Statistical considerations

The entropy of mixing of very similar substances, i.e. the ideal solution law, can be derived from the simplest of statistical considerations. It too is a limiting law, of which the most nearly perfect example is the entropy of mixing of two isotopic species. 

The approach outlined here will describe a viewpoint which leads to the standard calculational rules used in various applications to systems in themiodynamic (themial, mechanical and chemical) equilibrium. Some applications to ideal and weakly interacting systems will be made, to illustrate how one needs to think in applying statistical considerations to physical problems. 

Attempts have been made to devise mathematical functions to represent the distributions that are found experimentally. The mathematical treatment is necessarily based on the assumption that the number of particles in the sample is large enough for statistical considerations to be applicable. With the SOO-member sample of the previous section one could not expect any more than approximate agreement between mathematical prediction and experiment. 

In Chaps. 5 and 6 we shall examine the distribution of molecular weights for condensation and addition polymerizations in some detail. For the present, our only concern is how such a distribution of molecular weights is described. The standard parameters used for this purpose are the mean and standard deviation of the distribution. Although these are well-known quantities, many students are familiar with them only as results provided by a calculator. Since statistical considerations play an important role in several aspects of polymer chemistry, it is appropriate to digress into a brief examination of the statistical way of describing a distribution. 

Statistical considerations make it possible to test the assumption of independent additions. Let us approach this topic by considering an easier problem coin tossing. Under conditions where two events are purely random-as in tossing a fair coin-the probability of a specific sequence of outcomes is given by the product of the probabilities of the individual events. The probability of tossing a head followed by a head-indicated HH-is given by... 

This kind of statistical consideration is used to detect oudiers, ie, when a sample does not belong to any known group. It is also the basis of a variation of SIMCA called asymmetric classification, where only one category is modelable and distinguished from all others, which spread randomly through hyperspace. This type of problem is commonly encountered in materials science, product quaUty, and stmcture—activity studies. 

The first equation expresses a scaling idea as in Eq. (18), introducing the arbitrary factor 2/3 instead of unity for numerical convenience—the problem of statistical accuracy becomes easier, and numerical prefactors should not matter in statistical considerations. 

We recommend that consideration be given to the statistical considerations involved in both the design and analysis of proposed research studies prior to commencing the study. 

Zaykin DV, Young SS. Large recursive partitioning analysis of complex disease pharmacogenetic studies. II. Statistical considerations. Pharmacogenomics 2005 6(l) 77-89. 

This paper discusses the role that statistics can play In environmental sampling. The primary difference between an Investigation based on statistical considerations and one that Is not Is the degree of objectivity that can be Incorporated Into the evaluation of the quality and uncertainty of the study results. Statistical methods In the planning stage can also aid In optimizing allocation of resources. 

To the users of CRMs, the concept of traceabDity" is very closely related to the statistical considerations in the measurement process and the quality of the measurements in the users laboratory. Traceability is defined in the international vocabulary on metrology (VIM) as ... 

Before undertaking a discussion of the mathematics involved in the determination of reaction rates is undertaken, it is necessary to point out the importance of proper data acquisition in stability testing. Applications of rate equations and predictions are meaningful only if the data utilized in such processes are collected using valid statistical and analytical procedures. It is beyond the scope of this chapter to discuss the proper statistical treatments and analytical techniques that should be used in a stability study. Some perspectives in these areas can be obtained by reading the comprehensive review by Meites [84], the paper by P. Wessels et al. [85], and the section on statistical considerations in the stability guidelines published by FDA in 1987 [86] and in the more recent Guidance for Industry published in June 1998 [87],... 

Currently the main application of interest for parametric release is to replace the sterility test as a control method in appropriate cases (given the limited value of that test to predict sterility assurance due to statistical considerations, although it is also pointed out that a sterility test provides a final opportunity to identify a major failure, although other means should provide a more reliable way of detecting such failures). The concept is applicable to well-founded methods of sterilization where the product stability is known and development data have identified the critical process parameters. The measured parameters should be such as to ensure that correct processing of the batch provides sufficient assurance that the sterility assurance level intended has been achieved. 

George, S. L. Statistical considerations and modeling of clinical utility of tumor markers. Hema-tol. Oncol. Clin. North Am. 8,457-468 (1994). 

Since the counting of radioactive disintegration events is subject to statistical considerations, the dates obtained with this method are inherently subject to some degree of uncertainty. 

The existence of active sites on surfaces has long been postulated, but confidence in the geometric models of kink and step sites has only been attained in recent years by work on high index surfaces. However, even a lattice structure that is unreconstructed will show a number of random defects, such as vacancies and isolated adatoms, purely as a result of statistical considerations. What has been revealed by the modern techniques described in chapter 2 is the extraordinary mobility of surfaces, particularly at the liquid-solid interface. If the metal atoms can be stabilised by coordination, very remarkable atom mobilities across the terraces are found, with reconstruction on Au(100), for example, taking only minutes to complete at room temperature in chloride-containing electrolytes. It is now clear that the... 

The chemical structure of any heteropolymer macromolecule is defined by its configuration o / where ol = 1,2 denotes the type of monomeric unit. A special feature of this structure in the system in hand is that its formation is largely conditioned by the conformational state of macromolecules involved. Hence, under the statistical consideration of these macromolecules it was suggested [22] to differentiate polymer chains apart from configuration oq also... 

In the model of micelle proposed by Dill et al. [21], the hydrocarbon chains of the surfactants are more randomly distributed, bearing in mind statistical considerations (Fig. 4). A considerable number of hydrocarbon chains are exposed to the water at the surface. 

While the collision theory of reactions is intuitive, and the calculation of encounter rates is relatively straightforward, the calculation of the cross-sections, especially the steric requirements, from such a dynamic model is difficult. A very different and less detailed approach was begun in the 1930s that sidesteps some of the difficulties. Variously known as absolute rate theory, activated complex theory, and transition state theory (TST), this class of model ignores the rates at which molecules encounter each other, and instead lets thermodynamic/statistical considerations predict how many combinations of reactants are in the transition-state configuration under reaction conditions. 

We saw in Chapter 2 how Coulomb s law states that charges of the opposite sign attract each other. We call this an electrostatic attraction. In consequence, the charges on the dust and tabletop are mutually attractive, so the dust no longer floats past the table but forms a weak electrostatic bond with it. The amount of dust on the table will, therefore, be much greater than would be suggested by a mere statistical consideration of dust accumulation. 

The change in the nature of the adsorption with increasing coverage (dissociative followed by associative) has been explained by a statistical consideration of the reaction mechanism shown above120). Associative adsorption is expected to occur at vacant sites for which all adjacent olefin binding sites are occupied by earlier dissociation products (or carbon monoxide, as shown by Fig. 6b), because dissociative adsorption (formation of vinyl and hydride species, followed by hydride migration to another alkene) requires two adjacent vacant sites. 

The single most important statistical consideration in the design of bioassays in the past was based on the point of view that what was being observed and evaluated was a simple quantal response (cancer occurred or it didn t), and that a sufficient number of animals needed to be used to have reasonable expectations of detecting such an effect. Though the single fact of whether or not the simple incidence of neoplastic tumors is increased due to an agent of concern is of interest, a much more complex model must now be considered. The time-to-tumor, patterns of tumor incidence, effects on survival rate, and age of first tumor all must now be captured in a bioassay and included in an evaluation of the relevant risk to humans. 

Second, Reinhardt and Hubbard (1998) performed a prediction using neural networks. From some statistical consideration, they selected three locations for prokaryotes (cytoplasmic, extracellular, and periplasmic) and four locations for eukaryotes, excluding plants (cytoplasmic, extracellular, mitochondrial, and nuclear). They did not include the membrane proteins because they can be distinguished rather reliably using existing methods. One potential problem of their analysis is that they only excluded sequence pairs with more than 90% identity. Nevertheless, the distinctions between pairs of groups were rather clear. The high accuracy between nuclear and cytoplasmic proteins was especially impressive. 

Vial, J., Jardy, A., Anger, P., Brun, A., Menet, J. M. Methodology for the transfer of liquid chromatography methods based on statistical considerations. J. Chromatogr. A, 815, 1998, 173-182. 

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