Comparison of Several Multivariate Means ShyhKang Jeng Department
- Slides: 103
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 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
Single Response (Univariate) Case 4
Multivariate Extension: Notations 5
Result 6. 1 6
Test of Hypotheses and Confidence Regions 7
Example 6. 1: Check Measurements from Two Labs 8
Experiment Design for Paired Comparisons 1 2 3 n . . . Treatments 1 and 2 assigned at random 9
Alternative View 10
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 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
Contrast Matrix 14
Test for Equality of Treatments in a Repeated Measures Design 15
Example 6. 2: Contrast Matrix 16
Example 6. 2: Test of Hypotheses 17
Example 6. 2: Simultaneous Confidence Intervals 18
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
Pooled Estimate of Population Covariance Matrix 21
Result 6. 2 22
Proof of Result 6. 2 23
Wishart Distribution 24
Test of Hypothesis 25
Example 6. 3: Comparison of Soaps Manufactured in Two Ways 26
Example 6. 3 27
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 30
Example 6. 4: 95% Confidence Ellipse 31
Bonferroni Simultaneous Confidence Intervals 32
Result 6. 4 33
Proof of Result 6. 4 34
Remark 35
Example 6. 5 36
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
Confidence Region 39
Example 6. 6 Example 6. 4 data 40
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
Assumptions about the Data 43
Univariate ANOVA 44
Univariate ANOVA 45
Univariate ANOVA 46
Univariate ANOVA 47
Concept of Degrees of Freedom 48
Concept of Degrees of Freedom 49
Examples 6. 7 & 6. 8 50
MANOVA 51
MANOVA 52
MANOVA 53
Distribution of Wilk’s Lambda 54
Test of Hypothesis for Large Size 55
Popular MANOVA Statistics Used in Statistical Packages 56
Example 6. 9 57
Example 6. 8 58
Example 6. 9 59
Example 6. 9 60
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 63
Example 6. 10 64
Bonferroni Intervals for Treatment Effects 65
Result 6. 5: Bonferroni Intervals for Treatment Effects 66
Example 6. 11: Example 6. 10 Data 67
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
Example 6. 12 Example 6. 10 - nursing home data 70
Example 6. 13: Plastic Film Data 71
Two-Way ANOVA 72
Effect of Interactions 73
Two-Way ANOVA 74
Two-Way ANOVA 75
Two-Way MANOVA 76
Two-Way MANOVA 77
Two-Way MANOVA 78
Two-Way MANOVA 79
Bonferroni Confidence Intervals 80
Example 6. 13: MANOVA Table 81
Example 6. 13: Interaction 82
Example 6. 13: Effects of Factors 1 & 2 83
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
Population Profile 86
Profile Analysis 87
Test for Parallel Profiles 88
Test for Coincident Profiles 89
Test for Level Profiles 90
Example 6. 14 91
Example 6. 14: Test for Parallel Profiles 92
Example 6. 14: Sample Profiles 93
Example 6. 14: Test for Coincident Profiles 94
Example 6. 15: Ulna Data, Control Group 95
Example 6. 15: Ulna Data, Treatment Group 96
Comparison of Growth Curves 97
Comparison of Growth Curves 98
Example 6. 15 99
Example 6. 16: Comparing Multivariate and Univariate Tests 100
Example 6. 14: Comparing Multivariate and Univariate Tests 101
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 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
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