## Materials and Methods
### Multiple-Parameter Complex Allometry (MCA) Identification Protocols
#### 2.2.1 Identification in Log Scales  
In order to place analysis of MCA in geometrical scales we consider a log transformation \(v=lny\), and \(u=lnx\). This sets domains \(U\) and \(V\) for \(u\) and \(v\) respectively. Thus, Equation (3) leads to the regression model:  
\[v=z(u)+\epsilon \tag{4}\]  
\[z(u)=ln\beta(e^{u})+\alpha(e^{u})u \tag{5}\]Then, Equation (4) provides the geometrical space MCA regression protocol. The resulting mean response function denoted through the symbol \(E_{z}(v|u)\) becomes \(E_{z}(v|u)=z(u)\). A retransformation \(y=\exp(z(x(u))+\epsilon)\) of Equation (4) to direct scales yields a mean response function \(E_{z}(y|x)\) namely,  
\[E_{z}(y|x)=\exp(z(x(u))\delta(\epsilon) \tag{6}\]  
where \(\delta(\epsilon)=E(e^{\epsilon})\) is the required correction factor (CF) for bias of retransformation of the regression error. Notice that a \(z\) subscript in Equation (6) is intended to signpost association to the mean response function \(z(u)\) fitted in geometrical space. This notation convention will be maintained throughout.  
Generally, the efficiency of [PERSON] methods in geometrical space centers on suitability of the retransformation step entailed by Equation (6). For instance, [PERSON] et al. [88] asserted that biased results that Packard [66] blamed on a TAMA fit can be explained by a missing CF. However, in spite of such a factor being taken into account the suitability of its form becomes crucial in determining reliable reproducibility of back transformed forms. What is more, there are restraints to be considered about CF appropriateness. Properly, if \(\epsilon\) represents error in the regression, then CF expresses as the mean of the exponential \(\epsilon\) random variable, that is \(\text{CF}=E(e^{\epsilon})\). Furthermore, assuming that \(\epsilon\) is normally distributed CF takes on a lognormal-mean form [105; 106]. But, if \(\epsilon\) fails to be normally distributed, two possibilities arise. If \(\epsilon\) has known distribution then \(E(e^{\epsilon})\) can be obtained and CF can be represented in a closed form. Otherwise, if distribution of \(\epsilon\) is not identified a priori, one commonly taken approach is setting CF as given by Duan's smearing estimator of bias [61; 106; 107; 108]. Yet, there are provisions for this since the chosen form can fail to appropriately compensate for downward bias intrinsic to retransformation of logged data [61; 109; 110]. Thus, whenever \(\epsilon\) turns unspecified, picking of a suitable CF seems subtle. In order to offer an appropriate form for \(\delta(\epsilon)\), [PERSON] et al. [92] suggested an arrangement that generalizes CF as introduced by [PERSON] and [PERSON] [52]. Adaptation of \(\delta(\epsilon)\) first considers an approximation \(\delta_{n}(\epsilon)\) given by the \(n\)-term partial sum of series representation of expected value of retransformed error \(\epsilon\), that is,  
\[\delta_{n}(\epsilon)=\sum\
olimits_{0}^{n}\frac{E\left(e^{k}\right)}{k!} \tag{7}\]  
then, [PERSON]'s concordance correlation coefficient (CCC) [111] between observed values and those projected by \(E_{z}(y|x)\) as determined by \(\delta_{n}(\epsilon)\) is obtained. If a value \(nmax\) for \(n\) entails maximum reproducibility of projections we take \(\delta(\epsilon)=\delta_{nmax}(\epsilon)\). We will keep this criterion to choose \(\delta(\epsilon)\) in retransformation tasks through. The explicit CCC formulae is provided by Equation (A44).