Transform-Both-Sides Approach

Carroll and Ruppert (1988) suggest that one should not transform either the model or the data, but one should transform the model AND the data. This leads to the model 

Equation (4.63) is called the transform-both-sides (TBS) approach. The transformation of Y is used to remove both skewness in the distribution of the data and to remove any dependence the variance may show on the mean response. The transformation of f(x 0) maintains the nature of the relationship between x and Y in the transformed domain allowing the parameter estimates to have the same meaning they would have in the original domain. 

Early applications of the transform-both-sides approach generally were done to transform a nonlinear problem into a linear one. One of the most common examples is found in enzyme kinetics. Given the Michae-lis-Menten model of enzyme kinetics 

One particularly useful transform, however, is the natural-log transform when the error model is the exponential 

In fact, the proportional error model and exponential error model are equal for small values of a. To see this, the function exp(x) if first written as a MacLaurin series 

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Transform-Both-Sides Approach with Accompanying Residual Variance Model... 

EXAMPLE APPLICATION OF THE TRANSFORM-BOTH-SIDES APPROACH TO A PHARMACODYNAMIC MODEL... 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

This system of difference equations is set-up just in analogy to the system of differential equations. It can be transformed to those if both sides of eq. (5.13) are divided by A and if approaches 0. 

I realize that this is a side track issue, but relevant all the same. Cycle protocols were an approach intended to facilitate optimal growth of muscle tissue. Remember there are two main muscle fiber types Type I, which is endurance orientated, and Type II which is strength orientated. Type "Ha", "Hb", and Type "He" are responsible for most musculature size and have the greatest potential for growth. Testosterone increases the number of Type II fibers at the expense of the Type I transformation. Growth hormone, Insulin, IGF-1, and thyroid hormones effect growth and hyperplasia of both fiber types. This should be another key relating to protocols that were utilized and why. 

In this section, we review the development of organocatalytic ylide-based epox-idation methods which allow a one-step route from carbonyl compounds, and therefore compete with the more traditional two-step approach of olefination followed by epoxidation of the resulting alkene (see Scheme 10.1) [13]. Indeed, ylide-based methodologies side-step the construction of a C=C double bond and achieve the whole transformation in one step thus, they are potentially more atom-efficient (see Scheme 10.1). However, there are greater challenges as both the absolute and relative stereochemistries must be controlled in one step [14-22]. 

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