Multifactor Experiments

1. Multifactor Experiments November 26, 2013 Gui Citovsky, Julie Heymann, Jessica Sopp, Jin Lee, Qi Fan, Hyunhwan Lee, JinzhuYu, Lenny Horowitz, Shuvro Biswas

2. Outline • Two-Factor Experiments with Fixed Crossed Factors • 2k Factorial Experiments • Other Selected Types of Two-Factor Experiments

3. Two-Factor Experiments with Fixed Crossed Factors

4. First, single factor • Comparison of two or more treatments (groups) • Single treatment factor • Example: A study to compare the average flight distances for three types of golf balls differing in the shape of dimples on them: circular, fat elliptical, thin elliptical • Treatments circular, fat elliptical, thin elliptical • Treatment factor type of ball

5. Single factor continued

6. Two-Factor Experiments With Fixed Crossed Factors • Two fixed factors, A with a ≥ 2 levels and B with b ≥ 2 levels • abtreatment combinations • If there are n observations obtained under each treatment combination (nreplicates), then there is a total of abn experimental units

7. Two-Factor Experiments With Fixed Crossed Factors • Example: Heat treatment experiment to evaluate the effects of a quenching medium (two levels: oil and water) and quenching temperature (three levels: low, medium, high) on the surface hardness of steel • 2 x 3 = 6 treatment combinations • If 3 steel samples are treated for each combination, we have N = 18 observations

8. Model and Estimates of its Parameters Let yijk=kth observation on the (i,j)thtreatment combination, i=1,2,…,a , j=1,2,…,b, and k=1,2,…,n. Let random variable Yijk correspond to observed outcome yijk. Basic Model: and independent where

9. Table format

10. Parameters Grand Mean: ithRow Average: jthColumn Average: jthColumn Main Effect: ithRow Main Effect: (i,j)thRow Column Interaction

11. Least Squares Estimates

12. Variance • Sample variance for (i, j)th cell is: • Pooled estimate for σ2:

13. Example Experiment to study how mechanical bonding strength of capacitors depends on the type of substrate (factor A) and bonding material (factor B). 3 substrates: Al2O3 with bracket, Al2O3 no bracket, BeO no bracket 4 types of bonding material: Epoxy I, Epoxy II, Solder I and Solder II Four capacitors were tested at each factor level combination

14. Example continued Pooled sample variance:

15. Example continued: Sample Means

16. Example continued: Other Model Parameters

17. Two- Way Analysis of Variance We define the following sum of squares:

18. Analysis of Variance • Degrees of Freedom: • SST: N– 1 • SSA: a – 1 • SSB: b – 1 • SSAB: (a – 1)(b– 1) • SSE: N– ab • SST = SSA + SSB + SSAB + SSE. • Similarly, the degrees of freedom also follow this identity, i.e.

19. Analysis of Variance • Mean squares =

20. Hypothesis Test We test three hypotheses: Not all Not all Not all If all interaction terms are equal to zero, then the effect of one factor on the mean response does not depend on the level of the other factors.

21. When do we reject H0? • Use F-statistics to test our hypotheses by taking the ratio of the mean squares to the MSE. Reject Reject Reject • We test the interaction hypothesis H0AB first.

22. Summary (Table 13.5)

23. Example: Bonding Strength of Capacitors DataCapacitors; input Bonding $ Substrate $ Strength @@; Datalines; Epoxy1 Al203 1.51 Epoxy1 Al203 1.96 Epoxy1 Al203 1.83 Epoxy1 Al203 1.98 … ; procGLM plots=diagnostics data=Capacitors; TITLE"Analysis of Bonding Strength of Capacitors"; CLASS Bonding Substrate; Model Strength = Bonding | Substrate; run;

24. Bonding Strength of Capacitors ANOVA Table • At α=0.05, we can reject H0Band H0ABbut fail to reject H0A. • The main effect of bonding material and the interaction between the bonding material and the substrate are both significant. • The main effect of substrate is not significant at our α.

25. Main Effects Plot • Definition: A main effects plot is a line plot of the row means of factor and A and the column means of factor B.

26. Interaction Plot

27. Model Diagnostics with Residual Plots • Why do we look at residual plots? • Is our constant variance assumption true? • Is our normality assumption true?

28. 2kFactorial Experiments

29. 2k Factorial Experiments • 2k factorial experiments is a class of multifactor experiments consists of design in which each factor is studied at 2 levels. • If there are k factors, then we have 2k treatment combinations • 2-factor and 3-factor experiments can be generalized to >3-factor experiments

30. 22 experiment • 22 Experiment: experiment with factors A and B, each at two levels. ab = (A high, B high) a= (A high, B low) b = (A low, B high) (1) = (A low, B low)

31. 22 experiment cont’d Assume a balanced design with n observations for each treatment combinations, denote these observations by yij Yij~ N(µi, σ2) i = (1), a, b, ab j = 1, 2, … , n

32. 22 experiment cont’d • Main effect of factor A (): difference in the mean response between the high level of A and the low level of A, averaged over the levels of B • Main effect of factor B (: difference in the mean response between the high level of B and the low level of B, averaged over the levels of A • Interaction effect of AB (): difference between the mean effect of A at the high level of B and at the low level of B = = ( =

33. 22 experiment cont’d The least square estimates of the main effects and the interaction effects are obtained by replacing the treatment means by the corresponding cell sample means. Est. Main Effect A = = Est. Main Effect B = = Est. Interaction AB = =

34. 22 experiment cont’d *Notice that the term-by-term products of any two contrast vectors equal the third one

35. 23 experiment • 23 Experiment: experiment with factors A, B, and C with n observations. Yij~ N(µi, σ2), i= (1), a, b, ab, c, ac, bc, abcj = 1, 2, … , n. Est. Main Effect A = Est. Main Effect B = Est. Main Effect C =

36. 23 experiment cont’d Est. Interaction Effect AB = Est. Interaction Effect BC = Est. Interaction Effect AC = Est. Interaction Effect ABC =

37. 23 experiment cont’d

38. 23 experiment example Factors affecting bicycle performance: Seat height (Factor A): 26" (-), 30" (+) Generator (Factor B): Off (-), On(+) Tire Pressure (Factor C): 40 psi (-), 55 psi (+)

39. 23 experiment example cont’d

40. 23 experiment example cont’d A = = -10.875 B = = 3.125 C = = -3.125 AB = = -0.625 AC = = 1.125 BC = = 0.125 ABC = = 0.875 significant

41. 2kexperiment • 2kexperiments, where k>3. • niid observations yij(j = 1,2,…n) at the ithtreatment combination and its sample mean yi(i = 1,2,…, 2k) has the following estimated effect. Est. Effect =

42. Statistical Inference for 2k Experiments Basic Notations and Derivations d.f.

43. CI and Hypotheses Test with t Test • Therefore a CI for any population effect is given by • The t-statistic for testing the significance of any estimated effect is

44. Hypotheses Test with F Test • Equivalently, we can use F test to do it • The estimated effect is significant at level if

45. Sums of Squares for Effects The effects are mutually orthogonal contrasts.

46. Regression Approach to 2k Experiments • a 22experiment • Multiple regression model

47. Regression Approach to 2k Experiments • 23 experiment • If all interactions are dropped from the model, the new fitted model is

48. Regression Approach to 2k Experiments • The interpolation formula

49. Bicycle Example: Main Effects Model A(seat height)= -10.875 B(generator) = 3.125 C(tire pressure) = -3.125 = 47.1875 • main effects model • minimum travel time

50. Bicycle Example: Main Effects Model Sums of squares for omitted interactions effects = 1.56+5.0625+0.0625+4.1875 = 10.875 d.f. = 4 Pure SSE = 33.5 d.f = 8 pooled SSE = 33.5 + 10.875 d.f. = 12 (total) MSE = = 3.698