1. Multidimensional Scaling
For
CONTINUOUS or DISCRETE data

2. 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.

3. They give similar results-2 -1 1 2
Factor 1 -3 3
PCA
-1.0
-0.5
0.0
0.5
1.0
Dimension 1
MDS VS
Factor 2
Dimension 2
Analysis of Limpet Shell Shape (Length, Width & Height)
Note: scales are reversed in MDS to facilitate comparison

4. Dimensions and Factors are correlated!

5. If MDS does the same thing as PCA, why not just use one of them?
Requires
a linear
relationship
between
variables
Does NOT
require
Choice of two
measures for
assessing
association
Use any
measure for
Can only use
Continuous
data
Can use
Continuous OR
Discrete data
Better
Resolution
Weaker

6. 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)

7. 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

8. 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)

9. Terminology Differences
PCA
MDS
Axes on graph
Factors
Dimensions
Values plotted
Scores
Coordinates

10. 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
4.2 3
99.0
21.7
1.7 4
66.8
88.3

11. Types of data Discrete e.g. character states – binary Site Rock Grass
Shade 1 2 3 4

12. Types of data Direct similarities E.g. response to questions
I like romantic movies.
Strongly
Agree
Strongly
Disagree

13. How does it work?
The idea is to develop a two dimensional representation that accurately reflects differences, distances or degree of similarity between subjects.

14. How does it work?
Let’s 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

15. How does it work? 12.1 km Scotsdale Guernyville 25.7 km 28.4 km
Aptos
25.7 km
28.4 km
Aptos
25.7 km
28.4 km

16. How does it work? Dissimilarity Euclidean Distance
With MDS, these distances are:
Dissimilarity or
Euclidean
Distance

17. 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

18. Why approximately?
Because there is some error associated with the original measurements, the fit is rarely exact.

19. 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 <

20. 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

21. How good is the ordination?
Shepard Diagram
Shepard Diagram – is a plot of the DERIVED Distances versus the ORIGINAL Distances (transformed to Disparities).

22. 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.

23. 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.

24. 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.