Biological diversity estimation and comparison: problems and solutions

W.B. Batista, S.B. Perelman and L.E. Puhl

• 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

S, total diversity

d, local diversity

Conceptual model

a, arrival rate

e, local-extinction rate

ddSd

eat

d-S , among-location diversity(heterogeneity)

S, total diversity

d, local diversity

Conceptual model

ddSd

eat

Sdeaa

Sd-Seae

Conceptual model

Species’ contribution to:

a ehigh low

high

low low

high

Conceptual model

Species’ contribution to:

a ehigh low frequent species

Increase d (local diversity)

high moderately frequent species

low low rare species

Increase S-d (heterogeneity)

high extremely rare species

Increase S-d (heterogeneity)

Conceptual model

density

frequency

URBAN

CORE

SATELLITE

RURAL d

S-d

Diversity estimation

S, total diversity

N, quadrat number

D, total number of observed species

ni, frequency of species i

q(k), number of species for which

ni, = k

N

q(k)S0k

Diversity estimation

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Observed frequency, k

Sp

ec

ies

nu

mb

er,

q(k

)

N

q(k)D1k

Diversity estimation

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Observed frequency, k

Sp

ec

ies

nu

mb

er,

q(k

)

N

q(k)D1k

Diversity estimation

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Observed frequency, k

Sp

ec

ies

nu

mb

er,

q(k

)

q(0)DS

Diversity estimation

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Observed frequency, k

Sp

ec

ies

nu

mb

er,

q(k

)

q(0)DS

q(0)DS Eˆ

Diversity-estimator functionsS, total diversity

D, total number of observed species

ni, frequency of species I

q(k), number of species for which ni, = k

q(0)DS Eˆ

S

Probq(0)1

0i

inE

• Decreases with increasing density

• Increases with increasing aggregation

• High for urban and satellite species• Low for rural and core species

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.

q(2)2

q(1)q(0)

2E

Chao estimator 1First order Jackknife

q(1)N

1-Nq(0) E

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

SD

C CD

Sˆ

ˆ

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

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

Modeling exercise

S=100, few rare species, no aggregation pattern

Scenario 1

0

0.1

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0.3

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0.9

1

0 20 40 60 80 100

Species Rank

Exp

ecte

d F

req

uen

cy

0

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0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

DensityE

xpec

ted

Fre

qu

ency

Modeling exercise

Scenario 2

0

0.1

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0 2 4 6 8 10

DensityE

xpec

ted

Fre

qu

ency

0

0.1

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0.3

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0 20 40 60 80 100

Species Rank

Exp

ecte

d F

req

uen

cy

S=100, many rare species, no aggregation pattern

Modeling exercise

Scenario 3

S=100, few rare species, with aggregation pattern

0

0.1

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0.3

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Species Rank

Exp

ecte

d F

req

uen

cy

0

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0.7

0.8

0.9

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0 1 2 3 4 5 6

DensityE

xpec

ted

Fre

qu

ency

Modeling exercise

Scenario 4

S=100, many rare species, with aggregation pattern

0

0.1

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0.3

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0 20 40 60 80 100

Species Rank

Exp

ecte

d F

req

uen

cy

0

0.1

0.2

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0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

DensityE

xpec

ted

Fre

qu

ency

Modeling exercise

d N 1 N 2 N 3 N 4 C 1 C 2 C 3 C 4 B5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

1 5 0E

sti

ma

ted

Div

ers

ity

Scenario 1

No aggregation Aggregation

Weak dominance

Few rare species

Strong Dominance

Many rare species

D J k 1 C h IC E B

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

1 5 0

Es

tim

ate

d D

ive

rsit

y

D J k 1 C h IC E B

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

1 5 0

D J k 1 C h IC E B

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

1 5 0

Es

tim

ate

d D

ive

rsit

y

D J k 1 C h IC E B

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

1 5 0

Observed species number Jackknife

Chao

ICE

Bayesian

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.

• Diversity estimation is a delicate task.

• It should be aided by assessment of the patterns of species density and aggregation.