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Quantitative Methods in Palaeoecology and Palaeoclimatology PAGES Valdivia October 2010 Ordination Analysis II – Direct Gradient Analysis John Birks.

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Presentation on theme: "Quantitative Methods in Palaeoecology and Palaeoclimatology PAGES Valdivia October 2010 Ordination Analysis II – Direct Gradient Analysis John Birks."— Presentation transcript:

1 Quantitative Methods in Palaeoecology and Palaeoclimatology PAGES Valdivia October 2010 Ordination Analysis II – Direct Gradient Analysis John Birks

2 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)

3 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)

4 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. Vegetational data Environmental data DUNE-MEADOW DATA

5 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=b 0 + b 1 x 1 + b 2 x 2 Angle (  )with axis 1 = arctan(b 2 / b 1 ) DCA axis 2 DCA axis 1 Indirect analysis

6 Site scores of the second DCA axis plotted against the amount of manure. Indirect analysis

7 Correlation coefficients (100  r) of the environmental variables for the four first DCA axes for the Dune Meadow Data Variable Axes A moisture use manure SF BF HF NM Eigenvalue Indirect analysis

8 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. TermParameterEstimates.e. t constant c 0 – –4.62 A1 c moisture c use c manure c 4 – –0.01 ANOVA table d.f.s.s.m.s.F Regression ,6 Residual Total R 2 = 0.75R 2 adj = 0.66 Ey 1 = b 0 + b 1 x 1 + b 2 x b n x n CA axis 1environmental variables x = environmental variables Indirect analysis

9 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.

10 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

11 PRIMARY DATA IN GRADIENT ANALYSIS Indirect GA Direct GA Abundances or +/- variables Response variables Values Classes Predictor or explanatory variables Species Env. vars PLUS

12 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: Moisture b: Linear combination of moisture and phosphate, chosen a priori c: 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

13 Combinations of environmental variables e.g.3 x moisture + 2 x phosphate e.g.all possible linear combinations z j = environmental variable at site j c = weights x j = resulting ‘compound’ environmental variable CCA selects linear combination of environmental variables that maximises dispersion of species scores, i.e. chooses the best weights (c i ) of the environmental variables.

14 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

15 1)Start with arbitrary, but unequal, site scores x i. 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]. CANONICAL CORRESPONDENCE ANALYSIS Algorithm REF

16 4)Obtain regression coefficients of site scores on the environmental variables by weighted multiple regression. whereb and x* are column vectors Z 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. REF

17 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)

18 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.

19 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).

20 12.81SF BF SF SF HF HF HF HF HF BF BF SF22* SF NM NM SF NM *1NM NM NM10 Sample number A1 horizon Moisture class Management type Use Manure class Environmental data of 20 relevées from the dune meadows Use categories: 1 = hay 2 = intermediate 3 = grazing * = mean value of variable

21 DCA ordination diagram of the Dune Meadow Data DCA axis 2 DCA axis 1 DCA

22 Correlations of environmental variables with DCA axes 1 and 2 Axis One 1 = 0.54 Axis Two 2 = 0.29 DCA

23 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. 1 2 R axis 1 R axis R axis 1 R axis 2 DCA CCA CCA

24 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. A Moisture Use Manure SF BF HF NM VariableCoefficientsCorrelations Axis 1 Axis 2 CANONICAL CORRESPONDENCE ANALYSIS

25 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. a b

26 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

27 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

28 Summary Axes Axes Total inertia Eigenvalues Species-environment correlations Cumulative percentage variance of species data of species-environment relation Sum of all unconstrained eigenvalues = inertia Sum of all canonical eigenvalues = species-environment relation 'Fitted' species data Rules of thumb: >0.30 strong gradient >0.40 good niche separation of species CANOCO

29 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

30 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)

31 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.

32 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 – active environmental 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

33 Passive ‘fossil’ samples added into CCA of modern data

34 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.

35 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 data unequal sampling designs highly intercorrelated environmental variables situations when not all environmental factors are known

36 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

37 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

38 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

39 McCune (1997) Ecology 78, 2617–2623 Simulated artificial data 10 x 10 grid. 40 species following Gaussian response model. 2environmental variablesX and Y co-ordinatesTENxTEN 2environmental variableswith added noise NOISMOD (random number mean = 0, variance 17%) added to each cell 10random environmental variablesNOIS1O 2environmental variables with added noise from NOISMOD + 10random environmental variables from NOIS 10 NOISBOTH 99random environmental variablesNOISFULL 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)(2)(3)(4)(5)

40 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

41 Monte Carlo tests 1 2 r 1 r 2 TENxTEN* NOISMOD* NOISE 10ns NOISBOTH* NOISFULL ns (99 env vars) Linear combination site best fit of species abundances to scores the environmental data WA site scoresbest represent the assemblage structure LC scores WA scores Sensitive to noise+– True direct gradient analysis +– (multivariate regression) Aim to describe biological +– variation in relation to environment Assemblage 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.

42 LC OR WA SCORES? M IKE P ALMER "Use LC scores, because they give the best fit with the environment and WA scores are a step from CCA towards CA." B RUCE M C C UNE "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)

43 DATA ORDERINGS

44 CCA DIAGRAM TEN SETS OF DISTANCES TO REPRESENT, EMPHASIS ON 5, 8, AND 1 (FITTED ABUNDANCES OF SPECIES AND SITES)

45 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)

46 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.

47 Hill scaling Default scaling –1 2 Emphasis on SITES SPECIES 1Species x sitesRel abundancesFitted abundances (rel) 2Species x species –Chi-squared distances 3Sites x sitesTurnover distances – Quant env vars 4Sites x env vars 3 –Values of env vars 5Species x env varsWeighted averagesWeighted averages 6Env vars x env varsEffects 2 Correlations 2 Qualit env vars 7Sites x env classes 4 Membership 1 Membership 1 8Species x env classes Rel total abundRel total abund 9Env vars x env classes –Mean values of env vars 10Env classes x env Turnover distances – classes fitted by least squares 1 by centroid principle 2 change in site scores if env variable changes are one standard deviation 3 inter-set correlations 4 group centroids SCALING IN CCA

48 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). Note that symmetric scaling (=3) has many optimal properties (Gabriel, 2002; ter Braak, personal communication) REF

49 Scaling -1: focus on sites Hill's scaling Interpreta- tion 2: focus on species biplot scaling of CCA Interpreta- tion 1. species x sites a Rel. Abundances b,c CENTROID Fitted rel. abund. b BIPLOT rule or CENTROID rule 2. species x species- UNKNOWN  -square distances d DISTANCE rule 3. sites x sitesTurnover distances c,e DISTANCE f DISTANCE rule Quantitative env. vars 4. sites x env. vars g - UNKNOWN Values of env.vars BIPLOT rule 5. species x env. varsWeighted averages BIPLOT Weighted averages BIPLOT rule 6. env.vars x env. varsEffects h ? BIPLOT Correlations BIPLOT rule Qualitative env. vars 7. sites x env. classes i Membership k CENTROID Membership k 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 distances c,e DISTANCE f DISTANCE rule

50 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 (y i+ ). 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

51 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 1  2 INTERPRETATION OF CCA PLOTS

52 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 of seasonal variation }

53 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) jVariable 1 Pj a P Cum ( a ) 1Shrubs (1/0)0.25(0.01)1Shrubs (1/0)0.25(0.01)0.25 2Source distance0.22(0.01)2Source distance0.19(0.01)0.44 3EC0.20(0.01)3EC0.19(0.01)0.63 4Discharge0.17(0.01)4Discharge0.14(0.03)0.75 5Total veg cover0.16(0.01) 6Shading0.15(0.01)-Cover emergent0.11(0.10)- 7Soil grain size0.14(0.02)-Cover bank veg0.11(0. 12)- 8Stream width0.14(0.05)-Soil grain size0.10(0.13)- 9High weedy veg0.14(0.08) 10Cover bank veg0.13(0.11) -U vs L stream0.22(0.01)-U vs L stream0.09(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

54 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 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.

55 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

56 CCADirectly CAIndirectly Gradient length estimation 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.

57 REDUNDANCY ANALYSIS – CONSTRAINED PCA Short (< 2SD) compositional gradients Linear or monotonic responses Reduced-rank regression PCA of y with respect to x Two-block mode C PLS PCA of instrumental variablesRao (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

58 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 units for the species and to 0.4 units for the environmental variables.

59 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. VariableCoefficientsCorrelations Axis1Axis2Axis1Axis2 A Moisture Use Manure SF BF HF NM

60 Axis 1Axis 2 PCA% RDA% PCA Correlation RDA Correlation PCA and RDA comparisons Important to do the check that the environmental variables relate to the major gradients in composition detected by the PCA.

61 Cosine of angle  correlation Long arrows of species and environmental variables most important Euclidean distance biplot Covariance (correlation) biplot RDA covariance or correlation matrix of species RDA – constrained form of multiple regression Uses 2 (q + m) + m parameters (q env variables, m species) Multiple regression m (q + 1) e.g.40 species 10 envir variables RDA140 parameters MR440 parameters RDA is thus reduced rank regression (RR) species unconstrained Goodness of fit sum of eigenvalues constrained fitted species BIPLOT INTERPRETATION

62 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.

63 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 scaling1: focus on sites2: focus on species distance biplotcorrelation biplot (a) principal components analysis species x sitesabundancesabundances sitesEuclidean distances- species-correlations a (b) redundancy analysis species x sites b fitted abundancesfitted abundances sites b Euclidean distances c - species-correlations a,c Quantitative env. vars.: species x env. vars. d correlationscorrelations sites x env. vars. d -values of env. vars env. vars. d effects e correlations Qualitative env. vars: species x env. classes f meansmeans sites x env. classes fgg env. classes f Euclidean distances- env. vars. x env. classes-means

64 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

65 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.

66 PROPOSED NEW SCALING FOR CCA AND RDA Gabriel, K.R. (2002) Biometrika 89, 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 sitesSpecies data (PCA) Fitted species data (RDA) Relative species data (CA) Fitted relative species data (CCA) Species x environmental variablesCorrelations of species (RDA) Optima (WA) of species (CCA) Species x environmental classesMean abundances of species (RDA) Relative abundances of species across classes (CCA)

67 Data tables with 95% preservation of proportional fit: Species x speciesCorrelations (PCA, RDA) Chi-square distances (CA, CCA) Sites x sitesEuclidean distances (PCA, RDA) Chi-square distances (CA, CCA) Env. classes x env. classesEuclidean distances (RDA) Chi-square distances (CCA) Env. variables x env. variablesCorrelations (RDA, CCA) Sites x env. variablesValues (RDA, CCA) Sites x env. classesMeans (RDA, CCA) Env. variables x env. classesMean values of env. variables (RDA, CCA)

68 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. ContoursBubble plots GAM J. Oksanen (2002)

69 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

70 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)

71 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)

72 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, O 2 = 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

73 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)

74 e.g. partial out the effects of some covariables prior to indirect gradient analysis within-plot changePRIMARY INTEREST between-plot differencesNOT 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

75 Background variables or 'covariables' Partial CCA Partial RDA Vegetation Environmental variables or 'constraints' Vegetation (residual) CCA RDA CA DCA PCA 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

76 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

77 Total inertia = total variance Sum canonical eigenvalues = % Explained variance57%  Unexplained variance = T – E43% What of explained variance component? Soil variables (pH, Ca, LOI) Land-use variables (e.g. grazing, mowing) Not independent Do CCA/RDA using 1)Soil variables only  canonical eigenvalues )Land-use variables only  canonical eigenvalues )Partial analysis SoilLand-use covariables )Partial analysis Land-useSoil covariables0.142 a)Soil variation independent of land-use (3) % b)Land-use structured (covarying) soil variation (1–3) % c)Land-use independent of soil (4) % Total explained variance56.9% d)Unexplained43.1% unexplainedunique covariance a b cd CANOCO

78 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 PredictorsCovariablesSum of canonical G+C+D D G+C G+C G+C D D Joint effect D  G+C= = =0.652 C D+G G+D G+D C C Joint effect C  D+G= = = VARIATION PARTITIONING OR DECOMPOSITION WITH 3 OR MORE SETS OF PREDICTOR (EXPLANATORY) VARIABLES

79 PredictorsCovariablesSum of canonical G D+G D+C D+C G G Joint effect G  D+C= = = Canonical eigenvalues All predictors0.811 Pure deposition0.027PD Pure climate0.106PC Pure geography0.034PG Joint G + C0.132 Joint G + D0.074 Joint D + C0.228 Unexplained variance1 – = PD DG CD CG CDG PG PC D GC Covariance terms CD DG CG CDG

80 CD + DG + CDG =0.652 CD + CG + CDG =0.631 DG + CG + CDG =0.549 PD + PC + CD = CD = – = PD PC (D+C) (DG + CG + CDG) PD + PG + DG = DG = – = PD PG (G+D) (CD + CG + CDG) PC + PG + CG = CG = – = PC PG (G+C) (CD + DG + CDG)  CD = 0.095DG = 0.013CG = –0.008 CDG= – – 0.095= = – (–0.008) – 0.095= = – (–0.008) – 0.013= 0.544

81 Total explained variance consists of: Common climate + deposition0.095 Unique climate PC Common deposition + geography0.013 Unique geography PG Common climate + geography0.008 Unique deposition PD Common climate + geography + deposition0.544 Unexplained variance See also Qinghong Liu (1997) – Environmetrics 8, 75–85 Anderson & Gribble (1998) – Australian J. Ecology 23, 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'.

82 '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.

83 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)

84 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 RegressionMultiple regressionGaussian regression Weighted averaging of site scores Calibration Linear calibration 'inverse regression' Gaussian calibration Weighted averaging of species scores (WA) Ordination Principal components analysis (PCA) Gaussian ordination Correspondence analysis (CA); detrended correspondence analysis (DCA) Constrained ordination 1 Redundancy analysis (RDA) 4 Gaussian canonical ordination Canonical correspondence analysis (CCA); detrended CCA Partial ordination 2 Partial component analysis Partial Gaussian ordination Partial correspondence analysis; partial DCA Partial constrained ordination 3 Partial redundancy analysis Partial Gaussian canonical ordination Partial canonical correspondence analysis; partial detrended CCA 1 = constrained multivariate regression 2 = ordination after regression on covariables 3 = constrained ordination after regression on covariables = constrained partial multivariate regression 4 = 'reduced rank regression' = “PCA of y with respect to x”

85 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

86 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 Gabbro Plot )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 = |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.

87 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 (pH 2 ) (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’.

88 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)

89 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!

90 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 < )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 = t = 3.  1 = t = 4,  1 = With 999 permutations (i.e. p of 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).

91 MINIMAL ADEQUATE MODEL IN CCA J. Oksanen (2002) 13 environmental variables3 environmental variables

92 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.

93 Variance of estimated regression (= canonical) coefficients (c j ) are proportional to their VIF. number of predictors number of samples VIF is related to the (partial) multiple correlation coefficient R j 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). VIFVIF pH Ca 2.6 – Mg VARIANCE INFLATION FACTORS (VIF)

94 'AIC' FOR MODEL SELECTION IN CCA AND RDA Jari Oksanen (2004) VEGAN 1.7-6Rdeviance.cca deviance.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.

95 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

96 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.

97 'AIC' MODEL SELECTION Godínez-Domínguez & Freire 2003 Marine Ecology Progress Series 253, 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.

98 From estimated residual sum of squares (RSS) in CCA where h = number of predictor variables in model, where log = log e n = sample size, and = RSS/n

99 To avoid bias in AIC due to links between sample size and number of parameters, corrected AIC is As in GLM, interested in differences in AIC between models  i = AIC ci – min AIC c

100 Data – 5 cruises (DEM-1 – DEM-5) - 8 models Godinez-Dominguez & Freire, 2003

101 CCA permutation tests40 models 27 p < 0.05 (global test) 20 p < 0.05 (first CCA axis) AIC c approach to find most parsimonious model

102 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.

103 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 RDA species scores are simply weighted sums of site scores CANCOR species scores are b parameters estimated by multiple regression of site scores on species variables  number of species << number of sites In factnumber 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

104 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 CANCORno effect no effect

105 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)

106 DISTPCOAPierre Legendre & Marti Anderson (1999) Ecol. Monogr. 69, 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 composition Varying e.g. ridge  snow bed gradient Other dissimilarities Bray & Curtisnon-Euclideansemi-metric Jaccard +/-non-Euclideansemi-metric Gower mixed datanon-Euclideansemi-metric Basic idea Reduce 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

107 Correction for negative eigenvalues where c 1 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

108 Raw data (replicates x species) Distance matrix (Bray-Curtis, etc) Principal coordinate analysis (PCoA) Correction for negative eigenvalues Matrix Y (replicates x principal coordinates) Matrix X (dummy variables for the factor) Test of one factor in a single-factor model Redundancy analysis (RDA) F # statistic Partial redundancy analysis (partial RDA) F # statistic Matrix Y (replicates x principal coordinates) Matrix X (dummy variables for the interaction) Matrix X C (dummy variables for the main effects) Test of F # by permutation under the full model Test of F # by permutation Test of interaction term in multifactorial model

109 Correspondence between the various components of the univariate F-statistics and the multivariate RDA statistics in the one-factor case. Univariate ANOVAMultivariate RDA statistic Total sum of squaressum of all eigenvalues of Y Treatment sum of squares = SS Tr trace = sum of all canonical eigenvalues of Y on X Treatment degrees of freedom = df Tr q Residual sum of squares = SS Res rss = sum of all eigenvalues – trace Residual degrees of freedom = df Res n T – q – 1 Treatment mean square = SS tr /df Tr = MS Tr trace/q Residual mean square SS Res /df Res = MS Res rss/(n T – q - 1) Tr = treatment, q = number of linearly independent dummy variables, n T = number of replicates REF

110 PERMUTATION TESTS in RDA (also CCA) in CANOCO. What is shuffled? Y = Z B + X C + E B & C - unknown but fixed regression coefficients responses covariables 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.

111 To test H 0 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 E r of regression Y on Z (covariables) REDUCED MODEL OR NULL MODELCANOCO 3 & 4 4. Permute residuals E f of regression Y on Z and X (covariables and predictors) FULL MODELCANOCO 3 & 4 1 & 2DESIGN-BASED PERMUTATIONS 1Wrong type I error, low power 2OK but no basis for testing interaction effects 3 & 4MODEL-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)

112 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).

113 Distance-based RDA offers special advantages to ecological researchers not shared by any other single multivariate method. These are: 1The researcher has the flexibility to choose an appropriate dissimilarity measure, including those with semi-metric qualities, such as the Bray-Curtis measure 2PCoA puts the information on dissimilarities among the replicates into a Euclidean framework which can then be assessed using linear models 3A 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) 4By 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 5Multivariate test statistics for any term on a linear model can be calculated, with regard to analogous univariate expected mean squares 6Statistical 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 7Permutations 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 8The significance of multivariate interaction terms can be tested Shares characteristics with: MAN-RDAANO-MAN- OVASIMTEL* ** *** **

114 DISTANCE-BASED MULTIVARIATE ANALYSIS FOR A LINEAR MODEL McArdle, B.H. & Anderson, M.J. (2001) Ecology 82; DISTLM, DISTLMforward - 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).

115 CANONICAL ANALYSIS OF PRINCIPAL CO-ORDINATES Anderson, M.J. & Willis, T.J. (2003) Ecology 84, CAP – 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.

116 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 Q t, 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 Q t must not exceed m or n and is chosen so that the proportion of variance explained by the first Q t 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 Q t 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 R

117 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).

118 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: (cryptic reef fish assemblages) Edgar et al Journal of Biogeography31: (shallow reef fish and invertebrate assemblages)

119 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 Q t ; unknown with Y (depends on distance measure)... among variables in X, and among principal coordinates Q t ^ ^

120 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 Q t 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

121 PRINCIPAL RESPONSE CURVES (PRC) van der Brink, P. & ter Braak, C.J.F. (1999) Environmental Toxicology & Chemistry 18, van der Brink, P. & ter Braak, C.J.F. (1998) Aquatic Ecology 32, 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.

122 PRC MODEL Y d(i)tk =Y otk + b k c dt +  d(i)tk where Y d(i)tk = abundance counts of taxon k at time t in replicate i of treatment d Y otk = mean abundance of taxon k in controls (o) at time t c dt = principal response of treatment d at time t (PRC) b k = weight of species k with respect to c dt  d(i)tk = error term with mean of zero and variance  2 k 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.

123 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.

124 PRC MODEL (continued) * * * * * * * * deleted in the RDA

125 PRC PLOTS One curve for each treatment expressed as deviation from the control. Species weights (b k ) allow species interpretation. Higher the weight, more the actual species response is likely to follow the PRC pattern, because the response pattern = b k c dt. 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.

126 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 (b k ) can be interpreted as the affinity of the taxon with the principal response curves.

127 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, 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:

128 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 (b k ) can be interpreted as the affinity of the variables with the Principal Response Curves (c dt ).

129 See biggest differences for the two upstream sites, with lower NO x, 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.

130 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.

131 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 Y m and X m  Model m where subscript m = modern 'Palaeo' CCA/RDA – modelling and reconstruction Y m and X m  Model m Y o and Model m  X o where subscript o = fossil or 'palaeo' data (Lecture 8 on Environmental Reconstruction) Predictive CCA/RDA – modelling and prediction Y m and X m  Model m X f and Model m  Y f where subscript f = future (predicted) data 

132 Gottfried et al Arctic and Alpine Research 30: Schrankogel (3497 m) Tyrol, eastern central Alps x1 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.

133 Gottfried et al. 1998

134 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

135 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).

136 Model performance Axis 1Axis 2Species variance CA % CCA % Species fell into five groups of kappa and other performance values Total inertia 1.79 Constrained inertia % explained by topography Kappa > 0.5e.g. Carex curvula, Veronica alpina Kappa > 0.4e.g. Androsace alpina, Oreochloa disticha Kappa > 0.3e.g. Saxifraga oppositifolia, Primula glutinosa Kappa > 0.25e.g. Ranunculus glacialis, Cerastium uniflorum Kappa < 0.1e.g Poa laxa

137 Predicted distributions Gottfried et al. 1998

138 Predicted vegetation types Gottfried et al. 1998

139 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 Diversity and Distributions 5: 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.

140 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 Predicted distribution patterns at +0º, +1º and +2º

141 Gottfried et al Predicted distribution patterns at +0º, +1º and +2º

142 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 Applied Vegetation Science 6: 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.

143 Dirnböck et al. 2003

144 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

145 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

146 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.

147 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 Seeter Braak C.J.F. & Verdonschot P.F.M (1995) Aquatic Sciences 57, 255–289 Canonical correspondence analysis and related multivariate methods in aquatic ecology

148 CANOCO + CANODRAW [CANCOR, CAP] canonical correlation analysis constrained principal co-ordinates analysis DISTPCOA distance-based redundancy analysis via principal co- ordinates analysis SOFTWARE FOR CONSTRAINED ORDINATIONS R(VEGAN) (CCA, RDA) R (Non-linear CAP)

149

150 CANODRAW 4 Pie symbols plot

151 CANODRAW 4

152 Isolines in RDA ordination diagram Biplot with environmental variables & sites Attribute plot T-value biplot Sample diagram with principal response curves Response curves fitted using GAM

153

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155 Mark Hill Cajo ter Braak Petr Šmilauer Marti Anderson


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