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Taylor's law

 

Taylor's law

Taylor's law is an empirical law in

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References

See also

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.

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

For large N E(θ) is approximately 1 and

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

The expectation and variance of θ are

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

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

A refinement on this test has also been published[77] 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

Note
\theta = am^{ b - 1 }

If the population obeys Taylor's law then

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

V_{ \theta } = \frac{ 2 } { n - 1 }

The derivation of the variance was re analysed by Bartlett[76] who considered it to be

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

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

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

In 1936 Clapham proposed using the ratio of the variance to the mean as a test statistic (the relative variance).[74] In symbols

Clapham's test

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[72] showed that this estimate of the variance was incorrect. Bohning's correction is given in the equations above.

Note

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.

O_T = \sqrt { \frac { n - 1 } { 2 } } \frac { s^2 - m } { m }

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

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.[73]

var( s^2 - m ) = \frac{ 2t^2 } { n - 1 }

A related statistic suggested by de Oliveria[71] is the difference of the variance and the mean.[72] If the population is Poisson distributed then

de Oliveria's statistic

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

Related statistics

where θ is the parameter of the distribution.[69]

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

If the data can be fitted with a beta-binomial distribution then[70]

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

D is also related to intraclass correlation ( ρ ) which is defined as[70]

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.

C = \frac { D( n N - 1 ) - n N } { ( 2 N ( n^2 - n ) ) ^{ 1 / 2 } }

An alternative test is the C test.[69]

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.

D = \frac {var_{ obs } } { var_{ bin } } = \frac{ 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

s^2 = n p ( 1 - p )

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

Binary dispersal index

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

If the population obeys Taylor's law

The distribution of Green's index is not currently known so statistical tests have been difficult to devise for it.

This index equals 0 if the distribution is random, 1 if it is maximally aggregated and -1 / ( nm - 1 ) if it is uniform.

C_x = \frac { s^2 / m - 1 } { nm - 1 }

Green’s index (GI) is a modification of the index of cluster size that is independent of n the number of sample units.[68]

Green’s index

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.

\mathrm{ ICS } = a m^{ b - 1 } - 1

If the population obeys Taylor's law

where s2 is the variance and m is the mean.

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

The index of cluster size (ICS) was created by David and Moore.[67] Under a random (Poisson) distribution ICS is expected to equal 0. Positive values indicate a clumped distribution; negative values indicate a uniform distribution.

Index of cluster size

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

If the population obeys Taylor's law then

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.

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.[66] It is equal to the scale parameter when the population obeys the gamma distribution.

\frac { \sum x } { n } > 3

This index may be used to test for over dispersion of the population. It is recommended that in applications n > 5[66] and that the sample total divided by the number of samples is > 3. In symbols

\mathrm{ID} = \frac{( n - 1 ) s^2 }{ m }

Fisher's index of dispersion[55][65] is

Fisher's index of dispersion

where m is the mean of the sample and m* is Lloyd's index of crowding.[35]

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

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

Southwood's index of spatial aggregation

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.

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

When 1 > Mu > Id

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 Mc > Id ≥ 1

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

When Id ≥ Mc > 1

The standardised index ( Ip ) is then calculated from one of the formulae below

where χ2 is the chi square value for n - 1 degrees of freedom at the 97.5% and 2.5% levels of confidence.

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

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

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[64]

Standardised Morisita’s index

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

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.

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

A alternative significance test for this index has been developed for large samples.[63]

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

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

Morisita showed that the statistic[62]

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

where IMC is Lloyd's index of crowding.[56]

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

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 \frac{ \sum x^2 - \sum x } { ( \sum x )^2 - \sum x }

An alternative formulation is

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

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

Morisita’s index of dispersion

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.

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

Parrella and Jones have proposed an alternative but related stop line[61]

Kuno's test is subject to the condition that n ≥ (b - 1) / D2

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.

T_n = \frac { a + 1 } { D^2 - \frac { b - 1 } { n } }

Kuno has proposed an alternative sequential stopping test also based on this regression.[60]

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.

N_l = im_c - t( i ( a + 1 ) m_c + ( b - 1 ) m_c^2 )^{ 1 / 2 }
N_u = im_c + t( i ( a + 1 ) m_c + ( b - 1 ) m_c^2 )^{ 1 / 2 }

Iawo has proposed a sequential sampling test based on this regression.[59] 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 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.

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


The sample size (n) for a given degree of precision (D) for this regression is given by[58]

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.[58]

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.

yi here is Lloyd's index of mean crowding.[56] Perform an ordinary least squares regression of mi against y.

y_i = m_i + m_i / s^2 - 1

Let

Iwao proposed a patchiness regression to test for clumping[43][57]

Patchiness regression test

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

If the population obeys Taylor's law then

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.

\mathrm{ IP } = IMC / m

Lloyd's index of patchiness (IP)[56] is

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

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

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.[56]

Lloyd's indexes

An alternative method was proposed by Elliot who suggested plotting ( s2 - m ) against ( m2 - s2 / n ).[55] kc is equal to 1/slope of this regression.

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.

k_i = a + b m_i

Southwood has recommended regressing k against the mean and a constant[35]

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.

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

The dispersion parameter (k)[26] is

Tests for a common dispersion parameter

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.

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


Note: The negative binomial is actually a family of distributions defined by the relation of the mean to the variance

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.

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 }

The variance of U is[26]

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

U = s^2 - m + m^2 / k

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 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.

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

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[26]

1/k = am^{ b - 2 } - 1 / m

Perry and Taylor have proposed an alternative estimator of k based on Taylor's law.[54]

A negative binomial model has also been proposed.[53] 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.

Negative binomial distribution model

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.

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

A variant of this equation was proposed by Shiyomi et al[52] who suggested testing the regression

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.

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

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

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

Hughes and Madden have proposed testing a similar relationship also applicable to binary sampling (presence/absence in a sampled unit)[50][51]

Hughes-Madden equation

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

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:

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

where

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

This model may also be used to estimate stop lines for enumerative (sequential) sampling. The variance of the estimated means is[49]

where MSE is the mean square error of the regression.

m_a = m e^{ ( MSE / 2 ) }

An alternative adjustment to the mean estimates is[48]

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

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

The predicted estimates of m from this equation are subject to bias[47] and it is recommended that the adjusted mean ( ma ) be used instead[48]

Uses

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

P = 1 - a e^{ b m }

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

Note

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.

\log( m ) = \log( a ) + b \log( - \log( p_0 ) )

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.[46]

Kono-Sugino equation

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.

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

where

N = n P_1

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 } ) )


Allsop used this relationship along with Taylor's law to derive an expression for the proportion of infested units in a sample[45]

\log m = c + d \log p_0

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

p_0 = \exp( -a m^b )

Nachman proposed a relationship between the mean density and the proportion of samples with zero counts:[44]

Nachman model

This alternative formulation has not been found to be as good a fit as Taylor's law in most studies.

When the population follows a negative binomial distribution, a = 1 and b = k (the exponent of the negative binomial distribution).

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

s_i^2 = am_i + bm_i^2 \,

Barlett in 1936[9] and later Iawo independently in 1968[43] both proposed an alternative relationship between the variance and the mean. In symbols

Barlett-Iawo model

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.[42] 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.

Related analyses

The authors recommended that D be set at 0.1 for studies of population dynamics and D = 0.25 for pest control.

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.

T_n \ge \frac{ \alpha - 1 }{ D^2 - \frac{ \beta - 1 }{ n } }

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

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.

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

Serra et al have proposed a stopping rule based on Taylor's law.[41]

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.

D = ( a n^{ 1 - b } T^{ b - 2 } )^ { 1 / 2 }

Green derived another sampling formula for sequential sampling based on Taylor's law[40]

where p is the probability of finding a sample with pests present and q = 1 - p.

n = t | m - T |^{ -2 } p q

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} a m^b

As an aid to pest control Wilson et al developed a test that incorporated a threshold level where action should be taken.[39] The required sample size is

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.

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

A formula for fixed precision in serial sampling to test Taylor's law was derived by Green in 1970.[38]

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.

Sequential sampling

d_p = \frac { CI } { 2p }

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

n = ( t / d_p )^2 p^{ -1 } q

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

d_m = \frac { CI } { 2m }

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

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

Karandinos proposed two similar estimators for n.[36] The first was modified by Ruesink to incorporate Taylor's law.[37]

where n is the required sample size, a and b are the Taylor's law coefficients and D is the desired degree of precision.

n = a m^b / D^2 \,

An alternative has been proposed by Southwood[35]

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

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

A more general sample size estimator has also been proposed[34]

where k is the parameter of the negative binomial distribution.

n = ( t / D )^2 ( m + k ) / ( mk )

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

n = ( t / D )^2 / m


Where the population is Poisson distributed the sample size (n) needed is

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%.[33] 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.

Sampling size estimators

The assumption of a lognormal distribution appears to apply to about half of a sample of 544 species.[32] suggesting that it is at least a plausible assumption.

is the minimum size of population for the species to persist.

m > a^{ \frac{ 1 }{ 2 - b } }

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

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

If a population is lognormally distributed then the harmonic mean of the population size (H) is related to the arithmetic mean (m)[31]

Minimum population size required to avoid extinction

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

The probability of extinction by time t is

where TE is the mean time to local extinction.

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

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

where var( r ) is the variance of r.

var(r) = s^2 ( \log( r ) )

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

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.[30]

Time to extinction

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


If the population obeys Taylor's law then

which is known to have an asymptotic chi squared distribution with n − 1 degrees of freedom when the population is Poisson distributed.

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

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

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

This statistic is related to the Neyman-Scott statistic

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.

J_n = \sqrt { \frac { n } { 2 } } \frac { s^2 - m } { m }

Katz also introduced a statistical test[28]

b = 1
a = -\log ( 1 - w_2 )

If the population obeys a Katz distribution then the coefficients of Taylor's law are

The only members of the Sundt-Jewel family are the Poisson, binomial, negative binomial (Pascal), extended truncated negative binomial and logarithmic series distributions.

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

The Katz family is related to the Sundt-Jewel family of distributions:[29]

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.

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

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

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

Katz proposed a family of distributions (the Katz family) with 2 parameters ( w1, w2 ).[28] This family of distributions includes the Bernoulli, Geometric, Pascal and Poisson distributions as special cases. The mean and variance of a Katz distribution are

Katz family of distributions

where CI is the confidence interval, t is the critical value taken from the t distribution and N is the total sample size.

\mathrm{ CI } = t ( \frac { P( x ) ( 1 - P( x ) ) } { N } )^{ 1 / 2 }

Jones also gives confidence intervals for these probabilities.

P( 0 ) = ( 1 + m / k )^{ -k }

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( x ) = P( x - 1 ) \frac { k + x - 1 } { x } \frac { m k^{ -1 } } { m k^{ -1 } - 1 }

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

p = 1 - e^{ - m \log( a m^{ b - 1 } )( a m^{ b - 1 } - 1 )^{ -1 } }

Jones[27] 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[25]

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

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

Wilson and Room assuming that Taylor's law applied to the population gave an alternative estimator for k:[25]

where m is the sample mean and s2 is the variance.[26] 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.

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

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

Dispersion parameter estimator

p = 1 - e^{ - m \log( a m^{ b - 1 } )( a m^{ b - 1 } - 1 )^{ -1 } }

Incorporating Taylor's law this relationship becomes

where the log is taken to the base e.

p = 1 - e^{ - m \log( s^2 / m )( s^2 / m - 1 )^{ -1 } }

Wilson and Room developed a binomial model that incorporates Taylor's law.[25] The basic relationship 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.

p = 1 - e^{ -m }

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

Randomly distributed populations

(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

It has been recommended based on simulation studies[24] in applications testing the validity of Taylor's law to a data sample that:

Recommendations as to use

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

Applications

where var() is the variance of the random variable X, E() is the expectation operator and a and b are parameters.

var( X ) = a [ E( X ) ]^b

The Tweedie distribution family is a family of probability distributions which obey the following relation

Relation to the Tweedie distribution family

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 overdispersion with no dependence on the mean incidence (p). When both a and b are > 1, the degree of aggregation varies with p.

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

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

s^2 = n p ( 1 - p )

A form of Taylor's law applicable to binary sampling (presence/absence of at least one individual in a sample unit) has been proposed.[23] In a binomial distribution the theoretical variance is

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).

Extension to binary sampling

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.[22]

It is known that both a and b are subject to change due to age-specific dispersal, mortality and sample unit size.[21]

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 (>1000) this question remains open.

Notes

Most populations that have been studied have b < 2 (usually 1.5–1.6) but values of 2 have been reported.[5] Occasionally cases with b > 2 have been reported.[19] b values below 1 are uncommon but have also been reported ( b = 0.93 ).[20]

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.

In Poisson distributed data b = 1.[18] If the population follows a lognormal or gamma distribution then b = 2.

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

Interpretation

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.

s^2 = ( c n^d ) ( m^b )

A extension of Taylor's law has been proposed by Ferris et al when multiple samples are taken[17]

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.

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.

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

A refinement in the estimation of the slope b has been proposed by Rayner.[16]

Extensions and refinements

\log s_i^2 = \log a + b\log m_i

In logarithmic form

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.

s_i^2 = am_i^b

In symbols

Mathematical formulation

It appears that Taylor's law is an example of Stigler's law of eponymy.

The law itself is named after the ecologist L. R. Taylor (1924–2007). The name 'Taylor's law' was coined by Southwood in 1966.[15] Taylor's original name for this relationship was the law of the mean.

Fracker and Brischle in 1944[13] and Hayman and Lowe in 1961[14] independently described relationships between the mean and variance that are now known as Taylor's law.

This relationship was used by Yates and Finney in 1942.[12] The Bliss and Yates and Finney studies were later cited by Taylor as examples of this relationship.[2]

s^2 = a m^b

Bliss while studying Japanese beetles found the relationship[11]

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.

\log V_x = \log V_1 + b\log x \,

Smith while studying crop yields proposed a relationship in 1938 similar to Taylor's.[10] Smith proposed the relationship

in 1936.

s^2 = a m + b m^2

Barlett proposed a relationship between the sample mean and variance[9]

History

Contents

  • History 1
  • Mathematical formulation 2
    • Extensions and refinements 2.1
    • Interpretation 2.2
    • Notes 2.3
    • Extension to binary sampling 2.4
    • Relation to the Tweedie distribution family 2.5
  • Applications 3
    • Recommendations as to use 3.1
    • Randomly distributed populations 3.2
    • Dispersion parameter estimator 3.3
    • Katz family of distributions 3.4
    • Time to extinction 3.5
    • Minimum population size required to avoid extinction 3.6
    • Sampling size estimators 3.7
    • Sequential sampling 3.8
  • Related analyses 4
    • Barlett-Iawo model 4.1
    • Nachman model 4.2
    • Kono-Sugino equation 4.3
    • Hughes-Madden equation 4.4
    • Negative binomial distribution model 4.5
    • Tests for a common dispersion parameter 4.6
    • Lloyd's indexes 4.7
    • Patchiness regression test 4.8
    • Morisita’s index of dispersion 4.9
    • Standardised Morisita’s index 4.10
    • Southwood's index of spatial aggregation 4.11
    • Fisher's index of dispersion 4.12
    • Index of cluster size 4.13
    • Green’s index 4.14
    • Binary dispersal index 4.15
  • Related statistics 5
    • de Oliveria's statistic 5.1
    • Clapham's test 5.2
  • See also 6
  • References 7

It is possible to derive this law if it is assumed that the organisms of interest form clusters that obey a Poisson distribution.[7] Alternative suggestions for its origin have also been proposed.[8]

literature). physics (in the fluctuation scaling law (in the biological literature) or the power law This law is also known in the literature as the [6][5][4] and gene structures.single nucleotide polymorphisms, blood flow heterogeneity, genomic distributions of metastases cancer, leukemia, human sexual behavior, childhood infectious diseases It has also been found to be true in other areas including transmission of [3] and it has been found to be true for many species since.[2] Taylor described this relationship in 1961[1]

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