1. Biological diversity estimation and comparison: problems and solutions W.B. Batista, S.B. Perelman and L.E. Puhl

2. A simple conceptual model of plant-species diversity The rationale of diversity estimation Some essential diversity-estimator functions –Parametric –Non-parametric –Coverage based Assessment of diversity estimators: a modeling exercise

3. S, total diversity d, local diversity Conceptual model a, arrival rate e, local-extinction rate, among-location diversity (heterogeneity)

4. S, total diversity d, local diversity Conceptual model

5. Species’ contribution to: ae highlow high low high

6. Conceptual model Species’ contribution to: ae highlow frequent species Increase d (local diversity) high moderately frequent species low rare species Increase S-d (heterogeneity) high extremely rare species Increase S-d (heterogeneity)

7. Conceptual model density frequency URBAN CORE SATELLITE RURAL d S-d

8. Diversity estimation S, total diversity N, quadrat number D, total number of observed species n i, frequency of species i q(k), number of species for which n i, = k

9. Diversity estimation

13. Diversity-estimator functions S, total diversity D, total number of observed species n i, frequency of species I q(k), number of species for which ni, = k Decreases with increasing density Increases with increasing aggregation High for urban and satellite species Low for rural and core species

14. Diversity-estimator functions Parametric Estimation Based on specific assumptions about the probability distributions of species densities Maximize the Likelihood of the observed q(k) as a function of S and the parameters of the probability distributions of species densities. Non- Parametric Estimation Depend on no assumptions about the probability distributions of species densities e.g. Chao estimator 1First order Jackknife

15. Diversity-estimator functions Coverage-based Estimation Coverage is the sum of the proportions of total density accounted for by all species encountered in the sample. Anne Chao has developed coverage-based estimators by for the general case of unequal densities based on the coverage of infrequent species If all species had equal density, and therefore

16. Diversity-estimator functions A panoply of diversity estimators Parametric –Beta binomial CMLE –Beta binomial UMLE Non-Parametric –Chao 2 –Chao 2 bias corrected –1st order Jackknife –2nd order Jackknife Coverage-based –Model(h) Incidence Coverage Estimator –Model(h)-1 or ICE1 –Model(th) –Model(th)1 Bayesian estimators

17. Assessment of diversity estimators: a modeling exercise 4 scenarios of species density distribution 20 samples of size N=20 per scenario Using program SPADE by Anne Chao to calculate different diversity estimators Summary of estimator performance under all 4 scenarios

18. Modeling exercise S =100, few rare species, no aggregation pattern Scenario 1

19. Modeling exercise Scenario 2 S =100, many rare species, no aggregation pattern

20. Modeling exercise Scenario 3 S =100, few rare species, with aggregation pattern

21. Modeling exercise Scenario 4 S=100, many rare species, with aggregation pattern

22. Modeling exercise Scenario 1

23. No aggregationAggregation Weak dominance Few rare species Strong Dominance Many rare species Observed species number Jackknife Chao ICE Bayesian

24. Modeling exercise Parametric estimators either failed to converge or produced extremely biased results. When no species were very rare and no species had aggregation pattern most estimators worked well, but then so did the naïve estimator D. Some of the coverage-based estimators were relatively robust to the differences among the scenarios we tested.

25. Diversity estimation is a delicate task. It should be aided by assessment of the patterns of species density and aggregation.