1. Chapter 7 Design of Engineering Experiments Part I

2. 7-1 The Strategy of Experimentation Every experiment involves a sequence of activities: 1.Conjecture – the original hypothesis that motivates the experiment. 2.Experiment – the test performed to investigate the conjecture. 3.Analysis – the statistical analysis of the data from the experiment. 4.Conclusion – what has been learned about the original conjecture from the experiment. Often the experiment will lead to a revised conjecture, and a new experiment, and so forth.

3. 3 “Best-guess” experiments –Select (guess) a level combination of factors based on experience and technical or theoretical knowledge –Used a lot in practice and often giving a good result –Disadvantages: no guarantee for an optimal (or satisfactory ) solution in a reasonable time One-factor-at-a-time (OFAT) experiments –Selecting a baseline set of levels for each factor and then successively varying each factor over its range with the other factors held at the baseline level –Disadvantage: not considering any interaction between factors (“Interaction”: failure of one factor to produce the same effect on the response at different level of another factor) Statistically designed experiments –Based on Fisher’s factorial concept –Factors are varied together, instead of one at a time => correct approach to dealing with several factors

4. 4 Objective: lowering golf score Response variable: score (per round) Possible Factors 1. Type of driver (oversized or regular-sized) 2. Type of ball used (balata or three piece) 3. Walking or riding in a golf cart 4. Drinking while playing (water or beer) 5. Playing time (morning or afternoon) 6. Weather temperature (cool or hot) 7. Type of golf shoe 8. Windy or calm day Example Ignored by experience Factors for experiment

5. 5 Figure 1-2 Results of the one-factor-at-a-time strategy for the golf experiment OFAT Baseline: oversized driver, balata ball, walking and water drinking (O, B, W, W) Result suggests that - the combination of regular sized driver, riding, drinking water is optimal (R, -, R, W) - the ball type is not an important factor

6. 6 Figure 1-3 Interaction between type of driver and type of beverage for the golf experiment

7. 7 Factorial Design In a factorial experiment, all possible combinations of factor levels are tested Suppose that only two factors are of interested: type of ball and type of driver

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9. 7-2 Factorial Experiments In a factorial experiment all possible combinations of the levels of the factors are investigated in each complete replicate of the experiment. If there are two factors A and B with a levels of factor A and b levels of factor B, each replicate contains all ab treatments.

11. 11 Main effect (Factor effect): The change in the mean response when the factor is changed from low to high

12. 12 Interaction between factors: The average difference in one factor’s effect at all levels of the other factor 52 30 40 20 40 12 20 50

13. Factorial experiments are the only way to discover interactions between variables. A significant interaction can mask the significance of main effects. When interaction is present, the main effects of the factors involved in the interaction may not have much meaning.

14. Problem with One Factor at a Time 14

15. 7-3 2 k Factorial Design Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high (they could be either quantitative or qualitative) Very widely used in industrial experimentation Form a basic “building block” for other very useful experimental designs Particularly useful for factor screening experiments

16. 7-3.1 2 2 Design Two factors each having two levels Completely randomized experiment The effects model

17. Consider an investigation into the effect of the concentration of the reactant and the amount of the catalyst on the yield in a chemical process The reactant concentration is factor A having two levels of 15 and 25 percent The catalyst is factor B with two levels of 1 and 2 pounds The data obtained are as follows Example 17 A: reactant concentration ; 15 and 25 percent B: catalyst ; 1 and 2 pounds y: yield

18. 18 “-” and “+” denote the low and high levels of a factor, respectively Low and high are arbitrary terms Factors can be quantitative or qualitative, although their treatment in the final model will be different Geometrically, the four runs form the corners of a square Notations for treatment combinations a : A +, B- / b: A -, B + / ab: A +, B + / (1): A -, B -

19. The averaged effect of a factor: the change in response produced by a change in the level of that factor averaged over the levels of the other factor (1),a, b, ab: total of the response observation at all n replicates taken at the treatment combination Main effect of A (denoted by A): A= Average of the effect of A at the low level of B and the effect of A at the high level of B or A = Difference in the average response of the combinations at A+ and A- 19 2

20. Main effect of B (denoted by B): Interaction effect AB (denoted by AB): AB = average difference between the effect of A at the high level of B and the effect of A at the low level of B or AB = the average of the right-to left diagonal treatment combinations minus the average of the left –to right diagonal treatment combination 20

22. 22 Estimation of Factor Effects The effect estimate are: = (90+100-60-80)/(2x3) = 8.33 = (90+60-100-80)/(2X3) = -5.00 = (90+80-60-100)/(2*3) = 1.67 The effect of A is positive, implying that increasing A from the low level to the high level will increase the yield The effect of B is negative, suggesting that increasing the amount of catalyst added to the process will decrease the yield The interaction effect appears to be small relative to the two main effects

23. 23 ANOVA Contrast A =ab+a-b-(1): total effect of A Contrast B =ab+b-a-(1): total effect of B Contrast AB =ab+(1)-a-b: total effect of AB Note: Let C be a contrast E[C]=0 Var[C]=4*n*  2 C 2 /4n  2 ~  2 (1) SS=  2 (1)*  2 = C 2 /4n

24. 24 Standard order: it is convenient to write down the treatment combinations in the order (1), a,b, ab Main effect A, B Interaction effect AB The total of the entire experiment I

25. 25 Regression Model For the 2 2 design, the regression model is Then where x 1 is a coded variable for the factor A and x 2 is a coded variable for the factor B and The regression coefficients are and

26. 26 For the chemical process experiment,

27. 27 The model contains only the main effects.=> The fitted response surface is plane. The yield increases as reactant concentration increases and catalyst amount decreases. => We use a fitted surface to find a direction of potential improvement for a process  Response Surface

28. 28  Residuals and Model Adequacy Residual =observed y value – Fitted value ex) x 1 =-1, x 2 =-1 => 27.5+(8.33/2)*(-1) +(-5.0/2)(-1) = 25.835 e1=28-25.835=2.165, e2=25-25.835=-0.835, e3= 27-5.835=1.165

29. Problem 29 Temperature ( 0 C) Copper Content (%) 40%80% 50 0 C 17 20 24 22 100 0 C 16 12 25 23 A.State the effects model. B.Compute the estimates of the effects in the model. C.Construct two factor interaction plots D.State and test hypotheses related with the ANOVA table E.Give a regression model for the data

30. 7-4 2 k Design for k  3 Factors Consider a 2 3 Design Label (1) a b ab c ac bc abc

31. Except for column I, every column has an equal number of + and – signs The sum of the product of signs in any two columns is zero; the columns in the table are orthogonal. Multiplying any column by I leaves that column unchanged (identity element) The product of any two columns yields a column in the table:

32. Estimation of Factor Effects (1) Main effects

33. (2) Interaction Effects

35. A 2 3 factorial design was used to develop a nitride etch process on a single wafer plasma etching tool The gap between the electrodes is factor A having two levels of 0.80 and 1.20 cm The gas flow is factor B with two levels of 125 and 200 SCCM The RF power applied to the cathode is factor C with two levels of 275 and 325 W The data obtained are as follows Example 35

36. 36 Main Effect Estimate A= Average of the effects of A at the low level of B, at the high level of B, at the low level of C and at the high level of C

37. 37 or A = Difference in the average response of the combinations at A+ and A- Interaction Effect Estimate AB = One-half of the difference between the average A effects at the two levels of B ABC = the average difference between the AB interactions at the two different levels of C

38. 38

39. 39 The regression model for predicting etch rate is, where x 1 and x 3 are the coded variables representing A and C, respectively

40. Figure 6.7 Response surface and contour plot of etch rate for Example 6-1. 40

41. 41 2 k Factorial Design

42. 7-5 Single Replicate of a 2 k Design Resources are usually limited and only allow a single replicate. These are 2 k factorial designs with one observation at each corner of the “cube” (also called a “single replicate” of the 2 k ) This is widely used in screening experiments when there are relatively many factors under consideration. Risks…if there is only one observation at each corner, there is a chance of unusual response observations spoiling the results.

43. 43 Lack of replication causes potential problems in statistical testing; With only one replicate, there’s no internal estimate of error (or”pure error”). With no replication, fitting the full model results in zero degrees of freedom for error => F- test cannot be applied. Potential solutions to this problem: pooling high-order interactions to estimate error (i.e. assuming that certain high order interactions are negligible and combining their mean squares to estimate the error). => Most systems are dominated by some of the main effect and low order interactions. The three-factor and higher-order interactions are usually negligible. How to select the significant effects? => Normal probability plotting of effects (Daniels, 1959) : The negligible effects ~ N(0,  2 ) Significant effects will have nonzero means and will not lie along the straight line in the normal probability plot.

44. A chemical product is produced in a pressure vessel A factorial experiment is carried out in the pilot plant to study the factors that influence the filtration rate of this product Factor A: temperature Factor B: pressure Factor C: concentration of formaldehyde Factor D: stirring rate The data obtained are as follows Example 44

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49. 49 The main effects of A, C and D are positive and if we considered only these main effects, A+, C+, D+ will be optimal. However, there are significant interactions AC and AD !! From AC interaction, effect A is large when C= -, and small when C=+ with the best results with C-, A+. From ASD interaction, A+, D+ is the best. Thus, A=+, C= -, D=+ is the best combination.

50. 50 Design Projection In the previous example, B (pressure) is not significant and all interactions involving B are negligible We may discard B from the experiment so that the design becomes a 2 3 factorial in A,C and D in two replicates In general, if we have a single replicate of a 2 k design, and if h (h<k) factors are negligible, then the original data correspond to a full two-level factorial in the remaining k-h factors (2 k-h factorial design) with 2 h replicates.