*Ecological Archives* A016-009-A4

**Belinda Barnes, Huiquan Bi, and Michael L. Roderick. Year. Application of an ecological framework linking scales based on self thinning. ***Ecological Applications * VOL: pp-pp.

Appendix D. Establishing an allometric relationship for .

We show that, for 0 < a < 1, the empirically derived allometric for individuals,

From the binomial expansion.

Thus, from above,

where is the variance of the data set {m_{di}}. Clearly, in the case is close to zero, the RHS can be estimated by the first term alone. Alternatively, assuming Further, although may be large, for all data sets considered in this paper it was found that . Thus, if *n* is sufficiently large so that is small (Chebychev's Theorem,
see, for example Kotz et al. (1982)), and if we ignore all higher order terms that we expect (for large *n*) to be smaller than the second order term given here, then,

We note, however, that many empirically based studies applied to establish this allometric for individuals use data from Cannell (1982) (for example, Niklas and Enquist (2002)). This data is recorded per unit of ground area (number of individuals and total mass per hectare) and thus, by default, such relationships refer to the average individual.

LITERATURE CITED

Cannell, M. 1982. World forest biomass and primary production data.Academic Press, London, UK.

Kotz, S., N. Johnson, and C. Read. 1982. Encyclopaedia of Statistical Sciences. John Wiley and Sons, New York, New York, USA.

Niklas, K., and B. Enquist. 2002. Canonical rules for plant organ biomass partitioning and annual allocation. American Journal of Botany **89(5)**:812–819.

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