Chapter 13 Analysis of Multi-factor Experiments Dec 6 th 2007

1. Chapter 13 Analysis of Multi-factor ExperimentsDec 6th 2007

2. Our Group Member • Part I: Background Introduction: • 1.Ruirui Pan: Why do we work on this topic? • 2.Xuanti Ying: Introduction to related technology

3. Part II: Theoretical Derivation • 3. Parameter Estimation: Ji-Young Yun • 4. Theory of two factor experiments: Mingyi Hong • 5. Theory of 2^k experiments Zheng Zhao

4. Part III: Data Analysis • 6. Data analysis of 2^2 experiment Wei Hu • 7. Data analysis of 2^3 experiment Hao Zhang • 8. Data analysis of 2^k experiment Ti Zhou

5. Part IV: Model analysis and Conclusion • 9.Model diagnostic and SAS programming Jun Huang • 10.Regression approach and conclusion Wenbin Zhang

6. Why do we work on this topic? by Ruirui Pan

7. What is multifactor experiment? • In statistics, a multifactor experiment (also called factorial experiment) is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors.

8. Basic Concepts • The primary purpose of an experiment is to evaluate how a set of predictor variables (called factors in experimental design jargon) affect a response variable. • The different possible values of a factor are called its levels. • Each treatment is a particular combination of the levels of different treatment factors. Example

9. Factors • A Factor is a linked set of experimental conditions we may wish to compare e.g. Levels of temperature Different methods of solving a problem Pressure of bicycle tire • Two types: Treatment factors Nuisance factors

10. Example • Suppose an engineer wishes to study the total power used by each of two different motors, A and B, running at each of two different speeds, 2000 or 3000 RPM. So the factorial experiment would consist of 8 experimental units: motor A at 2000 RPM, motor B at 2000 RPM, motor A at 3000 RPM, and motor B at 3000 RPM. Each combination of a single level selected from every factor is present twice.

11. Single factor experiment ---one-way ANOVA Two-factor Experiments with Fixed Crossed Factors--- Factor A with a levels and B with b levels are crossed, there are a*b treatment combinations 2^3 Factorial Experiments---3 factors with 2 levels each, so there are 8 treatment combinations 2^k Factorial Experiments--- k factors with 2 levels each, so there are 2^k treatment combinations The importance of multifactor experiments

12. Introduction to related technology By Xuanti Ying

13. related technology • ANOVA

14. Introduction to ANOVA • Analysis of variance (ANOVA) is used to test hypothesis about differences between two or more means. The t-test based on the standard error of the difference between two means can only be used to test differences between two means. When there are more than two means, it is possible to compare each mean with each other mean using t-tests. However, conducting multiple t-tests can lead to severe inflation of the Type I error rate. Analysis of variance can be used to test differences among several means for significance without increasing the Type I error rate.

15. Who Developed this Technology • The initial techniques of the analysis of variance were developed by the statistician and geneticist R.A.Fisher in the 1920s and 1930s.

16. The Significance of ANOVA One important reason for using ANOVA methods rather than multiple two-group studies analyzed via t-tests is that the former method is more efficient, and with fewer observations we can gain more information. • Controlling for factors • Detects interactive effects (The term interaction was first used by Fisher, 1926.)

17. Logic of ANOVA • Partitioning of the sum of squares The fundamental technique is a partitioning of the total sum of squares into components related to the effects used in the model. • The F-test The F-test is used for comparisons of the components of the total deviation. the F-test is the mean square for each main effect and the interaction effect divided by the within variance.

18. Several Types of ANOVA • One way ANOVA is used to test for differences among two or more independent groups. • Factorial ANOVA or Two way ANOVA is used when want to study the effects of two or more treatment variables. (our case) • Mixed-design ANOVA

19. Tests Supplementing ANOVA • All pairwise t-test • Fisher’s LSD (Least Significant Difference Method) • Tukey’s HSD (Honestly Significantly Different Test proposed by the statistician John Tukey) • Newman-Keuls method • Duncan’s Procedure (similar to the Newman-Keuls method)

20. Parameter Estimation by   Ji-Young Yun            • EXAMPLE • Consider a grade treatment experiment to evaluate the effects of sleeping hours and the percentage of attendance of the class

21. Factor A : sleeping hours Levels enough (if sleeping hours ≥ 8) normal (if 6 ≤ sleeping hours ˂ 8) lack (if sleeping hours ˂ 6 )

22. Factor B:the percentage of attendance High (the percentage ≥ 50%) Low (the percentage ˂ 50%)

23. A grade treatment experiment to evaluate the effects A and B

24. Each student numbers of treatment combination is 3 • a = 3 • b = 2 • n = 3 • N = (3)(2)(3) =18

25. A grade treatment experiment to evaluate the effects A and B

26. Parameters & Estimates yijk: the kthobservation on the (i, j)th treatment combination the mean of cell (i, j) i.i.d random error, normal distribution : i th row main effect : j th column main effect : (i, j)th row-column interaction

27. Parameters & Estimates : sample mean of the (i, j)th cell; least square estimate of

28. Analysis of Variance ANOVA Table for Crossed Two-Way Layout

29. SST = 2223.92 SSA = 788.64 SSB = 414.72 SSAB = 18.9 SSE = 1001.66 SST = SSA + SSB + SSAS + SSE

30. MSA = 394.43 FA = MSA/MSE=4.33 MSB = 414.72 FB = MSB/MSE=4.55 MSAB = 9.45 FAB = MSAB/MSE = 0.103 MSE = 91.06 The main effect of sleeping hours and the percentage of attendance are both highly significant, but the interaction between the sleeping hours and the percentage of attendance is NOT significant at the .1 level.

31. Theory Derivation for Two Factor ExperimentMingyi Hong

32. Overview • Sometimes a researcher might want to simultaneously examine the effects of two treatments. • Examples The effect of sex and race on wage The effect of the level of pollution and the level of city services on housing prices

33. Data from a Balanced Two-way Layout

34. The Model

35. The Sum of Squares

36. The Chi Square Distributions • Actually, each of the previous sum of squares divides the variance is a chi square distribution. For example,

37. The ANOVA Identity Total DF = Row DF + Column DF + Interaction DF + Error DF

38. The Tests

42. Multiple Comparisons Between Rows and/or Columns • Pairwise comparisons between the row main effects and/or between the column main effects are generally of interest only when the interactions are nonsignificant. • Tukey method to determine 100(1-a)% simultaneous confidence intervals is as follows.

43. Theory derivation about 2^k experiment Zheng Zhao

44. One Factor Experiment with Two levels

45. 2² Experiment—The Introduction of Factor B with Two levels • Factor A = High (+) or Low (-) • Factor B = High (+) or Low (-) Four Treatment Combinations • ab = (A High, B High) • a = (A High, B Low) • b = (A Low, B High) • (1) = (A Low, B Low)

46. Yij ~ N (μi, σ²), i = 1, 2, …,2^k; j = 1,2,…,n

47. Main Effect and Interaction Effect of 2² Experiment

48. Estimated Effects • Est. Main Effect A • Est. Main Effect B • Est. Interaction AB

49. Factor A Factor B Factor C 2³ = 8 Treatment Combinations (1): Low, Low, Low a: High, Low, Low b: Low, High, Low ab: High, High, Low c: Low, Low, High ac: High, Low, High bc: Low, High, High abc: High, High, High 2³ Experiment—One More Factor Considered