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2 The 2k Factorial: Calculation of Effects and Analysis of Variance

# 2 The 2k Factorial: Calculation of Effects and Analysis of Variance

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15.2 The 2k Factorial: Calculation of Eﬀects and Analysis of Variance

599

for the response data at the design point in question. As an introduction to the
calculation of important eﬀects that aid in the determination of the inﬂuence of
the factors and sums of squares that are incorporated into analysis of variance
computations, we have Table 15.1.
Table 15.1: A 22 Factorial Experiment
A
b

B
Mean

Mean
ab

(1)

a

(1)+b
2n

a+ab
2n

b+ab
2n
(1)+a
2n

In this table, (1), a, b, and ab signify totals of the n response values at the
individual design points. The simplicity of the 22 factorial lies in the fact that
apart from experimental error, important information comes to the analyst in
single-degree-of-freedom components, one each for the two main eﬀects A and B
and one degree of freedom for interaction AB. The information retrieved on all
these takes the form of three contrasts. Let us deﬁne the following contrasts
among the treatment totals:
A contrast = ab + a − b − (1),
B contrast = ab − a + b − (1),
AB contrast = ab − a − b + (1).
The three eﬀects from the experiment involve these contrasts and appeal to common sense and intuition. The two computed main eﬀects are of the form
eﬀect = y¯H − y¯L ,
where y¯H and y¯L are average response at the high, or “+” level and average
response at the low, or “−” level, respectively. As a result,
Calculation of
Main Eﬀects

A=

ab + a − b − (1)
A contrast
=
2n
2n

B=

ab − a + b − (1)
B contrast
=
.
2n
2n

and

The quantity A is seen to be the diﬀerence between the mean responses at the
low and high levels of factor A. In fact, we call A the main eﬀect of factor A.
Similarly, B is the main eﬀect of factor B. Apparent interaction in the data is
observed by inspecting the diﬀerence between ab − b and a − (1) or between ab − a
and b − (1) in Table 15.1. If, for example,
ab − a ≈ b − (1)

or

ab − a − b + (1) ≈ 0,

Chapter 15 2k Factorial Experiments and Fractions

600

a line connecting the responses for each level of factor A at the high level of factor
B will be approximately parallel to a line connecting the responses for each level of
factor A at the low level of factor B. The nonparallel lines of Figure 15.1 suggest
the presence of interaction. To test whether this apparent interaction is signiﬁcant,
a third contrast in the treatment totals orthogonal to the main eﬀect contrasts,
called the interaction eﬀect, is constructed by evaluating
Interaction Eﬀect
AB =

ab − a − b + (1)
AB contrast
=
.
2n
2n

High Level of B

ab

Response

b

a

f B

lo
eve

L
Low
(1)
Low

High
Level of A

Figure 15.1: Response suggesting apparent interaction.
Example 15.1: Consider the data in Tables 15.2 and 15.3 with n = 1 for a 22 factorial experiment.
Table 15.2: 22 Factorial with No Interaction

Table 15.3: 22 Factorial with Interaction

B
A
+

50
80

B
+
70
100

A
+

50
80

+
70
40

The numbers in the cells in Tables 15.2 and 15.3 clearly illustrate how contrasts
and the resulting calculation of the two main eﬀects and resulting conclusions can
be highly inﬂuenced by the presence of interaction. In Table 15.2, the eﬀect of A
is −30 at both the low and high levels of factor B and the eﬀect of B is 20 at both
the low and high levels of factor A. This “consistency of eﬀect” (no interaction)
can be very important information to the analyst. The main eﬀects are
70 + 50 100 + 80

= 60 − 90 = −30,
2
2
100 + 70 80 + 50
B=

= 85 − 65 = 20,
2
2
A=

while the interaction eﬀect is
AB =

100 + 50 80 + 70

= 75 − 75 = 0.
2
2

15.2 The 2k Factorial: Calculation of Eﬀects and Analysis of Variance

601

On the other hand, in Table 15.3 the eﬀect of A is once again −30 at the low level
of B but +30 at the high level of B. This “inconsistency of eﬀect” (interaction)
also is present for B across levels of A. In these cases, the main eﬀects can be
meaningless and, in fact, highly misleading. For example, the eﬀect of A is
A=

50 + 70 80 + 40

= 0,
2
2

since there is a complete “masking” of the eﬀect as one averages over levels of B.
The strong interaction is illustrated by the calculated eﬀect
AB =

70 + 80 50 + 40

= 30.
2
2

80

Here it is convenient to illustrate the scenarios of Tables 15.2 and 15.3 with interaction plots. Note the parallelism in the plot of Figure 15.2 and the interaction
that is apparent in Figure 15.3.
100

B ϭ ϩ1
70

60

80

Response

70

B ϭ ϩ1

B ϭ Ϫ1
50

Response

90

B ϭ Ϫ1
50
Ϫ1

A

1

Figure 15.2: Interaction plot for data of
Table 15.2.

40

60

Ϫ1

1
A

Figure 15.3: Interaction plot for data of
Table 15.3.

Computation of Sums of Squares
We take advantage of the fact that in the 22 factorial, or for that matter in the
general 2k factorial experiment, each main eﬀect and interaction eﬀect has an associated single degree of freedom. Therefore, we can write 2k − 1 orthogonal
single-degree-of-freedom contrasts in the treatment combinations, each accounting
for variation due to some main or interaction eﬀect. Thus, under the usual independence and normality assumptions in the experimental model, we can make
tests to determine if the contrast reﬂects systematic variation or merely chance or
random variation. The sums of squares for each contrast are found by following
the procedures given in Section 13.5. Writing
Y1.. = b + (1),

Y2.. = ab + a,

c1 = −1,

and

c2 = 1,

Chapter 15 2k Factorial Experiments and Fractions

602

where Y1.. and Y2.. are the total of 2n observations, we have
2

2

ci Yi..
SSA = SSwA =

i=1
2

2n
i=1

=
c2i

[ab + a − b − (1)]2
(A contrast)2
=
,
2
2 n
22 n

with 1 degree of freedom. Similarly, we ﬁnd that
SSB =

(B contrast)2
[ab + b − a − (1)]2
=
2
2 n
22 n

and
SS(AB) =

[ab + (1) − a − b]2
(AB contrast)2
=
.
2
2 n
22 n

Each contrast has l degree of freedom, whereas the error sum of squares, with
22 (n − 1) degrees of freedom, is obtained by subtraction from the formula
SSE = SST − SSA − SSB − SS(AB).
In computing the sums of squares for the main eﬀects A and B and the interaction eﬀect AB, it is convenient to present the total responses of the treatment
combinations along with the appropriate algebraic signs for each contrast, as in
Table 15.4. The main eﬀects are obtained as simple comparisons between the low
and high levels. Therefore, we assign a positive sign to the treatment combination
that is at the high level of a given factor and a negative sign to the treatment
combination at the low level. The positive and negative signs for the interaction
eﬀect are obtained by multiplying the corresponding signs of the contrasts of the
interacting factors.
Table 15.4: Signs for Contrasts in a 22 Factorial Experiment
Treatment
Combination
(1)
a
b
ab

Factorial Eﬀect
A
B AB

+
+

+

+
+
+

The 23 Factorial
Let us now consider an experiment using three factors, A, B, and C, each with
levels −1 and +1. This is a 23 factorial experiment giving the eight treatment
combinations (1), a, b, c, ab, ac, bc, and abc. The treatment combinations and
the appropriate algebraic signs for each contrast used in computing the sums of
squares for the main eﬀects and interaction eﬀects are presented in Table 15.5.

15.2 The 2k Factorial: Calculation of Eﬀects and Analysis of Variance

603

Table 15.5: Signs for Contrasts in a 23 Factorial Experiment
Treatment
Combination
(1)
a
b
c
ab
ac
bc
abc

Factorial Eﬀect (symbolic)
A

+

+
+

+

B

+

+

+
+

C

+

+
+
+

AB
+

+
+

+

AC
+

+

+

+

BC
+
+

+
+

bc

ABC

+
+
+

+

abc

+1 c

ac

C
+1

B
(1)

−1

−1
−1

ab

b
a
A

+1

Figure 15.4: Geometric view of 23 .
It is helpful to discuss and illustrate the geometry of the 23 factorial much as
we illustrated that of the 22 factorial in Figure 15.1. For the 23 , the eight design
points represent the vertices of a cube, as shown in Figure 15.4.
The columns of Table 15.5 represent the signs that are used for the contrasts
and thus computation of seven eﬀects and corresponding sums of squares. These
columns are analogous to those given in Table 15.4 for the case of the 22 . Seven
eﬀects are available since there are eight design points. For example,
a + ab + ac + abc − (1) − b − c − bc
,
4n
(1) + c + ab + abc − a − b − ac − bc
,
AB =
4n
A=

and so on. The sums of squares are merely given by
SS(eﬀect) =

(contrast)2
.
23 n

An inspection of Table 15.5 reveals that for the 23 experiment all contrasts

604

Chapter 15 2k Factorial Experiments and Fractions
among the seven are mutually orthogonal, and therefore the seven eﬀects are assessed independently.

Eﬀects and Sums of Squares for the 2k
For a 2k factorial experiment the single-degree-of-freedom sums of squares for the
main eﬀects and interaction eﬀects are obtained by squaring the appropriate contrasts in the treatment totals and dividing by 2k n, where n is the number of
replications of the treatment combinations.
As before, an eﬀect is always calculated by subtracting the average response at
the “low” level from the average response at the “high” level. The high and low
for main eﬀects are quite clear. The symbolic high and low for interactions are
evident from information as in Table 15.5.
The orthogonality property has the same importance here as it does for the
material on comparisons discussed in Chapter 13. Orthogonality of contrasts implies that the estimated eﬀects and thus the sums of squares are independent. This
independence is readily illustrated in the 23 factorial experiment if the responses,
with factor A at its high level, are increased by an amount x in Table 15.5. Only
the A contrast leads to a larger sum of squares, since the x eﬀect cancels out in
the formation of the six remaining contrasts as a result of the two positive and two
negative signs associated with treatment combinations in which A is at the high
level.
out when we discuss the 2k factorial experiment in regression situations.

15.3

Nonreplicated 2k Factorial Experiment
The full 2k factorial may often involve considerable experimentation, particularly
when k is large. As a result, replication of each factor combination is often not
feasible. If all eﬀects, including all interactions, are included in the model of the
experiment, no degrees of freedom are allowed for error. Often, when k is large,
the data analyst will pool sums of squares and corresponding degrees of freedom
for high-order interactions that are known or assumed to be negligible. This will
produce F-tests for main eﬀects and lower-order interactions.

Diagnostic Plotting with Nonreplicated 2k Factorial Experiments
Normal probability plotting can be a very useful methodology for determining the
relative importance of eﬀects in a reasonably large two-level factored experiment
when there is no replication. This type of diagnostic plot can be particularly
useful when the data analyst is hesitant to pool high-order interactions for fear
that some of the eﬀects pooled in the “error” may truly be real eﬀects and not
merely random. The reader should bear in mind that all eﬀects that are not real
(i.e., they are independent estimates of zero) follow a normal distribution with
mean near zero and constant variance. For example, in a 24 factorial experiment,
we are reminded that all eﬀects (keep in mind that n = 1) are of the form
AB =

contrast
= y¯H − y¯L ,
8

15.3 Nonreplicated 2k Factorial Experiment

605

where y¯H is the average of eight independent experimental runs at the high, or
“+,” level and y¯L is the average of eight independent runs at the low, or “−,”
level. Thus, the variance of each contrast is Var(¯
yH − y¯L ) = σ 2 /4. For any
real eﬀects, E(¯
yH − y¯L ) = 0. Thus, normal probability plotting should reveal
“signiﬁcant” eﬀects as those that fall oﬀ the straight line that depicts realizations
of independent, identically distributed normal random variables.
The probability plotting can take one of many forms. The reader is referred to
Chapter 8, where these plots were ﬁrst presented. The empirical normal quantilequantile plot may be used. The plotting procedure that makes use of normal
probability paper may also be used. In addition, there are several other types
of diagnostic normal probability plots. In summary, the procedure for diagnostic
eﬀect plots is as follows.
Probability Eﬀect
Plots for
Nonreplicated 24
Factorial
Experiments

1. Calculate eﬀects as
eﬀect =

contrast
.
2k−1

2. Construct a normal probability plot of all eﬀects.
3. Eﬀects that fall oﬀ the straight line should be considered real eﬀects.
Further comments regarding normal probability plotting of eﬀects are in order.
First, the data analyst may feel frustrated if he or she uses these plots with a small
experiment. On the other hand, the plotting is likely to give satisfying results when
there is eﬀect sparsity—many eﬀects that are truly not real. This sparsity will be
evident in large experiments where high-order interactions are not likely to be real.

Case Study 15.1: Injection Molding: Many manufacturing companies in the United States and
abroad use molded parts as components. Shrinkage is often a major problem.
Often, a molded die for a part is built larger than nominal to allow for part shrinkage. In the following experimental situation, a new die is being produced, and
ultimately it is important to ﬁnd the proper process settings to minimize shrinkage. In the following experiment, the response values are deviations from nominal
(i.e., shrinkage). The factors and levels are as follows:

A.
B.
C.
D.

Injection velocity (ft/sec)
Mold temperature (◦ C)
Mold pressure (psi)
Back pressure (psi)

Coded Levels
−1
+1
2.0
1.0
150
100
1000
500
120
75

The purpose of the experiment was to determine what eﬀects (main eﬀects and
interaction eﬀects) inﬂuence shrinkage. The experiment was considered a preliminary screening experiment from which the factors for a more complete analysis
might be determined. Also, it was hoped that some insight might be gained into
how the important factors impact shrinkage. The data from a nonreplicated 24
factorial experiment are given in Table 15.6.

Chapter 15 2k Factorial Experiments and Fractions

606

Table 15.6: Data for Case Study 15.1
Factor
Combination
(1)
a
b
ab
c
ac
bc
abc

Response
(cm × 104 )
72.68
71.74
76.09
93.19
71.25
70.59
70.92
104.96

Factor
Combination
d
bd
abd
cd
acd
bcd
abcd

Response
(cm × 104 )
73.52
75.97
74.28
92.87
79.34
75.12
79.67
97.80

Initially, eﬀects were calculated and placed on a normal probability plot. The
calculated eﬀects are as follows:
BD = −2.2787,

B = 12.4463,

C = 2.4138,
AC = 1.2613,
CD = 1.4088,

D = 2.1438,
ABC = 2.8588,

AB = 11.4038,
BC = 1.8163,
ABD = −1.7813,

ACD = −3.0438,

BCD = −0.4788,

ABCD = −1.3063.

A = 10.5613,

The normal quantile-quantile plot is shown in Figure 15.5. The plot seems to
imply that eﬀects A, B, and AB stand out as being important. The signs of the
important eﬀects indicate the preliminary conclusions.
2
B

Theoretical Quantiles

AB

1

A
ABC
C
D
BC
CD
AC

0
BCD
ABCD
ABD

−1

BD

ACD

−2

−3

−1

1

3

5

7

9

11

13

Effects Quantiles

Figure 15.5: Normal quantile-quantile plot of eﬀects for Case Study 15.1.

15.3 Nonreplicated 2k Factorial Experiment

607

1. An increase in injection velocity from 1.0 to 2.0 increases shrinkage.
2. An increase in mold temperature from 100◦ C to 150◦ C increases shrinkage.
3. There is an interaction between injection velocity and mold temperature; although both main eﬀects are important, it is crucial that we understand the
impact of the two-factor interaction.

Interpretation of Two-Factor Interaction
As one would expect, a two-way table of means provides ease in interpretation of
the AB interaction. Consider the two-factor situation in Table 15.7.
Table 15.7: Illustration of Two-Factor Interaction
A (velocity)
2
1

B (temperature)
100
150
73.355
97.205
74.1975
75.240

Notice that the large sample mean at high velocity and high temperature created the signiﬁcant interaction. The shrinkage increases in a nonadditive
manner. Mold temperature appears to have a positive eﬀect despite the velocity
level. But the eﬀect is greatest at high velocity. The velocity eﬀect is very slight
at low temperature but clearly is positive at high mold temperature. To control
shrinkage at a low level, one should avoid using high injection velocity and high
mold temperature simultaneously. All of these results are illustrated graphically in
Figure 15.6.
100
2

Shrinkage

95
90

Velocity

85
80
1

75
70

100

150
Temperature

Figure 15.6: Interaction plot for Case Study 15.1.

Chapter 15 2k Factorial Experiments and Fractions

608

Analysis with Pooled Mean Square Error: Annotated Computer Printout
It may be of interest to observe an analysis of variance of the injection molding data
with high-order interactions pooled to form a mean square error. Interactions of
order three and four are pooled. Figure 15.7 shows a SAS PROC GLM printout.
The analysis of variance reveals essentially the same conclusion as that of the
normal probability plot.
The tests and P-values shown in Figure 15.7 require interpretation. A significant P-value suggests that the eﬀect diﬀers signiﬁcantly from zero. The tests on
main eﬀects (which in the presence of interactions may be regarded as the eﬀects
averaged over the levels of the other factors) indicate signiﬁcance for eﬀects A and
B. The signs of the eﬀects are also important. An increase in the level from low
The GLM Procedure
Dependent Variable: y
Sum of
Source
DF
Squares Mean Square F Value Pr > F
Model
10 1689.237462
168.923746
9.37 0.0117
Error
5
90.180831
18.036166
Corrected Total
15 1779.418294
R-Square
Coeff Var
Root MSE
y Mean
0.949320
5.308667
4.246901
79.99938
Source
DF
Type III SS
Mean Square F Value
Pr > F
A
1
446.1600062
446.1600062
24.74
0.0042
B
1
619.6365563
619.6365563
34.36
0.0020
C
1
23.3047563
23.3047563
1.29
0.3072
D
1
18.3826563
18.3826563
1.02
0.3590
A*B
1
520.1820562
520.1820562
28.84
0.0030
A*C
1
6.3630063
6.3630063
0.35
0.5784
A*D
1
13.3042562
13.3042562
0.74
0.4297
B*C
1
13.1950562
13.1950562
0.73
0.4314
B*D
1
20.7708062
20.7708062
1.15
0.3322
C*D
1
7.9383063
7.9383063
0.44
0.5364
Standard
Parameter
Estimate
Error t Value
Pr > |t|
Intercept
79.99937500
1.06172520
75.35
<.0001
A
5.28062500
1.06172520
4.97
0.0042
B
6.22312500
1.06172520
5.86
0.0020
C
1.20687500
1.06172520
1.14
0.3072
D
1.07187500
1.06172520
1.01
0.3590
A*B
5.70187500
1.06172520
5.37
0.0030
A*C
0.63062500
1.06172520
0.59
0.5784
A*D
-0.91187500
1.06172520
-0.86
0.4297
B*C
0.90812500
1.06172520
0.86
0.4314
B*D
-1.13937500
1.06172520
-1.07
0.3322
C*D
0.70437500
1.06172520
0.66
0.5364
Figure 15.7: SAS printout for data of Case Study 15.1.

Exercises

609
to high of A, injection velocity, results in increased shrinkage. The same is true for
B. However, because of the signiﬁcant interaction AB, main eﬀect interpretations
may be viewed as trends across the levels of the other factors. The impact of the
signiﬁcant AB interaction is better understood by using a two-way table of means.

Exercises
15.1 The following data are obtained from a 23 factorial experiment replicated three times. Evaluate the
sums of squares for all factorial eﬀects by the contrast
method. Draw conclusions.
Treatment
Combination Rep 1 Rep 2 Rep 3
(1)
12
19
10
a
15
20
16
b
24
16
17
ab
23
17
27
c
17
25
21
ac
16
19
19
bc
24
23
29
abc
28
25
20
15.2 In an experiment conducted by the Mining Engineering Department at Virginia Tech to study a particular ﬁltering system for coal, a coagulant was added to
a solution in a tank containing coal and sludge, which
was then placed in a recirculation system in order that
the coal could be washed. Three factors were varied in
the experimental process:
Factor A:
Factor B:
Factor C:

percent solids circulated initially
in the overﬂow
ﬂow rate of the polymer
pH of the tank

The amount of solids in the underﬂow of the cleansing system determines how clean the coal has become.
Two levels of each factor were used and two experimental runs were made for each of the 23 = 8 combinations. The response measurements in percent solids
by weight in the underﬂow of the circulation system
are as speciﬁed in the following table:
Treatment
Response
Combination Replication 1 Replication 2
5.81
4.65
(1)
21.35
21.42
a
12.56
12.66
b
16.62
18.27
ab
7.88
7.93
c
12.87
13.18
ac
6.26
6.51
bc
17.83
18.23
abc
Assuming that all interactions are potentially impor-

tant, do a complete analysis of the data. Use P-values
15.3 In a metallurgy experiment, it is desired to test
the eﬀect of four factors and their interactions on the
concentration (percent by weight) of a particular phosphorus compound in casting material. The variables
are A, percent phosphorus in the reﬁnement; B, percent remelted material; C, ﬂuxing time; and D, holding
time. The four factors are varied in a 24 factorial experiment with two castings taken at each factor combination. The 32 castings were made in random order. The
following table shows the data and an ANOVA table is
given in Figure 15.8 on page 610. Discuss the eﬀects of
the factors and their interactions on the concentration
of the phosphorus compound.
Weight
Treatment
% of Phosphorus Compound
Combination
Rep 1
Rep 2
Total
(1)
30.3
28.6
58.9
a
28.5
31.4
59.9
b
24.5
25.6
50.1
ab
25.9
27.2
53.1
c
24.8
23.4
48.2
ac
26.9
23.8
50.7
bc
24.8
27.8
52.6
abc
22.2
24.9
47.1
d
31.7
33.5
65.2
24.6
26.2
50.8
bd
27.6
30.6
58.2
abd
26.3
27.8
54.1
cd
29.9
27.7
57.6
acd
26.8
24.2
51.0
bcd
26.4
24.9
51.3
abcd
26.9
29.3
56.2
Total
428.1
436.9
865.0
15.4 A preliminary experiment is conducted to study
the eﬀects of four factors and their interactions on the
output of a certain machining operation. Two runs are
made at each of the treatment combinations in order to
supply a measure of pure experimental error. Two levels of each factor are used, resulting in the data shown
next page. Make tests on all main eﬀects and interactions at the 0.05 level of signiﬁcance. Draw conclusions.