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Chapter 5
Factorial Designs
5
...

Advantages of Factorial Designs:
1) Economy: Generally to study two or more factors simultaneously will require fewer
experimental units than a ‘one-at-a-time’ approach
...
Analysing the interactions may
yield a valuable insight into the particular conditions under which the experimental
variables are most effective
...

Balanced Design: Each of the treatment combinations is observed the same number of times
...
Similarly if three Factors are included, A with 3
levels, B with 4 levels and C with 2 levels, the design is 3 x 4 x 2
...
2 Two Way Completely Crossed Balanced Design
General Notation: Factor A with a levels
Factor B with b levels

44

The model
y ijk     i   j   ij   ijk

where  ij are random errors with mean zero and variance  2
...
n
...

2
...


No difference between levels of Factor A
No difference between levels of Factor B
No interaction

5
...
3
...
The
study was conducted on nine men and nine women and three levels of the drug were
administered: a placebo, a moderate dose and a large dose
...

(Replication necessary in order to obtain an independent estimate of error variance)

45

5
...
2 The SPSS output
The data in SPSS are as follows:

Use Analyze/General Linear Model/Univariate to declare your Y-variable and the factors
...
Deviation
35
...
52
30
...
79
23
...
52
29
...
83
30
...
52
34
...
15
32
...
53
32
...
35
33
...
41
32
...
45
27
...
27
31
...
56

N

3
3
3
9
3
3
3
9
6
6
6
18

a
Levene's Test of Equality of Error Variances

Dependent Variable: depression score
F

1
...



...

a
...

Corrected Model
279
...
956
9
...
001
Intercept
17422
...
222 2825
...
000
SEX
37
...
556
6
...
030
DRUG
97
...
722
7
...
006
SEX * DRUG
144
...
389
11
...
001
Error
74
...
167
Total
17776
...
778
17
a
...
791 (Adjusted R Squared =
...
Grand Mean
Dependent Variable: depression score
95% Confidence Interval
Mean
Std
...
111

...
836
32
...
SEX
Dependent Variable: depression score
SEX
women
men

95% Confidence Interval
Mean
Std
...
667

...
863
31
...
556

...
752
34
...
DRUG
Dependent Variable: depression score
DRUG
placebo
moderate dose
high dose

95% Confidence Interval
Mean
Std
...
000
1
...
791
35
...
500
1
...
291
34
...
833
1
...
624
30
...
SEX * DRUG

Dependent Variable: depression score
SEX
women

men

DRUG
placebo
moderate dose
high dose
placebo
moderate dose
high dose

95% Confidence Interval
Mean
Std
...
333
1
...
210
38
...
333
1
...
210
33
...
333
1
...
210
26
...
667
1
...
543
33
...
667
1
...
543
37
...
333
1
...
210
35
...
3
...

From the initial descriptive statistics table, the standard deviations look fairly similar so
that the assumption of homogeneity of variance should be valid
...
The null hypothesis is accepted, that is, all the variances
are equal
...


51

From the (Test of Between-Subjects Effects) ANOVA table
Test of difference due to drug:

F

MS ( drug )
48
...
17 = 7
...
90 > = 3
...

ii
...
56

MS ( residual ) 6
...
09

F

Reject since 6
...
05 = 4
...

iii

F

Test of Interaction:

MS ( interaction ) 72
...
17 = 11
...
73 > F2 ,12 ,0
...
89
Conclude there is a significant interaction
...
A
particular dosage is not uniformly ‘best’
...
Looking at the interaction plot, there are
significant interactions between, Drug 1 and sex, i
...
, there appears to be a difference
between the sexes on the depression score when the placebo is given
...
The men do not show this effect
...


52

From the Residual Plots:
The assumptions appear to be satisfactory- the normal Q-Q plot looks suitably like a straight
line plot; the histogram looks relatively symmetric - Hence Normality assumption
satisfactory
...


5
...
4
...
(from Box, G
...
P
...
R
...

1
...
2

1
...
8

TREAT


...
00

...
00

Y


...
00
4
...
0
N=

4

4

4

1
...
00

4

4

4

4

4

3
...

The new y-variable is calculated as newy=1/y
...
00

NEWY

2
...
00
4
...
00

4

4

4

4

2
...
00

POISON

Now the data looks more homogeneous
...
It is difficult to tell if interaction is present
...
4
...

Tests of Between-Subjects Effects
Dependent Variable: NEWY
Type III
Sum of
Squares
a
56
...
089
34
...
414
1
...
643
395
...
505

Mean
Source
df
Square
Corrected Model
11
5
...
089
POISON
2
17
...
805
POISON * TREAT
6

...
240
Total
48
Corrected Total
47
a
...
868 (Adjusted R Squared =
...
531
1374
...
635
28
...
090

From the (Test of Between-Subjects Effects) ANOVA table
i
...



...
000

...
000

...
439

MS ( residual )
0
...
635

Reject since 72
...
05

=3
...

iii
...
805

MS ( residual )
0
...
343

F
Reject since 28
...
05 =2
...

iii
...
 43  0 No Interaction
F

MS ( interaction ) 0
...
24 =1
...
09 < 6,36, 0
...
34
Conclude that there is no significant interaction
...

This can be seen also from the profile plots

55

5
...
3 Checking the assumptions
Tests the null hypothesis that the error variance of the dependent variable is equal across
a
Levene's Test of Equality of Error Variances

Dependent Variable: NEWY
F

1
...



...

a
...
025
...


Std
...
Residual

Model: Intercept + poison + treat + poison * treat

56

The histogram of the residuals looks reasonable and the Q-Q is relatively on the straight line
14

12

Frequency

10

8

6

4

2
Mean = 6
...
Dev
...
87519
N = 48
0
-2
...
00

0
...
00

2
...
5

Unequal sample sizes

In a balanced factorial design the treatment combinations have equal sample sizes
...
Quite often data go missing which give rise to
unequal sample sizes
...

There are 3 possible methods of analysis
...

Type III SS: The sum of squares of each effect is calculated by adjusting the effect for all
other effects in the design
...

Source
A
B
AB
Error
Total

df
a-1
b-1
(a – 1)(b – 1)
N - ab
N-1

SS
SS(ModelA,B) - SS(ModelB)
SS(ModelA,B,) - SS(ModelA)
SS(ModelA,B,AB) - SS(ModelA,B,)
SS(ResidualA,B,AB)

Type I SS: The sum of squares of each effect is calculated by adjusting only for those effects
preceding it in a hierarchical sequence
...

The three methods can lead to completely different conclusions with respect to the main
effects
...
They were
classified into two groups: those who had had no previous day care experience and those who
had had extensive day care experience
...
Each child was given a battery of role taking tasks and a score based
on the outcome of these tasks, with a higher score representing better performance, was
recorded
...
139, -2
...
631,
-2
...
179, -0
...
503, 0
...
934,
-1
...
470, -1
...
137, -2
...
412, -0
...
638,
-0
...
668, -0
...
464, -1
...
096,
0
...
167, -0
...
851,
-0
...
351, -0
...
160, -0
...
102,
0
...
277, 0
...
859, 0
...
851,
-0
...
728
2
...
640
11
...
037
21
...
111
36
...
243
2
...
640
11
...
037

...

R Squared =
...
380)

59

F

8
...
201
4
...
016

...



...
048

...
000

...
728
9
...
171
14
...
037
21
...
111
36
...
243
9
...
171
14
...
037

...

R Squared =
...
380)

8
...
962
5
...
577

...



...
000

...
000

...
728
9
...
320
14
...
037
21
...
111
36
...
243
9
...
320
14
...
037

...

R Squared =
...
380)

F

8
...
962
2
...
577

...



...
000

...
000

...
The different SS used to analyse
the data resulted in different conclusions for the hypothesis test of main effect of Daycare,
where the p-value based on Type I SS was 0
...
026 for Type II and 0
...


60

With Type I SS, if the order of the main effects of Daycare and Age is reversed, it will
produce another different conclusion
...
728
9
...
519
3
...
037
21
...
111
36
...
243
9
...
519
3
...
037

...

R Squared =
...
380)

8
...
962
21
...
424

...



...
000

...
026

...
This is the
default option in SPSS General Linear Model procedure
...
6 Higher Order Factorial Designs
Several factors may be combined in an experiment
...
However, as
the number of factors included increases, so too does the number of experimental units
required to measure each treatment combination
...


Add to this replication in order to provide an independent estimate of experimental error and
the number of measurements required becomes quite large
...


5
...
1 Three-factor Design
The three factor model can be written as
EMBED Equation y ijkl     i   j   k   ij   ik   kl   ijk   ijkl

61

In this model there are first-order interactions between AB, between AC and between BC, (
 ij  ik
 jk
 ijk
,
and
)
...

The same model formulation is used with higher order designs
...


Example:
A three factor experiment examined the effects of Drug (3 levels), Diet (2 levels) and
Biofeed (2 levels)
...
(six replicates over the 12 treatment combinations) The analysis is shown below:
DATA:
diet
biofeed
1

2

62

1
170
175
165
180
160
158

1
drug
2
186
194
201
215
219
209

173
194
197
190
176
198

189
194
217
206
199
195

3
180
187
199
170
204
194
202
228
190
206
224
204

1
161
173
157
152
181
190

2
drug
2
164
166
159
182
187
174

3
162
184
183
156
180
173

164
190
169
164
176
173

171
173
196
199
180
203

205
199
170
160
179
179

5
...
2 SPSS Output

Univariate Analysis of Variance

Descriptive Statistics
Dependent Variable: blood pressure
diet
1

biofeed
1

2

Total

2

1

2

Total

Total

1

2

Total

drug
1
2
3
Total
1
2
3
Total
1
2
3
Total
1
2
3
Total
1
2
3
Total
1
2
3
Total
1
2
3
Total
1
2
3
Total
1
2
3
Total

Mean
Std
...
00
8
...
00
12
...
00
12
...
00
18
...
00
10
...
00
10
...
00
14
...
00
14
...
00
14
...
00
11
...
00
16
...
00
17
...
00
14
...
00
10
...
00
11
...
33
11
...
67
9
...
00
14
...
00
17
...
56
14
...
83
12
...
50
14
...
50
14
...
94
13
...
50
11
...
00
20
...
00
14
...
17
17
...
33
12
...
50
13
...
50
20
...
78
16
...
42
13
...
75
17
...
25
18
...
47
17
...
790

df1

11

df2

60

Sig
...
649

Tests the null hypothesis that the error variance of the
dependent variable is equal across groups
...

Design: Intercept+diet+biofeed+drug+diet *
biofeed+diet * drug+biofeed * drug+diet * biofeed * drug

63

Tests of Between-Subjects Effects
Dependent Variable: blood pressure
Type III Sum
Source
of Squares a
df
Mean Square
F
Sig
...
611
11
1203
...
687

...
056
1
2450160
...
088

...
056
1
5236
...
438

...
722
1
2026
...
943

...
111
2
1857
...
863

...
722
1
34
...
222

...
111
2
441
...
820

...
444
2
128
...
819

...
444
2
544
...
475

...
333
60
156
...
000
72
Corrected Total
22635
...

R Squared =
...
509)

Estimated Marginal Means
1
...
Error Lower Bound Upper Bound
184
...
475
181
...
422

2
...
Error Lower Bound Upper Bound
193
...
086
188
...
172
175
...
086
171
...
116

3
...
Error Lower Bound Upper Bound
179
...
086
174
...
338
189
...
086
185
...
950

4
...
Error Lower Bound Upper Bound
174
...
554
169
...
526
190
...
554
185
...
859
188
...
554
183
...
359

5
...
Error Lower Bound Upper Bound
187
...
949
181
...
900
199
...
949
193
...
900
171
...
949
165
...
233
180
...
949
174
...
455

6
...
Error Lower Bound Upper Bound
178
...
612
170
...
226
202
...
612
194
...
226
199
...
612
191
...
226
170
...
612
163
...
059
179
...
612
172
...
726
177
...
612
170
...
726

7
...
Error Lower Bound Upper Bound
168
...
612
161
...
726
188
...
612
180
...
226
181
...
612
173
...
226
180
...
612
173
...
559
193
...
612
186
...
726
195
...
612
188
...
726

8
...
Error Lower Bound Upper Bound
168
...
109
157
...
219
204
...
109
193
...
219
189
...
109
178
...
219
188
...
109
177
...
219
200
...
109
189
...
219
209
...
109
198
...
219
169
...
109
158
...
219
172
...
109
161
...
219
173
...
109
162
...
219
172
...
109
162
...
885
187
...
109
176
...
219
182
...
109
171
...
219

1

2

Profile Plots
Estimated Marginal Means of blood pressure

192

188

189

Estimated Marginal Means

Estimated Marginal Means

Estimated Marginal Means

190

195

190

186

186

185

183

184

180

182

180

177

180

175

174

178
1

2

diet

66

Estimated Marginal Means of blood pressure

Estimated Marginal Means of blood pressure

1

2

biofeed

1

2

drug

3

DRUG * BIOFEED * DIET
Histogram
12

10

Frequency

8

6

4

2
Mean = 3
...
Dev
...
91928
N = 72
0
-2
...
00

0
...
00

2
...
Residual

2

0

Observed

Predicted

Std
...
6
...
438
Biofeed
F=12
...
863

p=0
...
001
p=0
...
819
p=0
...
222
p=0
...
820
p=0
...
475

67

p=0
...

The presence of a significant second order interaction means that the results cannot be
interpreted as simply as this
...

Thus the normality assumption appears to be justified
...


5
...
4 No Replication
Because of the high number of treatment combinations with the higher order factorial
designs, it is quite common to have no replication within the treatment combinations at all
...
To make any
progress it is necessary to simplify the fitted model and usually it is assumed that the higher
order interactions are ‘negligible’ and the sums of squares associated with these effects is
used to estimate the error
...
7

Random and Fixed Effects

Definitions:
A factor is fixed if its levels consist of the entire population of possible levels or its levels
are selected by a non-random process
...

A model is a fixed effects model if all the factors in the design are fixed effects
A model is a random effects model if all the factors in the design are random effects
...


5
...
1 Example from a mixed model
The factors that influence the breaking strength of a synthetic fibre were studied
...

The experiment was run using fibre from the same production batch
...
7
...
042
160
...
458
44
...
333
44
...
667
45
...

302626
...
961

...
167
4
...
558

...
444
80
...
769

...
444
7
...
963

...
792

a
...

MS(MACHINE * OPERATOR)
c
...
MACHINE
Dependent Variable: breaking strength

MACHINE
1
2
3
4

69

Mean
Std
...
833

...
167

...
667

...
500

...
101
113
...
435
113
...
935
113
...
768
115
...
OPERATOR
Dependent Variable: breaking strength

OPERATOR
1
2
3

95% Confidence Interval
Lower
Upper
Mean
Std
...
875

...
375
111
...
125

...
625
112
...
875

...
375
117
...

The model is a mixed model since machine is a fixed effect and operator a random effect
...
The form of the
hypothesis on test is now different
...
Test of difference between MACHINES
H 0 : 1   2   3   4 No difference between the breaking strength, on average, due to machines
F = 0
...
662
Do not reject the null hypothesis
...
Test of the random effect OPERATOR
2
2
Ho : O  0 where  O is the variance of the random effect OPERATOR
F = 10
...
010
Reject the null hypothesis and conclude there is a difference between operators
...
Test of the interaction effect MACHINE * OPERATOR
2
2
Ho : MO  0 where  MO is the variance of the interaction effect MACHINE * OPERATOR
F = 1
...
151
Do not reject the null hypothesis
...


70

Practical 5
...
Four of
the groups are incidental learning groups and the fifth is an intentional learning group
...
The groups were as follows:
Counting : Task -read through a list of words and count the number of letters in each word
Rhyming : Task - read each word and think of one that rhymed
...

Imagery : Task- try to form a vivid image of each word
Intentional : Task - read the list and memorise them for future recall
The list had 27 words
...

The experiment was then repeated with a random sample of 50 younger subjects, aged 18-30
years
...


71

b)

Create two grouping variables age and task
...

1
...

3
...

5
...

7
...

2
...

4
...

6
...

8
...

10
...

12
...

14
...

16
...

18
...

20
...

22
...


e)

72

Click on Graphs
Boxplot
to open the Boxplot dialog box
...

Click on Define to open the Define dialog box
...

Click on task, then click > to move it to the Category Axis box
...

Click OK
...

Hold down the control key and click on age and task, then click on > to move
them to the Fixed Factor(s) box
...

In the Specify Model box check that the option Full Factorial is selected
...

Click on Options
In the Factor(s) and Factor Interactions dialog box, click on age, task and
age*task and then click on > to move them to the Display Means for box
...

In the Display box, click on Descriptive Statistics to obtain a table of means
and standard deviations of words for all combinations of age and task
...

Click on Save
Click on Standardized to save residuals
...

Click on Plots to open the GLM - General Factorial: Profile Plots dialog box
...

Click on Add
...

To obtain a plot of the interaction effect, click on task in the Factors box and
then on > to move it to the Horizontal Axis box
...

Click on Add
...

Click OK
...


Oxygen uptake Experiment
Nine men took part in an experiment
...
They each performed three types of stress test
- bicycle (1), Treadmill (2) and Step (3)
...

Time to Maximum oxygen uptake in minutes
Exercise
Smoking
1
2
1
12
...
2
13
...
1
11
...
8

3
22
...
3
18
...
9
11
...
8

15
...
8
16
...
1
21
15
...
7
9
...
5

14
...
2
8
...
2
16
...
8

Input the data and analyse the data
...
1
General questions
i)
ii)
iii)
iv)

What do we mean by factorial designs? What are the advantages of the factorial
designs compared to the simple one way analysis?
How many factors and how many levels in each factor there are in a 4x3x2 factorial
design?
What are the three main questions which can be answered in a two way analysis of
variance table?
Define what is meant by the term interaction between two factors
...
Can you say
anything about whether different tasks or age are important or whether there is
interaction between task and age?

ii)

From the ANOVA table (Tests of Between –Subjects Effects):
a)
Is the variation between the two ages statistical significant?
b)
Is the variation between different tasks statistically significant?

73

c)

Is the interaction between age and test statistical significant?

iii)

What are the implications from your results: in terms of
a) younger and older subjects?
b) tasks involving greater depth of processing?
c) interaction between age and different tasks?

iv)

Checking the model:
a) Are the variances homogeneous?
b) Is the assumption of Normality a reasonable one?

From practical 5:Oxygen uptake
i) From the ANOVA table (Tests of Between –Subjects Effects):
a) Is the oxygen uptake different for the different types of stress test?
b) Does smoking habit affect the oxygen uptake?
c) Is there any interaction between smoking habit and type of stress?
ii)

What are the conclusions that you can draw from your experiment?

iii)

What are the model assumptions that are necessary in creating the ANOVA table
...
2: More Factorial Designs
Three-way Factorial Design
Example from Howell: In an experiment on the driving ability of drivers, two types of
drivers – inexperienced and experienced, were taken
...
In all there were 48 drivers and during the experiment the number
of steering corrections was counted
...


inexperienced

experienced

A
4
18
8
10

day
B
23
15
21
13

dirt
16
27
23
14

A
21
14
19
26

night
B
25
33
30
20

dirt
32
42
46
40

6
4
13
7

2
6
8
12

20
15
8
17

11
7
6
16

23
14
13
12

17
16
25
12

a) Enter the data on the response variable in the first column and the levels of the three
factors in the next three columns
...

Click on Data
Split File
2
...

Click on the variable condition and then on  to move it to
the Groups Based on box
...

Click OK
...

d)
Now we need to cancel the split file status:
1
...

Click on Analyze all cases
3
...
Check the
validity of the fitted model using homogeneity of variance test and residual plots
...
Three
operators were selected at random and four cycle times were used
...
The results are:

Operator

1

2

3

a)
b)

40
37
23
38
32
44
26
32
20
25
14
34
40
30
27
45

50
32
20
14
29
19
36
33
45
41
26
33
38
45
28
35

Cycle time
60
34
28
42
41
22
30
33
25
44
24
22
31
26
19
25

70
29
32
37
21
24
33
16
32
13
25
45
37
34
40
24

Input the data and analyse
...


Questions for practical 5
...

From the ANOVA table,
a)
Is the effect of driving experience significant?
b)
Is the effect of road type significant?
c)
Is the effect of driving condition significant?
d)
Which of the first and second order interactions are significant?
e)
What conclusions can you draw from your results?
Are the assumptions of the fitted model valid?

From cloth dyeing experiment
i)
ii)
iii)
iv)

76

What are the hypotheses we can test using the ANOVA table?
State clearly the null and the alternative hypotheses
...




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