  RSS  model3 
Type III SS
• The type III sum of squares is calculated by
comparing the full model, to the full model
without the effect.
Comments
• When using The type I sum of squares the
effects are tested in a specified sequence
resulting in a increasingly simpler model.
The test is valid only the null Hypothesis (H0)
has been accepted in the previous tests.
• When using The type II sum of squares the
test for a k-factor effect is valid only the all kfactor model can be assumed.
• When using The type III sum of squares the
tests require neither of these assumptions.
An additional Comment
• When the completely randomized design is
balanced (equal number of observations per
treatment combination) then type I sum of
squares, type II sum of squares and type III
sum of squares are equal.
Example
• A two factor (A and B) experiment, response
variable y.
• The SPSS data file
Using ANOVA SPSS package
Select the type of SS using model
ANOVA table – type I S.S
Tests of Between-Subjects Effects
Dependent Variable: Y
Ty pe I Sum
Source
of Squares
Correc ted Model 11545. 858a
Intercept
61603. 201
A
3666.552
B
809.019
A*B
7070.287
Error
760.361
Total
73909. 420
Correc ted Tot al
12306. 219
df
8
1
2
2
4
24
33
32
Mean
Square
1443.232
61603. 201
1833.276
404.509
1767.572
31.682
a. R Squared = .938 (Adjusted R Squared = .918)
F
45.554
1944.440
57.865
12.768
55.792
Sig.
.000
.000
.000
.000
.000
ANOVA table – type II S.S
Tests of Between-Subjects Effects
Dependent Variable: Y
Ty pe II
Sum of
Source
Squares
Correc ted Model 11545. 858a
Intercept
61603. 201
A
3358.643
B
809.019
A*B
7070.287
Error
760.361
Tot al
73909. 420
Correc ted Tot al 12306. 219
df
8
1
2
2
4
24
33
32
Mean
Square
1443.232
61603. 201
1679.321
404.509
1767.572
31. 682
a. R Squared = .938 (Adjusted R Squared = .918)
F
45. 554
1944.440
53. 006
12. 768
55. 792
Sig.
.000
.000
.000
.000
.000
ANOVA table – type III S.S
Tests of Between-Subjects Effects
Dependent Variable: Y
Ty pe III
Sum of
Source
Squares
Correc ted Model 11545. 858a
Intercept
52327. 002
A
2812.027
B
1010.809
A*B
7070.287
Error
760.361
Tot al
73909. 420
Correc ted Tot al 12306. 219
df
8
1
2
2
4
24
33
32
Mean
Square
1443.232
52327. 002
1406.013
505.405
1767.572
31. 682
a. R Squared = .938 (Adjusted R Squared = .918)
F
45. 554
1651.647
44. 379
15. 953
55. 792
Sig.
.000
.000
.000
.000
.000
Next Topic
Other Experimental Designs