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Comparison of Several Multivariate Means

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Title: Comparison of Several Multivariate Means


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
1
2
3
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

Present
Halothane
Absent
Low
High
CO2 pressure
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 Wilks 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
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