Transcript The Variance of the Estimator

Chapter 14 Nonmetric Scaling
Measurement, perception and preference are the main themes of this section. The
sequence of topics is
 Reversing statistical reasoning
 Additive Conjoint Measurement
 Multidimensional Scaling
 Geometric Preference Models
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Slide 14.1
Nonmetric Scaling
Let’s Turn Things On Their Heads
 ANOVA: Assume metric data, test for additivity
 Nonmetric Conjoint: Assume additivity, test for metric data
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Slide 14.2
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Additive Conjoint Measurement Algorithmic Steps
0. Inititialize a metric version of y
1. Fit an additive model to the metric version
2. Rescale the metric version to improve the fit
3. If the model does not yet fit, go back to Step 2
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Slide 14.3
Nonmetric Scaling
The Steps in Detail: Step 0.
Copy the dependent variable,
y* = y
where y contains ordinal factorial data.
Since the yi are ordinal, I can apply any monotonic
transformation to the yi*
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Slide 14.4
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Step 1 – The Usual Least Squares
We use least squares to fit an additive model:
y *  Xβ  e  yˆ *  e
where X is a design matrix for example,
1 1 1
1 1  1

x
1  1 1


1

1
1


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Slide 14.5
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Step 1 Scalar Representation
Looking at row effect i and column effect j we might have
y  αi  β j  eij  yˆ  eij
*
ij
*
ij
Note that since the yij* are optimally rescaled anyway we do not have to worry about the grand
mean.
We modify I and j so as to improve the fit of the model
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Slide 14.6
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Step 2 – Optimal Scaling
Find the monotone transformation that will improve the fit of the above model as much as
possible.
y  H m [yij, yˆ ]
*
ij
*
ij
We modify y* so as to improve the fit of the model
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Slide 14.7
Nonmetric Scaling
Constraints on Hm
 Assuming no ties, we arrange the original (ordinal) data in sequence:
y1  y 2    yn
 To honor the ordinal scale assumption, we impose the following constraints on the optimally
rescaled (transformed) data:
y  y  y
*
1
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*
2
*
n
Slide 14.8
Nonmetric Scaling
The Shephard Diagram
Scaled
Values
Transformed Data y*
ˆ
Model data y
*
Ranked Values
1
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2
3
4
5
Slide 14.9
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Minimize Stress to Optimize Hm
STRESS 
*
* 2
ˆ
(
y

y
 i i)
i
*
*)2
ˆ
ˆ
(
y

y
 i
i
where
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yˆ *
is the average of the
yˆ *i
Slide 14.10
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Multidimensional Scaling – Proximity Data Collection
The respondent’s job is to rank (or rate) pairs of brands as to how similar they are.
Assume three brands A, B and C. Respondent ranks the three unique pairs as to similarity:
___ AB
___ AC
___ BC
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The Geometric Model
 Brands judged highly similar (proximal) are represented near each other in
a perceptual space.
 dij - similarity judgment between brand i and brand j
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Slide 14.12
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Nonmetric MDS proceeds identically, but the model is a distance model, rather
than an additive model
3
j = [2 2]
dˆ *ij 
r
 (xim  x jm )
2
dˆ *ij
2
m
1
i = [1 1]
(dˆ *ij ) 2  (1  2) 2  (1  2) 2
0
0
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1
2
3
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In individual differences scaling (INDSCAL), each subject’s data are analyzed as follows:
Typical element:
Consumer i rating
brands j and k
d (jki )
Brands
r


(i)
*
dˆ jk   w im (x jm  x km ) 2 
m

1/2
Repondents
Brands
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Nonmetric Scaling
Each subject’s perceptual map might be different:
A
B
C
D
E
F
G
H
I
Subject 1
Subject 2
Group Space
A
B
C
D
E
F
G
H
C
Subject 1’s Space
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Subject Weights
A
D
G
B
E
H
C
F
I
Subject 2’s Space
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Geometric Models of Preference: The Vector Model
Subject i
Brand A
Brand B
Subject i
Isopreference lines
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Subject i
Two consumers who disagree
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The Ideal Brand Model
Is also called the Unfolding Model
i
Subjects
i
i
C
A
B
C
D
B
D
Brands
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A
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The Ideal Brand Model in Two Dimensions
i
A
B

D
C
i
Isopreference circles
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Two consumers who disagree
Slide 14.18
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