1. Design Of Experiments With Several Factors
CHAPTER 14
Design Of Experiments With Several Factors

2. 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

3. 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

4. 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

5. Effect of Factor Effect of a factor Main effect of factor A
Called a main effect
Consider the following data
Two factors, A and B, each at two levels
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
Factor B
Factor A
Blow
Bhigh
Alow 20 30
Ahigh 40 60

6. 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
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
Factor B
Factor A
Blow
Bhigh
Alow 20 30
Ahigh 40 10

7. TWO-FACTOR FACTORIAL EXPERIMENTS
Involves only two factors
a levels of factor A and b levels of factor B
The format of wo-factor factorial
Experiment has n replicates
Observation is denoted by yijk
abn observations would be run in a random order
Two-factor factorial is a completely randomized design

8. Linear Statistical Model
Described by the linear statistical model
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
Page 511 Eq 14-1

9. 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

10. Notation
Define y... yi.., y.j., yij., and as the row, column, cell, and grand averages
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

11. Hypotheses Hypotheses that we will test No main effect of factor A
No main effect of factor B
No interaction
Pg 512 Eq 14-2

12. 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
Sum of squares decomposition
Pg 512

13. 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

14. 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
Pg 513

15. Test Statistics Test that the row factor effects are all equal to zero
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
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
Null hypothesis is rejected if fo > fo > fα,(a-1)(b-1),ab(n-1)

16. ANOVA Table
ANOVA table

17. 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

18. 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

19. 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

20. 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

21. 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
Drying
paint*drying
Error
Total

22. 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

23. ANOVA Table

24. 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.

25. Solution ANOVA table Source DF SS MS F P
Hardwood
Cooking time
freeness
hardwood*cookingt
hardwood*freeness
cookingt*freeness
Error
Total
All main factors are significant
Interaction of hardwood*freeness is also significant

26. 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

27. 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

28. 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)

29. 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

30. 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

31. Analysis of Variance
Effect estimates and the sums of squares for A, B, and the AB interaction
Sums of squares
Analysis of variance
SST with 4n - 1 d.of.
SSE with 4(n -1) d.o.f
Pg 526 Eq 14-14

32. 2k 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

33. 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

34. 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

35. Effect Estimates and SS
Effect estimates are computed from
Sum of squares for any effect

36. 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.

37. Calculations SSSPEED =(146)2/16 = 1332 Main effects are estimated
A=1/4n [ a+ab+ac+abc-(1)-b-c-bc]
=1/8( )+( )+( )+( )-( ) )-( ) ( )=146
Sum of squares for A
SSSPEED =(146)2/16 = 1332
Easy to verify that the other effects
Source
dof SS MS F P
Speed 1
1332
1322
0.49
0.502
Hardness
28932
10.42
0.010
Angle
20592
7.56
0.023
Speed,
506
0.19
0.677
56882
20.87
0.000
Hardness,
2352
0.86
0.377
Error 9
24530
2726
Total 15
134588

38. Solution ANOVA table Source DF SS MS F P Speed 1 1332 1332 0.49 0.502
Hardness
Angle
speed*hardness
speed*angle
hardness*angle
Error
Total

39. Next Agenda
Methods and applications of nonparametric statistics