II.2 Four Factors in Eight Runs
Introduction
 Confounding

– Confounding/Aliasing
– Alias Structure
Examples and Exercises
 A Demonstration of the Effects of
Confounding

II.2 Four Factors in Eight Runs: Introduction
Figure 2 - 23 Design Signs Table and
Four Factors in Eight Runs Design Matrix
Standard
Order
1
2
3
4
5
6
7
8
oLet’s
A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
Compare
–23 Design Signs Table
–Four Factors in Eight
Runs Design Matrix
AB
1
-1
-1
1
1
-1
-1
1
AC
1
-1
1
-1
-1
1
-1
1
BC ABC
1
-1
1
1
-1
1
-1
-1
-1
1
-1
-1
1
-1
1
1
A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
D
-1
1
1
-1
1
-1
-1
1
II.2 Four Factors in Eight Runs: Introduction
-
Exercise Four Factors in Eight Runs Signs Table

A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
To compute estimates, create columns for a signs table by
multiplying columns as before; some are done for you.
C
-1
-1
-1
-1
1
1
1
1
D
-1
1
1
-1
1
-1
-1
1
AB AC AD BC
1
1
1
-1 -1
1
-1
1
-1
1 -1
-1
1 -1
-1
-1
1
-1
-1 -1
1
1
1
1
BD CD ABC
-1
1
1
-1
1
-1
-1
1
ABD ACD
BCD ABCD
II.2 Four Factors in Eight Runs: Introduction
-
Exercise Solution Four Factors in Eight Runs Signs Table

A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
The completed signs table is below
C
-1
-1
-1
-1
1
1
1
1
D
-1
1
1
-1
1
-1
-1
1
AB AC AD BC
1
1
1
1
-1 -1
1
1
-1
1 -1 -1
1 -1 -1 -1
1 -1 -1 -1
-1
1 -1 -1
-1 -1
1
1
1
1
1
1
BD CD ABC
1
1
-1
-1 -1
1
1 -1
1
-1
1
-1
-1
1
1
1 -1
-1
-1 -1
-1
1
1
1
ABD ACD
-1
-1
-1
-1
-1
1
-1
1
1
-1
1
-1
1
1
1
1
BCD ABCD
-1
1
1
1
-1
1
1
1
-1
1
1
1
-1
1
1
1
II.2 Four Factors in Eight Runs: Introduction
-
Exercise Four Factors in Eight Runs Signs Table Solution

By plan the column for D = column for ABC, so we say
that for this design "D=ABC." Also, we can see from
above
•
•
•
•
•
•
•

A = BCD
B = ACD
C = ABD
AB = CD
AC = BD
BC = AD
I = ABCD
A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
D
-1
1
1
-1
1
-1
-1
1
Where "I" is a column of ones.
AB AC AD BC
1
1
1
1
-1 -1
1
1
-1
1 -1 -1
1 -1 -1 -1
1 -1 -1 -1
-1
1 -1 -1
-1 -1
1
1
1
1
1
1
BD CD ABC
1
1
-1
-1 -1
1
1 -1
1
-1
1
-1
-1
1
1
1 -1
-1
-1 -1
-1
1
1
1
ABD ACD
-1
-1
-1
-1
-1
1
-1
1
1
-1
1
-1
1
1
1
1
BCD ABCD
-1
1
1
1
-1
1
1
1
-1
1
1
1
-1
1
1
1
II.2 Four Factors in Eight Runs: Confounding



If we use the signs table to estimate D, what we
really get is an estimate of D + ABC. (Exactly the
same estimate we’d get if we had done a full 24
Design, computed D and ABC and added them.)
The two effects are “stuck” together; hence, we say
they are confounded with each other (on purpose
here).
Similarly, in this design,
•
•
•
•
A is confounded with BCD
B is confounded with ACD
AB is confounded with CD
ETC!
II.2 Four Factors in Eight Runs: Confounding

To Illustrate
– We want to know if method 1 is better than
method 2 for a task. Ann does method 1,
Dan does method 2. If Ann’s results are
better, is it because method 1 is better than
method 2? Or, is Ann better than Dan? Or,
is it both? The factor worker is confounded
with the factor method . We can’t separate
their effects.

Confounding can sometimes be a very
dumb thing to do (but not always).
II.2 Four Factors in Eight Runs: Confounding



When we get the data and compute “D”, the result is really an
estimate of D + ABC.
So, (another new word coming - duck!) “D” is a false name for
the estimate - an alias. When two effects are confounded, we
say they are aliases of each other.
The Alias Structure (also called the confounding structure) of
the design is this table you’ve already seen (rearranged here):
•
•
•
•
•
•
•
•
I = ABCD
A = BCD
B = ACD
C = ABD
D = ABC
AB = CD
AC = BD
BC = AD
II.2 Four Factors in Eight Runs: An Example
Revisit Examples 2 and 4 of Part I


Response y: Throughput (KB/sec)
The Original Experiment was a 24 Design (16 Runs)
– Four Factors: A, B, C, D, performance tuning parameters such as
 number of buffers
 size of unix inode tables for file handling
– Two Levels

In Example 2 an 8 Run Design with only Three Factors
was Considered for Illustrative Purposes. The Numbers
were Rounded Off for Ease of Calculation
– Original Data Was In Tenths and Involved Four Factors
– The Estimate of the Three-way Interaction ABC was also Estimating
the Effect of D. (D and ABC are confounded/aliased.)
II.2 Four Factors in Eight Runs: An Example
Revisit Examples 2 and 4 of Part I
ABCD = I determines runs in half fraction
 D = ABC for these runs

(Complementary half fraction is determined
by ABCD = -I, or D = -ABC, for these runs)
Main Effects
y
66
70
64
71
67
73
68
74
67
70
66
70
66
75
67
72
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
AB
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
Interaction Effects
AC
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
AD
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
BC
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
BD
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
CD ABC ABD
1
-1
-1
1
1
1
1
1
1
1
-1
-1
-1
1
-1
-1
-1
1
-1
-1
1
-1
1
-1
-1
-1
1
-1
1
-1
-1
1
-1
-1
-1
1
1
1
1
1
-1
-1
1
-1
-1
1
1
1
ACD
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
B CD ABCD
-1
1
-1
-1
1
-1
1
1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
-1
1
-1
-1
1
-1
1
1
II.2 Four Factors in Eight Runs: An Example
Design Matrix
A
Lo
Hi
Lo
Hi
Lo
Hi
Lo
Hi
Factors
B
Lo
Lo
Hi
Hi
Lo
Lo
Hi
Hi
C
Lo
Lo
Lo
Lo
Hi
Hi
Hi
Hi
D
Lo
Hi
Hi
Lo
Hi
Lo
Lo
Hi
Response
y
66
70
66
71
66
73
68
72
II.2 Four Factors in Eight Runs: An Example
Signs Table



Use Eight Run Signs Table to Estimate Effects
Factor D is Assigned to the Last Column, ABC
Use Alias Structure to Determine What These
Quantities Are Estimating
y
66
70
66
71
66
73
68
72
552
8
69
A
B
+BCD +ACD
-1
-1
1
-1
-1
1
1
1
-1
-1
1
-1
-1
1
1
1
20
2
4
4
5
.5
C
+ABD
-1
-1
-1
-1
1
1
1
1
6
4
1.5
AB
+CD
1
-1
-1
1
1
-1
-1
1
-2
4
-.5
AC
+BD
1
-1
1
-1
-1
1
-1
1
2
4
.5
AD
+BC
1
1
-1
-1
-1
-1
1
1
0
4
0
D
+ABC
-1
1
1
-1
1
-1
-1
1
-4
4
-1
II.2 Four Factors in Eight Runs: An Example
Normal Probability Plot

Effect A+BCD is
Statistically Significant
Normal 1.0
S cores
A+BC D
0.5
0.0
-0.5
-1.0
-1.5
-1
0
1
2
E ffects
3
4
5
II.2 Four Factors in Eight Runs: An Example
Interpretation
Response y: Throughput (KB/sec)
 Assuming BCD is negligible, you should
choose A Hi (A = +) to maximize y

Caution: ASSUME
II.2 Four Factors in Eight Runs
U-Do-It Exercise: Violin Example

For the Violin Data, Pretend That a Half-fraction of
the Full 24 Was Run. For your convenience, the
violin data and signs table is on the next slide as
well as an eight run signs table with the aliasing
structure that determines the half-fraction
– Find the Levels of Factors A, B, C and D that Would
Have Been Run
– Pick out the observed y’s for these runs. Enter these
into an eight-run response table and compute the
observed effects.
– Compare these effects to those which were computed
from the full 24
II.2 Four Factors in Eight Runs
U-Do-It Exercise: Violin Example - Signs Tables
Main Effects
Actual
Order
Sum
Divisor
Effect
y
69.3
75.3
75.9
79.3
67.4
74.9
74.4
78.8
73.4
77.5
78.8
81.6
72.3
78.8
78.1
81.5
A
B
C
D
AB
-1
-1
-1
-1
1
1
-1
-1
-1
-1
-1
1
-1
-1
-1
1
1
-1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
1
1
-1
1
-1
-1
-1
1
1
1
-1
-1
1
-1
-1
1
-1
1
-1
1
1
-1
1
1
-1
-1
1
1
1
1
-1
1
1
-1
-1
1
1
1
-1
1
1
1
1
1
1 2 1 7 .3 38.1 39.5 -4.9 26.7 -10.1
16
8
8
8
8
8
76.1 4.8 4.9 -.6 3.34 -1.3
Interaction Effects
AC AD
1
1
-1
-1
1
1
-1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
-1
1
1
-1
-1
1
1
5.5 -4.5
8
8
.7
-.6
BC BD
1
1
1
1
-1
-1
-1
-1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
-1
-1
-1
1
1
1
1
-.7 -3.5
8
8
-.1
-.4
CD ABC ABD ACD B CD ABCD
1
-1
-1
-1
-1
1
1
1
1
1
-1
-1
1
1
1
-1
1
-1
1
-1
-1
1
1
1
-1
1
-1
1
1
-1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
-1
-1
-1
-1
-1
1
1
1
-1
-1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
-1
1
1
1
-1
-1
1
1
-1
-1
1
-1
-1
1
-1
-1
-1
1
-1
1
1
1
1
1
1
3.7 -2.3
1.3
.5
-1.3
-1.3
8
8
8
8
8
8
.5
-.3
.2
0
-.2
-.2
y
8
A+BCD B+ACD C+ABD AB+CD AC+BD AD+BC D+ABC
-1
-1
-1
1
1
1
-1
1
-1
-1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
1
-1
1
-1
-1
-1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-1
1
1
1
1
1
1
1
4
4
4
4
4
4
4
II.2 Four Factors in Eight Runs
U-Do-It Exercise: Violin Solution - Completed Design Matrix



The recommended runs used for the half-fraction would
assign D to column 7 of an eight run signs table
The completed eight run design matrix (with runs
rearranged to standard order) is shown below
The completed signs table is shown on the next page
A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
D
-1
1
1
-1
1
-1
-1
1
y
69.3
77.5
78.8
79.3
72.3
74.9
74.4
81.5
II.2 Four Factors in Eight Runs
y
69.3
75.3
75.9
79.3
67.4
74.9
74.4
78.8
73.4
77.5
78.8
81.6
72.3
78.8
78.1
81.5
U-Do-It Exercise: Violin Solution - Completed Signs
Main Effects
Interaction Effects
Tables
 The completed signs table
A
B
C
D
AB
-1
-1
-1
-1
1
1
-1
-1
-1
-1
-1
1
-1
-1
-1
1
1
-1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
1
1
-1
1
-1
-1
-1
1
1
1
-1
-1
1
-1
-1
1
-1
1
-1
1
1
-1
1
1
-1
-1
1
1
1
1
-1
1
1
-1
-1
1
1
1
-1
1
1
1
1
1
1217.3 38.1 39.5 -4.9 26.7 -10.1
16
8
8
8
8
8
76.1 4.8 4.9 -.6 3.34 -1.3
o
AC AD
1
1
-1
-1
1
1
-1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
-1
1
1
-1
-1
1
1
5.5 -4.5
8
8
.7 -.6
BC BD
1
1
1
1
-1
-1
-1
-1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
-1
-1
-1
1
1
1
1
-.7 -3.5
8
8
-.1
-.4
CD ABC ABD ACD B CD ABCD
1
-1
-1
-1
-1
1
1
1
1
1
-1
-1
1
1
1
-1
1
-1
1
-1
-1
1
1
1
-1
1
-1
1
1
-1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
-1
-1
-1
-1
-1
1
1
1
-1
-1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
-1
1
1
1
-1
-1
1
1
-1
-1
1
-1
-1
1
-1
-1
-1
1
-1
1
1
1
1
1
1
3.7 -2.3
1.3
.5 -1.3
-1.3
8
8
8
8
8
8
.5
-.3
.2
0
-.2
-.2
The half-fraction we used
corresponds to those runs in
the sixteen run experiment
when ABCD = I
y
69.3
77.5
78.8
79.3
72.3
74.9
74.4
81.5
608
8
76

is below
The responses that go in
standard order on A, B, C
in the half-fraction are
runs 1, 10, 11, 4, 13, 6, 7,
and 16 (in standard order)
A+BCD B+ACD C+ABD AB+CD AC+BD AD+BC D+ABC
-1
-1
-1
1
1
1
-1
1
-1
-1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
1
-1
1
-1
-1
-1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-1
1
1
1
1
1
1
1
18.4
20
-1.8
-3.2
1
-2.6
12.2
4
4
4
4
4
4
4
4.6
5
-.45
-.80
.25
-.65
3.05
II.2 Four Factors in Eight Runs
U-Do-It Exercise: Violin Solution
Table Comparing Estimated Effects

A table comparing estimated effect:
24 effect
A
B
C
D
AB
AC
BC

Estimate
4.8
4.9
-.6
3.34
-1.3
.7
-.1
Half-Fraction Effect
A+BCD
B+ACD
C+ABD
D+ABC
AB+CD
AC+BD
BC+AD
Estimate
4.60
5.00
-.45
3.05
-.80
.25
-.65
There is strong agreement between the two results. With the
half-fraction, we would have come to essentially the same
conclusions as the full 24, with half the data (and half the
work.)
II.2 Four Factors in Eight Runs
Some Notation
 24-1 is
Shorthand for a Half Fraction of the
24 Design (Four Factors in Eight Runs)
• The 4 stands for four factors
• 24-1 = (24)(2-1 ) = half of the 24 experiment
 2k-p is
Shorthand for k Factors in 2k-p Runs
• k stands for number of factors
• The 2-p stands for the fractionation of the 2k
experiment
(p=1 for a half fraction, p=2 for a quarter
fraction, etc)