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An exploration of the relationship between productivity and diversity in British Grasslands

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.

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An exploration of the relationship between productivity and diversity in British Grasslands

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  1. 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

  2. The unimodal relationship

  3. Oksanen’s intervention

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

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

  6. Example: recording

  7. Example: species richness 8 7 7 6 5

  8. Example: Ellenberg scores

  9. 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 ?

  10. 1. Is the relationship unimodal ?

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

  12. 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

  13. 2. The effect of plot size

  14. Species-area curves

  15. 3. Plot heterogeneity

  16. Example: review

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

  18. Heterogeneity

  19. 4. Parametric modelling

  20. Oksanen’s “no-interaction” model

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

  22. Conclusions

  23. 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

  24. Conclusions Summary of findings • Impact of plot size • Plot heterogeneity • Parametric modelling Problems Adequacy of Ellenberg scores ? Extensions • Mechanistic models ? • Changes over time ? • Can results upon variation be applied to manipulation ?

  25. Acknowledgements Thanks Peter Rothery, David Roy, David Elston, Andy Scott Sources for images Landscapes: The Perthshire 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/

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