Productivity and Diversity in British Grasslands - An Exploration

2. 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

3. The unimodal relationship

4. Oksanen’s intervention

5. 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

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

7. Example: recording

8. Example: species richness 8 7 7 6 5

9. Example: Ellenberg scores

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

11. General statistical approach • Formulation of explicit statistical models. • Likelihood-based inference, using R. • Assessment of variability using resampling methods. • Use of diagnostics.

12. The effects of plot size

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

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

17. Assessment of uncertainty

18. 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 • It may not exist • The assumptions underlying it may not be reasonable Alternative • Assume that the null hypothesis is true, simulate data under this hypothesis, and calculate the statistic for each of these simulated datasets.

19. 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)

21. Plot heterogeneity

22. Example: review

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

24. Heterogeneity

25. Parametric modelling

26. No-interaction model

27. Oksanen’s fit

28. Piecewise polynomial model

29. 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

31. Conclusions

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

33. 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: • http://www.pkc.gov.uk/herbarium/ • http://www.pitt.edu/~medart/