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

2 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 Example 6.1: Effluent Data from Two Labs

4 Single Response (Univariate) Case

5 Multivariate Extension: Notations

6 Result 6.1

7 Test of Hypotheses and Confidence Regions

8 Example 6.1: Check Measurements from Two Labs

9 Experiment Design for Paired Comparisons 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 Alternative View

11 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 Example 6.2: Treatments in an Anesthetics Experiment 19 dogs were initially given the drug pentobarbitol followed by four treatments Halothane Present Absent CO2 pressure LowHigh 12 34

13 Example 6.2: Sleeping-Dog Data

14 Contrast Matrix

15 Test for Equality of Treatments in a Repeated Measures Design

16 Example 6.2: Contrast Matrix

17 Example 6.2: Test of Hypotheses

18 Example 6.2: Simultaneous Confidence Intervals

19 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 Assumptions Concerning the Structure of Data

21 Pooled Estimate of Population Covariance Matrix

22 Result 6.2

23 Proof of Result 6.2

24 Wishart Distribution

25 Test of Hypothesis

26 Example 6.3: Comparison of Soaps Manufactured in Two Ways

27 Example 6.3

28 Result 6.3: Simultaneous Confidence Intervals

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

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

31 Example 6.4: 95% Confidence Ellipse

32 Bonferroni Simultaneous Confidence Intervals

33 Result 6.4

34 Proof of Result 6.4

35 Remark

36 Example 6.5

37 Example 6.9: 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

38 One-Way MANOVA

39 Assumptions about the Data

40 Univariate ANOVA

41 Univariate ANOVA

42 Univariate ANOVA

43 Univariate ANOVA

44 Concept of Degrees of Freedom

45 Concept of Degrees of Freedom

46 Examples 6.6 & 6.7

47 MANOVA

48 MANOVA

49 MANOVA

50 Distribution of Wilk’s Lambda

51 Test of Hypothesis for Large Size

52 Popular MANOVA Statistics Used in Statistical Packages

53 Example 6.8

54 Example 6.8

55 Example 6.8

56 Example 6.8

57 Example 6.9: 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

58 Example 6.9

59 Example 6.9

60 Example 6.9

61 Bonferroni Intervals for Treatment Effects

62 Result 6.5: Bonferroni Intervals for Treatment Effects

63 Example 6.10: Example 6.9 Data

64 Example 6.11: Plastic Film Data

65 Two-Way ANOVA

66 Two-Way ANOVA

67 Two-Way ANOVA

68 Two-Way MANOVA

69 Effect of Interactions

70 Two-Way MANOVA

71 Two-Way MANOVA

72 Two-Way MANOVA

73 Bonferroni Confidence Intervals

74 Example 6.11: MANOVA Table

75 Example 6.11: Interaction

76 Example 6.11: Effects of Factors 1 & 2

77 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?

78 Example 6.12: Love and Marriage Data

79 Population Profile

80 Profile Analysis

81 Test for Parallel Profiles

82 Test for Coincident Profiles

83 Test for Level Profiles

84 Example 6.12

85 Example 6.12: Test for Parallel Profiles

86 Example 6.12: Sample Profiles

87 Example 6.12: Test for Coincident Profiles

88 Example 6.13: Ulna Data, Control Group

89 Example 6.13: Ulna Data, Treatment Group

90 Comparison of Growth Curves

91 Comparison of Growth Curves

92 Example 6.13

93 Example 6.14: Comparing Multivariate and Univariate Tests

94 Example 6.14: Comparing Multivariate and Univariate Tests

95 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

96 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