1. Constrained ordinations Dependence of multivariate response on one or many predictors

2. Linear regression

3. 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

4. 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.

5. 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”.

6. Linear regression

7. 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.

8. Monte Carlo permutation test

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

10. Permutation types

11. Permutation within blocks

12. Line transects or grids - spatial dependence Problem of autocorrelation

13. N N Roses

14. Permutation restrictions

15. Time series or line transect ORIG12345678 PERM167812345 PERM234567812 MIRORP65432187 Remove the trend by using position as a covariable

16. Regression (correlation): r=0.48, P=0.0073 R 2 =0.230 F=8.368 Permutation test: Test of significance of all canonical axes : Trace = 0.230 F-ratio = 8.368 P-value = 0.1540

17. 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.

18. With trend removal Test of significance of all canonical axes : Trace = 0.231 F-ratio = 8.134 P-value = 0.2340

20. 12 87 4 3 56 Hierarchical desing (split plot)

21. The main plots are permuted Treatment00001111 ORIG12345678 PERM134781256 PERM234567812 PERM378345612 The subplots of the same main plot have always the same treatment level of the variable tested.

22. Stepwise selection (forward)

23. FS summary