The allometric hypothesis which relates the shape (y) of biological organs to the size of the plant or animal (x), as a function of the relative growth rates, is ubiquitous in biology. This concept has been especially useful in studies of carcass composition of farm animals, and is the basis for the definition of maintenance requirements in animal nutrition.
When the size variable is random the differential equation describing the relative growth rates of organs becomes a stochastic differential equation, with a solution different from that of the deterministic equation normally used to describe allometry. This is important in studies of carcass composition where animals are slaughtered in different sizes and ages, introducing variance between animals into the size variable.
This paper derives an equation that relates values of the shape variable to the expected values of the size variable at any point. This is the most easily interpreted relationship in many applications of the allometric hypothesis such as the study of the development of carcass composition in domestic animals by serial slaughter. The change in the estimates of the coefficients of the allometric equation found through the usual deterministc equation is demonstrated under additive and multiplicative errors. The inclusion of a factor based on the reciprocal of the size variable to the usual log - log regression equation is shown to produce unbiased estimates of the parameters when the errors can be assumed to be multiplicative.
The consequences of stochastic size variables in the study of carcass composition are discussed.