4/4/2018. Stepwise model fitting. CCA with first three variables only Call: cca(formula = community ~ env1 + env2 + env3, data = envdata)

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1 0 Correlation matrix for ironmental matrix vif.cca (Variance Inflation Scores) > vif.cca(base_cca) spatialb spatialc Eliminate variable 10: spatialb spatialc Eliminate variable spatial: cca(formula = community ~ spatial + 7, data = data) Total Constrained Unconstrained Eigalues, and their contribution to the mean squared contingency coefficient CCA3 CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8 CA9 Eigalue Proportion Explained Cumulative Proportion Accumulated constrained eigalues CCA3 Eigalue Proportion Explained Cumulative Proportion sample16 spatialb spatialc > vif.cca(model_fit) spatialb spatialc CCA with first three variables only Call: cca(formula = community ~ , data = data) Total Constrained Unconstrained Eigalues, and their contribution to the mean squared contingency coefficient CCA3 CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8 CA9 Eigalue Prop Explain Cum. Prop Accumulated constrained eigalues CCA Eigalue Proportion Explained Cumulative Proportion sample Stepwise model fitting Start: community ~ ** + spatial ** ** Step: community ~ spatial ** - spatial ** Step: community ~ spatial + 7 Note that exact output changes each time you run it (P values permutational). Adjust Pin option to 0.2 and Pout to 0.3: Step: community ~ spatial ** spatial ** Step: community ~ spatial ** - spatial **

2 sample16 sample3 Tests of significance What to Present > anova(model_fit) ** Model > anova(model_fit, by = "terms", perm = 1000) Terms added sequentially (first to last) ** spatial ** > anova(model_fit, by = "axis", perm = 1000) ** ** CCA CCA CCA aligned with CA, species relationships assumed to be unimodal A CCA with no constraints = CA RDA aligned with PCA, species relationships assumed to be linear An RDA with no constraints = PCA CCA is more common for the same reasons PCA not usually used for species data. Function rda (vegan package) works similarly to cca. rda(community ~ spatial + 7, data) Total inertia differs because PCA begins with variance/covariance matrix, not X 2 Total Constrained Unconstrained Eigalues, and their contribution to the correlations RDA spatialb RDA1 RDA2 RDA3 PC1 PC2 PC3 PC4 PC5 PC6 PC7 Eigalue Proportion Explained Cumulative Proportion RDA1 spatialc

3 sample3 0 PDA without ironmental matrix = PCA rda(x = community, scale = T) Partitioning of correlations: Total 10 1 Unconstrained 10 1 Eigalues, and their contribution to the correlations PC sample3 sample PC1 Example RDA application How do various ironmental variables affect morphology or behavior (plasticity question)? Assumption is that you know and have measured the ironmental variables that are going to impact the morphology/behavior measured. Morphology/Behavior Env. Variables PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 Eigalue Proportion Explained Cumulative Proportion Individuals Samples Constrained Analysis of Principal Coordinates (capscale) Function capscale (vegan package) Similar to RDA but based on non-euclidean distances Function works with raw data (indicate which distance metric to use) or with a distance matrix (e.g. genetic distance) Same anova function tests for significance Various options for dealing with data standardization Also called distance based redundancy analysis in literature Anderson, M.J. & Willis, T.J. (2003). Canonical analysis of principal coordinates: a useful method of constrained ordination for ecology. Ecology 84, CAP spatialb sample16 spatialc Call: CAP1 capscale(formula = community ~ spatial + 7, data = data, distance = "bray") Partitioning of squared Bray distance: Total Constrained Unconstrained Gradients unknown, want to describe data or imply gradients from species data. No yes Important gradients are known and measured. Goal is to describe variability in species data as it relates to the measured gradients. Variability due to other unmeasured factors is ignored. Direct gradient analysis: CCA or RDA. Data are thought to be distinctly structured. Goal is to put in distinct groups. no yes Cluster hierarchical (agglomerative or divisive) or non hierarchial (k means). Goal is to look at the relationship among samples. Indirect gradient analysis: NMDS or CA preferred, PCA or PCoA for noncommunity datasets. 3

4 Unconstrained NMDS approach (exploratory) NMDS sample sample Used when you want to control (or partial out) a variable. Eg. larger spatial variable or something traditionally considered a covariate. Typically not something you re interested in quantifying as much as controlling for. Recall that CCA (and RDA, capscale) use multiple regression to optimize constraints to species or sample scores. Partial Regression Regression coefficient of X after controlling for influence of Z on both X and Y. NMDS1 Dementia Score Dementia Score Age Wanted relationship between X (age) and y (degree of dementia). Numerous other variables confound the relationship, such as IQ and mean number of hours spent reading daily. Residualized dementia score, effects of IQ have been partialed out. IQ Residualized Dimentia Age 4

5 0 0 All the same precautions apply as with CCA. Output will be nearly identical with the inclusion of the variance accounted for by the partialed variable. Total inertial in the ordination same as CA Inertia partitioned out comparison of the model with and without partial variable Inertia constrained by variables in model Unconstrained inertia Uses the same function as CCA or RDA Code cca_example<-cca(community ~ Condition(spatial),ironmental) There is no significance test for the partialed variable(s), just the % variance accounted for. cca(formula = community ~ 7 + Condition(spatial), data = data) Total Conditioned Constrained Unconstrained CA II a b c sample16 spatialb spatialc Test for influence of 7 while controlling for spatial variable. Eigalues, and their contribution to the mean squared contingency coefficient after removing the contribution of conditiniong variables CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8 CA9 Eigalue Proportion Explained Cumulative Proportion CA sample > anova(partial_cca) Model: cca(formula = community ~ 7 + Condition(spatial), data = data) Model ** Residual > anova(partial_cca, by = "terms", perm = 1000) Terms added sequentially (first to last) Model: cca(formula = community ~ 7 + Condition(spatial), data = data) ** Residual CA I 5