2. 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

3. The unimodal relationship

4. Oksanen’s intervention

5. 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  Plot-averaged Ellenberg fertility scores

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

7. Example: recording

8. Example: species richness 5 6 7 8 7

9. Example: Ellenberg scores

10. 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 ?

11. 1. Is the relationship unimodal ?

13. 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

14. 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

16. 2. The effect of plot size

18. Species-area curves

20. 3. Plot heterogeneity

21. Example: review

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

23. Heterogeneity

24. 4. Parametric modelling

25. Oksanen’s “no-interaction” model

26. 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

28. z Conclusions

29. 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

30. 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

31. Acknowledgements Thanks Peter Rothery, David Roy, David Elston, Andy Scott Sources for images Landscapes: The Perthshire Herbarium http://www.pkc.gov.uk/herbarium/ http://www.pkc.gov.uk/herbarium/ No-interaction model: Homepage of Jari Oksanen http://cc.oulu.fi/~jarioksa/ Species-area curves: University of Oklahoma, BISC3034 website http://www.okstate.edu/artsci/botany/bisc3034/