Multidimensional scalingMDS

G. Quinn, M. Burgman & J. Carey 2003

Aim

• Graphical representation of dissimilarities between objects in as few dimensions (axes) as possible

• Graphical representation is termed an “ordination” in ecology

• Axes of graph represent new variables which are summaries of original variables

Approximate distances by air (km) between Australian Capital cities

CAN SYD MELB BRIS ADEL PER HOB DAR

CAN 0 . . . . . . .

SYD 246 0 . . . . . .

MELB 506 727 0 . . . . .

BRIS 1021 775 1393 0 . . . .

ADEL 976 1185 651 1961 0 . . .

PER 3126 3339 2804 4114 2152 0 . .

HOB 1120 1075 613 1852 1264 3417 0 .

DAR 3409 3163 3355 2886 2727 2951 4186 0

-2 -1 0 1 2Dimension 1

-2

-1

0

1

2

Dim

ensi

on 2

Stress = 0.014

-2 -1 0 1 2Dimension 1

-2

-1

0

1

2

Dim

ensi

on 2

x -

1

Darwin

Perth

AdelaideHobart

Melbourne

Canberra

Sydney

Brisbane

http://www.boardtheworld.com/resorts/country.php?cc=AU

Darwin

Perth

Adelaide

Hobart

Melbourne

Canberra

Sydney

Brisbane

Haynes & Quinn (unpublished)

• Four sites along Morwell River– site 1 upstream from planned sewage

outfall– sites 2, 3 and 4 downstream– site 3 below fish farm

• Abundance of all species of invertebrates recorded from 3 stations at each site

• 12 objects (sampling units):– 4 sites by 3 stations at each site

• 94 variables (species)

Do invertebrate communities (or assemblages) differ between stations and sites?– Is Site 1 different from rest?

Multidimensional scaling

1. Set up a raw data matrix

Species 1 2 3 4 5 etc.

Site/sampleS11 54 0 0 5 0S12 37 1 0 4 0S13 68 2 0 2 0S21 60 0 0 0 1S22 47 0 0 2 0S23 60 0 0 0 0etc.

2. Calculate a dissimilarity (Bray-Curtis) matrix

S11 S12 S13 S21 S22 S23 etc.

S11 .000

S12 .203 .000

S13 .666 .652 .000

S21 .216 .331 .759 .000

S22 .328 .410 .796 .191 .000

S23 .336 .432 .796 .183 .054 .000

etc.

3. Decide on number of dimensions (axes) for the ordination:– suspected number of underlying ecological

gradients– match distances between objects on plot

and dissimilarities between objects as closely as possible

– more dimensions means better match– usually between 2 and 4 dimensions

4. Arrange objects (eg. sampling units) initially on ordination plot in chosen number of dimensions– starting configuration– usually generated randomly

Starting configuration

-2 -1 0 1 2-2

-1

0

1

2

Axis I

Axis II

Site 1 Site 3Site 2 Site 4

5. Compare distances between objects on ordination plot and Bray-Curtis dissimilarities between objects– strength of relationship measured by

Kruskal’s stress value– measures “badness of fit” so lower values

indicate better match– plot is called Shepard plot

Starting configuration

-2 -1 0 1 2-2

-1

0

1

2

Axis I

Axis II

Site 1

Site 3

Site 2

Site 4

0 0.5 10

1

2

3

Dissimilarity

Distance

Shepard plotStress = 0.394

6. Move objects on ordination plot iteratively by method of steepest descent– each step improves match between

dissimilarities and distances between objects on ordination plot

– lowers stress value

0 0.5 10

1

2

3

Dissimilarity

Distance

-2 -1 0 1 2-2

-1

0

1

2

Axis I

Axis II

After 20 iterations

Stress = 0.119

7. Final configuration

• further moving of objects on ordination plot cannot improve match between dissimilarities and distances

• stress as low as possible

0 0.5 10

1

2

3

Dissimilarity

Distance

-2 -1 0 1 2-2

-1

0

1

2

Axis II

Axis I

Final configuration - 50 iterations

Stress = 0.069

Iteration Stress1 0.3942 0.3683 0.3574 0.351... ...20 0.119... ...49 0.06950 0.069

Stress of final configuration is 0.069

Iteration history

How low should stress be?

Clarke (1993) suggests:

• > 0.20 is basically random

• < 0.15 is good

• < 0.10 is ideal– configuration is close to actual

dissimilarities

How many dimensions?

• Increasing no. of dimensions above 4 usually offers little reduction in stress

• 2 or 3 dimensions usually adequate to get good fit (ie. low stress)

• 2 dimensions straightforward to plot

Types of MDS

• Based on how stress is measured

• Relationship between distance and dissimilarity Dissimilarity

Dis

tanc

e

Metric MDS

• stress measured from relationship between actual dissimilarities and distances

• but relationship often non-linear

• inefficient?

Non-metric MDS

• stress measured from relationship between ranks of dissimilarities and ranks of distances

• similar to Spearman rank correlation• better for ecological data

Anderson et al. (1994)

• Effects of substratum type on recruitment of intertidal estuarine fouling assemblage

• Six replicate panels of 4 substrata placed in estuary for 1 month at 2 times of the year

• 14 species in total recorded

• MDS to examine relationship between panel– do substrata appear different in spp

composition?• Bray-Curtis dissimilarity• Non-metric MDS

concretealuminium

plywoodfibreglass

Stress = 0.126 Stress = 0.116

January October

Comparing groups in MDS

Haynes & Quinn data

• 4 groups (sites) - must be a priori groups• 3 replicate stations per site (n = 3)

• Are sites significantly different in species composition?

• Is there an ANOVA-like equivalent for MDS?

Analysis of similarities - ANOSIM

• Uses (dis)similarity matrix• Because dissimilarities are not normally distributed,

uses ranks of pairwise dissimilarities• Because dissimilarities are not independent of each

other, uses randomisation test rather than usual significance testing procedure

• Generates own test statistic (called R) by randomisation of rank dissimilarities

• Available through PRIMER package– Not SYSTAT nor SPSS

Null hypothesis

Average of rank dissimilarities between objects within groups = average of rank dissimilarities between objects between groups

rB = rW

No difference in species composition between groups

Within group dissimilaritiesBetween group dissimilarities

Test statistic

R average of rank dissimilarities between objects between groups - average of rank dissimilarities between objects within groups

R = (rB - rW) / (M / 2) where M = n(n-1)/2

• R between -1 and +1.• Use randomization test to generate probability

distribution of R when H0 is true.

Haynes & Quinn ANOSIM

• R = 0.583, P = 0.002 so reject Ho.• Significant differences between sites

• Followed by pairwise ANOSIM comparisons• Adjusted significance levels

ANOSIM

• Available also for 2 level nested and factorial designs.

• Primer package.

• Limited to total of 125 objects (e.g. SU’s).

• If 2 groups, n must be > 4 for randomization procedure.

• Alternative is to use ANOVA on NMDS axis scores - ANOSIM is better.

Which variables (species) most important?

• For MDS-type analyses, three methods:– correlate individual variables (species

abundances) with axis scores– SIMPER (similarity percentages) to determine

which species contribute most to Bray-Curtis dissimilarity

– CA and/or CANOCO to simultaneously ordinate objects and species - biplots

SIMPER (similarity percentages)

|yij - yik|Bray-Curtis dissimilarity =yij + yik)

Note is summing over each species, 1 to p.

The contribution of species i is: |yij - yik|

i =yij + yik)

Which species discriminate groups of objects?

• Calculate average i over all pairs of objects between groups– larger values indicate species contribute more to group

differences

• Calculate standard deviation of i – smaller values indicate species contribution is consistent

across all pairs of objects

• Calculate ratio of i / SD(i)– larger values indicate good discriminating species between 2

groups

Linking biota MDS to environmental variables

• Are differences between SU’s in species abundances related to differences in environmental variables?

• Correlate MDS axis scores with environmental variables

• BIO-ENV procedure - correlates dissimilarities from biota with dissimilarities from environmental variables

BIO-ENV procedure

Samples

Speciesabundances

Envvariables

Euclidean

Bray-Curtis

Subsets ofvariables

Rank correlation- Spearman- Weighted Spearman

Dissimilarity matrix

BIO-ENV correlations

• Exploratory rather than hypothesis testing procedure.

• Tries to find best combination of environmental variables, ie. combination most correlated with biotic dissimilarities.

• A priori chosen correlations can be tested with RELATE procedure - randomization test of correlation.

Vector fitting

• Uses final NMDS configuration rather than dissimilarity matrix - dependent on dimension number.

• Calculates vector (direction) through configuration of samples along which sample scores have max. correlation with environmental variable (one at a time).

• Significance testing (Ho: no correlation) done with randomization (Monte-Carlo) test.

• Available in DECODA and PATN.