CHAPTER 14
Design Of Experiments With
Several Factors
LEARNING OBJECTIVES
• Design and conduct engineering experiments
involving several factors
• Analyze and interpret main effects and
interactions
• How the ANOVA is used to analyze the data
• Use the two-level series of factorial designs
• Design and conduct two-level fractional
factorial designs
Factorial Experimental Design
• Performed in all engineering disciplines
– Learn about how systems and processes work
• Focus on experiments that include two or
more factors
• Experimental trials are performed at all
combinations of factor levels
• Single-factor experiments can be extended
to the factorial experiments
• ANOVA as the primary tools
FACTORIAL EXPERIMENTS
• Factorial experimental design should be used
• Mean that in each complete trial all possible
combinations of the levels
• Two factors A and B with a levels of factor A and b
levels of factor B
Effect of Factor
• Effect of a factor
• Called a main effect
• Consider the following
data
Factor B
Factor A
Blow
Bhigh
Alow
20
30
Ahigh
40
60
• Main effect of factor A
• Using the data
A=(40+60)/2–(20+30)/2= 25
• Changing factor A from
the low level to the high
level causes an
average response
increase of 25
• Main effect of B
B=(30+60)/2–(20+40)/2= 15
• Two factors, A and B,
each at two levels
Interaction Effect
• Difference in response
is not the same at all
levels
• When this occurs,
there is an interaction
between the factors
• Consider the following
data
Factor B
Factor A
Blow
Bhigh
Alow
20
30
Ahigh
40
10
• At low level of factor B,
A effect
A=40–20=20
• At high level of factor
B, A effect
A=10–30=-20
• There is interaction
between A and B
• Knowledge of the AB
interaction is more
useful
TWO-FACTOR FACTORIAL EXPERIMENTS
• Involves only two factors • Experiment has n
replicates
• a levels of factor A and
b levels of factor B
• Observation is
denoted by yijk
• The format of wo-factor
factorial
• abn observations
would be run in a
random order
• Two-factor factorial is
a completely
randomized design
Linear Statistical Model
•
Described by the linear statistical model
Page 511 Eq 14-1
•
•
•
•
•
Where µ is the overall mean effect
i is the effect of the i th level of factor A
βj is the effect of the jth level of factor B
(β)ij is the effect of interaction between A and B
εijk is a random error component
Testing The Hypotheses
• Interested in testing the hypotheses
• Analysis of variance (ANOVA) will be used to
test these hypotheses
• Test procedure is sometimes called the two-way
analysis of variance
• A and B are fixed factors
• Chosen by the experimenter
• Test hypotheses about the main factor effects of
A and B and the AB
• Need some symbols
Notation
• Let yi.. denote the total of
the observations taken at
the ith level of factor A
• y.j. denote the total of the
observations taken at the
jth level of factor B
• yij. denote the total of the
observations in the ij th
• y… denote the grand total
of all the observations
Pg 511
•
Define y... yi.., y.j., yij.,
and as the row,
column, cell, and grand
averages
Hypotheses
• Hypotheses that we will test
– No main effect of factor A
H o :  1   2  ...   a  0
H1 : at least one  i  0
– No main effect of factor B
H o : 1   2  ...   b  0
H1 : at least one  j  0
– No interaction
H o : ( )11  ( )12  ...  ( ) ab  0
H1 : at least one ( )ij  0
Pg 512 Eq 14-2
Total Variability
• ANOVA tests these hypotheses by decomposing
the total variability into component parts
• Total variability is measured by the total sum of
squares of the observations
a
b
n
SST   ( yijk  y...)2
i 1 j 1 k 1
• Sum of squares decomposition
Pg 512
Decomposition of SST
• SST is partitioned
– Sum of squares for the row factor A (SSA)
– Sum of squares for the column factor B (SSB)
– Sum of squares for the interaction between A and B
(SSAB)
– an error sum of squares (SSE)
•
•
•
•
abn - 1 total degrees of freedom
A and B have a - 1 and b - 1 degrees of freedom
AB has (a - 1)(b - 1) degrees of freedom
ab(n-1) degrees of freedom for error
Mean Squares
• If we divide each of the sum of squares by the
corresponding number of degrees of freedom
– Obtain the mean squares for A, B, the interaction,
and error
SS A
MS A 
a 1
SS B
MS B 
b 1
SS AB
MS AB 
(a  1)( b  1)
SS E
MS E 
ab( n  1)
Pg 513
Test Statistics
• Test that the row factor effects are all equal to zero
FO 
MS A
MS E
– F-distribution with a -1 and ab(n - 1) d.o.f
– Null hypothesis is rejected if fo > fα,a-1,ab(n-1)
• Test the hypothesis that all the column factor
effects are equal to zero
FO 
MS B
MS E
– F-distribution with b -1 and ab(n - 1) d.o.f
– Null hypothesis is rejected if fα,b-1,ab(n-1)
• Test that all interaction effects are zero
FO 
MS AB
MS E
– F-distribution with a -1 and ab(n - 1) d.o.f
– Null hypothesis is rejected if fo > fo > fα,(a-1)(b-1),ab(n-1)
ANOVA Table
• ANOVA table
Interaction or Main Effects?
• Conduct the test for interaction first
• Interpretation of the tests on the main
effects
• When interaction is significant
– Main effects of the factors may not have much
practical interpretative value
MINITAB Output
• Shows some of the output from the Minitab
• Upper portion gives factor name and level information
• Lower portion presents the analysis of variance
Pg 517 Table 14- 7
MINITAB Output
• Shows some of the output from the Minitab
• Upper portion gives factor name and level information
• Lower portion presents the analysis of variance
Example
• An engineer who suspects that the surface finish of metal
parts is influenced by the type of paint used and the drying
time. He selected three drying times—20, 25, and 30
minutes—and used two types of paint. Three parts are
tested with each combination of paint type and drying
time. The data are as follows:
• State and test the appropriate hypotheses using the
analysis of variance with α=0.05
Analysis of Variance
•
•
•
•
•
•
•
•
ANOVA Table
Test statistic for the factors are and 0.07
f0=1.90<f0.05,1,12 =4.75 and f0=0.07<f0.05,2,12 =3.89
Main effects do not affect surface finish
f0=5.03>f0.05,2,12= 3.89
Indication of interaction between these factors
Last column shows the P-value
P-values for the main effects >0.05, while the P-value for the
interaction <0.05
Source
DF
SS
MS
F
P
Paint
1
355.6
355.6
1.90 0.193
Drying
2
27.4
13.7
0.07 0.930
paint*drying 2
1878.8
939.4
5.03 0.026
Error
12
2242.7
186.9
Total
17
4504.4
More Than Two Factors
• Involve more than two factors
• a levels of factor A, b levels of factor B, c
levels of factor C, and so on
• abc n total observations
• Three main effects, three two-factor
interactions, a three-factor interaction, and
an error term
• Must be at least two replicates (n - 2) to
compute an error sum of squares
ANOVA Table
Example
• The percentage of hardwood concentration in raw pulp, the
freeness, and the cooking time of the pulp are being
investigated for their effects on the strength of paper
• The data from a three-factor factorial experiment are shown
in the following table
• Analyze the data using the analysis of variance assuming
that all factors are fixed. Use α=0.05.
Solution
• ANOVA table
Source
DF
SS
MS F
Hardwood
2
8.37
4.18 7.64
Cooking time
1
17.36
17.36 31.66
freeness
2
21.85
10.92 19.92
hardwood*cookingt 2
3.20
1.60 2.92
hardwood*freeness 4
6.51
1.62 2.97
cookingt*freeness 2
1.05
0.52 0.96
Error
22
12.06
0 .5484
Total
35
70.42
• All main factors are significant
• Interaction of hardwood*freeness is also significant
P
0.003
0.000
0.000
0.075
0.042
0.399
2k FACTORIAL DESIGNS
• Experiments involving several factors are
widely employed in research work
• k factors each at only two levels
• Levels may be quantitative or they may be
qualitative
• A complete replicate of such a design
requires 2 x 2 x …x 2 = 2k observations
• 2k design is particularly useful
• Provides the smallest number of runs
22 Design
• Simplest type of 2k design is the 22
• Think of these levels as the low and high levels of the factor
• Represented geometrically as a square with the 22=4 runs
• Denote the levels of the factors A and B by the signs - and +
• Called the geometric notation for the design
Main and Interaction Effects
• Effects of interest are A
and B and AB
• Let (1), a, b, and ab
represent the totals of all
n observations
• Main effect of A
• Main effect of B
• AB interaction
• Quantities in
brackets are called
contrasts
• A Contrast
Contrast A=a+ab-b-(1)
How to set up contrasts
• Make a table where the rows are treatment
combinations ((1), a, b, ab, etc.) and the columns
are factorial effects (A, B, AB, etc.)
• For each treatment combination, write a plus sign
for that factorial effect if it's high and a minus if it's
low
• To get interaction effects, multiply the signs
together like arithmetic
– if two signs are the same, their product is a plus sign,
and if not, their product is a minus sign
Signs for Effects
• A table of plus and minus
signs can be used to
determine the sign on each
treatment
• Column headings are the
main effects A and B, the
AB interaction, and I
• Row headings are the
treatment combinations
• Note that the signs in the
AB column
• Multiply the signs in the
appropriate column by the
treatment combinations
• Note that the signs in the
AB column
• Multiply the signs in the
appropriate column by the
treatment combinations
Analysis of Variance
• Effect estimates and the sums of squares for A, B,
and the AB interaction
• Sums of squares
Pg 526 Eq 14-14
• Analysis of variance
– SST with 4n - 1 d.of.
– SSE with 4(n -1) d.o.f
k
2 Design
for k>3 Factors
• Methods presented in the
previous section can be easily
extended to more than two
factors
• Consider k=3 factors, each at
two levels
• 23 factorial design and it has
eight runs
• Geometrically, the design is a
cube with the eight runs
• Lists the eight runs with each
row representing one of the
runs
Developing the Signs
• Denote factors with capital letters as usual: A, B, C, etc.
• Denote the "high" level by its associated letter (a, b, c, etc.)
and the "low" level as (1)
• Make a table where the rows are (1), a, b, ab, etc. and the
columns are A, B, AB, etc.
• “+” for the factorial effect if it's high and “-” if it's low
• Get interaction effects, multiply the signs together
• AB column are the products of the A and B column signs
• If two signs are the same, their product is a plus sign
Calculating the Effects
• Letters (1), a, b, ab, c, ac, bc, and abc
• Main and interaction effects
A = 1/4n [a+ab+ac+abc-(1)-b-c-bc]
B = 1/4n [b+ab+bc+abc-(1)-a-c-ac]
C = 1/4n [c+ac+bc+abc-(1)-a-b-ab]
AB = 1/4n [ abc-bc+ab-b-ac+c-a+(1)]
AC = 1/4n [(1)-a+b-ab-c+ac-bc+abc]
BC = 1/4n [(1)+a-b-ab-c-ac+bc+abc]
ABC= 1/4n[abc-bc-ac+c-ab+b+a-(1)]
• Quantities in brackets are contrasts
Effect Estimates and SS
• Effect estimates are computed from
• Sum of squares for any effect
Example
• An engineer is interested in the effect of cutting speed (A),
metal hardness (B), and cutting angle (C) on the life of a
cutting tool
• Two levels of each factor are chosen, and two replicates
of a 23 factorial design are run
• The tool life data (in hours) are shown in the following
table:
• Analyze the data from this experiment.
Calculations
• Main effects are estimated
Source
dof
SS
MS
F
P
A=1/4n [ a+ab+ac+abc-(1)-b-c-bc]
=1/8(325+435)+(552+472)+(406
+377)+(392+419)-(221+311)354+348)-(440+453)
(605+550)=146
Speed
1
1332
1322
0.49
0.502
Hardness
1
28932
28932
10.42
0.010
Angle
1
20592
20592
7.56
0.023
Speed,
Hardness
1
506
506
0.19
0.677
Speed,
Angle
1
56882
56882
20.87
0.000
Hardness,
Angle
1
2352
2352
0.86
0.377
Error
9
24530
2726
Total
15
134588
• Sum of squares for A
SSSPEED =(146)2/16 = 1332
• Easy to verify that the other
effects
Solution
• ANOVA table
Source
Speed
Hardness
Angle
speed*hardness
speed*angle
hardness*angle
Error
Total
DF
1
1
1
1
1
1
9
15
SS
1332
28392
20592
506
56882
2352
24530
134588
MS
1332
28392
20592
506
56882
2352
2726
F
0.49
10.42
7.56
0.19
20.87
0.86
P
0.502
0.010
0.023
0.677
0.000
0.377
Next Agenda
• Methods and applications of
nonparametric statistics