Hello, I am in trouble with relationships between various methods used to describe dispersion patterns of individuals among samples. In benthic ecology, the fisher index (I=variance to mean ratio) is in use to test for non-randomness in spatial dispersion. Another way to describe dispersion is the Taylor's power law where var=a*mean^b, with b=1 indicates randomness and b>2 aggregation. A combination of the two gives I=(a*mean^b)/mean and for b=1, I=a. A chi-square test is used to test the statistical significance of I*(n-1), with n the number of samples. Then for b=1, the test for I is a*(n-1). Surprisingly (to me) the test for dispersion becomes a function of the sample size. Furthermore, for a=2 and n>9 for example, there is a significant departure from randomness, despite b=1. So that the "power" of the test decreases when sample size increases. Taylor (1961) states that the term "a" in the power law is a sampling factor. Could we consider from the relation between fisher index and power law that "a" actually is a function of sample size? This is of interest for me because the power law is sometime used to deduce the sample size needed to achieve a given precision. Many thanks for any comment about this.
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