Choosing Between Choosing Between Diversity Indices Diversity Indices James A. Danoff-Burg Dept. Ecol., Evol., & Envir. Biol. Columbia University
Mar 31, 2015
Choosing Between Choosing Between Diversity IndicesDiversity Indices
James A. Danoff-Burg
Dept. Ecol., Evol., & Envir. Biol.
Columbia University
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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?
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
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H’ Shannon
S
Diversity Index Value
Sample Addition Sequence
1/D Simpson
1/d Berger-Parker
Pielou’s t
Pielou’s t
Pielou’s t
Pielou’s t
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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?
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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)
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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?
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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?
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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)
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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”
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Richness
Samples: AccumulationSamples: Accumulation
Samples: RarefactionSamples: Rarefaction
Individuals: AccumulationIndividuals: AccumulationIndividuals: Individuals: RarefactionRarefaction
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Gotelli & Colwell 2001Gotelli & Colwell 2001 Factors influencing richness estimates
Underlying species richness Relative abundance distributions Sampling effort
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Gotelli & Colwell 2001Gotelli & Colwell 2001 Solution?
Use rarefaction on your data to standardize sampling effort
Bring larger sample down to size of smallest one
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Lecture 5 – Choosing Between Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Hypothetical Model CurvesHypothetical Model Curves
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Geometric SeriesLog Series
Log-Normal Series
Broken Stick Model
Per Species
Abundance
Species Addition Sequence