1. Applied Multivariate Quantitative Methods
Multidimensional Scaling
By Jen-pei Liu, PhD
Division of Biometry, Department of Agronomy,
National Taiwan University
and
Wei-Chie Chie, MD, PhD
Department of Public Health
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

2. Multidimensional Scaling
Introduction
Procedures
Examples
Summary
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

3. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Introduction
Objectives
Given a distance matrix between objects
To construct a diagram (map) showing the relationships between a number of objects
Maps – one dimension, two dimensions, three dimensions or a high number of dimension (a simple geometrical representation is not possible)
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

4. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Introduction
The mirror image of a map has the same distance matrix
If n >=3, the objects may not lie on a plane
Multidimensional scaling is applied when the underlying relationship is unknown but the estimated distance matrix is available
A tool for reduction of dimensions and visulaization
An exploratory tool
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

5. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Introduction
Distance matrix for 4 objects
1 0
2019/2/16
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2019/2/16
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Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

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Procedures
Step 1: Start with a matrix of distance between n objects [n(n-1)/2] in p-dimensions, ij
Step 2: Determine the number of dimension for multidimensional scaling, say t
Step 3: A starting configuration is set up for the n objects in t dimensions
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

9. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Procedures
Step 4: The Euclidean distances between the objects are computed for the assumed configuration. Let dij be object i and object j for this configuration
Step 5: Fit a regression of dij on ij. Denote the fitted distances by d*ij which is called the disparities
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

10. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Procedures
Step 6: Compute the STRESS to check the fit
STRESS = ((dij – dij*)2/ d2ij)1/2
Repeat 3-6 until the stress can not be further reduced
Obtain the values of the n objects and to draw a map based on the new values
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

11. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Procedures
The word stress is used because the statistic is a measure of the extent to which the spatial configuration of points has to be stressed to obtained the data distances ij.
Small value of stress is desirable
Increase the number of dimension when stress is already less than 0.05
Reduce the number of dimension to the extent if stress exceeds 0.1
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

12. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Road distances between 13 New Zealand Towns
Road distances are proportional to geographical distances
Recover the true map by 2-dimensional analysis
Stress value is 0.043
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

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2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

14. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
New Values after multidimensional scaling (2)
Town 1 2 Town 1 2
Alexandra Invercargill
Balclutha Milford
Blenheim Nelson
Christchurch Queenstown
Dunedin Te Anau
Franz Josef Timaru
Greymouth
2019/2/16
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16. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Very successful in recovering the real map
Town are in correct relationship except Milford which can be reached only by road through Te Anau
2-dimension map: Milford is close to Te Anau
Geographical map: Milford is close to Queenstown
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

17. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Voting behavior of 15 New Jersey congressmen
Distances are based on the number of voting disagreement on 19 bills concerned with environmental matters
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

18. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Metric multidimensional scaling: doubling a distance value is equivalent to configuration distance between 2 objects is also doubled. Assume a linear relationship between dij and ij.
Nonmetric multidimensional scaling: only assume monotonic relationship between dij on ij.
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

19. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Stress
Dimmension Metric Nonmetric
2019/2/16
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20. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Values after multidimensional Scaling
Congressmen
Hunt (R)
Sand (R)
Howard (D)
Thompson (D)
Frelinghuysen (R)
Forsythe (R)
Windnall (R )
Roe (D)
Helstoski (D)
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

21. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Values after multidimensional Scaling
Congressmen
Rodino (D)
Minish (D)
Rinaldo (R)
Maraziti (R)
Daniels (D)
Pattern (D)
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22. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

23. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

24. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Voting pattern of 15 NJ congressmen
Dimension 1: party difference
Dimension 2: number of abstentions from voting
Dimension 3: no simple or obvious interprestation
Configuration distance vs. disparities
Not a straight linear relationship
Data distances do not increase with the configruation distances
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

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26. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Examples
Microarray data
32 patients with prostate cancers
23 patients from primary sites with no known metastatasis
9 patients from metastatic disease site
Expression data from genes
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

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2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

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2019/2/16
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2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

30. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Summary
Objectives
Procedures
Stress
Effect of scale
Real Data
Distance between cities
Voting pattern
Gene expression levels
2019/2/16
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD