Factorial Designs
Outlines:
2
 2 factorial design
 2k factorial design, k>=3
 Blocking and confounding in 2k factorial design
2
2



factorial design
The experiment consists of 2 factors- High (+) and Low(-)
The design can be represented as a square with 22=4 runs
Let the letters (1), a, b, and ab also represent the totals of all n
observations taken at these design points
2
2

factorial design
Main effect of A, B and Interaction effect AB
Contrast of A, B, AB
Used to calculate SS
2
2

factorial design
Sum of Square (SS)
2
2

factorial design
Ex. An article in the AT&T Technical Journal (Vol. 65, March/April 1986, pp. 39–50) describes
the application of two-level factorial designs to integrated circuit manufacturing. A basic
processing step in this industry is to grow an epitaxial layer on polished silicon wafers. The
wafers are mounted on a susceptor and positioned inside a bell jar. Chemical vapors are
introduced through nozzles near the top of the jar. The susceptor is rotated, and heat is
applied.

A deposition time and B arsenic flow rate.

two levels of deposition time are short (-) and long (+)

two levels of arsenic flow rate are 55% (-) and 59%(+)

n=4 replications
2
2

factorial design
Effect of A, B, AB
2
2

factorial design
SS of A, B, AB
2
2

factorial design
Model Adequacy Checking
k
2
factorial design, k>=3

The experiment consists of k factors, each factor consists of 2 level (+,-)

For example k=3;
k
2

factorial design, k>=3
Main & Interaction effects
k
2

factorial design, k>=3
Main & Interaction effects
The value in the brackets are
“Contrast”
k
2
factorial design, k>=3

Effects

SS
k
2

factorial design, k>=3
Ex. Consider the surface roughness experiment. This is a 23 factorial design
in the factors feed rate (A), depth of cut (B), and tool angle (C), with n 2
replicates.
k
2
factorial design, k>=3

Main and Interaction effects

SS:
k
2
factorial design, k>=3
k
2



factorial design, k>=3
Single replication of 2k design
Ex. Study the effects of Gap, pressure, C2F6 Flow rate and power to the
etch rate for silicon nitride
There are 4 factors, each factor has 2 level (+,-)
k
2
factorial design, k>=3
k
2
factorial design, k>=3
k
2

factorial design, k>=3
Main and Interaction effects
k
2
factorial design, k>=3
k
2
factorial design, k>=3
Y   0  1 x1   2 x4   3 x1 x4
101.625
306.125
153.625
 776.0625 (
) x1  (
) x2  (
) x1 x4
2
2
2
Total average
The regression coefficient is one-half the effect estimate because regression
coefficients measure the effect of a unit change in x1 on the mean
of Y, and the effect estimate is based on a two-unit change from low to high.
Blocking a replicated 2k design


Suppose that the 2k factorial design has been replicated n times. Each
replicate is run in one block.
The effect of block should be considered.
Block 1
(1)
a
b
ab
Block 2
(1)
a
b
ab
Block n
(1)
a
b
ab
Blocking a replicated

k
2
design
Ex. Consider the chemical process. Suppose that only four experimental
trials can be made from a single batch of raw material.
factor
A
Treatment
combination
B
replicate
1
2
total
3
-
-
A low, B low
28
25
27
80
+
-
A high, B low
36
32
32
100
-
+
A low, B high
18
19
23
60
+
+
A high, B high 31
30
29
90
Blocking a replicated

Low effect of blocks
k
2
design
Blocking and confounding in 2k design



Blocking: It is often impossible to run all the observations in a 2k factorial
design under homogeneous conditions. Blocking is the design technique that
is appropriate for this general situation.
Confounding: a useful procedure for running the 2k design in 2p blocks
where the number of runs in a block is less than the number of treatment
combinations, where p < k.
For 22 factors: there are 4 treatments
Blocking and confounding in 2k design

Contrast
these contrasts are unaffected by blocking since in
each contrast there is one plus and one minus
treatment combination from each block.
•two treatment combinations with the plus signs, ab
and (1), are in block 1 and the two with the minus
signs, a and b, are in block 2
•the block effect and the AB interaction are
identical.
•That is, the AB interaction is confounded with
blocks.
If {1,b} and {a, ab} then the main effect of A have been confounded with blocks.
Usually, we confound the highest order interaction with blocks!
Blocking and confounding in 2k design

Consider 23 design divided into 2 blocks
Blocking and confounding in 2k design

Consider 24 design divided into 2 blocks
Blocking and confounding in 2k design

Consider 23 design divided into 2 blocks with 4
replicates
R1
R2
R3
R4