1. 11 Comparison of Several Multivariate Means Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

2. 22 Paired Comparisons Measurements are recorded under different sets of conditions See if the responses differ significantly over these sets Two or more treatments can be administered to the same or similar experimental units Compare responses to assess the effects of the treatments

3. 33 Example 6.1: Effluent Data from Two Labs

4. 44 Single Response (Univariate) Case

5. 55 Multivariate Extension: Notations

6. 66 Result 6.1

7. 77 Test of Hypotheses and Confidence Regions

8. 88 Example 6.1: Check Measurements from Two Labs

9. 99 Experiment Design for Paired Comparisons... 123 n Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random

10. 1010 Alternative View

11. 1111 Repeated Measures Design for Comparing Measurements q treatments are compared with respect to a single response variable Each subject or experimental unit receives each treatment once over successive periods of time

12. 1212 Example 6.2: Treatments in an Anesthetics Experiment 19 dogs were initially given the drug pentobarbitol followed by four treatments Halothane Present Absent CO 2 pressure LowHigh 12 34

13. 1313 Example 6.2: Sleeping-Dog Data

14. 1414 Contrast Matrix

15. 1515 Test for Equality of Treatments in a Repeated Measures Design

16. 1616 Example 6.2: Contrast Matrix

17. 1717 Example 6.2: Test of Hypotheses

18. 1818 Example 6.2: Simultaneous Confidence Intervals

19. 1919 Comparing Mean Vectors from Two Populations Populations: Sets of experiment settings Without explicitly controlling for unit- to-unit variability, as in the paired comparison case Experimental units are randomly assigned to populations Applicable to a more general collection of experimental units

20. 2020 Assumptions Concerning the Structure of Data

21. 2121 Pooled Estimate of Population Covariance Matrix

22. 2222 Result 6.2

23. 2323 Proof of Result 6.2

24. 2424 Wishart Distribution

25. 2525 Test of Hypothesis

26. 2626 Example 6.3: Comparison of Soaps Manufactured in Two Ways

27. 2727 Example 6.3

28. 2828 Result 6.3: Simultaneous Confidence Intervals

29. 2929 Example 6.4: Electrical Usage of Homeowners with and without ACs

30. 3030 Example 6.4: Electrical Usage of Homeowners with and without ACs

31. 3131 Example 6.4: 95% Confidence Ellipse

32. 3232 Bonferroni Simultaneous Confidence Intervals

33. 3333 Result 6.4

34. 3434 Proof of Result 6.4

35. 3535 Remark

36. 3636 Example 6.5

37. 37 Multivariate Behrens-Fisher Problem Test H 0 :  1 -  2 =0 Population covariance matrices are unequal Sample sizes are not large Populations are multivariate normal Both sizes are greater than the number of variables 37

38. 38 Approximation of T 2 Distribution 38

39. 39 Confidence Region 39

40. 40 Example 6.6 Example 6.4 data 40

41. 4141 Example 6.10: Nursing Home Data Nursing homes can be classified by the owners: private (271), non-profit (138), government (107) Costs: nursing labor, dietary labor, plant operation and maintenance labor, housekeeping and laundry labor To investigate the effects of ownership on costs

42. 4242 One-Way MANOVA

43. 4343 Assumptions about the Data

44. 4444 Univariate ANOVA

45. 4545 Univariate ANOVA

46. 4646 Univariate ANOVA

47. 4747 Univariate ANOVA

48. 4848 Concept of Degrees of Freedom

49. 4949 Concept of Degrees of Freedom

50. 5050 Examples 6.7 & 6.8

51. 5151 MANOVA

52. 5252 MANOVA

53. 5353 MANOVA

54. 5454 Distribution of Wilk’s Lambda

55. 5555 Test of Hypothesis for Large Size

56. 5656 Popular MANOVA Statistics Used in Statistical Packages

57. 5757 Example 6.9

58. 5858 Example 6.8

59. 5959 Example 6.9

60. 6060 Example 6.9

61. 6161 Example 6.10: Nursing Home Data Nursing homes can be classified by the owners: private (271), non-profit (138), government (107) Costs: nursing labor, dietary labor, plant operation and maintenance labor, housekeeping and laundry labor To investigate the effects of ownership on costs

62. 6262 Example 6.10

63. 6363 Example 6.10

64. 6464 Example 6.10

65. 6565 Bonferroni Intervals for Treatment Effects

66. 6666 Result 6.5: Bonferroni Intervals for Treatment Effects

67. 6767 Example 6.11: Example 6.10 Data

68. 68 Test for Equality of Covariance Matrices With g populations, null hypothesis H 0 :  1 =  2 =... =  g =  Assume multivariate normal populations Likelihood ratio statistic for testing H 0

69. 69 Box’s M-Test

70. 70 Example 6.12 Example 6.10 - nursing home data

71. 7171 Example 6.13: Plastic Film Data

72. 7272 Two-Way ANOVA

73. 7373 Effect of Interactions

74. 7474 Two-Way ANOVA

75. 7575 Two-Way ANOVA

76. 7676 Two-Way MANOVA

77. 7777 Two-Way MANOVA

78. 7878 Two-Way MANOVA

79. 7979 Two-Way MANOVA

80. 8080 Bonferroni Confidence Intervals

81. 8181 Example 6.13: MANOVA Table

82. 8282 Example 6.13: Interaction

83. 8383 Example 6.13: Effects of Factors 1 & 2

84. 8484 Profile Analysis A battery of p treatments (tests, questions, etc.) are administered to two or more group of subjects The question of equality of mean vectors is divided into several specific possibilities –Are the profiles parallel? –Are the profiles coincident? –Are the profiles level?

85. 8585 Example 6.14: Love and Marriage Data

86. 8686 Population Profile

87. 8787 Profile Analysis

88. 8888 Test for Parallel Profiles

89. 8989 Test for Coincident Profiles

90. 9090 Test for Level Profiles

91. 9191 Example 6.14

92. 9292 Example 6.14: Test for Parallel Profiles

93. 9393 Example 6.14: Sample Profiles

94. 9494 Example 6.14: Test for Coincident Profiles

95. 9595 Example 6.15: Ulna Data, Control Group

96. 9696 Example 6.15: Ulna Data, Treatment Group

97. 9797 Comparison of Growth Curves

98. 9898 Comparison of Growth Curves

99. 9999 Example 6.15

100. 100100 Example 6.16: Comparing Multivariate and Univariate Tests

101. 101101 Example 6.14: Comparing Multivariate and Univariate Tests

102. 102102 Strategy for Multivariate Comparison of Treatments Try to identify outliers –Perform calculations with and without the outliers Perform a multivariate test of hypothesis Calculate the Bonferroni simultaneous confidence intervals –For all pairs of groups or treatments, and all characteristics

103. 103103 Importance of Experimental Design Differences could appear in only one of the many characteristics or a few treatment combinations Differences may become lost among all the inactive ones Best preventative is a good experimental design –Do not include too many other variables that are not expected to show differences