Statistical Derivation

An expression for can be derived from statistical arguments. First, a branching coefficient is introduced, defined as the probability that a multifunctional monomer (/ 2) is coimected either by a linear chain segment or directly to a second multifunctional monomer (or branch point) rather than to a chain segment terminating in 

The critical value necessary for incipient gelation can be calculated from the probability that at least one of the (/ - 1) chain segments extending from a branch unit will be coimected to another branch unit, and as this is simply f- 1) we have 

It follows that if a chain ends in a branch point, chains on the average 

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The LD q is the statistically derived dose of a substance which results ia death ia 50% of a population of animals. The is expressed ia milligrams of... 

Sampling is a statistically derived process—a small amount of material S is taken from a large quantity B for the purpose of estimating properties of B, If S is an accurate sample (or stated more correctly, is representative of B according to a defined statistical parameter), it is a suitable estimator for the properties of B,... 

Einally, structural properties that depend directly neither on the data nor on the energy parameters can be checked by comparing the structures to statistics derived from a database of solved protein structures. PROCHECK-NMR and WHAT IE [94] use, e.g., statistics on backbone and side chain dihedral angles and on hydrogen bonds. PROSA [95] uses potentials of mean force derived from distributions of amino acid-amino acid distances. 

The LD50 is the statistically derived single dosage of a substance that can be expected to cause death in 50% of the sample population. It is therefore an indicator of acute toxicity, usually determined by ingestion using rats or mice, although other animals may be used. LD50 is also determined by other routes, e.g. by skin absorption in rabbits. The values are affected by species, sex, age, etc. 

Median Effective Dose (ED) The statistically derived single dose of a substance that can be expected to cause a defined nonlethal effect in 50% of a given population of organisms under a defined set of e.xperimental conditions. 

Recently, a quantitative lateral interaction model for desorption kinetics has been suggested (103). It is based on a statistical derivation of a kinetic equation for the associative desorption of a heteronuclear diatomic molecule, taking into account lateral interactions between nearest-neighbor adatoms in the adsorbed layer. Thereby a link between structural and kinetic studies of chemisorption has been suggested. 

After phase separation, two sets of equations such as those in Table A-1 describe the polymerization but now the interphase transport terms I, must be included which couples the two sets of equations. We assume that an equilibrium partitioning of the monomers is always maintained. Under these conditions, it is possible, following some work of Kilkson (17) on a simpler interfacial nylon polymerization, to express the transfer rates I in terms of the monomer partition coefficients, and the iJolume fraction X. We assume that no interphase transport of any polymer occurs. Thus, from this coupled set of eighteen equations, we can compute the overall conversions in each phase vs. time. We can then go back to the statistical derived equations in Table 1 and predict the average values of the distribution. The overall average values are the sums of those in each phase. 

Statistics Derived from Structural Databases Are Skewed and Sparse... 

As early as 1952, Flory [5, 6] pointed out that the polycondensation of AB -type monomers will result in soluble highly branched polymers and he calculated the molecular weight distribution (MWD) and its averages using a statistical derivation. Ill-defined branched polycondensates were reported even earlier [7,8]. In 1972, Baker et al. reported the polycondensation of polyhydrox-ymonocarboxylic acids, (OH)nR-COOH, where n is an integer from two to six [ 9]. In 1982, Kricheldorf et al. [ 10] pubhshed the cocondensation of AB and AB2 monomers to form branched polyesters. However, only after Kim and Webster published the synthesis of pure hyperbranched polyarylenes from an AB2 monomer in 1988 [11-13], this class of polymers became a topic of intensive research by many groups. A multitude of hyperbranched polymers synthesized via polycondensation of AB2 monomers have been reported, and many reviews have been published [1,2,14-16]. 

The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

In this contribution, we report equilibrium modulus and sol fraction measurements on diepoxidet-monoepoxide-diamine networks and polyoxypropylene triol-diisocyanate networks and a comparison with calculated values. A practically zero (epoxides) or low (polyurethanes) Mooney-Rivlin constant C and a low and accounted for wastage of bonds in elastically inactive cycles are the advantages of the systems. Plots of reduced modulus against the gel fraction have been used, because they have been found to minimize the effect of EIC, incompleteness of the reaction, or possible errors in analytical characteristics (16-20). A full account of the work on epoxy and polyurethane networks including the statistical derivation of various structural parameters will be published separately elsewhere. 

To evaluate inhalation toxicity in situations where workers are exposed to various concentrations and durations of a drug vapor, aerosol, or powder in the work environment during manufacturing or packaging, a more comprehensive determination of E(COso or L(Ct)so values are used. The E(Ct)50 or L(Ct)so values are statistically derived values that represent the magnitude of exposure, expressed as a function of the product of C and t, that is expected to affect or kill less than 50% and more than 50% of the animals. The other curve represents exposures that kill 50% or more than 50% of each group of animals (Irish and Adams, 1940). 

While he was not able to supply equal numbers of male and female rats for these studies, there were sufficient rats of both sexes to warrant Hopkins statistically derived conclusions that a small milk addendum is a growth promoter at a rate far in excess of what might be expected from the addition of the milk fat, proteins, carbohydrate and salts contained in 2-5 cc of whole milk. 

Cluster analysis is simply a method to group entities, for which a number of properties or parameters exist, by similarity [292, 308-313]. Various distance measurements are used, and the analysis is performed in a sequential manner, reducing the number of clusters at each step. Such a procedure has been described for use in drug design and environmental engineering research as a way to group substituents that have the most similarity when various combinations of the electronic, steric, and statistically derived parameters are considered. 

All statistics derived from the data should be presented in the form of tables and diagrams, including confidence intervals. 

SSDs are being routinely used for the display and interpretation of effects data (Parkhurst et al. 1996 Posthuma et al. 2002). An SSD for atrazine (shown in Figure 7.3) displays the typical S-shaped curve associated with many chemical dose-response relationships. Each point on the curve represents an LC50 for a particular species exposed to atrazine under standard toxicity test protocols. The SSD approach uses only a single statistically derived endpoint from each available toxicity test (e.g., the LC50 or EC50). In contrast, all data collected during any specific toxicity test can be used in a hierarchical model. The ability to use all available data to make inferential decisions is a marked improvement over the standard SSD effects distribution. 

F. Fowler. A Statistical Derivation of Langmuir s Adsorption Isotherm. Proc. Camb. Phil. Soc., 31 260-280,1935. 

This is a very important result. By this expression we give a statistical derivation of the dielectric function of the atomic gas, which is determined by the properties of the atoms. 

Figure 11 Comparison of results observed for total related substances (TRS) from stability testing of different lots and the predicted values using statistically derived models.

Meylan et al. (1992) described another attempt to extend MCI-Koc relationships to polar compounds. This method uses the first order molecular connectivity index (Jy) and a series of statistically derived fragment contribution factors for polar compounds. To develop the model, they performed two separate regression analyses. The first related log Koc to for... 

Similar to the PARCC parameters, acceptance criteria are expressed in qualitative and quantitative terms. Some of them are statistically derived values, while others are purely qualitative and represent industry standards and accepted practices. Quantitative parameters (precision, accuracy, and completeness) are evaluated mathematically and compared to numerical acceptance criteria representativeness, which is a qualitative parameter, is established by comparing documented field and laboratory procedures to applicable standards and specifications. Comparability is estimated as the closeness of analytical results obtained at two different laboratories, and is usually expressed qualitatively. In environmental project work, acceptance criteria for the PARCC parameters are documented in the SAP. 

For GC/MS analyses, some laboratories use surrogate standard recovery limits from outdated versions of EPA Methods 8260 and 8270. These recovery limits, shown in Example 4.18, are fairly close to the statistically derived control limits at most laboratories and can be safely used in the evaluation of data quality. The surrogate... 

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