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Adapted from Ecological Statistical Workshop, FLC, Daniel Laughlin Properties of Community Data in Ecology 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? Community data matrices or Molecular marker Independent sample units (abundance or presence/absence used as a measure of species performance) Traits SPARSE Full Community Dataset n = # of sample units (plots) p = # of species t = # of traits e = # of environmental variables or factors d = # of dimensions plots in species space nxp nxe plots in envir space nxt plots in trait space traits in species space txp used for species in environmental space (A’E) exp dxp nxd txe plots in reduced species space traits in envir space species in reduced plot space Ordination can address more questions than how plots differ in composition… Species on environmental gradients Gaussian ideal - peak abundances, nonlinear - this is challenging to analyze Linear responses to gradients - okay for short gradients 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 Major Problems with Community Data 1. Zero truncation • species responses truncated at zero • only zeros are possible beyond limits • no info on how unfavorable the environment is for a species 2. “Solid” curves • “curves” are typically solid envelopes rather than curves • species is usually less abundant than its potential (even zeros are possible) Major Problems with Community Data 3. Complex curves -polymodal, asymmetric, discontinuous Average lichen cover on twigs in shore pine bogs in SE Alaska. High beta diversity • Beta diversity = the difference in community composition between communities along an environmental gradient or among communities within a landscape 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: βw < 1 are low, βw > 5 are high 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) Univariate normality Skew = 0 Kurtosis = 0 Normality can be assessed by: skewness (asymmetry), and kurtosis (peakiness) 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) Positively skewed distribution typical of community data HYVI PLHE -0.1 0 .1 .2 .3 .4 .5 .6 .7 0 .05 HYIN .1 .15 -0.2 0 .2 .4 .6 .8 1 Bivariate Normality Views from above Bivariate Species Distributions positive association bivariate distribution is non-linear negative association dust bunny distribution-plotting one species against another (lots of points near orgin and along axes) Idealized Gaussian species response curves Bivariate Species Distributions positive association dust bunny distribution negative association dust bunny distribution Realistic data with “solid” response curves Bi- and Tri-variate Distributions Bivariate normal distribution forms elliptical cloud Multivariate normal distribution (hyperellipsoid) Bivariate distribution with most points lying near one or two axes Multivariate dust bunny distribution Dust bunny in 3-D species space Environmental gradients form strong non-linear shape in species space 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. 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