2DS01

1. 2DS01 Statistics 2 for Chemical Engineering lecture 2 http://www.win.tue.nl/~sandro/2DS01

2. Contents • why is design of experiments useful? • regression analysis and effects • 2p-experiments • blocks • 2p-k-experiments (fractional factorial experiments) • software • literature

3. Traditional approach to experimentation • change setting of one factor • perform measurement(s) • change setting of another factor • perform measement(s) • ... This is called a One-Factor-At-a-Time (OFAT) or Change-One-Separate-factor-at-a-Time (COST) strategy.

4. Use of statistics in design of experiments Design of Experiments (short: DOE) is a general term for a collection of statistical techniques for systematic experimentation. The most important benefits of using DOE are: • lessexperiments are needed • more precise estimates of parameter effects • interactions between factors are taken into account

5. Interactions Factors mayinfluence each other. E.g, the optimal setting of a factor may depend on the settings of the other factors. When factors are optimised separately, the overall result (as function of all factors) may be suboptimal ...

6. The real maximum 30 40 50 60 factor B has been optimised The apparent maximum factor A has been optimised

7. Types of experimental designs • “screening designs” These designs are used to investigate which factors are important (“significant”). • “response surface designs” These designs are used to determine the optimal settings of the significant factors.

8. Three factors: example Response: deviation filling height bottles Factors: carbon dioxide level (%) A pressure (psi) B speed (bottles/min) C

9. Effects How do we determine whether an individual factor is of importance? Measure the outcome at 2 different settings of that factor. Scale the settings such that they become the values +1 and -1.

10. measurement -1 +1 setting factor A

11. measurement -1 +1 setting factor A

12. effect measurement -1 +1 setting factor A

13. effect slope measurement -1 +1 setting factor A N.B. effect = 2 * slope

14. Effect factor A = 50 – 35 = 15 50 measurement 35 -1 +1 setting factor A

15. More factors We denote factors with capitals: A, B,… Each factor only attains two settings: -1 and +1 The joint settings of all factors in one measurement is called a level combination.

16. More factors Level Combination

17. Notation A level combination consists of small letters. The small letters denote which factors are set at +1; the letters that do not appear are set at -1. Example: ac means: A and C at 1, the remaining factors at -1 N.B. (1) means that all factors are set at -1.

18. An experiment consists of performing measurements at different level combinations. A run is a measurement at one level combination. Suppose that there are 2 factors, A and B. • We perform 4 measurements with the following settings: • A -1 and B -1 (short: (1) ) • A +1 and B -1 (short: a ) • A -1 and B +1 (short: b ) • A +1 and B +1 (short: ab )

19. A 22 Experiment with 4 runs

20. Note: CAPITALS for factors and effects (A, BC, CDEF) small letters for level combinations ( = settings of the experiments) (a, bc, cde, (1))

21. Graphical display b ab +1 B -1 (1) a -1 +1 A

22. 40 60 +1 B -1 35 50 A -1 +1

23. 40 60 +1 B -1 35 50 A -1 +1 2 estimates for effect A:

24. 40 60 +1 B -1 35 50 A -1 +1 2 estimates for effect A: 50 - 35 = 15

25. 40 60 +1 B -1 35 50 A -1 +1 2 estimates for effect A: 50 - 35 = 15 60 - 40 = 20

26. 40 60 +1 B -1 35 50 A -1 +1 2 estimates for effect A: 50 - 35 = 15 60 - 40 = 20 Which estimate is superior?

27. 40 60 +1 B -1 35 50 A -1 +1 2 estimates for effect A: 50 - 35 = 15 60 - 40 = 20 Combine both estimates: ½(50-35) + ½(60-40) = 17.5

28. 40 60 +1 B -1 35 50 A -1 +1 In the same way we estimate the effect B(note that all 4 measurements are used!): ½(40-35) ½(60-50) = 7.5 +

29. 40 60 +1 B -1 35 50 A -1 +1 The interaction effect AB is the difference between the estimates for the effect A: ½(60-40) - ½(50-35) = 2.5

30. Interaction effects Cross terms in linear regression models cause interaction effects: Y = 3 + 2 xA + 4 xB + 7 xA xB xA xA +1 YY + 2 + 7 xB, so increase depends on xB. Likewise for xB xB+1 This explains the notation AB .

31. No interaction 55 B low 50 B high Output 25 20 low high Factor A

32. Interaction I 55 50 B low B high Output 45 20 low high Factor A

33. Interaction II 55 50 B low B high Output 45 20 low high Factor A

34. Interaction III 55 B high Output 45 20 B low 20 low high Factor A

35. Trick to Compute Effects (coded) measurement settings

36. Trick to Compute Effects Effect estimates

37. Trick to Compute Effects Effect estimates Effect A = ½(-35 - 40 + 50 + 60) = 17.5 Effect B = ½(-35 + 40 – 50 + 60) = 7.5

38. Trick to Compute Effects Effect AB = ½(60-40) - ½(50-35) = 2.5

39. Trick to Compute Effects × = × = × = × = AB equals the product of the columns A and B Effect AB = ½(60-40) - ½(50-35) = 2.5

40. Trick to Compute Effects Computational rules: I×A = A, I×B = B, A×B=AB etc. This holds true in general (i.e., also for more factors).

41. 3 Factors: a 23 Design

42. 3 Factors: a 23 Design

43. scheme 23 design bc=9 abc=7 · · ac=6 c=7 · · C effect A = b=7 ab=1 · · B ¼(+16-28)=-3 (1)=5 a=2 · · A

44. scheme 23 design bc=9 abc=7 · · ac=6 c=7 · · C effect AB = b=7 ab=1 · · ¼(+20-24)=-1 B (1)=5 a=2 · · A

45. Back to 2 factors – Blocking day 1 day 2 Suppose that we cannot perform all measurements at the same day. We are not interested in the difference between 2 days, but we must take the effect of this into account. How do we accomplish that?

46. Back to 2 factors – Blocking “hidden” block effect Suppose that we cannot perform all measurements at the same day. We are not interested in the difference between 2 days, but we must take the effect of this into account. How do we accomplish that?

47. Back to 2 factors – Blocking We note that the columns A and day are the same. Consequence: the effect of A and the day effect cannot be distinguished. This is called confounding or aliasing).

48. Back to 2 factors – Blocking A general guide-line is to confound the day effect with an interaction of highest possible order. How can we accomplish that here?

49. Back to 2 factors – Blocking Solution: day 1: a, b day 2: (1), ab or interchange the days!

50. Back to 2 factors – Blocking Choose within the days by drawing lots which experiment must be performed first. In general, the order of experiments must be determined by drawing lots. This is called randomisation. Solution: day 1: a, b day 2: (1), ab or interchange the days!