Comparison of Several Multivariate Means ShyhKang Jeng Department

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Comparison of Several Multivariate Means Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of

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

Paired Comparisons Measurements are recorded under different sets of conditions See if the responses

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 2

Example 6. 1: Effluent Data from Two Labs 3

Example 6. 1: Effluent Data from Two Labs 3

Single Response (Univariate) Case 4

Single Response (Univariate) Case 4

Multivariate Extension: Notations 5

Multivariate Extension: Notations 5

Result 6. 1 6

Result 6. 1 6

Test of Hypotheses and Confidence Regions 7

Test of Hypotheses and Confidence Regions 7

Example 6. 1: Check Measurements from Two Labs 8

Example 6. 1: Check Measurements from Two Labs 8

Experiment Design for Paired Comparisons 1 2 3 n . . . Treatments 1

Experiment Design for Paired Comparisons 1 2 3 n . . . Treatments 1 and 2 assigned at random 9

Alternative View 10

Alternative View 10

Repeated Measures Design for Comparing Measurements q treatments are compared with respect to a

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 11

Example 6. 2: Treatments in an Anesthetics Experiment 19 dogs were initially given the

Example 6. 2: Treatments in an Anesthetics Experiment 19 dogs were initially given the drug pentobarbitol followed by four treatments Present 3 4 2 1 Low High Halothane Absent CO 2 pressure 12

Example 6. 2: Sleeping-Dog Data 13

Example 6. 2: Sleeping-Dog Data 13

Contrast Matrix 14

Contrast Matrix 14

Test for Equality of Treatments in a Repeated Measures Design 15

Test for Equality of Treatments in a Repeated Measures Design 15

Example 6. 2: Contrast Matrix 16

Example 6. 2: Contrast Matrix 16

Example 6. 2: Test of Hypotheses 17

Example 6. 2: Test of Hypotheses 17

Example 6. 2: Simultaneous Confidence Intervals 18

Example 6. 2: Simultaneous Confidence Intervals 18

Comparing Mean Vectors from Two Populations: Sets of experiment settings Without explicitly controlling for

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

Assumptions Concerning the Structure of Data 20

Assumptions Concerning the Structure of Data 20

Pooled Estimate of Population Covariance Matrix 21

Pooled Estimate of Population Covariance Matrix 21

Result 6. 2 22

Result 6. 2 22

Proof of Result 6. 2 23

Proof of Result 6. 2 23

Wishart Distribution 24

Wishart Distribution 24

Test of Hypothesis 25

Test of Hypothesis 25

Example 6. 3: Comparison of Soaps Manufactured in Two Ways 26

Example 6. 3: Comparison of Soaps Manufactured in Two Ways 26

Example 6. 3 27

Example 6. 3 27

Result 6. 3: Simultaneous Confidence Intervals 28

Result 6. 3: Simultaneous Confidence Intervals 28

Example 6. 4: Electrical Usage of Homeowners with and without ACs 29

Example 6. 4: Electrical Usage of Homeowners with and without ACs 29

Example 6. 4: Electrical Usage of Homeowners with and without ACs 30

Example 6. 4: Electrical Usage of Homeowners with and without ACs 30

Example 6. 4: 95% Confidence Ellipse 31

Example 6. 4: 95% Confidence Ellipse 31

Bonferroni Simultaneous Confidence Intervals 32

Bonferroni Simultaneous Confidence Intervals 32

Result 6. 4 33

Result 6. 4 33

Proof of Result 6. 4 34

Proof of Result 6. 4 34

Remark 35

Remark 35

Example 6. 5 36

Example 6. 5 36

Multivariate Behrens-Fisher Problem Test H 0: m 1 -m 2=0 Population covariance matrices are

Multivariate Behrens-Fisher Problem Test H 0: m 1 -m 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

Approximation of T 2 Distribution 38

Approximation of T 2 Distribution 38

Confidence Region 39

Confidence Region 39

Example 6. 6 Example 6. 4 data 40

Example 6. 6 Example 6. 4 data 40

Example 6. 10: Nursing Home Data Nursing homes can be classified by the owners:

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 41

One-Way MANOVA 42

One-Way MANOVA 42

Assumptions about the Data 43

Assumptions about the Data 43

Univariate ANOVA 44

Univariate ANOVA 44

Univariate ANOVA 45

Univariate ANOVA 45

Univariate ANOVA 46

Univariate ANOVA 46

Univariate ANOVA 47

Univariate ANOVA 47

Concept of Degrees of Freedom 48

Concept of Degrees of Freedom 48

Concept of Degrees of Freedom 49

Concept of Degrees of Freedom 49

Examples 6. 7 & 6. 8 50

Examples 6. 7 & 6. 8 50

MANOVA 51

MANOVA 51

MANOVA 52

MANOVA 52

MANOVA 53

MANOVA 53

Distribution of Wilk’s Lambda 54

Distribution of Wilk’s Lambda 54

Test of Hypothesis for Large Size 55

Test of Hypothesis for Large Size 55

Popular MANOVA Statistics Used in Statistical Packages 56

Popular MANOVA Statistics Used in Statistical Packages 56

Example 6. 9 57

Example 6. 9 57

Example 6. 8 58

Example 6. 8 58

Example 6. 9 59

Example 6. 9 59

Example 6. 9 60

Example 6. 9 60

Example 6. 10: Nursing Home Data Nursing homes can be classified by the owners:

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 61

Example 6. 10 62

Example 6. 10 62

Example 6. 10 63

Example 6. 10 63

Example 6. 10 64

Example 6. 10 64

Bonferroni Intervals for Treatment Effects 65

Bonferroni Intervals for Treatment Effects 65

Result 6. 5: Bonferroni Intervals for Treatment Effects 66

Result 6. 5: Bonferroni Intervals for Treatment Effects 66

Example 6. 11: Example 6. 10 Data 67

Example 6. 11: Example 6. 10 Data 67

Test for Equality of Covariance Matrices With g populations, null hypothesis H 0: S

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

Box’s M-Test 69

Box’s M-Test 69

Example 6. 12 Example 6. 10 - nursing home data 70

Example 6. 12 Example 6. 10 - nursing home data 70

Example 6. 13: Plastic Film Data 71

Example 6. 13: Plastic Film Data 71

Two-Way ANOVA 72

Two-Way ANOVA 72

Effect of Interactions 73

Effect of Interactions 73

Two-Way ANOVA 74

Two-Way ANOVA 74

Two-Way ANOVA 75

Two-Way ANOVA 75

Two-Way MANOVA 76

Two-Way MANOVA 76

Two-Way MANOVA 77

Two-Way MANOVA 77

Two-Way MANOVA 78

Two-Way MANOVA 78

Two-Way MANOVA 79

Two-Way MANOVA 79

Bonferroni Confidence Intervals 80

Bonferroni Confidence Intervals 80

Example 6. 13: MANOVA Table 81

Example 6. 13: MANOVA Table 81

Example 6. 13: Interaction 82

Example 6. 13: Interaction 82

Example 6. 13: Effects of Factors 1 & 2 83

Example 6. 13: Effects of Factors 1 & 2 83

Profile Analysis A battery of p treatments (tests, questions, etc. ) are administered to

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? 84

Example 6. 14: Love and Marriage Data 85

Example 6. 14: Love and Marriage Data 85

Population Profile 86

Population Profile 86

Profile Analysis 87

Profile Analysis 87

Test for Parallel Profiles 88

Test for Parallel Profiles 88

Test for Coincident Profiles 89

Test for Coincident Profiles 89

Test for Level Profiles 90

Test for Level Profiles 90

Example 6. 14 91

Example 6. 14 91

Example 6. 14: Test for Parallel Profiles 92

Example 6. 14: Test for Parallel Profiles 92

Example 6. 14: Sample Profiles 93

Example 6. 14: Sample Profiles 93

Example 6. 14: Test for Coincident Profiles 94

Example 6. 14: Test for Coincident Profiles 94

Example 6. 15: Ulna Data, Control Group 95

Example 6. 15: Ulna Data, Control Group 95

Example 6. 15: Ulna Data, Treatment Group 96

Example 6. 15: Ulna Data, Treatment Group 96

Comparison of Growth Curves 97

Comparison of Growth Curves 97

Comparison of Growth Curves 98

Comparison of Growth Curves 98

Example 6. 15 99

Example 6. 15 99

Example 6. 16: Comparing Multivariate and Univariate Tests 100

Example 6. 16: Comparing Multivariate and Univariate Tests 100

Example 6. 14: Comparing Multivariate and Univariate Tests 101

Example 6. 14: Comparing Multivariate and Univariate Tests 101

Strategy for Multivariate Comparison of Treatments Try to identify outliers – Perform calculations with

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 102

Importance of Experimental Design Differences could appear in only one of the many characteristics

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 103