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Taylor's power law is an empirical law in
ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ...
that relates the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of the number of individuals of a species per unit area of habitat to the corresponding
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
by a
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...
relationship. It is named after the ecologist who first proposed it in 1961, Lionel Roy Taylor (1924–2007). Taylor's original name for this relationship was the law of the mean. The name ''Taylor's law'' was coined by Southwood in 1966.


Definition

This law was originally defined for ecological systems, specifically to assess the spatial clustering of organisms. For a population count Y with mean \mu and variance \operatorname (Y), Taylor's law is written : \operatorname (Y) = a\mu^b, where ''a'' and ''b'' are both positive constants. Taylor proposed this relationship in 1961, suggesting that the exponent ''b'' be considered a species specific index of aggregation. This power law has subsequently been confirmed for many hundreds of species. Taylor's law has also been applied to assess the time dependent changes of population distributions. Related variance to mean power laws have also been demonstrated in several non-ecological systems: *cancer metastasis *the numbers of houses built over the Tonami plain in Japan. *measles epidemiology *
HIV The human immunodeficiency viruses (HIV) are two species of '' Lentivirus'' (a subgroup of retrovirus) that infect humans. Over time, they cause acquired immunodeficiency syndrome (AIDS), a condition in which progressive failure of the im ...
epidemiology, *the geographic clustering of childhood
leukemia Leukemia ( also spelled leukaemia; pronounced ) is a group of blood cancers that usually begin in the bone marrow and produce high numbers of abnormal blood cells. These blood cells are not fully developed and are called ''blasts'' or '' ...
*blood flow heterogeneity *the genomic distributions of
single-nucleotide polymorphism In genetics and bioinformatics, a single-nucleotide polymorphism (SNP ; plural SNPs ) is a germline substitution of a single nucleotide at a specific position in the genome. Although certain definitions require the substitution to be present in a ...
s (SNPs) *gene structures *in
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
with sequential values of the
Mertens function In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive r ...
and also with the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s *from the eigenvalue deviations of Gaussian orthogonal and unitary ensembles of
random matrix In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the ...
theory


History

The first use of a double log-log plot was by Reynolds in 1879 on thermal aerodynamics. Pareto used a similar plot to study the proportion of a population and their income.Pareto V (1897) Cours D'économie Politique. Volume 2. Lausanne: F. Rouge The term ''
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
'' was coined by Fisher in 1918.


Biology

Pearson in 1921 proposed the equation (also studied by Neyman) : s^2 = a m + b m^2 Smith in 1938 while studying crop yields proposed a relationship similar to Taylor's. This relationship was : \log V_x = \log V_1 + b\log x \, where ''V''''x'' is the variance of yield for plots of ''x'' units, ''V''1 is the variance of yield per unit area and ''x'' is the size of plots. The slope (''b'') is the index of heterogeneity. The value of ''b'' in this relationship lies between 0 and 1. Where the yield are highly correlated ''b'' tends to 0; when they are uncorrelated ''b'' tends to 1. Bliss in 1941, Fracker and Brischle in 1941 and Hayman & Lowe in 1961 also described what is now known as Taylor's law, but in the context of data from single species. Taylor's 1961 paper used data from 24 papers, published between 1936 and 1960, that considered a variety of biological settings:
virus A virus is a submicroscopic infectious agent that replicates only inside the living Cell (biology), cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Viruses are ...
lesions, macro-zooplankton,
worm Worms are many different distantly related bilateria, bilateral animals that typically have a long cylindrical tube-like body, no limb (anatomy), limbs, and usually no eyes. Worms vary in size from microscopic to over in length for marine ...
s and symphylids in
soil Soil, also commonly referred to as earth, is a mixture of organic matter, minerals, gases, water, and organisms that together support the life of plants and soil organisms. Some scientific definitions distinguish dirt from ''soil'' by re ...
,
insect Insects (from Latin ') are Hexapoda, hexapod invertebrates of the class (biology), class Insecta. They are the largest group within the arthropod phylum. Insects have a chitinous exoskeleton, a three-part body (Insect morphology#Head, head, ...
s in soil, on
plant Plants are the eukaryotes that form the Kingdom (biology), kingdom Plantae; they are predominantly Photosynthesis, photosynthetic. This means that they obtain their energy from sunlight, using chloroplasts derived from endosymbiosis with c ...
s and in the air,
mite Mites are small arachnids (eight-legged arthropods) of two large orders, the Acariformes and the Parasitiformes, which were historically grouped together in the subclass Acari. However, most recent genetic analyses do not recover the two as eac ...
s on
leaves A leaf (: leaves) is a principal appendage of the stem of a vascular plant, usually borne laterally above ground and specialized for photosynthesis. Leaves are collectively called foliage, as in "autumn foliage", while the leaves, stem, ...
,
tick Ticks are parasitic arachnids of the order Ixodida. They are part of the mite superorder Parasitiformes. Adult ticks are approximately 3 to 5 mm in length depending on age, sex, and species, but can become larger when engorged. Ticks a ...
s on
sheep Sheep (: sheep) or domestic sheep (''Ovis aries'') are a domesticated, ruminant mammal typically kept as livestock. Although the term ''sheep'' can apply to other species in the genus '' Ovis'', in everyday usage it almost always refers to d ...
and
fish A fish (: fish or fishes) is an aquatic animal, aquatic, Anamniotes, anamniotic, gill-bearing vertebrate animal with swimming fish fin, fins and craniate, a hard skull, but lacking limb (anatomy), limbs with digit (anatomy), digits. Fish can ...
in the
sea A sea is a large body of salt water. There are particular seas and the sea. The sea commonly refers to the ocean, the interconnected body of seawaters that spans most of Earth. Particular seas are either marginal seas, second-order section ...
.; the ''b'' value lay between 1 and 3. Taylor proposed the power law as a general feature of the spatial distribution of these species. He also proposed a mechanistic hypothesis to explain this law. Initial attempts to explain the spatial distribution of animals had been based on approaches like Bartlett's stochastic population models and the
negative binomial distribution In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...
that could result from
birth–death process The birth–death process (or birth-and-death process) is a special case of continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the stat ...
es. Taylor's explanation was based the assumption of a balanced migratory and congregatory behavior of animals. His hypothesis was initially qualitative, but as it evolved it became semi-quantitative and was supported by simulations. Many alternative hypotheses for the power law have been advanced. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction. Hanski's model predicted that the power law exponent would be constrained to range closely about the value of 2, which seemed inconsistent with many reported values. Anderson ''et al'' formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function. As a response to this model Taylor argued that such a
Markov process In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, ...
would predict that the power law exponent would vary considerably between replicate observations, and that such variability had not been observed. Adrienne W. Kemp reviewed a number of discrete stochastic models based on the negative binomial, Neyman type A, and Polya–Aeppli distributions that with suitable adjustment of parameters could produce a variance to mean power law. Kemp, however, did not explain the parameterizations of her models in mechanistic terms. Other relatively abstract models for Taylor's law followed. Statistical concerns were raised regarding Taylor's law, based on the difficulty with real data in distinguishing between Taylor's law and other variance to mean functions, as well the inaccuracy of standard regression methods. Taylor's law has been applied to
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
data, and Perry showed, using simulations, that
chaos theory Chaos theory is an interdisciplinary area of Scientific method, scientific study and branch of mathematics. It focuses on underlying patterns and Deterministic system, deterministic Scientific law, laws of dynamical systems that are highly sens ...
could yield Taylor's law. Taylor's law has been applied to the spatial distribution of plants and bacterial populationsRamsayer J, Fellous S, Cohen JE & Hochberg ME (2011) Taylor's Law holds in experimental bacterial populations but competition does not influence the slope. ''Biology Letters'' As with the observations of Tobacco necrosis virus mentioned earlier, these observations were not consistent with Taylor's animal behavioral model. A variance to mean power function had been applied to non-ecological systems, under the rubric of Taylor's law. A more general explanation for the range of manifestations of the power law a hypothesis has been proposed based on the Tweedie distributions, a family of probabilistic models that express an inherent power function relationship between the variance and the mean. Several alternative hypotheses for the power law have been proposed. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction. Hanski's model predicted that the power law exponent would be constrained to range closely about the value of 2, which seemed inconsistent with many reported values. Anderson ''et al'' formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function. The Lewontin Cohen growth model. is another proposed explanation. The possibility that observations of a power law might reflect more mathematical artifact than a mechanistic process was raised. Variation in the exponents of Taylor's Law applied to ecological populations cannot be explained or predicted based solely on statistical grounds however. Research has shown that variation within the Taylor's law exponents for the North Sea fish community varies with the external environment, suggesting ecological processes at least partially determine the form of Taylor's law.


Physics

In the physics literature Taylor's law has been referred to as ''fluctuation scaling''. Eisler ''et al'', in a further attempt to find a general explanation for fluctuation scaling, proposed a process they called ''impact inhomogeneity'' in which frequent events are associated with larger impacts. In appendix B of the Eisler article, however, the authors noted that the equations for impact inhomogeneity yielded the same mathematical relationships as found with the Tweedie distributions. Another group of physicists, Fronczak and Fronczak, derived Taylor's power law for fluctuation scaling from principles of equilibrium and non-equilibrium
statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. Their derivation was based on assumptions of physical quantities like free energy and an external field that caused the clustering of biological organisms. Direct experimental demonstration of these postulated physical quantities in relationship to animal or plant aggregation has yet to be achieved, though. Shortly thereafter, an analysis of Fronczak and Fronczak's model was presented that showed their equations directly lead to the Tweedie distributions, a finding that suggested that Fronczak and Fronczak had possibly provided a maximum entropy derivation of these distributions.


Mathematics

Taylor's law has been shown to hold for
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s not exceeding a given real number. This result has been shown to hold for the first 11 million primes. If the Hardy–Littlewood twin primes conjecture is true then this law also holds for twin primes.


The Tweedie hypothesis

About the time that Taylor was substantiating his ecological observations, MCK Tweedie, a British statistician and medical physicist, was investigating a family of probabilistic models that are now known as the Tweedie distributions.Tweedie MCK (1984) An index which distinguishes between some important exponential families. In: ''Statistics: Applications and New Directions Proceedings of the Indian Statistical Institute Golden Jubilee International Conference'' pp 579-604 Eds: JK Ghosh & J Roy, Indian Statistical Institute, Calcutta As mentioned above, these distributions are all characterized by a variance to mean power law mathematically identical to Taylor's law. The Tweedie distribution most applicable to ecological observations is the compound Poisson-gamma distribution, which represents the sum of ''N'' independent and identically distributed random variables with a gamma distribution where ''N'' is a random variable distributed in accordance with a Poisson distribution. In the additive form its
cumulant generating function In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
(CGF) is: : K^*_b(s;\theta,\lambda)=\lambda\kappa_b(\theta)\left left(1+\right)^\alpha-1\right where ''κ''''b''(''θ'') is the cumulant function, : \kappa_b(\theta) = \frac \alpha \left( \frac \right)^\alpha, the Tweedie exponent : \alpha = \frac, ''s'' is the generating function variable, and ''θ'' and ''λ'' are the canonical and index parameters, respectively. These last two parameters are analogous to the scale and
shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. th ...
s used in
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. The
cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s of this distribution can be determined by successive differentiations of the CGF and then substituting ''s=0'' into the resultant equations. The first and second cumulants are the mean and variance, respectively, and thus the compound Poisson-gamma CGF yields Taylor's law with the proportionality constant : a=\lambda^. The compound Poisson-gamma cumulative distribution function has been verified for limited ecological data through the comparison of the theoretical distribution function with the
empirical distribution function In statistics, an empirical distribution function ( an empirical cumulative distribution function, eCDF) is the Cumulative distribution function, distribution function associated with the empirical measure of a Sampling (statistics), sample. Th ...
. A number of other systems, demonstrating variance to mean power laws related to Taylor's law, have been similarly tested for the compound Poisson-gamma distribution. The main justification for the Tweedie hypothesis rests with the mathematical
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
properties of the Tweedie distributions. The Tweedie convergence theorem requires the Tweedie distributions to act as foci of convergence for a wide range of statistical processes. As a consequence of this convergence theorem, processes based on the sum of multiple independent small jumps will tend to express Taylor's law and obey a Tweedie distribution. A limit theorem for independent and identically distributed variables, as with the Tweedie convergence theorem, might then be considered as being fundamental relative to the ''ad hoc'' population models, or models proposed on the basis of simulation or approximation. This hypothesis remains controversial; more conventional population dynamic approaches seem preferred amongst ecologists, despite the fact that the Tweedie compound Poisson distribution can be directly applied to population dynamic mechanisms. One difficulty with the Tweedie hypothesis is that the value of ''b'' does not range between 0 and 1. Values of ''b'' < 1 are rare but have been reported.


Mathematical formulation

In symbols : s_i^2 = am_i^b, where ''s''''i''2 is the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of the density of the ''i''th sample, ''m''''i'' is the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
density of the ''i''th sample and ''a'' and ''b'' are constants. In logarithmic form : \log s_i^2 = \log a + b\log m_i


Scale invariance

The exponent in Taylor's law is scale invariant: If the unit of measurement is changed by a constant factor c, the exponent (b) remains unchanged. To see this let ''y'' = ''cx''. Then : \mu_1 = \operatorname( x ) : \mu_2 = \operatorname( y ) =\operatorname( cx ) = c \operatorname(x) = c\mu_1 : \sigma^2_1 = \operatorname (( x - \mu_1 )^2) : \sigma^2_2 = \operatorname((y - \mu_2)^2) = \operatorname((cx - c\mu_1)^2) = c^2 \operatorname ((x - \mu_1)^2) = c^2 \sigma^2_1 Taylor's law expressed in the original variable (''x'') is : \sigma_1^2 = a \mu_1^b and in the rescaled variable (''y'') it is : \sigma_2^2 = c^2 \sigma_1^2 = c^2 a \mu_1^b = c^ a (c\mu_1)^b = c^ a \mu_2^b Thus, \sigma_2^2 is still proportional to \mu_2^b (even though the proportionality constant has changed). It has been shown that Taylor's law is the only relationship between the mean and variance that is scale invariant.Jørgensen B (1997) The theory of exponential dispersion models. Chapman & Hall. London


Extensions and refinements

A refinement in the estimation of the slope ''b'' has been proposed by Rayner. : b = \frac where r is the Pearson moment correlation coefficient between \log (s^2) and \log m, f is the ratio of sample variances in \log (s^2) and \log m and \varphi'' is the ratio of the errors in \log (s^2) and \log m. Ordinary least squares regression assumes that ''φ'' = ∞. This tends to underestimate the value of ''b'' because the estimates of both \log (s^2) and \log m are subject to error. An extension of Taylor's law has been proposed by Ferris ''et al'' when multiple samples are taken : s^2 = c n^d m^b, where ''s''2 and ''m'' are the variance and mean respectively, ''b'', ''c'' and ''d'' are constants and ''n'' is the number of samples taken. To date, this proposed extension has not been verified to be as applicable as the original version of Taylor's law.


Small samples

An extension to this law for small samples has been proposed by Hanski.Hanski I(1982) On patterns of temporal and spatial variation in animal populations. ''Ann. zool. Fermici'' 19: 21—37 For small samples the Poisson variation (''P'') - the variation that can be ascribed to sampling variation - may be significant. Let ''S'' be the total variance and let ''V'' be the biological (real) variance. Then : S = V + P Assuming the validity of Taylor's law, we have : V = a m^b Because in the Poisson distribution the mean equals the variance, we have : P = m This gives us : S = V + P = a m^b + m This closely resembles Barlett's original suggestion.


Interpretation

Slope values (''b'') significantly > 1 indicate clumping of the organisms. In Poisson-distributed data, ''b'' = 1. If the population follows a
lognormal In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
or
gamma distribution In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the g ...
, then ''b'' = 2. For populations that are experiencing constant per capita environmental variability, the regression of log( variance ) versus log( mean abundance ) should have a line with ''b'' = 2. Most populations that have been studied have ''b'' < 2 (usually 1.5–1.6) but values of 2 have been reported. Occasionally cases with ''b'' > 2 have been reported. ''b'' values below 1 are uncommon but have also been reported ( ''b'' = 0.93 ). It has been suggested that the exponent of the law (''b'') is proportional to the skewness of the underlying distribution.Cohen J E, Xua M (2015) Random sampling of skewed distributions implies Taylor’s power law of fluctuation scaling.Proc. Natl. Acad. Sci. USA 2015 112 (25) 7749–7754 This proposal has criticised: additional work seems to be indicated.Reply to Chen: Under specified assumptions, adequate random samples of skewed distributions obey Taylor's law (2015) Proc Natl Acad Sci USA 112 (25) E3157-E3158Random sampling of skewed distributions does not necessarily imply Taylor's law (2015) Proc Natl Acad Sci USA 112 (25) E3156


Notes

The origin of the slope (''b'') in this regression remains unclear. Two hypotheses have been proposed to explain it. One suggests that ''b'' arises from the species behavior and is a constant for that species. The alternative suggests that it is dependent on the sampled population. Despite the considerable number of studies carried out on this law (over 1000), this question remains open. It is known that both ''a'' and ''b'' are subject to change due to age-specific dispersal, mortality and sample unit size. This law may be a poor fit if the values are small. For this reason an extension to Taylor's law has been proposed by Hanski which improves the fit of Taylor's law at low densities.


Extension to cluster sampling of binary data

A form of Taylor's law applicable to
binary data Binary data is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra. Binary data occurs in many different technical and scientific fields, wh ...
in clusters (e.q., quadrats) has been proposed. In a binomial distribution, the theoretical variance is : \text_\text = np(1 - p), where (varbin) is the binomial variance, ''n'' is the sample size per cluster, and ''p'' is the proportion of individuals with a trait (such as disease), an estimate of the probability of an individual having that trait. One difficulty with binary data is that the mean and variance, in general, have a particular relationship: as the mean proportion of individuals infected increases above 0.5, the variance deceases. It is now known that the observed variance (varobs) changes as a power function of (varbin). Hughes and Madden noted that if the distribution is Poisson, the mean and variance are equal. As this is clearly not the case in many observed proportion samples, they instead assumed a binomial distribution. They replaced the mean in Taylor's law with the binomial variance and then compared this theoretical variance with the observed variance. For binomial data, they showed that varobs = varbin with overdispersion, varobs > varbin. In symbols, Hughes and Madden's modification to Tyalor's law was : \text_\text = a( \text_\text )^b. In logarithmic form this relationship is : \log( \text_\text ) = \log a + b \log( \text_\text ). This latter version is known as the binary power law. A key step in the derivation of the binary power law by Hughes and Madden was the observation made by Patil and Stiteler that the variance-to-mean ratio used for assessing over-dispersion of unbounded counts in a single sample is actually the ratio of two variances: the observed variance and the theoretical variance for a random distribution. For unbounded counts, the random distribution is the Poisson. Thus, the Taylor power law for a collection of samples can be considered as a relationship between the observed variance and the Poisson variance. More broadly, Madden and Hughes considered the power law as the relationship between two variances, the observed variance and the theoretical variance for a random distribution. With binary data, the random distribution is the binomial (not the Poisson). Thus the Taylor power law and the binary power law are two special cases of a general power-law relationships for heterogeneity. When both ''a'' and ''b'' are equal to 1, then a small-scale random spatial pattern is suggested and is best described by the binomial distribution. When ''b'' = 1 and ''a'' > 1, there is over-dispersion (small-scale aggregation). When ''b'' is > 1, the degree of aggregation varies with ''p''. Turechek ''et al.'' have shown that the binary power law describes numerous data sets in plant pathology. In general, ''b'' is greater than 1 and less than 2. The fit of this law has been tested by simulations. These results suggest that rather than a single regression line for the data set, a segmental regression may be a better model for genuinely random distributions. However, this segmentation only occurs for very short-range dispersal distances and large quadrat sizes. The break in the line occurs only at ''p'' very close to 0. An extension to this law has been proposed. The original form of this law is symmetrical but it can be extended to an asymmetrical form. Using simulations the symmetrical form fits the data when there is positive correlation of disease status of neighbors. Where there is a negative correlation between the likelihood of neighbours being infected, the asymmetrical version is a better fit to the data.


Applications

Because of the ubiquitous occurrence of Taylor's law in biology it has found a variety of uses some of which are listed here.


Recommendations as to use

It has been recommended based on simulation studies in applications testing the validity of Taylor's law to a data sample that: (1) the total number of organisms studied be > 15
(2) the minimum number of groups of organisms studied be > 5
(3) the density of the organisms should vary by at least 2 orders of magnitude within the sample


Randomly distributed populations

It is common assumed (at least initially) that a population is randomly distributed in the environment. If a population is randomly distributed then the mean ( ''m'' ) and variance ( ''s''2 ) of the population are equal and the proportion of samples that contain at least one individual ( ''p'' ) is : p = 1 - e^ When a species with a clumped pattern is compared with one that is randomly distributed with equal overall densities, p will be less for the species having the clumped distribution pattern. Conversely when comparing a uniformly and a randomly distributed species but at equal overall densities, ''p'' will be greater for the randomly distributed population. This can be graphically tested by plotting ''p'' against ''m''. Wilson and Room developed a binomial model that incorporates Taylor's law. The basic relationship is : p = 1 - e^ where the log is taken to the base ''e''. Incorporating Taylor's law this relationship becomes : p = 1 - e^


Dispersion parameter estimator

The common dispersion parameter (''k'') of the negative binomial distribution is : k = \frac where m is the sample mean and s^2 is the variance. If 1 / ''k'' is > 0 the population is considered to be aggregated; 1 / ''k'' = 0 ( ''s''2 = ''m'' ) the population is considered to be randomly (Poisson) distributed and if 1 / ''k'' is < 0 the population is considered to be uniformly distributed. No comment on the distribution can be made if ''k'' = 0. Wilson and Room assuming that Taylor's law applied to the population gave an alternative estimator for ''k'': : k = \frac where ''a'' and ''b'' are the constants from Taylor's law. Jones using the estimate for ''k'' above along with the relationship Wilson and Room developed for the probability of finding a sample having at least one individual : p = 1 - e^ derived an estimator for the probability of a sample containing ''x'' individuals per sampling unit. Jones's formula is : P( x ) = P( x - 1 ) \frac x \frac where ''P''( ''x'' ) is the probability of finding ''x'' individuals per sampling unit, ''k'' is estimated from the Wilon and Room equation and ''m'' is the sample mean. The probability of finding zero individuals ''P''( 0 ) is estimated with the
negative binomial distribution In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...
: P( 0 ) = \left( 1 + \frac m k \right)^ Jones also gives confidence intervals for these probabilities. : \mathrm = t \left( \frac N \right)^ where ''CI'' is the confidence interval, ''t'' is the critical value taken from the t distribution and ''N'' is the total sample size.


Katz family of distributions

Katz proposed a family of distributions (the Katz family) with 2 parameters ( ''w''1, ''w''2 ).Katz L (1965) United treatment of a broad class of discrete probability distributions. ''in'' Proceedings of the International Symposium on Discrete Distributions. Montreal This family of distributions includes the Bernoulli,
Geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Pascal and Poisson distributions as special cases. The mean and variance of a Katz distribution are : m = \frac : s^2 = \frac where ''m'' is the mean and ''s''2 is the variance of the sample. The parameters can be estimated by the method of moments from which we have : \frac = m : \frac = \frac m For a Poisson distribution ''w''2 = 0 and ''w''1 = ''λ'' the parameter of the Possion distribution. This family of distributions is also sometimes known as the Panjer family of distributions. The Katz family is related to the Sundt-Jewel family of distributions: : p_n = \left( a + \frac b n \right) p_ The only members of the Sundt-Jewel family are the Poisson, binomial, negative binomial (Pascal), extended truncated negative binomial and logarithmic series distributions. If the population obeys a Katz distribution then the coefficients of Taylor's law are : a = -\log (1 - w_2) : b = 1 Katz also introduced a statistical test : J_n = \sqrt \frac m where ''J''n is the test statistic, ''s''2 is the variance of the sample, ''m'' is the mean of the sample and ''n'' is the sample size. ''J''n is asymptotically normally distributed with a zero mean and unit variance. If the sample is Poisson distributed ''J''n = 0; values of ''J''n < 0 and > 0 indicate under and over dispersion respectively. Overdispersion is often caused by latent heterogeneity - the presence of multiple sub populations within the population the sample is drawn from. This statistic is related to the Neyman–Scott statistic : NS = \sqrt \left( \frac m - 1 \right) which is known to be asymptotically normal and the conditional chi-squared statistic (Poisson dispersion test) : T = \frac m which is known to have an asymptotic chi squared distribution with ''n'' − 1 degrees of freedom when the population is Poisson distributed. If the population obeys Taylor's law then : J_n = \sqrt ( a m^ - 1 )


Time to extinction

If Taylor's law is assumed to apply it is possible to determine the mean time to local extinction. This model assumes a simple random walk in time and the absence of density dependent population regulation. Let N_ = r N_t where ''N''''t''+1 and ''N''''t'' are the population sizes at time ''t'' + 1 and ''t'' respectively and ''r'' is parameter equal to the annual increase (decrease in population). Then : \operatorname(r) = s^2 \log r where \text (r) is the variance of r. Let K be a measure of the species abundance (organisms per unit area). Then : T_E = \frac \left( \log K - \frac 2\right) where TE is the mean time to local extinction. The probability of extinction by time ''t'' is : P( t ) = 1 - e^


Minimum population size required to avoid extinction

If a population is lognormally distributed then the
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...
of the population size (''H'') is related to the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
(''m'') : H = m - am^ Given that ''H'' must be > 0 for the population to persist then rearranging we have : m > a^ is the minimum size of population for the species to persist. The assumption of a lognormal distribution appears to apply to about half of a sample of 544 species. suggesting that it is at least a plausible assumption.


Sampling size estimators

The degree of precision (''D'') is defined to be ''s'' / ''m'' where ''s'' is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
and ''m'' is the mean. The degree of precision is known as the
coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), percent RMS, and relative standard deviation (RSD), is a standardized measure of dispersion of a probability ...
in other contexts. In ecology research it is recommended that ''D'' be in the range 10–25%.Southwood TRE & Henderson PA (2000) Ecological methods. 3rd ed. Blackwood, Oxford The desired degree of precision is important in estimating the required sample size where an investigator wishes to test if Taylor's law applies to the data. The required sample size has been estimated for a number of simple distributions but where the population distribution is not known or cannot be assumed more complex formulae may needed to determine the required sample size. Where the population is Poisson distributed the sample size (''n'') needed is : n = \frac m where ''t'' is critical level of the t distribution for the type 1 error with the
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
that the mean (''m'') was calculated with. If the population is distributed as a
negative binomial distribution In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...
then the required sample size is : n = \frac where ''k'' is the parameter of the negative binomial distribution. A more general sample size estimator has also been proposed : n = \left( \frac t D \right)^2 a m^ where a and b are derived from Taylor's law. An alternative has been proposed by SouthwoodSouthwood TRE (1978) Ecological methods. Chapman & Hall, London, England : n = a \frac \, where ''n'' is the required sample size, ''a'' and ''b'' are the Taylor's law coefficients and ''D'' is the desired degree of precision. Karandinos proposed two similar estimators for ''n''. The first was modified by Ruesink to incorporate Taylor's law.Ruesink WG (1980) Introduction to sampling theory, in Kogan M & Herzog DC (eds.) Sampling Methods in Soybean Entomology. Springer-Verlag New York, Inc, New York. pp 61–78 : n = \left( \frac t \right)^2 a m^ where ''d'' is the ratio of half the desired confidence interval (''CI'') to the mean. In symbols : d_m = \frac The second estimator is used in binomial (presence-absence) sampling. The desired sample size (''n'') is : n = \left( t d_p \right)^2 p^ q where the ''d''p is ratio of half the desired confidence interval to the proportion of sample units with individuals, ''p'' is proportion of samples containing individuals and ''q'' = 1 − ''p''. In symbols : d_p = \frac For binary (presence/absence) sampling, Schulthess ''et al'' modified Karandinos' equation : N = \left( \frac t \right)^2 \frac p where ''N'' is the required sample size, ''p'' is the proportion of units containing the organisms of interest, ''t'' is the chosen level of significance and ''D''ip is a parameter derived from Taylor's law.


Sequential sampling

Sequential analysis In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data is evaluated as it is collected, and further sampling is stopped in accordance with a pre-defi ...
is a method of
statistical analysis Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of ...
where the sample size is not fixed in advance. Instead samples are taken in accordance with a predefined
stopping rule In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of "random time": a random variable whose value is interpre ...
. Taylor's law has been used to derive a number of stopping rules. A formula for fixed precision in serial sampling to test Taylor's law was derived by Green in 1970. : \log T = \frac + (\log n) \frac where ''T'' is the cumulative sample total, ''D'' is the level of precision, ''n'' is the sample size and ''a'' and ''b'' are obtained from Taylor's law. As an aid to pest control Wilson ''et al'' developed a test that incorporated a threshold level where action should be taken.Wilson LT, Gonzalez D & Plant RE(1985) Predicting sampling frequency and economic status of spider mites on cotton. Proc. Beltwide Cotton Prod Res Conf, National Cotton Council of America, Memphis, TN pp 168-170 The required sample size is : n = t , m - T , ^ a m^b where ''a'' and ''b'' are the Taylor coefficients, , , is the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, ''m'' is the sample mean, ''T'' is the threshold level and ''t'' is the critical level of the t distribution. The authors also provided a similar test for binomial (presence-absence) sampling : n = t , m - T , ^ p q where ''p'' is the probability of finding a sample with pests present and ''q'' = 1 − ''p''. Green derived another sampling formula for sequential sampling based on Taylor's law : D = ( a n^ T^ )^ where ''D'' is the degree of precision, ''a'' and ''b'' are the Taylor's law coefficients, ''n'' is the sample size and ''T'' is the total number of individuals sampled. Serra ''et al'' have proposed a stopping rule based on Taylor's law. : T_n \ge \left( \frac \right)^ where ''a'' and ''b'' are the parameters from Taylor's law, ''D'' is the desired level of precision and ''T''n is the total sample size. Serra ''et al'' also proposed a second stopping rule based on Iwoa's regression : T_n \ge \frac where ''α'' and ''β'' are the parameters of the regression line, ''D'' is the desired level of precision and ''T''''n'' is the total sample size. The authors recommended that ''D'' be set at 0.1 for studies of population dynamics and ''D'' = 0.25 for pest control.


Related analyses

It is considered to be good practice to estimate at least one additional analysis of aggregation (other than Taylor's law) because the use of only a single index may be misleading. Although a number of other methods for detecting relationships between the variance and mean in biological samples have been proposed, to date none have achieved the popularity of Taylor's law. The most popular analysis used in conjunction with Taylor's law is probably Iwao's Patchiness regression test but all the methods listed here have been used in the literature.


Barlett–Iwao model

Barlett in 1936 and later Iwao independently in 1968 both proposed an alternative relationship between the variance and the mean. In symbols : s_i^2 = am_i + bm_i^2 \, where ''s'' is the variance in the ''i''th sample and ''m''''i'' is the mean of the ''i''th sample When the population follows a
negative binomial distribution In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...
, ''a'' = 1 and ''b'' = ''k'' (the exponent of the negative binomial distribution). This alternative formulation has not been found to be as good a fit as Taylor's law in most studies.


Nachman model

Nachman proposed a relationship between the mean density and the proportion of samples with zero counts: : p_0 = \exp( -a m^b ) where ''p''0 is the proportion of the sample with zero counts, ''m'' is the mean density, ''a'' is a scale parameter and ''b'' is a dispersion parameter. If ''a'' = ''b'' = 1 the distribution is random. This relationship is usually tested in its logarithmic form : \log m = c + d \log p_0 Allsop used this relationship along with Taylor's law to derive an expression for the proportion of infested units in a sample : P_1 = 1 - \exp\left( -\exp\left( \frac d \right) \right) : N = n P_1 where : A^2 = \frac where ''D''2 is the degree of precision desired, ''z''α/2 is the upper α/2 of the normal distribution, ''a'' and ''b'' are the Taylor's law coefficients, ''c'' and ''d'' are the Nachman coefficients, ''n'' is the sample size and ''N'' is the number of infested units.


Kono–Sugino equation

Binary sampling is not uncommonly used in ecology. In 1958 Kono and Sugino derived an equation that relates the proportion of samples without individuals to the mean density of the samples. : \log( m ) = \log( a ) + b \log( - \log( p_0 ) ) where ''p''0 is the proportion of the sample with no individuals, ''m'' is the mean sample density, ''a'' and ''b'' are constants. Like Taylor's law this equation has been found to fit a variety of populations including ones that obey Taylor's law. Unlike the negative binomial distribution this model is independent of the mean density. The derivation of this equation is straightforward. Let the proportion of empty units be ''p''0 and assume that these are distributed exponentially. Then : p_0 = \exp( -A m^B ) Taking logs twice and rearranging, we obtain the equation above. This model is the same as that proposed by Nachman. The advantage of this model is that it does not require counting the individuals but rather their presence or absence. Counting individuals may not be possible in many cases particularly where insects are the matter of study. ;Note The equation was derived while examining the relationship between the proportion ''P'' of a series of rice hills infested and the mean severity of infestation ''m''. The model studied was : P = 1 - a e^ where ''a'' and ''b'' are empirical constants. Based on this model the constants ''a'' and ''b'' were derived and a table prepared relating the values of ''P'' and ''m'' ;Uses The predicted estimates of ''m'' from this equation are subject to bias and it is recommended that the adjusted mean ( ''m''a ) be used instead : m_a = m \left( 1 - \frac 2 \right) where var is the variance of the sample unit means ''m''''i'' and ''m'' is the overall mean. An alternative adjustment to the mean estimates is : m_a = m e^ where MSE is the mean square error of the regression. This model may also be used to estimate stop lines for enumerative (sequential) sampling. The variance of the estimated means is : \operatorname( m ) = m^2 ( c_1 + c_2 - c_3 + \text ) where : c_1 = \frac : c_2 = \frac + s_\beta^2 ( \log_e( \log_e( p_0 ) ) - p^2 ) : c_3 = \frac n where MSE is the
mean square error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
of the regression, ''α'' and ''β'' are the constant and slope of the regression respectively, ''s''β2 is the variance of the slope of the regression, ''N'' is the number of points in the regression, ''n'' is the number of sample units and ''p'' is the mean value of ''p''0 in the regression. The parameters ''a'' and ''b'' are estimated from Taylor's law: : s^2 = a + b \log_e( m )


Hughes–Madden equation

Hughes and Madden have proposed testing a similar relationship applicable to binary observations in cluster, where each cluster contains from 0 to n individuals. : var_\text = a p^b ( 1 - p )^c where ''a'', ''b'' and ''c'' are constants, ''varobs'' is the observed variance, and p is the proportion of individuals with a trait (such as disease), an estimate of the probability of an individual with a trait. In logarithmic form, this relationship is : \log( \operatorname_\text ) = \log( a ) + b \log( p ) + c \log( 1 - p ) . In most cases, it is assumed that ''b = c'', leading to a simple model : \operatorname_\text = a ( p( 1 - p ) )^b This relationship has been subjected to less extensive testing than Taylor's law. However, it has accurately described over 100 data sets, and there are no published examples reporting that it does not works. A variant of this equation was proposed by Shiyomi et al. (Shiyomi M, Egawa T, Yamamoto Y (1998) Negative hypergeometric series and Taylor's power law in occurrence of plant populations in semi-natural grassland in Japan. Proceedings of the 18th International Grassland Congress on grassland management. The Inner Mongolia Univ Press pp 35–43 (1998)) who suggested testing the regression : \log( \operatorname_\text / n^2 ) = a + b \log \frac n where varobs is the variance, ''a'' and ''b'' are the constants of the regression, ''n'' here is the sample size (not sample per cluster) and ''p'' is the probability of a sample containing at least one individual.


Negative binomial distribution model

A negative binomial model has also been proposed. The dispersion parameter (''k'') using the method of moments is ''m''2 / ( ''s''2 – ''m'' ) and ''p''''i'' is the proportion of samples with counts > 0. The ''s''2 used in the calculation of ''k'' are the values predicted by Taylor's law. ''p''''i'' is plotted against 1 − (''k''(''k'' + ''m'')−1)''k'' and the fit of the data is visually inspected. Perry and Taylor have proposed an alternative estimator of ''k'' based on Taylor's law.Perry JN & Taylor LR(1986). Stability of real interacting populations in space and time: implications, alternatives and negative binomial. ''J Animal Ecol'' 55: 1053–1068 : \frac 1 k = \frac m A better estimate of the dispersion parameter can be made with the method of
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
. For the negative binomial it can be estimated from the equation : \sum \frac = N \log\left(1 + \frac m k\right) where ''A''''x'' is the total number of samples with more than ''x'' individuals, ''N'' is the total number of individuals, ''x'' is the number of individuals in a sample, ''m'' is the mean number of individuals per sample and ''k'' is the exponent. The value of ''k'' has to be estimated numerically. Goodness of fit of this model can be tested in a number of ways including using the chi square test. As these may be biased by small samples an alternative is the ''U'' statistic – the difference between the variance expected under the negative binomial distribution and that of the sample. The expected variance of this distribution is ''m'' + ''m''2 / ''k'' and : U = s^2 - m + \frac k where ''s''2 is the sample variance, ''m'' is the sample mean and ''k'' is the negative binomial parameter. The variance of U is : \operatorname( U ) = 2m p^2 q \left( \frac \right) + p^4 \frac where ''p'' = ''m'' / ''k'', ''q'' = 1 + ''p'', ''R'' = ''p'' / ''q'' and ''N'' is the total number of individuals in the sample. The expected value of ''U'' is 0. For large sample sizes ''U'' is distributed normally. Note: The negative binomial is actually a family of distributions defined by the relation of the mean to the variance \sigma^2 = \mu + a \mu^p where ''a'' and ''p'' are constants. When ''a'' = 0 this defines the Poisson distribution. With ''p'' = 1 and ''p'' = 2, the distribution is known as the NB1 and NB2 distribution respectively. This model is a version of that proposed earlier by Barlett.


Tests for a common dispersion parameter

The dispersion parameter (''k'') is : k = \frac where ''m'' is the sample mean and ''s''2 is the variance. If ''k''−1 is > 0 the population is considered to be aggregated; ''k''−1 = 0 the population is considered to be random; and if ''k''−1 is < 0 the population is considered to be uniformly distributed. Southwood has recommended regressing ''k'' against the mean and a constant : k_i = a + b m_i where ''k''''i'' and ''m''''i'' are the dispersion parameter and the mean of the ith sample respectively to test for the existence of a common dispersion parameter (''k''''c''). A slope (''b'') value significantly > 0 indicates the dependence of ''k'' on the mean density. An alternative method was proposed by Elliot who suggested plotting ( ''s''2 − ''m'' ) against ( ''m''2 − ''s''2 / ''n'' ).Elliot JM (1977) Some methods for the statistical analysis of samples of benthic invertebrates. 2nd ed. Freshwater Biological Association, Cambridge, United Kingdom ''k''''c'' is equal to 1/slope of this regression.


Charlier coefficient

This coefficient (''C'') is defined as : C = \frac m If the population can be assumed to be distributed in a negative binomial fashion, then ''C'' = 100 (1/''k'')0.5 where ''k'' is the dispersion parameter of the distribution.


Cole's index of dispersion

This index (''I''c) is defined as : I_c = \frac The usual interpretation of this index is as follows: values of ''I''c < 1, = 1, > 1 are taken to mean a uniform distribution, a random distribution or an aggregated distribution. Because ''s''2 = Σ x2 − (Σx)2, the index can also be written : I_c = \frac = \frac \frac + 1 If Taylor's law can be assumed to hold, then : I_c = \frac + 1


Lloyd's indexes

Lloyd's index of mean crowding (''IMC'') is the average number of other points contained in the sample unit that contains a randomly chosen point. : \mathrm = m + \frac where ''m'' is the sample mean and ''s''2 is the variance. Lloyd's index of patchiness (''IP'') is : \mathrm = \text / m It is a measure of pattern intensity that is unaffected by thinning (random removal of points). This index was also proposed by Pielou in 1988 and is sometimes known by this name also. Because an estimate of the variance of ''IP'' is extremely difficult to estimate from the formula itself, LLyod suggested fitting a negative binomial distribution to the data. This method gives a parameter ''k'' : s^2 = m + \frac k Then : SE( IP ) = \frac \left \operatorname( k ) + \frac \right where SE( IP ) is the standard error of the index of patchiness, \text (k) is the variance of the parameter ''k'' and ''q'' is the number of
quadrat A quadrat is a frame used in ecology, geography, and biology to isolate a standard unit of area for study of the distribution of an item over a large area. Quadrats typically occupy an area of 0.25 m2 and are traditionally square, but modern quad ...
s sampled.. If the population obeys Taylor's law then : \mathrm = m + a^ m^ - 1 : \mathrm = 1 + a^ m^ - \frac 1 m


Patchiness regression test

Iwao proposed a patchiness regression to test for clumping Let : y_i = m_i + \frac - 1 ''y''''i'' here is Lloyd's index of mean crowding. Perform an ordinary least squares regression of ''m''''i'' against ''y''. In this regression the value of the slope (''b'') is an indicator of clumping: the slope = 1 if the data is Poisson-distributed. The constant (''a'') is the number of individuals that share a unit of habitat at infinitesimal density and may be < 0, 0 or > 0. These values represent regularity, randomness and aggregation of populations in spatial patterns respectively. A value of ''a'' < 1 is taken to mean that the basic unit of the distribution is a single individual. Where the statistic ''s''2/''m'' is not constant it has been recommended to use instead to regress Lloyd's index against ''am'' + ''bm''2 where ''a'' and ''b'' are constants. The sample size (''n'') for a given degree of precision (''D'') for this regression is given by : n = \left( \frac t D \right)^2 \left( \frac + b - 1 \right) where ''a'' is the constant in this regression, ''b'' is the slope, ''m'' is the mean and ''t'' is the critical value of the t distribution. Iwao has proposed a sequential sampling test based on this regression. The upper and lower limits of this test are based on critical densities mc where control of a pest requires action to be taken. : N_u = im_c + t( i ( a + 1 ) m_c + ( b - 1 ) m_c^2 )^ : N_l = im_c - t( i ( a + 1 ) m_c + ( b - 1 ) m_c^2 )^ where ''N''''u'' and ''N''''l'' are the upper and lower bounds respectively, ''a'' is the constant from the regression, ''b'' is the slope and ''i'' is the number of samples. Kuno has proposed an alternative sequential stopping test also based on this regression. : T_n = \frac where ''T''''n'' is the total sample size, ''D'' is the degree of precision, ''n'' is the number of samples units, a is the constant and b is the slope from the regression respectively. Kuno's test is subject to the condition that ''n'' ≥ (''b'' − 1) / ''D''2 Parrella and Jones have proposed an alternative but related stop line : T_n = \left( 1 - \frac n N \right) \frac where ''a'' and ''b'' are the parameters from the regression, ''N'' is the maximum number of sampled units and ''n'' is the individual sample size.


Morisita’s index of dispersion

Masaaki Morisita's index of dispersion ( ''I''''m'' ) is the scaled probability that two points chosen at random from the whole population are in the same sample. Higher values indicate a more clumped distribution. : I_m = \frac An alternative formulation is : I_m = n \frac where ''n'' is the total sample size, ''m'' is the sample mean and ''x'' are the individual values with the sum taken over the whole sample. It is also equal to : I_m = \frac where ''IMC'' is Lloyd's index of crowding. This index is relatively independent of the population density but is affected by the sample size. Values > 1 indicate clumping; values < 1 indicate a uniformity of distribution and a value of 1 indicates a random sample. Morisita showed that the statistic : I_m \left( \sum x - 1 \right) + n - \sum x is distributed as a chi squared variable with ''n'' − 1 degrees of freedom. An alternative significance test for this index has been developed for large samples.Pedigo LP & Buntin GD (1994) Handbook of sampling methods for arthropods in agriculture. CRC Boca Raton FL : z = \frac where ''m'' is the overall sample mean, ''n'' is the number of sample units and ''z'' is the normal distribution
abscissa In mathematics, the abscissa (; plural ''abscissae'' or ''abscissas'') and the ordinate are respectively the first and second coordinate of a point in a Cartesian coordinate system: : abscissa \equiv x-axis (horizontal) coordinate : ordinate \eq ...
. Significance is tested by comparing the value of ''z'' against the values of the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
. A function for its calculation is available in the statistical
R language R, or r, is the eighteenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ar'' (pronounced ), plural ''ars''. The lette ...
in th
vegan package
Note, not to be confused with Morisita's overlap index.


Standardised Morisita’s index

Smith-Gill developed a statistic based on Morisita's index which is independent of both sample size and population density and bounded by −1 and +1. This statistic is calculated as follows First determine Morisita's index ( ''I''''d'' ) in the usual fashion. Then let ''k'' be the number of units the population was sampled from. Calculate the two critical values : M_u = \frac : M_c = \frac where χ2 is the chi square value for ''n'' − 1 degrees of freedom at the 97.5% and 2.5% levels of confidence. The standardised index ( ''I''''p'' ) is then calculated from one of the formulae below. When ''I''''d'' ≥ ''M''''c'' > 1 : I_p = 0.5 + 0.5 \left( \frac \right) When ''M''''c'' > ''I''''d'' ≥ 1 : I_p = 0.5 \left( \frac \right) When 1 > ''I''''d'' ≥ ''M''''u'' : I_p = -0.5 \left( \frac \right) When 1 > ''M''''u'' > ''I''''d'' : I_p = -0.5 + 0.5 \left( \frac \right) ''I''''p'' ranges between +1 and −1 with 95% confidence intervals of ±0.5. ''I''''p'' has the value of 0 if the pattern is random; if the pattern is uniform, ''I''''p'' < 0 and if the pattern shows aggregation, ''I''''p'' > 0.


Southwood's index of spatial aggregation

Southwood's index of spatial aggregation (''k'') is defined as : \frac = \frac - 1 where ''m'' is the mean of the sample and ''m''* is Lloyd's index of crowding.


Fisher's index of dispersion

Fisher's
index of dispersion In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a pro ...
Elliot JM (1977) Statistical analysis of samples of benthic invertebrates. Freshwater Biological Association. AmblesideFisher RA (1925) Statistical methods for research workers. Hafner, New York is : \mathrm = \frac This index may be used to test for over dispersion of the population. It is recommended that in applications n > 5 and that the sample total divided by the number of samples is > 3. In symbols : \frac > 3 where ''x'' is an individual sample value. The expectation of the index is equal to ''n'' and it is distributed as the
chi-square distribution The term chi-square, chi-squared, or \chi^2 has various uses in statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analys ...
with ''n'' − 1 degrees of freedom when the population is Poisson distributed. It is equal to the scale parameter when the population obeys the
gamma distribution In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the g ...
. It can be applied both to the overall population and to the individual areas sampled individually. The use of this test on the individual sample areas should also include the use of a Bonferroni correction factor. If the population obeys Taylor's law then : \mathrm = ( n - 1 ) a m^


Index of cluster size

The index of cluster size (''ICS'') was created by David and Moore. Under a random (Poisson) distribution ''ICS'' is expected to equal 0. Positive values indicate a clumped distribution; negative values indicate a uniform distribution. : \mathrm = \frac where ''s''2 is the variance and ''m'' is the mean. If the population obeys Taylor's law : \mathrm = a m^ - 1 The ''ICS'' is also equal to Katz's test statistic divided by ( ''n'' / 2 )1/2 where ''n'' is the sample size. It is also related to Clapham's test statistic. It is also sometimes referred to as the clumping index.


Green’s index

Green's index (''GI'') is a modification of the index of cluster size that is independent of ''n'' the number of sample units. : C_x = \frac This index equals 0 if the distribution is random, 1 if it is maximally aggregated and −1 / ( ''nm'' − 1 ) if it is uniform. The distribution of Green's index is not currently known so statistical tests have been difficult to devise for it. If the population obeys Taylor's law : C_x = \frac


Binary dispersal index

Binary sampling (presence/absence) is frequently used where it is difficult to obtain accurate counts. The dispersal index (''D'') is used when the study population is divided into a series of equal samples ( number of units = ''N'': number of units per sample = ''n'': total population size = ''n'' x ''N'' ). The theoretical variance of a sample from a population with a binomial distribution is : s^2 = n p ( 1 - p ) where ''s''2 is the variance, ''n'' is the number of units sampled and ''p'' is the mean proportion of sampling units with at least one individual present. The dispersal index (''D'') is defined as the ratio of observed variance to the expected variance. In symbols : D = \frac = \frac where varobs is the observed variance and varbin is the expected variance. The expected variance is calculated with the overall mean of the population. Values of ''D'' > 1 are considered to suggest aggregation. ''D''( ''n'' − 1 ) is distributed as the chi squared variable with ''n'' − 1 degrees of freedom where ''n'' is the number of units sampled. An alternative test is the ''C'' test. : C = \frac where ''D'' is the dispersal index, ''n'' is the number of units per sample and ''N'' is the number of samples. C is distributed normally. A statistically significant value of C indicates
overdispersion In statistics, overdispersion is the presence of greater variability (statistical dispersion) in a data set than would be expected based on a given statistical model. A common task in applied statistics is choosing a parametric model to fit a giv ...
of the population. ''D'' is also related to
intraclass correlation In statistics, the intraclass correlation, or the intraclass correlation coefficient (ICC), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly uni ...
(''ρ'') which is defined asFleiss JL (1981) Statistical methods for rates and proportions. 2nd ed. Wiley, New York, USA : \rho = 1 - \frac where ''T'' is the number of organisms per sample, ''p'' is the likelihood of the organism having the sought after property (diseased, pest free, ''etc''), and xi is the number of organism in the ''i''th unit with this property. ''T'' must be the same for all sampled units. In this case with ''n'' constant : \rho = \frac If the data can be fitted with a
beta-binomial distribution In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Ber ...
then : D = 1 + \frac where ''θ'' is the parameter of the distribution.


Ma's population aggregation critical density

Ma has proposed a parameter (''m''0) − the population aggregation critical density - to relate population density to Taylor's law.Ma ZS (1991) Further interpreted Taylor’s Power Law and population aggregation critical density. Trans Ecol Soc China (1991) 284–288 : m_0 = \exp\left( \frac \right)


Related statistics

A number of statistical tests are known that may be of use in applications.


de Oliveria's statistic

A related statistic suggested by de Oliveriade Oliveria T (1965) Some elementary tests for mixtures of discrete distributions, ''in'' Patil, GP ''ed.'', Classical and contagious discrete distributions. Calcutá, Calcutta Publishing Society pp379-384 is the difference of the variance and the mean. If the population is Poisson distributed then : var( s^2 - m ) = \frac where ''t'' is the Poisson parameter, ''s''2 is the variance, ''m'' is the mean and ''n'' is the sample size. The expected value of ''s''2 - ''m'' is zero. This statistic is distributed normally. If the Poisson parameter in this equation is estimated by putting ''t'' = ''m'', after a little manipulation this statistic can be written : O_T = \sqrt \frac This is almost identical to Katz's statistic with ( ''n'' - 1 ) replacing ''n''. Again ''O''''T'' is normally distributed with mean 0 and unit variance for large ''n''. This statistic is the same as the Neyman-Scott statistic. ;Note de Oliveria actually suggested that the variance of ''s''2 - ''m'' was ( 1 - 2''t''1/2 + 3''t'' ) / ''n'' where ''t'' is the Poisson parameter. He suggested that ''t'' could be estimated by putting it equal to the mean (''m'') of the sample. Further investigation by Bohning showed that this estimate of the variance was incorrect. Bohning's correction is given in the equations above.


Clapham's test

In 1936 Clapham proposed using the ratio of the variance to the mean as a test statistic (the relative variance). In symbols : \theta = \frac For a Possion distribution this ratio equals 1. To test for deviations from this value he proposed testing its value against the chi square distribution with ''n'' degrees of freedom where ''n'' is the number of sample units. The distribution of this statistic was studied further by BlackmanBlackman GE (1942) Statistical and ecological studies on the distribution of species in plant communities. I. Dispersion as a factor in the study of changes in plant populations. Ann Bot N.s. vi: 351 who noted that it was approximately normally distributed with a mean of 1 and a variance ( ''V''θ ) of : V_ = \frac The derivation of the variance was re analysed by Bartlett who considered it to be : V_\theta = \frac 2 For large samples these two formulae are in approximate agreement. This test is related to the later Katz's ''J''''n'' statistic. If the population obeys Taylor's law then : \theta = am^ ;Note A refinement on this test has also been published These authors noted that the original test tends to detect overdispersion at higher scales even when this was not present in the data. They noted that the use of the multinomial distribution may be more appropriate than the use of a Poisson distribution for such data. The statistic ''θ'' is distributed : \theta = \frac m = \frac 1 n \sum \left( x_i - \frac n N \right)^2 where ''N'' is the number of sample units, ''n'' is the total number of samples examined and ''x''i are the individual data values. The expectation and variance of ''θ'' are : \operatorname E( \theta ) = \frac N : \operatorname(\theta) = \frac - \frac For large ''N'', E(''θ'') is approximately 1 and : \operatorname(\theta) \sim\ \frac 2 N \left( 1 - \frac 1 n \right) If the number of individuals sampled (''n'') is large this estimate of the variance is in agreement with those derived earlier. However, for smaller samples these latter estimates are more precise and should be used.


See also

* Morisita's overlap index * Natural exponential family *
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*
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* Density-mass allometry * Variance-mass allometry


References

{{Statistics, applications Ecology Biology laws Power Laws Statistical deviation and dispersion Statistical laws Environmental statistics