2. Multidimensional Scaling Analysis#

Multidimensional scaling analysis (MDS) provides a visual representation of similarity between samples that have multiple variables. MDS is built on the idea of a distance matrix that quantifies the dissimilarity between samples.

2.1. Euclidean Distance#

How far away two points are when plotted as \(x\),\(y\) coordinates.

For three variables, the distance between two samples (\(i\) and \(j\)) can be visualized in three-dimensional space.

\(d_{ij} =||X_i-X_j|| = \sqrt{(x_i-x_j)^2 + (y_i-y_j)^2 + (z_i-z_j)^2}\)

if \(X_i = (x_{i}, y_{i}, z_{i})\)

2.2. Bray Curtis Distance#

Often used for identifying differences in community composition based on abundance. If \(u\) and \(v\) represent two different samples of counts of different groups, the Bray Curtis distance is:

\(d= \frac{\Sigma|u_i-v_i|}{\Sigma(u_i+v_i)}\)

if \(u\) and \(v\) are positive, then 0 < d < 1

Example#

In this example, \(u\), \(v\) and \(q\) are three different samples, in which four different groups are counted.

import numpy as np
from scipy.spatial import distance

u = [415,200,310,411]
v = [615,100,330,203]
q = [614,101,331,202]

data = np.array([u,v,q])

print(data)

[[415 200 310 411]
 [615 100 330 203]
 [614 101 331 202]]

data.T # Transpose so that each column represents a sample

array([[415, 615, 614],
       [200, 100, 101],
       [310, 330, 331],
       [411, 203, 202]])

# compute distances between points
dist = distance.pdist(data,'braycurtis')
dist

array([0.20433437, 0.20433437, 0.00160256])

# represent distances as a matrix
distmatrix = distance.squareform(dist)
distmatrix

array([[0.        , 0.20433437, 0.20433437],
       [0.20433437, 0.        , 0.00160256],
       [0.20433437, 0.00160256, 0.        ]])

2.3. Other measures of distance#

Python can be used to compute a variety of different distance measures.

https://docs.scipy.org/doc/scipy/reference/spatial.distance.html

For an excellent resource on the applications of different distance calculations in ecology, including appropriate measures for binary (presence/absence) data, see A Primer of Ecological Statistics by Gotelli and Ellison.

2.4. Types of multidimensional scaling analysis#

  1. classical MDS

Also known as Torgerson MDS or principal coordinate analysis (PCoA)

The distance matrix is converted to a similarity matrix. Once this is done, the same steps as PCA are performed:

  • compute eigenvectors and eigenvalues

  • same as PCA for Euclidean distances

Steps:

  • create a data matrix

  • compute a dissimilarity matrix, D, with elements \(d_{ij}\)

  • transform the dissimilarity matrix \(d^*_{ij} = \frac{1}{2}d^2_{ij}\)

  • center the dissimilarity matrix \(\delta^*_{ij} = d^*_{ij}-\bar{d}^*_{i}-\bar{d}^*_{j}+\bar{d}^*\)

  • compute the eigenvectors and eigenvalues

  • if the dissimilarity index is euclidean distance, this is mathematically equivalent to PCA

  1. Non-metric (iterative) MDS

  • preserves rank of distance

  • minimizes stress

images/pcoa.png images/pcoa_iterative.png

[source] http://stats.stackexchange.com/questions/14002/whats-the-difference-between-principal-components-analysis-and-multidimensional

2.5. Summary of comparison of PCA and MDS#

  • PCA - based on Euclidean distances, good for data without strong skew & data without outliers

  • PCoA - use when other distance measures are appropriate equivalent to PCA when Euclidean distances used

  • Non-metric multidimensional scaling (NMDS) - preserves ran order of distance rather than actual values (similar to many non-parametric statistics. One reason for using this type of analysis is that it is less sensitive to outliers.

2.6. ANOSIM and PermANOVA#

  • determine whether groups of samples are significantly different

  • are distances WITHIN the groups smaller than the differences BETWEEN groups

2.7. Implementing MDS in Python#

http://scikit-bio.org

  • Compute distance matrices

  • Includes PCA functions

  • Includes PCoA (metric MDS based on distance matrix)

  • Does not include non-metric MDS

  • Includes PermANOVA for assessing statistically significant differences

https://scikit-learn.org

  • Computes distance matrices

  • Includes general MDS function (metric or non-metric)

  • Includes related analyses such as linear discriminant analysis