Statistical Analysis
Professor Lynne Stokes
Department of Statistical Science
Lecture 5
Complete Factorial Experiments,
Completely Randomized Designs,
Main Effects & Interactions
1
Complete Factorial Experiments
All combinations of the factor levels
appear in the design at least once


Does not require that each combination occur
an equal number of times
Does not state what type of statistical design
is being used
“Factorial Design”
is not a correct statistical term
2
Completely Randomized Design
Randomize the sequence of test runs,
assignment to experimental units

Can be any type of complete or
fractional factorial experiment
3
Randomization
Inexpensive Insurance



Bias due to changes in uncontrollable Factors
or in experimental conditions
Bias due to premature termination of the
experiment
Bias due to machine drift, fatigue, wear
Validates key assumptions
(Independence, Randomization
Distributions)
4
Changes in Experimental Conditions :
Oil Viscosity Tests
Experiment : Two Oils
One Test Run Per Day
Six Test Runs Per Week
Day
1
2
3
4
5
6
Average
Week #1
56.4 (A)
44.7 (A)
67.5 (A)
76.6 (A)
71.3 (A)
72.8 (A)
64.9
Week #2
60.6
54.5
62.8
65.2
78.6
80.3
67.0
t-Test: averages are not
significantly different
(B)
(B)
(B)
(B)
(B)
(B)
5
Changes in Experimental Conditions :
Oil Viscosity Tests
Environmental or equipment change between weeks :
add 5 unit bias to all measurements in Week #2
Day
1
2
3
4
5
6
Average
Week #1
56.4 (A)
44.7 (A)
67.5 (A)
76.6 (A)
71.3 (A)
72.8 (A)
64.9
Week #2
65.6
59.5
67.8
70.2
83.6
85.3
72.0
(B)
(B)
(B)
(B)
(B)
(B)
Average viscosity for Oil B
is significantly greater than for Oil A
6
Changes in Experimental Conditions :
Oil Viscosity Tests
Randomize the test sequence, 3 oils each week,
add 5 unit bias to all measurements in Week #2
Day
1
2
3
4
5
6
Average
Week #1
80.3 (B)
67.5 (A)
56.4 (A)
54.5 (B)
76.6 (A)
78.6 (B)
67.4
Oil B - Oil A
Original
2.1
Week #2
67.8
77.8
70.2
76.3
65.6
49.7
69.5
Biased
7.1
(B)
(A)
(B)
(A)
(B)
(A)
Randomized
2.1
7
Premature Termination :
Oil Viscosity Tests
Oil Type
A
B
Lubricant
#1
#2
#3
#4
#1
#2
#3
#4
Viscosity
54.6 54.3
44.7 47.0
67.5 67.2
76.6 77.4
60.6
Equipment
Failure
8
Response
Drift
50 Gallon
Drum of
Chemicals
Background
Noise
Time
Test Runs
Figure 4.7 Influence of machine drift; test runs indicated by arrows.
9
Torque Study
Goal : Investigate the effects of three factors on
torque forces on rotating shafts
Factor
Shaft Alloy
Sleeve Metal
Lubricant Type
Rotating Shaft
Levels
Steel, Aluminum
Porous, Nonporous
Lub 1, Lub 2, Lub 3, Lub 4
Stationary Sleeve
Lay Out
the Design
MGH Figure
10 5.1
Construction of Completely
Randomized Designs



List the factor-level combinations
 All combinations, if complete factorial; only those to be
tested, if a fractional factorial
 Include repeat tests, if any
Number the combinations (including repeats) from 1 to N
Obtain one or more random number sequences of
numbers from 1 to N
 Randomize the test run sequence, if testing is
performed sequentially (one after another)
 Randomize the assignment of factor-level
combinations to experimental units, if any
11
Completely Randomized Design for
Torque Study
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Alloy
Steel
Steel
Steel
Steel
Steel
Steel
Steel
Steel
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Sleeve
Porous
Porous
Porous
Porous
Nonporous
Nonporous
Nonporous
Nonporous
Porous
Porous
Porous
Porous
Nonporous
Nonporous
Nonporous
Nonporous
Lubricant
Lub 1
Lub 2
Lub 3
Lub 4
Lub 1
Lub 2
Lub 3
Lub 4
Lub 1
Lub 2
Lub 3
Lub 4
Lub 1
Lub 2
Lub 3
Lub 4
Random Number Sequence : 8, 13, 4, 7, 5, 1, 11, 15
9, 3, 12, 10, 6, 14, 16, 2
MGH Table 5.1
12
Completely Randomized
Design for Torque Study
Run No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Combination
8
13
4
7
5
1
11
15
9
3
12
10
6
14
16
2
Alloy
Steel
Aluminum
Steel
Steel
Steel
Steel
Aluminum
Aluminum
Aluminum
Steel
Aluminum
Aluminum
Steel
Aluminum
Aluminum
Steel
Sleeve
Nonporous
Nonporous
Porous
Nonporous
Nonporous
Porous
Porous
Nonporous
Porous
Porous
Porous
Porous
Nonporous
Nonporous
Nonporous
Porous
Repeats : Same Procedure
Lubricant
Lub 4
Lub 1
Lub 4
Lub 3
Lub 1
Lub 1
Lub 3
Lub 3
Lub 1
Lub 3
Lub 4
Lub 2
Lub 2
Lub 2
Lub 4
Lub 2
MGH Table 5.213
Completely Randomized
Design for Torque Study
cf. Table 5.3
CRD with 2 Repeats
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2
3
5
9
10
12
13
16
Alloy
Steel
Steel
Steel
Steel
Steel
Steel
Steel
Steel
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Steel
Steel
Steel
Aluminum
Aluminum
Aluminum
Aluminum
Aluminum
Sleeve
Porous
Porous
Porous
Porous
Nonporous
Nonporous
Nonporous
Nonporous
Porous
Porous
Porous
Porous
Nonporous
Nonporous
Nonporous
Nonporous
Porous
Porous
Nonporous
Porous
Porous
Porous
Nonporous
Nonporous
Lubricant
Lub 1
Lub 2
Lub 3
Lub 4
Lub 1
Lub 2
Lub 3
Lub 4
Lub 1
Lub 2
Lub 3
Lub 4
Lub 1
Lub 2
Lub 3
Lub 4
Lub 2
Lub 3
Lub 1
Lub 1
Lub 2
Lub 4
Lub 1
Lub 4
Randomly
Selected
Repeat Tests
14
Lubricant Deposit Study
Lubricant Repeats
1
9
2
9
3
9
4
9
5
9
6
9
7
9
8
9
Overall
Pooled
Average Std. Dev.
24.89
20.11
78.11
37.96
64.94
23.95
72.50
29.15
88.61
29.46
78.94
26.48
61.94
12.95
51.50
33.52
65.18
27.68
15 ?
What are the Main Factor Effects
Notation
(One-Factor Experiment)
Factor Level
Average
1
2
3
...
k
Overall Average
y1
y2
y3
...
yk
y
Data: yij
i = Factor Level
j = Repeat
16
Factor Main Effects
Change in the average response
due to
changes in levels of one factor
Factor Level
1
2
...
k
Mean
m1
m2
...
mk
Average
y1
y2
...
yk
17
Factor Main Effects
Fixed Effects
: Constant (mean) changes
(Pre-selected levels, systematic changes)
Random Effects : Random changes (Standard Deviation > 0)
(Effects sampled from a probability
distribution)
18
Factor Main Effects Models
One factor model
yij  mi  eij
 m  a i  eij
Averages
yi .  mi  ei .
 m  a i  ei .
Cell Means Model
Effects Model
Assumption
E(eij) = 0
Conventions
mi = m + ai
S ai = 0
19
Factor Main Effects Models
Main Effects
yi . - y j.  mi - m j  (ei . - e j.)
 a i - a j  (ei . - e j.)
yi .  y..  mi  m  (ei .  e.. )
 a i  (ei .  e.. )
Note: mi = mj  ai = aj
Note: mi = m
 ai = 0
Main Effects
Theoretical: changes in factor-level means
Empirical : changes in factor-level averages
20
Pilot Plant Experiment
45
80
C2
Catalyst
52
83
54
68
C1
40
60
160
Concentration
72
Temperature
180
20
MGH Figure 5.3
21
Pilot Plant Experiment
45
80
C2
Catalyst
52
83
54
68
C1
40
60
160
Concentration
72
Temperature
180
20
22
Pilot Plant Experiment
45
80
C2
Catalyst
52
83
54
68
C1
40
60
160
Concentration
72
Temperature
180
20
23
Pilot Plant Experiment
45
80
C2
Catalyst
52
83
54
68
C1
40
60
160
Concentration
72
Temperature
180
20
24
Pilot Plant Experiment
Main Effects
Change in the average response
due to changes in levels of one factor
Temperature
Concentration
Catalyst
Low
52.75
66.75
63.50
High
75.75
61.75
65.00
(High – Low)
Main Effect
23.0
-5.0
1.5
Main effects do not measure joint factor effects
Main effects are averaged across levels of the other factors
25
Interactions
Effects of the levels of one factor
on the response
depend on the levels of one or more other factors
26
No Interaction
Response
Factor #2
Change in
average
response
for factor #1
is constant
for all levels
of factor #2
Factor #1
27
Pilot Plant Experiment
45
52
62.5
Change with
concentration
roughly the
same for each
catalyst
C2
Catalyst
67.5
54
80
83
68
61
C1
40
60
160
66
Temperature
Concentration
72
180
20
28
Pilot Plant Experiment
45
80
C2
81.5
48.5
Catalyst
Change with
temperature
greater for
catalyst C2
than for
catalyst C1
52
83
54
57
60
160
68
C1
40
70
Concentration
72
Temperature
180
20
29
Average Chemical Yield, Conditioned on Catalyst and Concentration (%).
160
165
C2
20
170
175
180
C2
40
80
Average yield (%)
70
60
50
C1
20
C1
40
80
70
60
50
160
165
170
175
180
Temperature (deg C)
30
Pilot Plant Experiment
Interaction Plot
No Interaction
75
70
Average
Yield (%)
Catalyst C2
65
60
Catalyst C1
55
50
20
40
Concentration (%)
31
MGH Figure 5.4
Pilot Plant Experiment
Interaction Plot
80
Catalyst C2
Interaction
75
70
Average
Yield (%)
65
Catalyst C1
60
55
50
160
180
Temperature (deg C)
32
MGH Figure 5.4
Cutting Tool Life
50
No Interaction
40
Tool
Life
(hrs)
Tool Type B
30
}
20
10
Tool Type A
15 Hours
0
500
600
700
800
Lathe Speed (rpm)
900
1000
MGH Figure 5.2
33
Plasticity Study
Purpose : Study “Plastic-Like” Properties of
Friction-Reducing Lubricants and Additives
Moveable Test Sample
Stationary Platform
Lubricant + Additive
Responses: Plastic Viscosity
Gel
34
Plasticity Experiment

Design Factors

Lubricant


Additive


Acme XLT, Monarch1, Standard
None, 1%, 5%
Design

Factorial Experiment


2 Repeat Tests / Combination
Completely Randomized Design
35
Factor Effects on Plastic Viscosity
Effects of Lubricants
Acme XLT - Standard :
Monarch 1 - Standard :
6.22 - 4.81 = 1.41
4.79 - 4.81 = -0.02
Main Effects for
Lubricants
Effects of Additives
1% - None :
5% - None :
Lubricant
Acme XLT
Monarch 1
Standard
Average
Average
Viscosity
6.22
4.79
4.81
5.27
Main Effects for
Additives
4.04 - 5.39 = -1.35
6.39 - 5.39 = 1.00
Main
Effect
0.95
-0.48
-0.46
0.00
Additive
None
1%
5%
Average
Viscosity
5.39
4.04
6.39
5.27
Main
Effect
0.12
-1.23
1.12
0.00
36
Factor Effects on Plastic Viscosity
Primary Conclusion
Acme XLT has a greater
mean plastic viscosity,
especially with 5% concentration
If Verified
by statistical
analysis
Weak, if any,
interaction effects
Average
Plastic
Viscosity
8
Acme XLT
6
Monarch1
Standard
4
2
0
None
1%
5%
37
Additive Amount
Factor Effects on Gel
Effects of Lubricants
Acme XLT - Standard :
Monarch 1 - Standard :
11.45 - 10.01 = 1.44
10.10 - 10.01 = 0.09
Main Effects for
Lubricants
Effects of Additives
1% - None :
5% - None :
8.49 - 15.70 = -7.21
7.37 - 15.70 = -8.33
Main Effects for
Additives
38
Factor Effects on Gel
Primary Conclusion
Acme XLT has a lower
mean gel with no additive,
greater with additives
If verified
by statistical
analysis
Strong
Interaction Effects
20
Average
Gel
15
Acme XLT
10
Monarch1
Standard
5
0
None
1%
5%
39
Additive Amount
Completely Randomized Design for
Torque Study
Run No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Alloy
Steel
Aluminum
Steel
Steel
Steel
Steel
Aluminum
Aluminum
Aluminum
Steel
Aluminum
Aluminum
Steel
Aluminum
Aluminum
Steel
Sleeve
Nonporous
Nonporous
Porous
Nonporous
Nonporous
Porous
Porous
Nonporous
Porous
Porous
Porous
Porous
Nonporous
Nonporous
Nonporous
Porous
Lubricant
Lub 4
Lub 1
Lub 4
Lub 3
Lub 1
Lub 1
Lub 3
Lub 3
Lub 1
Lub 3
Lub 4
Lub 2
Lub 2
Lub 2
Lub 4
Lub 2
Average Torque
79.0
69.5
74.5
77.5
78.5
79.0
78.5
69.0
78.0
77.0
73.5
2.0
67.0
70.5
70.0
77.5
40
Interaction Effects :
Torque Study
Aluminum Shaft
85
Porous Sleeve
80
Average
Torque
(in-oz)
75
70
65
60
1
2
3
4
Lubricant Type
41
Interaction Effects :
Torque Study
Aluminum Shaft
85
Porous Sleeve
80
Average
Torque
(in-oz)
75
70
65
Nonporous Sleeve
60
1
2
3
4
Lubricant Type
42
Interaction Effects :
Torque Study
Average
Torque
(in-oz)
Average
Torque
(in-oz)
Aluminum Shaft
85
Steel Shaft
85
Porous Sleeve
80
75
75
70
70
65
Porous Sleeve
80
Nonporous Sleeve
60
65
Nonporous Sleeve
60
1
2
3
Lubricant Type
4
1
2
3
4
Lubricant Type
43
Interaction Effects
Interaction effects cannot be properly evaluated
if the design does not permit their estimation
Complete factorials permit the evaluation of
all main effects and all interaction effects
44