Chapter3. Multi Dimension Scaling: MDS

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Multivariate Analysis
Chapter3. Multi Dimension Scaling MDS
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Multidimensional Scaling MDS
Object
11 Unidimensuin Scaling
Attribute

Attribute
How to Combine? Summative Evaluation?

• Market Segmentation
• Product Positioning
• Effectiveness Evaluation
• New-Product Development
• Product-Concept Testing
• Ideal Point

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MDS? ??

• Concepts
• Scaling Origin and Unit
• - ??? ???? ???? ???? ????
• ex) ??( 5) ??(2)? ??? ?? Scaling? ??? ???
  ??? ??
• ?? ???????, ??, ?????, ????
• Mental Image of human JND (Just Noticeable
  Difference)
• - Physics vs. Psycho
• ??? ??? ????? Unidimensional Scaling
• ???? ?? ??? ?? Multidimensional Scaling

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MDS? ??

• Concepts
• Multidimensional Scaling? ???
• - ???? ?? ??? ??? ??? ??? ???? ??? ?? ??
• - ??? ?? ??
• - ????? ??? ???? ?? ??? ??? ??
• MDS
• - ???? ?? ??? ???? ?????? ??? ?? ??? ???? ??
• - ??? ??? ??? ??? ??? ??? ????? ??? ?, ???
  ????? ??? ??? ????? ????? ??? ?? ??
• Data of Attributes Similarity or Preference
  ? Perceptual Map (Mental Space)

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MDS? ??

• ??? ??
• ??
• ???(???/ ???) ??? ?? ?? ??? ?? ? Scaling (??? ?
• ??? ?? ?? ??) ? ?????? ?? ? ???? ???? ??
• ? ??? ?? ??
• ??
• - ??? ?? ???? ???, ???, ?, ??? ?? ???
• - ??? ? ?? ??? ??
• ??? ?? ???? ??? ? ????? ????? ???
• ?? ????? MDS
• ?????? ????
• MDS ??, ???? ??

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MDS? ??
Multidimension Scaling
Similarity
Individual Differences Scaling
DATA
Internal Analysis
Preference
External Analysis

• ?? ?? Clustering Analysis, Correspondence
  Analysis, Factor Analysis, Discriminat Analysis,
  Multivariate Analysis of Variance ?? ?? ??? ?? ??
• ?? ?? Similarity or Dissimliarity/ Preference ?
  ???? ? ?? ??? ??? ???? ??
• ??? ????? ??? ??? ??(Latent Dimension)?? ????
  ??? ??? ??? ????? ??

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MDS? ??
Multidimension Scaling
Similarity
Individual Differences Scaling
DATA
Internal Analysis
Preference
External Analysis

• Multidimensional Scaling(??? ???) ??? ?? ????
  ?
• Individual Differences Scaling(??? ???) ???? ???
  ?? ?? ? ??? ???? ??? ?? ??(?? ???????? ?????
  ??)
• Internal Analysis(???? ) ????? ??? ???? ???? ??
  ??? ??
• - ?????? ??? ?? ??? ??? ??? ???? ??? ??
• - ???? ?? ???? ???? ??? ??
• External Analysis(????) Similarity or Preference
  ??? ??? ???? ??? ???? ??(Mapping ??? ?? ??
  ??(Direction) ??, Ideal Point? ??

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MDS? ??
?? ?? ?? ????
??? KYST ????? ???? ????? ?? ??? ??
??? NDSCAL ????? ???? ? ??? ??? ??? ???
??? MDPREF ???? ? ???? ???? ????? ?? ??? ??
??? PROFIT ??? ???? ??? ???? ????? ??? ???? ??? ?? ?? ?? ?? ??
??? PREFMAP ??? ???? ??? ???? ??, ??? ? ??? ????
KYST (Kruskal, Young, Speperd, Togerson)
INDSCAL (Individual Difference SCALing) MDPREF
(Multi Dimensional PREFerence) PROFIT (PROperty
FITting) PREFMAP (PREFerence MAP)
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MDS ??
????? ??
??? ???? ???? ??? ? ??
??? ??
???? ????
??? ??
Goodness of Fitting ???? ??/??? ??
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MDS ??
????? ??

• ????
• ????? ??? ???? ??? ??
• ??? ? ??
• Klahr 8? ??
• Green and Wind ???3?

???? ? ??? ? ??
??? ??
???? ????
??? ??
Goodness of Fitting ???? ??/??? ??
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MDS ??

• ????
• Metric ?? Nonmetric ?? ?? ????
• ????? Nonmetric ??
• - ?? ??? ??
• - ???? (??, ???)
• ????
• Similarity
• - Paired Comparison (Similarity or
  Nonsimilarity)
• - Ranking or Numbering
• - n(n-1)/2 (ex ??? 10? 10(10-1) / 2 25
  
• Preference
• - Ranking or Rating
• - ????? ??? ??

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MDS ??

• Goodness of Fitting
• Stress Value 0.1? ???? ? ??? ?? ? Fitting ???
• R2 ?? ???? ??? ???? ????? ?? ????
• SAS Program
• ????
• ???? ??? ?? ???
• - 122, 183, 92, 133, 174
• ?? ?? ???? ??? ???? ??
• ????? 2?? ?? ?? ??? ?? ?? ??
• ??/???
• ???? ???? ??? ??, ? ??? ?? Clustring ?? ??
• (????? ???)
• ??? ??? ??? ?? ?? ???? ?????? ??? ??? ??? ?? ??
  (Vector? ???)
• ??? ?? ??? ???? ?? ??? Positioning ??
• ??
• End-Point Model ? Ideal-Point Model? Vector?
  Positioning ??

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Multidimensional Scaling SAS ??
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SAS? ??? MDS

• MDPREF (Multi-Dimensional PREFerence) - ??????
  ??? ??
• - ???? ? ???? ???? Positioning Map
• - ????? ?? ??? ??
• - Matrix (Row ????/ Column ???) ???
  ?? Principal Component Analysis.
• - MDPREF ?? ??
• (1) ????? ?? ??? ?? ??? ??? ?? ??
  ???, ?? ?????
• Positioning?? ???? ??, ??
• (2) ??? ?? ??? Positioning ?? ???
  ?? ???? ??? Positioning?
• ???, ? ??? ??
• ?? ??
• 1)??? ?? ??????? ?? ?? ?? (R-Square ?? ??
  ??)
• 2)???? ?????? ?? ??? ??? ??
• 3)???? ??? ??
• 4)??? ?????? ??? ??? ?? ????? ?? ???? ??,
  ??
• 5) Position ?? ?? ??? ?? ?? ???, ????? ???
  Position???? ??

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MDPREF (Multi-Dimensional PREFerence)

• SAS ??

title 'Perference Rating for Automobiles' data
carpref input make 1-10 model 12-22
judge1-judge12 cards Cadiliac Eldorado
8 0 0 7 9 9 0 4 9 1 2 4 Chevolet
Chevette 0 0 5 1 2 0 0 4 2 3 4
5 Chevolet Citation 4 0 5 3 3 0
5 8 1 4 1 6 Chevolet Malibu 6
0 2 7 4 0 0 7 2 3 1 2 Ford
Fairmont 2 0 2 4 0 0 6 7 1 5 0
2 Ford Mustang 5 0 0 7 1 9
7 7 0 5 0 2 Ford Pinto
0 0 2 1 0 0 0 3 0 3 0 3 Honda
Accord 5 9 5 6 8 9 7 6 0 9
6 9 Honda Civic 4 8 3 6
7 0 9 5 0 7 4 8 Proc print /??? ????
?? / run
???/ ???
? ? ? ?
Matrix
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??

/ factor analysis MDREF ???? 2? ????? ???
2???? ??? ??? ????? ?? / proc factor nfactors2
scree maxiter30 methodprincipal
rotatenone var judge1-judge12 run /
RotateVarimax / proc factor nfactors2 scree
maxiter30 methodprincipal rotateVarimax var
judge1-judge12 run / MDPREF ??? ????? ???
/ proc prinqual datacarpref outresults n2
maxiter30 methodmtv replace standard
scores correlation / ????? ???? ????,
???? ???? ?? / id model transform
monotone(judge1-judge12) run proc print
dataresults run / ?????? ?? / proc factor
dataresults nfactors2 scree var
judge1-judge12 where _type_'score'
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??

/ ????? ?? ?? / data biplot set
results if _type_ 'corr' then do
prin1 prin12 prin2 prin22 model
substr(model,6) end y prin2 x
prin1 xsys '2' ysys '2' text model size
.6 lable y 'dimension2' x
'dimension1' keep x y text xsys ysys size run
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MDPREF (Multi-Dimensional PREFerence)

• SAS ???? (???)

??.?? ????? ?? Factor 2?? ?? R-Square ??
0.6749?? 0.8177? ?? ????? ? ??? 2???? ??? ?? ???
?? ???? 67?? 83? ?? ??? ?? ???? ???, MDPREF? ??
??? ?? ??? ?? ??.
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??? ? ?? (???? ? ????)

eldorado
mustang
Accord
malibu
chevette
Pinto
civic
Citation
fairmont
X? (??? ??(civic, accord) ?? ?? ?? ???) Y?
(?? ??. ?? ??? ???.??? ??? ????? ?? 5?? ????
??? ? ??
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??? ? ??

??? ??? ?? 5?? ???? ??? ? ??
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??? ? ?? (???)

eldorado
Accord
civic
1) Accord ???? 36 ?? ???? ??????? ???? ?? ??,
?? ??? ?? ?? ???
??? ???? ????? ???? ???? ??? ??? 2) Eldorado
???? 9? ????? ????? ??/??? ???? ??? 3) Civic
???? 9? ?? ???? ??? ???? ???
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??? ? ?? (Positioning)

eldorado
B
2
3
5
D
Accord
C
A
6
E
4
civic
1
??? ????? ???? ??? ???? ????? ??? Positioning
???? ??? 1) E??? ?? ??? ?? ? ??? ??? 6? ??????
?? ? Positioning ??? 2) A??? ??? ?????? ?? ??
????? ?????? Positioning ? ???? 3) ???? ?? ??
???? ??? ?? ?? 5?? ??? ?? ??(Niche Market)?.
?????? ??? ??? ??? ?? ???? ???
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SAS? ??? MDS

• PREFMAP (PREFerence MAP)
• - Map ???? ????? ????? ???? ??? ?????
  ???????? Positioning? ?? ?? ???? ?? ???? ??
• - MDPREFSMS?????? ???? ??? ?? ?????? ???
  ?? ?????, ???? ?? ??. ?? ?? PREFMAP? ?? ??? ??
  ???? ??? ???? ????? ??????? Positioning? ?? ??
  ???? ?? ?????, ????? ??
• - ?? ??? ???? ??? End-Point?
  Ideal-Point?? Positioning Map ?? ????? ???? ?
• ?? ??
• 1)??? ?? ?? ??? ??? End-Point Model? Ideal-
  Point Model? ?? Propotion of Variance ??? ??.
• 2)???? ?????? ?? ??? ??? ??
• 3)???? Ideal- Point Model ??? ???? ???
  ??.???? ???? ??
• 4)????/??? ??? ?? ??, ??
• 5) Position ?? ?? ??? ?? ?? ???, ????? ???
  Position???? ??

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PREFMAP (PREFerence MAP)

• SAS ??

title 'Perference Rating for Automobiles' data
carpref input make 1-10 model 12-23 (j1-j12)
(2.) mpg reliable ride cards Cadiliac
Eldorado 8 0 0 7 9 9 0 4 9 1 2 4 3 2
4 Chevolet Chevette 0 0 5 1 2 0 0 4 2 3 4 5
5 3 2 Chevolet Citation 4 0 5 3 3 0 5
8 1 4 1 6 4 1 5 Chevolet Malibu 6 0 2
7 4 0 0 7 2 3 1 2 3 3 4 Ford Fairmont
2 0 2 4 0 0 6 7 1 5 0 2 3 3 4 Ford
Mustang 5 0 0 7 1 9 7 7 0 5 0 2 3 2
2 Ford Pinto 0 0 2 1 0 0 0 3
0 3 0 3 4 1 1 Honda Accord 5 9 5
6 8 9 7 6 0 9 6 9 5 5 3 Honda Civic
4 8 3 6 7 0 9 5 0 7 4 8 5 5 3 proc
print
???/???(???)
??
? ? ? ?
Matrix
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PREFMAP (PREFerence MAP)

• SAS ??

/ MDPREF TLFGOD / proc prinqual datacarpref
outresults (dropj1-j12) n2 maxiter30
methodmtv replace standard scores correlations
/???? ???, ??? ??? ??/ id model mpg
reliable ride /?? ??? ??? ?? model mpg reliable
ride? ?? ?? / transform monotone(j1-j12)
/????? ???? Monotone?? ??, ??? ???? ?? / data
plot set results if _type_ 'CORR' then do
prin1 prin12 /???? ?? ???? 2?? ????. /
prin2 prin22 end proc print proc
prinqual datacarpref outpresults(drop j1-j12)
/MDPREF??? ??? ??? ???? ??? / n2
maxiter30 methodmtv replace standard scores /
??? End Point? Ideal Point? ?? ????/ id
model mpg reliable ride transform
monotone(j1-j12) proc transreg datapresults
/PREFMAP??? ???? / model monotone(mpg
reliable ride) linear(prin1 prin2)
/maxiter20 output tstandardcenter
coefficients replace outtresult1 id
model proc print datatresults1
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PREFMAP (PREFerence MAP)

• SAS ??

/PREFMAP??? ???? / proc transreg
datapresults model monotone(mpg
reliable ride) point(prin1 prin2) /maxiter20
output tstandardcenter coefficients
replace outtresult2 id model run proc
print run proc print datatresult2 run
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PREFMAP (PREFerence MAP)

• SAS ??? ?? (???)

End-Point Model Mpg 45 ? 48 Reliable 65
? 85 Ride 1 ? 25
Ideal-Point Model Mpg 50 ? 51 Reliable
64 ? 85 Ride 5 ? 31
?Reliable? Ride? ???? ??? ???? ???, Mpg? ?? ????
????? Mpg? ?? ?? ?? ? ???? ?? ??? ??? ?? ??
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PREFMAP (PREFerence MAP)

• SAS ??? ? ?? (???)

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MDPREF (Multi-Dimensional PREFerence)

• SAS ??? ? ?? (???)

??? ??? ?? 5?? ???? ??? ? ??
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PREFMAP (PREFerence MAP)

• SAS ??? ?? (????)

Ride
Reliable
Mpg
Ideal Points X?? Y?? ??? ???? ???? ??? ?
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??? ? ??

eldorado
Accord
Ride
Reliable
civic
Mpg
MDPREF ??X? (??? ??(civic, accord) ?? ?? ??
???) Y? (?? ??. ?? ??? ???.??? ??? PREFMAP
?? ??(mpg), ???(reliable), ???(ride) ?? 3??
??? ? ?? ? ?? ??? ?? ???
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MDPREF (Multi-Dimensional PREFerence)

• SAS ??? ? ??

eldorado
B
2
3
5
D
Accord
C
A
6
E
4
Ride
Reliable
civic
Mpg
1
1) E??? ?? ??? ?? ? ??? ??? 6? ?????? ?? ?
Positioning ??? C? ??? ??? ??? ??? ?? ??? ??
?? 2) A??? ??? ?????? ?? ?? ????? ??????
Positioning ? ???????? ?? ?? ??? ??? 3) ?????
Ride ?? Reliable ? Mpg? ? ???? ??? ?????? A?????
?????? ?? ??? ??? ??? Positioning ?? ?? ????.
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Homework

Conjoint Analysis
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Chapter3. Multi Dimension Scaling MDS

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