Chapter 2: Multiple Factor Designs
and Blocking
2.1 Randomized Complete Block Design
2.2 Randomized Incomplete Block Design
2.3 Full Factorial Designs
2.4 The 2k Factorial Design
2.5 Split-Plot Designs
1
Chapter 2: Multiple Factor Designs
and Blocking
2.1 Randomized Complete Block Design
2.2 Randomized Incomplete Block Design
2.3 Full Factorial Designs
2.4 The 2k Factorial Design
2.5 Split-Plot Designs
2
Objectives



3
Define blocking effects.
Define a randomized complete block design.
Generate and analyze a randomized complete
block design.
Blocking



4
Blocks are groups of experimental units that are
formed such that units within blocks are as
homogeneous as possible.
Blocking is a statistical technique designed to identify
and control variation among groups of experimental
units.
Blocking is a restriction on randomization.
Randomized Block Design Model
yij     i  b j   ij
5
Drill Tip Experiment
6
Generating a Randomized
Complete Block Design
This demonstration illustrates the concepts discussed
previously.
7
The Design
P
B
G
O
O
B
8
P
G
O
P
B G
O P
B
G
Is the Blocking Factor Useful?
Did the blocking factor
help the experiment?
9
Analysis Concerns: Should the Blocking
Factor be Deleted from the Model?
10
...
Analysis Concerns: Should the Blocking
Factor be Deleted from the Model?
NEVER!
11
12
2.01 Multiple Choice Poll
Which of the following statements is true?
a. Blocking is a restriction on randomization.
b. A block effect is not always significant.
c. A blocking factor should never be removed from a
model.
d. All of the above are true.
13
2.01 Multiple Choice Poll – Correct Answer
Which of the following statements is true?
a. Blocking is a restriction on randomization.
b. A block effect is not always significant.
c. A blocking factor should never be removed from a
model.
d. All of the above are true.
14
Analyzing a Randomized
Complete Block Design
This demonstration illustrates the concepts discussed
previously.
15
Prospective versus Retrospective Power
Prospective
(CRD)
16
Retrospective
(CRD)
Retrospective
(RCBD)
Alpha (α)
.05
.05
.05
Error Standard
Deviation
.2
.275
.094
Mean for Orange
9.0
9.450
9.450
Mean for Purple
9.1
9.575
9.575
Mean for Green
9.4
9.600
9.600
Mean for Blue
9.6
9.875
9.875
Sample Size (n)
16
16
16
Power (1-β)
.9374
.3357
.9960
17
Exercise
This exercise reinforces the concepts discussed
previously.
18
19
2.02 Poll
In the exercise, the blocking factor Mill had an F statistic
of 5.0167. Did the blocking factor help this experiment?
 Yes
 No
20
2.02 Poll – Correct Answer
In the exercise, the blocking factor Mill had an F statistic
of 5.0167. Did the blocking factor help this experiment?
 Yes
 No
21
Chapter 2: Multiple Factor Designs
and Blocking
2.1 Randomized Complete Block Design
2.2 Randomized Incomplete Block Design
2.3 Full Factorial Designs
2.4 The 2k Factorial Design
2.5 Split-Plot Designs
22
Objectives




23
Define an incomplete block.
State the requirements for a balanced incomplete
block design.
Understand the differences between unbalanced and
balanced block designs.
Generate and analyze an incomplete block design.
What Is an Incomplete Block?
24
Properties of a Balanced Incomplete Block
Design
Each block has the same number of experimental
units.
 Each treatment occurs the same number of times
in the experiment.
 The number of times any two treatments occur
together in the same block is the same for all pairs
of treatments.
If the design is an incomplete block design (that is,
each treatment cannot be applied in each block an
equal number of times) and any one of these properties
is not met, the design is an unbalanced incomplete block
design.

25
Balanced Incomplete Block Design
BLOCK
26
TREATMENT
I
1, 2, 3
II
1, 2, 4
III
IV
1, 3, 4
2, 3, 4
Differences between Balanced and Unbalanced
Designs



27
A balanced design can require too many runs to be
practical.
All estimates of treatment means and all comparisons
between pairs of treatments have equal precision in a
balanced design.
A balanced design is more statistically efficient than an
unbalanced design.
Drill Tip Experiment
28
Generating and Analyzing an
Incomplete Block Design
This demonstration illustrates the concepts discussed
previously.
29
30
Exercise
This exercise reinforces the concepts discussed
previously.
31
32
2.03 Multiple Choice Poll
Which of the following is not a property of the balanced
incomplete block design?
a. Each block contains the same number of experimental
units.
b. Each treatment occurs the same number of times in
the experiment.
c. Each treatment is present in every block.
d. Each pair of treatments occurs together in the same
block the same number of times.
33
2.03 Multiple Choice Poll – Correct Answer
Which of the following is not a property of the balanced
incomplete block design?
a. Each block contains the same number of experimental
units.
b. Each treatment occurs the same number of times in
the experiment.
c. Each treatment is present in every block.
d. Each pair of treatments occurs together in the same
block the same number of times.
34
Chapter 2: Multiple Factor Designs
and Blocking
2.1 Randomized Complete Block Design
2.2 Randomized Incomplete Block Design
2.3 Full Factorial Designs
2.4 The 2k Factorial Design
2.5 Split-Plot Designs
35
Objectives



36
Explain the advantages of multiple factor designs.
Define common terms.
Generate and analyze a full factorial design.
Full Factorial



37
Most experiments involve two or more factors. These
factors have two or more levels and can be
quantitative or qualitative.
A design is said to be a full factorial design if all
possible combinations of factor levels are included in
the experiment.
A large number of runs is required for a full factorial
design as the number of factors increase.
– For example, for a full factorial design with f factors
such that factor i has li levels, the number of runs is
given by l1*l2*l3…*lf.
Advantages of Factorials



38
Factorials reveal interactions.
Factorials are more efficient than experiments that
use one factor at a time.
Combinations of factor levels provide replication for
individual factors, when factors are removed from
the design.
Full Factorial Designs
39
40
2.04 Multiple Choice Poll
How many runs would a full factorial experiment with 4
factors, each at 3 levels, require?
a. 12
b. 64
c. 81
41
2.04 Multiple Choice Poll – Correct Answer
How many runs would a full factorial experiment with 4
factors, each at 3 levels, require?
a. 12
b. 64
c. 81
42
Battery Life Experiment
Temperature
(categorical)
15, 70, 125
Battery
Life
Type of Plate
Material
(categorical)
1, 2, 3
43
Generating a Full Factorial
Design
This demonstration illustrates the concepts discussed
previously.
44
45
Exercise
This exercise reinforces the concepts discussed
previously.
46
47
2.05 Quiz
The output from the exercise on fuel use for the aircraft
fleet is below. Based on the output, what factors should
the company use in future experiments?
48
2.05 Quiz – Correct Answer
The output from the exercise on fuel use for the aircraft
fleet is below. Based on the output, what factors should
the company use in future experiments?
Weight and Speed; the experiment shows that only
these two factors are significant so future
experiments need only include these factors.
49
Chapter 2: Multiple Factor Designs
and Blocking
2.1 Randomized Complete Block Design
2.2 Randomized Incomplete Block Design
2.3 Full Factorial Designs
2.4 The 2k Factorial Design
2.5 Split-Plot Designs
50
Objectives



51
Understand the application of full factorial designs
such that each factor has only two levels.
Understand the reasons for using these designs.
Generate and analyze a 2k factorial design.
Factorial Designs
The 2k factorial design
 is a special case of the full factorial design, introduced
to reduce the number of design points
 is widely used in industrial experimentation
 forms basic building blocks for other designs
 assumes that the response is linear in terms of the
factors in the design.
52
Two-Level Full Factorial Design
22 = 4 runs
53
23 = 8 runs
Coded Levels in a 2k Factorial Design
A 2k design is one such that each factor has exactly two
levels. These two levels are usually called low and high
and are usually denoted by –1 and +1.
54
The 2k Factorial Design with k Factors
There will be k main effects and
k 
  two-factor
2
interactions
k 
 
 3  three-factor
interactions
..
.
1
k-factor interactions
For example, a design with 5 factors, or a 25 design,
will have 5 main effects, 10 two-factor interactions, 10
three-factor interactions, 5 four-factor interactions, and 1
five-factor interaction.
55
Two-Factor Factorial Experiment
22
The main effect
Catalyst =
(55+22)/2 –
(45+10)/2 = 11
Pressure
1
High
45
1
Low
10
1
Low
56
55
1
High
Catalyst
Interactions
Yes
Rate
Rate
No
-1
Catalyst
-1
+1
Low Pressure
High Pressure
57
Catalyst
+1
Completely Randomized Design for
Two-Level Full Factorial Design
Catalyst
Pressure
High
(-1-1)1
Low
High
(-1+1)1
(+1-1)1
(+1+1)1
(-1+1)2
(+1-1)2
(+1+1)2
(+1+1)1
(+1+1)2
Low
(-1-1)1
(-1+1)1
(+1-1)1
(-1-1)2
(-1+1)2
(-1-1)2
(+1-1)2
In a completely randomized design, the order of the
design points is selected at random.
58
The Model
59
2k Factorial with No Replicates




60
These are designs with one observation at each
corner of the cube.
One unusual value of the response could cause an
experimenter to draw incorrect conclusions.
The experimenter runs the risk of modeling only noise
and not the effects of the factors.
These designs are widely used.
2k Factorial with Replication


61
Replication allows for an estimation of pure error and
for a test of lack of fit.
Center points can be used for replication, if the factors
are quantitative.
Yield Experiment
62
Temperature
(continuous)
Pressure
(continuous)
225 & 250
14 & 18
Concentration
(continuous)
Time
(continuous)
60 & 80
2.5 & 3
63
2.06 Quiz
Match the Yield experiment component on the left with the
correct value on the right.
1. Number of factors
A. 4
B. 16
2. Number of levels (for each factor)
C. 2
3. Number of treatments
64
2.06 Quiz – Correct Answer
Match the Yield experiment component on the left with the
correct value on the right.
1. Number of factors
A. 4
B. 16
2. Number of levels (for each factor)
C. 2
3. Number of treatments
1-A, 2-C, 3-B
65
2k Factorial Experiment with
No Replication
This demonstration illustrates the concepts discussed
previously.
66
67
Exercise
This exercise reinforces the concepts discussed
previously.
68
69
2.07 Poll
In the exercise, Time*Pressure was found to be significant
but Pressure was not found to be significant. Should
Pressure remain in the model?
 Yes
 No
70
2.07 Poll – Correct Answer
In the exercise, Time*Pressure was found to be significant
but Pressure was not found to be significant. Should
Pressure remain in the model?
 Yes
 No
71
Chapter 2: Multiple Factor Designs
and Blocking
2.1 Randomized Complete Block Design
2.2 Randomized Incomplete Block Design
2.3 Full Factorial Designs
2.4 The 2k Factorial Design
2.5 Split-Plot Designs
72
Objectives



73
Define a split-plot design.
Define random and fixed effects.
Generate and analyze a split-plot design.
Carbon Anode Experiment
180 (°C)
2
1
180 (°C)
3
1
190 (°C)
3
1
2
2
1
74
1
2
3
3
2
1
2
2
1
3
2
1
2
3
1
3
3
2
3
1
3
2
1
200 (°C)
2
3
1
Weight
2
210 (°C)
210 (°C)
3
3
190 (°C)
200 (°C)
210 (°C)
2
1
180 (°C)
190 (°C)
200 (°C)
210 (°C)
3
2
190 (°C)
200 (°C)
1
3
180 (°C)
1
3
2
1
Randomized Experiment


75
The runs in every experiment so far have been
randomized.
– Treatment and sample are randomly selected.
– Conditions are reset before every run.
Randomization helps meet model assumptions and
reduces the chance that changes in an uncontrolled
variable will bias the model estimates.
Limited Randomization

Complete randomization might be difficult
or impossible; factor (such as Furnace
Temperature) is hard to change.

Restricted randomization should be part of the design
to maintain an optimal study.
Restricted randomization should be part of the
analysis so that the model accounts for multiple
components of variance.

76
Split-Plot Designs




77
Split-plot designs are used when factors are
impractical, inconvenient, or costly to change.
This methodology originated in agricultural studies.
– The hard-to-change factor was at the same level
for an entire plot of land (whole plot factor).
– The easy-to-change factor varied levels within
a plot of land (sub-plot factor).
Randomization is restricted in split-plot designs.
These designs have two components of variation:
between plots and within a plot.
Split-Plot Designs
Whole Plots
Subplot
78
Subplot
Subplot
Subplot
Whole Plot Effect

Restricted randomization is represented by a new
variable, Whole Plots.

The levels of this factor indicate when hard-to-change
factors are randomized.
This factor appears as a new term in the regression
model.

79
The Model
Yijk    i   j   ij  k (i )   ijk
k i  ~ N (0,   )
2
 ijk ~ N (0,  2 )
80
Hard-to-Change Oven
Temperature
This demonstration illustrates the concepts
discussed previously.
81
82
Fixed Effects

The levels of interest result from deliberate choice, not
sampling a distribution. For example, the levels of
Furnace Temperature and Material
Type are deliberately chosen and controlled.

83
Inferences are to be made only to those levels
included in the study. In this case, you are interested
in the mean response at a given factor level.
Random Effects


84
Levels represent a sample, for example, different runs
in the furnace
Inferences apply to the population of levels, not just
the subset of levels included in the study. In this
example, you are interested in the variation across
oven runs in general, not just the particular set of runs
in the study.
Mixed Models

Models in which some factors are treated as fixed
effects and other factors are treated as random
effects are called mixed effects models, or simply
mixed models.
– Whole plot variation is the random effect.
– Furnace Temperature and Material
Type are the fixed effects.


85
The Whole Plots term in the JMP model
specification includes the Random attribute.
Restricted Maximum Likelihood (REML) is the
recommended method to estimate the variance
components and parameters.
86
2.08 Quiz
Match the term on the left with the appropriate description
on the right.
A. receives the easy-to-change factor
1. Fixed effect
2. Random effect B. extends inferences to all population
levels
3. Whole plot
C. receives the hard-to-change factor
4. Subplot
D. extends inferences only to the levels
in the experiment
87
2.08 Quiz – Correct Answer
Match the term on the left with the appropriate description
on the right.
A. receives the easy-to-change factor
1. Fixed effect
2. Random effect B. extends inferences to all population
levels
3. Whole plot
C. receives the hard-to-change factor
4. Subplot
D. extends inferences only to the levels
in the experiment
1-D, 2-B, 3-C, 4-A
88
Split-Plot Analysis
This demonstration illustrates the concepts
discussed previously.
89
90
Exercise
This exercise reinforces the concepts discussed
previously.
91
92
2.09 Quiz
The output from the split-plot exercise is below. Based on
the output, which is the largest component of variance?
93
2.09 Quiz – Correct Answer
The output from the split-plot exercise is below. Based on
the output, which is the largest component of variance?
The larger component of variance is the Whole Plot
component. Therefore more of the error was
accounted for by the differences between the whole
plots rather than from random noise that was not
accounted for by the whole plots.
94