Constrained ordinations Dependence of multivariate response on one or many predictors

Linear regression

Covariables Just another type of explanatory variables, effect of which is not interesting at this moment and should be removed from the analysis Analogy with covariates in ANCOVA Marginal and partial effects in Multiple regression

Partial analyses: The effect of covariable(s) is first subtracted from the data, and then the analysis (usually constrained, but also unconstrained) is carried out on the residual variability. If X is covariable, the analysis is then carried out on e.

Two main reasons to apply partial analyses. 1. Covariable is real nuisance (e.g. meadows were sampled in the course of 3 weeks, because we did not managed to sample faster. The date of sampling is used as covariable.) 2. We want to separate effects of several (often correlated) predictors. Each explanatory variable will be used in different analyses both as “environmental variable” and “covariable”.

Linear regression

Test on the first axis and on the trace (=on all canonical axes) n=total no. of axes p=no. of expl. var. q=no. of covariab. If there is a strong univariate variation in the data, the test on the first axis is stronger than test on the trace.

Monte Carlo permutation test

What is permuted Reduced model: Residuals after fitting covariables Full model: Residuals after fitting all variables

Permutation types

Permutation within blocks

Line transects or grids - spatial dependence Problem of autocorrelation

N N Roses

Permutation restrictions

Time series or line transect ORIG PERM PERM MIRORP Remove the trend by using position as a covariable

Regression (correlation): r=0.48, P= R 2 =0.230 F=8.368 Permutation test: Test of significance of all canonical axes : Trace = F-ratio = P-value =

We do remove the trends - consequently, this is a good methods for testing of patchily distributed characteristics. Not to be used for trends. (If there is a trend, the data are necessarily autocorrelated.) Repeated measurements are analysed using split plot design.

With trend removal Test of significance of all canonical axes : Trace = F-ratio = P-value =

Hierarchical desing (split plot)

The main plots are permuted Treatment ORIG PERM PERM PERM The subplots of the same main plot have always the same treatment level of the variable tested.

Stepwise selection (forward)

FS summary