Bases, pairwise comparisons

Table 2. Stomatal density (SD), Epidermal cell density (ED) and stomatal index (SI) of modern Quercus kelloggii leaves, assigned to light regime during growth by degree of undulation, and p-values from a pairwise comparison using a nested mixed-model ANOVA based on a general linear model.

Keeney and Raiffa (1976, pp. 66-130) describe several ways of how to construct such value functions. If no directly measurable attributes exist for an objective, a scale that maps verbal descriptions with a numerical value indicating the decision maker s strength of preference has to be defined. The MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique) procedure offers a pairwise comparison approach to support constructing such a scale (cf. Bana e Costa et al. 2005). 

PROMETHEE evaluates discrete alternatives based on pairwise comparisons for each sub-objective. However, these comparisons are not performed "manually". Instead, partial preference functions, usually using the difference between two alternatives attribute values as data input, are used to perform the pairwise comparisons. These partial preference functions have to be defined for each sub-objective as a first step of using PROMETHEE. They are used to determine the decision maker s strength of preference for an alternative (with 0 assigned for indifference, 1 for strict preference and values between 0 and 1 for intermittent preference values). 

The majority of the location factors shown in Figure 29 are self-explanatory. While no uniform scale is required when using the pairwise comparison approach in a site selection situation, it has proven helpful to create a joint definition of each attribute that also contains an indication of minimum requirements and ideal state together with all stakeholders involved in the site selection process. To this end, verbal scale descriptions were defined for all sub-objectives. The global scale to use the model for site controlling (see below) was then created based on these definitions. 

Feature trees have been described by Rarey and Dixon [106] as a new way of analyzing the similarity of molecules. This approach is based on building trees that represent molecules. These trees describe the major building blocks of molecules, in addition to their overall arrangement. They are conformation independent. Different types of pairwise comparison algorithms are available to compare trees of different molecules. 

Based on the result of the pairwise comparison, the synthesized weight of the elements is calculated and the total evaluation value is determined. 

Phylogenetic analyses of sequences can be conducted by analyzing discrete characters (i.e., the nucleotides themselves) or by making pairwise comparisons of whole sequences (the distance approach). Deciding whether to use a distance-based or a character-based method depends on... 

The problem lies in the model. The Euclidean distance calculation is inappropriate for use with correlated variables because it is based only on pairwise comparisons, without regard to the elongation of data point swarms along particular axes. In effect, Euclidean distance imposes a spherical constraint on the data set (18). When correlation has been removed from the data, (by derivation of standardized characteristic vectors) Euclidean distance and average-linkage cluster analysis return the three groups. 

Fig. 16.4. Phylogenetic trees or cladograms based on the sequence differences (SEQ) and three-dimensional structural differences (STR) of immunoglobulin fragments. The structural distance metric is a function of both the rms distance difference between superposed structures and the number of topologically equivalent positions in each pairwise comparison (taken from [11]). The lower part of the diagram shows a multidimensional scaling analysis based on structural differences. The constant (C) and variable (V) domains cluster together for the light (L) and heavy (H) chains of the immunoglobulin fragments...

The paralinear or log-det transformation corrects for nonstationarity (see Swofford et al 1996). In this method, which is applicable only to distance tree building, the numbers of raw substitutions of each type and in each direction are tallied for each sequence pair in a fom-by-fom matrix as shown in Figure 14.7. Each matrix has an algebraic determinant, the log of which becomes a factor in estimating sequence divergence, hence the name log-det. Pairwise comparisons of sequences having various and assorted patterns of base frequencies will yield a variety of matrix patterns, giving a variety of determinant values. Thus, each estimated pairwise distance will be affected by the determinant particular to each pair, which effectively... 

Pairwise comparisons among 80 adult E. rufescens using a distribution-free multiple comparison test based upon the Kruskal-Wallis ranked sums test (Hollander Wolfe, 1973) indicate that the active surface area of paired males is significantly larger than that of single males and those of both paired and single females (P < 0.05 for each comparison)... 

For a large number of organic functionalities, significant protonation is only achieved in more concentrated acid solutions e.g. alcohols, ethers, ketones, esters, sulfides, sulfoxides). More concentrated acid solutions cannot be treated as ideal, and Ka values cannot be measured in terms of concentrations as in eqn (3.4). In strong acid media, the significantly decreased water concentration results in additional solvent effects on pA"a that are not accounted for by the pH scale. To account for acid-base behaviour in strong acid media, a number of acidity functions have been established. One of the earliest examples was the Hammett Ho acidity function based on a pairwise comparison of spectrophotometric changes in a series of aniline bases in concentrated acid solution. However, this scale could only be applied for structurally similar bases with similar protonation behaviour. Several other acidity functions have been proposed for other classes of bases such as the Hr acidity function for the ionisation of alcohols. As recently reviewed by Scorrano and More O Ferrall, later treatments by Bunnett and... 

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