Choosing Between Diversity Indices James A. Danoff-Burg Dept. Ecol., Evol., & Envir. Biol. Columbia University.

Choosing Between Choosing Between Diversity IndicesDiversity Indices

James A. Danoff-Burg

Dept. Ecol., Evol., & Envir. Biol.

Columbia University

Alpha Diversity IndicesAlpha Diversity Indices A diversity of diversities

Log Alpha Log-Normal Lambda Q-Statistic Simpson McIntosh Berger-Parker Shannon-Wiener Brillouin

How to choose between these?

Bases for ChoiceBases for Choice Appropriateness of each index for your data Discriminant ability of the index Statistical Comparability Widespread utility of the index Your Question

Appropriateness Appropriateness Index assumptions need to be met Abundance model of data Sensitivity to sample size Each index needs to be considered for all of

these aspects determines whether can be used for your data

AssumptionsAssumptions Alpha diversity indices do not make many

assumptions No assumptions made about species abundance

distributions• Cause of distribution not needed

– species abundance models have assumptions about these» Geometric – niche pre-emption, regular arrivals» Log – niche pre-emption, irregular arrival intervals» Log-Normal – successively apportioning available niche

space of all resources in proportion to abundance» Broken Stick – simultaneously apportioning available niche

space of one resource in proportion to abundance

• Shape of curve “Non-parametric”

• Normality is not needed

Abundance ModelAbundance Model Some indices perform better under a specific

abundance model Example: Simpson – probability that two

individuals are of the same species Geometric

• Underestimate Simpson diversity value Log

• Underestimate Simpson diversity value Log-Normal

• Best for Simpson diversity analysis analysis Broken Stick

• May overestimate Simpson diversity value

Abundance Model FittingAbundance Model Fitting Main problem: Often have multiple models

that fit the data Occasionally because of low number of

abundance classes Log2 has only 11 classes (octaves) even possible

• Most data have less than 11 • E.g., less than 256 individuals in a species

– Resulting in only 8 classes Fewer classes, mean fewer opportunities for

departures from fit Small data sets fit many models

• Few spp in each abundance class decreased discriminability

Abundance Model FittingAbundance Model Fitting Secondary problem: Log-Normal is a frequent

consequence Often because of the central limit tendency of

large data sets If a data set has many species often log-normal

distribution results Does not necessarily mean that the community

has assembled by a successive breaking of the available nice space

• As is the assumption with the log-normal distribution

Appropriateness – Sample Appropriateness – Sample EffortEffort

Some indices are tremendously sensitive to sample size Low replication skewed values

• Idiosyncratic results

• Not truly representative of the environment

Indices sensitive to inadequate sampling S = very sensitive to sampling effort Dominance indices (Simpson, Berger-Parker, McIntosh) Information statistics indices (Shannon) Evenness indices

Indices insensitive to sampling effort Always: Log series , 1/d (influenced by abd of most abd sp) If more than 50% of spp represented: Q

Appropriateness - Sampling Appropriateness - Sampling EffortEffort

How to determine when you have completely sampled the environment?

Assuming prior information• Leveling off of S with adding more samples

– If interest is largely richness

• Leveling off of Pielou’s t point– If interest is proportional abundance

Leveling off point = adequate sample size

Appropriateness, Sample Appropriateness, Sample Size - When is Enough Size - When is Enough

Enough?Enough? Leveling off of S with adding

more samples If interest is largely richness S is more sensitive to sample

size than diversity indices Need more samples

Leveling off of Pielou’s t point

If interest is proportional abundance

Any diversity index can be used

10 20 30 40

H’ Shannon

Diversity Index Value

Sample Addition Sequence

1/D Simpson

1/d Berger-Parker

Pielou’s t

Sampling EffortSampling Effort Need consistency in sampling effort

Need to use the same effort throughout experiment• Helps to ensure comparability of indices• All would then be equal(ly biased)

When sample sizes are unequal? Rarefy the larger sample to the smaller More next week on rarefaction (WE 1)

Better to have many small samples than few large samples

Increases replication Decreases thoroughness of each replicate

Discriminant Ability of the Discriminant Ability of the IndexIndex

Differences are usually very subtle Need analytical rigor to differentiate

Assuming differences exist, how best to see them?

No two sites are identical in terms of S, N, relative abundance

All sites will differ, how can we detect this?

Studies of Discriminability IStudies of Discriminability I Taylor (1978)

Using 8 indices on moths, 9 sites, over 4 years• Rothamsted Insect Survey, England

Best: Log (by far) Next: H’, S, log-normal , 1/D, log biomass Useless: log-normal S* and

Kempton (1979) Using 4 indices on same data, 14 sites, 7 years Best: S, H’ Useless: 1/D, 1/d

Studies of Discriminability IIStudies of Discriminability II Kempton & Taylor (1976)

Transformed indices > untransformed form exp H’ > H’ 1 / D > D

Kempton & Wedderburn (1978) and Q > any H’ and any D

Magurran (1981) Best: Margalef (Dmg = (S-1) / ln N); U, S Less well: HB > H’, exp H’ Worst: 1/d, D or 1/D, McIntosh D, H’ E, HB E

Morris & Lakhani (1979) H’ > D or 1/D

Diversity Aspects MeasuredDiversity Aspects Measured Not just discriminability, but what is the index

measuring? Richness, Dominance, Evenness, Abundance…

What most affects the index? Rare species or species richness?

• Type 1 Measures• log , log-normal , Q, S, H’, HB, Dmg, McU

Abundance of the most common species or dominance?

• Type 2 Measures• 1/D or D, 1/d, McD, H’E, HBE

Within each type significant correlation

Statistical ComparabilityStatistical Comparability Historically, statistical comparisons not made

Mostly descriptive comparisons in past

Statistical Options H’, Var H’, t-test Replication

• Most sets of replicated estimates are normally distributed• Non-normal data can be transformed for normality

Jackknife data• Provides standard error and confidence limits as well

Widespread Utility of the Widespread Utility of the IndexIndex

Important for ensuring comparability Between studies Between sites Between researchers

Most commonly used S, H’, 1/D, log

Less commonly used Log-Normal , Q

• Even though highly valuable as discussed above

Dmg, McU, McE, HB, 1/d

Widespread Utility - CautionsWidespread Utility - Cautions Be careful with the H’ Shannon

Heavily criticized, despite widespread use “no direct biological interpretation” (Goodman 1975)

Be careful with the log Based only on S & N

• Insensitive to changes when both stay constant• Uncommon situation

“there can be no universal best buy but there are rich opportunities for inappropriate usages” (Southwood 1978)

Indices Performance SummaryIndices Performance Summary

IndexDiscriminant Ability

Sample Size Sensititivy

Richness, Evenness, Dominance Calculation

Widely used?

Sensitivity to Abd models

Log Good Low Richness Simple Yes N (?)

Log Normal Good Moderate Richness Complex N Yes

Q Good Low Richness Complex N N

S Good High Richness Simple Yes N

Margalef Good High Richness Simple N N

Shannon Moderate Moderate Richness Intermediate Yes N

Brillouin Moderate Moderate Richness Complex N N

McIntosh U Good Moderate Richness Intermediate N N

Simpson Moderate Low Dominance Intermediate Yes Yes

Berger-Parker Poor Low Dominance Simple N N

Shannon E Poor Moderate Evenness Simple N N

Brillouin E Poor Moderate Evenness Complex N N

McIntosh D Poor Moderate Dominance Simple N N

= Desirable Traits in an Index

Index Choice GuidelinesIndex Choice Guidelines1. Clearly formulate question you are studying

2. Ensure equal sample sizes

3. Draw a Rank Abundance graph

4. Calculate Margalef (Richness) and Berger-Parker (Dominance) indices

5. Determine Log and Q

6. Test fit to abundance models

7. Use ANOVA to test for treatment differences

8. Use Jackknife to improve estimate of indexes

9. Be consistent in choice of index across your studies

Your QuestionYour Question What you want to know determines how you

analyze your data How important is each aspect of diversity?

Richness? Evenness? Dominance? Abundance? Per-species (relative) abundance? Taxon diversity? Trophic structuring? Guild diversity?

Your QuestionYour Question What answers your question?

What are the most important aspects of diversity? What data directly addresses your question?

• How should it be presented?• How should it be emphasized?

Your QuestionYour Question Work through the gardens example

Abundance model? Diversity aspects? Form of the data? Appropriate analyses? Answer for a few subcomponent questions

Assignment: Do above using your thesis (or the gardens data) 3-5 pages Due 2nd April before class

Studies of Discriminability III Studies of Discriminability III – Gotelli & Colwell 2001– Gotelli & Colwell 2001

Purpose / Goal To discuss the different ways of presenting richness To discuss the ways in which we can approximate total

species richness To discuss the difficulties encountered when using

richness• Proportional abundances• Species Density• Standardizing number of species across different sized areas• Species / Genus

– Important in biogeography

Gotelli & Colwell 2001Gotelli & Colwell 2001 Differentiates between Individual-based and

Sample-based assessment methods Individual: life lists, Christmas bird counts, collector’s

curves Sample: replicated quadrats, mist nets, trap data Hybrid: m-species lists (observing to a point)

Differentiates between accumulation and rarefaction curves (either individual or sample based)

Accumulation – total # of spp during process of data collection

Rarefaction – repeatedly subsampling without replacement from the data (progressive data reduction)

Gotelli & Colwell 2001Gotelli & Colwell 2001 Samples always below

Individuals Individuals are clumped Random assortments of traps

through time unclumped

Rarefaction smoothed curves

Replicated data removal process (like Pielou)

Built “right to left”

Accumulation stepped line Observed fact Built “left to right”

10 20 30 40Sample Addition Sequence

Richness

Samples: AccumulationSamples: Accumulation

Samples: RarefactionSamples: Rarefaction

Individuals: AccumulationIndividuals: AccumulationIndividuals: Individuals: RarefactionRarefaction

Gotelli & Colwell 2001Gotelli & Colwell 2001 Factors influencing richness estimates

Underlying species richness Relative abundance distributions Sampling effort

Gotelli & Colwell 2001Gotelli & Colwell 2001 Cautions when using richness

Scaling different sized areas to species density• Susceptible to non-linearity of increasing area and

richness• Leads to an inability to extrapolate from smaller to larger

areas Species per comparable unit area

• Problem with non-linear relationship between sampling efforts and richness

Species per genus

Gotelli & Colwell 2001Gotelli & Colwell 2001 Solution?

Use rarefaction on your data to standardize sampling effort

Bring larger sample down to size of smallest one

Hypothetical Model CurvesHypothetical Model Curves

10 20 30 40

Geometric SeriesLog Series

Log-Normal Series

Broken Stick Model

Per Species

Abundance

Species Addition Sequence

Choosing Between Diversity Indices James A. Danoff-Burg Dept. Ecol., Evol., & Envir. Biol. Columbia University.

diversity indices james

data slide

environment indices

lognormal distribution

proportional abundance

q slide

species abundance models

discriminability slide

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