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. Experiments with Blocking Factors
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 The Randomized Complete Block Design The Latin Square Design
The Graeco-Latin Square Design
Balanced Incomplete Block Design

4. The Randomized Complete Block Design
In some experiment, the variability may arise from factors that we are not interested in.
A nuisance factor (擾亂因子)is a factor that probably has some effect on the response, but it’s of no interest to the experimenter … however, the variability it transmits to the response needs to be minimized
These nuisance factor could be unknown and uncontrolled  use randomization

5. The Randomized Complete Block Design
If the nuisance factor are known but uncontrollable  use the analysis of covariance.
If the nuisance factor are known but controllable  use the blocking technique
Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units

6. The Randomized Complete Block Design
Many industrial experiments involve blocking (or should)
Failure to block is a common flaw in designing an experiment (consequences?)

7. The Randomized Complete Block Design-example
We wish determine whether or not four different tips produce different readings on a hardness testing machine.
One factor to be consider  tip type
Completely Randomized Design could be used with one potential problem  the testing block could be different
The experiment error could include both the random and coupon errors.

8. The Randomized Complete Block Design-example
To reduce the error from testing coupon, randomize complete block design(RCBD) is used

9. The Randomized Complete Block Design-example
Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips
“complete” indicates each testing coupon (BLOCK) contains all treatments
Variability between blocks can be large, variability within a block should be relatively small
In general, a block is a specific level of the nuisance factor

10. The Randomized Complete Block Design-example
A complete replicate of the basic experiment is conducted in each block
A block represents a restriction on randomization
All runs within a block are randomized
Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)

11. The Randomized Complete Block Design– Extension from ANOVA
Suppose that there are a treatments (factor levels) and b blocks

12. The Randomized Complete Block Design– Extension from ANOVA
Suppose that there are a treatments (factor levels) and b blocks

13. The Randomized Complete Block Design– Extension from ANOVA
A statistical model (effects model) for the RCBD is
The relevant (fixed effects) hypotheses are

14. The Randomized Complete Block Design– Extension from ANOVAOr

15. The Randomized Complete Block Design– Extension from ANOVA
Partitioning the total variability

16. The Randomized Complete Block Design– Extension from ANOVA
The degrees of freedom for the sums of squares in
are as follows:
Therefore, ratios of sums of squares to their degrees of freedom result in mean squares and the ratio of the mean square for treatments to the error mean square is an F statistic that can be used to test the hypothesis of equal treatment means

17. The Randomized Complete Block Design– Extension from ANOVA
Mean squares

18. The Randomized Complete Block Design– Extension from ANOVA
F-test with (a-1), (a-1)(b-1) degree of freedom
Reject the null hypothesis if
F0>F α,a-1,(a-1)(b-1)

19. The Randomized Complete Block Design– Extension from ANOVA
ANOVA Table

20. The Randomized Complete Block Design– Extension from ANOVA
Manual computing:

21. The Randomized Complete Block Design– Extension from ANOVA
Meaning of F0=MSBlocks/MSE?
The randomization in RBCD is applied only to treatment within blocks
The Block represents a restriction on randomization
Two kinds of controversial theories

22. The Randomized Complete Block Design– Extension from ANOVA
Meaning of F0=MSBlocks/MSE?
General practice, the block factor has a large effect and the noise reduction obtained by blocking was probably helpful in improving the precision of the comparison of treatment means if the ration is large

23. The Randomized Complete Block Design– Example

24. The Randomized Complete Block Design– Example
To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin
Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures

25. The Randomized Complete Block Design– Example—Minitab
Vascular-Graft.MTW
The Randomized Complete Block Design– Example—Minitab
StatANOVATwo-way
Two-way ANOVA: Yield versus Pressure, Batch
Source DF SS MS F P
Pressure
Batch
Error
Total
S = R-Sq = 77.12% R-Sq(adj) = 64.92%

26. The Randomized Complete Block Design– Example—Minitab
Vascular-Graft.MTW
The Randomized Complete Block Design– Example—Minitab

27. The Randomized Complete Block Design– Example —Residual Analysis
Basic residual plots indicate that normality, constant variance assumptions are satisfied
No obvious problems with randomization
No patterns in the residuals vs. block
Can also plot residuals versus the pressure (residuals by factor)
These plots provide more information about the constant variance assumption, possible outliers

28. The Randomized Complete Block Design– Example—Minitab
Vascular-Graft.MTW
The Randomized Complete Block Design– Example—Minitab

29. The Randomized Complete Block Design– Example —No Blocking
Vascular-Graft.MTW
The Randomized Complete Block Design– Example —No Blocking
StatANOVAOne-way
One-way ANOVA: Yield versus Pressure
Source DF SS MS F P
Pressure
Error
Total
S = R-Sq = 37.10% R-Sq(adj) = 27.66%

30. The Randomized Complete Block Design– Example—No Blocking-Residual
Vascular-Graft.MTW
The Randomized Complete Block Design– Example—No Blocking-Residual

31. The Randomized Complete Block Design– Other Example

32. The Randomized Complete Block Design– Other Example

33. The Randomized Complete Block Design– Other Example
Blocking effect
Without blocking effect
Two-way ANOVA: 濃度 versus 化學品類別, 樣品
Source DF SS MS F P
化學品類別 樣品
Error
Total
S = R-Sq = 96.30% R-Sq(adj) = 94.14%
One-way ANOVA: 濃度 versus 化學品類別
Source DF SS MS F P
化學品類別
Error
Total
S = R-Sq = 70.24% R-Sq(adj) = 64.66%

34. The Randomized Complete Block Design– Other Example
Blocking effect
Without blocking effect

35. The Randomized Complete Block Design– Other Aspects
The RCBD utilizes an additive model – no interaction between treatments and blocks
Treatments and/or blocks as random effects
Missing values
What are the consequences of not blocking if we should have?

36. The Randomized Complete Block Design– Other Aspects
Sample sizing in the RCBD? The OC curve approach can be used to determine the number of blocks to run..see page 133

37. The Latin Square Design
These designs are used to simultaneously control (or eliminate) two sources of nuisance variability
Those two sources of nuisance factors have exactly same levels of factor to be considered
A significant assumption is that the three factors (treatments, nuisance factors) do not interact
If this assumption is violated, the Latin square design will not produce valid results
Latin squares are not used as much as the RCBD in industrial experimentation

38. The Latin Square Design
The Latin square design systematically allows blocking in two directions
In general, a Latin square for p factors is a square containing p rows and p columns.
Each cell contain one and only one of p letters that represent the treatments.

39. A Latin Square Design – The Rocket Propellant
This is a

40. Statistical Analysis of the Latin Square Design
The statistical (effects) model is
The statistical analysis (ANOVA) is much like the analysis for the RCBD.

41. Statistical Analysis of the Latin Square Design

42. Statistical Analysis of the Latin Square Design

43. The Standard Latin Square Design
A square with first row and column in alphabetical order.

44. Other Topics Missing values in blocked designs RCBD Latin square
Estimated by

45. Other Topics Replication of Latin Squares
To increase the error degrees of freedom
Three methods
1. Use the same batches and operators in each replicate
2. Use the same batches but different operators in each replicate
3. Use different batches and different operator

46. Other Topics
Replication of Latin Squares
ANOVA in Case 1

47. Other Topics
Replication of Latin Squares
ANOVA n Case 2

48. Other Topics
Replication of Latin Squares
ANOVA n Case 3

49. Other Topics Crossover design
p treatments to be tested in p time periods using np experiment units.
Ex : 20 subjects to be assigned to two periods
First half of the subjects are assigned to period 1 (in random) and the other half are assigned to period 2.
Take turn after experiments are done.

50. Other Topics
Crossover design
ANOVA

51. Graeco-Latin Square
For a pxp Latin square, one can superimpose a second pxp Latin square that treatments are denoted by Greek letters.

52. Graeco-Latin Square
If the two squares have the property that each Greek letter appears once and only once with each Latin letter, the two Latin squares are to be orthogonal and this design is named as Graeco-Latin Square.
It can control three sources of extraneous variability.

53. Graeco-Latin Square
ANOVA

54. Graeco-Latin Square --Example
In the rocket propellant problem, batch of material, operators, and test assemblies are important.
If 5 of them are considered, a Graeco-Latn square can be used.

55. Graeco-Latin Square --Example

56. Graeco-Latin Square --Example