1. 1 Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture #10 Testing the Statistical Significance of Factor Effects

2. 2 Fixed Effects Analysis of Variance Assumptions Levels of the factors are preselected and the only ones for which inferences are desired Levels of the factors are preselected and the only ones for which inferences are desired The ANOVA model contains parameters for all main effects and interactions that might change the mean response The ANOVA model contains parameters for all main effects and interactions that might change the mean response Experimental errors are statistically independent Experimental errors are statistically independent Experimental errors are normally distributed, with mean 0 and constant standard deviation Experimental errors are normally distributed, with mean 0 and constant standard deviation

3. 3 Flow Rate Experiment MGH Fig 6.1

4. 4 Flow Rate Experiment ABCD 0.30 0.25 0.20 Average Flow Rate Constant Standard Deviation Reasonable Constant Standard Deviation Reasonable Filter Type 0.35

5. 5 -0.04 -0.02 0.0 0.02 0.04 -1.5-0.50.00.51.01.5 Flowrate Residual Standard Normal Quantile Normal Quantile Plot of Flowrate Residuals Normality Reasonable

6. 6 Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials Distributional Properties SS A & SS E are statistically independent lecture8QF.ppt

7. 7 Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials iff  1 =... =  a = 0 lecture8QF.ppt

8. 8 Testing Factor Effects Simultaneous Test for Main Effects y ij =  +  i + e ij i = 1,..., a; j = 1,..., r Single-Factor Model H 0 :  1 =  2 =... =  a vs. H a :  i  j for some (i,j) Equivalent lecture8QF.ppt

9. 9 Analysis of Variance for the Flow Rate Data MGH Table 6.5 Conclusion ?

10. 10 Extra Sum of Squares Hierarchy Principle An interaction is included only if ALL main effects and lower-order interactions involving the interaction factors are included in the model and analysis Full Model All hierarchical model terms Reduced Model One or more model terms deleted All remaining model terms are hierarchical

11. 11 Hierarchical Models

12. 12 Extra Sum of Squares F-Test for Hierarchical Models For balanced designs, F Tests for fixed effects derived from QF, mean squares, and ESS are identical

13. 13 Where Are We? Multifactor Experiments Complete Factorial Experiments Completely Randomized Designs Check Model Assumptions – Carefully! Constant variance: scatterplots, boxplots, standard deviations vs. averages Normal Distribution: maybe quantile plots if r > 2 If r = 2, residuals are equal and opposite in sign Analysis of Variance to assess overall main effects and interactions Next: Comparison of factor level effects

14. 14 Mean Comparisons Which filter flowrate averages are significantly different from one another ? Which filter flowrate averages are significantly different from one another ? Which filter flowrate means are different from one another ? Which filter flowrate means are different from one another ?

15. 15 Mean Comparisons Which temperature and catalyst averages are significantly different from one another ? Which temperature and catalyst averages are significantly different from one another ? Which temperature and catalyst means are different from one another ? Which temperature and catalyst means are different from one another ?

16. 16 Flow Rate Data Pairwise Comparisons Problem 6 Tests, Each at  = 0.05 Problem 6 Tests, Each at  = 0.05

17. 17 General Method for Quantifying Factor Main Effects and Interactions Main Effects for Factor A Main Effects for Factor B Interaction Effects for Factors A & B Change in average response due to changes in the levels of Factor A Change in average response due to changes in the levels of Factor B Change in average response due joint changes in Factors A & B in excess of changes in the main effects

18. 18 Normal Distribution Theory Multivariate Normal Distribution lecture8QF.ppt

19. 19 Normal Distribution Theory Marginal Distributions P n - p Linear Transformations

20. 20 General Result

21. 21 Individual Prespecified Comparisons (Single-factor Model Used for Illustration) Ho:  =  0 vs Ha:   0 Reject Ho if | t | > t  /2 Very general result MGH Sec 6.2.2

22. 22 Pairwise Comparisons Specific application Specific application Ho:  i  i’ vs Ha:  i  i’ Reject Ho if | t | > t  /2 Ordinary t-Test

23. 23 Algebraic Main Effects Representation Two Factors, Two Levels (No Repeats for Simplicity)

24. 24 Algebraic Main Effects Representation y = (y 11 y 12 y 21 y 22 ) m A = ( -1 -1 +1 +1) Contrast Two Factors, Two Levels (No Repeats for Simplicity) MGH Sec 5.3

25. 25 Algebraic Main Effects Representation y = (y 11 y 12 y 21 y 22 ) m A = ( -1 -1 +1 +1) M(B) = m B ’y /2 m B = ( -1 +1 -1 +1) Two Factors, Two Levels (No Repeats for Simplicity)

26. 26 Algebraic Interaction Effects Representation y = (y 11 y 12 y 21 y 22 ) m AB = ( +1 -1 -1 +1) Two Factors, Two Levels (No Repeats for Simplicity)

27. 27 Effects Representation : Two-Level, Two-Factor Factorial Factor Levels: Lower = - 1 Upper = +1 Factor Levels Effects Representation Factor AFactor B A B AB Lower Upper Lower Upper Lower Upper 1 1 1 1 Note: AB = A x B MGH Table 5.6 Mutually Orthogonal Contrasts Mutually Orthogonal Contrasts

28. 28 Two-Level Complete Factorial Effects Representation Designate one factor level as -1, the other as +1 Lay out a table with column headings for each factor (e.g., A, B,...) n = 2 k (k = # factors) Lay out the main effects representation A: first n/2 values equal -1, last n/2 values equal +1 B: first n/4 values equal -1, next n/4 equal +1, next n/4 equal -1, last n/4 equal +1... K: alternate -1 and +1 values Interaction effects representation Multiply the corresponding main effects column values MGH Exhibit 5.5

29. 29 Pilot Plant Chemical-Yield Study MGH Table 5.7

30. 30 Calculated Effects y = Vector of averages across repeats (response vector if r = 1) m = Effects representation vector Effect = m’ y / 2 k-1

31. 31 Pilot Plant Chemical-Yield Study s e = 2.828 SE EFFECT = s e (8/[2 x {2 (3-1) } 2 ]) 1/2 = 2.8/2 = 1.414 MGH Table 5.7 Mutually Orthogonal Contrasts Mutually Orthogonal Contrasts

32. 32 Pilot Plant Chemical-Yield Study MGH Table 6.4 Conclusion ?

33. 33

34. 34 Note: 6 Treatments x 8 Blocks x ( 2 – 1 ) = 48 df for Error from Repeats

35. 35 Note: Block Averages and Block Variability Increase Together

36. 36 Note: Perfect Symmetry Since r = 2

37. 37 Note: Averages Change but Variability is RelativelyConstant

38. 38 Conclusions ?

39. 39 Should only the statistically significant effects be kept in the model ?

40. 40