Single comparison

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

Tests of statistical significance lead to statements concerning the probability that some observed difference could have been due to chance alone. Probabilities are computed as though only one significance test, or comparison, had been conducted. However, one is seldom faced with a single-comparison situation. As the number of comparisons increases, so does the probability that chance alone will lead to finding "significant" differences. 

Another approach to multiple comparisons is the Bonferroni test. The strategy in this case is to divide a (typically 0.05 when a single comparison is being made) by the number of tests conducted following a significant omnibus ANOVA test. Hence, if ten comparisons were to be made, the a-level used for each comparison would become 0.05/10, i.e., 0.005, a considerably more conservative value. 

This approach reduces the number of study participants necessary to achieve the required statistical power in two ways. If the three individual components of the MACE composite endpoint (or the higher number of individual components in a MACE-plus composite endpoint) were compared separately between treatment groups, the numbers of events in each case would be lower than the total number of composite endpoint events. Moreover, a statistical correction would need to be made to address the issue of multiplicity as more comparisons are made, the chances of finding a statistically significant difference that does not in fact exist, i.e., committing a type I error, increase (recall discussions in Chap. 4). To counter this possibility, the alpha level (typically 0.05 for a single comparison) used for each of the multiple comparisons must be lowered. 

Table 5 Comparison of wire IQI sensitivities obtained with Selenium and iridium for different pipe diameters and thicknesses (DW=double wall, SW=single wall)[2].

Figure Cl.5.4. Comparison of near-field and far-field fluorescence images, spectra and lifetimes for the same set of isolated single molecules of a carbocyanine dye at a PMMA-air interface. Note the much higher resolution of the near-field image. The spectmm and lifetime of the molecule indicated with the arrow were recorded with near-field excitation and with far-field excitation at two different excitation powers. Reproduced with pennission from Trautman and Macklin [125].

Brand L, Eggeling C, Zander C, Drexhage K FI and Seidel CAM 1997 Single-molecule identification of coumarin-120 by time-resolved fluorescence detection comparison of one- and two-photon excitation in solution J. Chem. Phys. A 101 4313-21... 

Zilker S J, Kador L, Friebel J, Vainer Y G, Kol chenko M A and Personov R I 1998 Comparison of photon echo, hole burning, and single molecule spectroscopy data on low-temperature dynamics of organic amorphous solids J. Phys. Chem 109 6780-90... 

As an example for an efficient yet quite accurate approximation, in the first part of our contribution we describe a combination of a structure adapted multipole method with a multiple time step scheme (FAMUSAMM — fast multistep structure adapted multipole method) and evaluate its performance. In the second part we present, as a recent application of this method, an MD study of a ligand-receptor unbinding process enforced by single molecule atomic force microscopy. Through comparison of computed unbinding forces with experimental data we evaluate the quality of the simulations. The third part sketches, as a perspective, one way to drastically extend accessible time scales if one restricts oneself to the study of conformational transitions, which arc ubiquitous in proteins and are the elementary steps of many functional conformational motions. 

Btamp/e Conformations of molecules like n-decane can be globally characterized by the end-to-end distance, R. In a comparison of single-molecule Brownian (Langevin) dynamics to molecular dynamics, the average end-to-end distance for n-decane from a 600 ps single-molecule Langevin dynamics run was almost identical to results from 19 ps of a 27-molecule molecular dynamics run. Both simulations were at 481K the time step and friction coeffi-... 

The most direct test is to compare the BET area with the geometrical area of the solid. Unfortunately, comparisons of this kind are relatively rare on account of experimental difficulties. The choices are to work with, say, single crystals having a well defined surface, when techniques of quite extraordinary sensitivity will be needed for measurement of the adsorption or, to obtain a larger surface area by use of thin sheets, narrow rods or small spheres, and run the risk that the surface will not be truly smooth so that the actual area will exceed the geometrical area. 

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