An exploration of the relationship between productivity and

An exploration of the relationship between productivity and diversity in British Grasslands Adam Butler & Jan Heffernan, Lancaster University Department of Mathematics & Statistics Simon Smart, CEH Merlewood

The unimodal relationship

Oksanen’s intervention

Our dataset Source of the data u CS 2000. u Modified form of stratfied random sampling. u Nested quadrats. u Grassland plots only. Variables u Species richness u Plot-averaged Ellenberg fertility scores

Example: nested quadrats 200 m 2 100 m 2 50 m 2 25 m 2 4 m 2

Example: recording

Example: species richness 8 7 7 6 5

Example: Ellenberg scores

Aims of the analysis 1 Is there a unimodal relationship ? Is the relationship maintained as we increase plot size ? 2 How can we quantify our uncertainty about this relationship ? 3 Do our large plots suffer from heterogeneity ? 4 Does the no-interaction model provide a reasonable fit ?

General statistical approach 1 Formulation of explicit statistical models. 2 Likelihood-based inference, using R. 3 Assessment of variability using resampling methods. 4 Use of diagnostics.

The effects of plot size

Non-parametric regression Possible approaches… l Local polynomial regression l l l Local average Local linear regression LOESS Smoothing splines GAMs Orthogonal projection approaches l l Fourier methods Wavelets Advantages l Data-driven l Assume no process knowledge Disadvantages l Cannot incorporate process information l Formal inference is difficult l Computationally-intensive to fit

Local linear regression Model l Linear regression at each point l Weighting function - the “kernel function” l Smoothness parameter - “bandwidth” Inference u Local log-likelihood Advantages : A generalization of simple linear regression : Degree of bias is independent of data density

Assessment of uncertainty

Resampling methods The situation We are interested in assessing the statistical properties of a statistic (i. e. a function of the data) under the assumption that some null hypothesis is true. Problems with standard theory l It may not exist l The assumptions underlying it may not be reasonable . Alternative l Assume that the null hypothesis is true, simulate data under this hypothesis, and calculate the statistic for each of these simulated datasets.

Minitab Macros for resampling methods Hypothesis tests Confidence intervals Regression ANOVA Spatial statistics Co-workers Peter Rothery David Roy Hannah Butler Downloadable from the CEH website (under products & services)

Plot heterogeneity

Example: review

Example: heterogeneity test (2, 1, 2, 2, 2) (1) (3)

Heterogeneity

Parametric modelling

No-interaction model

Oksanen’s fit

Piecewise polynomial model

Fitting parametric models Parametric models F Piecewise polynomial model F Poisson polynomial regression models F Beta response model F Huisman-Olff-Fresco (HOF) models Comparison of models T Likelihood ratio tests (nested models) T Akaike Information Criterion (non-nested models) Performance F Beta response model performs badly F Models with more parameters perform significantly better

Conclusions z

Conclusions Summary of findings 1. Impact of plot size 2. Plot heterogeneity 3. Parametric modelling Extensions 1. Mechanistic models ? 2. Changes over time ? 3. Can results upon variation be applied to manipulation ?

Further information E-mail: a. butler@lancaster. ac. uk Web: http: //www. maths. lancs. ac. uk/~butler/diversity Questions ? ? Sources for images used in the presentation: z http: //www. pkc. gov. uk/herbarium/ z http: //www. pitt. edu/~medart/