Linear relations in biomechanics: the statistics of scaling functions

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Abstract
The problem of fitting a linear relation to a bivariate data cluster obtained from morphometric measurement or from experiment is formulated rigorously, and a family of solutions (the general structural relation, g.s.r.) is derived. The regression, reduced major axis and major axis models are special cases of this model; it permits a more realistic treatment of the errors in the variates, in particular when the errors are correlated, which is particularly important in the many biological situations in which the variates contain uncontrolled real variation in addition to measurement errors. The analysis is particularly directed to the testing of hypotheses about scaling relations derived from biomechanical theory. By making different assumptions about the configuration of the errors, the g.s.r. can also be used to test for transposition allometry and for the significance of an estimated or hypothesized gradient. The model generalizes simply to multivariate problems. Application is demonstrated with examples drawn from the study of bird flight mechanics. Finally, it is demonstrated that since observed quantities correspond to peaks of adaptation or of selective fitness, scaling relations are determined primarily by scale variation of constraints on adaptation and behaviour, and are the result of a variety of interacting factors rather than a response to a single selective force described by one simple hypothesis.