Taylor's law

Not to be confused with Taylor Law or Taylor rule.

Taylor's law (also known as 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]

Definition

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

\text{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.[2][3]

Taylor’s law has also been applied to assess the time dependent changes of population distributions.[2] Related variance to mean power laws have also been demonstrated in several non-ecological systems:

History

Much controversy has surrounded Taylor’s law. Bliss[16] in 1941, Fracker and Brischle[17] in 1941 and Hayman & Lowe [18] in 1961 also described the power law, but in the context of data from single species. Taylor’s 1961 paper[1] was different in that the power law had been demonstrated for many different species; the power law was proposed here as a general feature of the spatial distribution of these species; and he provided a mechanistic hypothesis to explain the power 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.[19] Taylor’s novel 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.[20] In proposing that animal behavior was the principle mechanism behind the clustering of organisms, Taylor though appeared to have ignored his own report of clustering seen with tobacco necrosis virus plaques.[1]

Following Taylor’s initial publications several alternative hypotheses for the power law were advanced. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction.[21] 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.[2][3]

Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function.[22] 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.[23]

About this time concerns were, however, raised regarding the statistical variability with measurements of the power law exponent, and the possibility that observations of a power law might reflect more mathematical artefact than a mechanistic process.[24] Taylor et al responded with an additional publication of extensive observations which he claimed refuted Downing’s concerns.[25]

In addition, Thórarinsson published a detailed critique of the animal behavioral model, noting that Taylor had modified his model several times in response to concerns raised, and that some of these modifications were inconsistent with earlier versions.[26] Thórarinsson also claimed that Taylor confounded animal numbers with density and that Taylor had incorrectly interpreted simulations that had been constructed to demonstrate his models as validation.

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.[27] Kemp, however, did not explain the parameterizations of his models in mechanistic terms. Other relatively abstract models for Taylor’s law followed.[5][28]

A number of additional 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.[29][30]

Reports also began to accumulate where Taylor’s law had been applied to time series data. Perry showed how simulations based on chaos theory could yield Taylor’s law, and Kilpatrick & Ives provided simulations which showed how interactions between different species might lead to Taylor's law.[31][32]

Other reports appeared where Taylor’s law had been applied to the spatial distribution of plants[33] and bacterial populations[34] As with the observations of Tobacco necrosis virus mentioned earlier, these observations were not consistent with Taylor’s animal behavioral model.

Earlier it was mentioned that variance to mean power function had been applied to non-ecological systems, under the rubric of Taylor’s law. To provide a more general explanation for the range of manifestations of the power law a hypothesis was proposed based on the Tweedie distributions,[35] a family of probabilistic models that express an inherent power function relationship between the variance and the mean.[10][12][36] Details regarding this hypothesis will be provided in the next section.

A further alternative explanation for Taylor's law was proposed by Cohen et al, derived from the Lewontin Cohen growth model.[37] This model was successfully used to describe the spatial and temporal variability of forest populations.

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.[38] 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.[39] 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.[13]

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.[40][41] 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;\theta,\lambda)=\lambda\kappa_b(\theta)\left [\left(1+{s \over \theta}\right)^\alpha-1\right],

where κb(θ) is the cumulant function,

\kappa_b(\theta)={(\alpha-1)\over \alpha}\left({\theta \over (\alpha-1)}\right)^\alpha,

the Tweedie exponent

\alpha=(b-2)/(b-1),

s is the generating function variable, and θ and λ are the canonical and index parameters, respectively.[35] 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=\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.[36] 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.[11][12][13][15]

The main justification for the Tweedie hypothesis rests with the mathematical convergence properties of the Tweedie distributions.[12] The Tweedie convergence theorem requires the Tweedie distributions to act as foci of convergence for a wide range of statistical processes.[42] 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.[13][15]

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.[5]

Mathematical formulation

In symbols

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

Extensions and refinements

A refinement in the estimation of the slope b has been proposed by Rayner.[43]

b = \frac { f - \varphi + \sqrt{ ( f - \varphi )^2 - 4 r^2 f \varphi } }{ 2 r \sqrt{ f } }

where r is the Pearson moment correlation coefficient between log(s2) and log m, f is the ratio of sample variances in log(s2) and log m and φ is the ratio of the errors in log(s2) and log m.

Ordinary least squares regression assumes that φ = ∞. This tends to underestimate the value of b because the estimates of both log(s2) and log m are subject to error.

An extension of Taylor's law has been proposed by Ferris et al when multiple samples are taken[44]

s^2 = ( c n^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.

Interpretation

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

In Poisson-distributed data, b = 1.[23] 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.[45] Occasionally cases with b > 2 have been reported.[2] b values below 1 are uncommon but have also been reported ( b = 0.93 ).[46]

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.[47]

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.[48]

Extension to binary sampling

Binomial sampling is popular where there are large number of units (crops, trees) to be examined and where counts of individuals of interest (typically insects) may be difficult (frequently because the insects fly away before they can be accurately counted).

A form of Taylor's law applicable to binary sampling (presence/absence of at least one individual in a sample unit) has been proposed.[49] In a binomial distribution, the theoretical variance is

s^2 = n p ( 1 - p ) ,

where s2 is the variance, n is the sample size and p is the proportion of sample units with at least one individual. The proposed binary form of Taylor's law is

var_{ obs } = \log( a ) + b \log( var_{ bin } ) ,

where varobs is the observed variance and varbin is that expected from the binomial distribution. When both a and b are equal to 1, then a random spatial pattern is suggested and is best described by the binomial distribution. When b = 1 and a > 1, there is over-dispersion with no dependence on the mean incidence (p). When both a and b are > 1, the degree of aggregation varies with p.

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[50] 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

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.[51] The basic relationship is

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 } }

Dispersion parameter estimator

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

k = m^2 / ( s^2 - m )

where m is the sample mean and s2 is the variance.[52] 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:[51]

k = \frac { m } { a m^{ b - 1 } - 1 }

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

Jones[53] 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[51]

p = 1 - e^{ - m \log( a m^{ b - 1 } )( a m^{ 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 ) \frac { k + x - 1 } { x } \frac { m k^{ -1 } } { m k^{ -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 }

Jones also gives confidence intervals for these probabilities.

\mathrm{ CI } = t ( \frac { P( x ) ( 1 - P( x ) ) } { N } )^{ 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

Katz proposed a family of distributions (the Katz family) with 2 parameters ( w1, w2 ).[54] 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 = \frac { w_1 } { 1 - w_2 }
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

\frac { w_1 } { ( 1 - w_2 ) } = m
\frac { w_2 } { ( 1 - w_2 ) } = \frac { s^2 - m } { m }

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:[55]

p_n = ( a + \frac{ b }{ n } ) 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 )
b = 1

Katz also introduced a statistical test[54]

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 < 0 and > 0 indicate under and over dispersion respectively. Overdispersion is often caused by latent heterogeneity - the presence of multiple sub populations within the population the sample is drawn from.

This statistic is related to the Neyman-Scott statistic

NS = \sqrt { \frac { n - 1 } { 2 } } ( \frac { s^2} { m } - 1 )

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

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 = \sqrt { \frac { n } { 2 } } ( a m^{ b - 1 } - 1 )

Time to extinction

If Taylor's law is assumed to apply it is possible to determine the mean time to local extinction. This model assumes a simple random walk in time and the absence of density dependent population regulation.[56]

Let N_{ t + 1 } = r N_{ 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 ) )

where var( r ) is the variance of r.

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

T_E = \frac{ 2\log( N ) } { Var( r ) } ( \log( K ) - \frac{ \log( N ) } { 2 })

where TE is the mean time to local extinction.

The probability of extinction by time t is

P( t ) = 1 - e^{ \frac{ t } { T_E } }

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)[57]

H = m - am^{ ( b - 1 ) }

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

m > a^{ \frac{ 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.[58] 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%.[59] 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

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 ) / ( mk )

where k is the parameter of the negative binomial distribution.

A more general sample size estimator has also been proposed[60]

n = ( t / D )^2 a m^{( b - 2 )}

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

An alternative has been proposed by Southwood[61]

n = a 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.[62] The first was modified by Ruesink to incorporate Taylor's law.[63]

n = ( t / d_m )^2 a m^{( b - 2 )}

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

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

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 = \frac { CI } { 2p }

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.[64]

\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.[65] The required sample size is

n = t | m - T |^{ - 2} a m^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

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[66]

D = ( a n^{ 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.[67]

T_n \ge ( \frac{ a n^{ 1 - b } }{ D^2 } )^{ \frac{ 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 \ge \frac{ \alpha - 1 }{ D^2 - \frac{ \beta - 1 }{ n } }

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.[68] 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 Iowa's Patchiness regression test but all the methods listed here have been used in the literature.

Barlett-Iawo model

Barlett in 1936[69] and later Iawo independently in 1968[70] both proposed an alternative relationship between the variance and the mean. In symbols

s_i^2 = am_i + bm_i^2 \,

where s is the variance in the 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:[71]

p_0 = \exp( -a m^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

Allsop used this relationship along with Taylor's law to derive an expression for the proportion of infested units in a sample[72]

P_1 = 1 - exp( -exp( \frac{ \frac{ log_e ( \frac{ A^2 }{ a } ) }{ b - 2 } + log_e( n )( \frac{ b - 1 }{ b - 2 } - 1 ) - c }{ d } ) )

N = n P_1

where

A^2 = \frac{ D^2 }{ z^2_{ \frac{ \alpha } { 2 } } }

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.[73]

\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.

Note

The equation was derived while examining the relationship between the proportion ( P ) of a series of rice hills infested and the mean severity of infestation ( m ). The model studied was

P = 1 - a e^{ b m }

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[74] and it is recommended that the adjusted mean ( ma ) be used instead[75]

m_a = m ( 1 - \frac { var( \log( m_i ) ) } { 2 } )

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[75]

m_a = m e^{ ( 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[76]

Var( m ) = m^2 ( c_1 + c_2 - c_3 + MSE )

where

c_1 = \frac{ \beta^2 ( 1 - p_0 ) }{ n p_0 \log_e( p_0 )^2 }
c_2 = \frac{ MSE } { N } + s_{\beta}^2 ( \log_e( \log_e( p_0 ) ) - p^2 )
c_3 = \frac{ \exp( a + ( b - 2 )[\alpha - \beta \log_e( p_0 ) ] ) }{ n }

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:

s^2 = a + b \log_e( m )

Hughes-Madden equation

Hughes and Madden have proposed testing a similar relationship also applicable to binary sampling (presence/absence in a sampled unit)[77][78]

s^2 = a p^b ( 1 - p )^c

where a, b and c are constants, s2 is the variance and p is the proportion of units with at least one individual. In logarithmic form this relationship is

\log( s^2 ) = \log( a ) + b \log( p ) + c \log( 1 - p ) .

This relationship has not yet been subjected to the extensive testing that Taylor's law has been subjected to. For this reason its general applicability presently remains uncertain.

A variant of this equation was proposed by Shiyomi et al[79] who suggested testing the regression

\log( s^2 / n^2 ) = a + b \log( p ( 1 - p ) / n )

where s2 is the variance, a and b are the constants of the regression, n is the sample size 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.[80] The dispersion parameter (k) using the method of moments is m2 / ( s2 - m ) 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.[81]

1/k = 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[52]

\sum \frac { A_x } { k + x } = N \log( 1 + m / k )

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 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

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

The variance of U is[52]

Var( U ) = 2m p^2 q ( \frac { 1 - R^2 } {-\log( 1 - R ) - R } ) + 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 variance

\sigma^2 = \mu + a \mu^p

where a and p are constants. When a = 0 this defines the Poisson distribution. With p = 1 and p = 2, the distribution is known as the NB1 and NB2 distribution respectively.

Tests for a common dispersion parameter

The dispersion parameter (k)[52] is

k = m^2 / ( s^2 - m )

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[61]

k_i = a + b m_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 > 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 ).[82] kc is equal to 1/slope of this regression.

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.[83]

\mathrm{ IMC } = m + m / s^2 - 1

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

Lloyd's index of patchiness (IP)[83] is

\mathrm{ IP } = 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.

If the population obeys Taylor's law then

\mathrm{ IMC } = m + a^{ -1 } m^{ 1 - b } - 1
\mathrm{ IP } = 1 + a^{ -1 } m^{ - b } - \frac { 1 } { m }

Patchiness regression test

Iwao proposed a patchiness regression to test for clumping[70][84]

Let

y_i = m_i + m_i / s^2 - 1

yi here is Lloyd's index of mean crowding.[83] 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.[85]

The sample size (n) for a given degree of precision (D) for this regression is given by[85]

n = ( \frac{ t }{ D } )^2 ( \frac{ a + 1 } { m } + b - 1 )

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.

Iawo has proposed a sequential sampling test based on this regression.[86] The upper and lower limits of this test are based on critical densities mc where control of a pest requires action to be taken.

N_u = im_c + t( i ( a + 1 ) m_c + ( b - 1 ) m_c^2 )^{ 1 / 2 }
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.[87]

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) / D2

Parrella and Jones have proposed an alternative but related stop line[88]

T_n = ( 1 - \frac { n } { N } ) \frac { a + 1 } { D^2 - ( 1 - \frac { n } { N } ) \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

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.[89] Higher values indicate a more clumped distribution.

I_m = \frac { \sum x ( x - 1 ) } { n m ( m - 1 ) }

An alternative formulation is

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 = \frac { n IMC } { ( nm - 1 ) }

where IMC is Lloyd's index of crowding.[83]

This index is relatively independent of the population density but is affected by the sample size.

Morisita showed that the statistic[89]

I_m ( \sum x - 1 ) + n - \sum x

is distributed as a chi squared variable with n - 1 degrees of freedom.

A alternative significance test for this index has been developed for large samples.[90]

z = \frac { I_m - 1 } { ( 2 / n m^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 function

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[91]

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 = \frac { \chi^2_{ 0.975 } - k + \sum x } { \sum x - 1 }
M_c = \frac { \chi^2_{ 0.025 } - k + \sum x } { \sum x - 1 }

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 Id ≥ Mc > 1

I_p = 0.5 + 0.5 ( \frac { I_d - M_c } { k - M_c } )

When Mc > Id ≥ 1

I_p = 0.5 ( \frac { I_d - 1 } { M_u - 1 } )

When 1 > Id ≥ Mu

I_p = -0.5 ( \frac { I_d - 1 } { M_u - 1 } )

When 1 > Mu > Id

I_p = -0.5 + 0.5 ( \frac { I_d - M_u } { M_u } )

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

\frac {1}{k} = \frac{m^*}{m} - 1

where m is the mean of the sample and m* is Lloyd's index of crowding.[61]

Fisher's index of dispersion

Fisher's index of dispersion[82][92] is

\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 > 5[93] and that the sample total divided by the number of samples is > 3. In symbols

\frac { \sum x } { n } > 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.[93] 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

\mathrm{ ID } = ( n - 1 ) a m^{ b - 1 }

Index of cluster size

The index of cluster size (ICS) was created by David and Moore.[94] Under a random (Poisson) distribution ICS is expected to equal 0. Positive values indicate a clumped distribution; negative values indicate a uniform distribution.

\mathrm{ ICS } = s^2 / m - 1

where s2 is the variance and m is the mean.

If the population obeys Taylor's law

\mathrm{ ICS } = a m^{ 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

Green’s index (GI) is a modification of the index of cluster size that is independent of n the number of sample units.[95]

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 = \frac { a m^{ b - 1 } - 1 } { ( nm - 1 ) }

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

s^2 = n p ( 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 = \frac {var_{ obs } } { var_{ bin } } = \frac{ s^2 } { n p ( 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 > 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.[96]

C = \frac { D( n N - 1 ) - n N } { ( 2 N ( 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 ( ρ ) which is defined as[97]

\rho = 1 - \frac{ \sum x_i ( T - x_i ) } { p ( 1 - p ) N T ( 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

\rho = \frac{ D - 1 } { n - 1 }

If the data can be fitted with a beta-binomial distribution then[97]

D = 1 + \frac{ ( n - 1 ) \theta }{ 1 + \theta }

where θ is the parameter of the distribution.[96]

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[98] is the difference of the variance and the mean.[99] If the population is Poisson distributed then

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.[100]

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

O_T = \sqrt { \frac { 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.

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[99] 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).[101] In symbols

\theta = \frac{ s^2 }{ m }

For a Possion distribution this ratio equals 1. To test for deviations from this value he prosed 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[102] who noted that it was approximately normally distributed with a mean of 1 and a variance ( Vθ ) of

V_{ \theta } = \frac{ 2n } { ( n - 1 )^2 }

The derivation of the variance was re analysed by Bartlett[103] who considered it to be

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

\theta = am^{ b - 1 }
Note

A refinement on this test has also been published[104] These authors noted that this test tends to detect overdispersion at higher scales even when this was not present in the data. They noted that that the use of the multinomial distribution may be more appropriate than the use of a Poisson distribution for such data. The statistic θ is distributed

\theta = \frac{ s^2 }{ m } = \frac{ 1 } { n } \sum{ ( x_i - \frac{ n }{ N } )^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 θ are

E( \theta ) = \frac{ N } { N - 1}
Var( \theta ) = \frac{ ( N - 1 )^2 } { N^3 } - \frac { 2N - 3 } { nN^2 }

For large N E(θ) is approximately 1 and

Var( \theta ) \sim\ \frac{ 2 } { N } ( 1 - \frac{ 1 } { n } )

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

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