Designs for Experiments with
More Than One Factor
• When the experimenter is interested in the effect of
multiple factors on a response a factorial design
should be used.
• A factorial experiment means that all factor level
combinations are included in each replicate of the
experiment.
• For example, if the experimenter wanted to test 4
factors at 2 levels each, then all 24 = 16
combinations of factor levels would be included in
each replicate.
• The effect of a factor is defined as the change in the
response produced by a change in the level of a
factor. This is often termed a main effect.
20
40
B2
B1
10
A1
30
A2
• Consider the experiment depicted above. Here we have 2
factors each investigated at 2 levels.
• The main effect of each factor is calculated as the difference
between the average response at the first level of the factor
and the average response at the second level of the factor.
• It is often useful to graphically display the results of
the experiment. Note that we can easily see the
effects calculated earlier.
50
40
30
B1
B2
20
10
0
A1
A2
• This are the results from a similar experiment
where factors interact. When the interaction effects
are large the main effects have little meaning
20
0
10
30
B2
B1
A1
A2
35
30
25
20
15
10
5
0
B1
B2
A1
A2
General 2k Designs
• 2k designs are popular in industry particularly in the
exploratory phase of process and product
improvement. They consist of k factors each
studied at two levels (usually denoted as high and
low levels). Our first example was a 22 design.
• A simple notation has been developed to represent
the replicates.
• A run is represented by a series of lowercase letters.
If a letter is present it indicates that the
corresponding factor is set at its high level. The run
with all factors set at their low levels is denoted as
(1)
• This notation applied to our first example is as
follows:
b
ab
High(+)
Low(-)
(1)
a
Low(-)
High (+)
• The main effect is then just the difference between
the average of the observations on the right side of
the square and the average of the observations on
left side of the square, if n = the number of
replicates under each factor combination then the
main effect of factor A is:
A

a  ab
b  (1)

2n
2n
1
[a  ab  b  (1)]
2n
• The main effect of B is the difference between the
average of the observations at the top of the square
and the average at the bottom of the square:
B

b  ab a  (1)

2n
2n
1
[b  ab  a  (1)]
2n
• The interaction effect is found by taking the
difference in the diagonal averages:
AB 

ab  (1) a  b

2n
2n
1
[ab  (1)  a  b]
2n
• The terms in the brackets in each of these equations
are called contrasts. For example:
• ContrastA = a + ab - b - (1)
• Note that the coefficients in these contrasts are
always = to either -1 or + 1
• A table of + and - signs is helpful in determining
the sign on each run for developing the contrasts.
Run
(1)
a
b
ab
A
+
+
Effect
B
+
+
AB
+
+
• The sum of squares for each effect is then as
follows:
SS 
(contrast ) 2
n (contrast coefficients) 2
• The total sum of squares is obtained as usual and
the error sum of squares can be obtained by
subtraction
An Example
• A router is used to cut notches in printed circuit
boards. The process is in statistical control, the
average dimension is satisfactory, but there is too
much variability in the process which leads to
problems in assembly. The quality improvement
team identified two factors which may have an
impact: bit size (A) (tested at 1/8 inch and 1/16
inch) and the speed (B) (tested at 40 rpm and 80
rpm). It was felt that vibration of the boards during
the process was responsible for the excess variation.
An experiment was conducted using four boards at
each treatment level. The treatment levels were
randomly assigned to the 16 boards and the results
are as follows:
Run
(1)
a
b
ab
A
+
+
Effects
B
+
+
AB
+
+
18.2
27.2
15.9
41
Vibration
18.9
12.9
24
22.4
14.5
15.1
43.9
36.3
14.4
22.5
14.2
39.9
• Calculate the main and interaction effects and the
sum of squares.
Residual Analysis
• The residuals from a 2k design are easily obtained
through fitting a regression model to the data. For
our experiment the appropriate model is as follows:
y   0  1 x1   2 x2   3 x1 x2  
• This model can also be used for obtaining predicted
values for the four points in our experimental
design.
• Minitab Ouput:
General Linear Model: Vibration versus A, B
Factor
A
B
Type Levels Values
fixed
2 1 -1
fixed
2 1 -1
Analysis of Variance for Vibration, using Adjusted SS for Tests
Source
A
B
A*B
Error
Total
DF
1
1
1
12
15
Seq SS
1107.23
227.26
303.63
71.72
1709.83
Term
Constant
A1
B1
A* B
Coef
23.8313
8.3188
3.7688
4.3562
SE Coef
0.6112
0.6112
0.6112
0.6112
Adj SS
1107.23
227.26
303.63
71.72
T
38.99
13.61
6.17
7.13
Adj MS
1107.23
227.26
303.63
5.98
P
0.000
0.000
0.000
0.000
F
185.25
38.02
50.80
P
0.000
0.000
0.000
Interaction Plot - LS Means for Vibration
-1
1
-1
1
40
A
-1
30
20
1
40
B
-1
30
20
1
Designs for K  3
• The methods for 22 designs discussed can be easily
extended to designs involving more than two
factors. The effects are calculated similarly,
Effect 
contrast
n 2 k 1
• As are the sum of squares,
SS 
(contrast ) 2
n2 k
• An article in Solid State Technology describes an
experiment for improving the etch rate on a wafer
plasma etcher. The factors in the experiment were
gas flow (A), power applied to the cathode (B), gap
between the cathode and the anode (C), and
pressure (D). Each factor was tested at two levels.
We will assume a single replicate was performed
with the following results:
Run
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
Etch Rate
550
669
604
650
633
642
601
635
1037
749
1052
868
1075
860
1063
729
• Use Minitab to analyze the results
Fractional Factorial Designs
• Since the number of runs increases exponentially
with the number of factors investigated in a 2k
design it is desirable to limit the number of runs
while maintaining the ability to obtain information
on the factors of interest. If we can assume that the
higher order interactions are negligible, then a
fractional factorial design involving fewer than 2k
runs may be used. Consider the 23 design matrix
depicted below:
Run
a
b
c
abc
ab
ac
bc
(1)
A
+
+
+
+
-
B
+
+
+
+
-
C
+
+
+
+
-
Effect
AB
+
+
+
+
AC
+
+
+
+
BC
+
+
+
+
ABC
+
+
+
+
-
• Note that in the full factorial design, the sum of the
products of any two columns = 0. This indicates
that the columns are orthogonal, e.g, their
associated effects are statistically independent.
• If we were to assume that the high order interaction,
ABC were insignificant we might conduct the
experiment using just the top half of the design
matrix
• Notice that the 2(3-1) design is formed by selecting
only those runs that yield a + on the ABC effect.
The interaction ABC is termed the generator of this
fraction.
• The estimates of the main effects from this
fractional design are as follows
A = 1/2 [a - b - c + abc]
B = 1/2[-a + b - c + abc]
C = 1/2[-a - b + c + abc]
also,
BC = 1/2[a - b - c + abc]
AC = 1/2[-a + b - c + abc]
AB = 1/2[-a - b + c + abc]
Note that the linear combination of observations in
column A estimates A + BC. Therefore, if the
contrast is significant, we cannot tell whether it is
due to the main effect of A or the interaction effect
of B or a mixture of both. That is, the two columns
are no longer orthogonal. Two or more effects that
have this property are termed aliases.
The alias for any factor can be found by multiplying
the factor by the generator
• The alias of A is:
A x ABC = A2BC = BC
• likewise,
• the alias of B = AC
• the alias of C = AB
• If we had chosen the other half fraction, the
generator would have been -ABC and the aliases
would be:
• A = -BC
• B = -AC
• C =-AB
• That is the column associated with A really
estimates A - BC, the column associated with B
estimates B - AC and the column associated with C
estimates C - AB
• The fraction with the + sign is sometimes referred
to as the principle fraction while the fraction with
the - sign is termed the alternate fraction.
Using sequences of fractional
designs to estimate effects
• If we had chosen the first design and were
convinced that the two-way interactions were
insignificant then the design will produce estimates
of the main effects of the three factors. If, however,
after running the principle fraction we believe are
uncertain as to the interaction effects we can
estimate them by running the alternate fraction.
• It is easy to construct a 2 k-1 design. Simply write
down the treatment levels for the full factorial
experiment in k-1 factors. Then equate the column
associated with the kth factor with the product of the
signs of the k-1 factors. For example we would
construct a 24-1 fractional factorial for our plasma
etching experiment as follows:
Run
(1)
ad
bd
ab
cd
ac
bc
abcd
A
+
+
+
+
B
+
+
+
+
C
+
+
+
+
D = ABC
+
+
+
+
Etch Rate
550
749
1052
650
1075
642
601
729
• The minitab output below agrees substantially with
the output generated from the full factorial design.
Factor
A
D
Type Levels Values
fixed
2 -1 1
fixed
2 -1 1
Analysis of Variance for response, using Adjusted SS for Tests
Source
A
D
A*D
Error
Total
DF
1
1
1
4
7
Seq SS
32258
168780
78012
1797
280848
Adj SS
32258
168780
78012
1797
Adj MS
32258
168780
78012
449
Term
Constant
A
-1
D
-1
A* D
-1 -1
Coef
756.000
SE Coef
7.494
T
100.88
P
0.000
63.500
7.494
8.47
0.001
-145.250
7.494
-19.38
0.000
-98.750
7.494
-13.18
0.000
F
71.80
375.69
173.65
P
0.001
0.000
0.000