Statistical Methods
for Economists
Lecture 9
Full Factorial Design
of Experiments
•David Bartl
•Statistical Methods for Economists
•INM/BASTE

Outline of the lecture
•Motivation & Introduction
•Full Factorial Experiments
•Graphical assessment of factor / interaction significance
•Graphs of interaction effects

Full Factorial Experiments:  Motivation



Full Factorial Experiments:  Motivation  &  Introduction



Full Factorial Experiments:  Motivation  &  Introduction
−−
+−
−+
++
Factor
A
B
−
−
+
−
−
+
+
+

Full Factorial Experiments:  Motivation  &  Introduction
−−−
+−−
−−+
+−+
−+−
++−
−++
+++
Factor
A
B
C
−
−
−
+
−
−
−
+
−
+
+
−
−
−
+
+
−
+
−
+
+
+
+
+

Full Factorial Experiments:  Motivation  &  Introduction



Full Factorial Experiments:  Motivation  &  Introduction



Full Factorial Experiments:  Motivation  &  Introduction



Full Factorial Experiments:  Example



Full Factorial Experiments:  Example
•We consider the following levels of the three factors:
•Factor L = the length of the spring
“−” or “−1”  = 10 cm
“+” or “+1”  = 15 cm
•Factor G = the thickness of the wire of the spring
“−” or “−1” = 5 mm
“+” or “+1”  = 7 mm
•Factor T = the material of (the wire of) the spring
“−” or “−1”  = Material / alloy “A”
“+” or “+1”  = Material / alloy “B”

Full Factorial Experiments:  Example
The experiment was carried out 2× for each combination of the factors:


Full Factorial Experiments:  Example



Full Factorial Experiments:  Example



Full Factorial Experiments:  Example
Interaction = the product of the Factors.
Intercept
Factor
Interaction
0
L
G
T
LG
LT
GT
LGT
1
−
−
−
+
+
+
−
1
+
−
−
−
−
+
+
1
−
+
−
−
+
−
+
1
+
+
−
+
−
−
−
1
−
−
+
+
−
−
+
1
+
−
+
−
+
−
−
1
−
+
+
−
−
+
−
1
+
+
+
+
+
+
+
“−”  =  “−1”    &    “+”  =  “+1”

Full Factorial Experiments:  Example
Since we carried out each experiment 2× for each combination
of the factors, we double each row of the table:
Interaction = the product of the Factors.

Full Factorial Experiments



Full Factorial Experiments



Full Factorial Experiments:  Example



Full Factorial Experiments



Full Factorial Experiments:  Example



Full Factorial Experiments:  Example



Full Factorial Experiments:  Example



Full Factorial Experiments



Full Factorial Experiments



Full Factorial Experiments



Full Factorial Experiments



Full Factorial Experiments:  Example



Full Factorial Experiments



Full Factorial Experiments



Full Factorial Experiments:  Example



Full Factorial Experiments:  Example



Full Factorial Experiments:  Example



Graphical assessment of factor / interaction significance
•
•

Full Factorial Experiments



Full Factorial Experiments



Full Factorial Experiments:  Example



Full Factorial Experiments:  Example



Full Factorial Experiments:  Graphical assessment



Graphs of interaction effects
•
•

Full Factorial Experiments:  Interaction effect
•There is an interaction effect between two factors in a full factorial experiment
when the effect of one independent variable (factor) depends on the level
of another independent variable (factor).
•
•Considering two factors  A and B,  take
•all the observations when factors  A and B  were at the levels  −−,  respectively
•all the observations when factors  A and B  were at the levels  −+,  respectively
•all the observations when factors  A and B  were at the levels  +−,  respectively
•all the observations when factors  A and B  were at the levels  ++,  respectively
•and calculate the average  (sample mean)  for each of the four groups

Full Factorial Experiments:  Interaction effect
•We then have:
•
•
•
•
•
•We then depict the values in the form of a chart
•(here as the dependence of  Factor B  on  Factor A):
•If the lines are ≈ parallel,
then the interaction is not significant.
•If the lines are not parallel,
then the interaction is significant.

Full Factorial Experiments:  Example
•In our example, we have:
Factor G
+
76, 74, 72, 74
90, 94, 92, 88
74
91
−
77, 81, 63, 65
98, 96, 82, 86
71.5
90.5
−
+
Factor L
Factor T
+
63, 65, 72, 74
82, 86, 92, 88
68.5
87
−
77, 81, 76, 74
98, 96, 90, 94
77
94.5
−
+
Factor L

Full Factorial Experiments:  Example
G
+
74
91
−
71.5
90.5
−
+
L
Factor L
74
71.5
91
90.5
G+
G−
−
+
L
+
90.5
91
−
71.5
74
−
+
G
Factor G
74
71.5
91
90.5
L+
L−
−
+
≈ parallel   ð  the interaction is not significant

Full Factorial Experiments:  Example
T
+
68.5
87
−
77
94.5
−
+
L
Factor L
77
68.5
94.5
87
T+
T−
−
+
L
+
94.5
87
−
77
68.5
−
+
T
Factor T
68.5
77
87
94.5
L+
L−
−
+
≈ parallel   ð  the interaction is not significant

Full Factorial Experiments:  Example
T
+
74
83.5
−
88
81.5
−
+
G
Factor G
88
74
83.5
81.5
T+
T−
−
+
G
+
81.5
83.5
−
88
74
−
+
T
Factor T
74
81.5
83.5
88
G+
G−
−
+
NOT parallel   ð  the interaction is SIGNIFICANT