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Alpha Diversity IndicesAlpha Diversity Indices
James A. Danoff-Burg
Dept. Ecol., Evol., & Envir. Biol.
Columbia University
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Diversity of DiversitiesDiversity 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Diversity LevelsDiversity 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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)
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Q Statistic Visually Measures “inter-quartile slope” on the cumulative
species abundance curve
250
200
150
100
50
010 100 1,000 10,000
Cumulative Species
Species Abundance
S = 250S/4 = 62.5
1st = 62.52nd = 1253rd = 187.5
R1 = 62.5 = 0.25*S
R2 = 187.5 = 0.75*S
Q = slope
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Q StatisticQ Statistic Relationship to other indices
Similar to the value in the log series model Q = (0.371)(S*) /
Biases in Q May be biased in small samples
• Because we are including more of the rare and abundant species in the calculation
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Calculating QCalculating Q- Worked Example #6- 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 = [((nR1)/2) + nr + ((nR2)/2)] / [ln(R2/R1)]
• nR1 and nR2 = # species in each quartile class
nr = total number of species between the quartiles
• R1 and R2 = # of individuals at each quartile break point
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Two General CategoriesTwo 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Simpson Index ValuesSimpson 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 < 1.0• 0.0 < 1/D < 10+
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Simpson IndexSimpson 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 Stick10000
1000
100
10
10 20 30
Number of Species
D value
Log Series
Log Normal Series
Broken Stick Series
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Simpson IndexSimpson 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?
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Simpson Calculation – Simpson Calculation – Worked Example 9Worked Example 9
Calculate N and S Calculate D
D = (ni(ni-1)) / (N(N-1) Solve and then sum for all species in the sample
Calculate 1/D Increases with increasing diversity
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
McIntosh IndexMcIntosh 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
McIntosh ManipulationsMcIntosh Manipulations Base calculations (U metric)
Strongly influenced by sample size
Conversion to a dominance measure (D) Use Dm for our class Makes value independent of sample size
Derive a simple evenness index using McIntosh Most often used contribution of McIntosh
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
McIntosh Calculation – McIntosh Calculation – Worked Example 10Worked Example 10
Base calculations U = (ni
2)
• ni = abundance of ith species
• Different from Magurran’s definition
Conversion to a dominance measure Dm = (N-U) / (N-N)
Derive evenness index Em = (N-U) / ((N-(N/S))
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Berger-ParkerBerger-Parker Proposed by Berger and Parker (1970) and
developed by May (1975) Simple calculation = d Expresses proportional importance of most
abundant species
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Berger-ParkerBerger-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?
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Berger-Parker – An ExampleBerger-Parker – An Example
River Mouth #1 Intertidal #2 Intertidal #3 + Sewage #4 Fresh & Hot #5
Nereis 394 1642 90 126 32
Corophium 3487 5681 320 17 0
Gammarus 275 196 180 115 0
Tubifex 683 1348 46 436 5
Chironomids 22 12 2 27 0
Insect larvae 1 0 0 0 0
Arachnid 0 1 0 0 0
Carcinus 4 48 1 3 0
Cragnon 6 21 0 1 13
Neomysis 8 1 0 0 9
Sphaeroma 1 5 2 0 0
Flounder 1 7 1 1 0
Other fish 2 3 5 0 4
d 0.714 0.634 0.495 0.601 0.508
1 / d 1.41.4 1.58 2.022.02 1.67 1.96
Dominant species in flounder (Platichys flesus) diet across an Irish estuary (Wirjoatmodjo 1980)
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Berger-Parker Calculations –Berger-Parker Calculations –Worked Example 11Worked Example 11
Calculate N, S, Nmax
Calculate d and 1/d Very simple
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Two General CategoriesTwo 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Information TheoryInformation Theory Information Theory, described (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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Shannon-Wiener IndexShannon-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 here
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Shannon-WienerShannon-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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Shannon-Wiener Shannon-Wiener MathematicsMathematics
Equation H’ = -pi ln pi
• pi = proportion of individuals found in the ith species
• Unknowable, estimated using ni / N
– Flawed estimation, need more sophisticated equation (2.18 in Magurran)
Error • Mostly from inadequate sampling
• Flawed estimate of pi is negligible in most instances from this simple estimate
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Shannon-Wiener Mathematics
Need to convert data Log2 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Building on H’ Evenness measure (E)
Useful for determining the departure from maximal evenness and diversity
• Similar to the Exp H’ Hmax = maximal diversity which could occur if all species
collected were equally abundant E = H’ / Hmax = H’ / ln S 0 < E < 1
• H’ will always be less than Hmax Assumes all species have been sampled Some have criticized this as being biologically
unrealistic• Argue for best fit to the Broken Stick model
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Shannon-Wiener Calculation – Worked Example 7
Calculate proportion of individuals in each species (pi) and ln pi
Sum all (pi)(ln pi) values
Calculate E E = H’ / ln S
Calculate Var H’ Var H’= ([pi)(ln pi)2 – pi)(ln pi))2] N) – ((S-1)/(2N2))
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 / N1] + [(Var H’2)2 / N2])
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Brillouin IndexBrillouin 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Brillouin MathematicsBrillouin Mathematics HB
Rarely larger than 4.5 Ranges between 1 and 4 most commonly
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Brillouin vs. Shannon-WienerBrillouin 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Brillouin MathematicsBrillouin Mathematics Uses factorials throughout Equation
HB = (ln N! – ln ni!) / N
Evenness E = HB / HBmax
HBmax HBmax = [(1/n)][(ln {((N!) / (((N/S)!)s-r)*((((N/S)+1)!)r)}]
r r = N – S (N/S)
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Brillouin Calculations – Brillouin Calculations – Worked Example 8Worked Example 8
Calculate HB Calculate r Calculate HBmax
Calculate E
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Jack-Knifing Diversity IndicesJack-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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Jack-Knifing 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Data for Jack-KnifingData 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?
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Jack-Knifing ProcedureJack-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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Applications of the Jack-KnifeApplications 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Jack-Knifing – Jack-Knifing – Worked Example 12Worked Example 12
Overall diversity index including all data (V) Recalculate, excluding each sample in turn
Creates n number of VJi estimates
Convert VJi to pseudovalues VPi Use VPi = (nV) – [(n-1) (VJi)] n = number of samples
Calculate mean VP value Calculate Sample Influence Function
SIF = V – VP
Calculate standard error VP = stand dev Vpis / n
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Alpha Diversity IndicesAlpha 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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Pielou’s Pooled Quadrat Pielou’s Pooled Quadrat MethodMethod
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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Pielou’s Pooled QuadratPielou’s Pooled Quadrat Outputs
A graph that levels off when diversity has been best estimated in the community (Hpop)
Determine the minimal number of samples to achieve maximal diversity (t)
Quadrats
Diversity
t
Hpop
0
1
2
3
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Pielou’s Method UtilityPielou’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
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Pielou’s Pooled Quadrat – Pielou’s Pooled Quadrat – Worked Example 13Worked Example 13
Using Brillouin index, calculate all HBk From k = 0 k = z k = number of samples z = total samples Mk = total abundance in k number of samples
Estimate t t = Point at which HBk levels off
Calculate Hpop Using k+t number of samples Calculate mean Hpop
Calculate standard deviation
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Next week(s):Next week(s): Continuing Alpha Diversity Indices Read
Magurran Ch 2, pages 32-45 Magurran Worked Examples 6-13
We will continue conducting alpha diversity analyses next week
Lecture 4 – Alpha Diversity Indices © 2003 Dr. James A. Danoff-Burg, [email protected]
Hypothetical Model CurvesHypothetical Model Curves
100
10
1
0.1
0.01
0.001
10 20 30 40
Geometric SeriesLog Series
Log-Normal Series
Broken Stick Model
Per Species
Abundance
Species Addition Sequence