quantitative methods in palaeoecology and palaeoclimatology pages valdivia october 2010 ordination...
TRANSCRIPT
Quantitative Methods in Palaeoecology and Palaeoclimatology
PAGES Valdivia October 2010
Ordination Analysis II –
Direct Gradient AnalysisJohn Birks
Canonical correspondence analysis (CCA)
Introduction
Basic terms and ordination plots
Other topics in CCA
Robustness
Scaling and interpretation of CCA plots
Example
Redundancy analysis (RDA) (= constrained PCA)
Scaling and interpretation of RDA plots
Statistical testing of constrained ordination axes
DIRECT GRADIENT ANALYSIS
Partial constrained ordinations
Partial ordinations
Partitioning variance
Environmental (predictor) variables and their selection
Canonical correlation analysis
Distance-based redundancy analysis
Canonical analysis of principal co-ordinates
Principal response curves
CCA/RDA as predictive tools
CANODRAW
Interpretation of ordination axes with external data
Canonical or constrained ordination techniques (= direct gradient analysis)
BASIS OF CLASSICAL ORDINATION INTERPRETATION AND ENVIRONMENT
We tend to assume that biological assemblages are controlled by environment, so:
1. Two sites close to each other in an indirect ordination are assumed to have similar composition, and
2. if two sites have similar composition, they are assumed to have similar environment.
In addition:
3. Two sites far away from each other in ordination are assumed to have dissimilar composition, and thus
4. if two sites have different composition, they are assumed to have different environment.
J. Oksanen (2002)
Values of environmental variables and Ellenberg’s indicator values of species written alongside the ordered data table of the Dune Meadow Data, in which species and sites are arranged in order of their scores on the second DCA axis. A1: thickness of A1 horizon (cm), 9 meaning 9cm or more; moisture: moistness in five classes from 1 = dry to 5 = wet; use: 1 = hayfield, 2 = a mixture of pasture and hayfield, 3 = pasture; manure: amount applied in five classes from 0 = no manure to 5 = heavy use of manure. The meadows are classified by type of management: SF, standard farming; BF, biological farming; HF, hobby farming; NM, nature management; F, R, N refer to Ellenberg’s indicator values for moisture, acidity and nutrients, respectively .
Veg
eta
tion
al
data
En
vir
on
men
tal
data
DUNE-MEADOW DATA
The amount of manure written on the DCA ordination. The trend in the amount across the diagram is shown by an arrow, obtained by a multiple regression of manure on the site scores of the DCA axes. Also shown are the mean scores for the four types of management, which indicate, for example, that the nature reserves (NM) tend to lie at the top of the diagram. Ez=b0 + b1x1 + b2x2
Angle ()with axis 1 = arctan(b2 / b1)
DCA axis 2
DCA axis 1
Indirect analysis
Site scores of the second DCA axis plotted against the amount of manure.
Indirect analysis
Correlation coefficients (100 r) of the environmental variables for the four first DCA axes for the Dune Meadow Data
Variable Axes
1 2 3 4
1 A1 58 24 7 9
2 moisture 76 57 7 -7
3 use 35 -21 -3 -5
4 manure 6 -68 -7 -64
5 SF 22 -29 5 -60
6 BF -28 -24 39 22
7 HF -22 -26 -55 -14
8 NM 21 73 17 56
Eigenvalue 0.54 0.29 0.08 0.05
Indirect analysis
Multiple regression of the first CA axis on four environmental variables of the dune meadow data, which shows that moisture contributes significantly to the explanation of the first axis, whereas the other variables do not.Term Parameter Estimate s.e. tconstant c0 –2.32 0.50 –4.62
A1 c1 0.14 0.08 1.71
moisture c2 0.38 0.09 4.08
use c3 0.31 0.22 1.37
manure c4 –0.00 0.12 –0.01
ANOVA tabled.f. s.s. m.s. F
Regression 4 17.0 4.25 10,6Residual 15 6.2 0.41Total 19 23.2 1.22 R2 = 0.75 R2
adj = 0.66
Ey1 = b0 + b1x1 + b2x2 + ...bnxn
CA axis 1 environmental variables
x = environmental variables
Indirect analysis
TWO-STEP APPROACH OF INDIRECT GRADIENT ANALYSIS
Standard approach to about 1985: started by D.W. Goodall in 1954
Limitations: (1) environmental variables studied may turn out to be poorly related to the first few ordination axes.
(2) may only be related to 'residual' minor directions of variation in species data.
(3) remaining variation can be substantial, especially in large data sets with many zero values.
(4) a strong relation of the environmental variables with, say, axis 5 or 6 can easily be overlooked and unnoticed.
Limitations overcome by canonical or constrained ordination techniques = multivariate direct gradient analysis.
Ordination and regression in one technique – Cajo ter Braak 1986
Search for a weighted sum of environmental variables that fits the species best, i.e. that gives the maximum regression sum of squares Ordination diagram 1) patterns of variation in the species data 2) main relationships between species and each environmental variable
Redundancy analysis constrained or canonical PCA
Canonical correspondence analysis (CCA) constrained CA
(Detrended CCA) constrained DCA
Axes constrained to be linear combinations of environmental variables.
In effect PCA or CA with one extra step:
Do a multiple regression of site scores on the environmental variables and take as new site scores the fitted values of this regression.
Multivariate regression of Y on X.
CANONICAL ORDINATION TECHNIQUES
PRIMARY DATA IN GRADIENT ANALYSIS
Indir
ect
G
A
Dir
ect
GA
Abundances or +/-
variables
Response variables
Values
Classes
Predictor or explanatory variables
Speci
es
Env. vars
PLUS
Artificial example of unimodal response curves of five species (A-E) with respect to standard-ised environmental variables showing different degrees of separation of the species curves
a: Moistureb: Linear combination of moisture and phosphate, chosen a prioric: Best linear combination of environmental variables, chosen by CCA. Sites are shown as dots, at y = 1 if Species D is present and at y = 0 if Species D is absent
moisture
linear combination of moisture and phosphate
CCA linear combination
Combinations of environmental variables
e.g. 3 x moisture + 2 x phosphate
e.g. all possible linear combinations
zj = environmental variable at site j
c = weights
xj = resulting ‘compound’ environmental variable
CCA selects linear combination of environmental variables that maximises dispersion of species scores, i.e. chooses the best weights (ci) of the environmental variables.
..... jjjoj zczczccx 332211
Algorithms for (A) Correspondence Analysis, (B) Detrended Correspondence Analysis, and (C) Canonical Correspondence Analysis, diagrammed as flowcharts. LC scores are the linear combination site scores, and WA scores are the weighted averaging scores.
ALTERNATING REGRESSION ALGORITHMS
- CA - DCA - CCA
1) Start with arbitrary, but unequal, site scores xi.
2) Calculate species scores by weighted averaging of site
scores.
3) Calculate new site scores by weighted averaging of species
scores.
[So far, two-way weighted average algorithm of correspondence analysis].
n
iik
n
iiikk yxyu
11
/
m
kik
m
kkiki yuyx
11
/
CANONICAL CORRESPONDENCE ANALYSIS
Algorithm
REF
REF
REF
REF
4) Obtain regression coefficients of site scores on the environmental variables by weighted multiple regression.
where b and x* are column vectorsZ is environmental data n x (q +1)R is n x n matrix with site totals in diagonal
5) Calculate new site scores
or 6) Centre and standardise site scores so that:
and 7) Stop on convergence, i.e. when site scores are sufficiently close to
site scores of previous iteration. If not, go to 2.
xRZRZZb 111
bzbx 0
n
iii xy
1
0 11
2
n
iii xy
zbx
REF
REF
REF
REF
CANONICAL OR CONSTRAINED CORRESPONDENCE ANALYSIS (CCA)
Ordinary correspondence analysis gives:
1. Site scores which may be regarded as reflecting the underlying gradients.
2. Species scores which may be regarded as the location of species optima in the space spanned by site scores.
Canonical or constrained correspondence analysis gives in addition:
3. Environmental scores which define the gradient space.
These optimise the interpretability of the results.J. Oksanen (2002)
BASIC TERMS
Eigenvalue = Maximised dispersion of species scores along axis. In CCA usually smaller than in CA. If not, constraints are not useful.
Canonical coefficients = ‘Best’ weights or parameters of final regression.
Multiple correlation of regression = Species–environment correlation. Correlation between site scores that are linear combinations of the environmental variables and site scores that are WA of species scores. Multiple correlation from the regression. Can be high even with poor models. Use with care!
Species scores = WA optima of site scores, approximations to Gaussian optima along individual environmental gradients.
Site scores = Linear combinations of environmental variables (‘fitted values’ of regression) (1).
Can also be calculated as weighted averages of species scores that are themselves WA of site scores (2).
(1) LC scores are predicted or fitted values of multiple regression with constraining predictor variables 'constraints'.
(2) WA scores are weighted averages of species scores.
Generally always use (1) unless all predictor variables are 1/0 variables.
SUMMARY OF DUNE MEADOW DATA
Dune Meadow Data. Unordered table that contains 20 relevées (columns) and 30 species (rows). The right-hand column gives the abbreviation of the species names listed in the left-hand column; these abbreviations will be used throughout the book in other tables and figures. The species scores are according to the scale of van der Maarel (1979b).
1 2.8 1 SF 2 42 3.5 1 BF 2 23 4.3 2 SF 2 44 4.2 2 SF 2 45 6.3 1 HF 1 26 4.3 1 HF 2 27 2.8 1 HF 3 38 4.5 5 HF 3 39 3.7 4 HF 1 110 3.3 2 BF 1 111 3.5 1 BF 3 112 5.8 4 SF 22*13 6.0 5 SF 2 314 9.3 5 NM 3 015 11.5 5 NM 2 016 5.7 5 SF 3 317 4.0 2 NM 1 018 4.6* 1 NM 1 019 3.7 5 NM 1 020 3.5 5 NM 1 0
Sample number
A1 horizon
Moisture class
Management type
UseManure
class
Environmental data of 20 relevées
from the dune meadows
Use categories: 1 = hay 2 = intermediate 3 = grazing
* = mean value of variable
DCA ordination diagram of the Dune Meadow Data
DCAaxis 2
DCAaxis 1
DCA
Correlations of environmental variables with DCA axes 1 and 2
Axis One1 = 0.54
Axis Two2 = 0.29
DCA
CCA of the Dune Meadow Data. a: Ordination diagram with environmental variables represented by arrows. the c scale applies to environmental variables, the u scale to species and sites. the types of management are also shown by closed squares at the centroids of the meadows of the corresponding types of management.
11 22 R axis 1 R axis 2R axis 1 R axis 2
DCADCA 0.54 0.54 0.40 0.40 0.87 0.87 0.83 0.83CCACCA 0.46 0.46 0.29 0.29 0.96 0.96 0.89 0.89
CCA
Canonical correspondence analysis: canonical coefficients (100 x c) and intra-set correlations (100 x r) of environmental variables with the first two axes of CCA for the Dune Meadow Data. The environmental variables were standardised first to make the canonical coefficients of different environmental variables comparable. The class SF of the nominal variable 'type of management' was used as a reference class in the analysis.
A1 9 -37 57 -17Moisture 71 -29 93 -14Use 25 5 21 -41Manure -7 -27 -30 -79SF - - 16 -70BF -9 16 -37 15HF 18 19 -36 -12NM 20 92 56 76
Variable Coefficients Correlations
Axis 1 Axis 2 Axis 1 Axis 2
CANONICAL CORRESPONDENCE ANALYSIS
CCA of the Dune Meadow Data. a: Ordination diagram with environmental variables
represented by arrows. the c scale applies to environmental variables,
the u scale to species and sites. the types of management are also
shown by closed squares at the centroids of the meadows of the
corresponding types of management.
b: Inferred ranking of the species along the variable amount of manure, based on the biplot interpretation of Part a of this figure.
aa
bb
BIPLOT PREDICTION OF ENVIRONMENTAL VARIABLES
Modified from J. Oksanen (2002)
• Project a site point onto environmental arrow: predict its environmental value
• Exact with two constraints only
• Projections are exact only in the full multi-dimensional space. Often curved when projected onto a plane
• You may have in a same figure• WA scores of species• WA or LC scores of sites• Biplot arrows or class centroids of environmental variables
• In full space, the length of an environmental vector is 1: When projected
onto ordination space• Length tells the strength of the variable• Direction shows the gradient• For every arrow, there is an equal arrow to the opposite direction,
decreasing direction of the gradient• Project sample points onto a biplot arrow to get the expected value
• Class variables coded as dummy variables• Plotted as class centroids• Class centroids are weighted averages
• LC score shows the class centroid, WA scores show the dispersion of the centroid
• With class variables only: Multiple Correspondence Analysis or Analysis of Concentration
CCA: JOINT PLOTS AND TRIPLOTS
Summary Axes
Axes 1 2 3 4 Total inertiaEigenvalues .461 .298 .160 .134 2.115Species-environment .958 .902 .855 .889 correlationsCumulative percentage variance
of species data 21.8 35.9 43.5 49.8 of species-environment 37.8 62.3 75.4 86.3 relation
Sum of all unconstrained eigenvalues = inertia 2.115Sum of all canonical eigenvalues = species-environment 1.220 relation
'Fitted' species data
10021
seigenvalue canonical
Rules of thumb:>0.30 strong gradient>0.40 good niche separation of species
CANOCO
1) Environmental variables continuous – biplot arrows
classes – centroid (weighted average) of sites belonging to that class
2) CA approximates ML solution of Gaussian model
CCA approximates ML solution of Gaussian model if CA axis is close to the linear com-bination of environmental variables. [Johnson & Altman (1999) Environmetrics 10, 39-52]
In CCA species compositional data are explained through a Gaussian unimodal response model in which the explanatory variable is a linear combination of environmental variables.
3) CCA – very robust, major assumption is that response model is UNIMODAL.
(Tolerances, maxima, and location of optima can be violated - see Johnson & Altman 1999)
4) Constraints become less and less strict the more environmental variables there are. If q, number of environmental variables ≥ number of samples -1, no real constraints and CCA = CA.
5) Arch effect may crop up. Detrending (by polynomials) DCCA. Useful for estimating gradient lengths (use segments).
6) Arch effect can often be removed by dropping superfluous environmental variables, especially those highly correlated with the arched axis.
OTHER CCA TOPICS
REPRESENTATION OF CLASS VARIABLES (1/0) IN CCA
1. Make class centroids as distinct as possible
2. Make clouds about centroids as compact as possible
• Success
• LC scores are the class centroids: the expected locations, WA scores are the dispersion of the centroid
• If high , WA scores are close to LC scores
• With several class variables, or together with continuous variables, the simple structure can become blurred J. Oksanen (2002)
Canonical correspondence analysis
Unimodal curves for the expected abundance response (y) of four species against an environmental gradient or variable (x). The optima, estimated by weighted averages, (u) [k=1,2,3], of three species are indicated. The curve for the species on the left is truncated and therefore appears monotonic instead of unimodal; its optimum is outside the sampled interval but, its weighted average is inside. The curves drawn are symmetric, but this is no strict requirement for CCA.
7) t-values of canonical coefficients or forward selection option in CANOCO to find minimal set of significant variables that explain data about as well as full set.
8) Can be sensitive to deviant sites, but only if there are outliers in terms of both species composition and environment. CCA usually much more robust than CA.
9) Can regard CCA as a display of the main patterns in weighted averages of each species with respect to the environmental variables.
Intermediate between CA and separate WA regressions for each species. Separate WA regressions point in q-dimensional space of environmental
variables. NICHE. CCA attempts to provide a low-dimensional representation of this niche.
10) ‘Dummy’ variables (e.g. group membership or classes) as environmental variables. Shows maximum separation between pre-defined groups.
11) ‘Passive’ species or samples or environmental variables. Some environmental variables active, others passive
e.g. group membership – activeenvironmental variables – passive
12) CANOCO ordination diagnostics fit of species and samples
pointwise goodness of fit can be expressed either as residual distance from the ordination axis or plane, or as proportion of projection from the total chi-squared distance species tolerances, sample heterogeneity
Passive ‘fossil’ samples added into CCA of modern data
Canonical correspondence analysis (CCA) time-tracks of selected cores from the Round Loch of Glenhead; (a) K5, (b) K2, (c) K16, (d) k86, (e) K6, (f) environmental variables. Cores are presented in order of decreasing sediment accumulation rate.
13) Indicator species
14) Behaves well with simulated data.
M W Palmer (1993) Ecology 74, 2215–2230
Copes with skewed species distributions‘noise’ in species abundance dataunequal sampling designshighly intercorrelated environmental variablessituations when not all environmental factors are known
Site scores along the first two axes in CCA and DCA ordinations, with varying levels of quantitative noise in species abundance. Quantitative noise was not simulated. The top set represents CCA LC scores and environmental arrows, the middle represents CCA WA scores, and the bottom represents DCA scores. Sites with equal positions along the environmental gradient 2 are connected with lines to facilitate comparisons.
Palmer, M.W. (1993) Ecology 74, 2215–2230
Site scores along the first two axes in CCA and DCA ordinations, with varying levels of quantitative noise in species abundance. Quantitative noise was not simulated. The top set represents CCA LC scores and environmental arrows, the middle represents CCA WA scores, and the bottom represents DCA scores. Sites with equal positions along the environmental gradient 2 are connected with lines to facilitate comparisons.
Palmer, M.W. (1993) Ecology 74, 2215–2230
..continued
Like all numerical techniques, CCA makes certain assumptions, most particularly that the abundance of a species is a unimodal function of position along environmental gradient. Does not have to be symmetric unimodal function.
Simulated data Palmer 1993 – CCA performs well even with highly skewed species distributions.
‘Noise’ in ecological data – errors in data collection, chance variation, site-specific factors, etc. Noise is also regarded as ‘unexplained’ or ‘residual’ variance. Regardless of cause, noise does not affect seriously CCA.
‘Noise’ in environmental data is another matter. In regression, assumed that predictor variables are measured without error. CCA is a form of regression, so noise in environmental variables can affect CCA.
Highly correlated environmental variables, e.g. soil pH and Ca. Species distributions along Ca gradient may be identical to distributions along pH gradient, even if one is ecologically unimportant. Species and object arrangement in CCA plot not upset by strong inter-correlations. CCA (like all other regression techniques) cannot tell us which is the ‘real’ important variable.
Both may be statistically significant – small amount of variation in Ca at a fixed level of pH may cause differences in species composition.
Arch – very rarely occurs in CCA. Detrended CCA generally should not be used except in special cases.
ROBUSTNESS OF CANONICAL CORRESPONDENCE ANALYSIS
McCune (1997) Ecology 78, 2617–2623
Simulated artificial data 10 x 10 grid. 40 species following Gaussian response model.
2 environmental variables X and Y co-ordinatesTENxTEN
2 environmental variables with added noise NOISMOD
(random number mean = 0, variance 17%)
added to each cell10 random environmental variables NOIS1O 2 environmental variables with added noise from
NOISMOD+10 random environmental variables from NOIS
10
NOISBOTH99 random environmental variables NOISFULL
NOISFULL – ‘Species-environment’ correlation increases as number of random variables increases for axis 1 and 2.
Is in fact the correlation between the linear combination and WA site scores.
Poor criterion for evaluating success.Not interpreted as measure of strength of relationship.Monte Carlo permutation tests - NO STATISTICAL SIGNIFICANCE!
INFLUENCE OF NOISY ENVIRONMENTAL DATA ON CANONICAL CORRESPONDENCE
ANALYSIS
(1(1))
(2(2))
(3(3))
(4(4))
(5(5))
Dependence of the 'species-environment correlation,' the correlation between the LC and WA site scores, on a second matrix composed of from 1 to 99 random environmental variables. This correlation coefficient is inversely related to the degree of statistical constraint exerted by the environmental variables.
TEN x TEN
NOISMOD
NOISIO
NOISFULL
Monte Carlo tests 1 2 r1 r2
TENxTEN * 0.77 0.79 0.98 0.9NOISMOD * 0.65 0.65 0.90 0.8NOISE 10 ns 0.20 0.12 0.49 0.4NOISBOTH * 0.65 0.65 0.91 0.8NOISFULL ns 0.12 0.08 1.0 1.0(99 env vars)
Linear combination site best fit of species abundances to scores the environmental dataWA site scores best represent the assemblage structure
LC scores WA scoresSensitive to noise + –True direct gradient analysis + – (multivariate regression)Aim to describe biological + – variation in relation to environmentAssemblage structure – +Which to use depends on one's aims and the nature of the data.
‘Species-environmental correlation’ better called ‘LC-WA’ correlation. Better measure of the strength of the relationship is the proportion of the variance in the species data that is explained by the environmental data. Evaluation should always be by a Monte Carlo permutation test.
LC OR WA SCORES?
MIKE PALMER
"Use LC scores, because they give the best fit with the environment and WA scores are a step from CCA towards CA."
BRUCE MCCUNE
"LC scores are excellent, if you have no error in constraining variables. Even with small error, LC scores can become poor, but WA scores can be good even in noisy data."
LC scores are the default in CANODRAW.
Be aware of both - plot both to be sure.
J. Oksanen (2002)
DATA ORDERINGS
CCA DIAGRAMTEN SETS OF DISTANCES TO REPRESENT, EMPHASIS ON 5, 8, AND 1
(FITTED ABUNDANCES OF SPECIES AND SITES)
Data-tables in an ecological study on species environmental relations. Primary data are the sub-table 1 of abundance values of species and the sub-tables 4 and 7 of values and class labels of quantitative and qualitative environmental variables (env. var), respectively. The primary data are input for canonical correspondence analysis (CCA). The other sub-tables contain derived (secondary) data, as the arrows indicate, named after the (dis)similarity coefficient they contain. The coefficients shown in the figure are optimal when species-environmental relations are unimodal. The CA ordination diagram represents these sub-tables, with emphasis on sub-tables 5 (weighted averages of species with respect to quantitative environmental variables), 8 (totals of species in classes of qualitative environmental variables) and 1 (with fitted, as opposed to observed, abundance values of species). The sub-tables 6, 9, and 10 contain correlations among quantitative environmental variables, means of the quantitative environmental variables in each of the classes of the qualitative variables and chi-square distances among the classes, respectively. (Chis-sq = Chi-square; Aver = Averages; Rel = Relative)
DEFAULT CCA PLOT
J. Oksanen (2002)
• Like CA biplot, but now a triplot: vectors for linear constraints.
• Classes as weighted averages or centroids.
• Most use LC scores: these are the fitted values.
• Popular to scale species relative to eigenvalues, but keep sites unscaled. Species-conditional plot.
• Sites do not display their real configuration, but their projections onto environmental vectors are the estimated values.
Hill scaling Default scaling
–1 2
Emphasis on SITES SPECIES
1 Species x sites Rel abundances Fitted abundances (rel)
2 Species x species – Chi-squared distances
3 Sites x sites Turnover distances –
Quant env vars
4 Sites x env vars3 – Values of env vars
5 Species x env varsWeighted averages Weighted averages
6 Env vars x env vars Effects2 Correlations2
Qualit env vars
7 Sites x env classes4 Membership1
Membership1
8 Species x env classes Rel total abund Rel total abund
9 Env vars x env classes – Mean values of env vars
10 Env classes x env Turnover distances – classes
fitted by least fitted by least squaressquares
11 by centroid by centroid principleprinciple
22 change in site change in site scores if env scores if env variable variable changes changes are are one standard one standard deviationdeviation
33 inter-set inter-set correlationscorrelations
44 group centroids group centroids
SCALING IN CCA
Sub-tables (row numbers) that can be displayed by two differently scaled ordination diagrams in canonical correspondence analysis (CCA). Display is by the biplot rule unless noted otherwise. Hill's scaling (column 2) was the default in CANOCO 2.1, whereas the species-conditional biplot scaling (column 3) is the default in CANOCO 3.1 and 4. The weighted sum of squares of sites scores of an axis is equal to /(1-) with its eigenvalue and equal to 1 in scaling -1 and scaling 2, respectively. The weighted sum of squares of species scores of an axis is equal to 1/(1-) and equal to in scaling -1 and scaling 2, respectively. If the scale unit is the same of both species and sites scores, then sites are weighted averages of species scores in scaling -1 and species are weighted averages of site scores in scaling 2. Table in italics are fitted by weighted least-squares (rel. = relative; env. = environmental; cl. = classes; - = interpretation unknown).
NoteNote that that symmetric scalingsymmetric scaling (=3) has many optimal properties (=3) has many optimal properties (Gabriel, 2002; ter Braak, personal communication)(Gabriel, 2002; ter Braak, personal communication)
REF
REF
REF
REF
Scaling-1: focus on sites Hill's scaling
Interpreta-tion
2: focus on species biplot scaling of CCA
Interpreta-tion
1. species x sitesa Rel. Abundancesb,c
CENTROID Fitted rel. abund.b
BIPLOT rule or CENTROID rule
2. species x species
- UNKNOWN -square distancesd
DISTANCE rule
3. sites x sites Turnover distancesc,e
DISTANCE f DISTANCE rule
Quantitative env. vars
4. sites x env. varsg - UNKNOWN Values of env.vars
BIPLOT rule
5. species x env. vars
Weighted averages
BIPLOT Weighted averages
BIPLOT rule
6. env.vars x env. vars
Effectsh ? BIPLOT Correlations BIPLOT rule
Qualitative env. vars
7. sites x env. classesi
Membershipk CENTROID Membershipk CENTROID rule
8. species x env. cls. Rel. total abund.c,b
CENTROID Rel. total abund.b
CENTROID rule
9. env.vars x env. classes
- UNKNOWN Mean values of env. vars
BIPLOT rule
10. env. classes x env. classes.
Turnover distancesc,e
DISTANCE f DISTANCE rule
a Site scores are linear combinations of the environmental variables. The adjective "fitted" must be deleted if site scores are proportional to the weighted average of species scores.
b The centroid principle can be applied also if sites and species scores are plotted in the same units, i in scaling -1, species that occur in a site lie around it, whereas in scaling 2, the species' distribution is centred at the species point.
c The biplot rule cannot be applied
d In the definition of this coefficient, abundance must be replaced by fitted abundance values, because CCA is correspondence analysis of fitted abundance values
e No explicit formula known
f Chi-square distances, provided the eigenvalues of the axes are of the same magnitude
g Environmental scores are (intra-set) correlations in scaling 2; more precisely, the coordinate of an arrow head on an axis (i.e. the score) is the weighted product-moment coefficient of the environmental variable with the axis, the weights being the abundance totals of the sites (yi+). The scores in scaling -1 are {(1-)}½ times
those in scaling 2.
h Effect is defined as the change in site scores if the environmental variable changes one standard deviation in value (while neglecting the other variables).
i Environmental points are centroids of site points
k Via centroid principle, not via biplot
REF
REF
REF
REF
Centroid principle
Distance principle
Biplot principle (of relative abundances)
Small eigenvalues, short (< 4SD) gradients – Biplot principle
Large eigenvalues (> 0.40), long (> 4SD) gradients – Centroid and distance principles and some biplot principles
Note that the centroid and distance principle may approximate biplot principle if gradients are short and eigenvalues small.
Differences are least important if 12
INTERPRETATION OF CCA PLOTS
Example data: quantitative and qualitative environmental variables (a) and qualitative covariables (b) recorded at 40 sites along two tributaries from the Hierden stream (sd: standard deviation, min: minimum, max: maximum). Aquatic macro-fauna data
CCA EXAMPLE
Ordinal
4 classes
3 classes
7 binary class variables
Remove effect ofseasonal variation
}
Ranking environmental variables in importance by their marginal (left) and conditional (right) effects of the macrofauna in the example data-set, as obtained by forward selection. (1 = fit = eigenvalue with variable j only; a = additional fit = increase in eigenvalue; cum (a) = cumulative total of eigenvalues a; P = significance level of the effect, as obtained with a Monte Carlo permutation test under the null model with 199 random permutations; - additional variables tested; veg. = vegetation). Seasonal variation is partialled out by taking the month class variables as covariables.Marginal effects (forward: step 1)
Conditional effects (forward: continued)
j Variable 1P j Variable a
P Cum (a)
1 Shrubs (1/0) 0.25 (0.01) 1 Shrubs (1/0) 0.25 (0.01) 0.25
2 Source distance
0.22 (0.01) 2 Source distance
0.19 (0.01) 0.44
3 EC 0.20 (0.01) 3 EC 0.19 (0.01) 0.63
4 Discharge 0.17 (0.01) 4 Discharge 0.14 (0.03) 0.75
5 Total veg cover
0.16 (0.01)
6 Shading 0.15 (0.01) - Cover emergent
0.11 (0.10) -
7 Soil grain size 0.14 (0.02) - Cover bank veg
0.11 (0. 12)
-
8 Stream width 0.14 (0.05) - Soil grain size 0.10 (0.13) -
9 High weedy veg
0.14 (0.08)
10
Cover bank veg
0.13 (0.11)
- U vs L stream 0.22 (0.01) - U vs L stream 0.09 (0.01) -
EXTRA FIT
Each variable is the only env. var.
Change in eigenvalue if particular variable selected
MARGINAL EFFECTS i.e. ignoring all other variables
CONDITIONAL EFFECTS given other selected variables
Species-conditional triplot based on a canonical correspondence analysis of the example macro-invertebrate data displaying 13% of the inertia (=weighted variance) in the abundances and 69% of the variance in the weighted averages and class totals of species with respect to the environmental variables. The eigenvalue of axis 1 (horizontally) and axis 2 (vertically) are 0.35 and 0.17 respectively; the eigenvalue of the axis 3 (not displayed) is 0.13. Sites are labelled with stream code (U, L) and are ranked by distance from the source (rank number within the stream). Species (triangles) are weighted averages of site scores (circles). Quantitative environmental variables are indicated by arrows. The class variable shrub is indicated by the square points labelled Shrub and No shrub. The scale marks along the axes apply to the quantitative environmental variables; the species scores, site scores and class scores were
multiplied by 0.4 to fit in the coordinate system. Only selected species are displayed which have N2>4 and a small N2-adjusted root mean square tolerance for the first two axes. The species names are abbreviated to the part in italics as follows Ceratopogonidae, Dendrocoelum lacteum, Dryops luridus, Erpobdella testacea, Glossiphonia complanata, Haliplus lineatocollis, Helodidae, Micropsectra atrofasciata, Micropsectra fusca, Micropterna sequax, Prodiamesa olivacea, Stictochironomus sp.
• Unconstrained CA gives
• Species ordination which is derived from site ordination
• Site ordination which is derived from species ordination
• Fitted vectors for environmental variables (indirect gradient analysis)
• Constrained CA (Canonical CA) gives a direct gradient analysis
• Species ordination which is derived from site ordination
• Site scores which are linear combinations of environmental variables (LC scores)
• Site ordination which is derived from species ordination (WA scores) so that species-environment correlation is maximised with the LC scores
• Vectors of environmental variables that define the linear combination scores for sites
CANONICAL CORRESPONDENCE ANALYSIS (CCA) - A SUMMARY
CCA CCA DirectlyDirectly
CACAIndirectlyIndirectly
Gradient length Gradient length estimationestimation
Outline of ordination techniques present-ed here. DCA (detrended correspondence analysis) was applied for the determina-tion of the length of the gradient (LG). LG is important for choosing between ordination based on a linear or on an unimodal response model. Correspond-ence analysis (CA) is not considered any further because in “microcosm experi-ment discussed here LG < or = 1.5 SD units. LG < 3 SD units are considered to be typical in experimental ecotoxicology. In cases where LG < 3, ordination based on linear response models is considered to be most appropriate. PCA (principal component analysis) visualizes variation in species data in relation to best fitting theoretical variables. Environmental variables explaining this visualised variation are deduced afterwards, hence, indirectly. RDA ( redundancy analysis) visualises variation in species data directly in relation to quantified environ-
mental variables. Before analysis, covariables may be introduced in RDA to compensate for systematic differences in experimental units. After RDA, a permutation test can be used to examine the significance of effects.
REDUNDANCY ANALYSIS – CONSTRAINED PCA
Short (< 2SD) compositional gradientsLinear or monotonic responses Reduced-rank regressionPCA of y with respect to xTwo-block mode C PLSPCA of instrumental variables Rao (1964)
PCA - best hypothetical latent variable is the one that gives the smallest total residual sum of squares
RDA - selects linear combination of environmental variables that gives smallest total residual sum of squares
ter Braak (1994) Ecoscience 1, 127–140 Canonical community ordination Part I: Basic theory and linear methods
RDA ordination diagram of the Dune Meadow Data with environmental variables represen-ted as arrows. The scale of the diagram is: 1 unit in the plot corresponds to 1 unit for the sites, to 0.067 units for the species and to 0.4 units for the environmental variables.
Redundancy analysis: canonical coefficients (100 x c) and intra-set correlations (100 x r) of environmental variables with the first two axes of RDA for the Dune Meadow Data. The environmental variables were standardized first to make the canonical coefficients of different environmental variables comparable. The class SF of the nominal variable “type of management” was used as reference class in the analysis.
Variable Coefficients CorrelationsAxis1 Axis2 Axis1 Axis2
A1 -1 -5 54 -6Moisture 15 9 92 12Use 5 -6 15 29Manure -8 16 -26 86SF - - 25 76BF -10 0 -48 -11HF -10 -2 -40 13NM -4 -13 51 -79
Axis 1 Axis 2
PCA % 29 21
RDA % 26 17
PCA Correlation 0.90 0.82
RDA Correlation 0.95 0.89
PCA and RDA comparisons
Important to do the check that the environmental variables relate to the major gradients in composition detected by the PCA.
Cosine of angle correlation
Long arrows of species and environmental variables most important
Euclidean distance biplotCovariance (correlation) biplotRDA covariance or correlation matrix of speciesRDA – constrained form of multiple regressionUses 2 (q + m) + m parameters (q env variables, m species)Multiple regression m (q + 1)e.g. 40 species 10 envir variablesRDA 140 parametersMR 440 parametersRDA is thus reduced rank regression (RR)
speciesunconstrained
Goodness of fit 1 + 2 sum of
eigenvalues constrained
fitted species
BIPLOT INTERPRETATION
Primary and secondary data tables in a typical community ecological study of species-environment relations. Indirect methods of ordination use the tables under (a). Direct methods also use the tables under (b). The primary data are the table of abundance values and the tables of values and class labels of quantitative and qualitative environmental variables (env. var), respectively. The secondary tables are named after the (dis)similarity coefficients they contain. The appropriate coefficients must be chosen by the ecologist. The coefficients shown in the figure are optimal when species-environment relations are linear.
Tables that can be displayed by two differently scaled biplots in principal components analysis (a) and redundancy analysis (b).
The sum of squares of site scores of an axis is equal to its eigenvalue in scaling 1, and equal to 1 in scaling 2. The sum of squares of species scores of an axis is equal to 1 in scaling 1 and equal to its eigenvalue in scaling 2. Tables in bold are fitted by
(weighted) least-squares.Biplot scaling 1: focus on sites 2: focus on species
distance biplot correlation biplot(a) principal components analysisspecies x sites abundances abundancessites Euclidean distances -species - correlationsa
(b) redundancy analysisspecies x sitesb fitted abundances fitted abundancessitesb Euclidean distancesc -species - correlationsa,c
Quantitative env. vars.:species x env. vars. d correlations correlationssites x env. vars. d - values of env. varsenv. vars. d effectse correlationsQualitative env. vars:species x env. classesf means meanssites x env. classesf g g
env. classesf Euclidean distances -env. vars. x env. classes - means
a Automatic if abundance is standardised by species. If abundance is only centred by species, a post-hoc rescaling of the site scores is needed so as to account for the differences in variance amongst species. b Site scores are a linear combination of the environment variables instead of being a weighted sum of species abundances. c In the definition of this coefficient, abundance must be replaced by the fitted abundance. d Environmental scores are intraset correlations in scaling 2 and s
½ times those in scaling 1 with s the eigenvalue of axis . In CANOCO, the scores are termed biplot scores for environmental variables. e Effect of the environmental variable on the ordination scores, while neglecting the other environmental variables; length of arrow is the effect size, i.e. the variance explained by the variable. f Environmental classes are centroids of site points belonging to the class. g membership via centroid principle, not via the biplot rules.
REF
REF
REF
REF
The scale marks along the axes apply to the species and quantitative environmental variables; the site scores and class scores were multiplied by 0.46 to fit in the coordinate system. The abbreviations are given in Jongman et al. (1987).The rule for interpreting a biplot (projection on an imaginary axis) is illustrated for the species Pla lan and sites 11 and 12.
Correlation biplot based on a redundancy analysis of the Dune Meadow Data displaying 43% of the variance in the abundances and 71% of the variances in the fitted abun-dances. Quantitative environ-ment variables are indicated by arrows. The qualitative variable Management type is indicated by the square points labelled SF, BF, HF, and NM. The displayed species are selected on the basis that more than 30% of their variance is accounted for by the diagram. Eigenvalues of the first three axes are 0.26, 0.17,and 0.07; the sum of all canonical eigenvalues is 0.61.
PROPOSED NEW SCALING FOR CCA AND RDA
Gabriel, K.R. (2002) Biometrika 89, 423-436
Symmetric scaling (3) of biplots preserves the optimal fit to the species data table and preserves the (proportional) fit of at least 95% of the inter-species correlations/distances and inter-sample distances. It is a very good compromise.
Only recommended (ter Braak, pers. comm.) to deviate from symmetric scaling if the focus of study is strongly on either species (scaling 2) or on samples (scaling 1).
Data table unaffected by scaling:
Species x sites Species data (PCA)Fitted species data (RDA)Relative species data (CA)Fitted relative species data (CCA)
Species x environmental variables Correlations of species (RDA)Optima (WA) of species (CCA)
Species x environmental classes Mean abundances of species (RDA)Relative abundances of species across
classes (CCA)
Data tables with 95% preservation of proportional fit:
Species x species Correlations (PCA, RDA)
Chi-square distances (CA, CCA)
Sites x sites Euclidean distances (PCA, RDA)
Chi-square distances (CA, CCA)
Env. classes x env. classes Euclidean distances (RDA)
Chi-square distances (CCA)
Env. variables x env. variables Correlations (RDA, CCA)
Sites x env. variables Values (RDA, CCA)
Sites x env. classes Means (RDA, CCA)
Env. variables x env. classes Mean values of env. variables (RDA, CCA)
ALTERNATIVES TO ENVIRONMENTAL VECTORS IN CCA AND RDA
• Fitted vectors natural in constrained ordination, since these have linear constraints.
• Distant sites are different, but may be different in various ways: environmental variables may have a non-linear relation to ordination.
Contours Bubble plots GAM
J. Oksanen (2002)
Statistical significance of species-environmental relationships. Monte Carlo permutation tests.
Randomly permute the environmental data, relate to species data ‘random data set’. Calculate eigenvalue and sum of all canonical eigenvalues (trace). Repeat many times (99).
If species react to the environmental variables, observed test statistic (1 or trace) for observed data should be larger than most (e.g. 95%) of test statistics calculated from random data. If observed value is in top 5% highest values, conclude species are significantly related to the environmental variables.
STATISTICAL TESTING OF CONSTRAINED ORDINATION RESULTS
STATISTICAL SIGNIFICANCE OF CONSTRAINING VARIABLES
• CCA or RDA maximise correlation with constraining variables and eigenvalues.
• Permutation tests can be used to assess statistical significance:
- Permute rows of environmental data.
- Repeat CCA or RDA with permuted data many times.
- If observed higher than (most) permutations, it is regarded as statistically significant.
J. Oksanen (2002)
e.g. pollution effects
seasonal effects COVARIABLES
Eliminate (partial out) effect of covariables. Relate residual variation to pollution variables.
Replace environmental variables by their residuals obtained by regressing each pollution variable on the covariables.
Analysis is conditioned on specified variables or covariables. These conditioning variables may typically be 'random' or background variables, and their effect is removed from the CCA or RDA based on the 'fixed' or interesting variables.
PARTIAL CONSTRAINED ORDINATIONS
(Partial CCA, RDA, etc)
Natural variation due to sampling season and due to gradient from fresh to brackish water partialled out by partial CCA.
Variation due to pollution could now be assumed.
Ordination diagram of a partial canonical correspond-ence analysis of diatom species (A) in dykes with as explanatory variables 24 variables-of-interest (arrows) and 2 covariables (chloride concentration and season). The diagram is symmetrically scaled [23] and shows selected species and standardized variables and, instead of individual dykes, centroids (•) of dyke clusters. The variables-of-interest shown are: BOD = biological oxygen demand, Ca = calcium, Fe = ferrous compounds, N = Kjeldahl-nitrogen, O2 = oxygen, P = ortho-phosphate, Si= silicium-compunds, WIDTH = dyke width, and soil types (CLAY, PEAT). All variables except BOD, WIDTH, CLAY and PEAT were transformed to logarithms because of their skew distribution. The diatoms shown are: Ach hun = Achnanthes hungarica, Ach min = A. minutissima, Aph cas= Amphora castellata Giffen, Aph lyb = A. lybica, Aph ven = A. veneta, Coc pla = Cocconeis placentulata, Eun lun = Eunotia lunaris, Eun pec = E. pectinalis, Gei oli = Gomphoneis olivaceum, Gom par = Gomphonema parvulum, Mel jur = Melosira jürgensii, Nav acc = Navicula accomoda, Nav cus = N. cuspidata, Nav dis = N. diserta, Nav exi = N. exilis, Nav gre = N. gregaria, Nav per = N. permitis, Nav sem = N. seminulum, Nav sub= N. subminuscula,Nit amp = Nitzschia amphibia, Nit bre = N. bremensis v. brunsvigensis, Nit dis = N. dissipata, Nit pal = N. palea, Rho cur = Rhoicosphenia curvata.(Adapted from H. Smit, in prep)
PARTIAL CCA
There can be many causes of variation in ecological or other data. Not all are of major interest. In partial ordination, can ‘factor out’ influence from causes not of primary interest. Directly analogous to partial correlation or partial regression. Can have partial ordination (indirect gradient analysis) and partial constrained ordination (direct gradient analysis). Variables to be factored out are ‘COVARIABLES’ or ‘CONCOMITANT VARIABLES’. Examples are: 1) Differences between observers.
2) Time of observation.
3) Between-plot variation when interest is temporal trends within repeatedly sampled plots.
4) Uninteresting gradients, e.g. elevation when interest is on grazing effects.
5) Temporal or spatial dependence, e.g. stratigraphical depth, transect position, x and y co-ordinates. Help remove autocorrelation and make objects more independent.
6) Collecting habitat – outflow, shore, lake centre.
7) Everything – partial out effects of all factors to see residual variation in data. Given ecological knowledge of sites and/or species, can try to interpret residual variation. May indicate environmental variables not measured, may be largely random, etc.
PARTIAL ORDINATION ANALYSIS(Partial PCA, CA, DCA)
e.g. partial out the effects of some covariables prior to indirect gradient analysis
within-plot change PRIMARY INTERESTbetween-plot differences NOT OF INTEREST
Partial plot identity, ordination of residual variation, i.e. within-plot change. e.g. Swaine & Greig-Smith (1980) J Ecol 68, 33–41 Bakker et al. (1990) J Plankton Research 12, 947–972
PARTIAL ORDINATIONS
Background variables or 'covariables'
Partial CCAPartial RDA
Vegetation
Environmental variables or 'constraints'
Vegetation (residual)
CCARDA
CADCAPCA
Vegetation (residual)
"Nuisance" variables or other background factors can be removed before studying interesting factors. Partial CCA or partial RDA.
Permutation tests are for environmental variables only.
Residual variation can be analysed at any level. Can partition the variance.
Final residual shows what you cannot explain with available environmental variables.
Interpretation of final residual based on other correlates and/or ecological knowledge.
COVARIABLES IN CCA AND RDA
PARTITIONING VARIATION
ANOVA total SS = regression SS + residual SS
Two-way ANOVA between group (factor 1)
+ between treatments (factor 2)
+ interactions
+ error component
Borcard et al. (1992) Ecology 73, 1045–1055
Variance or variation decomposition into 4 components
Important to consider groups of environmental variables relevant at same level of ecological relevance (e.g. micro-scale, species-level, assemblage-level, etc.).
Variation = variance in RDA
Variation = inertia in CCA = chi-square statistic of data divided by the data’s total = sum of all eigenvalues of CA
Total inertia = total variance 1.164Sum canonical eigenvalues = 0.663 57%Explained variance 57%Unexplained variance = T – E 43%
What of explained variance component?
Soil variables (pH, Ca, LOI)Land-use variables (e.g. grazing, mowing)Not independentDo CCA/RDA using
1) Soil variables only canonical eigenvalues0.5212) Land-use variables only canonical eigenvalues0.5033) Partial analysis Soil Land-use covariables 0.1604) Partial analysis Land-use Soil covariables 0.142a) Soil variation independent of land-use (3) 0.160 13.7%b) Land-use structured (covarying) soil variation (1–3) 0.36131%c) Land-use independent of soil (4) 0.142 12.2%
Total explained variance 56.9%d) Unexplained 43.1%
unexplaineduniqueuniquecovariance
a b c d
CANOCO
Qinghong & Bråkenheim, (1995) Water, Air and Soil Pollution 85, 1587–1592 Three sets of predictors – Climate (C), Geography (G) and Deposition of Pollutants (D) Series of RDA and partial RDA
Predictors Covariables Sum of canonical G+C+D - 0.811D G+C 0.027G+C - 0.784G+C D 0.132D - 0.679Joint effectDG+C=0.784-0.132=0.679-0.027=0.652C D+G 0.106G+D - 0.706G+D C 0.074C - 0.737Joint effectCD+G=0.737-0.106=0.706-0.074=0.631
0.811
0.811
0.812
0.811
VARIATION PARTITIONING OR DECOMPOSITION WITH 3 OR MORE SETS OF
PREDICTOR (EXPLANATORY) VARIABLES
Predictors Covariables Sum of canonical G D+G 0.034D+C - 0.777D+C G 0.228G - 0.538Joint effectGD+C=0.777-0.228=0.538-0.034=0.549
0.811
0.811
Canonical eigenvaluesAll predictors 0.811Pure deposition 0.027 PDPure climate 0.106 PCPure geography 0.034 PGJoint G + C 0.132Joint G + D 0.074Joint D + C 0.228Unexplained variance 1 – 0.811 = 0.189
PD
DGCD
CG
CDG
PGPC
D
GC
Covariance terms
CDDGCGCDG
CD + DG + CDG = 0.652
CD + CG + CDG = 0.631
DG + CG + CDG = 0.549
PD + PC + CD = 0.027 + 0.106 + CD = 0.777 – 0.549 = 0.228
PD PC (D+C) (DG + CG + CDG)
PD + PG + DG = 0.027 + 0.034 + DG = 0.706 – 0.631 = 0.074
PD PG (G+D) (CD + CG + CDG)
PC + PG + CG = 0.106 + 0.034 + CG = 0.784 – 0.652 = 0.132
PC PG (G+C) (CD + DG + CDG)
CD = 0.095 DG = 0.013 CG = –0.008
CDG = 0.652 – 0.013 – 0.095 = 0.544
= 0.631 – (–0.008) – 0.095 = 0.544
= 0.054 – (–0.008) – 0.013 = 0.544
Total explained variance 0.811 consists of: Common climate + deposition 0.095 Unique climate PC 0.106Common deposition + geography 0.013 Unique geography PG 0.034Common climate + geography 0.008 Unique deposition PD 0.027Common climate + geography + deposition 0.544 Unexplained variance 0.189
See also Qinghong Liu (1997) – Environmetrics 8, 75–85Anderson & Gribble (1998) – Australian J. Ecology 23, 158-167 Total variation: 1) random variation 2) unique variation from a specific predictor variable or set of predictor variables 3) common variation contributed by all predictor variables considered together and in all possible combinations
Usually only interpretable with 2 or 3 'subsets' of predictors.
In CCA and RDA, the constraints are linear. If levels of the environmental variables are not uncorrelated (orthogonal), may find negative 'components of variation'.
'NEGATIVE' VARIANCES
In variance partitioning, the groups of predictor variables used should be non-linearly independent for unbiased partitioning or decomposition.
If the groups of variables have polynomial dependencies, some of the variance components may be negative. Negative variances are, in theory, impossible.
High-order dependencies commonly arise with high numbers of variables and high number of groups of variables.
Beware of inter-relationships between predictor variables and between groups of predictors. Problem common to all regression-based techniques, including (partial) CCA or RDA.
Careful model selection (minimal adequate model) is essential for many purposes, including variance partitioning.
ENVIRONMENTAL CONSTRAINTS AND CURVATURE IN ORDINATIONS
• Curvature often cured because axes are forced to be linear combination of environmental variables (constraints).
• High number of constraints = no constraint.
• Absolute limit: number of constraints = min (M, N) - 1, but release from the constraints can begin much earlier.
• Reduce environmental variables so that only the important remain: heuristic value better than statistical criteria.
• Reduces multicollinearity as well.
J. Oksanen (2002)
Classification of gradient analysis techniques by type of problem, response model and method of
estimation Method of estimation Type of problem
Linear Least Squares
Maximum Likelihood
Unimodal Weighted Averaging
Regression Multiple regression
Gaussian regression
Weighted averaging of site scores
Calibration
Linear calibration 'inverse regression'
Gaussian calibration
Weighted averaging of species scores (WA)
OrdinationPrincipal components analysis (PCA)
Gaussian ordination
Correspondence analysis (CA); detrended correspondence analysis (DCA)
Constrained ordination1
Redundancy analysis (RDA)4
Gaussian canonical ordination
Canonical correspondence analysis (CCA); detrended CCA
Partial ordination2
Partial component analysis
Partial Gaussian ordination
Partial correspondence analysis; partial DCA
Partial constrained ordination3
Partial redundancy analysis
Partial Gaussian canonical ordination
Partial canonical correspondence analysis; partial detrended CCA
1 = constrained multivariate regression2 = ordination after regression on covariables3 = constrained ordination after regression on covariables = constrained partial multivariate regression4 = 'reduced rank regression' = “PCA of y with respect to x”
1) Choice can greatly influence the results. Fewer the environmental variables, the more constrained the ordination is.
2) Possible to have one only – can evaluate its explanatory power.
3) Can always remove superfluous variables if they are confusing or difficult to interpret. Can often remove large number without any marked effect. Remember post-hoc removal of variables is not valid in a hypothesis-testing analysis.
4) Linear combinations – environmental variables cannot be linear combinations of other variables. If a variable is a linear combination of other variables, singular matrix results, leads to analogous process of dividing by zero.
Examples: - total cations, Ca, Mg, Na, K, etc. Delete total cations
- % clay, % silt, % sand- dummy variables (granite or limestone or basalt)
5) Transformation of environmental data – how do we scale environmental variables in such a way that vegetation ‘perceives’ the environment? Need educated guesses.
Log transformation usually sensible – 1 unit difference in N or P is probably more important at low concentrations than at high concentrations.
As statistical significance in CANOCO is assessed by randomisation tests, no need to transform data to fulfil statistical assumptions.
Transformations useful to dampen influence of outliers.
Environmental data automatically standardised in RDA and CCA.
ENVIRONMENTAL VARIABLES IN CONSTRAINED ORDINATIONS
6) Dummy variables – factors such as bedrock type, land-use history, management, etc, usually described by categorical or class variables. 1 if belongs to class, 0 if it does not. For every categorical variable with K categories, only need K – 1 dummy variables e.g.
Granite Limestone Basalt GabbroPlot 1 1 0 0 0
2 0 1 0 03 1 0 0 04 0 1 0 05 0 0 1 06 0 0 0 1
7) Circular data – some variables are circular (e.g. aspect) and large values are very close to small values. Aspect – transform to trigonometric functions.
northness = cosine (aspect)eastness = sine (aspect)
Northness will be near 1 if aspect is generally northward and –1 if southward. Close to 0 if west or east.Alternatively for aspect
southness = 180 - |aspect - 180| (S = 180, N =0)westness = |180 - |aspect - 270|| (W = 180, E = 0)
Day of year – usually not a problem unless dealing with sampling over whole year. Can create ‘winterness’ and ‘springness’ variables as for aspect.
8) Vegetation-derived variables – maximum height, total biomass, total cover, light penetration, % open ground can all be ‘environmental’ variables. Such variables SHOULD NOT BE USED in hypothesis testing, as danger of circular reasoning.
9) Interaction terms – e.g. elevation * precipitation. Easy to implement, difficult to interpret. If elevation and precipitation interact to influence species composition, easy to make term but the ecological meaning of where in environmental space the stands or species are is unclear. Huge number of possibilities N variables ½ N (N – 1) possible interactions. 5 variables 10 interactions.
AVOID quadratic terms [e.g. pH * pH (pH2) (cf. multiple regression and polynomial terms)]. Can create an ARCH effect or warpage of ordination space.
Try to avoid interaction terms except in clearly defined hypothesis-testing studies where the null hypothesis is that ‘variables c and d do not interact together to influence the species composition’.
For interaction to be significant, eigenvalue 1 of the analysis with product term should be considerably greater than 1 when there is no product term and the t-value associated with the product term should be greater than 2 in absolute value.
Avoid product variables to avoid ‘data dredging’.
1) The fewer the environmental variables, stronger the constraints are.
2) With q (number of samples – 1) environmental variables, the analysis is unconstrained.
3) Small numbers of environmental variables may remove any arch effect.
4) Want to try to find MINIMAL ADEQUATE SET of environmental variables that explain the species data about as well as the FULL SET.
5) Automatic selection (e.g. forward selection) can be dangerous:
a) Several sets can be almost equally good. Automatic selection finds one but may not be the best.
b) Selection order may change the result and important variables may not be selected.
c) Small changes in the data can change the selected variables. Difficult to draw reliable conclusions about relative importance of variables. Omission of a variable does not mean it is not ecologically important.
6) If you are lucky, there may only be one minimal adequate model but do not assume that there is only one such model.
7) How do we go about finding a minimal adequate model or set of environmental variables?
SELECTING ENVIRONMENTAL VARIABLES IN CON-STRAINED ORDINATION ANALYSIS
(e.g. CCA, RDA)
1) Start with all explanatory variables in the analysis, FULL MODEL. Consider sum of canonical eigenvalues (amount of explained variance), eigenvalues and species-environmental correlations.
2) Try to simplify full model by deleting variables but not reducing the model performance. May be impossible to remove variables without some loss of information.
Deletion criteria:
a) Deletion on external criteria – variables not relevant.
b) Deletion on correlation structure – variables may be highly correlated (e.g. pH, Ca, Mg, CEC). Any one could be used as a proxy for the others. Best to choose the one that is likely to be the most direct cause of vegetation response.
Can do a PCA of environmental variables to explore correlation structure of variables.
c) Interpretability – variables with short arrows.
d) Non-significant – delete any that are non-significant (p > 0.05) in analysis with one environmental variable only in CCA or RDA.
e) Ecological importance
f) Stepwise analysis – forward selection, add one variable at a time until no other variables ‘significantly’ explain residual variation in species data.
3) Final selection must be based on ecological and statistical criteria. The purpose of numerical data analysis is 'INSIGHT', not complex statistics!
WHAT IS DONE?
1) CCA (or RDA) is performed on each variable separately and marginal effects are listed in order.
2) Select the variable with largest marginal effect (= eigenvalue) and test its statistical significance by unrestricted Monte Carlo permutation tests and 999 permutations. Accept if p < 0.05.
3) This variable is now used as a covariable and the variables are now listed in order of their conditional effects (i.e. variance explained when allowing for effects of variable one selected). Evaluate its statistical significance and apply Bonferroni-type correction for simultaneous multiple tests,
namely 1 = /t where t =number of tests.
For = 0.05 t = 1, 1 = 0.05
t = 2, 1 = 0.025
t = 3. 1 = 0.0166
t = 4, 1 = 0.0125
With 999 permutations (i.e. p of 0.001 can be evaluated), becomes very slow. Required if you are to properly evaluate the Monte Carlo permutation probabilities.
These tests do not control for overall Type I error. In practical terms this means that too many variables will be judged ‘significant’.
Alternatively, can stop when the increase in fit when including a variable is less than 1.0% (EXTRA FIT).
MINIMAL ADEQUATE MODEL IN CCA
J. Oksanen (2002)
13 environmental variables 3 environmental variables
OTHER PROBLEMS
1) Selection of categorical variables coded as dummy variables. Suppose there are 5 bedrock types but only ‘granite’ is selected by forward selection. Should you select the other variables as well? If you consider the different bedrock types to be independent, the answer is NO. If you consider there to be one categorical variable (bedrock) with five states, the answer is YES.
2) Last two remaining variables within a category will always have identical fit because they contain identical information (if it is not z then it must be y). Does not matter which you choose. Select the commoner category.
3) No guarantee that forward selection (or any other stepwise procedure) will result in ‘best’ set of variables. Only way is perform constrained ordinations for every conceivable combination of variables. Currently impossible with current technology.
4) Accept that minimal adequate model is one possible solution only.
5) For exploratory, descriptive studies, do not be reluctant to use a priori ecological knowledge.
1 qncj / varianceresidual VIFVar
211
jRVIF
Variance of estimated regression (= canonical) coefficients (cj) are proportional to their VIF.
number of predictors
number of samples
VIF is related to the (partial) multiple correlation coefficient Rj between variable j and the other environmental variables.
If VIF > 20, that variable is almost perfectly correlated with other variables and has no unique contribution to the regression equation. Regression (= canonical) coefficient unstable, not worth considering.
Useful for finding minimal set of variables.
Not unique, e.g. pH and Ca (and other variables).
VIF VIF
pH 123.8 6.4
Ca 2.6 –
Mg 8.1 29.3
VARIANCE INFLATION FACTORS (VIF)
'AIC' FOR MODEL SELECTION IN CCA AND RDA
Jari Oksanen (2004) VEGAN 1.7-6 R deviance.ccadeviance.rda
Find statistics in CCA and RDA that resemble deviance and assess an AIC-like statistic as in regression model building.
Deviance of CCA = chi-square of the residual data matrix after fitting the constraints.
Deviance of RDA = average residual variance per species.
Can be used to help select between possible models in CCA or RDA.
AIC - index of fit that takes account of the parsimony of the model by penalising for the number of parameters.
smaller the values, better the fit.
here equals the residual deviance + 2x number of regression (canonical) coefficients fitted.
STAGES IN 'AIC' MODEL SELECTION IN CCA AND RDA
1. Define a null model into which variables are sequentially added in order of their statistical importance. Null model is unconstrained PCA or CA.
2. Now do stepping by either a forward selection of a backward elimination of the predictor variables. Need to define an upper and lower scope for the stepping to occur within.
Forward selection – lower scope = null model (no predictors)
- upper scope = full model (all predictors included)
Backward elimination – lower scope = full model
- upper scope = null model
3. At each step, the effect of adding or deleting a variable is evaluated in terms of the AIC criterion. Low AIC values are to be preferred.
4. If a lower AIC can be achieved by adding or deleting a variable at a stage, then this predictor variable is added/deleted.
5. Useful to use both backward elimination and forward selection at each step. Start with full model, eliminate first variable, then the next, try to add either variable back into the model, and so on.
6. After the final model is derived (lowest AIC), can test this model to see if the effects of the constraining predictor variables are statistically significant. Use Monte Carlo permutation test under the reduced model.
'AIC' MODEL SELECTION
Godínez-Domínguez & Freire 2003 Marine Ecology Progress Series 253, 17-24
1. Definition of set of a priori models
2. Statistical fit of models to data (e.g. CCA)
3. Selection of 'best' model – Akaike Information Criterion (AIC)
AIC =
where k = number of free parameters in the model
= model maximum likelihood
Rather than a statistical test of one null hypothesis, AIC provides a methodology for selecting an a priori set of alternative hypotheses.
kL 2ˆlog2
L
From estimated residual sum of squares (RSS) in CCA
where h = number of predictor variables in model,
where log = loge
n = sample size, and
= RSS/n
h
ii
1
)traceseigenvalue allof sum(
)ˆlog(21)ˆ(log 2 nL
2
To avoid bias in AIC due to links between sample size and number of parameters, corrected AIC is
1)1(2
AICAICc KnKK
As in GLM, interested in differences in AIC between models
i = AICci – min AICc
Data – 5 cruises (DEM-1 – DEM-5)
- 8 models
Godinez-Dominguez & Freire, 2003
CCA permutation tests 40 models
27 p < 0.05 (global test) 20 p < 0.05 (first CCA axis)
AICc approach to find most parsimonious model
Can determine not only the 'best' model but rank the different underlying hypotheses according to AIC criteria of parsimony. Spatial models 2 and 5, namely depth stratification but no difference between sheltered and exposed stations, are most appropriate for these data.
Standard linear technique for relating two sets of variables.
Similar to RDA – assumes linear response model.
Selects canonical coefficients for species and environmental variables to MAXIMISE species – environmental correlation canonical correlation
1
RDA species scores are simply weighted sums of site scores kikybx1
CANCOR species scores are b parameters estimated by multiple regression of site scores on species variables
number of species << number of sites
In fact number of species + number of environmental variables must be smaller than number of sites
i.e. m must be < n – q
CANCOR biplot
Differs from RDA also in error component.
van der Meer (1991) J. Expl. Mar. Biol. Ecol. 148, 105–120
CANONICAL CORRELATION ANALYSIS – CANCOR
RDA uncorrelated independent errors with equal variances (least-squares technique).
CANCOR correlated normal errors (maximum likelihood technique). Is in realty, a GLM.
residual correlations between errors are additional parameters in CANCOR. Many species. Cannot estimate them reliably from data from few sites.
Generalised variance minimised in CANCOR = product of eigenvalues of matrix ‘volume’ of hyperellipsoid.
Total variance minimised in RDA = sum of diagonal elements = sum of eigenvalues.
Linear transformation Non-linear transformation Linear transformation
of species data of species data of environmental data
CA, PCA affects results no effect
CCA, RDAaffects results no effect
CANCOR no effect no effect
• Canonical Correlation Analysis (CANCOR)
• Continuous environmental variables and vegetation
• Can be computed only if number of sites > number of species + number of env. vars +1
• Redundancy analysis (RDA)
• As CANCOR but assumes that error variance constant for all plant species
• Technically possible to estimate in vegetation data, unlike CANCOR
• Canonical Variates Analysis (CVA) or Discriminant Analysis – see lecture 9
• Predict class membership using continuous variables
• For instance, pre-determined vegetation type using vegetation data
• LC score shows the centroids, Weighted sum scores show the dispersion and overlap
CONSTRAINED LINEAR ORDINATION (PCA FRAMEWORK)
DISTPCOA Pierre Legendre & Marti Anderson (1999) Ecol. Monogr. 69, 1-24. RDA but with any distance coefficient RDA - Euclidean distance Absolute abundances Quantity dominated CCA - chi-square metric Relative abundances Shape/composition
dominated
Does it matter?Total biomass or cover and species compositionVarying e.g. ridge snow bed gradient
Other dissimilaritiesBray & Curtis non-Euclidean semi-metricJaccard +/- non-Euclidean semi-metricGower mixed data non-Euclidean semi-metric
Basic ideaReduce sample x sample DC matrix (any DC) to principal co-ordinates (principal co-ordinates analysis, classical scaling, metric scaling – Torgerson, Gower) but with correction for negative eigenvalues to preserve distances.
PCoA – embeds the Euclidean part of DC matrix, rest are negative eigenvalues for which no real axes exist. These correspond to variation in distance matrix, which cannot be represented in Euclidean space. If only use positive eigenvalues, RDA gives a biased estimate of the fraction of variance in original data.
DISTANCE-BASED REDUNDANCY ANALYSIS
Correction for negative eigenvalues
where c1 is equal to absolute value of largest negative eigenvalue of matrix used in PCoA 1
D
1
Use all principal co-ordinate sample scores (n - 1 or m, whichever is less) as RESPONSE (species) data in RDA. Use dummy variables for experimental design as predictors in X in RDA.
Now under framework of RDA and battery of permutation tests, can analyse structured experiments but WHOLE ASSEMBLAGE (cf. MANOVA but where m >N).
Now can test null hypothesis (as in MANOVA) that assemblages from different treatments are no more different than would be expected due to random chance at a given level of probability. BUT unlike non-parametric tests (ANOSIM, Mantel tests), can test for interactions between factors in multivariate data but using any DC (not only Euclidean as in ANOVA / MANOVA). Using permutation tests means we do not have to worry about multivariate normality or homogeneity of covariance matrices within groups, or abundance of zero values as in ecological data.
DISTPCoA www.umontreal.ca
jiforcdd ijij 501
22 .)('
2
21
ijij da
..aaaa jiijij
Raw dataRaw data(replicates x species)(replicates x species)
Distance matrixDistance matrix(Bray-Curtis, etc)(Bray-Curtis, etc)
Principal coordinate analysis(PCoA)
Correction for negative eigenvalues
Matrix YMatrix Y(replicates x (replicates x
principal coordinates)principal coordinates)
Matrix XMatrix X(dummy variables (dummy variables
for the factor)for the factor)
Test of one factorin a single-factor model
Redundancy analysis (RDA)F# statistic
Partial redundancy analysis (partial RDA) F# statistic
Matrix YMatrix Y(replicates x (replicates x
principal coordinates)principal coordinates)
Matrix XMatrix X(dummy (dummy variables variables
for the for the interaction)interaction)
Matrix XMatrix XCC
(dummy (dummy variables variables
for the main for the main effects)effects)
Test of F# by permutation under the full model
Test of F# by permutation
Test of interaction term in multifactorial model
Correspondence between the various components of the univariate F-statistics and the multivariate RDA statistics in the one-factor case.
Univariate ANOVA Multivariate RDA statistic
Total sum of squares sum of all eigenvalues of Y
Treatment sum of squares = SSTr trace = sum of all canonical eigenvalues of Y on X
Treatment degrees of freedom = dfTr q
Residual sum of squares = SSRes rss = sum of all eigenvalues – trace
Residual degrees of freedom = dfRes nT – q – 1
Treatment mean square = SStr/dfTr = MSTr trace/q
Residual mean square SSRes/dfRes = MSRes rss/(nT – q - 1)
Res
Tr
MSMSF
)1/(rss/trace#
qnq
FT
Tr = treatment, q = number of linearly independent dummy variables, nT = number of replicates
REF
REF
REF
REF
PERMUTATION TESTS in RDA (also CCA) in CANOCO. What is shuffled?
Y = Z B + X C + E
B & C - unknown but fixed regression coefficients
responsescovariables
random error
predictors
(Note Z = covariables and X = predictors here)
Placing non-metric distances into Euclidean space first, then use ANOVA/MANOVA within RDA with permutation tests-.
Builds on ANOVA as form of multiple regression with orthogonal dummy variables as predictors.
MATCH between ANOVA statistics and RDA statistics.
To test H0 C = 0 (i.e. the effect of X on Y)
1. Permute rows of Y
2. Permute rows of X (env. data) CANOCO 2
3. Permute residuals Er of regression Y on Z (covariables)
REDUCED MODEL OR NULL MODEL CANOCO 3 & 4
4. Permute residuals Ef of regression Y on Z and X (covariables and predictors)
FULL MODEL CANOCO 3 & 4 1 & 2 DESIGN-BASED PERMUTATIONS
1 Wrong type I error, low power
2 OK but no basis for testing interaction effects
3 & 4 MODEL-BASED PERMUTATIONS
3. Permute residuals of Y on Z (covariables) Default in CANOCO 3 & 4
Reduced model – maintains type I error in small data sets. Without covariables, gives exact Monte Carlo significance level. Retains structure in X and Z.
4. Permute residuals of Y on X and Z. Full model. Gives lower type II error, but only slightly so.
(If no covariables Y = XC + E, does not matter if samples in Y or X are permuted. CANOCO permutes X)
In DISTPCOA, do RDA with Y as principal co-ordinates scores, X defines dummy variables to code for interaction terms, and Z defines dummy variables for main effects (covariables) if interested in interactions. Can determine components of variation attributable to individual factors and interaction terms as in a linear model for multivariate data BUT using any DC that integrates both quantities and composition.
Can test the significance of individual terms and interaction terms for any complex multi-factorial experimental design. Cannot be applied to unbalanced data. If unbalanced because of missing or lost values, use missing data replacements (if other replicates).
Distance-based RDA offers special advantages to ecological researchers not shared by any other single multivariate method. These are:
1 The researcher has the flexibility to choose an appropriate dissimilarity measure, including those with semi-metric qualities, such as the Bray-Curtis measure
2 PCoA puts the information on dissimilarities among the replicates into a Euclidean framework which can then be assessed using linear models
3 A correction for negative eigenvalues in the PCoA, if needed, can be done such that probabilities obtained by a permutation test using the RDA F# -statistic are unaffected (correction method 1)
4 By using the multiple regression approach to analysis of variance, with dummy variables coding for the experimental design, RDA can be used to determine the components of variation attributable to individual factors and interaction terms in a linear model for multivariate data
5 Multivariate test statistics for any term on a linear model can be calculated, with regard to analogous univariate expected mean squares
6 Statistical tests of multivariate hypothesis using RDA statistics are based on permutations, meaning that there is no assumption of multinormality of response variables in the analysis. Also, there are no restrictions to the number of variables that can be included in RDA
7 Permutations of residuals using the method of ter Braak (1992) allows the permutation test to be structured precisely to the hypothesis and the full linear model of the design under consideration
8 The significance of multivariate interaction terms can be tested
Shares characteristics with:MAN- RDA ANO- MAN-OVA SIM TEL
* *
* *
* *
* * *
*
* *
DISTANCE-BASED MULTIVARIATE ANALYSIS FOR A LINEAR MODEL
McArdle, B.H. & Anderson, M.J. (2001) Ecology 82; 290-297
DISTLM, DISTLMforward - www.stat.auckland.ac.nz/~mja
DISTLM - multivariate multiple regression of any symmetric distance matrix with or without forward selection of individual predictors or sets of predictors and associated permutation tests.
Y response variables (n x m)
X predictor variables (n x q) (1/0 or continuous variables)
Performs a non-parametric test of the multivariate null hypothesis of no relationship between Y and X on the basis of any distance measure of choice, using permutations of the observations. X may contain the codes of an ANOVA model (design matrix) or it may contain one or more predictor variables (e.g., environmental variable) of interest.
Like Legendre and Anderson's (1999) distance-based redundancy analysis but with no correction for negative eigenvalues. Shown theoretically that partitioning the variability in X according to a design matrix or model can be achieved directly from the distance matrix itself, even if the distance measure is semi-metric (e.g., Bray-Curtis distance).
CANONICAL ANALYSIS OF PRINCIPAL CO-ORDINATES
Anderson, M.J. & Willis, T.J. (2003) Ecology 84, 511-525
CAP – www.stat.auckland.ac.nz/~mja
CAP - canonical analysis of principal co-ordinates based on any symmetric distance matrix including permutation tests.
Y response variables (n x m)
X predictor variables (n x q) (1/0 or continuous variables)
Performs canonical analysis of effects of X on Y on the basis of any distance measure of choice and uses permutations of the observations to assess statistical significance.
If X contains 1/0 coding of an ANOVA model (design matrix), result is a generalised discriminant analysis. If X contains one or more quantitative predictor variables, result is a generalised canonical correlation analysis.
Output
1. Eigenvalues and eigenvectors from the PCOORD. Can use latter to plot an indirect ordination of data.
2. Canonical correlations and squared canonical correlations.
3. Canonical axis scores.
4. Correlations of original variables (Y) with canonical axes.
5. Diagnostics to help determine appropriate value for Qt, number of eigenvectors. Select the lowest misclassification error (in the case of groups) or the minimum residual sum of squares (in the case of quantitative variables in X). Also Qt must not exceed m or n and is chosen so that the proportion of variance explained by the first Qt axes is more that 60% and less than 100% of the total variation in the original dissimilarity matrix.
6. In the case of groups, table of results for 'leave-one-out' classification of individual observations to groups.
7. Results of permutation test to test statistical significance of Qt axis model (trace and first eigenvalue).
8. Scores to construct constrained ordination diagram to compare with unconstrained ordination diagram.Very good at highlighting and testing for group differences (e.g. sampling times) as CAP finds axes that maximise separation between groups.
With quantitative predictors, CAP finds axes that maximise correlation with predictor variables.
'AIC' for model selection deviance-capscale Jari Oksanen VEGAN 1.7-6 R
Extensions by Jari Oksanen (capscale in Vegan R)
1. Axes are weighted by their corresponding eigenvalues so that the ordination distances are best approximations of the original dissimilarities.
2. Uses all axes with positive eigenvalues. Guarantees that the results are the best approximation of the original dissimilarities.
3. Adds species scores as weighted sums of the (residual) species data.
4. Negative eigenvalues are harmless and can be ignored. Often most sensible to use dissimilarity coefficients that do not have negative eigenvalues. Square-root transformation of the species data prior to calculating dissimilarities can drastically reduce the number of negative eigenvalues.Note that CAP with Euclidean distance is identical to RDA in
sample, species, and biplot scores (except for possible reversal of sign).
Possible uses of canonical analysis of principal co-ordinates
1. As in CCA or RDA with biological and environmental data.
2. Fit models to data with rare or unusual samples or species that may upset CCA.
3. Analyse many environmental variables in relation to external (e.g. geographical, geological, topographical) constraints with Monte Carlo permutation tests. In other words, do a multivariate linear regression but not have to worry about the data meeting the assumptions of least-squares estimation and models.
Examples:
Willis & Anderson 2003 Marine Ecology Progress Series 257: 209-221 (cryptic reef fish assemblages)
Edgar et al. 2003 Journal of Biogeography31: 1107-1124 (shallow reef fish and invertebrate assemblages)
SUMMARY OF CONSTRAINED ORDINATION METHODS
Methods of constrained ordination relating response variables, Y (species abundance variables) with predictor variables, X (such as quantitative environmental variables or qualitative variables that identify factors or groups as in ANOVA).Name of methods (acronyms, synonyms)
Distance measure preserved
Relationship of ordination axes with original variables
Takes into account correlation structure
Redundancy Analysis (RDA) Euclidean distance
Linear with X, linear with fitted values, Y = X(X'X)-1 X'Y
... among variables in X, but not among variables in Y
Canonical Correspondence Analysis (CCA)
Chi-square distance
Linear with X, approx unimodal with Y, linear with fitted values, Y*
... among variables in X, but not among variables in Y
Canonical Correlation Analysis (CCorA, COR)
Mahalanobis distance
Linear with X, linear with Y ... among variables in X, and among variables in Y
Canonical Discriminant Analysis (CDA; Canonical Variate Analysis CVA; Discriminant Function Analysis, DFA)
Mahalanobis distance
Linear with X, linear with Y ... among variables in X, and among variables in Y
Canonical Analysis of Principal Coordinates (CAP; Generalized Discriminant Analysis)
Any chosen distance or dissimilarity
Linear with X, linear with Qt; unknown with Y (depends on distance measure)
... among variables in X, and among principal coordinates Qt
^
^
CRITERION FOR DRAWING ORDINATION AXES
• Finds axis of maximum correlation between Y and some linear combination of variables in X (i.e., multivariate regression of Y on X, followed by PCA on fitted values, Y).
• Same as RDA, but Y are transformed to Y* and weights (square roots of row sums) are used in multiple regression.
• Finds linear combination of variables in Y and X that are maximally correlated with one another.
• Finds axis that maximises differences among group locations. Same as CCorA when X contains group identifiers. Equivalent analysis is regression of X on Y, provided X contains orthogonal contrast vectors.
• Finds linear combination of axes in Qt and in X that are maximally correlated, or (if X contains group identifiers) finds axis in PCO space that maximises differences among group locations.
^
RDA
CCA
CCorA
CVA
CAP
PRINCIPAL RESPONSE CURVES (PRC)
van der Brink, P. & ter Braak, C.J.F. (1999) Environmental Toxicology & Chemistry 18, 138-148
van der Brink, P. & ter Braak, C.J.F. (1998) Aquatic Ecology 32, 163-178
PRC is a means of analysing repeated measurement designs and of testing and displaying optimal treatment effects that change across time.
Based on RDA (= reduced rank regression) that is adjusted for changes across time in the control treatment. Allows focus on time-dependent treatment effects. Plot resulting principal component against time in PRC diagram.
Developed in ecotoxicology; also used in repeated measures in experimental ecology and in descriptive ecology where spatial replication is substituted for temporal replication.
Highlights differences in measurement end-points betweeen treatments and the reference control.
PRC MODEL
Yd(i)tk = Yotk + bk cdt + d(i)tk where
Yd(i)tk = abundance counts of taxon k at time t in replicate i of treatment d
Yotk = mean abundance of taxon k in controls (o) at time t
cdt = principal response of treatment d at time t (PRC)
bk = weight of species k with respect to cdt
d(i)tk = error term with mean of zero and variance 2k
Modelling the abundance of particular species as a sum of three terms, mean abundance in control, a treatment effect, and an error term.
Data input - species data (often log transformed) for different treatments at different times
- predictor variables of dummy variables (1/0) to indicate all combinations of treatment and sampling time ('indicator variables')
- covariables of dummy variables to indicate sampling time
Do partial RDA with responses, predictors, and covariables and delete the predictor variables that represent the control. This ensures that the treatment effects are expressed as deviations from the control.
PRC MODEL (continued)
Simple example - three treatments: C = control, L = low dosage (not rep-licated), H = high dosage sampled at four times (W0, W1, W2, W3), six species.
PRC MODEL (continued)
* ** * * * * * deleted in the RDA
PRC PLOTS
One curve for each treatment expressed as deviation from the control. Species weights (bk) allow species interpretation. Higher the weight, more the actual species response is likely to follow the PRC pattern, because the response pattern = bk cdt. Taxa with high negative weight are inferred to show opposite pattern. Taxa with near zero weight show no response.
Significance of PRC can be tested by Monte Carlo permutation of the whole time series within each treatment.
Can use the second RDA axis to generate a second PRC diagram to rank 2 model.
PRC PLOTS (continued)
See maximum deviation from control after 4 weeks, maximum effect larger for 150 g/l treatment than for 50 g/l treatment.
Chlamydomonas has high negative weight and this had highest abundances in high doses after treatment began.
Principal response curves resulting from the analysis of the example data set, indicating the effects of the herbicide linuron on the phyto-plankton community. Of all variance, 47% could be attributed to sampling date, and is on the horizontal axis. Of all variance, 30% could be attributed to treatment. Of the variance explained by treatment, 23% is displayed on the vertical axis. The lines represent the course of the treatment levels in time. The species weight (bk) can be interpreted as the affinity of the taxon with the principal response curves.
PRINCIPAL RESPONSE CURVES AND ANALYSIS OF MONITORING DATA
PRC usually used with experimental data. Can be used with (bio)monitoring data.
Samples at several dates at several sites of a river, some upstream of a sewage treatment plant (STP) (300 m, 100 m), in the STP outlet, and some downstream (100 m, 1 km). 795 samples, 5 sites, 1994-2002.
PRC using sampling month as covariable, product of sampling month and site as explanatory variables. Used STP outlet as the reference site.
Van der Brink, P. et al. (2003) Austr. J. Ecotoxicology 9: 141-156
Principal Response Curves indicating the effects of the outlet of a sewage treatment plant on some monthly averages of physico-chemical characteristics of a river. Of all variance, 24% could be attributed to between-month variation; this is displayed on the horizontal axis. 57% of all variance could be allocated to between-site differences, the remaining 19% to within-month variation. Of the between-site variation, 58% is displayed on the vertical axis. The lines represent the course of the sites in time with respect to the outlet. The weight of the physico-chemical variable (bk) can be interpreted as the affinity of the variables with the Principal Response Curves (cdt).
See biggest differences for the two upstream sites, with lower NOx, total N, conductivity, salinity, total P, and temperature and higher values of turbidity and faecal coliforms. STP outlet leads to increases in N, P, temperature, etc. Downstream values decrease but are not as low as upstream sites. STP successfully reduces faecal coliforms as their values are higher in the upstream sites due to pollution.
PRINCIPAL RESPONSE CURVES – A SUMMARY
Filters out mean abundance patterns across time in the control. Focuses on deviation between treatment and control. PRC displays major patterns in those deviations and provides good summary of response curves of individual taxa.
PRC helps to highlight 'signal' from 'noise' in ecological data in replicated experimental studies.
Simplified RDA - simplified by representing the time trajectory for the controls as a horizontal line and taking the control as the reference to which other treatments are compared.
PRC gives simple representation of how treatment effects develop over time at the assemblage level.
CCA/RDA AS PREDICTIVE TOOLS
Prediction is important challenge in environmental science.
• Given environmental shift, how will species respond?
• Given environmental data only (e.g. satellite image data), what biotic assemblages could be expected?
Conventional CCA/RDA – description and modelling
Ym and Xm Modelm where subscript m = modern
'Palaeo' CCA/RDA – modelling and reconstruction
Ym and Xm Modelm
Yo and Modelm Xo where subscript o = fossil or 'palaeo' data(Lecture 8 on Environmental Reconstruction)
Predictive CCA/RDA – modelling and prediction
Ym and Xm Modelm
Xf and Modelm Yf where subscript f = future (predicted) data
Gottfried et al. 1998. Arctic and Alpine Research 30: 207-211
Schrankogel (3497 m) Tyrol, eastern central Alps.
1000 1x1 m plots between 2830 and 3100 m, ecotonal transition between alpine zone (vegetation cover >50%) and nival zone (vegetation cover <10%).
Vegetation data - species +/- and relative abundance of 19 species
Environmental data – Digital Elevation Model (DEM) with pixel size of plots in GIS ARC/INFO
- gives 17 topographic descriptors at 10 resolutions plus altitude.
Gottfried et al. 1998
CCA with forward selection to give 37 predictor variables
Calculated CCA sample scores for 650,000 cells of DEM area as weighted linear combination of environmental variables times the canonical coefficients.
For each predicted environment of each cell, estimated which of the 1000 plots it is closest to CCA space.
Gottfried et al. 1999
Extrapolate vegetation data from those plots to the 650,000 cells to predict species distributions, vegetation types, species richness, etc.
To evaluate predictions, did 10-fold cross-validation, namely model with 90% of the plots, predict with left-out 10%, and repeat 10 times. Compare predictions with actual observed data. Also calculated Cohen's kappa statistic between observed and predicted distributions (0 = uncorrelated, 1 = perfect match).
Model performance
Axis 1 Axis 2 Species variance
CA 0.41 0.21 34.3%
CCA 0.28 0.11 21.8%
Species fell into five groups of kappa and other performance values
Total inertia 1.79 Constrained inertia 0.68 38% explained by topography
Kappa > 0.5 e.g. Carex curvula, Veronica alpina
Kappa > 0.4 e.g. Androsace alpina, Oreochloa disticha
Kappa > 0.3 e.g. Saxifraga oppositifolia, Primula glutinosa
Kappa > 0.25
e.g. Ranunculus glacialis, Cerastium uniflorum
Kappa < 0.1 e.g Poa laxa
Predicted distributions
Gottfried et al. 1998
Predicted vegetation types
Gottfried et al. 1998
Best predictors are topographic measures of roughness and curvature rather than simple elevation, slope, or exposure.
Modelled richness patterns decline with altitude but with a maximum richness at the alpine-nival ecotone.
What might happen under future climate warming of 1ºC or 2ºC?
Gottfried et al. 1999 Diversity and Distributions 5: 241-251.
Calculated altitudinal lapse rate using 33 temperature loggers.
Assume that the altitudinal limits are determined by temperature. Knowing the temperature lapse rate, predict species distributions for +1ºC and +2ºC temperature increases.
Done by increasing the values of the altitude variable in the environmental data for the sample plots corresponding to the lapse rate. Repeated the CCA/GIS interpolation/mapping.
Assumes that species growing at lower altitudes and hence warmer situations will occur in a future warmer climate in the same topographical conditions
Gottfried et al. 1999
Predicted distribution patterns at +0º, +1º and +2º
Gottfried et al. 1999
Predicted distribution patterns at +0º, +1º and +2º
Predictions - 19 species modelled, about 2 will become extinct
will be a reduction in genetic diversity as some 'topographical forms' will be lost
Dirnböck et al. 2003 Applied Vegetation Science 6: 85-96.
Same approach but with topographic descriptors and infra-red spectral data, and 3 nearest neighbours to predict rather than 1.
Predictive mapping of 17 vegetation types between 1600 m and 2277 m on Hochschwab, eastern Alps.
Dirnböck et al. 2003
69.4% accuracy, Cohen kappa of 0.64
Topography - good predictors of different alpine grasslands
Infra-red spectra - good predictors of different pioneer vegetation types
Unexplained variation - land-use history, soil variation especially nutrients like N, P, and K
Builds on:
1) Weighted averaging of indicator species and extends WA to the simultaneous analysis of many species and many environmental variables.
2) Reciprocal averaging (= correspondence analysis) by adding the statistical methodology of regression. General framework of estimation and statistical testing of the effects of explanatory variables on biological communities.
CANONICAL CORRESPONDENCE ANALYSIS
Major Uses:
1. Identify environmental gradients in ecological data-sets.
2. In palaeoecology, used as a preliminary to determine what variables influence present-day community compositions well enough to warrant palaeoenvironmental reconstruction.
3. Add 'fossil' samples into modern 'environmental' space.
4. Study seasonal and spatial and temporal variation in communities and how this variation can be explained by environmental variation. Variance can be decomposed into seasonal, temporal, spatial, environmental and random components.
5. Niche analysis – niche-space partitioning where species probability or abundance is unimodal function of environment.
6. Impact studies.
7. Predictive studies.
8. Experimental data analysis.
Powerful alternative to multivariate analysis of variance MANOVA.
e.g. analysis of BACI (before-after-control-impact) studies with and without replication of the impacted site
e.g. repeated measurement designs
e.g. experimental plot (= block) designs
e.g. split-plot designs
See ter Braak C.J.F. & Verdonschot P.F.M (1995) Aquatic Sciences 57, 255–289
Canonical correspondence analysis and related multivariate methods in aquatic ecology
CANOCO + CANODRAW
[CANCOR, CAP]
canonicalcorrelation
analysis
constrained principalco-ordinates analysis
DISTPCOA
distance-based redundancy analysis via principal co-ordinates analysis
SOFTWARE FOR CONSTRAINED ORDINATIONS
R(VEGAN) (CCA, RDA)
R (Non-linear CAP)
CANODRAW 4
Pie symbols plot
CANODRAW 4
CANODRAW 4
Isolines in RDA ordination diagram
Biplot with environmental variables
& sites
Attribute plot
T-value biplotSample diagram with principal response curves
Response curves fitted using GAM
1987
20032002
1987
Mark Hill
Cajo ter Braak
Petr ŠmilauerMarti Anderson