Rarefaction and Beta
Diversity
James A. Danoff-Burg
Dept. Ecol., Evol., & Envir. Biol.
Columbia University
Rarefaction
Used to standardize unequal sampling
sizes
First proposed by Sanders and modified by
Hurlbert (1971)
Goal is to scale the larger sample down to
the size of the smaller one
Know what you want to standardize and
rarefy
Number of individuals? Sampling time?
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Rarefaction
After rarefaction, can use a simple
comparison of richness or a simple
richness measure
Margalef (DMg) = (S – 1) / ln N
Menhinick (DMn) = S / √N
Both of these measures are sensitive to
sampling effort
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Rarefaction
Equation for expected # spp in larger
sample
E(S) = S {1 – [(N - Ni over n) / (N over n)]}
Terms
E(S) = expected # of spp in larger sample
n = standardized sample size
N = total # of indiv in sample to be rarefied
Ni = # of indiv in ith spp to be rarefied
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Rarefaction
Problems
Great loss of information in the larger
sample
After rarefaction of the larger sample
• All that is left is the expected number of species
per sample
• Not a real value or real data
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Rarefaction
Alternatives to rarefaction
Randomly select evenly-sized samples
from the larger sample
• Could be done iteratively to provide a
normalized distribution of the expected number
of species
• Akin to the Jackknife values
Kempton & Wedderburn (1978)
• Produce equal sized samples after fitting
species abundances to gamma distribution
• Not commonly used
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Rarefaction
– Worked Example 1
E(S) = S {1 – [(N - Ni combination n) /
(N combination n)]}
Calculate the E(S) for each species in the
larger sample
Sum the E(S) values
= total expected number of species in larger
sample when rarefied to the smaller sample
Calculate Margalef & Menhinick
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Diversity
Main goal of b diversity
Similarity along a range of samples or gradient
All samples are related to each other
Do not have the assumption of separate processes at
different samples
Pseudoreplication less of a concern here
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Diversity
Summary index significance
Higher similarity
•  few species differences between samples
•  lower b diversity values
Lower similarity
•  more species differences between samples
•  higher b diversity values
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Measures
Beta Diversity Indices
Decreasing index values with increasing similarity
Beta Similarity Indices
Increasing index values with increasing similarity
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Diversity
 Six main measures of b diversity
1.
2.
3.
4.
5.
6.
Whittaker’s measure bW
Cody’s measure bC
Routledge’s measure bR
Routledge’s measure bI
Routledge’s measure bE
Wilson & Shmida’s measure bT
More on each index next week
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Differentiating Between b
Diversity Indices
Four main characteristics
Number of community changes
• Or, how many community turnovers were measured
• Similar to differentiation ability
– we know a difference should be there, does the index
detect it?
Additivity
• Does b(a, b) + b (b, c) = b (a, c)?
Independence from Alpha Diversity
• Will two gradients give same b values, even though one
is twice as rich as the other (same abundance)
Independence of sample size
• Will same b value be obtained if have many more
identical samples from one one subsite
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Best b Diversity Measure?
Wilson & Shmida (1984) compared 6 diversity
measures
Number of community changes
 bW was the best of the lot, followed by bT
Additivity
 bC was purely additive, others less so (bT and bW next
best)
Independence from Alpha Diversity
• All but bC passed this test
Independence of sample size
• All but bI and bE passed this tests
Take home: Best index = bW and bT
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Similarity Measures
Other applications of b indexes
Can also use bW measures to look at pairs of
samples
Not necessarily along a gradient
Similarity Coefficients
Jaccard CJ
Sorenson CS
Sorenson Quantitative CN
Morisita-Horn CmH
Cluster Analyses
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Similarity Calculations
First two are simple to calculate and intuitive
Based only on the number of species present in
each sample
All species counted & weighted equally
Jaccard CJ
CJ = j / (a + b – j)
• a = richness in first site, b = richness in second site
• j = shared species
Sorenson CS
CS = 2j / (a + b)
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Similarity Calculations
Sorenson Quantitative CN
Makes an effort to weight shared species by their
relative abundance
CN = 2(jN) / (aN + bN)
jN = sum of the lower of the two abundances
recorded for species found in each site
Error in the text on p. 95 & 165
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Similarity Calculations
Morisita-Horn CmH
Not influenced by sample size & richness
• Only similarity measure that is nots
Highly sensitive to the abd of the most abd sp.
CmH = 2S(ani * bni) / (da + db)(aN)(bN)
• aN = total # of indiv in site A
• ani = # of individuals in ith species in site A
• da = Sani2 / aN2
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Comparing Beta Similarity
Measures
Taylor 1986
Simple, qualitative measures were generally
unsatisfactory
• Jaccard, Sorensen
Morisita-Horn index was among most robust and
useful available
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Beta Similarity Measures on
More than Two Sites
Cluster Analyses
Uses a similarity matrix of all sites
• All sites are on both axes of matrix
• Number of shared species is tallied
Two most similar sites are combined to form a
cluster
Repeated elsewhere until all sites are accounted
and included
Methods of Cluster Analyses
Group Average clustering
Centroid Clustering
Lecture 6 – Rarefaction and Beta Diversity
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu