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 CS 2000. Modified form of

stratfied random sampling.

Nested quadrats. Grassland plots only.

Variables Species richness Plot-averaged

Ellenberg fertility scores

Example: nested quadrats

4m2

25m2

100m2

200m2

50m2

Example: recording

Example: species richness

5

6

7

8

7

Example: Ellenberg scores

Species Nest level Ellenberg scoreCalluna vulgaris 1 2Erica tetralix 1 2Erica cinerea 1 1Ulex gallii 1 2Molinia caerulae 1 2

Potentilla erecta 2 2

Agrostis curtisii 3 1

Salix repens agg. 5 3

Plot Ellenberg score: 1.875

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 approachesLocal polynomial regression

Nadaraya-Watson estimator Local linear regressionLOESS

Smoothing splines / GAMs Orthogonal projection approaches

Fourier methodsWavelets

Inferenceo Local likelihoodo Penalized likelihood

Local polynomial regressionModel• 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 modelsParametric 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

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 Impact of plot size Plot heterogeneity Parametric modelling

ProblemsAdequacy of Ellenberg scores ?

Extensions Mechanistic models ? Changes over time ? Can results upon variation be applied to

manipulation ?

Conclusions

AcknowledgementsThanks

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/