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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Multivariate Extension: NotationsMultivariate Extension: Notations

2tment under trea variable

2tment under trea 2 variable


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1tment under trea variable



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

zero includesBoth

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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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Example 6.2: Simultaneous Example 6.2: Simultaneous Confidence IntervalsConfidence Intervals

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Assumptions Concerning the Assumptions Concerning the Structure of DataStructure of Data

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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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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

32323232

Bonferroni Simultaneous Bonferroni Simultaneous Confidence IntervalsConfidence Intervals

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Proof of Result 6.4Proof of Result 6.4

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Example 6.5 Example 6.5

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127.121.7,or 17.46499.54.74:

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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

3838

Approximation of Approximation of TT22 Distribution Distribution

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Example 6.4 dataExample 6.4 data

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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

42424242

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

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,,, :2 Population

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Assumptions about the DataAssumptions about the Data

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Univariate ANOVAUnivariate ANOVA

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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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Example 6.11: Example 6.10 DataExample 6.11: Example 6.10 Data

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Strategy for Multivariate Strategy for Multivariate Comparison of TreatmentsComparison of Treatments

Try to identify outliersTry to identify outliers– Perform calculations with and without Perform calculations with and without

the outliersthe outliers

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and all characteristicsand all characteristics

103103103103

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

that are not expected to show that are not expected to show differencesdifferences

11 comparison of several multivariate means shyh-kang jeng department of electrical engineering/...

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