1. L. M. LyeDOE Course1 Design and Analysis of Multi-Factored Experiments Two-level Factorial Designs

2. L. M. LyeDOE Course2 The 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 (DNA) Special (short-cut) methods for analysis We will make use of Design-Expert for analysis

3. L. M. LyeDOE Course3 Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery

4. L. M. LyeDOE Course4 The Simplest Case: The 2 2 “-” and “+” denote the low and high levels of a factor, respectively Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative, although their treatment in the final model will be different

5. L. M. LyeDOE Course5 Estimating effects in two-factor two-level experiments Estimate of the effect of A a 1 b 1 - a 0 b 1 estimate of effect of A at high B a 1 b 0 - a 0 b 0 estimate of effect of A at low B sum/2estimate of effect of A over all B Or average of high As – average of low As. Estimate of the effect of B a 1 b 1 - a 1 b 0 estimate of effect of B at high A a 0 b 1 - a 0 b 0 estimate of effect of B at high A sum/2estimate of effect of B over all A Or average of high Bs – average of low Bs

6. L. M. LyeDOE Course6 Estimating effects in two-factor two-level experiments Estimate the interaction of A and B a 1 b 1 - a 0 b 1 estimate of effect of A at high B a 1 b 0 - a 0 b 0 estimate of effect of A at low B difference/2estimate of effect of B on the effect of A called as the interaction of A and B a 1 b 1 - a 1 b 0 estimate of effect of B at high A a 0 b 1 - a 0 b 0 estimate of effect of B at low A difference/2estimate of the effect of A on the effect of B Called the interaction of B and A Or average of like signs – average of unlike signs

7. L. M. LyeDOE Course7 Note that the two differences in the interaction estimate are identical; by definition, the interaction of A and B is the same as the interaction of B and A. In a given experiment one of the two literary statements of interaction may be preferred by the experimenter to the other; but both have the same numerical value. Estimating effects, contd...

8. L. M. LyeDOE Course8 Remarks on effects and estimates Note the use of all four yields in the estimates of the effect of A, the effect of B, and the effect of the interaction of A and B; all four yields are needed and are used in each estimates. Note also that the effect of each of the factors and their interaction can be and are assessed separately, this in an experiment in which both factors vary simultaneously. Note that with respect to the two factors studied, the factors themselves together with their interaction are, logically, all that can be studied. These are among the merits of these factorial designs.

9. L. M. LyeDOE Course9 Remarks on interaction Many scientists feel the need for experiments which will reveal the effect, on the variable under study, of factors acting jointly. This is what we have called interaction. The simple experimental design discussed here evidently provides a way of estimating such interaction, with the latter defined in a way which corresponds to what many scientists have in mind when they think of interaction. It is useful to note that interaction was not invented by statisticians. It is a joint effect existing, often prominently, in the real world. Statisticians have merely provided ways and means to measure it.

10. L. M. LyeDOE Course10 Symbolism and language A is called a main effect. Our estimate of A is often simply written A. B is called a main effect. Our estimate of B is often simply written B. AB is called an interaction effect. Our estimate of AB is often simply written AB. So the same letter is used, generally without confusion, to describe the factor, to describe its effect, and to describe our estimate of its effect. Keep in mind that it is only for economy in writing that we sometimes speak of an effect rather than an estimate of the effect. We should always remember that all quantities formed from the yields are merely estimates.

11. L. M. LyeDOE Course11 Table of signs The following table is useful: Notice that in estimating A, the two treatments with A at high level are compared to the two treatments with A at low level. Similarly B. This is, of course, logical. Note that the signs of treatments in the estimate of AB are the products of the signs of A and B. Note that in each estimate, plus and minus signs are equal in number

12. L. M. LyeDOE Course12 Example 3 B LowHigh Low High A B Example 2 LowHigh Low High A Example 1 B LowHigh Low High A Example 4 B LowHigh Low High A A+A+ A-A- B B=2 A B+B+ B-B- A=3 9 10 11 12 13 14 15 -201 Y Example 3 A B+B+ B-B- Discussion of examples: Notice that in examples 2 & 3 interaction is as large as or larger than main effects. B+B+ B-B- A A=2.5 Example 2 B-,B+B-,B+ Example 4 A *A = [-(1) - b + a + ab]/2 = [-10 - 12 + 13 + 15]/2 = 3

13. L. M. LyeDOE Course13 Change of scale, by multiplying each yield by a constant, multiplies each estimate by the constant but does not affect the relationship of estimates to each other. Addition of a constant to each yield does not affect the estimates. The numerical magnitude of estimates is not important here; it is their relationship to each other.

14. L. M. LyeDOE Course14 Modern notation and Yates’ order Modern notation: a 0 b 0 = 1 a 0 b 1 = ba 1 b 0 = aa 1 b 1 = ab We also introduce Yates’ (standard) order of treatments and yields; each letter in turn followed by all combinations of that letter and letters already introduced. This will be the preferred order for the purpose of analysis of the yields. It is not necessarily the order in which the experiment is conducted; that will be discussed later. For a two-factor two-level factorial design, Yates’ order is 1abab Using modern notation and Yates’ order, the estimates of effects become: A=(-1 + a - b + ab)/2 B=(-1 - a + b +ab)/2 AB=(1 -a - b + ab)/2

15. L. M. LyeDOE Course15 Three factors each at two levels Example: The variable is the yield of a nitration process. The yield forms the base material for certain dye stuffs and medicines. Lowhigh A time of addition of nitric acid2 hours7 hours B stirring time1/2 hour4 hours C heelabsentpresent Treatments (also yields) (i) old notation (ii) new notation. (i) a 0 b 0 c 0 a 0 b 0 c 1 a 0 b 1 c 0 a 0 b 1 c 1 a 1 b 0 c 0 a 1 b 0 c 1 a 1 b 1 c 0 a 1 b 1 c 1 (ii) 1 c b bc a ac ab abc Yates’ order: 1 a b ab c ac bc abc

16. L. M. LyeDOE Course16 Effects in The 2 3 Factorial Design

17. L. M. LyeDOE Course17 Estimating effects in three-factor two-level designs (2 3 ) Estimate of A (1) a - 1estimate of A, with B low and C low (2) ab - bestimate of A, with B high and C low (3) ac - cestimate of A, with B low and C high (4) abc - bc estimate of A, with B high and C high = (a+ab+ac+abc - 1-b-c-bc)/4, = (-1+a-b+ab-c+ac-bc+abc)/4 (in Yates’ order)

18. L. M. LyeDOE Course18 Estimate of AB Note that interactions are averages. Just as our estimate of A is an average of response to A over all B and all C, so our estimate of AB is an average response to AB over all C. AB= {[(4)-(3)] + [(2) - (1)]}/4 = {1-a-b+ab+c-ac-bc+abc)/4, in Yates’ order or, = [(abc+ab+c+1) - (a+b+ac+bc)]/4 Effect of A with B high - effect of A with B low, all at C high plus effect of A with B high - effect of A with B low, all at C low

19. L. M. LyeDOE Course19 interaction of A and B, at C high minus interaction of A and B at C low ABC= {[(4) - (3)] - [(2) - (1)]}/4 =(-1+a+b-ab+c-ac-bc+abc)/4, in Yates’ order or,=[abc+a+b+c - (1+ab+ac+bc)]/4 Estimate of ABC

20. L. M. LyeDOE Course20 This is our first encounter with a three-factor interaction. It measures the impact, on the yield of the nitration process, of interaction AB when C (heel) goes from C absent to C present. Or it measures the impact on yield of interaction AC when B (stirring time) goes from 1/2 hour to 4 hours. Or finally, it measures the impact on yield of interaction BC when A (time of addition of nitric acid) goes from 2 hours to 7 hours. As with two-factor two-level factorial designs, the formation of estimates in three-factor two-level factorial designs can be summarized in a table.

21. L. M. LyeDOE Course21 Sign Table for a 2 3 design

22. L. M. LyeDOE Course22 Example A=main effect of nitric acid time = 1.25 B=main effect of stirring time= -4.85 AB=interaction of A and B= -0.60 C=main effect of heel= 0.60 AC=interaction of A and C= 0.15 BC=interaction of B and C= 0.45 ABC=interaction of A, B, and C= -0.50 Yield of nitration process discussed earlier: 1 a b ab c ac bc abc Y = 7.2 8.4 2.0 3.0 6.7 9.2 3.4 3.7 NOTE: ac = largest yield; AC = smallest effect

23. L. M. LyeDOE Course23 We describe several of these estimates, though on later analysis of this example, taking into account the unreliability of estimates based on a small number (eight) of yields, some estimates may turn out to be so small in magnitude as not to contradict the conjecture that the corresponding true effect is zero. The largest estimate is -4.85, the estimate of B; an increase in stirring time, from 1/2 to 4 hours, is associated with a decline in yield. The interaction AB = -0.6; an increase in stirring time from 1/2 to 4 hours reduces the effect of A, whatever it is (A = 1.25), on yield. Or equivalently

24. L. M. LyeDOE Course24 an increase in nitric acid time from 2 to 7 hours reduces (makes more negative) the already negative effect (B = -485) of stirring time on yield. Finally, ABC = -0.5. Going from no heel to heel, the negative interaction effect AB on yield becomes even more negative. Or going from low to high stirring time, the positive interaction effect AC is reduced. Or going from low to high nitric acid time, the positive interaction effect BC is reduced. All three descriptions of ABC have the same numerical value; but the chemist would select one of them, then say it better.

25. L. M. LyeDOE Course25 Number and kinds of effects We introduce the notation 2 k. This means a factor design with each factor at two levels. The number of treatments in an unreplicated 2 k design is 2 k. The following table shows the number of each kind of effect for each of the six two-level designs shown across the top.

26. L. M. LyeDOE Course26 Main effect 2 factor interaction 3 factor interaction 4 factor interaction 5 factor interaction 6 factor interaction 7 factor interaction 37153163127 In a 2 k design, the number of r-factor effects is C k r = k!/[r!(k-r)!]

27. L. M. LyeDOE Course27 Notice that the total number of effects estimated in any design is always one less than the number of treatments In a 2 2 design, there are 2 2 =4 treatments; we estimate 2 2 -1 = 3 effects. In a 2 3 design, there are 2 3 =8 treatments; we estimate 2 3 - 1 = 7 effects One need not repeat the earlier logic to determine the forms of estimates in 2 k designs for higher values of k. A table going up to 2 5 follows.

28. L. M. LyeDOE Course28 2 2323 2424 2525 TreatmentsTreatments E f f e c t s

29. L. M. LyeDOE Course29 Yates’ Forward Algorithm (1) A systematic method of calculating estimates of effects. For complete factorials first arrange the yields in Yates’ (standard) order. Addition, then subtraction of adjacent yields. The addition and subtraction operations are repeated until 2 k terms appear in each line: for a 2 k there will be k columns of calculations 1. Applied to Complete Factorials (Yates, 1937)

30. L. M. LyeDOE Course30 Yates’ Forward Algorithm (2) Example: Yield of a nitration process 17.215.6 20.6 43.6 Contrast of µ a8.4 5.0 23.0 5.0 Contrast of A b2.015.9 2.2-19.4 Contrast of B ab3.0 7.1 2.8 -2.4 Contrast of AB c6.7 1.2-10.6 2.4 Contrast of C ac9.2 1.0 -8.8 0.6 Contrast of AC bc3.4 2.5 -0.2 1.8 Contrast of BC abc3.7 0.3 -2.2 -2.0 Contrast of ABC Tr.Yield1 st Col2 nd Col3 rd Col Again, note the line-by-line correspondence between treatments and estimates; both are in Yates’ order.

31. L. M. LyeDOE Course31 Main effects in the face of large interactions Several writers have cautioned against making statements about main effects when the corresponding interactions are large; interactions describe the dependence of the impact of one factor on the level of another; in the presence of large interaction, main effects may not be meaningful.

32. L. M. LyeDOE Course32 low level (-1)high level (+1) N (A)bloodsulphate of ammonia P(B)superphosphatesteamed bone flower; Example (Adapted from Kempthorne) Yields are in bushels of potatoes per plot. The two factors are nitrate (N) and phosphate (P) fertilizers. The yields are 1 = 746.75n = 625.75p = 611.00np = 656.00 the estimates are N = -38.00P = -52.75NP = 83.00 In the face of such high interaction we now specialize the main effect of each factor to particular levels of the other factor. Effect of N at high level P = np-p = 656.00-611.00 = 45.0 Effect of N at low level P = n-1 = 625.71-746.75 = -121.0, which appear to be more valuable for fertilizer policy than the mean (-38.00) of such disparate numbers Y N P+P+ P-P- -38 -121 746.75 611.0 656 625.75 Keep both low is best

33. L. M. LyeDOE Course33 Note that answers to these specialized questions are based on fewer than 2 k yields. In our numerical example, with interaction NP prominent, we have only two of the four yields in our estimate of N at each level of P. In general we accept high interactions wherever found and seek to explain them; in the process of explanation, main effects (and lower-order interactions) may have to be replaced in our interest by more meaningful specialized or conditional effects.

34. L. M. LyeDOE Course34 Specialized or Conditional Effects Whenever there is large interactions, check: Effect of A at high level of B = A + = A + AB Effect of A at low level of B = A - = A – AB Effect of B at high level of A = B + = B + AB Effect of B at low level of A = B - = B - AB

35. L. M. LyeDOE Course35 Factors not studied In any experiment, factors other than those studied may be influential. Their presence is sometimes acknowledged under the dubious title “experimental error”. They may be neglected, but the usual cost of neglect is high. For they often have uneven impact, systematically affecting some treatments more than others, and thereby seriously confounding inferences on the studied factors. It is important to deal explicitly with them; even more, it is important to measure their impact. How?

36. L. M. LyeDOE Course36 1. Hold them constant. 2. Randomize their effects. 3. Estimate their magnitude by replicating the experiment. 4. Estimate their magnitude via side or earlier experiments. 5. Argue (convincingly) that the effects of some of these non-studied factors are zero, either in advance of the experiment or in the light of the yields. 6. Confound certain non-studied factors.

37. L. M. LyeDOE Course37 Simplified Analysis Procedure for 2-level Factorial Design Estimate factor effects Formulate model using important effects Check for goodness-of-fit of the model. Interpret results Use model for Prediction

38. L. M. LyeDOE Course38 Example: Shooting baskets Consider an experiment with 3 factors: A, B, and C. Let the response variable be Y. For example, Y = number of baskets made out of 10 Factor A = distance from basket (2m or 5m) Factor B = direction of shot (0° or 90 °) Factor C = type of shot (set or jumper) Factor Name Units Low Level (-1) High Level (+1) ADistance m 25 B Direction Deg. 090 CShot type Set Jump

39. L. M. LyeDOE Course39 Treatment Combinations and Results OrderABCCombinationY 1-1-1-1 (1)9 2+1-1-1 a5 3-1+1-1 b7 4+1+1-1 ab3 5-1-1+1 c6 6+1-1+1 ac5 7-1+1+1 bc4 8+1+1+1 abc2

40. L. M. LyeDOE Course40 Estimating Effects OrderABABCACBCABCCombY 1-1-1+1-1 +1+1 -1 (1)9 2+1-1-1-1 -1 +1+1 a5 3-1+1-1-1 +1 -1+1 b7 4+1+1+1-1 -1 -1-1ab3 5-1-1+1+1 -1-1+1 c6 6+1-1-1+1 +1 -1-1ac5 7-1+1-1+1 -1 +1-1bc4 8+1+1+1+1 +1 +1+1abc2 Effect A = (a + ab + ac + abc)/4 - (1 + b + c + bc)/4 = (5 + 3 + 5 + 2)/4 - (9 + 7 + 6 + 4)/4 = -2.75

41. L. M. LyeDOE Course41 Effects and Overall Average Using the sign table, all 7 effects can be calculated: Effect A = -2.75  Effect B = -2.25  Effect C = -1.75  Effect AC = 1.25  Effect AB = -0.25 Effect BC = -0.25 Effect ABC = -0.25 The overall average value = (9 + 5 + 7 + 3 + 6 + 5 + 4 + 2)/8 = 5.13

42. L. M. LyeDOE Course42 Formulate Model The most important effects are: A, B, C, and AC Model: Y =  0 +  1 X 1 +  2 X 2 +  3 X 3 +  13 X 1 X 3  0 = overall average = 5.13  1 = Effect [A]/2 = -2.75/2 = -1.375  2 = Effect [B]/2 = -2.25/2 = -1.125  3 = Effect [C]/2 = -1.75/2 = - 0.875  13 = Effect [AC]/2 = 1.25/2 = 0.625 Model in coded units: Y = 5.13 -1.375 X 1 - 1.125 X 2 - 0.875 X 3 + 0.625 X 1 X 3

43. L. M. LyeDOE Course43 Checking for goodness-of-fit ActualPredicted Value Value 9.00 9.13 5.00 5.13 7.00 6.88 3.00 2.87 6.00 6.13 5.00 4.63 4.00 3.88 2.00 2.37 Amazing fit!!

44. L. M. LyeDOE Course44 Interpreting Results # out of 10 2 6 10 8 4 B090 (9+5+6+5)/4=6.25 (7+3+4+2)/4=4 Effect of B=4-6.25= -2.25 A2m5m # out of 10 10 8 6 4 2 C: Shot type C(-1) C (+1) Interaction of A and C = 1.25 At 5m, Jump or set shot about the same BUT at 2m, set shot gave higher values compared to jump shots

45. L. M. LyeDOE Course45 Design and Analysis of Multi-Factored Experiments Analysis of 2 k Experiments Statistical Details

46. L. M. LyeDOE Course46 Errors of estimates in 2 k designs 1. Meaning of  2 Assume that each treatment has variance  2. This has the following meaning: consider any one treatment and imagine many replicates of it. As all factors under study are constant throughout these repetitions, the only sources of any variability in yield are the factors not under study. Any variability in yield is due to them and is measured by  2.

47. L. M. LyeDOE Course47 Errors of estimates in 2 k designs, Contd.. 2. Effect of the number of factors on the error of an estimate What is the variance of an estimate of an effect? In a 2 k design, 2 k treatments go into each estimate; the signs of the treatments are + or -, depending on the effect being estimated. So, any estimate = 1/2 k-1 [generalized (+ or -) sum of 2 k treatments]  2 (any estimate) = 1/2 2k-2 [2 k  2 ] =  2 /2 k-2 ; The larger the number of factors, the smaller the error of each estimate. Note:  2 (kx) = k 2  2 (x)

48. L. M. LyeDOE Course48 3. Effect of replication on the error of an estimate What is the effect of replication on the error of an estimate? Consider a 2 k design with each treatment replicated n times. Errors of estimates in 2 k designs, Contd.. -------- -------- -------- --- -------- -------- 1ababcd

49. L. M. LyeDOE Course49 Any estimate = 1/2 k-1 [sums of 2 k terms, all of them means based on samples of size n]  2 (any estimate) = 1/2 2k-2 [2 k  2 /n] =  2 /(n2 k-2 ); The larger the replication per treatment, the smaller the error of each estimate. Errors of estimates in 2 k designs, Contd..

50. L. M. LyeDOE Course50 So, the error of an estimate depends on k (the number of factors studied) and n (the replication per factor). It also (obviously) depends on  2. The variance  2 can be reduced holding some of the non-studied factors constant. But, as has been noted, this gain is offset by reduced generality of any conclusions.

51. L. M. LyeDOE Course51 Effects, Sum of Squares and Regression Coefficients

52. L. M. LyeDOE Course52 Judging Significance of Effects a) p- values from ANOVA Compute p-value of calculated F. IF p < , then effect is significant. b) Comparing std. error of effect to size of effect

53. L. M. LyeDOE Course53 Hence If effect ± 2 (se), contains zero, then that effect is not significant. These intervals are approximately the 95% CI. e.g. 3.375 ± 1.56 (significant) 1.125 ± 1.56 (not significant)

54. L. M. LyeDOE Course54 c) Normal probability plot of effects Significant effects are those that do not fit on normal probability plot. i. e. non-significant effects will lie along the line of a normal probability plot of the effects. Good visual tool - available in Design-Expert software.

55. L. M. LyeDOE Course55 Design and Analysis of Multi-Factored Experiments Examples of Computer Analysis

56. L. M. LyeDOE Course56 Analysis Procedure for a Factorial Design Estimate factor effects Formulate model –With replication, use full model –With an unreplicated design, use normal probability plots Statistical testing (ANOVA) Refine the model Analyze residuals (graphical) Interpret results

57. L. M. LyeDOE Course57 Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery

58. L. M. LyeDOE Course58 Estimation of Factor Effects The effect estimates are: A = 8.33, B = -5.00, AB = 1.67 Design-Expert analysis A = (a + ab - 1 - b)/2n = (100 + 90 - 60 - 80)/(2 x 3) = 8.33 B = (b + ab - 1 - a)/2n = -5.00 C = (ab + 1 - a - b)/2n = 1.67

59. L. M. LyeDOE Course59 Estimation of Factor Effects Form Tentative Model Term Effect SumSqr% Contribution Model Intercept Model A 8.33333 208.333 64.4995 Model B -5 75 23.2198 Model AB 1.66667 8.33333 2.57998 Error Lack Of Fit 0 0 Error P Error 31.3333 9.70072 Lenth's ME6.15809 Lenth's SME7.95671

60. L. M. LyeDOE Course60 Statistical Testing - ANOVA Response:Conversion ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model291.67397.2224.820.0002 A208.331208.3353.19< 0.0001 B75.00175.0019.150.0024 AB8.3318.332.130.1828 Pure Error31.3383.92 Cor Total323.0011 Std. Dev.1.98R-Squared 0.9030 Mean27.50Adj R-Squared0.8666 C.V.7.20Pred R-Squared0.7817 PRESS70.50Adeq Precision11.669 The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?

61. L. M. LyeDOE Course61 Statistical Testing - ANOVA Coefficient Standard95% CI95% CI Factor Estimate DFErrorLowHighVIF Intercept 27.50 10.5726.1828.82 A-Concent 4.17 10.572.855.481.00 B-Catalyst -2.50 10.57 -3.82 -1.181.00 AB 0.83 10.57 -0.482.151.00 General formulas for the standard errors of the model coefficients and the confidence intervals are available. They will be given later.

62. L. M. LyeDOE Course62 Refine Model Response:Conversion ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model283.332141.6732.14< 0.0001 A208.331208.3347.27< 0.0001 B75.00175.0017.020.0026 Residual39.6794.41 Lack of Fit8.3318.332.130.1828 Pure Error31.3383.92 Cor Total323.0011 Std. Dev.2.10R-Squared0.8772 Mean27.50Adj R-Squared0.8499 C.V.7.63Pred R-Squared0.7817 PRESS70.52Adeq Precision12.702 There is now a residual sum of squares, partitioned into a “lack of fit” component (the AB interaction) and a “pure error” component

63. L. M. LyeDOE Course63 Regression Model for the Process

64. L. M. LyeDOE Course64 Residuals and Diagnostic Checking

65. L. M. LyeDOE Course65 The Response Surface

66. L. M. LyeDOE Course66 An Example of a 2 3 Factorial Design A = carbonation, B = pressure, C = speed, y = fill deviation

67. L. M. LyeDOE Course67 Estimation of Factor Effects TermEffectSumSqr% Contribution Model Intercept Error A33646.1538 Error B2.2520.2525.9615 Error C1.7512.2515.7051 Error AB0.752.252.88462 Error AC0.250.250.320513 Error BC0.511.28205 Error ABC0.511.28205 Error LOF0 Error P Error 56.41026 Lenth's ME1.25382 Lenth's SME1.88156

68. L. M. LyeDOE Course68 ANOVA Summary – Full Model Response:Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model73.00710.4316.690.0003 A36.00136.0057.60< 0.0001 B20.25120.2532.400.0005 C12.25112.2519.600.0022 AB2.2512.253.600.0943 AC0.2510.250.400.5447 BC1.0011.001.600.2415 ABC1.0011.001.600.2415 Pure Error5.0080.63 Cor Total78.0015 Std. Dev.0.79R-Squared0.9359 Mean1.00Adj R-Squared0.8798 C.V.79.06Pred R-Squared0.7436 PRESS20.00Adeq Precision13.416

69. L. M. LyeDOE Course69 Model Coefficients – Full Model Coefficient Standard95% CI 95% CI Factor EstimateDFErrorLowHighVIF Intercept 1.0010.200.541.46 A-Carbonation 1.5010.201.041.96 1.00 B-Pressure 1.1310.200.671.58 1.00 C-Speed 0.8810.200.421.33 1.00 AB 0.3810.20 -0.0810.83 1.00 AC 0.1310.20 -0.330.58 1.00 BC 0.2510.20-0.210.71 1.00 ABC 0.2510.20-0.210.71 1.00

70. L. M. LyeDOE Course70 Refine Model – Remove Nonsignificant Factors Response:Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model70.75417.6926.84< 0.0001 A36.00136.0054.62< 0.0001 B20.25120.2530.720.0002 C12.25112.2518.590.0012 AB2.2512.253.410.0917 Residual7.25110.66 LOF2.2530.751.200.3700 Pure E5.0080.63 C Total78.0015 Std. Dev.0.81R-Squared0.9071 Mean1.00Adj R-Squared0.8733 C.V.81.18Pred R-Squared0.8033 PRESS15.34Adeq Precision15.424

71. L. M. LyeDOE Course71 Model Coefficients – Reduced Model Coefficient Standard 95% CI 95% CI Factor EstimateDFErrorLowHigh Intercept1.0010.200.551.45 A-Carbonation1.5010.201.051.95 B-Pressure1.1310.200.681.57 C-Speed0.8810.200.431.32 AB 0.3810.20-0.0720.82

72. L. M. LyeDOE Course72 Model Summary Statistics R 2 and adjusted R 2 R 2 for prediction (based on PRESS)

73. L. M. LyeDOE Course73 Model Summary Statistics Standard error of model coefficients Confidence interval on model coefficients

74. L. M. LyeDOE Course74 The Regression Model Final Equation in Terms of Coded Factors: Fill-deviation = +1.00 +1.50 * A +1.13 * B +0.88 * C +0.38 * A * B Final Equation in Terms of Actual Factors: Fill-deviation = +9.62500 -2.62500 * Carbonation -1.20000 * Pressure +0.035000 * Speed +0.15000 * Carbonation * Pressure

75. L. M. LyeDOE Course75 Residual Plots are Satisfactory

76. L. M. LyeDOE Course76 Model Interpretation Moderate interaction between carbonation level and pressure

77. L. M. LyeDOE Course77 Model Interpretation Cube plots are often useful visual displays of experimental results

78. L. M. LyeDOE Course78 Contour & Response Surface Plots – Speed at the High Level

79. L. M. LyeDOE Course79 Design and Analysis of Multi-Factored Experiments Unreplicated Factorials

80. L. M. LyeDOE Course80 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner of the “cube” An unreplicated 2 k factorial design is also sometimes called a “single replicate” of the 2 k These designs are very widely used Risks…if there is only one observation at each corner, is there a chance of unusual response observations spoiling the results? Modeling “noise”?

81. L. M. LyeDOE Course81 Spacing of Factor Levels in the Unreplicated 2 k Factorial Designs If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data More aggressive spacing is usually best

82. L. M. LyeDOE Course82 Unreplicated 2 k Factorial Designs Lack of replication causes potential problems in statistical testing –Replication admits an estimate of “pure error” (a better phrase is an internal estimate of error) –With no replication, fitting the full model results in zero degrees of freedom for error Potential solutions to this problem –Pooling high-order interactions to estimate error –Normal probability plotting of effects (Daniels, 1959)

83. L. M. LyeDOE Course83 Example of an Unreplicated 2 k Design A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate Experiment was performed in a pilot plant

84. L. M. LyeDOE Course84 The Resin Plant Experiment

85. L. M. LyeDOE Course85 The Resin Plant Experiment

86. L. M. LyeDOE Course86 Estimates of the Effects TermEffectSumSqr% Contribution Model Intercept Error A21.6251870.5632.6397 Error B3.12539.06250.681608 Error C9.875390.0626.80626 Error D14.625855.56314.9288 Error AB0.1250.06250.00109057 Error AC-18.1251314.0622.9293 Error AD16.6251105.5619.2911 Error BC2.37522.56250.393696 Error BD-0.3750.56250.00981515 Error CD-1.1255.06250.0883363 Error ABC1.87514.06250.245379 Error ABD4.12568.06251.18763 Error ACD-1.62510.56250.184307 Error BCD-2.62527.56250.480942 Error ABCD1.3757.56250.131959 Lenth's ME6.74778 Lenth's SME13.699

87. L. M. LyeDOE Course87 The Normal Probability Plot of Effects

88. L. M. LyeDOE Course88 The Half-Normal Probability Plot

89. L. M. LyeDOE Course89 ANOVA Summary for the Model Response:Filtration Rate ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb >F Model5535.8151107.1656.74< 0.0001 A1870.5611870.5695.86< 0.0001 C390.061390.0619.990.0012 D855.561855.5643.85< 0.0001 AC1314.0611314.0667.34< 0.0001 AD1105.5611105.5656.66< 0.0001 Residual195.121019.51 Cor Total5730.9415 Std. Dev.4.42R-Squared0.9660 Mean70.06Adj R-Squared0.9489 C.V.6.30Pred R-Squared0.9128 PRESS499.52Adeq Precision20.841

90. L. M. LyeDOE Course90 The Regression Model Final Equation in Terms of Coded Factors: Filtration Rate = +70.06250 +10.81250 * Temperature +4.93750 * Concentration +7.31250 * Stirring Rate -9.06250 * Temperature * Concentration +8.31250 * Temperature * Stirring Rate

91. L. M. LyeDOE Course91 Model Residuals are Satisfactory

92. L. M. LyeDOE Course92 Model Interpretation – Interactions

93. L. M. LyeDOE Course93 Model Interpretation – Cube Plot If one factor is dropped, the unreplicated 2 4 design will project into two replicates of a 2 3 Design projection is an extremely useful property, carrying over into fractional factorials

94. L. M. LyeDOE Course94 Model Interpretation – Response Surface Plots With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates

95. L. M. LyeDOE Course95 The Drilling Experiment A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill

96. L. M. LyeDOE Course96 Effect Estimates - The Drilling Experiment TermEffectSumSqr% Contribution Model Intercept Error A0.91753.367221.28072 Error B6.4375165.76663.0489 Error C3.292543.362216.4928 Error D2.2920.97647.97837 Error AB0.591.39240.529599 Error AC0.1550.09610.0365516 Error AD0.83752.805631.06712 Error BC1.519.12043.46894 Error BD1.592510.14423.85835 Error CD0.44750.8010250.30467 Error ABC0.16250.1056250.0401744 Error ABD0.762.31040.87876 Error ACD0.5851.36890.520661 Error BCD0.1750.12250.0465928 Error ABCD0.54251.177220.447757 Lenth's ME2.27496 Lenth's SME4.61851

97. L. M. LyeDOE Course97 Half-Normal Probability Plot of Effects

98. L. M. LyeDOE Course98 Residual Plots

99. L. M. LyeDOE Course99 The residual plots indicate that there are problems with the equality of variance assumption The usual approach to this problem is to employ a transformation on the response Power family transformations are widely used Transformations are typically performed to –Stabilize variance –Induce normality –Simplify the model Residual Plots

100. L. M. LyeDOE Course100 Selecting a Transformation Empirical selection of lambda Prior (theoretical) knowledge or experience can often suggest the form of a transformation Analytical selection of lambda…the Box-Cox (1964) method (simultaneously estimates the model parameters and the transformation parameter lambda) Box-Cox method implemented in Design-Expert

101. L. M. LyeDOE Course101 The Box-Cox Method A log transformation is recommended The procedure provides a confidence interval on the transformation parameter lambda If unity is included in the confidence interval, no transformation would be needed

102. L. M. LyeDOE Course102 Effect Estimates Following the Log Transformation Three main effects are large No indication of large interaction effects What happened to the interactions?

103. L. M. LyeDOE Course103 ANOVA Following the Log Transformation Response:adv._rate Transform: Natural log Constant: 0.000 ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model7.1132.37164.82< 0.0001 B5.3515.35371.49< 0.0001 C1.3411.3493.05< 0.0001 D0.4310.4329.920.0001 Residual0.17120.014 Cor Total7.2915 Std. Dev.0.12R-Squared 0.9763 Mean1.60Adj R-Squared0.9704 C.V.7.51Pred R-Squared0.9579 PRESS0.31Adeq Precision34.391

104. L. M. LyeDOE Course104 Following the Log Transformation Final Equation in Terms of Coded Factors: Ln(adv._rate) = +1.60 +0.58 * B +0.29 * C +0.16 * D

105. L. M. LyeDOE Course105 Following the Log Transformation

106. L. M. LyeDOE Course106 The Log Advance Rate Model Is the log model “better”? We would generally prefer a simpler model in a transformed scale to a more complicated model in the original metric What happened to the interactions? Sometimes transformations provide insight into the underlying mechanism

107. L. M. LyeDOE Course107 Other Analysis Methods for Unreplicated 2 k Designs Lenth’s method –Analytical method for testing effects, uses an estimate of error formed by pooling small contrasts –Some adjustment to the critical values in the original method can be helpful –Probably most useful as a supplement to the normal probability plot

108. L. M. LyeDOE Course108 Design and Analysis of Multi-Factored Experiments Center points

109. L. M. LyeDOE Course109 Addition of Center Points to a 2 k Designs Based on the idea of replicating some of the runs in a factorial design Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models:

110. L. M. LyeDOE Course110 The hypotheses are: This sum of squares has a single degree of freedom

111. L. M. LyeDOE Course111 Example Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature

112. L. M. LyeDOE Course112 ANOVA for Example Response:yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model2.8330.9421.920.0060 A2.4012.4055.870.0017 B0.4210.429.830.0350 AB2.500E-00312.500E-0030.0580.8213 Curvature2.722E-00312.722E-0030.0630.8137 Pure Error0.1740.043 Cor Total3.008 Std. Dev.0.21R-Squared 0.9427 Mean40.44Adj R-Squared0.8996 C.V.0.51Pred R-SquaredN/A PRESSN/AAdeq Precision14.234

113. L. M. LyeDOE Course113 If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model

114. L. M. LyeDOE Course114 Practical Use of Center Points Use current operating conditions as the center point Check for “abnormal” conditions during the time the experiment was conducted Check for time trends Use center points as the first few runs when there is little or no information available about the magnitude of error Can have only 1 center point for computer experiments – hence requires a different type of design