Comparison of Several Multivariate Means

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Comparison of Several Multivariate Means

• Shyh-Kang Jeng
• Department of Electrical Engineering/
• Graduate Institute of Communication/
• Graduate Institute of Networking and Multimedia

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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

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Example 6.1 Effluent Data from Two Labs
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Single Response (Univariate) Case
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Multivariate Extension Notations
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Result 6.1
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Test of Hypotheses and Confidence Regions
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Example 6.1 Check Measurements from Two Labs
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Experiment Design for Paired Comparisons
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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
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Alternative View
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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

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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
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Example 6.2 Sleeping-Dog Data
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Contrast Matrix
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Test for Equality of Treatments in a Repeated
Measures Design
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Example 6.2 Contrast Matrix
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Example 6.2 Test of Hypotheses
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Example 6.2 Simultaneous Confidence Intervals
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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

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Assumptions Concerning the Structure of Data
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Pooled Estimate of Population Covariance Matrix
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Result 6.2
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Proof of Result 6.2
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Wishart Distribution
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Test of Hypothesis
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Example 6.3 Comparison of Soaps Manufactured in
Two Ways
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Example 6.3
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Result 6.3 Simultaneous Confidence Intervals
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Example 6.4 Electrical Usage of Homeowners with
and without ACs
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Example 6.4 Electrical Usage of Homeowners with
and without ACs
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Example 6.4 95 Confidence Ellipse
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Bonferroni Simultaneous Confidence Intervals
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Result 6.4
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Proof of Result 6.4
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Remark
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Example 6.5
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Multivariate Behrens-Fisher Problem

• Test H0 m1-m20
• Population covariance matrices are unequal
• Sample sizes are not large
• Populations are multivariate normal
• Both sizes are greater than the number of
  variables

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Approximation of T2 Distribution
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Confidence Region
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Example 6.6

• Example 6.4 data

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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

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One-Way MANOVA
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Assumptions about the Data
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Univariate ANOVA
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Univariate ANOVA
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Univariate ANOVA
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Univariate ANOVA
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Concept of Degrees of Freedom
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Concept of Degrees of Freedom
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Examples 6.7 6.8
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MANOVA
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MANOVA
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MANOVA
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Distribution of Wilks Lambda
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Test of Hypothesis for Large Size
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Popular MANOVA Statistics Used in Statistical
Packages
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Example 6.9
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Example 6.8
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Example 6.9
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Example 6.9
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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

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Example 6.10
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Example 6.10
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Example 6.10
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Bonferroni Intervals for Treatment Effects
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Result 6.5 Bonferroni Intervals for Treatment
Effects
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Example 6.11 Example 6.10 Data
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Test for Equality of Covariance Matrices

• With g populations, null hypothesis
• H0 S1 S2 . . . Sg S
• Assume multivariate normal populations
• Likelihood ratio statistic for testing H0

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Boxs M-Test
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Example 6.12

• Example 6.10 - nursing home data

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Example 6.13 Plastic Film Data
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Two-Way ANOVA
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Effect of Interactions
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Two-Way ANOVA
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Two-Way ANOVA
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Two-Way MANOVA
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Two-Way MANOVA
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Two-Way MANOVA
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Two-Way MANOVA
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Bonferroni Confidence Intervals
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Example 6.13 MANOVA Table
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Example 6.13 Interaction
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Example 6.13 Effects of Factors 1 2
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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?

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Example 6.14 Love and Marriage Data
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Population Profile
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Profile Analysis
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Test for Parallel Profiles
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Test for Coincident Profiles
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Test for Level Profiles
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Example 6.14
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Example 6.14 Test for Parallel Profiles
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Example 6.14 Sample Profiles
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Example 6.14 Test for Coincident Profiles
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Example 6.15 Ulna Data, Control Group
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Example 6.15 Ulna Data, Treatment Group
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Comparison of Growth Curves
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Comparison of Growth Curves
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Example 6.15
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Example 6.16 Comparing Multivariate and
Univariate Tests
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Example 6.14 Comparing Multivariate and
Univariate Tests
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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

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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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