Design and Analysis of Experiments

1. Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC

2. Experiments with Blocking Factors Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC

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 nuisancefactor (擾亂因子)is a factor that probably has some effect on the response, but it’s of no interest to theexperimenter … 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 • Manyindustrial 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 bblocks

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

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 ANOVA • Or

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>Fα,a-1,(a-1)(b-1)

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

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

21. The Randomized CompleteBlock 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 CompleteBlock 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 CompleteBlock 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. Vascular-Graft.MTW • The Randomized Complete Block Design– Example—Minitab Two-way ANOVA: Yield versus Pressure, Batch Source DF SS MS F P Pressure 3 178.171 59.3904 8.11 0.002 Batch 5 192.252 38.4504 5.25 0.006 Error 15 109.886 7.3258 Total 23 480.310 S = 2.707 R-Sq = 77.12% R-Sq(adj) = 64.92% StatANOVATwo-way

26. 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 varianceassumptions are satisfied • No obvious problems with randomization • No patterns in theresiduals 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. Vascular-Graft.MTW • The Randomized Complete Block Design– Example—Minitab

29. Vascular-Graft.MTW • The Randomized Complete Block Design– Example —No Blocking One-way ANOVA: Yield versus Pressure Source DF SS MS F P Pressure 3 178.2 59.4 3.93 0.023 Error 20 302.1 15.1 Total 23 480.3 S = 3.887 R-Sq = 37.10% R-Sq(adj) = 27.66% StatANOVAOne-way

30. 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 3 18.044 6.01467 75.89 0.000 樣品 4 6.693 1.67325 21.11 0.000 Error 12 0.951 0.07925 Total 19 25.688 S = 0.2815 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 3 18.044 6.015 12.59 0.000 Error 16 7.644 0.478 Total 19 25.688 S = 0.6912 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 consequencesof not blockingif we should have?

36. The Randomized Complete Block Design– Other Aspects • Sample sizingin the RCBD? The OC curveapproach 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