1. Rarefaction and Beta Diversity James A. Danoff-Burg Dept. Ecol., Evol., & Envir. Biol. Columbia University

2. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu 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?

3. 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 (D Mg ) = (S – 1) / ln N  Menhinick (D Mn ) = S / √N  Both of these measures are sensitive to sampling effort

4. 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 - N i 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  N i = # of indiv in ith spp to be rarefied

5. 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

6. 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

7. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Rarefaction – Worked Example 1  E(S) = S {1 – [(N - N i 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

8. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Beta Diversity  Main goal of  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

9. 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  diversity values  Lower similarity  more species differences between samples  higher  diversity values

10. 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

11. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Beta Diversity  Six main measures of  diversity  Whittaker’s measure  W  Cody’s measure  C  Routledge’s measure  R  Routledge’s measure  I  Routledge’s measure  E  Wilson & Shmida’s measure  T More on each index next week

12. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Differentiating Between  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  (a, b) +  (b, c) =  (a, c)?  Independence from Alpha Diversity Will two gradients give same  values, even though one is twice as rich as the other (same abundance)  Independence of sample size Will same  value be obtained if have many more identical samples from one one subsite

13. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Best  Diversity Measure?  Wilson & Shmida (1984) compared 6 diversity measures  Number of community changes  W was the best of the lot, followed by  T  Additivity  C was purely additive, others less so (  T and  W next best)  Independence from Alpha Diversity All but  C passed this test  Independence of sample size All but  I and  E passed this tests  Take home: Best index =  W and  T

14. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Beta Similarity Measures  Other applications of  indexes  Can also use  W measures to look at pairs of samples  Not necessarily along a gradient  Similarity Coefficients  Jaccard C J  Sorenson C S  Sorenson Quantitative C N  Morisita-Horn C mH  Cluster Analyses

15. 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 C J  C J = j / (a + b – j) a = richness in first site, b = richness in second site j = shared species  Sorenson C S  C S = 2j / (a + b)

16. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Beta Similarity Calculations  Sorenson Quantitative C N  Makes an effort to weight shared species by their relative abundance  C N = 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

17. Lecture 6 – Rarefaction and Beta Diversity© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu Beta Similarity Calculations  Morisita-Horn C mH  Not influenced by sample size & richness Only similarity measure that is nots  Highly sensitive to the abd of the most abd sp.  C mH = 2  (an i * bn i ) / (da + db)(aN)(bN) aN = total # of indiv in site A an i = # of individuals in ith species in site A da =  an i 2 / aN 2

18. 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

19. 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