1. Design and Analysis of Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
This is a basic course blah, blah, blah…

2. Factorial Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
This is a basic course blah, blah, blah…

3. Outline Basic Definition and Principles The Advantages of Factorials
The Two Factors Factorial Design
The General Factorial Design
Fitting Response Curve and Surfaces
Blocking in Factorial Design

4. Basic Definitions and Principles
Factorial Design—all of the possible combinations of factors’ level are investigated
When factors are arranged in factorial design, they are said to be crossed
Main effects – the effects of a factor is defined to be changed
Interaction Effect – The effect that the difference in response between the levels of one factor is not the same at all levels of the other factors.

5. Basic Definitions and Principles
Factorial Design without interaction

6. Basic Definitions and Principles
Factorial Design with interaction

7. Basic Definitions and Principles
Average response – the average value at one factor’s level
Average response increase – the average value change for a factor from low level to high level
No Interaction:

8. Basic Definitions and Principles
With Interaction:

9. Basic Definitions and Principles
Another way to look at interaction:
When factors are quantitative
In the above fitted regression model, factors are coded in (-1, +1) for low and high levels
This is a least square estimates

10. Basic Definitions and Principles
Since the interaction is small, we can ignore it.
Next figure shows the response surface plot

11. Basic Definitions and Principles
The case with interaction

12. Advantages of Factorial design
Efficiency
Necessary if interaction effects are presented
The effects of a factor can be estimated at several levels of the other factors

13. The Two-factor Factorial Design
Two factors
a levels of factor A, b levels of factor B
n replicates
In total, nab combinations or experiments

14. The Two-factor Factorial Design – An example
Two factors, each with three levels and four replicates
32 factorial design

15. The Two-factor Factorial Design – An example
Questions to be answered:
What effects do material type and temperature have on the life the battery
Is there a choice of material that would give uniformly long life regardless of temperature?

16. The Two-factor Factorial Design
Statistical (effects) model:
Means model
Regression model
𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 12 𝑥 12 +𝜖

17. The Two-factor Factorial Design
Hypothesis
Row effects:
Column effects:
Interaction:

18. The Two-factor Factorial Design -- Statistical Analysis

19. The Two-factor Factorial Design -- Statistical Analysis
Mean square: A: B:
Interaction:

20. The Two-factor Factorial Design -- Statistical Analysis
Mean square:
Error:

21. The Two-factor Factorial Design -- Statistical Analysis
ANOVA table

22. The Two-factor Factorial Design -- Statistical Analysis
Example—Batery-life.mtw

23. The Two-factor Factorial Design -- Statistical Analysis
STATANOVAGeneral Linear Model
Fit General Linear Model

24. The Two-factor Factorial Design -- Statistical Analysis
Model….
Select Temp and Material then click Add
OK, OK

25. The Two-factor Factorial Design -- Statistical Analysis
Example
General Linear Model: Life versus Temp, Material
Method
Factor coding (-1, 0, +1)
Factor Information
Factor Type Levels Values
Temp Fixed , 70, 125
Material Fixed , 2, 3
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Temp
Material
Temp*Material
Error
Total

26. The Two-factor Factorial Design -- Statistical Analysis
Example
General Linear Model: Life versus Temp, Material
Model Summary
S R-sq R-sq(adj) R-sq(pred)
% % %
Coefficients
Term Coef SE Coef T-Value P-Value VIF
Constant
Temp
Material
Temp*Material

27. The Two-factor Factorial Design -- Statistical Analysis
Example
General Linear Model: Life versus Temp, Material
Regression Equation
Life = Temp_ Temp_ Temp_125
Material_ Material_ Material_3
Temp*Material_ Temp*Material_15 2
Temp*Material_ Temp*Material_70 1
Temp*Material_ Temp*Material_70 3
Temp*Material_ Temp*Material_125 2
Temp*Material_125 3
Fits and Diagnostics for Unusual Observations
Obs Life Fit Resid Std Resid R R
R Large residual

28. The Two-factor Factorial Design -- Statistical Analysis
Example

29. The Two-factor Factorial Design -- Statistical Analysis
Example

30. The Two-factor Factorial Design -- Statistical Analysis
Example STATANOVA--GLM

31. The Two-factor Factorial Design -- Statistical Analysis
Example STATANOVA--GLM

32. The Two-factor Factorial Design -- Statistical Analysis
Estimating the model parameters

33. The Two-factor Factorial Design -- Statistical Analysis
Choice of sample size
Row effects
Column effects
Interaction effects
D:difference, :standard deviation

34. The Two-factor Factorial Design -- Statistical Analysis

35. The Two-factor Factorial Design -- Statistical Analysis
Appendix Chart V
For n=4, giving D=40 on temperature, v1=2, v2=27, Φ 2 =1.28n. β =0.06 n Φ2 Φ υ1 υ2 β 2
2.56
1.6 9
0.45 3
3.84
1.96 18
0.18 4
5.12
2.26 27
0.06

36. The Two-factor Factorial Design -- Statistical Analysis – example with no interaction
Analysis of Variance for Life, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Material
Temp
Error
Total
S = R-Sq = 64.14% R-Sq(adj) = 59.51%

37. The Two-factor Factorial Design – One observation per cell
Single replicate
The effect model

38. The Two-factor Factorial Design – One observation per cell
ANOVA table

39. The Two-factor Factorial Design -- One observation per cell
The error variance is not estimable unless interaction effect is zero
Needs Tuckey’s method to test if the interaction exists.
Check page 183 for details.

40. The General Factorial Design
In general, there will be abc…n total observations if there are n replicates of the complete experiment.
There are a levels for factor A, b levels of factor B, c levels of factor C,..so on.
We must have at least two replicate (n≧2) to include all the possible interactions in model.

41. The General Factorial Design
If all the factors are fixed, we may easily formulate and test hypotheses about the main effects and interaction effects using ANOVA.
For example, the three factor analysis of variance model:

42. The General Factorial Design
ANOVA.

43. The General Factorial Design
where

44. The General Factorial Design --example
Three factors: pressure, percent of carbonation, and line speed.

45. The General Factorial Design --example
STATANOVAGeneral Linear Model
Fit General Linear Model
Model…three factors, up to 3 level

46. The General Factorial Design --example
General Linear Model: Deviation versus Speed, Carbonation, Pressure
Method
Factor coding (-1, 0, +1)
Factor Information
Factor Type Levels Values
Speed Fixed , 250
Carbonation Fixed , 12, 14
Pressure Fixed , 30
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Speed
Carbonation
Pressure
Speed*Carbonation
Speed*Pressure
Carbonation*Pressure
Speed*Carbonation*Pressure
Error
Total

47. The General Factorial Design --example
ANOVA

48. Fitting Response Curve and Surfaces
When factors are quantitative, one can fit a response curve (surface) to the levels of the factor so the experimenter can relate the response to the factors.
These surface could be linear or quadratic.
Linear regression model is generally used

49. Fitting Response Curve and Surfaces -- example
Battery life data
Factor temperature is quantitative

50. Fitting Response Curve and Surfaces -- example
Example STATANOVA—GLM
Response  Life
Factors: Material
Covariates  temp
Model … interactions, 2, add
terms, 2 , add
cross, Add
temp, material, temp*temp, material*temp, material*temp*temp

51. Fitting Response Curve and Surfaces -- example
General Linear Model: Life versus Temp, Material
Method
Factor coding (-1, 0, +1)
Factor Information
Factor Type Levels Values
Material Fixed , 2, 3
Analysis of Variance
Source DF Seq SS Seq MS F-Value P-Value
Temp
Material
Temp*Temp
Temp*Material
Temp*Temp*Material
Error
Total

52. Fitting Response Curve and Surfaces -- example
coding method: -1, 0, +1
General Linear Model: Life versus Material
Factor Type Levels Values
Material fixed , 2, 3
Analysis of Variance for Life, using Sequential SS for Tests
Source DF Seq SS Adj SS Seq MS F P
Temp
Material
Temp*Temp
Material*Temp
Material*Temp*Temp
Error
Total
S = R-Sq = 76.52% R-Sq(adj) = 69.56%
Term Coef SE Coef T P
Constant
Temp
Temp*Temp
Temp*Material
Temp*Temp*Material
Two kinds of coding methods:
1, 0, -1
0, 1, -1

53. Fitting Response Curve and Surfaces -- example
Model Summary
S R-sq R-sq(adj) R-sq(pred)
% % %
Coefficients
Term Coef SE Coef T-Value P-Value VIF
Constant
Temp
Material
Temp*Temp
Temp*Material
Temp*Temp*Material
Regression Equation
Life = Temp Temp*Temp
Life = Temp Temp*Temp
Life = Temp Temp*Temp

54. Fitting Response Curve and Surfaces -- example
Final regression equation:

55. Fitting Response Curve and Surfaces – example –32 factorial design
Tool life
Factors: cutting speed, total angle
Data are coded

56. Fitting Response Curve and Surfaces – example –32 factorial design

57. Fitting Response Curve and Surfaces – example –32 factorial design
STATRegressionRegression
Fit Regression Model
Response  Life
Continuous predictors: Speed, Angle
Model…select: Speed, Angle
 interaction, 2,Add
Terms, 2, Add
cross, Add
Speed, Angle, Speed*Speed, Angle*Angle, Speed*Angle, Speed*Speed*Angle, Speed*Angle*Angle Speed*Speed*Angle*Angle

58. Fitting Response Curve and Surfaces – example –32 factorial design
Results for: Tool-Life.MTW
Regression Analysis: Life versus Speed, Angle
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Regression
Speed
Angle
Speed*Speed
Angle*Angle
Speed*Angle
Speed*Speed*Angle
Speed*Angle*Angle
Speed*Speed*Angle*Angle
Error
Total

59. Fitting Response Curve and Surfaces – example –32 factorial design
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total
Source DF Seq SS
Speed
Angle
Angle*Angle
Speed*Speed
Angle*Speed
Angle*Speed*Speed
Angle*Angle*Speed
Angle*Angle*Speed*Speed

60. Fitting Response Curve and Surfaces – example –32 factorial design
StatRegressionregressioncontour plot

61. Fitting Response Curve and Surfaces – example –32 factorial design
StatRegressionregressionsurface plot

62. Blocking in a Factorial Design
We may have a nuisance factor presented in a factorial design
Original two factor factorial model:
Two factor factorial design with a block factor model:

63. Blocking in a Factorial Design

64. Blocking in a Factorial Design – example—target-detection.mtw
Response: intensity level
Factors: Ground cutter and filter type
Block factor: Operator

65. Blocking in a Factorial Design -- example
Results for: Target-Detection.MTW
General Linear Model: Intensity versus Blocks, Filter, Clutter
Method
Factor coding (-1, 0, +1)
Factor Information
Factor Type Levels Values
Blocks Fixed , 2, 3, 4
Filter Fixed , 2
Clutter Fixed High, Low, Medium
Analysis of Variance
Source DF Seq SS Seq MS F-Value P-Value
Blocks
Filter
Clutter
Filter*Clutter
Error
Total
Model Summary
S R-sq R-sq(adj) R-sq(pred)
% % %

66. Blocking in a Factorial Design -- example
Coefficients
Term Coef SE Coef T-Value P-Value VIF
Constant
Blocks
Filter
Clutter
High
Low
Filter*Clutter
1 High
1 Low
Regression Equation
Intensity = Blocks_ Blocks_ Blocks_3
Blocks_ Filter_ Filter_2
Clutter_High Clutter_Low Clutter_Medium
Filter*Clutter_1 High Filter*Clutter_1 Low
Filter*Clutter_1 Medium Filter*Clutter_2 High
Filter*Clutter_2 Low Filter*Clutter_2 Medium