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

1. 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 Monkswood

2. Introduction • Context: Productivity-diversity relationships • Motivation: reconstructing and preserving diverse ecosystems. • Previous work: A unimodal relationship is widespread.

3. The unimodal relationship • Maximum diversity is obtained at intermediate productivity. • Lower limb: An increase in productivity leads to an increase in diversity. • Upper limb: Beyond a certain point, increasing productivity leads to a decrease in diversity. • Explanations: Lower-limb easy to explain, upper limb far harder.

4. Oksanen’s intervention

5. Our aims • We try to overcome Oksanen’s objections: • Our dataset is large (n = 1647). • Our dataset uses large quadrats (200m2). • Our dataset contains information on nested quadrats. • We compare the no-interaction model to alternative parametric and nonparametric models. • We investigate heterogeneity.

6. Our dataset • Countryside Survey 2000. • Modified form of stratfied random sampling. • Grassland plots only. • Nested quadrats: 4m2,25m2,50m2 and 100m2 • Diversity: Species richness. • Productivity: Mean Ellenberg fertility score.

7. Statistical methods • The impact of quadrat area: nonparametric regression models. • Testing the no-interaction model: parametric models. • Identifying plot heterogeneity: ANOVA. Resampling methods are needed to assess variability in all 3 cases...

8. Statistical models

9. The no-interaction model

10. Generalizing the no-interaction model • We use a proxy for biomass. • We introduce an intercept term. • We use a full inverse quadratic term above the crowding point. • Increases number of parameters from 3 to 6. • Includes Oksanen’s model as a special case. • Enables a far wider range of behaviour. • Need not follow from Oksanen’s assumptions.

11. Other parametric models MODELS • Normal and Poisson polynomial regression models • Beta response model • Huisman-Olff-Fresco (HOF) models MODEL FITTING Maximum likelihood estimation

12. Non-parametric regression models • General model: • Advantage: Largely data-driven, make less stringent assumptions. • Disadvantage: Largely descriptive, difficult to draw formal inferences

13. The local linear regression estimator FORM OF ESTIMATOR ADVANTAGES • Can be viewed as a generalization of simple linear regression • Degree of bias is independent of data density • Does not suffer from excessive bias at the edge of the covariate space

14. Assessing variability

15. Application of resampling methods • Analytic results are not available, so resampling methods are used to assess variability. • They are used to construct confidence intervals for nonparametric regression estimators. • Also used to construct reference bands about fitted parametric models. • Comparing parametric reference bands to a nonparametric regression estimator gives an assessment of model adequacy for parametric models.

16. Confidence intervals for LLR estimators • Problems with bias • Pointwise confidence intervals • Bootstrap confidence intervals

17. Algorithm to construct reference bands [1] Fit an LLR estimator to the observed data. [2] Fit the parametric model to the data, and calculate fitted values, I, for each site. [3] For the ith site, simulate d times from a Poisson(I) distribution. [4] By simulating data for all sites, construct d simulated datasets. [5] Fit LLR estimators to each of the d resampled datasets. [6] Compare LLR estimator for observed data to that for resampled datasets.

18. Results and conclusions

25. Limitations • Use of mean Ellenberg scores as a proxy for fertility. • Lack of information on abundance. • Lack of scientifically motivated statistical models. • Can the results be used to infer the effect of changing productivity upon diversity ?

26. Conclusions • Unimodal relationship is present. • Overdispersion. • Strength of the relationship increases with quadrat area. • Impact of plot heterogeneity appears to be negligible. • Interpretation of model fit. • Statistical interest: a new application of resampling methods and nonparametric regression.

28. Minitab Macros for resampling methods Adam Butler, Lancaster University Peter Rothery & David Roy, CEH Monks Wood

29. Introduction • Minitab is a commonly used environment for teaching statistics • Minitab has a wide variety of built in functions, but in some areas is severely lacking • A library of Minitab macros to perform resampling versions of standard Minitab functions.

30. Content • Follow the approach in Manly (1997) • Focus on creating resampling versions of standard Minitab functions • Focus on randomization as a means of resampling • Focus on hypothesis testing • Devise some more sophisticated macros, related to bootstrap confidence intervals and spatial statistics.

31. Spatial statistics • Tests for spatial autocorrelation • Mantel test for correlation between two matrices • Construction of EDF plots • Monte Carlo tests using inter-event distances • Mead’s randomization test

32. Running the macros • Invoke from the session window. • Where possible, names are very similar to the names of existing Minitab functions • Names distinguish clearly between randomization, bootstrapping and Monte Carlo methods. • Output is reasonably short, and of a similar form to standard Minitab output. • Subcommands may be used to store additional output.

33. Availability • Macros can be accessed from the CEH products and services website at http://www.ceh.ac.uk/products-services • Macros have also been submitted to the Minitab macro library. • Documentation is provided online. • Sample data and worked examples are provided online.