11 Comparison of Several Multivariate Means Shyh-Kang Jeng Department of Electrical Engineering/...

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Comparison of Several Comparison of Several Multivariate MeansMultivariate Means

Shyh-Kang JengShyh-Kang JengDepartment of Electrical Engineering/Department of Electrical Engineering/

Graduate Institute of Communication/Graduate Institute of Communication/

Graduate Institute of Networking and Graduate Institute of Networking and MultimediaMultimedia

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Paired ComparisonsPaired ComparisonsMeasurements are recorded under Measurements are recorded under different sets of conditions different sets of conditions See if the responses differ See if the responses differ significantly over these setssignificantly over these setsTwo or more treatments can be Two or more treatments can be administered to the same or similar administered to the same or similar experimental unitsexperimental unitsCompare responses to assess the Compare responses to assess the effects of the treatmentseffects of the treatments

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Example 6.1: Example 6.1: Effluent Data from Two LabsEffluent Data from Two Labs

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Single Response (Univariate) CaseSingle Response (Univariate) Case

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Multivariate Extension: NotationsMultivariate Extension: Notations

2tment under trea variable

2tment under trea 2 variable

2tment under trea 1 variable

--------------------------------

1tment under trea variable

1tment under trea 2 variable

1tment under trea 1 variable

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Result 6.1Result 6.1

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Test of Hypotheses and Test of Hypotheses and Confidence RegionsConfidence Regions

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Example 6.1: Check Measurements Example 6.1: Check Measurements from Two Labsfrom Two Labs

zero includesBoth

32.255.71,-or 11/61.41847.927.13:

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Experiment Design for Experiment Design for Paired ComparisonsPaired Comparisons

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Repeated Measures Design for Repeated Measures Design for Comparing MeasurementsComparing Measurements

qq treatments are compared with treatments are compared with respect to a single response variablerespect to a single response variable

Each subject or experimental unit Each subject or experimental unit receives each treatment once over receives each treatment once over successive periods of timesuccessive periods of time

12121212

Example 6.2: Treatments in an Example 6.2: Treatments in an Anesthetics ExperimentAnesthetics Experiment

19 dogs were initially given the drug 19 dogs were initially given the drug pentobarbitol followed by four pentobarbitol followed by four treatmentstreatments

Halothane

Present

Absent

CO2 pressureLow High

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Example 6.2: Sleeping-Dog DataExample 6.2: Sleeping-Dog Data

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Contrast MatrixContrast Matrix

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contrast Halothane

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Example 6.2: Test of HypothesesExample 6.2: Test of Hypotheses

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Example 6.2: Simultaneous Example 6.2: Simultaneous Confidence IntervalsConfidence Intervals

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Comparing Mean Vectors from Comparing Mean Vectors from Two PopulationsTwo Populations

Populations: Sets of experiment Populations: Sets of experiment settingssettingsWithout explicitly controlling for unit-Without explicitly controlling for unit-to-unit variability, as in the paired to-unit variability, as in the paired comparison casecomparison caseExperimental units are randomly Experimental units are randomly assigned to populationsassigned to populationsApplicable to a more general Applicable to a more general collection of experimental unitscollection of experimental units

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Assumptions Concerning the Assumptions Concerning the Structure of DataStructure of Data

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Result 6.2Result 6.2

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Proof of Result 6.2Proof of Result 6.2

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Example 6.3: Comparison of Soaps Example 6.3: Comparison of Soaps Manufactured in Two WaysManufactured in Two Ways

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Example 6.3Example 6.3

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Result 6.3: Simultaneous Result 6.3: Simultaneous Confidence IntervalsConfidence Intervals

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

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

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intervals confidence ussimultaneo 95%

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Example 6.4: Example 6.4: 95% Confidence Ellipse95% Confidence Ellipse

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Bonferroni Simultaneous Bonferroni Simultaneous Confidence IntervalsConfidence Intervals

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2:

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Result 6.4Result 6.4

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Proof of Result 6.4Proof of Result 6.4

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Example 6.5 Example 6.5

063.0

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127.121.7,or 17.46499.54.74:

15.264208.886

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Data 6.4 Example

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Multivariate Behrens-Fisher Multivariate Behrens-Fisher ProblemProblem

Test Test HH00: : 11--22=0 =0

Population covariance matrices are Population covariance matrices are unequalunequal

Sample sizes are not largeSample sizes are not large

Populations are multivariate normalPopulations are multivariate normal

Both sizes are greater than the Both sizes are greater than the number of variablesnumber of variables

3737

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Approximation of Approximation of TT22 Distribution Distribution

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Confidence RegionConfidence Region

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Example 6.6Example 6.6

Example 6.4 dataExample 6.4 data

4040rejected is 0:,32.666.15

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1

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Example 6.10: Nursing Home DataExample 6.10: Nursing Home Data

Nursing homes can be classified by Nursing homes can be classified by the owners: private (271), non-profit the owners: private (271), non-profit (138), government (107)(138), government (107)Costs: nursing labor, dietary labor, Costs: nursing labor, dietary labor, plant operation and maintenance plant operation and maintenance labor, housekeeping and laundry labor, housekeeping and laundry laborlaborTo investigate the effects of To investigate the effects of ownership on costsownership on costs

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One-Way MANOVAOne-Way MANOVA

tlysignificandiffer components

mean which not, if and, same, theare vectors

mean population he whether teinvestigat toused is

VAriance) Of ANalysis ate(MultivariMANOVA

,,, : Population

,,, :2 Population

,,, :1 Population

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Assumptions about the DataAssumptions about the Data

normal temultivaria is populationEach

matrix

covariancecommon a have spopulation All

tindependen

are spopulationdifferent from sample Random

,,2,1,mean with

population a from sample random :,,,21

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Univariate ANOVAUnivariate ANOVA

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Univariate ANOVAUnivariate ANOVA

)SS()SS()SS()(SS

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Univariate ANOVAUnivariate ANOVA

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Univariate ANOVAUnivariate ANOVA

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Popular MANOVA Statistics Used Popular MANOVA Statistics Used in Statistical Packagesin Statistical Packages

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Example 6.9Example 6.9

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Example 6.8Example 6.8

14972792639:Total

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160548 :Mean

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Example 6.10: Nursing Home DataExample 6.10: Nursing Home Data

Nursing homes can be classified by Nursing homes can be classified by the owners: private (271), non-profit the owners: private (271), non-profit (138), government (107)(138), government (107)Costs: nursing labor, dietary labor, Costs: nursing labor, dietary labor, plant operation and maintenance plant operation and maintenance labor, housekeeping and laundry labor, housekeeping and laundry laborlaborTo investigate the effects of To investigate the effects of ownership on costsownership on costs

62626262

Example 6.10Example 6.10

63636363

Example 6.10Example 6.10

304.0230.0610.0584.0

235.0453.0821.0

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Example 6.10Example 6.10

analysesboth by Reject

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Bonferroni Intervals for Bonferroni Intervals for Treatment EffectsTreatment Effects

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Example 6.11: Example 6.10 DataExample 6.11: Example 6.10 Data

019.0,021.0,026.0,058.0: and

for intervals confidence ussimultaneo 95%

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for interval confidence ussimultaneo 95%

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Importance of Experimental DesignImportance of Experimental DesignDifferences could appear in only one Differences could appear in only one of the many characteristics or a few of the many characteristics or a few treatment combinationstreatment combinationsDifferences may become lost among Differences may become lost among all the inactive onesall the inactive onesBest preventative is a good Best preventative is a good experimental designexperimental design– Do not include too many other variables Do not include too many other variables

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