Using All of the Factors

Now you have gone through all of the factors yourself. Now you should summarize all of this information in the following chart (just like we did for substitution reactions). For each factor, indicate which mechanism is favored. The first column has been filled in for you  

EXERCISE 10.22 Using the information that you constructed in the chart above, predict whether the following reaction will proceed via an E2 mechanism or an El mechanism  

Answer The substrate is tertiary, so it could be El or E2. The base is a strong base (a negative charge that is not resonance stabilized), so the E2 mechanism will be faster. 

PROBLEMS For each of the reactions below, predict whether the reaction will proceed via an E2 mechanism, an El mechanism, or neither. For now, do not worry about drawing the products. We need to cover the next section before we can do that. Right now, just focus on determining which mechanism, if any, is possible. 

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When we examine the plots in Figure 56 we see that the PRESS decreases each time we add another factor to the basis space. When all of the factors are included, the PRESS drops all the way to zero. Thus, these fits cannot provide us with any information about the dimensionality of the data. The problem is that we are trying to use the same data for both the training and validation data. We lose the ability to assess the optimum rank for the basis space because we do not have independent validation samples that contain independent noise. So, the more factors we add, the better the calibration is able to model the particular noise in these samples. When we use all of the factors, we are able to model the noise completely. Thus, when we predict the concentrations for... 

The physician may consciously use all of the factors listed above in therapeutic practice. But it is still not enough that patients get better, it is essential to know why they do so. This is because potent drugs should only be given if their pharmacodynamic effects are needed many adverse reactions have been shown to be due to drugs that are not needed, including some severe enough to cause hospital admission. 

It is not always necessaiy to use all of the terms in the correlating factor, and those conditions which are held constant throughout the testing may be dropped from the correlating factor. Many times, air rate data are not av able and reasonable correlations can be obtained... 

The models you use to portray failures that lead to accidents, and the models you use to propagate their effects, are attempts to approximate reality. Models of accident sequences (although mathematically rigorous) cannot be demonstrated to be exact because you can never precisely identify all of the factors that contribute to an accident of interest. Likewise, most consequence models are at best correlations derived from limited experimental evidence. Even if the models are validated through field experiments for some specific situations, you can never validate them for all possibilities, and the question of model appropriateness will always exist. 

The materials on the shortlist may be further investigated by use of the associated data bank in the Plascams system, which not only provides a precis of the materials properties and a property data bank but also information on material cost. Such a system cannot, however, be expected to include all of the factors... 

HFAM has 20 groups of factors instead of the 10 general failure types of the TRIPOD approach. The reason for this is that all of the 10 TRIPOD GFTs would be applied in all situations, even though the actual questions that make up the factors may vary. In the case of HFAM, it would be rare to apply all of the factors unless an entire plant was being evaluated. HFAM uses a screening process to first identify the major areas vulnerable to human error. The generic factors and appropriate job specific factors are then applied to these areas. For example, control room questions would not be applied to maintenance jobs. 

Figure 56. Logarithmic plots of the PRESS values as a function of the number of factors (rank) using the same samples for calibration and validation. As factors are added, the errors continue to decrease. When all of the factors are used, the errors equal exactly zero.

The prediction step for PLS is also slightly different than for PCR. It is also done on a rank-by-rank basis using pairs of special and concentration factors. For each component, the projection of the unknown spectrum onto the first spectral factor is scaled by a response coefficient to become a corresponding projection on the first concentration factor. This yields the contribution to the total concentration for that component that is captured by the first pair of spectral and concentration factors. We then repeat the process for the second pair of factors, adding its concentration contribution to the contribution from the first pair of factors. We continue summing the contributions from each successive factor pair until all of the factors in the basis space have been used. 

The various copolymerization models that appear in the literature (terminal, penultimate, complex dissociation, complex participation, etc.) should not be considered as alternative descriptions. They are approximations made through necessity to reduce complexity. They should, at best, be considered as a subset of some overall scheme for copolymerization. Any unified theory, if such is possible, would have to take into account all of the factors mentioned above. The models used to describe copolymerization reaction mechanisms arc normally chosen to be the simplest possible model capable of explaining a given set of experimental data. They do not necessarily provide, nor are they meant to be, a complete description of the mechanism. Much of the impetus for model development and drive for understanding of the mechanism of copolymerization conies from the need to predict composition and rates. Developments in models have followed the development and application of analytical techniques that demonstrate the inadequacy of an earlier model. 

These examples and those in Scheme 2.6 illustrate the key variables that determine the stereochemical outcome of aldol addition reactions using chiral auxiliaries. The first element that has to be taken into account is the configuration of the ring system that is used to establish steric differentiation. Then the nature of the TS, whether it is acyclic, cyclic, or chelated must be considered. Generally for boron enolates, reaction proceeds through a cyclic but nonchelated TS. With boron enolates, excess Lewis acid can favor an acyclic TS by coordination with the carbonyl electrophile. Titanium enolates appear to be somewhat variable but can be shifted to chelated TSs by use of excess reagent and by auxiliaries such as oxazolidine-2-thiones that enhance the tendency to chelation. Ultimately, all of the factors play a role in determining which TS is favored. 

Numerical models are used to predict the performance and assist in the design of final cover systems. The availability of models used to conduct water balance analyses of ET cover systems is currently limited, and the results can be inconsistent. For example, models such as Hydrologic Evaluation of Landfill Performance (HELP) and Unsaturated Soil Water and Heat Flow (UNSAT-H) do not address all of the factors related to ET cover system performance. These models, for instance, do not consider percolation through preferential pathways may underestimate or overestimate percolation and have different levels of detail regarding weather, soil, and vegetation. In addition, HELP does not account for physical processes, such as matric potential, that generally govern unsaturated flow in ET covers.39 42 47... 

The degrees of freedom for lack of fit, f-p, must not be negative or the model cannot be fitted to the data (see Section 5.6 for example). However, it is possible to use all of the degrees of freedom from the factor combinations to estimate up to / parameters in a model. If p = /, there will be a perfect fit . 

It is also beyond the graphical representation capabilities commonly used. Factor analysis is one of the pattern recognition techniques that uses all of the measured variables (features) to examine the interrelationships in the data. It accomplishes dimension reduction by minimizing minor variations so that major variations may be summarized. Thus, the maximum information from the original variables is included in a few derived variables or factors. Once the dimen-... 

The approach discussed above can be applied to the ambient polycyclic aromatic hydrocarbons (PAH). Studies designed to take into account all of the factors discussed in the previous sections have not been conducted, but data available in the literature can be used to illustrate the method. 

One final point it should be noted that the experimenter is not constrained to use a resolution V design or to add star points for all of the factors. In particular, if it is believed that certain two-factor interactions... 

Although not all of the factors that influence homogeneous hydrogenation and hydroboration in sc C02 are fully understood, it is clear that the use of sc C02 can lead to an increase in selectivity for some reactions. Additional work is needed to understand the opportunities for further selectivity enhancements and catalyst separation/recycle strategies. Even sc C02 systems that exhibit similar selectivities to those obtained in organic solvents could offer a practical, environmentally responsible method for the production of many important chiral building blocks. 

We can use the same approach to expand the design and obtain a data set applicable for polynomials of third order. The respective full factorial design is constructed by combining all of the factors at five levels, giving a total of N = 5", or N = 25 for n = 2, N = 125 for n = 3, N = 625 for n = 4, etc. It is apparent that the number of the experiments grows geometrically with the number of factors, and there are not many applications where the performance of 625 experiments to explore four factors is reasonable. 

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