Multidimensional scaling MDS  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 SYD MELB BRIS ADEL PER HOB DAR

-2012 Dimension Dimension 2 Stress = 0.014

-2012 Dimension Dimension 2 x -1 Darwin Perth Adelaide Hobart Melbourne Canberra Sydney Brisbane

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Haynes & Quinn (unpublished) Four sites along Morwell River –site 1 upstream from planned sewage outfall –sites 2, 3 and 4 downstream –site 3 below fish farm Abundance of all species of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 Species12345etc. Site/sample S S S S S S etc.

2. Calculate a dissimilarity (Bray-Curtis) matrix S 11 S 12 S 13 S 21 S 22 S 23 etc. S S S S S S 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 Axis I Axis II Site 1Site 3Site 2Site 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 Axis I Axis II Site 1 Site 3 Site 2 Site Dissimilarity Distance Shepard plot Stress = 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

Dissimilarity Distance 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

Dissimilarity Distance Axis II Axis I Final configuration - 50 iterations Stress = 0.069

IterationStress Stress of final configuration is 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 Distance

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

concrete aluminium plywood fibreglass Stress = 0.126Stress = JanuaryOctober

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 r B = r W No difference in species composition between groups

Within group dissimilarities Between group dissimilarities

Test statistic Raverage of rank dissimilarities between objects between groups - average of rank dissimilarities between objects within groups R = (r B - r W ) / (M / 2) where M = n(n-1)/2 R between -1 and +1. Use randomization test to generate probability distribution of R when H 0 is true.

Haynes & Quinn ANOSIM R = 0.583, P = 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)  |y ij - y ik | Bray-Curtis dissimilarity =  y ij + y ik ) Note  is summing over each species, 1 to p. The contribution of species i is: |y ij - y ik |  i =  y ij + y ik )

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 Species abundances Env variables Euclidean Bray-Curtis Subsets of variables 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.