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Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 12 Analysis of Factor-Effects: Linear Combinations, Contrasts, Polynomials. Analysis of Ballistic Limit Velocities MGH Exercise #6.6. Model and Assumptions. y ijk = m + a i + b j + (ab) ij + e ijk I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 12 Analysis of Factor-Effects: Linear Combinations, Contrasts, Polynomials

2. Analysis of Ballistic Limit VelocitiesMGH Exercise #6.6

3. Model and Assumptions yijk = m + ai + bj+ (ab)ij+eijk where yij = ballistic velocity for the kth repeat of the ith nose cone and the jth angle m = overall mean ballistic velocity ai = fixed effect of the ith nose cone on mean velocity bi = fixed effect of the jth angle on mean velocity (ab)ij = fixed effect of the interaction between the ith nose cone and the jth angle on mean velocity eij = random experimental error, NID(0,s2)

4. Analysis of Variance Table Mason, Gunst, & Hess: Exercise 6.6 3 The ANOVA Procedure Dependent Variable: velocity Sum of Source DF Squares Mean Square F Value Pr > F Model 5 275139.6111 55027.9222 7861.13 <.0001 Error 12 84.0000 7.0000 Corrected Total 17 275223.6111 R-Square Coeff Var Root MSE velocity Mean 0.999695 0.258192 2.645751 1024.722 Source DF Anova SS Mean Square F Value Pr > F shape 2 7916.4444 3958.2222 565.46 <.0001 angle 1 266206.7222 266206.7222 38029.5 <.0001 shape*angle 2 1016.4444 508.2222 72.60 <.0001

5. Test Procedure Reject H0 if F = MSAB/MSE > F0.05(2,12)=3.89 Interaction Null Hypothesis The effect of differing nose cones on mean ballistic velocity is the same for all angles of launch.

6. Conclusion There is sufficient evidence (p = 0.0001) to conclude at a significance level of 0.05 that the effects of nose cone design and launch angle differ on the mean ballistic velocities of 10 mm rolled armor projectiles.

7. Interaction Averages

8. Fisher’s LSD = 5.53 Statistical vs. Practical Significance for the Interaction

9. Linear Combinations of Parameters Estimable Functions of Parameters Not Usually of Interest Estimator Standard Error t Statistic

10. Several individual comparisons could be of interest

11. Interest in comparing Wheelchair and crutches vs. amputee and hearing loss

12. Contrasts of Effects Estimable Factor Effects Contrasts Elimination of the overall mean requires contrasts of main effect averages. (Note: Want to compare factor effects.) Elimination of main effects from interaction comparisons requires contrasts of the interaction Aaverages. (Note: Want interaction effects to measure variability that is unaccounted for by or in addition to the main effects.)

13. Statistical Independence Orthogonal linear combinations are statistically independent Orthogonal contrasts are statistically independent

14. Analysis of Water Pump Prototypes Designs 1 & 3 are Nominally Identical Designs 2 & 4 are Nominally Identical MGH Ex 6.22

15. Analysis of Water Pump Prototypes Model and Assumptions yij = m + ai + eij where yij = jth mileage driven before failure using ith pump design m = overall mean mileage ai = fixed effect of the ith design on mileage eij = random experimental error, NID(0,s2)

16. Prespecified Contrasts Are these statistics optimal ? Companies Designs 1&3 vs. 2&4 Designs 1 vs. 3 Designs 2 vs. 4

17. t-test Comparison of Company Effects The TTEST Procedure Statistics Lower CL Upper CL Lower CL Variable company N Mean Mean Mean Std Dev Std Dev mileage A 20 26430 27896 29363 2382.8 3133.2 mileage B 20 24630 25431 26232 1301.4 1711.3 mileage Diff (1-2) 849.45 2465.5 4081.6 2063.1 2524.4 Statistics Upper CL Variable company Std Dev Std Err Minimum Maximum mileage A 4576.3 700.6 24215 31416 mileage B 2499.4 382.65 21613 27724 mileage Diff (1-2) 3253.4 798.29 T-Tests Variable Method Variances DF t Value Pr > |t| mileage Pooled Equal 38 3.09 0.0037 mileage Satterthwaite Unequal 29.4 3.09 0.0044

18. t-test Comparison of Designs 1&3 with Designs 2&4 Designs Recoded: 1 & 3 = 1, 2 & 4 = 2 The TTEST Procedure Statistics Lower CL Upper CL Lower CL Variable design N Mean Mean Mean Std Dev Std Dev mileage 2 10 24506 24870 25235 350.47 509.53 mileage 4 10 26158 26594 27030 419.17 609.41 mileage Diff (1-2) -2251 -1724 -1196 424.42 561.69 Statistics Upper CL Variable design Std Dev Std Err Minimum Maximum mileage 2 930.2 161.13 24215 25930 mileage 4 1112.5 192.71 25804 27724 mileage Diff (1-2) 830.65 251.2 T-Tests Variable Method Variances DF t Value Pr > |t| mileage Pooled Equal 18 -6.86 <.0001 mileage Satterthwaite Unequal 17.5 -6.86 <.0001

19. t-test Comparison of Design #1 with Design #3 The TTEST Procedure Statistics Lower CL Upper CL Lower CL Variable design N Mean Mean Mean Std Dev Std Dev mileage 1 10 30675 30922 31169 237.72 345.61 mileage 3 10 23070 24268 25465 1151.8 1674.5 mileage Diff (1-2) 5518.5 6654.5 7790.5 913.56 1209 Statistics Upper CL Variable design Std Dev Std Err Minimum Maximum mileage 1 630.95 109.29 30321 31416 mileage 3 3057 529.53 21613 26233 mileage Diff (1-2) 1787.9 540.69 T-Tests Variable Method Variances DF t Value Pr > |t| mileage Pooled Equal 18 12.31 <.0001 mileage Satterthwaite Unequal 9.77 12.31 <.0001

20. t-test Comparison of Design #2 with Design #4 The TTEST Procedure Statistics Lower CL Upper CL Lower CL Variable design N Mean Mean Mean Std Dev Std Dev mileage 2 10 24506 24870 25235 350.47 509.53 mileage 4 10 26158 26594 27030 419.17 609.41 mileage Diff (1-2) -2251 -1724 -1196 424.42 561.69 Statistics Upper CL Variable design Std Dev Std Err Minimum Maximum mileage 2 930.2 161.13 24215 25930 mileage 4 1112.5 192.71 25804 27724 mileage Diff (1-2) 830.65 251.2 T-Tests Variable Method Variances DF t Value Pr > |t| mileage Pooled Equal 18 -6.86 <.0001 mileage Satterthwaite Unequal 17.5 -6.86 <.0001

21. t-test Comparisons Incorrect Analyses: Standard deviations are inefficient and possibly biased

22. Linear Model Analysis The ANOVA Procedure Dependent Variable: mileage Sum of Source DF Squares Mean Square F Value Pr > F Model 3 270956900.1 90318966.7 101.64 <.0001 Error 36 31990423.8 888622.9 Corrected Total 39 302947323.9 R-Square Coeff Var Root MSE mileage Mean 0.894403 3.535431 942.6680 26663.45 Source DF Anova SS Mean Square F Value Pr > F design 3 270956900.1 90318966.7 101.64 <.0001

23. Analysis of Water Pump Prototypes Comparison of All 4 Designs H0: ai = 0 for all i vs Ha: ai 0 for some i Reject H0 and accept Ha if F > F0.05(3,36) = 2.87 From the ANOVA Table, F = 101.64. Conclusion On the basis of this analysis, there is sufficient evidence (p = 0.001) to conclude that the mean mileage before failure of these prototype pumps differs by design type.

24. Analysis of Water Pump Prototypes Multiple Comparisons of Pump Design Average Mileages Using Fisher’s Least Significant Difference procedure, average mileages for two designs are significantly different from each other if their difference exceeds 1,135.4 miles. As indicated by the line in the above table, the average mileages for Designs 2 and 3 are not significantly different. Mileages for all other pairwise comparisons of designs are significantly different; in particular, Design 1 has a significantly greater average mileage measurement than the other designs.

25. Prespecified Contrasts Companies Company A Company B

26. Contrasts: the Details Companies Start with difference of averages

27. Contrasts: the Details Companies Start with a contrast

28. Orthogonal Sets of Contrasts Many other sets of orthogonal contrasts

29. General Method for Forming Orthogonal Contrasts

30. Comparing the Mean of Two Factor Level Effects with a Third Factor Effect Some Contrasts for Comparing Factor Level Averages Comparing Two Factor Level Effects

31. One Set of Main Effects Contrasts :Qualitative Factor Levels

32. Main Effects Contrasts :Qualitative Factor Levels • Three statistically independent contrasts of the response averages • A partitioning of the main effects degrees of freedom into single degree-of-freedom contrasts (a = 4: df = 3)

33. Simultaneous Test Single degree-of- freedom contrasts Sums of Squares and Contrasts a-1 mutually orthonormal contrast vectors Orthonormal Basis Set ANY set of orthonormal contrast vectors

34. Possible Analyses of Water Pump Prototypes

35. Warping of Copper Plates:Quantitative Factor Levels MGH Table 6.7

36. Model and Assumptions yijk = m + ai + bj+ (ab)ij+eijk where yijk = warping measurement for the kth repeat at the ith temp. using a plate having the jth amount of copper m = overall mean warping measurement ai = fixed effect of the ith temperature on the mean warping bi = fixed effect of the jth copper content on the mean warping (ab)ij = fixed effect of the interaction between the ith temperature and the jth copper content on the mean warping eij = random experimental error, NID(0,s2)

37. Warping of Copper Plates Quantitative factor levels HOW does mean warping change with the factor levels ? MGH Table 6.7

38. Warping of Copper Plates Are there contrast vectors that quantify curvature ? 35 30 25 Average Warping 20 15 10 0 50 75 150 100 125 Temperature (deg F)

39. Warping of Copper Plates Are there contrast vectors that quantify curvature ? 35 30 25 Average Warping 20 15 10 0 20 40 60 80 100 Copper Content (%)

40. Main Effects Contrasts :Quantitative Factor Levels Assumption Mean response can be well approximated by a low-order polynomial function of the factor levels

41. Orthogonal Polynomials Coded Factor Levels xi = a0 + a1i Equally Spaced e.g., Temp = 50, 75, 100, 125 >>>> xi = 25 + 25i Orthogonal Polynomials Not Orthongonal Orthogonal With coded factor levels, orthogonal polynomials are only a function of n

42. Orthogonal Polynomials Linear Effect Orthogonality Constraint c1’1 = 0

43. Orthogonal Polynomials Linear Effect Orthogonality Constraint c1’1 = 0 Solution Centered Data Values

44. Orthogonal Polynomials Quadratic Effect Orthogonality Constraints c2’1 = 0 and c2’c1 = 0 Solution

45. Main Effects Contrasts :Equally Spaced Quantitative Factor Levels Recurrence Relation for Coded Levels

46. Main Effects Contrasts :Equally Spaced Quantitative Factor Levels n=4 q1 = Linear q2 = Quadratic q3 = Cubic

47. Linear Combinations of Parameters Estimable Functions of Parameters Estimator Same for Contrasts Standard Error t Statistic

48. Possible Trends in Average Warping Changes in Average Warping