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

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Presentation on theme: "Alpha Diversity Indices James A. Danoff-Burg Dept. Ecol., Evol., & Envir. Biol. Columbia University."— Presentation transcript:

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

2 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

3 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Diversity of Diversities  Difference between the diversities is usually one of relative emphasis of two main envir. aspects  Two key features  Richness  Abundance – our emphasis today  Each index differs in the mathematical method of relating these features  One is often given greater prominence than the other  Formulae significantly differ between indices

4 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Diversity Levels  Progress from local to regional levels  Point: diversity at a single point or microenvironment Our emphasis thus far  Alpha: within habitat diversity Usually consists of several subsamples in a habitat  Beta: species diversity along transects & gradients High Beta indicates number of spp increases rapidly with additional sampling sites along the gradient  Gamma: diversity of a larger geographical unit (island)  Epsilon: regional diversity

5 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Q Statistic Introduction  A bridge between the abundance models & diversity indices  Does not involve fitting a model as in the abundance models  Provides an indication of community diversity  No weighting towards very abundant or rare species They are excluded from the analysis Whittaker (1972) created earlier analysis including these –Thereby more influenced by the few rare / abundant species  Proposed by Kempton & Taylor (1976)

6 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Q Statistic Visually  Measures “inter-quartile slope” on the cumulative species abundance curve ,00010,000 Cumulative Species Species Abundance S = 250 S/4 = st = nd = 125 3rd = R1 = 62.5 = 0.25*S R2 = = 0.75*S Q = slope

7 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Q Statistic  Relationship to other indices  Similar to the a value in the log series model  Q = (0.371)(S*) / s  Biases in Q  May be biased in small samples Because we are including more of the rare and abundant species in the calculation

8 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Calculating Q - Worked Example #6  Assemble table with 3 columns  # Individuals, # Species, Summed # species  Determine R1 and R2  R1 should be > or = 0.25 * S  R2 should be > or = 0.75 * S  Calculate Q  Q = [((n R1 )/2) + Sn r + ((n R2 )/2)] / [ln(R2/R1)] n R1 and n R2 = # species in each quartile class Sn r = total number of species between the quartiles R1 and R2 = # of individuals at each quartile break point

9 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

10 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  All based on proportional species abundances  Species abundance models have drawbacks Tedious and repetitive Problems if the data do not violate more than one model –How to choose between?  Building upon the species abundance models  Allows for formal comparisons between sites / treatments

11 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  “Heterogeneity Indices”  Consider both evenness AND richness  Species abundance models only consider evenness  No assumptions made about species abundance distributions  Cause of distribution  Shape of curve  “Non-parametric”  Free of assumptions of normality

12 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Two General Categories  Information Theory (complicated computation)  Diversity (or information) of a natural system is similar to info in a code or message  Examples: Shannon-Wiener and Brillouin Indices  Species Dominance Measures (simple comput.)  Weighted towards abundance of the commonest species  Total species richness is downweighted relative to evenness  Examples: Simpson, McIntosh, and Berger-Parker

13 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

14 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Simpson Index Values  Derived by Simpson (1949)  Basis  Probability of 2 individuals being conspecifics  If drawn randomly from an infinitely large community  Summarized by letter D, 1-D, or 1/D  D decreases with increasing diversity Can go from 1 – 30+ Probability that two species are conspecifics  with  diversity  1-D and 1/D increases with increasing diversity 0.0 < 1-D < < 1/D < 10+

15 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Simpson Index  Heavily weighted towards most abundant species  Less sensitive to changes in species richness  Once richness > 10  underlying species abundance is important in determining the index value  Inappropriate for some models Log Series & Geometric  Best for Log-Normal Possibly Broken Stick Number of Species D value Log Series Log Normal Series Broken Stick Series

16 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Simpson Index  When would this weight towards most abundant species be desired?  Not just when the abundance model fits the Log-Normal  Conservation implications of index use?

17 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Simpson Calculation – Worked Example 9  Calculate N and S  Calculate D  D = S (n i (n i -1)) / (N(N-1)  Solve and then sum for all species in the sample  Calculate 1/D  Increases with increasing diversity

18 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

19 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, McIntosh Index  Proposed by McIntosh (1967)  Community is a point in an S dimensional hypervolume whose Euclidean distance from the origin is a measure of diversity  Paraphrased from Magurran  Origin is no diversity, distances from origin are more diverse  Not strictly a dominance index  Needs conversion to dominance index

20 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, McIntosh Manipulations  Base calculations (U metric)  Strongly influenced by sample size  Conversion to a dominance measure (D)  Use D m for our class  Makes value independent of sample size  Derive a simple evenness index using McIntosh  Most often used contribution of McIntosh

21 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, McIntosh Calculation – Worked Example 10  Base calculations  U =  (Sn i 2 ) n i = abundance of ith species Different from Magurran’s definition  Conversion to a dominance measure  D m = (N-U) / (N-  N)  Derive evenness index  E m = (N-U) / ((N-(N/  S))

22 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

23 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Berger-Parker  Proposed by Berger and Parker (1970) and developed by May (1975)  Simple calculation = d  Expresses proportional importance of most abundant species

24 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Berger-Parker  Decreasing d values  increasing diversity  Often use 1 / d Increasing 1 / d  increasing diversity And reduction in dominance of one species  Independent of S, influenced by sample size  Comparability between sites if sampling efforts standardized  Question may lead to use of Berger-Parker  Example: Change in dominant species in diet?

25 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Berger-Parker – An Example River Mouth #1Intertidal #2Intertidal #3+ Sewage #4Fresh & Hot #5 Nereis Corophium Gammarus Tubifex Chironomids Insect larvae10000 Arachnid01000 Carcinus Cragnon Neomysis81009 Sphaeroma15200 Flounder17110 Other fish23504 d / d Dominant species in flounder (Platichys flesus) diet across an Irish estuary (Wirjoatmodjo 1980)

26 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Berger-Parker Calculations – Worked Example 11  Calculate N, S, N max  Calculate d and 1/d  Very simple

27 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

28 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  “Heterogeneity Indices”  Consider both evenness AND richness  Species abundance models only consider evenness  No assumptions made about species abundance distributions  Cause of distribution  Shape of curve  “Non-parametric”  Free of assumptions of normality

29 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Two General Categories  Information Theory (complicated computation)  Diversity (or information) of a natural system is similar to info in a code or message  Examples: Shannon-Wiener and Brillouin Indices  Species Dominance Measures (simple comput.)  Weighted towards abundance of the commonest species  Total species richness is downweighted relative to evenness  Examples: Simpson, McIntosh, and Berger-Parker

30 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Information Theory  Information Theory, described (read more here)read more here  A system contains more information when it has many possible states E.g., large numbers of species, or high species richness  Also contains more information when the probability of encountering each state is high E.g., all species are equally abundant or have high evenness  Indices derived from this simple relationship between richness and evenness  Examples Shannon-Wiener and Brillouin

31 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

32 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Shannon-Wiener Index  Derived by Claude Shannon and Warren Weaver in late 40s  Developed a general model of communication and information theory  Initially developed to separate noise from information carrying signals  Subsequently  mathematician Norbert Wiener contributed to the model as part of his work in developing cybernetic technology  Called alternatively Shannon-Weaver, Shannon- Wiener, or Shannon Index – more info heremore info here

33 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Shannon-Wiener  Assumptions  All individuals are randomly sampled  Population is indefinitely large, or effectively infinite  All species in the community are represented  Result: difficult to justify for many communities  Particularly very diverse communities, guilds, functional groups  Incomplete sampling  significant error & bias Increasingly important as proportion of species sampled declines Simple mathematical consequence – see next slide

34 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Shannon-Wiener Mathematics  Equation  H’ = -  p i ln p i p i = proportion of individuals found in the ith species Unknowable, estimated using n i / N –Flawed estimation, need more sophisticated equation (2.18 in Magurran)  Error Mostly from inadequate sampling Flawed estimate of p i is negligible in most instances from this simple estimate

35 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Shannon-Wiener Mathematics  Need to convert data  Log 2 was historically used  Any Log base is acceptable Need consistency across samples  Currently, Ln is used more commonly What we will use  Range of S-W index  Usually between 1.5 and 3.5  Rarely surpasses 4.5  If underlying distribution is log-normal Need 100,000 species to have a H’ > 5.0

36 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Building on H’  Can also use Exp H’  = Number of equally common species required to produce a given H’ value Reduces S from the observed value Allows for an estimation of departures from maximal evenness and diversity  We won’t explore this here

37 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Building on H’  Evenness measure (E)  Useful for determining the departure from maximal evenness and diversity Similar to the Exp H’  H max = maximal diversity which could occur if all species collected were equally abundant  E = H’ / H max = H’ / ln S  0 < E < 1 H’ will always be less than H max  Assumes all species have been sampled  Some have criticized this as being biologically unrealistic Argue for best fit to the Broken Stick model

38 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Comparing H’ Values  Using Shannon for a t-test  Can use a simple t-test for differences between two samples  Need variance in H’ (Var H’) and to know the df Both have complicated equations (2.19, 2.21 in Magurran)  Shannon and ANOVA  H’ values tend to be normally distributed  Can use ANOVAs for differences between multiple sites Need to have real replication to do this Pseudoreplication introduces error, particularly in parametric statistics

39 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Shannon-Wiener Calculation – Worked Example 7  Calculate proportion of individuals in each species (p i ) and ln p i  Sum all (p i )(ln p i ) values  Calculate E  E = H’ / ln S  Calculate Var H’  Var H’= ([  p i )(ln p i ) 2 –  p i )(ln p i )) 2 ]  N) – ((S-1)/(2N 2 ))  Calculate t  t = (H’ 1 - H’ 2 ) / (Var H’ 1 + Var H’ 2 ) 1/2  Calculate df  df = (Var H’ 1 + Var H’ 2 ) 2 / ([(Var H’ 1 ) 2 / N 1 ] + [(Var H’ 2 ) 2 / N 2 ])

40 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

41 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Brillouin Index  Useful when  The randomness of a sample is not guaranteed Light traps, baited traps, attractive traps in general  Community is completely (thoroughly) censused Similar to Shannon-Wiener index  Assumes  Community is completely sampled  Does not assume: Randomness of sampling Equal attractiveness of traps

42 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Brillouin Mathematics  HB  Rarely larger than 4.5  Ranges between 1 and 4 most commonly

43 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Brillouin vs. Shannon-Wiener  Give similar values – significantly correlated  Brillouin < Shannon-Wiener  Brillouin has no uncertainty about all species present in sample  Does not estimate those that were not sampled, as in Shannon  When relative proportions of spp are consistent, totals differ  Shannon stays constant  Brillouin will decrease with fewer total individuals  Brillouin is more sensitive to overall sample size  Collections are compared, not samples  Disallows statistical comparisons, as all collections are different

44 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Brillouin Mathematics  Uses factorials throughout  Equation  HB = (ln N! –  ln n i !) / N  Evenness  E = HB / HB max  HB max  HB max = [(1/n)][(ln {((N!) / (((N/S)!) s-r )*((((N/S)+1)!) r )}]  r  r = N – S (N/S)

45 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Brillouin Calculations – Worked Example 8  Calculate HB  Calculate r  Calculate HB max  Calculate E

46 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

47 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Jack-Knifing Diversity Indices  Improves the accuracy of any estimate  First proposed in 1956 (Quenouille) and refined by Tukey in 1958  Theoretical biostatisticians  First applied to diversity by Zahl in 1977

48 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Jack-Knifing  Assumptions:  None made about underlying distribution  Does not attempt to estimate actual number of species present As in Shannon-Wiener  Random sampling is not necessary  Repeated measures overcome the biases  Jack-Knifing can determine the impact of biased sampling

49 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Data for Jack-Knifing  Need multiple samples to conduct this procedure  Some debate exists about this, may be able to do a single sample  For our data  Can use each tray  to create an estimate for what?  Can use each garden  to create an estimate for what?

50 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Jack-Knifing Procedure  Procedure  Create the overall pooled index estimate (  Subsamples with replacement from the actual data  Creates pseudovalues of the statistic  Pseudovalues are normally distributed about the mean  Mean value is best estimate of the statistics  Confidence limits  Also possible to attach these to the estimate  Consequence of normal distribution of pseudovalues

51 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Applications of the Jack-Knife  Most commonly used for the most common indices  Shannon and Simpson in particular  Also useful for other indices  Variance in the pseudovalues  More useful than the Var H’ of the Shannon  Gives a better estimate of the accuracy and impact of non-random sampling

52 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Jack-Knifing – Worked Example 12  Overall diversity index including all data (V)  Recalculate, excluding each sample in turn  Creates n number of VJ i estimates  Convert VJ i to pseudovalues VP i  Use VP i = (nV) – [(n-1) (VJ i )]  n = number of samples  Calculate mean VP value  Calculate Sample Influence Function  SIF = V – VP  Calculate standard error VP = stand dev Vp i s /  n

53 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Alpha Diversity Indices  Q-Statistic  Intro to Alpha Diversity Indices  Simpson  McIntosh  Berger-Parker  Shannon-Wiener  Brillouin  Jack-Knifing Diversity Indices  Pielou’s Hierarchical Diversity Index Week 1 Week 2

54 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Pielou’s Pooled Quadrat Method  Similar to Jack-Knifing  Improves the estimate of diversity  Also not influenced by non-random sampling  Provides the best estimate of the value, given the data  Can be calculated using either of the information statistic indexes  Shannon  Brillouin

55 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Pielou’s Pooled Quadrat  Outputs  A graph that levels off when diversity has been best estimated in the community (H pop )  Determine the minimal number of samples to achieve maximal diversity (t) Quadrats Diversity t H pop

56 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Pielou’s Method Utility  Stability of an index  Evaluating the stability of a diversity index and its relationship to sample size  Determining an adequate sample size  Produces a graph of the indices  When the line levels out, you have adequate samples Adequately estimated biodiversity locally  Can create confidence limits  Then, can compare values between habitats  Use standard parametric statistics

57 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Pielou’s Pooled Quadrat – Worked Example 13  Using Brillouin index, calculate all HB k  From k = 0  k = z  k = number of samples  z = total samples  M k = total abundance in k number of samples  Estimate t  t = Point at which HB k levels off  Calculate H pop  Using k+t number of samples  Calculate mean H pop  Calculate standard deviation

58 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Next week(s):  Continuing Alpha Diversity Indices  Read  Magurran Ch 2, pages  Magurran Worked Examples 6-13  We will continue conducting alpha diversity analyses next week

59 Lecture 4 – Alpha Diversity Indices© 2003 Dr. James A. Danoff-Burg, Hypothetical Model Curves Geometric Series Log Series Log-Normal Series Broken Stick Model Per Species Abundance Species Addition Sequence


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