Lecture 9: Factorial Design Montgomery: chapter 5

Transcription

1 Lecture 9: Factorial Design Montgomery: chapter 5 Page 1

2 Examples Example I. Two factors (A, B) each with two levels (, +) Page 2

3 Three Data for Example I Ex.I-Data 1 A B ,33 51,51 18,22 39,41 EX.I-Data 2 A B ,42 10,14 19,21 53,47 EX.I-Data 3 A B ,33 62,68 21,21 38,42 Page 3

4 Example II: Battery life experiment An engineer is studying the effective life of a certain type of battery. Two factors, plate material and temperature, are involved. There are three types of plate materials (1, 2, 3) and three temperature levels (15, 70, 125). Four batteries are tested at each combination of plate material and temperature, and all 36 tests are run in random order. The experiment and the resulting observed battery life data are given below. temperature material ,155,74,180 34,40,80,75 20,70,82, ,188,159, ,122,106,115 25,70,58, ,110,168, ,120,150,139 96,104,82,60 Page 4

5 Example III: Bottling Experiment A soft drink bottler is interested in obtaining more uniform fill heights in the bottles produced by his manufacturing process. An experiment is conducted to study three factors of the process, which are the percent carbonation (A): the operating pressure (B): the line speed (C): 10, 12, 14 percent 25, 30 psi 200, 250 bpm The response is the deviation from the target fill height. Each combination of the three factors has two replicates and all 24 runs are performed in a random order. The experiment and data are shown below. Page 5

6 pressure(b) 25 psi 30 psi LineSpeed(C) LineSpeed(C) Carbonation(A) ,-1-1,0-1,0 1, , 1 2,1 2,3 6,5 14 5,4 7,6 7,9 10,11 Page 6

7 Factorial Design a number of factors: F 1,F 2,...,F r. each with a number of levels: l 1, l 2,..., l r number of all possible level combinations (treatments): l 1 l 2... l r interested in (main) effects, 2-factor interactions (2fi), 3-factor interactions (3fi), etc. One-factor-a-time design as the opposite of factorial design. Advantages of factorial over one-factor-a-time more efficient (runsize and estimation precision) able to accommodate interactions results are valid over a wider range of experimental conditions Page 7

8 Statistical Model (Two Factors: A and B) Statistical model is y ijk = µ + τ i + β j +(τβ) ij + ɛ ijk µ - grand mean i =1, 2,...,a j =1, 2,...,b k =1, 2,...,n τ i - ith level effect of factor A (ignores B) (main effects of A) β j - jth level effect of factor B (ignores A) (main effects of B) (τβ) ij - interaction effect of combination ij (Explain variation not explained by main effects) ɛ ijk N(0,σ 2 ) Over-parameterized model: must include certain parameter constraints. Typically i τ i =0 j β j =0 i (τβ) ij =0 j (τβ) ij =0 Page 8

9 Estimates Rewrite observation as: y ijk = y... +(y i.. y... )+(y.j. y... )+(y ij. y i.. y.j. + y... )+(y ijk y ij. ) result in estimates µ = y... τ i = y i.. y... β j = y.j. y... (τβ) ij = y ij. y i.. y.j. + y... predicted value at level combination ij is ŷ ijk = y ij. Residuals are ˆɛ ijk = y ijk y ij. Page 9

10 Partitioning the Sum of Squares Based on y ijk = y... +(y i.. y... )+(y.j. y... )+(y ij. y i.. y.j. + y... )+(y ijk y ij. ) Calculate SS T = (y ijk y... ) 2 Right hand side simplifies to SS A : bn i (y i.. y... )2 + df = a 1 SS B : an j (y.j. y... )2 + df = b 1 SS AB : n i j (y ij. y i. y.j + y.. )2 + df =(a 1)(b 1) SS E : i j k (y ijk y ij. ) 2 df = ab(n 1) SS T =SS A +SS B +SS AB +SS E Using SS/df leads to MS A,MS B, MS AB and MS E. Page 10

11 Testing Hypotheses 1 Main effects of A: H 0 : τ 1 =...= τ a =0vs H 1 : at least one τ i 0. 2 Main effects of B: H 0 : β 1 =...= β b =0vs H 1 : at least one β j 0. 3 Interaction effects of AB: E(MS E )=σ 2 H 0 :(αβ) ij =0for all i, j vs H 1 : at least one (τβ) ij 0. E(MS A )=σ 2 + bn τ 2 i /(a 1) E(MS B )=σ 2 + an βj 2 /(b 1) E(MS AB )=σ 2 + n (τβ) 2 ij /(a 1)(b 1) Use F-statistics for testing the hypotheses above: 1: F 0 = SS A/(a 1) SS E /(ab(n 1)) 2: F 0 = SS B/(b 1) SS E /(ab(n 1)) 3: F 0 = SS AB/(a 1)(b 1) SS E /(ab(n 1)) Page 11

12 Analysis of Variance Table Source of Sum of Degrees of Mean F 0 Variation Squares Freedom Square Factor A SS A a 1 MS A F 0 Factor B SS B b 1 MS B F 0 Interaction SS AB (a 1)(b 1) MS AB F 0 Error SS E ab(n 1) MS E Total SS T abn 1 SS T = y 2 ijk y2.../abn; SS A = 1 bn y 2 i.. y 2.../abn SS B = 1 an y 2.j. y 2.../abn; SS subtotal = 1 n y 2 ij. y 2.../abn SS AB =SS subtotal SS A SS B ; SS E =Subtraction df E > 0 only if n>1. When n =1,noSS E is available so we cannot test the effects. If we can assume that the interactions are negligible ((τβ) ij =0), MS AB becomes a good estimate of σ 2 and it can be used as MS E. Caution: if the assumption is wrong, then error and interaction are confounded and testing results can go wrong. Page 12

13 Battery Life Example data battery; input mat temp life; datalines; : : : : : : ; proc glm; class mat temp; model life=mat temp mat*temp; output out=batnew r=res p=pred; run; Page 13

14 Dependent Variable: life Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Cor Total R-Square Coeff Var Root MSE life Mean Source DF Type I SS Mean Square F Value Pr > F mat temp <.0001 mat*temp Page 14

15 Checking Assumptions 1 Errors are normally distributed Histogram or QQplot of residuals 2 Constant variance Residuals vs ŷ ij. plot, Residuals vs factor A plot and Residuals vs factor B 3 If n=1, no interaction. Tukey s Test of Nonadditivity Assume (τβ) ij = γτ i β j. H 0 : γ =0. SS N = [ y ij y i. y.j y.. (SS A +SS B + y 2../ab)] 2 abss A SS B F 0 = SS N /1 (SS E SS N )/((a 1)(b 1) 1) F 1,(a 1)(b a) 1 the convenient procedure used for RCBD can be employed. Page 15

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18 Effects Estimation (Battery Experiment) 0. ˆµ=ȳ... = Treatment mean response, or cell mean, or predicted value, ŷ ij =ˆµ ij =ȳ ij. =ˆµ +ˆτ i + ˆβ j + ˆ (τβ) ij temperature material Factor level means row means ȳ i.. for A; column means ȳ.j. for B. material : ȳ 1.. =83.166, ȳ 2.. = , ȳ 3.. = temperature : ȳ.1. = , ȳ.2. = , ȳ.3. = Page 18

19 3. Main effects estimates ˆτ 1 = , ˆτ 2 =2.8055, ˆτ 3 = ˆβ 1 = , ˆβ 2 =2.0555, ˆβ 3 = Interactions ( ˆ (τβ) ij ) temperature material Page 19

20 Understanding Interactions Example I Data 1: A B resp; Dependent Variable: resp Sum of Source DF Squares Mean Square F Value Pr > F A B A*B Error Cor Total Page 20

21 Interaction plot for A and B (No Interaction) mean of y B A Difference between level means of B (with A fixed at a level) does not depend on the level of A; demonstrated by two parallel lines. Page 21

22 Example I Data 2 A B resp Sum of Source DF Squares Mean Square F Value Pr > F A B A*B Error Cor Total Page 22

23 Interaction Plot for A and B (Antagonistic Interaction from B to A) mean of y B A Difference between level means of B (with A fixed at a level) depends on the level of A. If the trend of mean response over A reverses itself when B changes from one level to another, the interaction is said to be antagonistic from B to A. Demonstrated by two unparallel lines with slopes of opposite signs. Page 23

24 Example I Data 3 A B resp Sum of Source DF Squares Mean Square F Value Pr > F A B A*B Error Co Total Page 24

25 Interaction Plot for A and B (Synergistic Interaction from B to A) B mean of y A Difference between level means of B (with A fixed at a level) depends on the level of A. If the trend of mean response over A do not change when B changes from one level to another, the interaction is said to be synergistic; demonstrated by two unparallel lines with slopes of the same sign. Page 25

26 Interaction Plot: Battery Experiment data battery; input mat temp life; datalines; ; proc means noprint; var life; by mat temp; output out=batterymean mean=mn; symbol1 v=circle i=join; symbol2 v=square i=join; symbol3 v=triangle i=join; proc gplot; plot mn*temp=mat; run; Page 26

27 Interaction Plot for Material and Temperature Page 27

28 Multiple comparison when factors dont interact When factors dont interact, i.e., the F test for interaction is not significant in the ANOVA, factor level means can be compared to draw conclusions regarding their effects on response. Var(ȳ i.. )= σ2 nb,var(ȳ.j.) = σ2 na For A or rows : Var(ȳ i.. ȳ i..) = 2σ2 nb ; For B or columns : Var(ȳ.j. ȳ.j.)= 2σ2 na Tukey s method For rows: CD = q α(a,ab(n 1)) 2 MSE 2 nb For columns: CD = q α(b,ab(n 1)) 2 MSE 2 na Bonferroni method: CD = t α/2m,ab(n 1) S.E., where S.E. depends on whether for rows or columns. Page 28

29 Level mean comparison when A and B interact: An Example B A ,21 38, ,47 10,14 Compare factor level means of A: ȳ 1.. = ( )/4 =25 ȳ 2.. = ( )/4 =25=ȳ 1.. Does Factor A have effect on the response? Page 29

30 mean of y B A When interactions are present, be careful interpreting factor level means (row or column) comparisons because it can be misleading. Usually, we will directly compare treatment means (or cell means) instead. Page 30

31 Multiple comparisons when factors interact: treatment (cell) mean comparison When factors interact, multiple comparison is usually directly applied to treatment means µ ij = µ + τ i + β j +(τβ) ij vs µ i j = µ + τ i + β j +(τβ) i j ˆµ ij =ȳ ij. and ˆµ ij =ȳ i j. Var(ȳ ij. ȳ i j.)= 2σ2 n there are ab treatment means and m 0 = ab(ab 1) 2 pairs. Tukey s method: CD = q α(ab, ab(n 1)) 2 MSE 2 n Bonferroni s method. CD = t α/2m,ab(n 1) MSE 2 n Page 31

32 SAS Code and Output proc glm data=battery; class mat temp; model life=mat temp mat*temp; means mat temp /tukey lines; lsmeans mat temp/tdiff adjust=tukey; run; ============================================================ The GLM Procedure Least Squares Means Adjustment for Multiple Comparisons: Tukey LSMEAN mat life LSMEAN Number Page 32

33 Least Squares Means for Effect mat t for H0: LSMean(i)=LSMean(j) / Pr > t Dependent Variable: life i/j Page 33

34 Output (continued) Least Squares Means Adjustment for Multiple Comparisons: Tukey LSMEAN temp life LSMEAN Number Least Squares Means for Effect temp t for H0: LSMean(i)=LSMean(j) / Pr > t Dependent Variable: life i/j < < Page 34

35 Output (continued) The GLM Procedure Least Squares Means Adjustment for Multiple Comparisons: Tukey LSMEAN mat temp life LSMEAN Number Page 35

36 Output i/j Page 36

37 Output (continued) i/j Page 37

38 Fitting Response Curves or Surfaces Battery Experiment: Goal: Model the functional relationship between lifetime and temperature at every material level. Page 38

39 Material is qualitative while temperature is quantitative Want to fit the response using effects of material, temperature and their interactions Temperature has quadratic effect. Could use orthogonal polynomials as before. Here we will simply t and t 2. Levels of material need to be converted to dummy variables denoted by x 1 and x 2 as follows. mat x 1 x For convenience, convert temperature to -1,0 and 1 using t = temperature Page 39

40 Fitting Response Curve: Model matrix mat temp ==> x1 x2 t tˆ2 x1*t x2*t x1*tˆ2 x2*tˆ The following model is used: y ijk = β 0 + β 1 x 1 + β 2 x 2 + β 3 t + β 4 x 1 t + β 5 x 2 t + β 6 t 2 + β 7 x 1 t 2 + β 8 x 2 t 2 + ɛ ijk Want to estimate the coefficients: β 0, β 1, β 2,..., β 8 using regression Page 40

41 SAS Code: Battery Life Experiment data life; input mat temp if mat=1 then x1=1; if mat=1 then x2=0; if mat=2 then x1=0; if mat=2 then x2=1; if mat=3 then x1=-1; if mat=3 then x2=-1; t=(temp-70)/55; t2=t*t; x1t=x1*t; x2t=x2*t; x1t2=x1*t2; x2t2=x2*t2; datalines; ; proc reg; model y=x1 x2 t x1t x2t t2 x1t2 x2t2; run; Page 41

42 SAS output Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error CorrectedTotal Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 x <.0001 x t <.0001 x1t x2t t x1t x2t Page 42

43 Results From the SAS output, the fitted model is ŷ = x x t +1.71x 1 t 12.79x 2 t 3.08t x 1 t x 2 t 2 Note that terms with insignificant coefficients are still kept in the fitted model here. In practice, model selection may be employed to remove unimportant terms and choose the best fitted model. But we will not pursue it in this course. The model above are in terms of both x 1, x 2 and t. We can specify the level of material, that is, the values of dummy variable x 1 and x 2, to derive fitted response curves for material at different levels. Page 43

44 Fitted Response Curves Three response curves: Material at level 1 (x 1 =1,x 2 =0) E(y 1t )= t t 2 Material at level 2 (x 1 =0,x 2 =1) E(y 2t ) = t 17.12t 2 Material at level 3 (x 1 = 1,x 2 = 1) Where t = temperature E(y 3t ) = t 31t 2 These curves can be used to predict lifetime of battery at any temperature between 15 and 125 degree. But one needs to be careful about extrapolation. For example, the fitted curve at Material level 1 suggests that lifetime of a battery can be infinity when temperature goes to infinity, which is clearly false. Page 44

45 General Factorial Design and Model Factorial Design - including all possible level combinations a levels of Factor A, b levels of Factor B,... (Straightforward ANOVA if all fixed effects) In 3 factor model nabc observations Need n>1to test for all possible interactions Statistical Model (3 factor) y ijkl = µ + τ i + β j + γ k +(τβ) ij +(βγ) jk +(τγ) ik +(τβγ) ijk + ɛ ijkl i =1, 2,...,a j =1, 2,...,b k =1, 2,...,c l =1, 2,...,n Page 45

46 Analysis of Variance Table Source of Sum of Degrees of Mean F 0 Variation Squares Freedom Square Factor A SS A a 1 MS A F 0 Factor B SS B b 1 MS B F 0 Factor C SS C c 1 MS C F 0 AB SS AB (a 1)(b 1) MS AB F 0 AC SS AC (a 1)(c 1) MS AC F 0 BC SS BC (b 1)(c 1) MS BC F 0 ABC SS ABC (a 1)(b 1)(c 1) MS ABC F 0 Error SS E abc(n 1) MS E Total SS T abcn 1 Page 46

47 option nocenter data bottling; input carb pres spee devi; datalines; : : : : ; proc glm; class carb pres spee; model devi=carb pres spee; run; Bottling Experiment: SAS Code Page 47

48 Bottling Experiment: SAS Output Dependent Variable: devi Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Co Total R-Square Coeff Var Root MSE devi Mean Source DF Type I SS Mean Square F Value Pr > F carb <.0001 pres <.0001 carb*pres spee carb*spee pres*spee carb*pres*spee Page 48

49 Interaction Plot for Carb and Pressure Page 49

50 General Factorial Model Usual assumptions and diagnostics Multiple comparisons: simple extensions of the two-factor case Often higher order interactions are negligible. Beyond three-way interactions difficult to picture. Pooled together with error (increase df E ) Page 50

51 Battery Life Experiment: Blocking in Factorial Design: Example An engineer is studying the effective lifetime of some battery. Two factors, plate material and temperature, are involved. There are three types of plate materials (1, 2, 3) and three temperature levels (15, 70, 125). Four batteries are tested at each combination of plate material and temperature, and all 36 tests are run in random order. The experiment and the resulting observed battery life data are given below. Page 51

52 temperature material ,155,74,180 34,40,80,75 20,70,82, ,188,159, ,122,106,115 25,70,58, ,110,168, ,120,150,139 96,104,82,60 If we assume further that four operators (1,2,3,4) were hired to conduct the experiment. It is known that different operators can cause systematic difference in battery lifetime. Hence operators should be treated as blocks The blocking scheme is every operator conduct a single replicate of the full factorial design For each treatment (treatment combination), the observations were in the order of the operators 1, 2, 3, and 4. This is a blocked factorial design Page 52

53 Statistical Model for Blocked Factorial Experiment y ijk = µ + τ i + β j +(τβ) ij + δ k + ɛ ijk i =1, 2,...,a, j =1, 2,...,band k =1, 2,...,n, δ k is the effect of the kth block. randomization restriction is imposed. (complete block factorial design). interactions between blocks and treatment effects are assumed to be negligible. The previous ANOVA table for the experiment should be modified as follows: Add: Block Sum of Square SS Blocks = 1 ab Modify: Error Sum of Squares: k y 2..k y2... abn D.F. n 1 (new)ss E = (old)ss E SS Blocks D.F. (ab 1)(n 1) other inferences should be modified accordingly. Page 53

54 data battery; input mat temp oper life; dataline; ; proc glm; class mat temp oper; model life=oper mat temp; output out=new1 r=resi p=pred; SAS Code and Output Page 54

55 ==================================================== Dependent Variable: life Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error CorTotal Source DF Type I SS Mean Square F Value Pr > F oper mat temp <.0001 mat*temp Page 55

56 Factorial Experiment with Two blocking factors Use Latin square as blocking scheme 1. Suppose the experimental factors are F1 and F2. A has three levels (1,2, 3) and B has 2 levels. There are 3*2=6 treatment combinations. These treatments can be represented by Latin letters F1 F2 Treatment 1 1 A 1 2 B 2 1 C 2 2 D 3 1 E 3 2 F Page 56

57 Two blocking factors are Block1 and Block2, each with 6 blocks. 2. A 6 6 Latin square can be used as the blocking scheme: Block1 Block A B C D E F 2 B C D E F A 3 C D E F A B 4 D E F A B C 5 E F A B C D 6 F A B C D E 3. Statistical Model y ijkl = µ + α i + τ j + β k +(τβ) jk + θ l + ɛ ijkl where, α i and θ l are blocking effects, τ j, β k and (τβ) jk are the treatment main effects and interactions Page 57