Rarefaction (ecology)

In ecology, rarefaction is a technique to assess species richness from the results of sampling. Rarefaction allows the calculation of species richness for a given number of individual samples, based on the construction of so-called rarefaction curves. This curve is a plot of the number of species as a function of the number of samples. Rarefaction curves generally grow rapidly at first, as the most common species are found, but the curves plateau as only the rarest species remain to be sampled. The issue that occurs when sampling various species in a community is that the larger the number of individuals sampled, the more species that will be found. Rarefaction curves are created by randomly re-sampling the pool of N samples multiple times and then plotting the average number of species found in each sample (1,2, ... N). "Thus rarefaction generates the expected number of species in a small collection of n individuals (or n samples) drawn at random from the large pool of N samples.".

History The technique of rarefaction was developed in 1968 by Howard Sanders in a biodiversity assay of marine benthic ecosystems, as he sought a model for diversity that would allow him to compare species richness data among sets with different sample sizes; he developed rarefaction curves as a method to compare the shape of a curve rather than absolute numbers of species. Following initial development by Sanders, the technique of rarefaction has undergone a number of revisions. In a paper criticizing many methods of assaying biodiversity, Stuart Hurlbert refined the problem that he saw with Sanders' rarefaction method, that it overestimated the number of species based on sample size, and attempted to refine his methods. The issue of overestimation was also dealt with by Daniel Simberloff, while other improvements in rarefaction as a statistical technique were made by Ken Heck in 1975. Today, rarefaction has grown as a technique not just for measuring species diversity, but for understanding diversity at higher taxonomic levels as well. Most commonly, the number of species is sampled to predict the number of genera in a particular community; similar techniques had been used to determine this level of diversity in studies several years before Sanders quantified his individual to species determination of rarefaction. Rarefaction techniques are used to quantify species diversity of newly studied ecosystems, including human microbiomes, as well as in applied studies in community ecology, such as understanding pollution impacts on communities and other management applications.

Derivation Deriving rarefaction: N = total number of items K = total number of groups Ni = the number of items in group i (i = 1, ..., K). Mj = number of groups consisting in j elements From these definitions, it therefore follows that:

∑ i = 1 K N i = N {\displaystyle \sum _{i=1}^{K}N_{i}=N} ∑ j = 1 ∞ M j = K {\displaystyle \sum _{j=1}^{\infty }M_{j}=K} ∑ j = 1 ∞ j M j = N {\displaystyle \sum _{j=1}^{\infty }jM_{j}=N}

In a rarefied sample we have chosen a random subsample n from the total N items. The relevance of a rarefied sample is that some groups may now be necessarily absent from this subsample. We therefore let:

X n = {\displaystyle X_{n}=} the number of groups still present in the subsample of "n" items It is true that X n {\displaystyle X_{n}} is less than K whenever at least one group is missing from this subsample. Therefore the rarefaction curve, f n {\displaystyle f_{n}} is defined as:

f n = E [ X n ] = K − ( N n ) − 1 ∑ i = 1 K ( N − N i n ) {\displaystyle f_{n}=E[X_{n}]=K-{\binom {N}{n}}^{-1}\sum _{i=1}^{K}{\binom {N-N_{i}}{n}}}

From this it follows that 0 ≤ f(n) ≤ K. Furthermore, f ( 0 ) = 0 , f ( 1 ) = 1 , f ( N ) = K {\displaystyle f(0)=0,f(1)=1,f(N)=K} . Despite being defined at discrete values of n, these curves are most frequently displayed as continuous functions.

… excerpt ends here. Continue reading the full article.