Multidimensional%20Scaling

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Multidimensional Scaling

• For
• CONTINUOUS or DISCRETE data

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What is it?

• MDS is an ordination procedure like PCA and
  performs the same functions as PCA.
• Like PCA, it provides a single value to represent
  values of several measured variables.

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They give similar results
Analysis of Limpet Shell Shape (Length, Width
Height)
Note scales are reversed in MDS to facilitate
comparison
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Dimensions and Factorsare correlated!
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If MDS does the same thing as PCA, why not just
use one of them?
PCA
MDS
Requires a linear relationship between variables Does NOT require a linear relationship between variables
Can only use Continuous data Can use Continuous OR Discrete data
Better Resolution Weaker Resolution
Choice of two measures for assessing association Use any measure for assessing association
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MDS has weaker resolution
PCA MDS
Range of Factor 1 3.631 Range of Dimension 1 1.874
Range of Factor 2 4.320 Range of Dimension 2 1.256
Analysis of Limpet Shell Shape (Length, Width
Height)
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When should you use MDS?

• When you have discrete data
• e.g.
• Characterize Species Composition
• Species presence/absence data
• Characterize gene sequences
• Marker gene presence/absence
• Numerical Taxonomy
• Presence or absence of specific morphological
  characters
• Characterize Habitats
• Discrete habitat measures

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When should you use MDS?

• When you have want to use a particular measure of
  association
• e.g.
• Characterize Species Composition with a known
  similarity index
• When the relationship between variables is not
  linear (e.g. quadratic)

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Terminology Differences
PCA MDS
Axes on graph Factors Dimensions
Values plotted Scores Coordinates
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Types of data

• Continuous
• e.g. species densities

Site Sp A Sp B Sp C
1 10.1 0.0 100.0
2 20.7 0.0 4.2
3 99.0 21.7 1.7
4 0.0 66.8 88.3
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Types of data

• Discrete
• e.g. character states binary

Site Rock Grass Shade
1 0 1 1
2 0 0 0
3 1 0 1
4 0 1 1
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Types of data

• Direct similarities
• E.g. response to questions
• I like romantic movies.

Strongly Agree
Strongly Disagree
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How does it work?

• The idea is to develop a two dimensional
  representation that accurately reflects
  differences, distances or degree of similarity
  between subjects.

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How does it work?
Lets assume that you have distances between
three towns and you would like to create a map
showing their relative positions.
Towns Distance (km)
Guernyville to Scotsdale 12.1
Guernyville to Aptos 28.4
Scotsdale to Aptos 25.7
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How does it work?
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How does it work?
With MDS, these distances are
Dissimilarity or
Euclidean Distance
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How does it work?

• More points are added and the iterative process
  continues until the distance derived by the
  computer between all pairs of points is
  essentially equal to the original distances.

Distances derived by computer
?
Original distances
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Why approximately?

• Because there is some error associated with the
  original measurements, the fit is rarely exact.

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How do you know when you have the best possible
fit?

• The measure of fit is called STRESS
• As the value of STRESS decreases, fit increases.
• Typically, the point at which you have the best
  fit is when the STRESS index is less than or
  equal to 0.001

STRESS lt 0.001
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How good is the ordination?

• If the ordination procedure was successful, there
  should be a positive linear relationship between
  the derived distances and the original distances.
• Since the non-metric procedure employs distance
  between ranks rather than actual values, the
  original distances need to be transformed to
  DISPARITIES

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How good is the ordination?

• Shepard Diagram is a plot of the DERIVED
  Distances versus the ORIGINAL Distances
  (transformed to Disparities).

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Types of MDS

• Metric vs. Non-metric
• METRIC assumes that the distances or
  dissimilarities have interval or ratio scale
  properties NOT often used as it has many of the
  same assumptions as PCA.
• NON-METRIC assumes that the distances or
  similarities are merely rank order.

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Types of MDS

• Weighted MDS Individual Difference Scaling
• Allows you to use situations in which you have
  more than one (dis)similarity or distance matrix.
• E.g. You have 10 people and each person is to
  record the degree of similarity between 5
  quadrats, There will be a separate matrix for
  each person.

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Types of MDS

• Joint-Space Analysis or Multidimensional
  Unfolding
• This allows you to ordinate both the column and
  rows of the matrix
• Typically used when measure is rank order
• E.g. 10 People have ordered their preference of 5
  brands of tea. This analysis will ordinate the
  teas and then place the people next to their most
  preferred brand on the plot.

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