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

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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

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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

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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

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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)

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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

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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

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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

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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

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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

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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

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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

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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

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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+

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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

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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?

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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

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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

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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

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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

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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))

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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

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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

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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?

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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)

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 ])

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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

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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

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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

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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

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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)

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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

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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

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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

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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

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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?

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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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