Shyh-In Hwang in Yuan Ze University1 Chapter 6 Consistency and Replication.
1 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 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
2tment under trea 1 variable
--------------------------------
1tment under trea variable
1tment under trea 2 variable
1tment under trea 1 variable
2
22
12
1
21
11
pX
X
X
pX
X
X
jp
j
j
jp
j
j
66
Result 6.1Result 6.1
n
jjjd
n
jj
pnpd
dpj
jpjjj
jpjpjp
jjj
jjj
nn
Fpn
pnnT
njN
DDD
XXD
XXD
XXD
11
,12
21
21
22212
12111
'1
1,
1
)(
)1(:'
,,2,1),,(:
',,,
DDDDSDD
δDSδD
ΣδD
D
77
Test of Hypotheses and Test of Hypotheses and Confidence RegionsConfidence Regions
n
s
ptd
n
sF
pn
pnd
Fpn
pn
Fpn
pnnT
HH
dddd
ii dnii
dpnpii
pnpd
pnpd
jpjjj
2
1
2
,
,1
,12
10
21'
2:,)(
)1(:
)()1(
':regions Confidence
)()1(
'
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
61.41838.88
38.8826.199,
27.13
36.9
2
1
0
9,2
2
2
1
δ
Sd
H
F
T
d
dd
99
Experiment Design for Experiment Design for Paired ComparisonsPaired Comparisons
. . .
. . .
1 2 3 n
Treatments1 and 2assigned atrandom
Treatments1 and 2assigned atrandom
Treatments1 and 2assigned atrandom
Treatments1 and 2assigned atrandom
1010
Alternative ViewAlternative View
xCCSCCxCSCSxCdCxd
C
SS
SSS
x
12
)2(
2221
1211
2222111211
''',',,
100|100
|
010|010
001|001
,,,,,,,'
nT
xxxxxx
djj
pp
pp
1111
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
j
jq
j
j
j Enj
X
X
X
2
1
1
31
21
2
1
1001
0101
0011
)(,,2,1,
1515
Test for Equality of Treatments in a Test for Equality of Treatments in a Repeated Measures DesignRepeated Measures Design
)()1(
)1)(1(''
if Reject
0: vs.0: ofTest
matrixcontrast :),,(:
1,112
0
10
qnq
q
Fqn
qnnT
H
HH
N
xCCSCxC
CμCμ
CΣμX
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
',
79.12
05.60
31.209
99.487863.449944.406535.2295
32.685198.530349.2943
14.796342.3568
29.2819
,
89.502
26.479
63.404
21.368
0
1,1
12
Cμ
xCCSCxC
CSCxC
Sx
H
Fqn
qn
nT
qnq
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
2222111211
22
22221
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
variate from sample random:,,,
21
2
1
ΣΣ
XXXXXX
Σμ
XXX
Σμ
XXX
nn
p
p
nn
n
n
2121
Pooled Estimate of Pooled Estimate of Population Covariance MatrixPopulation Covariance Matrix
221
21
21
1
21
12222
11111
21
2222
11
1111
2
1
2
1
2
''
1'
1'
21
2
1
SS
xxxxxxxx
S
Σxxxx
Σxxxx
nn
n
nn
n
nn
n
n
n
jjj
n
jjj
pooled
n
jjj
n
jjj
2222
Result 6.2Result 6.2
1,
21
21
2121
1
212121
2
222221
111211
21
2
1
1
2
as ddistribute is
11'
),(:,,,
),(:,,,
pnnp
pooled
pn
pn
Fpnn
pnn
nnT
N
N
μμXX
SμμXX
ΣμXXX
ΣμXXX
2323
Proof of Result 6.2Proof of Result 6.2
1,
21
21
1
21
2
2121
2/1
21
12121
2/1
21
2
22211
122111
2121
22
212
11
111
21
21
21
21
21
21
1
2:),(
2
)()',0(
11
'11
)(:11
:1,:1
)11
,(:
1111
pnnppnn
p
pooled
nn
nn
p
nn
Fpnn
pnnN
nn
WN
nn
nnT
Wnn
WnWn
nnN
nnnn
Σ0Σ
Σ
μμXX
SμμXX
ΣSS
ΣSΣS
Σμμ
XXXXXX
2424
Wishart DistributionWishart Distribution
)'|'(:')|(:
)|(:
)|(:),|(:
:Properties
definite positive:
21
2
)|(
2121
2211
1
2/)1(4/)1(2/)1(
2/tr2/)2(
1
21
21
1
CΣΣCACCACΣAA
ΣAAAA
ΣAAΣAA
A
Σ
AΣA
AΣ
mm
mm
mm
p
i
nppnp
pn
n
WW
W
WW
in
ew
2525
Test of HypothesisTest of Hypothesis
ΣXX
XXXXXX
XX
μμXX
δxxSδxx
δμμδμμ
2121
212211
21
2121
1,21
21
021
1
21021
2
02110210
11)Cov()Cov(
)Cov(),Cov(),Cov()Cov(
)Cov(
)( Note
)(1
2
11'if
: offavor in :Reject
21
nn
E
Fpnn
pnn
nnT
HH
pnnp
pooled
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
'957.0290.0,303.5
: of rseigenvecto and sEigenvalue
2.0
9.1,
51
12
98
49
98
49
41
12,
9.3
2.10,
61
12,
1.4
3.8
50
21
1,21
21
21
12
11
2121
2211
21
21
pnnp
pooled
pooled
Fpnn
pnn
nn
nn
e
e
S
xxSSS
SxSx
2727
Example 6.3Example 6.3
2828
Result 6.3: Simultaneous Result 6.3: Simultaneous Confidence IntervalsConfidence Intervals
poolediiii
ii
pooled
pnnp
snn
cXX
μ
nnc
Fpnn
pnnc
,21
21
21
21
2121
1,21
212
11)(
by covered be will ,particularIn
allfor )('cover will
11')('
)(1
221
aμμa
aSaXXa
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
1
55,5.559647.19616
7.196160.8632,
0.355
0.130
45,4.731074.23823
4.238233.13825,
6.556
4.204
97,22
221
21
21
1
122
111
Fc
nn
n
nn
n
n
n
pooled SSS
Sx
Sx
3030
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
poolediinnii snnp
txx ,21
22121
11
2:
21
3333
Result 6.4Result 6.4
aSSaxxa
μμa
μμxx
SSμμxx
μμ
22
11
221
21
22121
1
22
11
2121
21
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
2121
22
11
2121
2121
~,~
:
11'
11,nearly :
11CovCovCov
SΣSΣ
μμXX
ΣΣμμXX
ΣΣμμXX
ΣΣXXXX
μμXX
p
p
nn
nnN
nn
E
3535
RemarkRemark
nnnnnn
nn
nnn
nn
n
nnn
pooled
1111
2
11
1112
1
2
1
If
21
2122
11
21
SSS
SSSS
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
2
1
ggngg
n
n
g XXX
XXX
XXX
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
n
4040
Univariate ANOVAUnivariate ANOVA
xxxxxx
nNeeX
H
H
g
NXXX
jj
g
jjj
g
g
n
0),,0(:,
0:
rizationReparamete
: hypothesis Null
,,2,1
),( from sample random:,,,
1
2
210
210
221
4141
Univariate ANOVAUnivariate ANOVA
)SS()SS()SS()(SS
)SS()SS()(SS
0
2
1 1
2
1
2221
1 1
2
1 1
2
1
2
1 1
2
1
22
1
2
1
222
restrmeanobs
g n
jj
gg n
jj
restrcor
g n
jj
gg n
jj
n
jj
n
jj
n
jj
jjj
xxxxnxnnnx
xxxxnxx
xxxxnxx
xx
xxxxxxxxxx
4242
Univariate ANOVAUnivariate ANOVA
4343
Univariate ANOVAUnivariate ANOVA
trres
res
restr
gngg
res
tr
g
F
gn
gF
H
SSSS
SS
SS/SS1
1
)(
/SS
)1/SS
if levelat 0:Reject
,1
1
210
4444
Concept of Degrees of FreedomConcept of Degrees of Freedom
gg
g
ggnnn
xxxxxx
xxxxxx
nnnnxxxxxg
uuu
y
2211
21
21221111
1
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
1
vectorTreatment
d.f. :,,,,,,,'21
4545
Concept of Degrees of FreedomConcept of Degrees of Freedom
gn
xxxxxxx
gx
g
g
gg
g
g
: of d.f.
,,,by spanned hyperplane thelar toperpendicu
vector Residual
1 d.f. : r mean vecto
vector treatment thelar toperpendicu is
d.f.:,,,by spanned
hyperplane on the all are andvector Treatment
1,,1
21
2211
21
21
e
uuu
uuu1ye
1
1
uuu
1
uuu1
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
/SS
)1/(SS
53-3)2(3 d.f.,10SS
213 d.f. ,78SS
128SS,216SS
011
11
121
222
33
444
444
44
444
213
20
969
3210
5,2
H
Fgn
gF
res
tr
res
tr
meanobs
4747
MANOVAMANOVA
g n
jjj
g n
j
g
jj
jjj
g
pj
jj
n
n
N
gnj
1 1
1 1 1
1
'
''
ˆˆˆ
effect,nt th treatme:
(level)mean overall:),,(:
,,2,1;,,2,1;
WBxxxx
xxxxxxxx
eτμxxxxxx
0ττ
μΣ0e
eτμX
4848
MANOVAMANOVA
4949
MANOVAMANOVA
small toois
'
'
lambda s Wilk'if 0:Reject
111
'
1 1
1 1*
210
2211
1 1
g n
jjj
g n
jjj
g
gg
g n
jjj
WB
W
H
nnn
xxxx
xxxx
τττ
SSS
xxxxW
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
:ln2
1
large, is and trueis If
2)1(
0
2)1(
*
0
gp
gp
gpn-
H
gpn-
nnH
WB
W
5252
Popular MANOVA Statistics Used Popular MANOVA Statistics Used in Statistical Packagesin Statistical Packages
1
1
1
*
)( of eigenvalue maximum
rootlargest sRoy'
)( tracePillai
traceHotelling-Lawley
lambda sWilk'
WBW
WBB
BW
WB
W
tr
tr
5353
Example 6.8Example 6.8
2722448200SSSSSSSS,
798
04
723
2161078128SSSSSSSS,
213
20
969
5
4,
5
4,
2
1,
4
8
7
2
9
1
8
30
2
4
07
9
2
6
3
9
321
restrmeanobs
restrmeanobs
xxxx
5454
Example 6.8Example 6.8
14972792639:Total
1(-1)031(-2)(-2)(-1)1:Residual
-123(-2)3(-3)(-3)2(-1)43 :Treatment
160548 :Mean
products Cross
110
22
321
333
33
111
555
55
555
798
04
723
011
11
121
222
33
444
444
44
444
213
20
969
5555
Example 6.8Example 6.8
5656
Example 6.8Example 6.8
0
8,412),1(2
*
*
*
Reject
01.7)01.0()01.0(19.8
13
138
0385.0
0385.01
1
11
0385.0
7211
1188
241
110
H
FF
g
gn
gng
WB
W
5757
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
Example 6.9Example 6.9
5959
Example 6.9Example 6.9
304.0230.0610.0584.0
235.0453.0821.0
225.1111.1
475.3
'
'380.0102.0519.0136.2
538.6394.0428.2581.9
484.1633.0695.1
200.8408.4
962.182
)1()1()1(
1
321
332211
332211
g
n
nnn
nnn
nnn
xxxxB
xxxx
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6060
Example 6.9Example 6.9
analysesboth by Reject
09.20)01.0()01.0(
76.132ln)2/)(1(
analysis eapproximat or,
51.28/)01.0()01.0(
67.1712
7714.0
0
28
2)1(
285102,42
*
*
*
H
WB
Wgpn
F
p
pn
WB
W
gp
6161
Bonferroni Intervals for Bonferroni Intervals for Treatment EffectsTreatment Effects
2/)1(
)(
11ˆˆVar
)()(
111
11VarˆˆVar
ˆˆ,ˆ
2211
gpgm
gn
w
nn
gngn
nnn
nnxx
xxxx
ii
kiki
pooled
gg
iik
ikiiki
ikiikiikiki
ΣS
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6262
Result 6.5: Bonferroni Intervals for Result 6.5: Bonferroni Intervals for Treatment EffectsTreatment Effects
nngn
w
gpgtxx
k
iigniki
iki
11
)1(
tobelongs
)1(least at confidenceWith
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%
025.0,061.0or 0.006142.870.043-
for interval confidence ussimultaneo 95%
87.2)234/05.0(
00614.03516
484.1
107
1
271
111
516,043.0023.020.0ˆˆ
'003.0023.0002.0137.0ˆ
'020.0020.0039.0070.0ˆ
33232313
3313
513
33
31
3313
33
11
t
gn
w
nn
n
xxτ
xxτ
6464
Example 6.11: Plastic Film DataExample 6.11: Plastic Film Data
6565
Two-Way ANOVATwo-Way ANOVA
resintfacfaccor
g b
k
n
rkkr
g b
kkk
b
kk
gg b
k
n
rkr
kkrkkkkr
kkkr
kr
b
kk
g
k
b
kk
g
krkkkr
xxxxxxn
xxgnxxbnxx
xxxxxxxxxxxx
XE
Ne
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eX
SSSSSSSSSS
),0(:,0
,,2,1;,,2,1;,,2,1
21
1 1 1
2
1 1
2
1
2
1
2
1 1 1
2
2
1111
6666
Two-Way ANOVATwo-Way ANOVA
6767
Two-Way ANOVATwo-Way ANOVA
nintercatio 2factor -1factor
of effectsfor :)1(/SS
)1)(1/(SS
2factor of effectsfor :)1(/SS
)1/(SS
1factor of effectsfor :)1(/SS
)1/(SS
testsratio
2
1
ngb
bg
ngb
b
ngb
g
F
res
int
res
fac
res
fac
6868
Two-Way MANOVATwo-Way MANOVA
g b
k
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