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Taylor's law

Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship.[1] It is named after the ecologist who first proposed it in 1961, Lionel Roy Taylor (1924–2007).[2] Taylor's original name for this relationship was the law of the mean.[1] The name Taylor's law was coined by Southwood in 1966.[2]

Definition

This law was originally defined for ecological systems, specifically to assess the spatial clustering of organisms. For a population count   with mean   and variance  , Taylor's law is written

 

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.[1] This power law has subsequently been confirmed for many hundreds of species.[3][4]

Taylor's law has also been applied to assess the time dependent changes of population distributions.[3] Related variance to mean power laws have also been demonstrated in several non-ecological systems:

History

The first use of a double log-log plot was by Reynolds in 1879 on thermal aerodynamics.[17] Pareto used a similar plot to study the proportion of a population and their income.[18]

The term variance was coined by Fisher in 1918.[19]

Biology

Pearson[20] in 1921 proposed the equation (also studied by Neyman[21])

 

Smith in 1938 while studying crop yields proposed a relationship similar to Taylor's.[22] This relationship was

 

where Vx is the variance of yield for plots of x units, V1 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[23] in 1941, Fracker and Brischle[24] in 1941 and Hayman & Lowe [25] 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 lesions, macro-zooplankton, worms and symphylids in soil, insects in soil, on plants and in the air, mites on leaves, ticks on sheep and fish in the sea.;[1] 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 that could result from birth–death processes.[26] Taylor's explanation was based the assumption of a balanced migratory and congregatory behavior of animals.[1] His hypothesis was initially qualitative, but as it evolved it became semi-quantitative and was supported by simulations.[27]

Many alternative hypotheses for the power law have been advanced. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction.[28] 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.[3][4]

Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function.[29] As a response to this model Taylor argued that such a Markov process would predict that the power law exponent would vary considerably between replicate observations, and that such variability had not been observed.[30]

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.[31] Kemp, however, did not explain the parameterizations of his models in mechanistic terms. Other relatively abstract models for Taylor's law followed.[6][32]

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.[33][34]

Taylor's law has been applied to time series data, and Perry showed, using simulations, that chaos theory could yield Taylor's law.[35]

Taylor's law has been applied to the spatial distribution of plants[36] and bacterial populations[37] 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,[38] a family of probabilistic models that express an inherent power function relationship between the variance and the mean.[11][13][39]

Several alternative hypotheses for the power law have been proposed. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction.[28] 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.[3][4] Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function.[29] The Lewontin Cohen growth model.[40] is another proposed explanation. The possibility that observations of a power law might reflect more mathematical artifact than a mechanistic process was raised.[41] Variation in the exponents of Taylor's Law applied to ecological populations cannot be explained or predicted based solely on statistical grounds however.[42] 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.[43]

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.[44] 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.[45] 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.[14]

Mathematics

Taylor's law has been shown to hold for prime numbers not exceeding a given real number.[46] 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.

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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.[47][48] 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 (CGF) is:

 

where κb(θ) is the cumulant function,

 

the Tweedie exponent

 

s is the generating function variable, and θ and λ are the canonical and index parameters, respectively.[38]

These last two parameters are analogous to the scale and shape parameters used in probability theory. The cumulants 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

 

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.[39] 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.[12][13][14][16]

The main justification for the Tweedie hypothesis rests with the mathematical convergence properties of the Tweedie distributions.[13] The Tweedie convergence theorem requires the Tweedie distributions to act as foci of convergence for a wide range of statistical processes.[49] 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.[14][16]

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.[6]

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.[50]

Mathematical formulation

In symbols

 

where si2 is the variance of the density of the ith sample, mi is the mean density of the ith sample and a and b are constants.

In logarithmic form

 

Scale invariance

The exponent in Taylor's law is scale invariant: If the unit of measurement is changed by a constant factor  , the exponent ( ) remains unchanged.

To see this let y = cx. Then

 
 
 
 

Taylor's law expressed in the original variable (x) is

 

and in the rescaled variable (y) it is

 

Thus,   is still proportional to   (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.[51]

Extensions and refinements

A refinement in the estimation of the slope b has been proposed by Rayner.[52]

 

where   is the Pearson moment correlation coefficient between   and  ,   is the ratio of sample variances in   and   and   is the ratio of the errors in   and  .

Ordinary least squares regression assumes that φ = ∞. This tends to underestimate the value of b because the estimates of both   and   are subject to error.

An extension of Taylor's law has been proposed by Ferris et al when multiple samples are taken[53]

 

where s2 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.[54] 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

 

Assuming the validity of Taylor's law, we have

 

Because in the Poisson distribution the mean equals the variance, we have

 

This gives us

 

This closely resembles Barlett's original suggestion.

Interpretation

Slope values (b) significantly > 1 indicate clumping of the organisms.

In Poisson-distributed data, b = 1.[30] If the population follows a lognormal or gamma distribution, 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.[55] Occasionally cases with b > 2 have been reported.[3] b values below 1 are uncommon but have also been reported ( b = 0.93 ).[50]

It has been suggested that the exponent of the law (b) is proportional to the skewness of the underlying distribution.[56] This proposal has criticised: additional work seems to be indicated.[57][58]

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.[59]

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.[54]

Extension to cluster sampling of binary data

A form of Taylor's law applicable to binary data in clusters (e.q., quadrats) has been proposed.[60] In a binomial distribution, the theoretical variance is

 

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).[60]

Hughes and Madden noted that if the distribution is Poisson, the mean and variance are equal.[60] 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

 

In logarithmic form this relationship is

 

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[61] 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[60] 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[62] have showed 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.[63] 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.[62] The break in the line occurs only at p very close to 0.

An extension to this law has been proposed.[64] The original form of this law is symmetrical but it can be extended to an asymmetrical form.[64] 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[65] 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 ( s2 ) of the population are equal and the proportion of samples that contain at least one individual ( p ) is

 

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.[66] The basic relationship is

 

where the log is taken to the base e.

Incorporating Taylor's law this relationship becomes

 

Dispersion parameter estimator

The common dispersion parameter (k) of the negative binomial distribution is

 

where   is the sample mean and   is the variance.[67] If 1 / k is > 0 the population is considered to be aggregated; 1 / k = 0 ( s2 = 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:[66]

 

where a and b are the constants from Taylor's law.

Jones[68] 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[66]

 

derived an estimator for the probability of a sample containing x individuals per sampling unit. Jones's formula is

 

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

 

Jones also gives confidence intervals for these probabilities.

 

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 ( w1, w2 ).[69] This family of distributions includes the Bernoulli, Geometric, Pascal and Poisson distributions as special cases. The mean and variance of a Katz distribution are

 
 

where m is the mean and s2 is the variance of the sample. The parameters can be estimated by the method of moments from which we have

 
 

For a Poisson distribution w2 = 0 and w1 = 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:[70]

 

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

 
 

Katz also introduced a statistical test[69]

 

where Jn is the test statistic, s2 is the variance of the sample, m is the mean of the sample and n is the sample size. Jn is asymptotically normally distributed with a zero mean and unit variance. If the sample is Poisson distributed Jn = 0; values of Jn < 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

 

which is known to be asymptotically normal and the conditional chi-squared statistic (Poisson dispersion test)

 

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

 

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.[71]

Let   where Nt+1 and Nt are the population sizes at time t + 1 and t respectively and r is parameter equal to the annual increase (decrease in population). Then

 

where   is the variance of  .

Let   be a measure of the species abundance (organisms per unit area). Then

 

where TE is the mean time to local extinction.

The probability of extinction by time t is

 

Minimum population size required to avoid extinction

If a population is lognormally distributed then the harmonic mean of the population size (H) is related to the arithmetic mean (m)[72]

 

Given that H must be > 0 for the population to persist then rearranging we have

 

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.[73] 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 and m is the mean. The degree of precision is known as the coefficient of variation in other contexts. In ecology research it is recommended that D be in the range 10–25%.[74] 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

 

where t is critical level of the t distribution for the type 1 error with the degrees of freedom that the mean (m) was calculated with.

If the population is distributed as a negative binomial distribution then the required sample size is

 

where k is the parameter of the negative binomial distribution.

A more general sample size estimator has also been proposed[75]

 

where a and b are derived from Taylor's law.

An alternative has been proposed by Southwood[76]

 

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.[77] The first was modified by Ruesink to incorporate Taylor's law.[78]

 

where d is the ratio of half the desired confidence interval (CI) to the mean. In symbols

 

The second estimator is used in binomial (presence-absence) sampling. The desired sample size (n) is

 

where the dp 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

 

For binary (presence/absence) sampling, Schulthess et al modified Karandinos' equation

 

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 Dip is a parameter derived from Taylor's law.[79]

Sequential sampling

Sequential analysis is a method of statistical analysis where the sample size is not fixed in advance. Instead samples are taken in accordance with a predefined stopping rule. 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.[80]

 

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.[81] The required sample size is

 

where a and b are the Taylor coefficients, || is the absolute value, 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

 

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[82]

 

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.[83]

 

where a and b are the parameters from Taylor's law, D is the desired level of precision and Tn is the total sample size.

Serra et al also proposed a second stopping rule based on Iwoa's regression

 

where α and β are the parameters of the regression line, D is the desired level of precision and Tn 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.[84] 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[85] and later Iwao independently in 1968[86] both proposed an alternative relationship between the variance and the mean. In symbols

 

where s is the variance in the ith sample and mi is the mean of the ith sample

When the population follows a negative binomial distribution, 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:[87]

 

where p0 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 = 0 the distribution is random. This relationship is usually tested in its logarithmic form

 

Allsop used this relationship along with Taylor's law to derive an expression for the proportion of infested units in a sample[88]

 
 

where

 

where D2 is the degree of precision desired, zα/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.[89]

 

where p0 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 p0 and assume that these are distributed exponentially. Then

 

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

 

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[90] and it is recommended that the adjusted mean ( ma ) be used instead[91]

 

where var is the variance of the sample unit means mi and m is the overall mean.

An alternative adjustment to the mean estimates is[91]

 

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[92]

 

where

 
 
 

where MSE is the mean square error of the regression, α and β are the constant and slope of the regression respectively, sβ2 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 p0 in the regression. The parameters a and b are estimated from Taylor's law:

 

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.[60]

 

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

 

In most cases, it is assumed that b = c, leading to a simple model

 

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.[62]

A variant of this equation was proposed by Shiyomi et al. ([93]) who suggested testing the regression

 

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.[94] The dispersion parameter (k) using the method of moments is m2 / ( s2m ) and pi is the proportion of samples with counts > 0. The s2 used in the calculation of k are the values predicted by Taylor's law. pi 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.[95]

 

A better estimate of the dispersion parameter can be made with the method of maximum likelihood. For the negative binomial it can be estimated from the equation[67]

 

where Ax 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 + m2 / k and

 

where s2 is the sample variance, m is the sample mean and k is the negative binomial parameter.

The variance of U is[67]

 

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

 

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)[67] is

 

where m is the sample mean and s2 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[76]

 

where ki and mi are the dispersion parameter and the mean of the ith sample respectively to test for the existence of a common dispersion parameter (kc). 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 ( s2m ) against ( m2s2 / n ).[96] kc is equal to 1/slope of this regression.

Charlier coefficient

This coefficient (C) is defined as

 

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 (Ic) is defined as[97]

 

The usual interpretation of this index is as follows: values of Ic < 1, = 1, > 1 are taken to mean a uniform distribution, a random distribution or an aggregated distribution.

Because s2 = Σ x2 − (Σx)2, the index can also be written

 

If Taylor's law can be assumed to hold, then

 

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.[98]

 

where m is the sample mean and s2 is the variance.

Lloyd's index of patchiness (IP)[98] is

 

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

 

Then

 

where   is the standard error of the index of patchiness,   is the variance of the parameter k and q is the number of quadrats sampled..

If the population obeys Taylor's law then

 
 

Patchiness regression test

Iwao proposed a patchiness regression to test for clumping[99][100]

Let

 

yi here is Lloyd's index of mean crowding.[98] Perform an ordinary least squares regression of mi 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 s2/m is not constant it has been recommended to use instead to regress Lloyd's index against am + bm2 where a and b are constants.[101]

The sample size (n) for a given degree of precision (D) for this regression is given by[101]

 

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.[102] The upper and lower limits of this test are based on critical densities mc where control of a pest requires action to be taken.

 
 

where Nu and Nl 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.[103]

 

where Tn 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) / D2

Parrella and Jones have proposed an alternative but related stop line[104]

 

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

Morisita's index of dispersion ( Im ) is the scaled probability that two points chosen at random from the whole population are in the same sample.[105] Higher values indicate a more clumped distribution.

 

An alternative formulation is

 

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

 

where IMC is Lloyd's index of crowding.[98]

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[105]

 

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.[106]

 

where m is the overall sample mean, n is the number of sample units and z is the normal distribution abscissa. Significance is tested by comparing the value of z against the values of the normal distribution.

A function for its calculation is available in the statistical R language. R function

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[107]

First determine Morisita's index ( Id ) in the usual fashion. Then let k be the number of units the population was sampled from. Calculate the two critical values

 
 

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 ( Ip ) is then calculated from one of the formulae below.

When IdMc > 1

 

When Mc > Id ≥ 1

 

When 1 > IdMu

 

When 1 > Mu > Id

 

Ip ranges between +1 and −1 with 95% confidence intervals of ±0.5. Ip has the value of 0 if the pattern is random; if the pattern is uniform, Ip < 0 and if the pattern shows aggregation, Ip > 0.

Southwood's index of spatial aggregation

Southwood's index of spatial aggregation (k) is defined as

 

where m is the mean of the sample and m* is Lloyd's index of crowding.[76]

Fisher's index of dispersion

Fisher's index of dispersion[108][109] is

 

This index may be used to test for over dispersion of the population. It is recommended that in applications n > 5[110] and that the sample total divided by the number of samples is > 3. In symbols

 

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 with n − 1 degrees of freedom when the population is Poisson distributed.[110] It is equal to the scale parameter when the population obeys the gamma distribution.

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

 

Index of cluster size

The index of cluster size (ICS) was created by David and Moore.[111] Under a random (Poisson) distribution ICS is expected to equal 0. Positive values indicate a clumped distribution; negative values indicate a uniform distribution.

 

where s2 is the variance and m is the mean.

If the population obeys Taylor's law

 

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.[112]

 

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

 

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 ).[113] The theoretical variance of a sample from a population with a binomial distribution is

 

where s2 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

 

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.[114]

 

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 of the population.

D is also related to intraclass correlation (ρ) which is defined as[115]

 

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 ith unit with this property. T must be the same for all sampled units. In this case with n constant

 

If the data can be fitted with a beta-binomial distribution then[115]

 

where θ is the parameter of the distribution.[114]

Ma's population aggregation critical density

Ma has proposed a parameter (m0) − the population aggregation critical density - to relate population density to Taylor's law.[116]

 

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 Oliveria[117] is the difference of the variance and the mean.[118] If the population is Poisson distributed then

 

where t is the Poisson parameter, s2 is the variance, m is the mean and n is the sample size. The expected value of s2 - m is zero. This statistic is distributed normally.[119]

If the Poisson parameter in this equation is estimated by putting t = m, after a little manipulation this statistic can be written

 

This is almost identical to Katz's statistic with ( n - 1 ) replacing n. Again OT 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 s2 - m was ( 1 - 2t1/2 + 3t ) / 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[118] 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).[120] In symbols

 

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 Blackman[121] who noted that it was approximately normally distributed with a mean of 1 and a variance ( Vθ ) of

 

The derivation of the variance was re analysed by Bartlett[122] who considered it to be

 

For large samples these two formulae are in approximate agreement. This test is related to the later Katz's Jn statistic.

If the population obeys Taylor's law then

 
Note

A refinement on this test has also been published[123] 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

 

where N is the number of sample units, n is the total number of samples examined and xi are the individual data values.

The expectation and variance of θ are

 
 

For large N, E(θ) is approximately 1 and

 

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

References

  1. ^ a b c d e Taylor, L. R. (1961). "Aggregation, variance and the mean". Nature. 189 (4766): 732–735. Bibcode:1961Natur.189..732T. doi:10.1038/189732a0. S2CID 4263093.
  2. ^ a b Thomas R. E. Southwood (1966). Ecological methods, with particular reference to the study of insect populations. Methuen. ISBN 9780416289305.
  3. ^ a b c d e Taylor, L. R.; Woiwod, I. P. (1980). "Temporal stability as a density-dependent species characteristic". Journal of Animal Ecology. 49 (1): 209–224. doi:10.2307/4285. JSTOR 4285.
  4. ^ a b c Taylor, LR; Woiwod (1982). "Comparative Synoptic Dynamics. I. Relationships Between Inter- and Intra-Specific Spatial and Temporal Variance/Mean Population Parameters". J Anim Ecol. 51 (3): 879–906. doi:10.2307/4012. JSTOR 4012.
  5. ^ Kendal, WS; Frost, P (1987). "Experimental metastasis: a novel application of the variance-to-mean power function". J Natl Cancer Inst. 79 (5): 1113–1115. PMID 3479636.
  6. ^ a b c Kendal, WS (1995). "A probabilistic model for the variance to mean power law in ecology". Ecological Modelling. 80 (2–3): 293–297. doi:10.1016/0304-3800(94)00053-k.
  7. ^ Keeling, M; Grenfell, B (1999). "Stochastic dynamics and a power law for measles variability". Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences. 354 (1384): 769–776. doi:10.1098/rstb.1999.0429. PMC 1692561. PMID 10365402.
  8. ^ Anderson, RM; May, RM (1989). "Epidemiological parameters of HIV transmission". Nature. 333 (6173): 514–519. doi:10.1038/333514a0. PMID 3374601. S2CID 43491211.
  9. ^ Philippe, P (1999). "The scale-invariant spatial clustering of leukemia in San Francisco". J Theor Biol. 199 (4): 371–381. Bibcode:1999JThBi.199..371P. doi:10.1006/jtbi.1999.0964. PMID 10441455.
  10. ^ Bassingthwaighte, JB (1989). "Fractal nature of regional myocardial blood flow heterogeneity". Circ Res. 65 (3): 578–590. doi:10.1161/01.res.65.3.578. PMC 3361973. PMID 2766485.
  11. ^ a b Kendal, WS (2001). "A stochastic model for the self-similar heterogeneity of regional organ blood flow". Proc Natl Acad Sci U S A. 98 (3): 837–841. Bibcode:2001PNAS...98..837K. doi:10.1073/pnas.98.3.837. PMC 14670. PMID 11158557.
  12. ^ a b Kendal, WS (2003). "An exponential dispersion model for the distribution of human single nucleotide polymorphisms". Mol Biol Evol. 20 (4): 579–590. doi:10.1093/molbev/msg057. PMID 12679541.
  13. ^ a b c d Kendal, WS (2004). "A scale invariant clustering of genes on human chromosome 7". BMC Evol Biol. 4 (1): 3. doi:10.1186/1471-2148-4-3. PMC 373443. PMID 15040817.
  14. ^ a b c d Kendal, WS; Jørgensen, B (2011). "Taylor's power law and fluctuation scaling explained by a central-limit-like convergence". Phys. Rev. E. 83 (6): 066115. Bibcode:2011PhRvE..83f6115K. doi:10.1103/physreve.83.066115. PMID 21797449.
  15. ^ Kendal, WS; Jørgensen, B (2015). "A scale invariant distribution of the prime numbers". Computation. 3 (4): 528–540. doi:10.3390/computation3040528.
  16. ^ a b c Kendal, WS; Jørgensen, BR (2011). "Tweedie convergence: a mathematical basis for Taylor's power law, 1/f noise and multifractality". Phys. Rev. E. 84 (6): 066120. Bibcode:2011PhRvE..84f6120K. doi:10.1103/physreve.84.066120. PMID 22304168. S2CID 22585727.
  17. ^ Reynolds, O (1879). "On certain dimensional properties of matter in the gaseous state. Part I. Experimental researches on thermal transpiration of gases through porous plates and on the laws of transpiration and impulsion, including an experimental proof that gas is not a continuous plenum. Part II. On an extension of the dynamical theory of gas, which includes the stresses, tangential and normal, caused by a varying condition of gas, and affords an explanation of the phenomena of transpiration and impulsion". Philosophical Transactions of the Royal Society of London. 170: 727–845. doi:10.1098/rstl.1879.0078.
  18. ^ Pareto V (1897) Cours D'économie Politique. Volume 2. Lausanne: F. Rouge
  19. ^ Fisher, RA (1918). "The correlation between relatives on the supposition of Mendelian inheritance". Transactions of the Royal Society of Edinburgh. 52 (2): 399–433. doi:10.1017/S0080456800012163.
  20. ^ Pearson, K (1921). "On a General Method of determining the successive terms in a Skew Regression Line". Biometrika. 13 (2–3): 296–300. doi:10.2307/2331756.
  21. ^ Neyman, J (1926). "On the correlation of the mean and the variance in samples drawn from an "infinite" population". Biometrika. 18 (3/4): 401–413. doi:10.2307/2331958. JSTOR 2331958.
  22. ^ Smith, HF (1938). "An empirical law describing heterogeneity in the yield of agricultural crops". J Agric Sci. 28: 1–23. doi:10.1017/s0021859600050516. S2CID 85867752.
  23. ^ Bliss, CI (1941). "Statistical problems in estimating populations of Japanese beetle larve". J Econ Entomol. 34 (2): 221–232. doi:10.1093/jee/34.2.221.
  24. ^ Fracker, SB; Brischle, HA (1944). "Measuring the local distribution of Ribes". Ecology. 25 (3): 283–303. doi:10.2307/1931277. JSTOR 1931277.
  25. ^ Hayman, BI; Lowe, AD (1961). "The transformation of counts of the cabbage aphid (Brevicovyne brassicae (L.))". NZ J Sci. 4: 271–278.
  26. ^ Taylor, LR (1984). "Anscombe's hypothesis and the changing distributions of insect populations". Antenna. 8: 62–67.
  27. ^ Taylor, LR; Taylor, RAJ (1977). "Aggregation, migration and population mechanics". Nature. 265 (5593): 415–421. Bibcode:1977Natur.265..415T. doi:10.1038/265415a0. PMID 834291. S2CID 6504396.
  28. ^ a b Hanski, I (1980). "Spatial patterns and movements in coprophagous beetles". Oikos. 34 (3): 293–310. doi:10.2307/3544289. JSTOR 3544289.
  29. ^ a b Anderson, RD; Crawley, GM; Hassell, M (1982). "Variability in the abundance of animal and plant species". Nature. 296 (5854): 245–248. Bibcode:1982Natur.296..245A. doi:10.1038/296245a0. S2CID 4272853.
  30. ^ a b Taylor, LR; Taylor, RAJ; Woiwod, IP; Perry, JN (1983). "Behavioural dynamics". Nature. 303 (5920): 801–804. Bibcode:1983Natur.303..801T. doi:10.1038/303801a0. S2CID 4353208.
  31. ^ Kemp, AW (1987). "Families of discrete distributions satisfying Taylor's power law". Biometrics. 43 (3): 693–699. doi:10.2307/2532005. JSTOR 2532005.
  32. ^ Yamamura, K (1990). "Sampling scale dependence of Taylor's power law". Oikos. 59 (1): 121–125. doi:10.2307/3545131. JSTOR 3545131.
  33. ^ Routledge, RD; Swartz, TB (1991). "Taylor's power law re-examined". Oikos. 60 (1): 107–112. doi:10.2307/3544999. JSTOR 3544999.
  34. ^ Tokeshi, M (1995). "On the mathematical basis of the variance–mean power relationship". Res Pop Ecol. 37: 43–48. doi:10.1007/bf02515760. S2CID 40805500.
  35. ^ Perry, JN (1994). "Chaotic dynamics can generate Taylor's power law". Proceedings of the Royal Society B: Biological Sciences. 257 (1350): 221–226. Bibcode:1994RSPSB.257..221P. doi:10.1098/rspb.1994.0118. S2CID 128851189.
  36. ^ Clark, S; Perry, JJN; Marshall, JP (1996). "Estimating Taylor's power law parameters for weeds and the effect of spatial scale". Weed Research. 36 (5): 405–417. doi:10.1111/j.1365-3180.1996.tb01670.x.
  37. ^ Ramsayer 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
  38. ^ a b Jørgensen, Bent (1997). The theory of dispersion models. [Chapman & Hall]. ISBN 978-0412997112.
  39. ^ a b Kendal, WS (2002). "Spatial aggregation of the Colorado potato beetle described by an exponential dispersion model". Ecol Model. 151 (2–3): 261–269. doi:10.1016/s0304-3800(01)00494-x.
  40. ^ Cohen, J E; Xu, m; Schuster, W S (2013). "Stochastic multiplicative population growth predicts and interprets Taylor's power law of fluctuation scaling". Proc R Soc Lond B Biol Sci. 280 (1757): 20122955. doi:10.1098/rspb.2012.2955. PMC 3619479. PMID 23427171.
  41. ^ Downing, JA (1986). "Spatial heterogeneity: evolved behaviour or mathematical artefact?". Nature. 323 (6085): 255–257. Bibcode:1986Natur.323..255D. doi:10.1038/323255a0. S2CID 4323456.
  42. ^ Xiao, X., Locey, K. & White, E.P. (2015). "A process-independent explanation for the general form of Taylor's law". The American Naturalist. 186 (2): 51–60. arXiv:1410.7283. doi:10.1086/682050. PMID 26655161. S2CID 14649978.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  43. ^ Cobain, M.R.D., Brede, M. & Trueman, C. N. (2018). "Taylor's power law captures the effects of environmental variability on community structure: An example from fishes in the North Sea" (PDF). Journal of Animal Ecology. 88 (2): 290–301. doi:10.1111/1365-2656.12923. PMID 30426504. S2CID 53306901.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  44. ^ Eisler, Z; Bartos, I; Kertesz (2008). "Fluctuation scaling in complex systems: Taylor's law and beyond". Adv Phys. 57 (1): 89–142. arXiv:0708.2053. Bibcode:2008AdPhy..57...89E. doi:10.1080/00018730801893043. S2CID 119608542.
  45. ^ Fronczak, A; Fronczak, P (2010). "Origins of Taylor's power law for fluctuation scaling in complex systems". Phys Rev E. 81 (6): 066112. arXiv:0909.1896. Bibcode:2010PhRvE..81f6112F. doi:10.1103/physreve.81.066112. PMID 20866483. S2CID 17435198.
  46. ^ Cohen, JE (2016). "Statistics of primes (and probably twin primes) satisfy Taylor's Law from ecology". The American Statistician. 70 (4): 399–404. doi:10.1080/00031305.2016.1173591. S2CID 13832952.
  47. ^ 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
  48. ^ Jørgensen, B (1987). "Exponential dispersion models". J R Stat Soc Ser B. 49 (2): 127–162. doi:10.1111/j.2517-6161.1987.tb01685.x.
  49. ^ Jørgensen, B; Marinez, JR; Tsao, M (1994). "Asymptotic behaviour of the variance function". Scandinavian Journal of Statistics. 21: 223–243.
  50. ^ a b Wilson, LT; Room, PM (1982). "The relative efficiency and reliability of three methods for sampling arthropods in Australian cotton fields". Australian Journal of Entomology. 21 (3): 175–181. doi:10.1111/j.1440-6055.1982.tb01786.x.
  51. ^ Jørgensen B (1997) The theory of exponential dispersion models. Chapman & Hall. London
  52. ^ Rayner, JMV (1985). "Linear relations in biomechanics: the statistics of scaling functions". Journal of Zoology. 206 (3): 415–439. doi:10.1111/j.1469-7998.1985.tb05668.x.
  53. ^ Ferris, H; Mullens, TA; Foord, KE (1990). "Stability and characteristics of spatial description parameters for nematode populations". J Nematol. 22 (4): 427–439. PMC 2619069. PMID 19287742.
  54. ^ a b Hanski I(1982) On patterns of temporal and spatial variation in animal populations. Ann. zool. Fermici 19: 21—37
  55. ^ Boag, B; Hackett, CA; Topham, PB (1992). "The use of Taylor's power law to describe the aggregated distribution of gastro-intestinal nematodes of sheep". Int J Parasitol. 22 (3): 267–270. doi:10.1016/s0020-7519(05)80003-7. PMID 1639561.
  56. ^ 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
  57. ^ 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-E3158
  58. ^ Random sampling of skewed distributions does not necessarily imply Taylor's law (2015) Proc Natl Acad Sci USA 112 (25) E3156
  59. ^ Banerjee, B (1976). "Variance to mean ratio and the spatial distribution of animals". Experientia. 32 (8): 993–994. doi:10.1007/bf01933930. S2CID 7687728.
  60. ^ a b c d e Hughes, G; Madden, LV (1992). "Aggregation and incidence of disease". Plant Pathology. 41 (6): 657–660. doi:10.1111/j.1365-3059.1992.tb02549.x.
  61. ^ Patil, GP; Stiteler, WM (1974). "Concepts of aggregation and their quantification: a critical review with some new results and applications". Researches on Population Ecology. 15: 238–254. doi:10.1007/bf02510670. S2CID 30108449.
  62. ^ a b c Turechek, WW; Madden, LV; Gent, DH; Xu, XM (2011). "Comments regarding the binary power law for heterogeneity of disease incidence". Phytopathology. 101 (12): 1396–1407. doi:10.1094/phyto-04-11-0100. PMID 21864088.
  63. ^ Gosme, Marie; Lucas, Philippe (2009-06-12). "Disease Spread Across Multiple Scales in a Spatial Hierarchy: Effect of Host Spatial Structure and of Inoculum Quantity and Distribution". Phytopathology. 99 (7): 833–839. doi:10.1094/phyto-99-7-0833. ISSN 0031-949X. PMID 19522581.
  64. ^ a b Xu, X-M; Madden, LV (2013). "The limits of the binary power law describing spatial variability for incidence data". Plant Pathology. 63 (5): 973–982. doi:10.1111/ppa.12172.
  65. ^ Clark, SJ; Perry, JN (1994). "Small sample estimation for Taylor's power law". Environment Ecol Stats. 1 (4): 287–302. doi:10.1007/BF00469426. S2CID 20054635.
  66. ^ a b c Wilson, LT; Room, PM (1983). "Clumping patterns of fruit and arthropods in cotton with implications for binomial sampling". Environ Entomol. 12: 50–54. doi:10.1093/ee/12.1.50.
  67. ^ a b c d Bliss, CI; Fisher, RA (1953). "Fitting the negative binomial distribution to biological data (also includes note on the efficient fitting of the negative binomial)". Biometrics. 9 (2): 177–200. doi:10.2307/3001850. JSTOR 3001850.
  68. ^ Jones, VP (1991). "Binomial sampling plans for tentiform leafminer (Lepidoptera: Gracillariidae) on apple in Utah". J Econ Entomol. 84 (2): 484–488. doi:10.1093/jee/84.2.484.
  69. ^ a b Katz L (1965) United treatment of a broad class of discrete probability distributions. in Proceedings of the International Symposium on Discrete Distributions. Montreal
  70. ^ Jewel, W; Sundt, B (1981). "Improved approximations for the distribution of a heterogeneous risk portfolio". Bull Assoc Swiss Act. 81: 221–240.
  71. ^ Foley, P (1994). "Predicting extinction times from environmental stochasticity and carrying capacity". Conserv Biol. 8: 124–137. doi:10.1046/j.1523-1739.1994.08010124.x.
  72. ^ Pertoldi, C; Bach, LA; Loeschcke, V (2008). "On the brink between extinction and persistence". Biol Direct. 3: 47. doi:10.1186/1745-6150-3-47. PMC 2613133. PMID 19019237.
  73. ^ Halley, J; Inchausti, P (2002). "Lognormality in ecological time series". Oikos. 99 (3): 518–530. doi:10.1034/j.1600-0706.2002.11962.x. S2CID 54197297.
  74. ^ Southwood TRE & Henderson PA (2000) Ecological methods. 3rd ed. Blackwood, Oxford
  75. ^ Service, MW (1971). "Studies on sampling larval populations of the Anopheles gambiae complex". Bull World Health Organ. 45 (2): 169–180. PMC 2427901. PMID 5316615.
  76. ^ a b c Southwood TRE (1978) Ecological methods. Chapman & Hall, London, England
  77. ^ Karandinos, MG (1976). "Optimum sample size and comments on some published formulae". Bull Entomol Soc Am. 22 (4): 417–421. doi:10.1093/besa/22.4.417.
  78. ^ 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
  79. ^ Schulthess, F; Bosque-Péreza, NA; Gounoua, S (1991). "Sampling lepidopterous pests on maize in West Africa". Bull Entomol Res. 81 (3): 297–301. doi:10.1017/s0007485300033575.
  80. ^ Bisseleua, DHB; Yede; Vida, S (2011). "Dispersion models and sampling of cacao mirid bug Sahlbergella singularis (Hemiptera: Miridae) on theobroma cacao in southern Cameroon". Environ Entomol. 40 (1): 111–119. doi:10.1603/en09101. PMID 22182619. S2CID 46679671.
  81. ^ 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
  82. ^ Green, RH (1970). "On fixed precision level sequential sampling". Res Pop Ecol. 12 (2): 249–251. doi:10.1007/BF02511568. S2CID 35973901.
  83. ^ Serraa, GV; La Porta, NC; Avalos, S; Mazzuferi, V (2012). "Fixed-precision sequential sampling plans for estimating alfalfa caterpillar, Colias lesbia, egg density in alfalfa, Medicago sativa, fields in Córdoba, Argentina". J Insect Sci. 13 (41): 41. doi:10.1673/031.013.4101. PMC 3740930. PMID 23909840.
  84. ^ Myers, JH (1978). "Selecting a measure of dispersion". Environ Entomol. 7 (5): 619–621. doi:10.1093/ee/7.5.619.
  85. ^ Bartlett, M (1936). "Some notes on insecticide tests in the laboratory and in the field". Supplement to the Journal of the Royal Statistical Society. 3 (2): 185–194. doi:10.2307/2983670. JSTOR 2983670.
  86. ^ Iwao, S (1968). "A new regression method for analyzing the aggregation pattern of animal populations". Res Popul Ecol. 10: 1–20. doi:10.1007/bf02514729. S2CID 39807668.
  87. ^ Nachman, G (1981). "A mathematical model of the functional relationship between density and spatial distribution of a population". J Anim Ecol. 50 (2): 453–460. doi:10.2307/4066. JSTOR 4066.
  88. ^ Allsopp, PG (1991). "Binomial sequential sampling of adult Saccharicoccus sacchari on sugarcane". Entomologia Experimentalis et Applicata. 60 (3): 213–218. doi:10.1111/j.1570-7458.1991.tb01540.x. S2CID 84873687.
  89. ^ Kono, T; Sugino, T (1958). "On the Estimation of the Density of Rice Stems Infested by the Rice Stem Borer". Japanese Journal of Applied Entomology and Zoology. 2 (3): 184. doi:10.1303/jjaez.2.184.
  90. ^ Binns, MR; Bostonian, NJ (1990). "Robustness in empirically based binomial decision rules for integrated pest management". J Econ Entomol. 83 (2): 420–442. doi:10.1093/jee/83.2.420.
  91. ^ a b Nachman, G (1984). "Estimates of mean population density and spatial distribution of Tetranychus urticae (Acarina: Tetranychidae) and Phytoseiulus persimilis (Acarina: Phytoseiidae) based upon the proportion of empty sampling units". J Appl Ecol. 21 (3): 903–991. doi:10.2307/2405055. JSTOR 2405055.
  92. ^ Schaalje, GB; Butts, RA; Lysyk, TL (1991). "Simulation studies of binomial sampling: a new variance estimator and density pre&ctor, with special reference to the Russian wheat aphid (Homoptera: Aphididae)". J Econ Entomol. 84: 140–147. doi:10.1093/jee/84.1.140.
  93. ^ 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)
  94. ^ Wilson, L T; Room, PM (1983). "Clumping patterns of fruit and arthropods in cotton, with implications for binomial sampling". Environ Entomol. 12: 50–54. doi:10.1093/ee/12.1.50.
  95. ^ 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
  96. ^ Elliot JM (1977) Some methods for the statistical analysis of samples of benthic invertebrates. 2nd ed. Freshwater Biological Association, Cambridge, United Kingdom
  97. ^ Cole, LC (1946). "A theory for analyzing contagiously distributed populations". Ecology. 27 (4): 329–341. doi:10.2307/1933543. JSTOR 1933543.
  98. ^ a b c d Lloyd, M (1967). "Mean crowding". J Anim Ecol. 36 (1): 1–30. doi:10.2307/3012. JSTOR 3012.
  99. ^ Iwao, S; Kuno, E (1968). "Use of the regression of mean crowding on mean density for estimating sample size and the transformation of data for the analysis of variance". Res Pop Ecology. 10 (2): 210–214. doi:10.1007/bf02510873. S2CID 27992286.
  100. ^ Ifoulis, AA; Savopoulou-Soultani, M (2006). "Developing optimum sample size and multistage sampling plans for Lobesia botrana (Lepidoptera: Tortricidae) larval infestation and injury in northern Greece". J Econ Entomol. 99 (5): 1890–1898. doi:10.1093/jee/99.5.1890. PMID 17066827.
  101. ^ a b Ho, CC (1993). "Dispersion statistics and sample size estimates for Tetranychus kanzawai (Acari: Tetranychidae) on mulberry". Environ Entomol. 22: 21–25. doi:10.1093/ee/22.1.21.
  102. ^ Iwao, S (1975). "A new method of sequential sampling to classify populations relative to a critical density". Res Popul Ecol. 16 (2): 281–28. doi:10.1007/bf02511067. S2CID 20662793.
  103. ^ Kuno, E (1969). "A new method of sequential sampling to obtain the population estimates with a fixed level of precision". Res. Pooul. Ecol. 11 (2): 127–136. doi:10.1007/bf02936264. S2CID 35594101.
  104. ^ Parrella, MP; Jones, VP (1985). "Yellow traps as monitoring tools for Liriomyza trifolii (Diptera: Agromyzidae) in chrysanthemum greenhouses". J Econ Entomol. 78: 53–56. doi:10.1093/jee/78.1.53.
  105. ^ a b Morisita, M (1959). "Measuring the dispersion and the analysis of distribution patterns". Memoirs of the Faculty of Science, Kyushu University Series e. Biol. 2: 215–235.
  106. ^ Pedigo LP & Buntin GD (1994) Handbook of sampling methods for arthropods in agriculture. CRC Boca Raton FL
  107. ^ Smith-Gill, SJ (1975). "Cytophysiological basis of disruptive pigmentary patterns in the leopard frog Rana pipiens. II. Wild type and mutant cell specific patterns". J Morphol. 146 (1): 35–54. doi:10.1002/jmor.1051460103. PMID 1080207. S2CID 23780609.
  108. ^ Elliot JM (1977) Statistical analysis of samples of benthic invertebrates. Freshwater Biological Association. Ambleside
  109. ^ Fisher RA (1925) Statistical methods for research workers. Hafner, New York
  110. ^ a b Hoel, P (1943). "On the indices of dispersion". Ann Math Statist. 14 (2): 155. doi:10.1214/aoms/1177731457.
  111. ^ David, FN; Moore, PG (1954). "Notes on contagious distributions in plant populations". Annals of Botany. 18: 47–53. doi:10.1093/oxfordjournals.aob.a083381.
  112. ^ Green, RH (1966). "Measurement of non-randomness in spatial distributions". Res Pop Ecol. 8: 1–7. doi:10.1007/bf02524740. S2CID 25039063.
  113. ^ Gottwald, TR; Bassanezi, RB; Amorim, L; Bergamin-Filho, A (2007). "Spatial pattern analysis of citrus canker-infected plantings in São Paulo, Brazil, and augmentation of infection elicited by the Asian leafminer". Phytopathology. 97 (6): 674–683. doi:10.1094/phyto-97-6-0674. PMID 18943598.
  114. ^ a b Hughes, G; Madden, LV (1993). "Using the beta-binomial distribution to describe aggregated patterns of disease incidence". Phytopathology. 83 (9): 759–763. doi:10.1094/phyto-83-759.
  115. ^ a b Fleiss JL (1981) Statistical methods for rates and proportions. 2nd ed. Wiley, New York, USA
  116. ^ Ma ZS (1991) Further interpreted Taylor’s Power Law and population aggregation critical density. Trans Ecol Soc China (1991) 284–288
  117. ^ de 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
  118. ^ a b Bohning, D (1994). "A note on a test for Poisson overdispersion". Biometrika. 81 (2): 418–419. doi:10.2307/2336974. JSTOR 2336974.
  119. ^ Ping, S (1995). "Further study on the statistical test to detect spatial pattern". Biometrical Journal. 37 (2): 199–203. doi:10.1002/bimj.4710370211.
  120. ^ Clapham, AR (1936). "Overdispersion in grassland communities and the use of statistical methods in plant ecology". J Ecol. 14 (1): 232–251. doi:10.2307/2256277. JSTOR 2256277.
  121. ^ Blackman 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
  122. ^ Greig-Smith, P (1952). "The use of random and contiguous quadrats in the study of the structure of plant communities". Ann. Bot. 16 (2): 293–316. doi:10.1093/oxfordjournals.aob.a083317.
  123. ^ Gosset, E; Louis, B (1986). "The binning analysis - Towards a better significance test". Astrophysics Space Sci. 120 (2): 263–306. Bibcode:1986Ap&SS.120..263G. doi:10.1007/BF00649941. hdl:2268/88597. S2CID 117653758.
taylor, confused, with, taylor, taylor, rule, taylor, power, empirical, ecology, that, relates, variance, number, individuals, species, unit, area, habitat, corresponding, mean, power, relationship, named, after, ecologist, first, proposed, 1961, lionel, taylo. Not to be confused with Taylor Law or Taylor rule Taylor s power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship 1 It is named after the ecologist who first proposed it in 1961 Lionel Roy Taylor 1924 2007 2 Taylor s original name for this relationship was the law of the mean 1 The name Taylor s law was coined by Southwood in 1966 2 Contents 1 Definition 2 History 2 1 Biology 2 2 Physics 2 3 Mathematics 3 The Tweedie hypothesis 4 Mathematical formulation 4 1 Scale invariance 4 2 Extensions and refinements 4 3 Small samples 4 4 Interpretation 4 5 Notes 4 6 Extension to cluster sampling of binary data 5 Applications 5 1 Recommendations as to use 5 2 Randomly distributed populations 5 3 Dispersion parameter estimator 5 4 Katz family of distributions 5 5 Time to extinction 5 6 Minimum population size required to avoid extinction 5 7 Sampling size estimators 5 8 Sequential sampling 6 Related analyses 6 1 Barlett Iwao model 6 2 Nachman model 6 3 Kono Sugino equation 6 4 Hughes Madden equation 6 5 Negative binomial distribution model 6 6 Tests for a common dispersion parameter 6 7 Charlier coefficient 6 8 Cole s index of dispersion 6 9 Lloyd s indexes 6 10 Patchiness regression test 6 11 Morisita s index of dispersion 6 12 Standardised Morisita s index 6 13 Southwood s index of spatial aggregation 6 14 Fisher s index of dispersion 6 15 Index of cluster size 6 16 Green s index 6 17 Binary dispersal index 6 18 Ma s population aggregation critical density 7 Related statistics 7 1 de Oliveria s statistic 7 2 Clapham s test 8 See also 9 ReferencesDefinition EditThis law was originally defined for ecological systems specifically to assess the spatial clustering of organisms For a population count Y displaystyle Y with mean m displaystyle mu and variance var Y displaystyle operatorname var Y Taylor s law is written var Y a m b displaystyle operatorname var 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 1 This power law has subsequently been confirmed for many hundreds of species 3 4 Taylor s law has also been applied to assess the time dependent changes of population distributions 3 Related variance to mean power laws have also been demonstrated in several non ecological systems cancer metastasis 5 the numbers of houses built over the Tonami plain in Japan 6 measles epidemiology 7 HIV epidemiology 8 the geographic clustering of childhood leukemia 9 blood flow heterogeneity 10 11 the genomic distributions of single nucleotide polymorphisms SNPs 12 gene structures 13 in number theory with sequential values of the Mertens function 14 and also with the distribution of prime numbers 15 from the eigenvalue deviations of Gaussian orthogonal and unitary ensembles of random matrix theory 16 History EditThe first use of a double log log plot was by Reynolds in 1879 on thermal aerodynamics 17 Pareto used a similar plot to study the proportion of a population and their income 18 The term variance was coined by Fisher in 1918 19 Biology Edit Pearson 20 in 1921 proposed the equation also studied by Neyman 21 s 2 a m b m 2 displaystyle s 2 am bm 2 Smith in 1938 while studying crop yields proposed a relationship similar to Taylor s 22 This relationship was log V x log V 1 b log x displaystyle log V x log V 1 b log x where Vx is the variance of yield for plots of x units V1 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 23 in 1941 Fracker and Brischle 24 in 1941 and Hayman amp Lowe 25 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 lesions macro zooplankton worms and symphylids in soil insects in soil on plants and in the air mites on leaves ticks on sheep and fish in the sea 1 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 that could result from birth death processes 26 Taylor s explanation was based the assumption of a balanced migratory and congregatory behavior of animals 1 His hypothesis was initially qualitative but as it evolved it became semi quantitative and was supported by simulations 27 Many alternative hypotheses for the power law have been advanced Hanski proposed a random walk model modulated by the presumed multiplicative effect of reproduction 28 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 3 4 Anderson et al formulated a simple stochastic birth death immigration and emigration model that yielded a quadratic variance function 29 As a response to this model Taylor argued that such a Markov process would predict that the power law exponent would vary considerably between replicate observations and that such variability had not been observed 30 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 31 Kemp however did not explain the parameterizations of his models in mechanistic terms Other relatively abstract models for Taylor s law followed 6 32 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 33 34 Taylor s law has been applied to time series data and Perry showed using simulations that chaos theory could yield Taylor s law 35 Taylor s law has been applied to the spatial distribution of plants 36 and bacterial populations 37 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 38 a family of probabilistic models that express an inherent power function relationship between the variance and the mean 11 13 39 Several alternative hypotheses for the power law have been proposed Hanski proposed a random walk model modulated by the presumed multiplicative effect of reproduction 28 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 3 4 Anderson et al formulated a simple stochastic birth death immigration and emigration model that yielded a quadratic variance function 29 The Lewontin Cohen growth model 40 is another proposed explanation The possibility that observations of a power law might reflect more mathematical artifact than a mechanistic process was raised 41 Variation in the exponents of Taylor s Law applied to ecological populations cannot be explained or predicted based solely on statistical grounds however 42 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 43 Physics Edit 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 44 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 45 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 14 Mathematics Edit Taylor s law has been shown to hold for prime numbers not exceeding a given real number 46 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 3 title for komonis 05 05 1080 00053051 2023 net for The Tweedie hypothesis EditAbout 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 47 48 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 CGF is K b s 8 l l k b 8 1 s 8 a 1 displaystyle K b s theta lambda lambda kappa b theta left left 1 s over theta right alpha 1 right where kb 8 is the cumulant function k b 8 a 1 a 8 a 1 a displaystyle kappa b theta frac alpha 1 alpha left frac theta alpha 1 right alpha the Tweedie exponent a b 2 b 1 displaystyle alpha frac b 2 b 1 s is the generating function variable and 8 and l are the canonical and index parameters respectively 38 These last two parameters are analogous to the scale and shape parameters used in probability theory The cumulants 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 l 1 a 1 displaystyle a lambda 1 alpha 1 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 39 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 12 13 14 16 The main justification for the Tweedie hypothesis rests with the mathematical convergence properties of the Tweedie distributions 13 The Tweedie convergence theorem requires the Tweedie distributions to act as foci of convergence for a wide range of statistical processes 49 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 14 16 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 6 One difficulty with the Tweedie hypothesis is that the value of b does not range between 0 and 1 Values of b lt 1 are rare but have been reported 50 Mathematical formulation EditIn symbols s i 2 a m i b displaystyle s i 2 am i b where si2 is the variance of the density of the ith sample mi is the mean density of the ith sample and a and b are constants In logarithmic form log s i 2 log a b log m i displaystyle log s i 2 log a b log m i Scale invariance Edit The exponent in Taylor s law is scale invariant If the unit of measurement is changed by a constant factor c displaystyle c the exponent b displaystyle b remains unchanged To see this let y cx Then m 1 E x displaystyle mu 1 operatorname E x m 2 E y E c x c E x c m 1 displaystyle mu 2 operatorname E y operatorname E cx c operatorname E x c mu 1 s 1 2 E x m 1 2 displaystyle sigma 1 2 operatorname E x mu 1 2 s 2 2 E y m 2 2 E c x c m 1 2 c 2 E x m 1 2 c 2 s 1 2 displaystyle sigma 2 2 operatorname E y mu 2 2 operatorname E cx c mu 1 2 c 2 operatorname E x mu 1 2 c 2 sigma 1 2 Taylor s law expressed in the original variable x is s 1 2 a m 1 b displaystyle sigma 1 2 a mu 1 b and in the rescaled variable y it is s 2 2 c 2 s 1 2 c 2 a m 1 b c 2 b a c m 1 b c 2 b a m 2 b displaystyle sigma 2 2 c 2 sigma 1 2 c 2 a mu 1 b c 2 b a c mu 1 b c 2 b a mu 2 b Thus s 2 2 displaystyle sigma 2 2 is still proportional to m 2 b displaystyle 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 51 Extensions and refinements Edit A refinement in the estimation of the slope b has been proposed by Rayner 52 b f f f f 2 4 r 2 f f 2 r f displaystyle b frac f varphi sqrt f varphi 2 4r 2 f varphi 2r sqrt f where r displaystyle r is the Pearson moment correlation coefficient between log s 2 displaystyle log s 2 and log m displaystyle log m f displaystyle f is the ratio of sample variances in log s 2 displaystyle log s 2 and log m displaystyle log m and f displaystyle varphi is the ratio of the errors in log s 2 displaystyle log s 2 and log m displaystyle log m Ordinary least squares regression assumes that f This tends to underestimate the value of b because the estimates of both log s 2 displaystyle log s 2 and log m displaystyle log m are subject to error An extension of Taylor s law has been proposed by Ferris et al when multiple samples are taken 53 s 2 c n d m b displaystyle s 2 cn d m b where s2 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 Edit An extension to this law for small samples has been proposed by Hanski 54 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 displaystyle S V P Assuming the validity of Taylor s law we have V a m b displaystyle V am b Because in the Poisson distribution the mean equals the variance we have P m displaystyle P m This gives us S V P a m b m displaystyle S V P am b m This closely resembles Barlett s original suggestion Interpretation Edit Slope values b significantly gt 1 indicate clumping of the organisms In Poisson distributed data b 1 30 If the population follows a lognormal or gamma distribution 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 lt 2 usually 1 5 1 6 but values of 2 have been reported 55 Occasionally cases with b gt 2 have been reported 3 b values below 1 are uncommon but have also been reported b 0 93 50 It has been suggested that the exponent of the law b is proportional to the skewness of the underlying distribution 56 This proposal has criticised additional work seems to be indicated 57 58 Notes Edit 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 59 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 54 Extension to cluster sampling of binary data Edit A form of Taylor s law applicable to binary data in clusters e q quadrats has been proposed 60 In a binomial distribution the theoretical variance is var bin n p 1 p displaystyle text var text bin 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 60 Hughes and Madden noted that if the distribution is Poisson the mean and variance are equal 60 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 gt varbin In symbols Hughes and Madden s modification to Tyalor s law was var obs a var bin b displaystyle text var text obs a text var text bin b In logarithmic form this relationship is log var obs log a b log var bin displaystyle log text var text obs log a b log text var text bin 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 61 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 60 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 gt 1 there is over dispersion small scale aggregation When b is gt 1 the degree of aggregation varies with p Turechek et al 62 have showed 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 63 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 62 The break in the line occurs only at p very close to 0 An extension to this law has been proposed 64 The original form of this law is symmetrical but it can be extended to an asymmetrical form 64 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 EditBecause 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 Edit It has been recommended based on simulation studies 65 in applications testing the validity of Taylor s law to a data sample that 1 the total number of organisms studied be gt 15 2 the minimum number of groups of organisms studied be gt 5 3 the density of the organisms should vary by at least 2 orders of magnitude within the sample Randomly distributed populations Edit 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 s2 of the population are equal and the proportion of samples that contain at least one individual p is p 1 e m displaystyle p 1 e m 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 66 The basic relationship is p 1 e m log s 2 m s 2 m 1 1 displaystyle p 1 e m log s 2 m s 2 m 1 1 where the log is taken to the base e Incorporating Taylor s law this relationship becomes p 1 e m log a m b 1 a m b 1 1 1 displaystyle p 1 e m log am b 1 am b 1 1 1 Dispersion parameter estimator Edit The common dispersion parameter k of the negative binomial distribution is k m 2 s 2 m displaystyle k frac m 2 s 2 m where m displaystyle m is the sample mean and s 2 displaystyle s 2 is the variance 67 If 1 k is gt 0 the population is considered to be aggregated 1 k 0 s2 m the population is considered to be randomly Poisson distributed and if 1 k is lt 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 66 k m a m b 1 1 displaystyle k frac m am b 1 1 where a and b are the constants from Taylor s law Jones 68 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 66 p 1 e m log a m b 1 a m b 1 1 1 displaystyle p 1 e m log am b 1 am b 1 1 1 derived an estimator for the probability of a sample containing x individuals per sampling unit Jones s formula is P x P x 1 k x 1 x m k 1 m k 1 1 displaystyle P x P x 1 frac k x 1 x frac mk 1 mk 1 1 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 P 0 1 m k k displaystyle P 0 left 1 frac m k right k Jones also gives confidence intervals for these probabilities C I t P x 1 P x N 1 2 displaystyle mathrm CI t left frac P x 1 P x N right 1 2 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 Edit Katz proposed a family of distributions the Katz family with 2 parameters w1 w2 69 This family of distributions includes the Bernoulli Geometric Pascal and Poisson distributions as special cases The mean and variance of a Katz distribution are m w 1 1 w 2 displaystyle m frac w 1 1 w 2 s 2 w 1 1 w 2 2 displaystyle s 2 frac w 1 1 w 2 2 where m is the mean and s2 is the variance of the sample The parameters can be estimated by the method of moments from which we have w 1 1 w 2 m displaystyle frac w 1 1 w 2 m w 2 1 w 2 s 2 m m displaystyle frac w 2 1 w 2 frac s 2 m m For a Poisson distribution w2 0 and w1 l 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 70 p n a b n p n 1 displaystyle p n left a frac b n right p n 1 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 displaystyle a log 1 w 2 b 1 displaystyle b 1 Katz also introduced a statistical test 69 J n n 2 s 2 m m displaystyle J n sqrt frac n 2 frac s 2 m m where Jn is the test statistic s2 is the variance of the sample m is the mean of the sample and n is the sample size Jn is asymptotically normally distributed with a zero mean and unit variance If the sample is Poisson distributed Jn 0 values of Jn lt 0 and gt 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 N S n 1 2 s 2 m 1 displaystyle NS sqrt frac n 1 2 left frac s 2 m 1 right which is known to be asymptotically normal and the conditional chi squared statistic Poisson dispersion test T n 1 s 2 m displaystyle T frac n 1 s 2 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 n 2 a m b 1 1 displaystyle J n sqrt frac n 2 am b 1 1 Time to extinction Edit 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 71 Let N t 1 r N t displaystyle N t 1 rN t where Nt 1 and Nt are the population sizes at time t 1 and t respectively and r is parameter equal to the annual increase decrease in population Then var r s 2 log r displaystyle operatorname var r s 2 log r where var r displaystyle text var r is the variance of r displaystyle r Let K displaystyle K be a measure of the species abundance organisms per unit area Then T E 2 log N Var r log K log N 2 displaystyle T E frac 2 log N operatorname Var r left log K frac log N 2 right where TE is the mean time to local extinction The probability of extinction by time t is P t 1 e t T E displaystyle P t 1 e t T E Minimum population size required to avoid extinction Edit If a population is lognormally distributed then the harmonic mean of the population size H is related to the arithmetic mean m 72 H m a m b 1 displaystyle H m am b 1 Given that H must be gt 0 for the population to persist then rearranging we have m gt a 1 2 b displaystyle m gt a 1 2 b 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 73 suggesting that it is at least a plausible assumption Sampling size estimators Edit The degree of precision D is defined to be s m where s is the standard deviation and m is the mean The degree of precision is known as the coefficient of variation in other contexts In ecology research it is recommended that D be in the range 10 25 74 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 t D 2 m displaystyle n frac t D 2 m where t is critical level of the t distribution for the type 1 error with the degrees of freedom that the mean m was calculated with If the population is distributed as a negative binomial distribution then the required sample size is n t D 2 m k m k displaystyle n frac t D 2 m k mk where k is the parameter of the negative binomial distribution A more general sample size estimator has also been proposed 75 n t D 2 a m b 2 displaystyle n left frac t D right 2 am b 2 where a and b are derived from Taylor s law An alternative has been proposed by Southwood 76 n a m b D 2 displaystyle n a frac m b D 2 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 77 The first was modified by Ruesink to incorporate Taylor s law 78 n t d m 2 a m b 2 displaystyle n left frac t d m right 2 am b 2 where d is the ratio of half the desired confidence interval CI to the mean In symbols d m C I 2 m displaystyle d m frac CI 2m The second estimator is used in binomial presence absence sampling The desired sample size n is n t d p 2 p 1 q displaystyle n left td p right 2 p 1 q where the dp 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 C I 2 p displaystyle d p frac CI 2p For binary presence absence sampling Schulthess et al modified Karandinos equation N t D p i 2 1 p p displaystyle N left frac t D pi right 2 frac 1 p 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 Dip is a parameter derived from Taylor s law 79 Sequential sampling Edit Sequential analysis is a method of statistical analysis where the sample size is not fixed in advance Instead samples are taken in accordance with a predefined stopping rule 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 80 log T log D 2 a b 2 log n b 1 b 2 displaystyle log T frac log D 2 a b 2 log n frac b 1 b 2 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 81 The required sample size is n t m T 2 a m b displaystyle n t m T 2 am b where a and b are the Taylor coefficients is the absolute value 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 2 p q displaystyle n t m T 2 pq 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 82 D a n 1 b T b 2 1 2 displaystyle D an 1 b T b 2 1 2 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 83 T n a n 1 b D 2 1 2 b displaystyle T n geq left frac an 1 b D 2 right 1 2 b where a and b are the parameters from Taylor s law D is the desired level of precision and Tn is the total sample size Serra et al also proposed a second stopping rule based on Iwoa s regression T n a 1 D 2 b 1 n displaystyle T n geq frac alpha 1 D 2 frac beta 1 n where a and b are the parameters of the regression line D is the desired level of precision and Tn 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 EditIt 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 84 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 Edit Barlett in 1936 85 and later Iwao independently in 1968 86 both proposed an alternative relationship between the variance and the mean In symbols s i 2 a m i b m i 2 displaystyle s i 2 am i bm i 2 where s is the variance in the ith sample and mi is the mean of the ith sampleWhen the population follows a negative binomial distribution 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 Edit Nachman proposed a relationship between the mean density and the proportion of samples with zero counts 87 p 0 exp a m b displaystyle p 0 exp am b where p0 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 0 the distribution is random This relationship is usually tested in its logarithmic form log m c d log p 0 displaystyle 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 88 P 1 1 exp exp log e A 2 a b 2 log e n b 1 b 2 1 c d displaystyle P 1 1 exp left exp left frac frac log e left frac A 2 a right b 2 log e n left frac b 1 b 2 1 right c d right right N n P 1 displaystyle N nP 1 where A 2 D 2 z a 2 2 displaystyle A 2 frac D 2 z alpha 2 2 where D2 is the degree of precision desired za 2 is the upper a 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 Edit 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 89 log m log a b log log p 0 displaystyle log m log a b log log p 0 where p0 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 p0 and assume that these are distributed exponentially Then p 0 exp A m B displaystyle p 0 exp Am 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 NoteThe 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 b m displaystyle P 1 ae bm 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 UsesThe predicted estimates of m from this equation are subject to bias 90 and it is recommended that the adjusted mean ma be used instead 91 m a m 1 var log m i 2 displaystyle m a m left 1 frac operatorname var log m i 2 right where var is the variance of the sample unit means mi and m is the overall mean An alternative adjustment to the mean estimates is 91 m a m e MSE 2 displaystyle m a me text MSE 2 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 92 var m m 2 c 1 c 2 c 3 MSE displaystyle operatorname var m m 2 c 1 c 2 c 3 text MSE where c 1 b 2 1 p 0 n p 0 log e p 0 2 displaystyle c 1 frac beta 2 1 p 0 np 0 log e p 0 2 c 2 MSE N s b 2 log e log e p 0 p 2 displaystyle c 2 frac text MSE N s beta 2 log e log e p 0 p 2 c 3 exp a b 2 a b log e p 0 n displaystyle c 3 frac exp a b 2 alpha beta log e p 0 n where MSE is the mean square error of the regression a and b are the constant and slope of the regression respectively sb2 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 p0 in the regression The parameters a and b are estimated from Taylor s law s 2 a b log e m displaystyle s 2 a b log e m Hughes Madden equation Edit Hughes and Madden have proposed testing a similar relationship applicable to binary observations in cluster where each cluster contains from 0 to n individuals 60 v a r obs a p b 1 p c displaystyle var text obs ap 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 var obs log a b log p c log 1 p displaystyle log operatorname var text obs log a b log p c log 1 p In most cases it is assumed that b c leading to a simple model var obs a p 1 p b displaystyle operatorname var text obs 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 62 A variant of this equation was proposed by Shiyomi et al 93 who suggested testing the regression log var obs n 2 a b log p 1 p n displaystyle log operatorname var text obs n 2 a b log frac p 1 p 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 Edit A negative binomial model has also been proposed 94 The dispersion parameter k using the method of moments is m2 s2 m and pi is the proportion of samples with counts gt 0 The s2 used in the calculation of k are the values predicted by Taylor s law pi 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 95 1 k a m b 2 1 m displaystyle frac 1 k frac am b 2 1 m A better estimate of the dispersion parameter can be made with the method of maximum likelihood For the negative binomial it can be estimated from the equation 67 A x k x N log 1 m k displaystyle sum frac A x k x N log left 1 frac m k right where Ax 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 m2 k and U s 2 m m 2 k displaystyle U s 2 m frac m 2 k where s2 is the sample variance m is the sample mean and k is the negative binomial parameter The variance of U is 67 var U 2 m p 2 q 1 R 2 log 1 R R p 4 1 R k 1 k R N log 1 R R 2 displaystyle operatorname var U 2mp 2 q left frac 1 R 2 log 1 R R right p 4 frac 1 R k 1 kR N log 1 R R 2 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 variances 2 m a m p displaystyle 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 Edit The dispersion parameter k 67 is k m 2 s 2 m displaystyle k frac m 2 s 2 m where m is the sample mean and s2 is the variance If k 1 is gt 0 the population is considered to be aggregated k 1 0 the population is considered to be random and if k 1 is lt 0 the population is considered to be uniformly distributed Southwood has recommended regressing k against the mean and a constant 76 k i a b m i displaystyle k i a bm i where ki and mi are the dispersion parameter and the mean of the ith sample respectively to test for the existence of a common dispersion parameter kc A slope b value significantly gt 0 indicates the dependence of k on the mean density An alternative method was proposed by Elliot who suggested plotting s2 m against m2 s2 n 96 kc is equal to 1 slope of this regression Charlier coefficient Edit This coefficient C is defined as C 100 s 2 m 0 5 m displaystyle C frac 100 s 2 m 0 5 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 Edit This index Ic is defined as 97 I c x 2 x 2 displaystyle I c frac sum x 2 sum x 2 The usual interpretation of this index is as follows values of Ic lt 1 1 gt 1 are taken to mean a uniform distribution a random distribution or an aggregated distribution Because s2 S x2 Sx 2 the index can also be written I c s 2 n m 2 n m 2 1 n 2 s 2 m 2 1 displaystyle I c frac s 2 nm 2 nm 2 frac 1 n 2 frac s 2 m 2 1 If Taylor s law can be assumed to hold then I c a m b 2 n 2 1 displaystyle I c frac am b 2 n 2 1 Lloyd s indexes Edit 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 98 I M C m s 2 m 1 displaystyle mathrm IMC m frac s 2 m 1 where m is the sample mean and s2 is the variance Lloyd s index of patchiness IP 98 is I P IMC m displaystyle mathrm IP text IMC 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 m 2 k displaystyle s 2 m frac m 2 k Then S E I P 1 k 2 var k k k 1 k m m q displaystyle SE IP frac 1 k 2 left operatorname var k frac k k 1 k m mq right where S E I P displaystyle SE IP is the standard error of the index of patchiness var k displaystyle text var k is the variance of the parameter k and q is the number of quadrats sampled If the population obeys Taylor s law then I M C m a 1 m 1 b 1 displaystyle mathrm IMC m a 1 m 1 b 1 I P 1 a 1 m b 1 m displaystyle mathrm IP 1 a 1 m b frac 1 m Patchiness regression test Edit Iwao proposed a patchiness regression to test for clumping 99 100 Let y i m i s 2 m i 1 displaystyle y i m i frac s 2 m i 1 yi here is Lloyd s index of mean crowding 98 Perform an ordinary least squares regression of mi 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 lt 0 0 or gt 0 These values represent regularity randomness and aggregation of populations in spatial patterns respectively A value of a lt 1 is taken to mean that the basic unit of the distribution is a single individual Where the statistic s2 m is not constant it has been recommended to use instead to regress Lloyd s index against am bm2 where a and b are constants 101 The sample size n for a given degree of precision D for this regression is given by 101 n t D 2 a 1 m b 1 displaystyle n left frac t D right 2 left frac a 1 m 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 102 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 i m c t i a 1 m c b 1 m c 2 1 2 displaystyle N u im c t i a 1 m c b 1 m c 2 1 2 N l i m c t i a 1 m c b 1 m c 2 1 2 displaystyle N l im c t i a 1 m c b 1 m c 2 1 2 where Nu and Nl 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 103 T n a 1 D 2 b 1 n displaystyle T n frac a 1 D 2 frac b 1 n where Tn 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 D2Parrella and Jones have proposed an alternative but related stop line 104 T n 1 n N a 1 D 2 1 n N b 1 n displaystyle T n left 1 frac n N right frac a 1 D 2 left 1 frac n N right frac b 1 n 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 Edit Morisita s index of dispersion Im is the scaled probability that two points chosen at random from the whole population are in the same sample 105 Higher values indicate a more clumped distribution I m x x 1 n m m 1 displaystyle I m frac sum x x 1 nm m 1 An alternative formulation is I m n x 2 x x 2 x displaystyle I m n frac sum x 2 sum x sum x 2 sum x 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 n IMC n m 1 displaystyle I m frac n operatorname IMC nm 1 where IMC is Lloyd s index of crowding 98 This index is relatively independent of the population density but is affected by the sample size Values gt 1 indicate clumping values lt 1 indicate a uniformity of distribution and a value of 1 indicates a random sample Morisita showed that the statistic 105 I m x 1 n x displaystyle 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 106 z I m 1 2 n m 2 displaystyle z frac I m 1 2 nm 2 where m is the overall sample mean n is the number of sample units and z is the normal distribution abscissa Significance is tested by comparing the value of z against the values of the normal distribution A function for its calculation is available in the statistical R language R functionNote not to be confused with Morisita s overlap index Standardised Morisita s index Edit 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 107 First determine Morisita s index Id in the usual fashion Then let k be the number of units the population was sampled from Calculate the two critical values M u x 0 975 2 k x x 1 displaystyle M u frac chi 0 975 2 k sum x sum x 1 M c x 0 025 2 k x x 1 displaystyle M c frac chi 0 025 2 k sum x sum x 1 where x2 is the chi square value for n 1 degrees of freedom at the 97 5 and 2 5 levels of confidence The standardised index Ip is then calculated from one of the formulae below When Id Mc gt 1 I p 0 5 0 5 I d M c k M c displaystyle I p 0 5 0 5 left frac I d M c k M c right When Mc gt Id 1 I p 0 5 I d 1 M u 1 displaystyle I p 0 5 left frac I d 1 M u 1 right When 1 gt Id Mu I p 0 5 I d 1 M u 1 displaystyle I p 0 5 left frac I d 1 M u 1 right When 1 gt Mu gt Id I p 0 5 0 5 I d M u M u displaystyle I p 0 5 0 5 left frac I d M u M u right Ip ranges between 1 and 1 with 95 confidence intervals of 0 5 Ip has the value of 0 if the pattern is random if the pattern is uniform Ip lt 0 and if the pattern shows aggregation Ip gt 0 Southwood s index of spatial aggregation Edit Southwood s index of spatial aggregation k is defined as 1 k m m 1 displaystyle frac 1 k frac m m 1 where m is the mean of the sample and m is Lloyd s index of crowding 76 Fisher s index of dispersion Edit Fisher s index of dispersion 108 109 is I D n 1 s 2 m displaystyle mathrm ID frac n 1 s 2 m This index may be used to test for over dispersion of the population It is recommended that in applications n gt 5 110 and that the sample total divided by the number of samples is gt 3 In symbols x n gt 3 displaystyle frac sum x n gt 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 with n 1 degrees of freedom when the population is Poisson distributed 110 It is equal to the scale parameter when the population obeys the gamma distribution 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 I D n 1 a m b 1 displaystyle mathrm ID n 1 am b 1 Index of cluster size Edit The index of cluster size ICS was created by David and Moore 111 Under a random Poisson distribution ICS is expected to equal 0 Positive values indicate a clumped distribution negative values indicate a uniform distribution I C S s 2 m 1 displaystyle mathrm ICS frac s 2 m 1 where s2 is the variance and m is the mean If the population obeys Taylor s law I C S a m b 1 1 displaystyle mathrm ICS am b 1 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 Edit Green s index GI is a modification of the index of cluster size that is independent of n the number of sample units 112 C x s 2 m 1 n m 1 displaystyle C x frac s 2 m 1 nm 1 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 a m b 1 1 n m 1 displaystyle C x frac am b 1 1 nm 1 Binary dispersal index Edit 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 113 The theoretical variance of a sample from a population with a binomial distribution is s 2 n p 1 p displaystyle s 2 np 1 p where s2 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 var obs var bin s 2 n p 1 p displaystyle D frac text var text obs text var text bin frac s 2 np 1 p 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 gt 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 114 C D n N 1 n N 2 N n 2 n 1 2 displaystyle C frac D nN 1 nN 2N n 2 n 1 2 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 of the population D is also related to intraclass correlation r which is defined as 115 r 1 x i T x i p 1 p N T T 1 displaystyle rho 1 frac sum x i T x i p 1 p NT T 1 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 ith unit with this property T must be the same for all sampled units In this case with n constant r D 1 n 1 displaystyle rho frac D 1 n 1 If the data can be fitted with a beta binomial distribution then 115 D 1 n 1 8 1 8 displaystyle D 1 frac n 1 theta 1 theta where 8 is the parameter of the distribution 114 Ma s population aggregation critical density Edit Ma has proposed a parameter m0 the population aggregation critical density to relate population density to Taylor s law 116 m 0 exp log a 1 b displaystyle m 0 exp left frac log a 1 b right Related statistics EditA number of statistical tests are known that may be of use in applications de Oliveria s statistic Edit A related statistic suggested by de Oliveria 117 is the difference of the variance and the mean 118 If the population is Poisson distributed then v a r s 2 m 2 t 2 n 1 displaystyle var s 2 m frac 2t 2 n 1 where t is the Poisson parameter s2 is the variance m is the mean and n is the sample size The expected value of s2 m is zero This statistic is distributed normally 119 If the Poisson parameter in this equation is estimated by putting t m after a little manipulation this statistic can be written O T n 1 2 s 2 m m displaystyle O T sqrt frac n 1 2 frac s 2 m m This is almost identical to Katz s statistic with n 1 replacing n Again OT is normally distributed with mean 0 and unit variance for large n This statistic is the same as the Neyman Scott statistic Notede Oliveria actually suggested that the variance of s2 m was 1 2t1 2 3t 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 118 showed that this estimate of the variance was incorrect Bohning s correction is given in the equations above Clapham s test Edit In 1936 Clapham proposed using the ratio of the variance to the mean as a test statistic the relative variance 120 In symbols 8 s 2 m displaystyle theta frac s 2 m 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 Blackman 121 who noted that it was approximately normally distributed with a mean of 1 and a variance V8 of V 8 2 n n 1 2 displaystyle V theta frac 2n n 1 2 The derivation of the variance was re analysed by Bartlett 122 who considered it to be V 8 2 n 1 displaystyle V theta frac 2 n 1 For large samples these two formulae are in approximate agreement This test is related to the later Katz s Jn statistic If the population obeys Taylor s law then 8 a m b 1 displaystyle theta am b 1 NoteA refinement on this test has also been published 123 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 8 is distributed 8 s 2 m 1 n x i n N 2 displaystyle theta frac s 2 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 xi are the individual data values The expectation and variance of 8 are E 8 N N 1 displaystyle operatorname E theta frac N N 1 Var 8 N 1 2 N 3 2 N 3 n N 2 displaystyle operatorname Var theta frac N 1 2 N 3 frac 2N 3 nN 2 For large N E 8 is approximately 1 and Var 8 2 N 1 1 n displaystyle operatorname Var 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 EditMorisita s overlap index Natural exponential family Scaling pattern of occupancy Spatial ecology Watson s power law Density mass allometry Variance mass allometryReferences Edit a b c d e Taylor L R 1961 Aggregation variance and the mean Nature 189 4766 732 735 Bibcode 1961Natur 189 732T doi 10 1038 189732a0 S2CID 4263093 a b Thomas R E Southwood 1966 Ecological methods with particular reference to the study of insect populations Methuen ISBN 9780416289305 a b c d e Taylor L R Woiwod I P 1980 Temporal stability as a density dependent species characteristic Journal of Animal Ecology 49 1 209 224 doi 10 2307 4285 JSTOR 4285 a b c Taylor LR Woiwod 1982 Comparative Synoptic Dynamics I Relationships Between Inter and Intra Specific Spatial and Temporal Variance Mean Population Parameters J Anim Ecol 51 3 879 906 doi 10 2307 4012 JSTOR 4012 Kendal WS Frost P 1987 Experimental metastasis a novel application of the variance to mean power function J Natl Cancer Inst 79 5 1113 1115 PMID 3479636 a b c Kendal WS 1995 A probabilistic model for the variance to mean power law in ecology Ecological Modelling 80 2 3 293 297 doi 10 1016 0304 3800 94 00053 k Keeling M Grenfell B 1999 Stochastic dynamics and a power law for measles variability Philosophical Transactions of the Royal Society of London Series B Biological Sciences 354 1384 769 776 doi 10 1098 rstb 1999 0429 PMC 1692561 PMID 10365402 Anderson RM May RM 1989 Epidemiological parameters of HIV transmission Nature 333 6173 514 519 doi 10 1038 333514a0 PMID 3374601 S2CID 43491211 Philippe P 1999 The scale invariant spatial clustering of leukemia in San Francisco J Theor Biol 199 4 371 381 Bibcode 1999JThBi 199 371P doi 10 1006 jtbi 1999 0964 PMID 10441455 Bassingthwaighte JB 1989 Fractal nature of regional myocardial blood flow heterogeneity Circ Res 65 3 578 590 doi 10 1161 01 res 65 3 578 PMC 3361973 PMID 2766485 a b Kendal WS 2001 A stochastic model for the self similar heterogeneity of regional organ blood flow Proc Natl Acad Sci U S A 98 3 837 841 Bibcode 2001PNAS 98 837K doi 10 1073 pnas 98 3 837 PMC 14670 PMID 11158557 a b Kendal WS 2003 An exponential dispersion model for the distribution of human single nucleotide polymorphisms Mol Biol Evol 20 4 579 590 doi 10 1093 molbev msg057 PMID 12679541 a b c d Kendal WS 2004 A scale invariant clustering of genes on human chromosome 7 BMC Evol Biol 4 1 3 doi 10 1186 1471 2148 4 3 PMC 373443 PMID 15040817 a b c d Kendal WS Jorgensen B 2011 Taylor s power law and fluctuation scaling explained by a central limit like convergence Phys Rev E 83 6 066115 Bibcode 2011PhRvE 83f6115K doi 10 1103 physreve 83 066115 PMID 21797449 Kendal WS Jorgensen B 2015 A scale invariant distribution of the prime numbers Computation 3 4 528 540 doi 10 3390 computation3040528 a b c Kendal WS Jorgensen BR 2011 Tweedie convergence a mathematical basis for Taylor s power law 1 f noise and multifractality Phys Rev E 84 6 066120 Bibcode 2011PhRvE 84f6120K doi 10 1103 physreve 84 066120 PMID 22304168 S2CID 22585727 Reynolds O 1879 On certain dimensional properties of matter in the gaseous state Part I Experimental researches on thermal transpiration of gases through porous plates and on the laws of transpiration and impulsion including an experimental proof that gas is not a continuous plenum Part II On an extension of the dynamical theory of gas which includes the stresses tangential and normal caused by a varying condition of gas and affords an explanation of the phenomena of transpiration and impulsion Philosophical Transactions of the Royal Society of London 170 727 845 doi 10 1098 rstl 1879 0078 Pareto V 1897 Cours D economie Politique Volume 2 Lausanne F Rouge Fisher RA 1918 The correlation between relatives on the supposition of Mendelian inheritance Transactions of the Royal Society of Edinburgh 52 2 399 433 doi 10 1017 S0080456800012163 Pearson K 1921 On a General Method of determining the successive terms in a Skew Regression Line Biometrika 13 2 3 296 300 doi 10 2307 2331756 Neyman J 1926 On the correlation of the mean and the variance in samples drawn from an infinite population Biometrika 18 3 4 401 413 doi 10 2307 2331958 JSTOR 2331958 Smith HF 1938 An empirical law describing heterogeneity in the yield of agricultural crops J Agric Sci 28 1 23 doi 10 1017 s0021859600050516 S2CID 85867752 Bliss CI 1941 Statistical problems in estimating populations of Japanese beetle larve J Econ Entomol 34 2 221 232 doi 10 1093 jee 34 2 221 Fracker SB Brischle HA 1944 Measuring the local distribution of Ribes Ecology 25 3 283 303 doi 10 2307 1931277 JSTOR 1931277 Hayman BI Lowe AD 1961 The transformation of counts of the cabbage aphid Brevicovyne brassicae L NZ J Sci 4 271 278 Taylor LR 1984 Anscombe s hypothesis and the changing distributions of insect populations Antenna 8 62 67 Taylor LR Taylor RAJ 1977 Aggregation migration and population mechanics Nature 265 5593 415 421 Bibcode 1977Natur 265 415T doi 10 1038 265415a0 PMID 834291 S2CID 6504396 a b Hanski I 1980 Spatial patterns and movements in coprophagous beetles Oikos 34 3 293 310 doi 10 2307 3544289 JSTOR 3544289 a b Anderson RD Crawley GM Hassell M 1982 Variability in the abundance of animal and plant species Nature 296 5854 245 248 Bibcode 1982Natur 296 245A doi 10 1038 296245a0 S2CID 4272853 a b Taylor LR Taylor RAJ Woiwod IP Perry JN 1983 Behavioural dynamics Nature 303 5920 801 804 Bibcode 1983Natur 303 801T doi 10 1038 303801a0 S2CID 4353208 Kemp AW 1987 Families of discrete distributions satisfying Taylor s power law Biometrics 43 3 693 699 doi 10 2307 2532005 JSTOR 2532005 Yamamura K 1990 Sampling scale dependence of Taylor s power law Oikos 59 1 121 125 doi 10 2307 3545131 JSTOR 3545131 Routledge RD Swartz TB 1991 Taylor s power law re examined Oikos 60 1 107 112 doi 10 2307 3544999 JSTOR 3544999 Tokeshi M 1995 On the mathematical basis of the variance mean power relationship Res Pop Ecol 37 43 48 doi 10 1007 bf02515760 S2CID 40805500 Perry JN 1994 Chaotic dynamics can generate Taylor s power law Proceedings of the Royal Society B Biological Sciences 257 1350 221 226 Bibcode 1994RSPSB 257 221P doi 10 1098 rspb 1994 0118 S2CID 128851189 Clark S Perry JJN Marshall JP 1996 Estimating Taylor s power law parameters for weeds and the effect of spatial scale Weed Research 36 5 405 417 doi 10 1111 j 1365 3180 1996 tb01670 x Ramsayer J Fellous S Cohen JE amp Hochberg ME 2011 Taylor s Law holds in experimental bacterial populations but competition does not influence the slope Biology Letters a b Jorgensen Bent 1997 The theory of dispersion models Chapman amp Hall ISBN 978 0412997112 a b Kendal WS 2002 Spatial aggregation of the Colorado potato beetle described by an exponential dispersion model Ecol Model 151 2 3 261 269 doi 10 1016 s0304 3800 01 00494 x Cohen J E Xu m Schuster W S 2013 Stochastic multiplicative population growth predicts and interprets Taylor s power law of fluctuation scaling Proc R Soc Lond B Biol Sci 280 1757 20122955 doi 10 1098 rspb 2012 2955 PMC 3619479 PMID 23427171 Downing JA 1986 Spatial heterogeneity evolved behaviour or mathematical artefact Nature 323 6085 255 257 Bibcode 1986Natur 323 255D doi 10 1038 323255a0 S2CID 4323456 Xiao X Locey K amp White E P 2015 A process independent explanation for the general form of Taylor s law The American Naturalist 186 2 51 60 arXiv 1410 7283 doi 10 1086 682050 PMID 26655161 S2CID 14649978 a href Template Cite journal html title Template Cite journal cite journal a CS1 maint multiple names authors list link Cobain M R D Brede M amp Trueman C N 2018 Taylor s power law captures the effects of environmental variability on community structure An example from fishes in the North Sea PDF Journal of Animal Ecology 88 2 290 301 doi 10 1111 1365 2656 12923 PMID 30426504 S2CID 53306901 a href Template Cite journal html title Template Cite journal cite journal a CS1 maint multiple names authors list link Eisler Z Bartos I Kertesz 2008 Fluctuation scaling in complex systems Taylor s law and beyond Adv Phys 57 1 89 142 arXiv 0708 2053 Bibcode 2008AdPhy 57 89E doi 10 1080 00018730801893043 S2CID 119608542 Fronczak A Fronczak P 2010 Origins of Taylor s power law for fluctuation scaling in complex systems Phys Rev E 81 6 066112 arXiv 0909 1896 Bibcode 2010PhRvE 81f6112F doi 10 1103 physreve 81 066112 PMID 20866483 S2CID 17435198 Cohen JE 2016 Statistics of primes and probably twin primes satisfy Taylor s Law from ecology The American Statistician 70 4 399 404 doi 10 1080 00031305 2016 1173591 S2CID 13832952 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 amp J Roy Indian Statistical Institute Calcutta Jorgensen B 1987 Exponential dispersion models J R Stat Soc Ser B 49 2 127 162 doi 10 1111 j 2517 6161 1987 tb01685 x Jorgensen B Marinez JR Tsao M 1994 Asymptotic behaviour of the variance function Scandinavian Journal of Statistics 21 223 243 a b Wilson LT Room PM 1982 The relative efficiency and reliability of three methods for sampling arthropods in Australian cotton fields Australian Journal of Entomology 21 3 175 181 doi 10 1111 j 1440 6055 1982 tb01786 x Jorgensen B 1997 The theory of exponential dispersion models Chapman amp Hall London Rayner JMV 1985 Linear relations in biomechanics the statistics of scaling functions Journal of Zoology 206 3 415 439 doi 10 1111 j 1469 7998 1985 tb05668 x Ferris H Mullens TA Foord KE 1990 Stability and characteristics of spatial description parameters for nematode populations J Nematol 22 4 427 439 PMC 2619069 PMID 19287742 a b Hanski I 1982 On patterns of temporal and spatial variation in animal populations Ann zool Fermici 19 21 37 Boag B Hackett CA Topham PB 1992 The use of Taylor s power law to describe the aggregated distribution of gastro intestinal nematodes of sheep Int J Parasitol 22 3 267 270 doi 10 1016 s0020 7519 05 80003 7 PMID 1639561 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 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 E3158 Random sampling of skewed distributions does not necessarily imply Taylor s law 2015 Proc Natl Acad Sci USA 112 25 E3156 Banerjee B 1976 Variance to mean ratio and the spatial distribution of animals Experientia 32 8 993 994 doi 10 1007 bf01933930 S2CID 7687728 a b c d e Hughes G Madden LV 1992 Aggregation and incidence of disease Plant Pathology 41 6 657 660 doi 10 1111 j 1365 3059 1992 tb02549 x Patil GP Stiteler WM 1974 Concepts of aggregation and their quantification a critical review with some new results and applications Researches on Population Ecology 15 238 254 doi 10 1007 bf02510670 S2CID 30108449 a b c Turechek WW Madden LV Gent DH Xu XM 2011 Comments regarding the binary power law for heterogeneity of disease incidence Phytopathology 101 12 1396 1407 doi 10 1094 phyto 04 11 0100 PMID 21864088 Gosme Marie Lucas Philippe 2009 06 12 Disease Spread Across Multiple Scales in a Spatial Hierarchy Effect of Host Spatial Structure and of Inoculum Quantity and Distribution Phytopathology 99 7 833 839 doi 10 1094 phyto 99 7 0833 ISSN 0031 949X PMID 19522581 a b Xu X M Madden LV 2013 The limits of the binary power law describing spatial variability for incidence data Plant Pathology 63 5 973 982 doi 10 1111 ppa 12172 Clark SJ Perry JN 1994 Small sample estimation for Taylor s power law Environment Ecol Stats 1 4 287 302 doi 10 1007 BF00469426 S2CID 20054635 a b c Wilson LT Room PM 1983 Clumping patterns of fruit and arthropods in cotton with implications for binomial sampling Environ Entomol 12 50 54 doi 10 1093 ee 12 1 50 a b c d Bliss CI Fisher RA 1953 Fitting the negative binomial distribution to biological data also includes note on the efficient fitting of the negative binomial Biometrics 9 2 177 200 doi 10 2307 3001850 JSTOR 3001850 Jones VP 1991 Binomial sampling plans for tentiform leafminer Lepidoptera Gracillariidae on apple in Utah J Econ Entomol 84 2 484 488 doi 10 1093 jee 84 2 484 a b Katz L 1965 United treatment of a broad class of discrete probability distributions in Proceedings of the International Symposium on Discrete Distributions Montreal Jewel W Sundt B 1981 Improved approximations for the distribution of a heterogeneous risk portfolio Bull Assoc Swiss Act 81 221 240 Foley P 1994 Predicting extinction times from environmental stochasticity and carrying capacity Conserv Biol 8 124 137 doi 10 1046 j 1523 1739 1994 08010124 x Pertoldi C Bach LA Loeschcke V 2008 On the brink between extinction and persistence Biol Direct 3 47 doi 10 1186 1745 6150 3 47 PMC 2613133 PMID 19019237 Halley J Inchausti P 2002 Lognormality in ecological time series Oikos 99 3 518 530 doi 10 1034 j 1600 0706 2002 11962 x S2CID 54197297 Southwood TRE amp Henderson PA 2000 Ecological methods 3rd ed Blackwood Oxford Service MW 1971 Studies on sampling larval populations of the Anopheles gambiae complex Bull World Health Organ 45 2 169 180 PMC 2427901 PMID 5316615 a b c Southwood TRE 1978 Ecological methods Chapman amp Hall London England Karandinos MG 1976 Optimum sample size and comments on some published formulae Bull Entomol Soc Am 22 4 417 421 doi 10 1093 besa 22 4 417 Ruesink WG 1980 Introduction to sampling theory in Kogan M amp Herzog DC eds Sampling Methods in Soybean Entomology Springer Verlag New York Inc New York pp 61 78 Schulthess F Bosque Pereza NA Gounoua S 1991 Sampling lepidopterous pests on maize in West Africa Bull Entomol Res 81 3 297 301 doi 10 1017 s0007485300033575 Bisseleua DHB Yede Vida S 2011 Dispersion models and sampling of cacao mirid bug Sahlbergella singularis Hemiptera Miridae on theobroma cacao in southern Cameroon Environ Entomol 40 1 111 119 doi 10 1603 en09101 PMID 22182619 S2CID 46679671 Wilson LT Gonzalez D amp 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 Green RH 1970 On fixed precision level sequential sampling Res Pop Ecol 12 2 249 251 doi 10 1007 BF02511568 S2CID 35973901 Serraa GV La Porta NC Avalos S Mazzuferi V 2012 Fixed precision sequential sampling plans for estimating alfalfa caterpillar Colias lesbia egg density in alfalfa Medicago sativa fields in Cordoba Argentina J Insect Sci 13 41 41 doi 10 1673 031 013 4101 PMC 3740930 PMID 23909840 Myers JH 1978 Selecting a measure of dispersion Environ Entomol 7 5 619 621 doi 10 1093 ee 7 5 619 Bartlett M 1936 Some notes on insecticide tests in the laboratory and in the field Supplement to the Journal of the Royal Statistical Society 3 2 185 194 doi 10 2307 2983670 JSTOR 2983670 Iwao S 1968 A new regression method for analyzing the aggregation pattern of animal populations Res Popul Ecol 10 1 20 doi 10 1007 bf02514729 S2CID 39807668 Nachman G 1981 A mathematical model of the functional relationship between density and spatial distribution of a population J Anim Ecol 50 2 453 460 doi 10 2307 4066 JSTOR 4066 Allsopp PG 1991 Binomial sequential sampling of adult Saccharicoccus sacchari on sugarcane Entomologia Experimentalis et Applicata 60 3 213 218 doi 10 1111 j 1570 7458 1991 tb01540 x S2CID 84873687 Kono T Sugino T 1958 On the Estimation of the Density of Rice Stems Infested by the Rice Stem Borer Japanese Journal of Applied Entomology and Zoology 2 3 184 doi 10 1303 jjaez 2 184 Binns MR Bostonian NJ 1990 Robustness in empirically based binomial decision rules for integrated pest management J Econ Entomol 83 2 420 442 doi 10 1093 jee 83 2 420 a b Nachman G 1984 Estimates of mean population density and spatial distribution of Tetranychus urticae Acarina Tetranychidae and Phytoseiulus persimilis Acarina Phytoseiidae based upon the proportion of empty sampling units J Appl Ecol 21 3 903 991 doi 10 2307 2405055 JSTOR 2405055 Schaalje GB Butts RA Lysyk TL 1991 Simulation studies of binomial sampling a new variance estimator and density pre amp ctor with special reference to the Russian wheat aphid Homoptera Aphididae J Econ Entomol 84 140 147 doi 10 1093 jee 84 1 140 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 Wilson L T Room PM 1983 Clumping patterns of fruit and arthropods in cotton with implications for binomial sampling Environ Entomol 12 50 54 doi 10 1093 ee 12 1 50 Perry JN amp Taylor LR 1986 Stability of real interacting populations in space and time implications alternatives and negative binomial J Animal Ecol 55 1053 1068 Elliot JM 1977 Some methods for the statistical analysis of samples of benthic invertebrates 2nd ed Freshwater Biological Association Cambridge United Kingdom Cole LC 1946 A theory for analyzing contagiously distributed populations Ecology 27 4 329 341 doi 10 2307 1933543 JSTOR 1933543 a b c d Lloyd M 1967 Mean crowding J Anim Ecol 36 1 1 30 doi 10 2307 3012 JSTOR 3012 Iwao S Kuno E 1968 Use of the regression of mean crowding on mean density for estimating sample size and the transformation of data for the analysis of variance Res Pop Ecology 10 2 210 214 doi 10 1007 bf02510873 S2CID 27992286 Ifoulis AA Savopoulou Soultani M 2006 Developing optimum sample size and multistage sampling plans for Lobesia botrana Lepidoptera Tortricidae larval infestation and injury in northern Greece J Econ Entomol 99 5 1890 1898 doi 10 1093 jee 99 5 1890 PMID 17066827 a b Ho CC 1993 Dispersion statistics and sample size estimates for Tetranychus kanzawai Acari Tetranychidae on mulberry Environ Entomol 22 21 25 doi 10 1093 ee 22 1 21 Iwao S 1975 A new method of sequential sampling to classify populations relative to a critical density Res Popul Ecol 16 2 281 28 doi 10 1007 bf02511067 S2CID 20662793 Kuno E 1969 A new method of sequential sampling to obtain the population estimates with a fixed level of precision Res Pooul Ecol 11 2 127 136 doi 10 1007 bf02936264 S2CID 35594101 Parrella MP Jones VP 1985 Yellow traps as monitoring tools for Liriomyza trifolii Diptera Agromyzidae in chrysanthemum greenhouses J Econ Entomol 78 53 56 doi 10 1093 jee 78 1 53 a b Morisita M 1959 Measuring the dispersion and the analysis of distribution patterns Memoirs of the Faculty of Science Kyushu University Series e Biol 2 215 235 Pedigo LP amp Buntin GD 1994 Handbook of sampling methods for arthropods in agriculture CRC Boca Raton FL Smith Gill SJ 1975 Cytophysiological basis of disruptive pigmentary patterns in the leopard frog Rana pipiens II Wild type and mutant cell specific patterns J Morphol 146 1 35 54 doi 10 1002 jmor 1051460103 PMID 1080207 S2CID 23780609 Elliot JM 1977 Statistical analysis of samples of benthic invertebrates Freshwater Biological Association Ambleside Fisher RA 1925 Statistical methods for research workers Hafner New York a b Hoel P 1943 On the indices of dispersion Ann Math Statist 14 2 155 doi 10 1214 aoms 1177731457 David FN Moore PG 1954 Notes on contagious distributions in plant populations Annals of Botany 18 47 53 doi 10 1093 oxfordjournals aob a083381 Green RH 1966 Measurement of non randomness in spatial distributions Res Pop Ecol 8 1 7 doi 10 1007 bf02524740 S2CID 25039063 Gottwald TR Bassanezi RB Amorim L Bergamin Filho A 2007 Spatial pattern analysis of citrus canker infected plantings in Sao Paulo Brazil and augmentation of infection elicited by the Asian leafminer Phytopathology 97 6 674 683 doi 10 1094 phyto 97 6 0674 PMID 18943598 a b Hughes G Madden LV 1993 Using the beta binomial distribution to describe aggregated patterns of disease incidence Phytopathology 83 9 759 763 doi 10 1094 phyto 83 759 a b Fleiss JL 1981 Statistical methods for rates and proportions 2nd ed Wiley New York USA Ma ZS 1991 Further interpreted Taylor s Power Law and population aggregation critical density Trans Ecol Soc China 1991 284 288 de Oliveria T 1965 Some elementary tests for mixtures of discrete distributions in Patil GP ed Classical and contagious discrete distributions Calcuta Calcutta Publishing Society pp379 384 a b Bohning D 1994 A note on a test for Poisson overdispersion Biometrika 81 2 418 419 doi 10 2307 2336974 JSTOR 2336974 Ping S 1995 Further study on the statistical test to detect spatial pattern Biometrical Journal 37 2 199 203 doi 10 1002 bimj 4710370211 Clapham AR 1936 Overdispersion in grassland communities and the use of statistical methods in plant ecology J Ecol 14 1 232 251 doi 10 2307 2256277 JSTOR 2256277 Blackman 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 Greig Smith P 1952 The use of random and contiguous quadrats in the study of the structure of plant communities Ann Bot 16 2 293 316 doi 10 1093 oxfordjournals aob a083317 Gosset E Louis B 1986 The binning analysis Towards a better significance test Astrophysics Space Sci 120 2 263 306 Bibcode 1986Ap amp SS 120 263G doi 10 1007 BF00649941 hdl 2268 88597 S2CID 117653758 span, wikipedia, wiki, book, books, library,

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