Multivariate analysis in
community ecology
Gerry Quinn
Deakin University
Data sets in community ecology
• Multivariate abundance data
• Sampling or experimental units
– plots, cores, panels, quadrats ……
– usually in hierarchical spatial or temporal structure
• Abundances recorded for multiple taxa in each unit
– simple counts, densities, % cover, presence-absence ……
• Environmental variables recorded in each unit
– pH, salinity, temperature, nutrients, sediment load, elevation …..
Typical aims
• Examine spatial and temporal patterns in species composition
– assemblage/community “structure”, more than simply biodiversity
(e.g. taxon richness/diversity)
– test formal hypotheses about spatial and temporal differences in
composition
• Relate patterns to unit (or higher) level environmental
predictors
– typical linear model type question
• Determine which taxa are most important in “driving” the
patterns
– which taxa most typify differences across spatial and temporal
hierarchies
Why multivariate?
• Individual taxa of main interest
– concern over multiple univariate hypothesis testing (Type 1 error
rates)
– referees and editors won’t accept paper with 50-100 ANOVAs
• Community (assemblage) structure interest
– recognition of limitations of univariate biodiversity (richness, diversity,
evenness) measures
– hypotheses about community/assemblage composition
• Most multivariate analyses in community ecology also
incorporate univariate (individual taxa or environmental
predictors) models
Forest bird communities
• Does bird community
composition vary between forest
types?
– 5 types (box-ironbark, river redgum,
Gippsland manna gum etc.) plus
mixed
• Maximum bird abundance
(across 4 seasons)
beechworthonline.com.au
– 102 species across 37 sites
• Mac Nally (1989)
Swift parrot - Wikipedia
Estuary nematode communities
• Does nematode community
composition vary between sites
and with environmental variables?
• Nematode abundance (6 seasonal
“replicates)
– 182 species across 19 “sites”
• Environmental variables
Exe estuary - Wikipedia
– 6 (sediment particle size, % organic
matter etc.) at each site
• Clarke & Warwick (1993)
Marine nematodes
http://www.ipm.iastate.edu
Site
Sp1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Sp2
90
54
47
52
0
5
8
3
51
0
0
0
0
1
0
0
0
0
1
Sp3
187
158
117
27
0
0
14
18
2
0
0
0
0
0
0
0
0
0
0
Sp4
90
66
28
6
0
0
145
35
206
0
0
0
0
0
0
0
0
0
0
Sp5
23
51
97
72
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Sp6
123
22
9
1
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
Sp7
28
10
26
3
0
0
4
17
1
0
0
0
0
0
0
0
0
0
0
etc.
5 etc.
5 etc.
3 etc.
1 etc.
0 etc.
1 etc.
120 etc.
94 etc.
76 etc.
0 etc.
0 etc.
0 etc.
0 etc.
0 etc.
0 etc.
0 etc.
0 etc.
0 etc.
0 etc.
Part
WTab H2S
Shore %
size
depth depth height organ Salinity
0.06
0 2.167
4
6.43 24.833
0.06
0 3.183
3
7.06 22.833
0.06
0 1.817
2
7.99 17.833
0.06
0
2.02
1
7.15
16.2
1.275
20
20
5
0.24
10
0.562 3.417
2.95
4
0.37
76.6
0.06
0 2.167
3
1.98
76
0.177
0 2.683
2
2.22
81.2
0.06
0
2.66
1
5.88
71.2
0.451
20
20
5
0.09
10
0.205 4.417
7.25
4
0.39
88
0.528
20
20
3
0.09
88
0.598
20
20
2
0.06
88
0.769
0
20
1
0.09
88.5
0.468 14.917
20
5
0.06
89
0.837 6.333
20
4
0.04 90.875
0.797
6.75
20
3
0.06 91.667
1.141 3.667
20
2
0.07
89.4
0.223
0
20
1
0.09 90.833
Impact assessment
• Does sessile marine animal
community composition vary between
sewage impact and control sites?
– 3 control and 1 impact locations
– 4 randomly chosen times
– replicate sites and photographic quadrats
at each location
• Percent cover of 58 taxa
• Classical “beyond” BACI design
– split-plot type linear model
• Terlizzi et al (2005)
http://www.conisma.it/total/t_aim.html
Three broad approaches
• Eigenanalyses
– distance measure implied
• Distance-based analyses
– distance measure explicit and user-selected
• Multi-species linear models
– combine taxon-specific univariate (linear) models
– no distance measure required
Eigenanalysis methods
• Principal components analysis (PCA)
– implied Euclidean distance
• Correspondence analysis (CA)
– implied chi-square distance
• Canonical correspondence analysis (CCA/CANOCO)
– constrains ordination based on linear modelling with
environmental variables
• Strengths
– biplots of sample and species ordinations
– CCA provides measures of fit with covarying environmental
variables
Cajo ter Braak
Rodents in habitat fragments
SITE
Florida
Sandmark
34street
Balboaterr
Katesess
Altalajolla
Laurel
Canon
Zena
AREA
25
84.1
53.8
51.8
25.6
32.1
9.7
8.7
8.5
DISTX
AGE
2100
914
1676
243
822
121
1554
1219
2865
etc.
Bolger et al (1997)
50
20
34
34
16
14
79
58
36
RRATTUS MMUS
0
0
0
0
0
0
0
0
3
PCALIF
13
1
4
4
2
1
11
16
8
PEREM
3
57
36
53
63
48
0
0
0
1
65
0
1
21
35
0
0
0
RMEGAL NFUSC
NLEPID
1
2
9
16
2
9
5
30
11
16
12
8
0
0
0
0
0
0
0
8
0
0
0
12
0
0
0
PFALLAX MCALIF
0
2
0
18
0
2
0
0
0
0
3
0
3
0
2
0
0
0
Rodent data – CA biplot
Axis 2
Rr
Acuna
El mac
54th Street
Baja
Zena
32nd Street Sth
Oakcrest
Axis 1
Florida
Mm
7 fragments
Rodent data – CCA triplot
Axis 2
Mc
Pe
Sandmark
34th Street
Nl
Area
Laurel
Balboa
Mm
Dist
Spruce
Age
Axis 1
El mac
Edison
Acuna
54th Street
Montanosa
Rr
Issues
• Both methods “compress”
distances at ends of axes (socalled arch or horseshoe
effect)
Comp 2
– detrended CA brute force “fix”
for this effect
• CA and CCA implicitly upweight rarer taxa by use of chisquare distance
• No choice of distance measure
Comp 1
PCA bird community data
Distance-based methods
• Include principal coordinates analysis
(PCoA), multidimensional scaling (MDS),
generalised dissimilarity modelling (GDS)
• Hypothesis testing
– compare groups using multi-response
permutation procedure (MRPP), analysis of
similarities (ANOSIM), permutational
multivariate ANOVA (PERMANOVA)
– relate to environmental variables with Mantel
test, BIO-ENV
Marti Anderson
John Curtis
Bob Clarke
Distance-based methods
• Strengths
– flexibility of distance/dissimilarity measure, standardisation and
transformation
– consistency in that ordination and subsequent analyses based on
original dissimilarities
– some dissimilarities can be “decomposed” into relative taxon
contributions (similarity percentages - SIMPER)
nMDS – bird community data
PERMANOVA – bird community data
nMDS – subtidal reef data
PERMANOVA – subtidal reef data
Issues
• Flexible choice of distance/dissimilarity measure
– ecologists nearly always default to Bray-Curtis
– does B-C represent ecological differences of interest?
• Modelling dissimilarities tricky
– appropriate probability distributions – permutation
procedures usually applied – robustness for complex
models?
– PERMANOVA only partitions SS not likelihoods
– lack of independence – rely on permutation robustness
• Limited predictive capacity
• Distance-based methods cannot easily separate
location and dispersion effects
• Location vs dispersion
• Warton et al (2012)
Location vs dispersion
• Transformation of abundances may help BUT many taxa have
very skewed distributions
• Issue recognised by PRIMER/PERMANOVA
– “we can consider the homogeneity of dispersions to be included as
part of the general null hypothesis of "no differences" among groups
being tested by PERMANOVA (even though the focus of the
PERMANOVA test is to detect location effects)” (PERMANOVA manual
p.22)
• On going debate PRIMER/PERMANOVA vs mvabund
“Univariate” linear model approach
• Fit separate generalised linear models to each taxon
– based on –ve binomial distribution (over-dispersed counts)
• Testing overall group or covariate effects
– sum likelihood ratio (LR) tests across taxa
– use permutation (resampling) methods to generate test statistic
• Relative taxon contribution to patterns
– LR statistic as measure of strength of individual taxon contributions
• Strengths
– linear models framework, univariate predictive capacity
– handles mean-variance relationship
• Issues
– not an “ordination” method
David Warton
Methods in community ecology
• Journals searched 2011-2012
– Austral Ecology
– Oikos
• Analyses of community/assemblage
(species abundance incl. pres-abs data)
– 62 papers found
• Methods used
–
–
–
–
–
overall multivariate “philosophy”
choice of dissimilarity measure (if relevant)
transformation/standardisation used
modeling (hypothesis testing) method
choice of “ordination” plot
Multivariate approach
Approach
Eigenanalysis
Distance-based
Combined taxonspecific linear
models
# papers
15
% papers
24
47
0
76
0
Eigenanalyses
Approach
MANOVA / DFA
# papers
3
PCA
Correspondence analysis (incl.
detrended)
0
8
Constrained (canonical)
correspondence analysis
4
Majority of “ordinations” based on biplots, many with
vectors fitted for environmental predictors (triplots)
Distance-based
Dissimilarity measure
Bray-Curtis
Sorensen
Jaccard
Gower
# papers
31
4
2
2
Distance/dissimilarity
• Why do ecologists default to Bray-Curtis?
– Faith et al (1987 – Vegetatio) strongly recommended B-C as robust
indicator of ecological gradients
– ranges between 0 (identical samples) and 1 (no species in common)
– handles joint absences (taxa missing from both samples)
– default in PRIMER/PERMANOVA, PC-ORD
• Does B-C represent patterns ecologists are really interested
in?
Distance-based
Approach
Comparing groups
ANOSIM / PERMANOVA / dbRDA
MRPP
ANOVA on MDS axis scores
# papers
24
6
2
Majority of “ordinations” based on non-metric MDS, 3
papers used cluster analysis
Distance-based
Approach
Relating to env predictors
BIO-ENV/ Relate
Mantel tests
Regression/correlation with MDS axis scores
Generalised dissimilarity modelling
Determining taxa driving group differences
SIMPER
# papers
24
6
2
1
9
Transformations
• Transformations of abundances common in ecology
– log (y+1) or square/fourth root
– original PRIMER program had 4th root as default!
• Most common reason - to reduce the influence of most
abundant (dominant) taxa and give relatively greater
weighting to rarer taxa
– each taxon will be affected differently depending on its distribution?
– effects on interaction terms almost never considered
• Issues of unequal dispersions almost never raised in ecological
papers
– “it is not at all difficult to understand that transformations will also
affect relative dispersions in multivariate space” (PERMANOVA manual
p. 97)
Standardisations
None
Sample
• Invertebrate assemblages in
lake (Quinn et al 1996)
• Four site-season
combinations
• nMDS on Bray-Curtis
• Four standardisations:
•
•
•
•
None
By sample totals
By taxa totals
Double
• Bray-Curtis vs Canberra
Taxa
Double
To Bayes or not to Bayes….
Bayesian approaches
• Detecting transitions between upslope
and riparian vegetation
– management of stream riparian zones
• Based on plant assemblage data (%
cover) along transects away from
stream
– pairwise Canberra distances between
quadrats along each transect
• Aim - to find the model with the
highest probability of being the break
between riparian and upslope
vegetation
– usual MCMC estimation of models
Acheron River
Bayes factors > 10
Higher elevation sites
Lower elevation sites
Mac Nally et al (2008) Plant Ecology
Bayesian approaches
• Maybe more robust than ML for complex models
– already being used for variance estimation and confidence (credible)
intervals in some mixed model software
• Straightforward(?) under mvabund generalised linear model
approach
– select suitable probability distributions for parameters
– use uninformative prior if appropriate
• More difficult with distance-based methods
– but can be adapted (see Mac Nally 2005 Divers & Distr)
– other examples using MDS and clustering (Oh & Raftery 2007 J Comp
Graph Stat) focus on graphical representation (“ordination”)
Questions for discussion
• Is the confounding of location and dispersion a “fatal”
flaw for distance-based measures?
– more direct comparisons between distance-based and linear
model approaches needed
• Comparison to other new methods
– generalised dissimilarity modelling (Ferrier et al 2007)
– gradient forests (Ellis et al 2012)
• If distance-based measures are used:
– what does Bray-Curtis actually measure ecologically?
• What do multivariate models actually predict?
Questions for discussion
• Should ecologists re-think their use of transformations?
– NOT just a multivariate issue!
• How do ecologists determine optimum sample sizes for
community ecology
– power characteristics will vary between taxa in linear models approach
– power for distance-based permutation analyses?