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

11

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 MultiGraduate Institute of Networking and Multimediamedia

22

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

33

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

44

Single Response (Univariate) CaseSingle Response (Univariate) Case

n

std

n

std

ttHH

tns

Dt

ND

njXXD

dn

dn

n

n

d

dj

jjj

)2/()2/(

for interval confidence )%1(100

)2/( if0:offavor in 0:Reject

:/

),(:

,,2,1,

11

110

1

2

21

55

Multivariate Extension: NotationsMultivariate Extension: Notations

2tment under trea variable

2tment under trea 2 variable


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

1tment under trea variable



2

22

12

1

21

11

pX

X

X

pX

X

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j

j

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j

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66

Result 6.1Result 6.1

n

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n

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pnpd

dpj

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jpjpjp

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njN

DDD

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12111

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)1(:'

,,2,1),,(:

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DDDDSDD

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77

Test of Hypotheses and Test of Hypotheses and Confidence RegionsConfidence Regions

n

s

ptd

n

sF

pn

pnd

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':regions Confidence

)()1(

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if 0: offavor in 0:Reject

sdifference observed:,,,

δdSδd

dSd

δδ

88

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:

74.3,46.22or 11/26.19947.936.9:

0:Reject

47.9)05.0(9

1026.13

27.13

36.9

0026.00012.0

0012.00055.027.1336.911

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F

T

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99

Experiment Design for Experiment Design for Paired ComparisonsPaired Comparisons

. . .

. . .

1 2 3 n

Treatments1 and 2assigned atrandom




1010

Alternative ViewAlternative View

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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

1212

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

19 dogs were initially given the drug pen19 dogs were initially given the drug pentobarbitol followed by four treatmentstobarbitol followed by four treatments

Halothane

Present

Absent

CO2 pressureLow High

12

3 4

1313

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

1414

Contrast MatrixContrast Matrix

Cμ

XμX

qq

j

jq

j

j

j Enj

X

X

X

2

1

1

31

21

2

1

1001

0101

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)(,,2,1,

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Test for Equality of Treatments in a Test for Equality of Treatments in a Repeated Measures DesignRepeated Measures Design

)()1(

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if Reject

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matrixcontrast :),,(:

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10

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q

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CμCμ

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1616

Example 6.2: Contrast MatrixExample 6.2: Contrast Matrix

1111

1111

1111

ninteractio COH

contrast CO

contrast Halothane

23241

24231

2143

C

1717

Example 6.2: Test of HypothesesExample 6.2: Test of Hypotheses

0:Reject

94.1005.0)1(

)1)(1(

116''

44.755754.91462.927

54.91484.519592.1098

62.92792.109832.9432

',

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05.60

31.209

99.487863.449944.406535.2295

32.685198.530349.2943

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,

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0

1,1

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Sx

H

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1818

Example 6.2: Simultaneous Example 6.2: Simultaneous Confidence IntervalsConfidence Intervals

97.6579.1219

44.755794.1079.12

"ninteractio"CO-H

70.545.6019

84.519594.1005.60

influence pressure CO

70.7331.209)05.0(16

)3(18

influence halothane ofContrast

2

2

16,32143

Fxxxx

1919

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

2020

Assumptions Concerning the Assumptions Concerning the Structure of DataStructure of Data

21

21

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22

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11

11211

normal temultivaria are spopulationBoth

:small and when sassumptionFurther

,,, oft independen are ,,,

covariance and r mean vecto with population

variate from sample random:,,,

covariance and r mean vecto with population

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21

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1

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p

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2121

Pooled Estimate of Pooled Estimate of Population Covariance MatrixPopulation Covariance Matrix

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2323

Proof of Result 6.2Proof of Result 6.2

1,

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2424

Wishart DistributionWishart Distribution

)'|'(:')|(:

)|(:

)|(:),|(:

:Properties

definite positive:

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mm

mm

mm

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2525

Test of HypothesisTest of Hypothesis

ΣXX

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XX

μμXX

δxxSδxx

δμμδμμ

2121

212211

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)( Note

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nn

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2626

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

65.025.0,15.125.0

25.0)05.0(1

211

'290.0957.0,697.1

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: of rseigenvecto and sEigenvalue

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e

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2727

Example 6.3Example 6.3

2828

Result 6.3: Simultaneous Result 6.3: Simultaneous Confidence IntervalsConfidence Intervals

poolediiii

ii

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pnnp

snn

cXX

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aμμa

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2929

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

26.6)05.0(97

)2(98

3.636615.21505

5.215057.10963

2

1

2

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55,5.559647.19616

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0.355

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

5.3287.47or

3.6366155

1

45

126.6)0.3556.556(:

1.1277.21or

7.1096355

1

45

126.6)0.1304.204(:

intervals confidence ussimultaneo 95%

2212

2212

2111

2111

3131

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

3232

Bonferroni Simultaneous ConfidencBonferroni Simultaneous Confidence Intervalse Intervals

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txx ,21

22121

11

2:

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3333


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μμa

μμxx

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11

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21

11')('

:'for intervals confidence usSimultaneo

)(

11'

:for ellipsoid confidence 100%

large are and

nn

nn

pnpn

p

p

3434

Proof of Result 6.4Proof of Result 6.4

2211

22121

1

22

11

2121

22

11

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22

11

2121

2121

~,~

:

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11,nearly :

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SΣSΣ

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ΣΣμμXX

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RemarkRemark

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n

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pooled

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If

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3636

Example 6.5 Example 6.5

063.0

041.011 :ncombinatiolinear Critical

99.5)05.0(66.1511

'

0:

4.27375.8,or 15.264299.56.201:

127.121.7,or 17.46499.54.74:

15.264208.886

08.88617.46411

Data 6.4 Example

21

1

22

11

2221

1

22

11

212

210

2212

2111

22

11

xxSS

xxSSxx

μμ

SS

nn

nnT

H

nn

3737

Example 6.9: Nursing Home DataExample 6.9: 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

3838

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

21

22221

11211

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1

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n

n

g XXX

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3939

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

Σ

μ

XXX

g

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4040

Univariate ANOVAUnivariate ANOVA

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g

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g

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: hypothesis Null

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Concept of Degrees of FreedomConcept of Degrees of Freedom

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4545

Concept of Degrees of FreedomConcept of Degrees of Freedom

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1,,1

21

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4646

Examples 6.6 & 6.7Examples 6.6 & 6.7

level 1% at the rejected is 0:

27.13)01.0(5.195/10

2/78

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4747

MANOVAMANOVA

g n

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n

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1 1

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WBxxxx

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4848

MANOVAMANOVA

4949

MANOVAMANOVA

small toois

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2211

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g n

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H

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5050

Distribution of Wilk’s LambdaDistribution of Wilk’s Lambda

5151

Test of Hypothesis for Large SizeTest of Hypothesis for Large Size

)(ln2

1

if level cesignificanat Reject

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1

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5252

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

1

1

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Example 6.9: Nursing Home DataExample 6.9: 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

5858


5959


304.0230.0610.0584.0

235.0453.0821.0

225.1111.1

475.3

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6161

Bonferroni Intervals for Bonferroni Intervals for Treatment EffectsTreatment Effects

2/)1(

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Result 6.5: Bonferroni Intervals for Result 6.5: Bonferroni Intervals for Treatment EffectsTreatment Effects

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6363

Example 6.10: Example 6.9 DataExample 6.10: Example 6.9 Data

019.0,021.0,026.0,058.0: and

for intervals confidence ussimultaneo 95%

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6464

Example 6.11: Plastic Film DataExample 6.11: Plastic Film Data

6565

Two-Way ANOVATwo-Way ANOVA

resintfacfaccor

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Two-Way MANOVATwo-Way MANOVA

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7575

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7878

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7979

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Test for Parallel ProfilesTest for Parallel Profiles

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Example 6.12: Example 6.12: Test for Coincident Profiles Test for Coincident Profiles

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8989

Example 6.13: Ulna Data, Example 6.13: Ulna Data, Treatment GroupTreatment Group

9090

Comparison of Growth CurvesComparison of Growth Curves

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9191

Comparison of Growth CurvesComparison of Growth Curves

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Example 6.14: Comparing MultivariExample 6.14: Comparing Multivariate and Univariate Testsate and Univariate Tests

9494

Example 6.14: Comparing MultivariExample 6.14: Comparing Multivariate and Univariate Testsate and Univariate Tests

21

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9595

Strategy for Multivariate Strategy for Multivariate Comparison of TreatmentsComparison of Treatments

Try to identify outliersTry to identify outliers– Perform calculations with and without the oPerform calculations with and without the o

utliersutliersPerform a multivariate test of hypothesiPerform a multivariate test of hypothesissCalculate the Bonferroni simultaneous cCalculate the Bonferroni simultaneous confidence intervalsonfidence intervals– For all pairs of groups or treatments, and all For all pairs of groups or treatments, and all

characteristicscharacteristics

9696

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

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

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