Multidimensional scaling MDS

Transcription

1 Multidimensional scaling MDS And other permutation based analyses MDS Aim Graphical representation of dissimilarities between objects in as few dimensions (axes) as possible

2 Graphical representation is termed an ordination in ecology Axes of graph represent new variables which are summaries of original variables Haynes & Quinn (unpublished) Four sites along Morwell River site upstream from planned sewage outfall sites, 3 and 4 downstream site 3 below fish farm Abundance of all species of invertebrates recorded from 3 stations at each site

3 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 different from rest? Multidimensional scaling. Set up a raw data matrix Species etc. Site/sample S S S S S S etc.

4 . Calculate a dissimilarity (Bray-Curtis) matrix S S S 3 S S S 3 etc. S.000 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 and 4 dimensions

5 4. Arrange objects (eg. sampling units) initially on ordination plot in chosen number of dimensions starting configuration usually generated randomly Starting configuration Axis II Axis I Site Site Site 3 Site 4

6 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 II Distance Axis I Site Site 3 Site Site Dissimilarity Shepard plot Stress = 0.394

7 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 After 0 iterations Axis II 0 - Distance Axis I Stress = Dissimilarity

8 7. Final configuration further moving of objects on ordination plot cannot improve match between dissimilarities and distances stress as low as possible Final configuration - 50 iterations Axis I 0 - Distance Axis II Stress = Dissimilarity

9 Iteration history Iteration Stress Stress of final configuration is How low should stress be? Clarke (993) suggests: > 0.0 is basically random < 0.5 is good < 0.0 is ideal configuration is close to actual dissimilarities

10 How many dimensions? Increasing no. of dimensions above 4 usually offers little reduction in stress or 3 dimensions usually adequate to get good fit (ie. low stress) dimensions straightforward to plot Lonhart (unpublished data) Effects of depth and piling location on marine fouling assemblage Two pilings, four sides of each panel, two depths, sampled 4 times 40 species in total recorded

11 MDS to examine relationship piling location and depth on invertebrate community Does the community vary as a function of depth? Does the community vary as a function of pilling location? Does the effect of depth on the community vary as a function of piling location? Bray-Curtis dissimilarity Non-metric MDS ANOSIM / PERMANOVA SIMPER MDS Plot Resemblance: S7 Bray Curtis similarity D Stress: 0.7 Date 00 3_05_00 3_8_00 4_0_00

12 Resemblance: S7 Bray Curtis similarity D Stress: 0.7 Piling Resemblance: S7 Bray Curtis similarity D Stress: 0.7 PilingDepth 838Shallow 838Deep 879Shallow 879Deep Resemblance: S7 Bray Curtis similarity D Stress: 0.7 Depth Shallow Deep Comparing groups in MDS Piling locations Depths 8 replicates per treatment combination (4 sides x samples) Are sites significantly different in species composition? Is there an ANOVA-like equivalent for MDS?

13 Procedure :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 randomization test rather than usual significance testing procedure Generates own test statistic (called R) by randomization of rank dissimilarities Available through PRIMER package Lonhart ANOSIM Depth effect R = 0.305, P = 0.00 so reject H o. - Significant differences between depths Piling location R = 0.76, P = 0.00 so reject H o - Significant difference by Piling

14 Permanova (permutation ANOVA) Run just like an ANOVA Sums of Squares can be partitioned in multivariate space (based on distances to multidimensional centroids) P values based on permutations of the analysis Permanova (permutation ANOVA) PERMANOVA table of results Unique Source df SS MS Pseudo-F P(perm) perms Depth Piling DepthxPiling Res 4.778E Total 7.4E5

15 Resemblance: S7 Bray Curtis similarity D Stress: 0.7 Piling Resemblance: S7 Bray Curtis similarity D Stress: 0.7 PilingDepth 838Shallow 838Deep 879Shallow 879Deep Resemblance: S7 Bray Curtis similarity D Stress: 0.7 Depth Shallow Deep Interaction effect Which variables (species) most important? For MDS-type analyses, three methods: correlate individual variables (species abundances) with axis scores like PCA loadings SIMPER (similarity percentages) to determine which species contribute most to Bray-Curtis dissimilarity CA (Correspondence Analyis)to simultaneously ordinate objects and species - biplots

16 SIMPER (similarity percentages) Bray-Curtis dissimilarity = y ij - y ik y ij + y ik ) Note is summing over each species, to p. The contribution of species i is: y ij - y ik i = y ij + y ik ) Simper results comparing deep depths between Pilings Groups 838Deep & 879Deep Average dissimilarity = Group Group 838Deep 879Deep Species Av.Abund Av.Abund Av.Diss Diss/SD Contrib% Cum.% Watersipora, live Detritus Corynactis californica Burgundy crust Diplosoma listerianum CaCO Dead bryozoan Orange bryozoan Dead Watersipora Ascidia ceratodes Rhynchozoon (brwn bryo)

17 Are these results interpretable graphically? Resemblance: S7 Bray Curtis similarity D Stress: 0.7 Watersipora, live 0 Resemblance: S7 Bray Curtis similarity D Stress: 0.7 PilingDepth 838Shallow 838Deep 879Shallow 879Deep Linking biota MDS to environmental variables Are differences in species composition 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

18 BIO-ENV procedure Samples Dissimilarity matrix Species abundances Bray-Curtis Env variables Euclidean Subsets of variables Rank correlation - Spearman - Weighted Spearman 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.

19 Example Bristol Bay Zooplankton 57 stations 5 species sampled Salinity measures taken at the same time Question: is zooplankton community related to salinity Zooplankton community data

20 Community Matrix NMDS plot Bristol Channel zooplankton Non-metric MDS Resemblance: S7 Bray-Curtis similarity D Stress:

21 NMDS plot with Salinity Bubbles Salinity Bristol Channel zooplankton Non-metric MDS Resemblance: S7 Bray-Curtis similarity D Stress: Salinity data

22 Salinity Matrix RELATE procedure Samples Dissimilarity matrix Species abundances Env variables Bray-Curtis Euclidean All variables Rank correlation - Spearman - Weighted Spearman

23 Frequency RELATE the matrices 56 Bristol Channel salinity group (-9 in increasing salinity) RELATE Parameters Correlation method: Spearman rank Sample statistic (Rho): 0.74 Significance level of sample statistic: 0. % (=<0.00) Number of permutations: 999 Number of permuted statistics greater than or equal to Rho: Rho A more complicated example linking multivariate biological data to multivariate environmental data Biological data: Nematode species (>00) abundance at 9 sites in Exe estuary Environmental: MPD: mean particle diameter % Org: Percent organic matter WT: water table depth H S: depth of Hydrogen sulfide layer Sal: interstitial salinity Ht: Intertidal range

24 Environmental NMDS Exe estuary Non-metric MDS 9 4 Normalise Resemblance: D Euclidean distance 8 D Stress: Biological NMDS Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity D Stress: 0.05 site

25 Linking Environment to Community Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity Med Part Diam D Stress: 0.05 site Interstit Salinity D Stress: 0.05 site Dep Water Tab Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity D Stress: 0.05 site %Organics Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity D Stress: 0.05 site Formally First: use RELATE to determine relationship between the biological community and the environmental community

26 Frequency RELATE procedure Samples Dissimilarity matrix Species abundances Env variables Bray-Curtis Euclidean All variables Rank correlation - Spearman - Weighted Spearman 67 Exe estuary RELATE Parameters Correlation method: Spearman rank Sample statistic (Rho): 0.79 Significance level of sample statistic: 0. % (<0.00) Number of permutations: 999 Number of permuted statistics greater than or equal to Rho: Rho

27 Formally First: use RELATE to determine relationship between the biological community and the environmental community Second: Use BIO ENV to determine best fit of environmental variables to Biological Community BIO-ENV procedure Samples Dissimilarity matrix Species abundances Bray-Curtis Env variables Euclidean Subsets of variables Rank correlation - Spearman - Weighted Spearman

28 Select best model Best result for each number of variables No.Vars Corr. Selections Dep HS layer Dep HS layer,interstit Salinity Med Part Diam,Dep HS layer,interstit Salinity Med Part Diam,Dep HS layer,%organics,interstit Salinity Med Part Diam,Dep HS layer,shore height,%organics,interstit Salinity Med Part Diam,Dep Water Tab,Dep HS layer,shore height,%organics,interstit Salinity Linking Environment to Community model results Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity Med Part Diam D Stress: 0.05 site Dep HS layer D Stress: 0.05 site Best result for each number of variables No.Vars Corr. Selections Dep HS layer Dep HS layer,interstit Salinity Med Part Diam,Dep HS layer,interstit Salinity Med Part Diam,Dep HS layer,%organics,interstit Salinity Med Part Diam,Dep HS layer,shore height,%organics,interstit Salinity Med Part Diam,Dep Water Tab,Dep HS layer,shore height,%organics,interstit Salinity Interstit Salinity Exe nematodes (9 sites averaged over season) Non-metric MDS Resemblance: S7 Bray-Curtis similarity D Stress: 0.05 site

29 Procedure :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 randomization test rather than usual significance testing procedure Generates own test statistic (called R) by randomization of rank dissimilarities Available through PRIMER package

30 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

31 Test statistic R average of rank dissimilarities between objects between groups - average of rank dissimilarities between objects within groups R = (r B - r W ) / (M / ) where M = n(n-)/ R between - and +. Use randomization test to generate probability distribution of R when H 0 is true. Lonhart ANOSIM Depth effect R = 0.305, P = 0.00 so reject H o. - Significant differences between depths Piling location R = 0.76, P = 0.00 so reject H o - Significant difference by Piling

32 SIMPER (similarity percentages) Bray-Curtis dissimilarity = y ij - y ik y ij + y ik ) Note is summing over each species, 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 groups