Class
1. Logical and philosophical frameworks for the analysis of ecological complexity
- Falsification and the hypothetic-deductive approach
- Strong inference
- Bayesian inference

2. Sampling populations
- Ecological variables
- Frequency distributions
- Parameters and their estimates
- Precision and accuracy of sample estimates

3. Relationships among ecological models, statistical models and data
- Estimation and hypothesis testing
- Linear statistical models
- Methods of estimation: OLS and ML
- Statistical hypothesis testing: general hints

4. Experimental design
- Basic concepts: replication, randomisation, independence
- Choosing levels for predictor variables: fixed vs. random factors
- Relationships among predictor variables: hierarchical and factorial designs
- Extension to multifactorial designs

5. Applications
- Hierarchical designs: solution to spatial and temporal confounding
- Hierarchical designs: sampling at multiple scales in space and time
- Factorial designs: understanding interactions among predictor variables
- Assessing the generality (or lack thereof) of ecological processes: multifactorial experiments

Lab (computer- based exercises)
A- Sampling populations: influence of variance and sample size on sample estimates
B- Estimating spatial and temporal variance: the analysis of pattern using simulated and real data sets
C- Understanding ecological processes: analysis and interpretation of real ecological experiments

MULTIVARIATE COMMUNITY ANALYSES USING PRIMER
(Dr. K.R. Clarke, Dr. R.N. Gorley, Plymouth Marine Laboratory, UK)

Class
1. Measures of resemblance (similarity/dissimilarity/distance) in multivariate structure for assemblage & environmental data, including pre-treatment options (standardisation, transformation, normalisation) and the effects of different coefficient choices
2. Hierarchical clustering of samples, including different linkage options (CLUSTER)
3. Ordination (of environmental data) by Principal Components Analysis (PCA)
4. Ordination (of assemblage data) by non-metric Multi-Dimensional Scaling (MDS)
5. Multivariate testing for differences between groups of samples (1- and 2-way crossed and nested ANOSIM), and comments on power
6. Determining variables which discriminate groups of samples (1- and 2-way similarity percentages, SIMPER), both for species and environmental variables
7. Diversity measures (DIVERSE) and comments on sampling properties and multivariate treatment of multiple indice. Dominance plots and tests for differences between sets of curves (DOMSIM), particle-size distributions etc
8. Taxonomic (or phylogenetic) diversity and distinctness for quantitative data, or simple species lists, as valid biodiversity measures (DIVERSE) over broad spatial and temporal scales; comments on sampling properties and testing structures (TAXDTEST)
9. Linking potential environmental drivers to an observed assemblage pattern, via bubble plots, the matching of multivariate structures (the BIO-ENV procedure), and linkage trees (LINKTREE, a 'classification and regression tree' approach)
10. Global hypothesis tests I: of no agreement between two resemblance matrices (RELATE), comparing assemblage (or environmental) structure with linear (seriation) or cyclic models in space and time; also of no evidence for a biota-environment link, allowing for the selection effects in finding an optimum match (the global BIO-ENV test)
11. Stepwise form of the BIO-ENV routine (BVSTEP) generalised to other comparisons, e.g. species subsets determining overall assemblage pattern, species best delineating modelled or observed environmental gradients, environmental variables best acting as 'proxy' for the full set
12. Second-stage analysis (2STAGE) to compare taxonomic levels and transformation or coefficient choices; also for a possible testing framework in some repeated measures designs
13. Widening the scope of assemblage resemblance measures: improving the signal to noise ratio by variable down-weighting of species whose individuals arrive in the sample in clusters (Dispersion weighting); exploiting ideas of taxonomic distinctness to define new similarity coefficients appropriate for data with few, or no, species in common (Taxonomic dissimilarity)

Lab (computer- based exercises)
A- Clustering and ordination (simple hierarchical clustering and Principal Components Analysis/non-metric Multi-Dimensional Scaling)
B- Multivariate ANOSIM tests (simple 1- and 2-way layouts) for determining and quantifying differences between groups of samples
C- 1- and 2-way SIMPER
D- DIVERSE, dominance plots and testing sets of curves (DOMSIM)
E- DIVERSE and TAXDTEST
F- Draftsman plots (to assess variable transforms), PCA, BIO-ENV and LINKTREE
G- ANOSIM for 2-way without replication, RELATE and the global BIO-ENV test
H- BEST (the combined BIO-ENV/BVSTEP routine) and 'own data' session
I- Lab session on 2STAGE, and 'own data' session
L- SIMPROF structure tests in CLUSTER, and using 2STAGE to compare different similarity coefficients (including dispersion weighted, hierarchy based etc)


ANALYSING MULTI-SPECIES RESPONSES TO COMPLEX EXPERIMENTAL DESIGNS
(Dr. M. J. Anderson, University of Auckland, New Zealand)

Class
1. Partitioning variation for a linear model based on any distance measure: permutational multivariate analysis of variance (PERMANOVA) and permutational multivariate multiple regression (PERMREG).
- Analysing complex multi-factor models and analyses using permutational and Monte Carlo approaches; Principal coordinate analysis (PCO, or metric MDS) as an unconstrained ordination method for viewing patterns; Generalised solution for multiple regression based on linear partitioning of principal coordinates for any linear model; Solution for MANOVA based on inter-point distances; Interpreting multivariate interaction terms; Comparison with other methods, including assumptions; Estimating pseudo multivariate variance components based on distances.

2. Permutation tests for complex experimental designs (ANOVA, multiple regression).
- Permutation of residuals under a reduced or a full model; Restricted permutations and exact versus asymptotically exact tests; Choosing the correct units to permute by reference to expected mean squares; What to do when there aren't enough permutations; Monte Carlo approximation; Permutational multivariate multiple regression (PERMREG); Testing for relationships between species data and one or more predictor variables; Building a linear model, using sets of predictor variables and covariables and choosing correct permutation strategies.

3. Canonical analysis of principal coordinates (CAP), a constrained ordination method.
- Unconstrained versus constrained ordination techniques - distinctions and potential pitfalls; Generalised discriminant analysis based on distances; Cross-validation check on arbitrariness of results; Correlations with species variables; Comparison with other methods, including PERMANOVA; Monte Carlo approximation; Generalised canonical correlation analysis based on distances; Using canonical axes to place new observations along an environmental gradient.

4. Permutational test for differences in multivariate dispersions among groups (PERMDISP).
- Testing for differences in group dispersions versus tests of location in multivariate space; Distances from centroids and non-independence; Defining the multivariate space for the hypothesis of interest; The effects of dissimilarity measures and transformations on relative dispersions; Interpretations by reference to PERMANOVA.

5. Putting it all together - a general approach for the analysis of species assemblages in complex experimental designs.
- Unconstrained and constrained ordination; Hypothesis-testing and multivariate models; Finding individual species related to significant multivariate patterns.

6. Environmental impact assessment and monitoring
- Environmental impact assessment; Building models and analysing multivariate data from BACI and beyond-BACI experimental designs using PERMANOVA and PERMREG; Environmental monitoring; Multivariate control charts based on distances, with bootstrap confidence bounds.

Lab (computer- based exercises)
A- PCO (metric MDS) and two-way PERMANOVA and PERMDISP.
B- PERMREG and building multivariate linear models.
C- CAP; Generalised discriminant analysis and canonical correlation analysis based on distances.
D- PERMANOVA for complex designs and multiple comparisons; A general strategy for analysis.
E- Environmental impact assessment using PERMANOVA and PERMREG; using the CONTROLCHART program for environmental monitoring.