# anova.cca: Permutation Test for Constrained Correspondence Analysis,... In pattakosn/Rworkshop: Community Ecology Package

## Description

The function performs an ANOVA like permutation test for Constrained Correspondence Analysis (`cca`), Redundancy Analysis (`rda`) or distance-based Redundancy Analysis (dbRDA, `capscale`) to assess the significance of constraints.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```## S3 method for class 'cca' anova(object, alpha=0.05, beta=0.01, step=100, perm.max=9999, by = NULL, ...) permutest(x, ...) ## S3 method for class 'cca' permutest(x, permutations = 99, model = c("reduced", "direct", "full"), first = FALSE, strata, ...) ```

## Arguments

 `object,x` A result object from `cca`. `alpha` Targeted Type I error rate. `beta` Accepted Type II error rate. `step` Number of permutations during one step. `perm.max` Maximum number of permutations. `by` Setting `by = "axis"` will assess significance for each constrained axis, and setting `by = "terms"` will assess significance for each term (sequentially from first to last), and setting `by = "margin"` will assess the marginal effects of the terms (each marginal term analysed in a model with all other variables). `...` Parameters passed to other functions. `anova.cca` passes all arguments to `permutest.cca`. In `anova` with `by = "axis"` you can use argument `cutoff` (defaults `1`) which stops permutations after exceeding the given level. `permutations` Number of permutations for assessing significance of constraints. `model` Permutation model (partial match). `first` Assess only the significance of the first constrained eigenvalue; will be passed from `anova.cca`. `strata` An integer vector or factor specifying the strata for permutation. If supplied, observations are permuted only within the specified strata.

## Details

Functions `anova.cca` and `permutest.cca` implement an ANOVA like permutation test for the joint effect of constraints in `cca`, `rda` or `capscale`. Functions `anova.cca` and `permutest.cca` differ in printout style and in interface. Function `permutest.cca` is the proper workhorse, but `anova.cca` passes all parameters to `permutest.cca`.

The default test is for the sum of all constrained eigenvalues. Setting `first = TRUE` will perform a test for the first constrained eigenvalue. Argument `first` can be set either in `anova.cca` or in `permutest.cca`. It is also possible to perform significance tests for each axis or for each term (constraining variable) using argument `by` in `anova.cca`. Setting `by = "axis"` will perform separate significance tests for each constrained axis. All previous constrained axes will be used as conditions (“partialled out”) and a test for the first constrained eigenvalues is performed (Legendre et al. 2011). You can stop permutation tests after exceeding a given significance level with argument `cutoff` to speed up calculations in large models. Setting `by = "terms"` will perform separate significance test for each term (constraining variable). The terms are assessed sequentially from first to last, and the order of the terms will influence their significances. Setting `by = "margin"` will perform separate significance test for each marginal term in a model with all other terms. The marginal test also accepts a `scope` argument for the `drop.scope` which can be a character vector of term labels that are analysed, or a fitted model of lower scope. The marginal effects are also known as “Type III” effects, but the current function only evaluates marginal terms. It will, for instance, ignore main effects that are included in interaction terms. In calculating pseudo-F, all terms are compared to the same residual of the full model. Permutations for all axes or terms will start from the same `.Random.seed`, and the seed will be advanced to the value after the longest permutation at the exit from the function.

In `anova.cca` the number of permutations is controlled by targeted “critical” P value (`alpha`) and accepted Type II or rejection error (`beta`). If the results of permutations differ from the targeted `alpha` at risk level given by `beta`, the permutations are terminated. If the current estimate of P does not differ significantly from `alpha` of the alternative hypothesis, the permutations are continued with `step` new permutations (at the first step, the number of permutations is `step - 1`). However, with `by="terms"` a fixed number of permutations will be used, and this is given by argument `permutations`, or if this is missing, by `step`.

Community data are permuted with choice `model="direct"`, residuals after partial CCA/ RDA/ dbRDA with choice `model="reduced"` (default), and residuals after CCA/ RDA/ dbRDA under choice `model="full"`. If there is no partial CCA/ RDA/ dbRDA stage, `model="reduced"` simply permutes the data and is equivalent to `model="direct"`. The test statistic is “pseudo-F”, which is the ratio of constrained and unconstrained total Inertia (Chi-squares, variances or something similar), each divided by their respective ranks. If there are no conditions (“partial” terms), the sum of all eigenvalues remains constant, so that pseudo-F and eigenvalues would give equal results. In partial CCA/ RDA/ dbRDA, the effect of conditioning variables (“covariables”) is removed before permutation, and these residuals are added to the non-permuted fitted values of partial CCA (fitted values of `X ~ Z`). Consequently, the total Chi-square is not fixed, and test based on pseudo-F would differ from the test based on plain eigenvalues. CCA is a weighted method, and environmental data are re-weighted at each permutation step using permuted weights.

## Value

Function `permutest.cca` returns an object of class `"permutest.cca"`, which has its own `print` method. The distribution of permuted F values can be inspected with `density.permutest.cca` function. The function `anova.cca` calls `permutest.cca` and fills an `anova` table.

## Note

Some cases of `anova` need access to the original data on constraints (at least `by = "term"` and `by = "margin"`), and they may fail if data are unavailable.

The default permutation `model` changed from `"direct"` to `"reduced"` in vegan version 1.15-0, and you must explicitly set `model = "direct"` for compatibility with the old version.

Tests `by = "terms"` and `by = "margin"` are consistent only when `model = "direct"`.

Jari Oksanen

## References

Legendre, P. and Legendre, L. (2012). Numerical Ecology. 3rd English ed. Elsevier.

Legendre, P., Oksanen, J. and ter Braak, C.J.F. (2011). Testing the significance of canonical axes in redundancy analysis. Methods in Ecology and Evolution 2, 269–277.

`cca`, `rda`, `capscale` to get something to analyse. Function `drop1.cca` calls `anova.cca` with `by = "margin"`, and `add1.cca` an analysis for single terms additions, which can be used in automatic or semiautomatic model building (see `deviance.cca`).
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```data(varespec) data(varechem) vare.cca <- cca(varespec ~ Al + P + K, varechem) ## overall test anova(vare.cca) ## Test for axes anova(vare.cca, by="axis", perm.max=500) ## Sequential test for terms anova(vare.cca, by="terms", permu=200) ## Marginal or Type III effects anova(vare.cca, by="margin") ## Marginal test knows 'scope' anova(vare.cca, by = "m", scope="P") ```