1. More complex (multidimensional) methods
brief outline of possibilities of selected methods

2. Path analysis
See also SEM (Structural Equation Modelling [e.g. in program Statistica; it is a bit wider conception]), or causal modelling

3. Classic (multidimensional) regression
Many predictors, one response
In reality - long causal chains – in nature: many variable are influenced and influencing at the same time – leads to causal networks

4. E.g. typical hydrobiological model
Carnivorous fishes
Other random factors (e.g. temperature, chemism of water etc.
Planktonophagous fishes
Zooplankton
Phytoplankton

5. Example with representation of Oxalis

6. “Causal modelling”
But causality is introduced by our assumptions about system functioning and not proved by experimental manipulation
Approaches differ according to extend by which our initial model could be “corrected” by our data

7. The method is useful especially there,
where we cannot manipulate (at least some) variables experimentally
Popular in evolutionary biology
But also in ecology (especially on the level of ecosystems and populations in greater spatial scales)
Be careful when interpreting causality

8. Described + intelligible for biologists
Bill Shipley 2004 Cause and Correlation in Biology: A User's Guide to Path Analysis, Structural Equations and Causal Inference. Cambridge University Press.
James B. Grace 2006 Structural Equation Modeling and Natural Systems. Cambridge University Press.

9. (Hierarchical) classifications
“Cluster analysis”

10. Goal for classification
Form groups of objects being internally homogenous, but different from other groups

11. Typical data (matrix)
Vegetation sample number

12. I can classify
vegetation samples according to their similarity in species structure (I get groups of similar vegetation samples – and I can call them somehow then [Seslerietum])
species, according to similarity to each other (correlation) in distribution (I get groups of species with similar ecological requirements)

13. Typical data
I want to obtain groups of similar individual - attention, data are on different scales - and have to be standardized prior to analyzes

14. Classification
Numerical taxonomy, numerical phenetics, cladistic methods
Numeric taxonomy (earlier mainly phenetics), nowadays much broader approach
Cladistics - phylogenetics – construction of phylogenetic trees - in fact, nowadays independent field

15. Classification With learning vs. without learning
Hierarchical vs. non-hierarchical
Hierarchical – divizive vs. agglomerative

16. Cluster analysis
= Hierarchical, agglomerative method, result is a tree:
Principle – first I compute similarity matrix among all pairs, then I construct tree

17. In cluster analysis keep in mind:
It is considerably influenced by so called (dis)similarity measure (also resemblance function). If I have data measured in different scales I have to standartize. Resemblance functions are usually specific for different fields of biology.

18. In cluster analysis keep in mind:
Linkage rule is very important
Defaults in program Statistica (particularly the single linkage, i.e. nearest neighbour) are mostly unsuitable for biological purposes

19. Cluster analysis always forms groups (clusters)
if I don’t want groups, I want just to visualize similarity structure in community structure

20. Ordination:
ordination diagram, where similar vegetation samples are close to each other, similar species close each other and species have their optimums near vegetation samples, where they are present

21. Proximity means similarity
Ordination diagram
Urtica
Chenopodium
Cactus
Nymphea
Menyanthes
Comarum
Aira
Drosera
Proximity means similarity

22. Ordination diagram
Nutrients
Urtica
Chenopodium
Cactus
Nymphea
Menyanthes
Water
Comarum
Aira
Drosera
I can have independent variables – either shown ex post, or as so called constrained ordinations.

23. Various methods
Correspondence analysis, Principal component analysis, factor analysis
Popular in ecology, but also in taxonomy (e.g., could suggest hybridization among species), and in psychology too

24. Constrained ordinations even for analysis of experiments
removed
moss and gob removed
gob removed
control

25. Discrimination analysis
Example: some populations are diploid and some tetraploid – but I can´t count chromosomes every time – I ask – am I able to find some rule on the basis of measured morphological characters (as their linear combination), which discriminate the two ploidities?

26. Discriminating function Z
Ratio of high and length of shell X 100
Discriminating function Z
Ratio of apico-frontal distance and length of shell X 100

27. When applied
beware of circular reasoning (an expert has determined two species for me [mainly on the basis of anther length, but I don’t know this] and then I prove that the two species exist and are well differentiated by their anther length).

28. Similar thing can be achieved by classification trees
Based on another principle (without effect additivity)