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

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

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

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

Example with representation of Oxalis

“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

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

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.

(Hierarchical) classifications “Cluster analysis”

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

Typical data (matrix) Vegetation sample number

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)

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

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

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

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

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.

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

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

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

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

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.

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

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

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?

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

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).

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