An exploration of the relationship between productivity and diversity in British Grasslands Adam Butler & Janet 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 u Modified form of stratfied random sampling. u Nested quadrats. u Grassland plots only. Variables u Species richness  Plot-averaged Ellenberg fertility scores

Example: nested quadrats 4m2 25m2 100m2 200m2 50m2

Example: recording

Example: species richness

Example: Ellenberg scores

Aims of the analysis  Is there a unimodal relationship ?  Is the relationship maintained as we increase plot size ?  Do our large plots suffer from heterogeneity ?  Does the no-interaction model provide a reasonable fit ?

1. Is the relationship unimodal ?

Non-parametric regression Possible approaches l Local polynomial regression l Nadaraya-Watson estimator l Local linear regression l LOESS l Smoothing splines / GAMs l Orthogonal projection approaches l Fourier methods l Wavelets Inference o Local likelihood o Penalized likelihood

Local polynomial regression Model Evaluation points Locally weighted polynomial regression Weighing: kernel function Complexity of kernel function: bandwidth Issues: bias Local linear regression A generalization of simple linear regression Degree of bias is independent of data density Inference Local likelihood Bandwidth selection Confidence intervals

2. The effect of plot size

Species-area curves

3. Plot heterogeneity

Example: review

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

Heterogeneity

4. Parametric modelling

Oksanen’s “no-interaction” model

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

z Conclusions

Statistical extensions Nonparametric regression models Alternative plot level Ellenberg fertility scores Bias correction Poisson local likelihood estimation Formal test for parallelism Parametric regression models Pseudo likelihood ratio test Formal test for smooth v sharp transition

Summary of findings lImpact of plot size lPlot heterogeneity lParametric modelling Problems Adequacy of Ellenberg scores ? Extensions lMechanistic models ? lChanges over time ? lCan results upon variation be applied to manipulation ? Conclusions

Acknowledgements Thanks Peter Rothery, David Roy, David Elston, Andy Scott Sources for images Landscapes: The Perthshire Herbarium No-interaction model: Homepage of Jari Oksanen Species-area curves: University of Oklahoma, BISC3034 website