an exploration of the relationship between productivity and diversity in british grasslands
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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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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The unimodal relationship
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Oksanen’s intervention
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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
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Example: nested quadrats
4m2
25m2
100m2
200m2
50m2
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Example: recording
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Example: species richness
5
6
7
8
7
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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
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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 ?
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1. Is the relationship unimodal ?
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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
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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
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2. The effect of plot size
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Species-area curves
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3. Plot heterogeneity
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Example: review
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Example: heterogeneity test
(2,1,2,2,2) (2)
(1)
(3)
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Heterogeneity
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4. Parametric modelling
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Oksanen’s “no-interaction” model
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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
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Conclusions
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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
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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
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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/