1. Properties of Community Data in Ecology Adapted from Ecological Statistical Workshop, FLC, Daniel Laughlin

2. Community Data Summary Community data matrices Species on gradients Problems with community data Normality assumptions Key questions to keep in the back of your mind: 1. How do species abundances relate to each other? 2. How do species relate to environmental gradients?

3. Community data matrices or Molecular marker (abundance or presence/absence used as a measure of species performance) Independent sample units Traits SPARSE

4. Full Community Dataset n = # of sample units (plots) p = # of species t = # of traits e = # of environmental variables or factors d = # of dimensions n x pn x en x tn x d t x pt x e e x p plots in species space plots in envir space plots in trait space plots in reduced species space traits in species space used for species in environmental space (A’E) traits in envir space d x p species in reduced plot space Ordination can address more questions than how plots differ in composition…

5. Species on environmental gradients Gaussian ideal - peak abundances, nonlinear - this is challenging to analyze Linear responses to gradients - okay for short gradients

6. Major Problems with Community Data 1.Species responses have the “zero truncation problem” 2.Curves are “solid” due to the action of many other factors 3.Response curves can be complex 4.High beta diversity 5.Nonnormal species distributions

7. Major Problems with Community Data species responses truncated at zero only zeros are possible beyond limits no info on how unfavorable the environment is for a species “curves” are typically solid envelopes rather than curves species is usually less abundant than its potential (even zeros are possible) 1. Zero truncation 2. “Solid” curves

8. Major Problems with Community Data 3. Complex curves -polymodal, asymmetric, discontinuous Average lichen cover on twigs in shore pine bogs in SE Alaska.

9. High beta diversity Beta diversity = the difference in community composition between communities along an environmental gradient or among communities within a landscape

10. Whittaker’s (1972) Beta Diversity γ = number of species in composite sample (total number of species) ά = average species richness in the sample units No formal units, but can be thought of as ‘number of distinct communities” The one is subtracted to make zero beta diversity correspond to zero variation in species turnover. Rule of thumb: β w 5 are high

11. Are species distributions normal? Univariate normality (it’s what we’re used to) Bivariate normality (it’s easy to visualize) –Idealized community data –Real community data Multivariate normality (straightforward extension of bivariate normality to multiple dimensions)

12. Univariate normality Normality can be assessed by: skewness (asymmetry), and kurtosis (peakiness) Skew = 0 Kurtosis = 0

13. Skewness Community data will nearly always be positively skewed due to lots of zeroes Linear models require |skew| < 1 Assess skewness of data in PCORD (Row and Column Summary)

14. Positively skewed distribution typical of community data PLHE HYVI HYIN

15. Bivariate Normality Views from above

16. Bivariate Species Distributions Idealized Gaussian species response curves positive association negative association bivariate distribution is non-linear dust bunny distribution- plotting one species against another (lots of points near orgin and along axes)

17. Bivariate Species Distributions Realistic data with “solid” response curves positive association negative association dust bunny distribution

18. Bi- and Tri-variate Distributions Bivariate normal distribution forms elliptical cloud Bivariate distribution with most points lying near one or two axes Multivariate normal distribution (hyperellipsoid) Multivariate dust bunny distribution

19. Dust bunny in 3-D species space Environmental gradients form strong non-linear shape in species space

20. A: cluster within the cloud of points (stands) occupying vegetation space. B: 3 dimensional abstract vegetation space: each dimension represents an element (e.g. proportion of a certain species) in the analysis (X Y Z axes). A, the results of a classification approach (here attempted after ordination) in which similar individuals are grouped and considered as a single cell or unit. B, the results of an ordination approach in which similar stands nevertheless retain their unique properties and thus no information is lost (X1 Y1 Z1 axes). Key Point: Abstract space has no connection with real space from which the records were initially collected.

21. Multivariate Normality Linear algebra easily extends these concepts into multiple dimensions Most multivariate methods assume multivariate normality (linear ordination methods) Ecological data are seriously abnormal Thus, we will often require different methods