Subset, statistical term

The macroscopic property of interest, e.g., heat of vaporization, is represented in terms of some subset of the computed quantities on the right side of Eq. (3.7). The latter are measures of various aspects of a molecule s interactive behavior, with all but surface area being defined in terms of the electrostatic potential computed on the molecular surface. Vs max and Fs min, the most positive and most negative values of V(r) on the surface, are site-specific they indicate the tendencies and most favorable locations for nucleophilic and electrophilic interactions. In contrast, II, a ot and v are statistically-based global quantities, which are defined in terms of the entire molecular surface. II is a measure of local polarity, °fot indicates the degree of variability of the potential on the surface, and v is a measure of the electrostatic balance between the positive and negative regions of V(r) (Murray et al. 1994 Murray and Politzer 1994). 

Another condition, similar to conditions SI to S3, which also guarantees that a stepdown method strongly controls the family wise error rate is the subset pivotality condition proposed by Westfall and Young (1993, page 42). Their original subset pivotality condition is given in terms of adjusted -values. For comparability with S1 to S3, we have paraphrased that condition here in terms of test statistics ... 

Statistical manipulations on the USDA database (cluster analysis, principal component analysis with varimax rotation e.g., Everitt, 1980) revealed subsets of represent ve species, as idealized in Fig 2b, but with dif ent variables (orthogonal principal components) than traditional fractions as measured by USDA. A set cf species from each orthogonal subset appears in Table 1. The Latin names, and where available, the common names of the biomass species are given. The extractives ranges are ash content, 4 to 17% protein content, 5 to 14% polyphenol, 3 to 11% and oil content, 1 to 4%. However no species contains extremes of all 4 variables. Nor can species be found, retaining native compositions, at extremes of just one extractive composition, while the other fractions are present at constant levels. Thus we use orthogonal but non-intuitive compositions in this work, then rank pyrolysis effects in terms of traditional extractives content to get an understanding of their impact on biomass pyrolysis. 

Further subset evaluation of GBM patients revealed a median survival advantage of 53.3 weeks vs. 39.9 weeks with placebo ip = 0.008). Interestingly, three of the study s four long-term survivors (still alive 3 years after study termination) received BCNU-loaded polymers. Age and KPS score significantly impacted survival while the survival effect of tumor type failed to achieve statistical significance. The phase III trial reinforced earlier research documenting no local or systemic morbidities with pol5rmer implantation. 

Clearly then, projection operator techniques are useful for the derivation of relaxation equations and for the derivation of formulas for transport coefficients in terms of microscopic properties. The various symmetry properties discussed here are useful for reducing a set of dynamical variables into subsets that are statistically independent of each other. 

The formalism of statistical mechanics is generally expressed in terms of sums over states instead of sums over a subset of the degrees of freedom for the system, in this case the constraints C. It is convenient to reconsider the reformulation of Section IXA for the case in which the entropy and free energy a -e to be expressed in terms of sums over microscopic states. Let an initial microscopic state of the system at be designated as i, while a final microscopic state at S is/. The existence of internal constraints implies that / and/must be correlated. There then must be some set of joint probabiL ties which describes the constraints in terms of the... 

More recently, new quantitative structure-property relationships for Tg have been developed (1) they are based on the statistical analysis of experimental data for 320 linear (uncross-linked) polymers collected from many different sources, containing a vast variety of compositions and structural features. The Tg of the atactic form was used, whenever available, for polymers manifesting different tac-ticities. The Tg values of a subset of the polymers listed in this extensive tabulation are reproduced (with some minor revisions) in Table 1. (It is important to caution the reader here that these data were assembled from a wide variety of sources. Many different experimental techniques were used in obtaining these data.) The resulting relationship for Tg has the form of a weighted sum of structural terms mainly taking the effects of chain stiffness into account plus a term proportional to the solubility parameter S which takes the effects of cohesive interchain interactions in an explicit manner, as shown in equation (1) ... 

In the statistical treatment of data, it is assumed that the handful of replicate experimental results obtained in the laboratory is a minute fraction of the infinite number of results that could, in principle, be obtained gi en infinite time and an infinite amount of sample. Statisticians call the handful of data a sample and view it as a subset of an infinite population, or universe, of data that exists in principle. The laws of statistics apply strictly to populations only when applying t hese laws to a sample of laboratory data, we must assume that the sample is truly representative of the population. Because there is no assurance that this assumption is valid, statements about random errors are necessarily uncertain and must be couched in terms of probabilities. 

The study of molecular diversity is a new and incompletely understood field. It is closely related to molecular similarity, structure-activity correlation, and statistical series design, which are thoroughly reviewed in numerous texts. This article records our current awareness, and outlines the most important advances in terms of theory, methodology, and practice. It is divided into four sections that address issues related to molecular representation, dimensionality reduction, quantification and subset selection, and visualization. The reader is also referred to excellent reviews by Martin et al., Blaney et al., and Martin et al. ... 

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