[3] Related variance to mean power laws have also been demonstrated in several non-ecological systems: The first use of a double log-log plot was by Reynolds in 1879 on thermal aerodynamics.
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.
[3][4] Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function.
[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.
A variance to mean power function had been applied to non-ecological systems, under the rubric of Taylor's law.
[3][4] Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function.
[41] Variation in the exponents of Taylor's Law applied to ecological populations cannot be explained or predicted based solely on statistical grounds however.
[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.
[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.
[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.
In logarithmic form The exponent in Taylor's law is scale invariant: If the unit of measurement is changed by a constant factor
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.
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.
This can be graphically tested by plotting p against m. Wilson and Room developed a binomial model that incorporates Taylor's law.
[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.
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.
As an aid to pest control Wilson et al developed a test that incorporated a threshold level where action should be taken.
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.
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.
Lloyd's index of patchiness (IP)[98] is It is a measure of pattern intensity that is unaffected by thinning (random removal of points).
If the population obeys Taylor's law then Iwao proposed a patchiness regression to test for clumping[99][100] Let yi here is Lloyd's index of mean crowding.
[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.
Masaaki 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.
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.
If the population obeys Taylor's law then The index of cluster size (ICS) was created by David and Moore.
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.
If the population obeys Taylor's law Binary sampling (presence/absence) is frequently used where it is difficult to obtain accurate counts.