Effects classical factor

Traditional tabular presentation of data for calculating the classical factor effects in a 2 factorial design. 

Table 14.4 shows a typical regression analysis output for the 2 factorial design in Table 14.3. Most of the output is self-explanatory. For the moment, however, note the regression analysis estimates for the parameters of the model given by Equation 14.5 and compare them to the estimates obtained in Equations 14.8-14.15 above. The mean is the same in both cases, but the other non-zero parameters (the factor effects and interactions) in the regression analysis are just half the values of the classical factor effects and interaction effects How can the same data set provide two different sets of values for these effects ...

If Xj is temperature and two experiments are carried out, one at a coded level of -1 and one at a coded level of -t-1, and we get a classical factor effect of -i-3.6% yield, it tells us that working at the -i-l coded temperature gives more yield than working at the -1 coded temperature. But this classical factor effect by itself doesn t tell us very much about how sensitive the reaction is to temperature because 5x, isn t included in the factor effect. 

What is the equivalent four-parameter linear model expressing y, as a function of jci and xfl Use matrix least squares (regression analysis) to fit this linear model to the data. How are the classical factor effects and the regression factor effects related. Draw the sums of squares and degrees of freedom tree. How many degrees of freedom are there for SS, 55, and SS 7... 

The Fourier coefficient average in the denominator of the last term is added to make the numerator and denominator symmetrical. It has no effect on the classical average. The classical factor AUa(x) signifies that the potential is evaluated at the centroid of the path... 

In the classical factorial design literature, a factor effect is defined as the difference in average response between the experiments carried out at the high level of the factor and the experiments carried out at the low level of the factor. Thus, in a 2 full factorial design, the main effect of A would be calculated as ... 

To illustrate this classical approach to the calculation of factor effects, consider the following 2 full factorial design ... 

The classical main effects of factors B and C can also be calculated in this manner ... 

In a similar way, the classical interaction effects AB, AC, BC, and ABC can be defined as the difference in average response between the experiments carried out at the high level of the interaction and the experiments carried out at the low level of the interaction. Again, the high level of an interaction is indicated by a plus sign in its column in Table 14.3 (either both of the individual factors are at a high level, or both of the individual factors are at a low level). The low level of a two-factor interaction is indicated by a minus sign in its column in Table 14.3 (one but not both of the individual factors is at a low level). Thus, the classical two-factor interaction effects are easily calculated ... 

Thus, in modem research using interval and ratio scales the 5x usually shouldn t be ignored. Let s add 8x, to the calculation to obtain b, as would be done with regression analysis. Because Xj went from a coded level of -1 to a coded level of -t-1, 5x, = 2. Thus, b (the factor effect in the coded factor space) = 8y,/8xJ = -i-3.6% per 2 coded units = -i-1.8% per coded unit. The fact that 8x is equal to 2 with this system of coding is why regression analysis of coded data gives results that are smaller by Vi from the results obtained from the classical approach ... 

Calculate the grand average (MEAN), the two classical main effects (A and B), and the single two-factor interaction (AB) for the two-factor two-level full factorial design shown in the square plot in Section 14.1. (Assume coded factor levels of -1 and +1). 

What clue is there in Equation 14.7 that suggests that there will be a difference between the classical calculation of factor effects and the regression analysis calculation of factor effects ... 

In Section 14.3, a coding of -1 and +1 gave linear model main effects (b, b, and fcj) that differed by a factor of Vi from the classical main effects (A, B, and C). If the coding had been -2 and +2 instead, by how much would they have differed ... 

An inductive effect which involves the successive polarization of the bonds between X and Y. The decrease in the effect with increasing number of bonds is due to a falloff factor /, which decreases the effect for each successive polarization. The value of / is 0.33-0.36 . This is the classical inductive effect (CIE) model. 

In the classical chelate effect, the enthalpic changes are supposed not to be altered by linking X and Y as shown in Eq. 3, but the enthalpic factor also contributes to the enhanced ligation, giving much increased complex stabilities in the case of linked ligand X-Y ... 

Optical activity in solution, unlike the same effect in crystals, is an isotropic effect. This interaction between a polarized photon and a molecule therefore implicates a chiral factor that is independent of direction, such as the molecular wave function, and in particular, its complex phase. It is a non-classical factor and hence cannot be attributed directly to a classical three-dimensional structure. In a crystal where optical activity arises from three-dimensional... 

If the classical dynamic effects are so small that = 1 or 1, then, and only then, the factor Pe or can be consi-... 

If the condition (72.Ill) of a very fast motion along the classical reaction path is realized, the factor 32 for an electronically adiabatic reaction is the actual tunneling correction, since the classical dynamical effects are then small (32 1).A dependence of 32 on... 

We are now prepared to outline a rational method for unlocking the secrets of C-Li bonding and their implications for the stereochemistry of lithiated hydrocarbons. As a first step, the general principles of MOVB theory are applied to the problem at hand, starting with application of the ID model and considering additional electronic factors, such as "classical" interaction effects and low lying vacant orbital participatipn. As a second step, quantum chemical computations are carried out in order to test the validity of the MOVB analysis. 

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

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