Robust methods - Introduction

Unlike most non-parametric methods, Theil s method involves fairly tedious calculations, so a computer program such as a spreadsheet macro is necessary in practice. It should be noted that non-parametric methods for fitting curves are also available, but these are beyond the scope of the present book. 

Robust statistical methods can be applied to samples from symmetrical but heavy-tailed distributions, or when outliers may occur. They should not be applied in situations where the underlying distribution is bi-modal, multimodal, or very asymmetrical, e.g. log-normal distributions. 

There are some very simple robust methods that do not require such iterations, because they arbitrarily eliminate, rather than downweight, a proportion of the data. For example, the trimmed mean for any set of data is found by omitting r observations at the top and at the bottom of the range of measurements. This principle can be applied to the set of data in Example 3.7.2 in Section 3.7. This example considered seven replicate measurements of nitrite ion in river water (mg 1 )  

---

The rhodium-catalyzed asymmetric 1,4-addition of arylboronic acids to a, 3-unsaturated carbonyl compounds is one of the most versatile and robust methods for the stereoselective introduction of aryl groups (Scheme 8.17). In contrast, copper-catalyzed processes generally transfer alkyl groups with high selectivities, but perform only poorly in aryl transfer [43]. 

An extensive introduction into robust statistical methods is given in Ref. 134 a discussion of non-linear robust regression is found in Ref. 135. An example is worked in Section 3.4. 

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

The fact that the EP wants to replace old TEC methods with more selective, efficient, and sensitive separation methods provides the chance for the introduction of more CE methods. The continuous development of analytical methods is reflected in the national and international pharmacopoeias. This might be demonstrated for atropine sulfate. Whereas the Deutsches Arzneibuch, 7th Edition (DAB 7) only limits the tropic acid by extraction and titration with NaOH and phenolphthalein indication, the 4th edition of the EP looked for foreign alkaloids and decomposition products by means of TEC with a potassium iodobismuthate for detection. By intensity comparison of the obtained spots, it was possible to limit these impurities to 0.5%. The EP 5 utilizes an ion-pair HPLC method that is able to limit most of the impurities to less than 0.2%. To make the method more robust, an HPLC method using a polar embedded was applied, which might be the next step for the EP. However, recently the same authors have reported on a MEEKC method being as robust and precise as the latter HPLC method (see Eigure 6) but far more sensitive and, therefore, a future perspective for the EP. 

Introduction Binary and Ternary Phase Equilibria 9.2.2.1 Analytical Solutions 92.2.2 General Solutions Calculation Methods for Multi-Component Systems Stepping and Mapping Robustness and Speed of Calculation... 

Amylose brushes (a layer consisting of polymer chains dangling in a solvent with one end attached to a surface is frequently referred to as a polymer brush) on spherical and planar surfaces can have several advantageous uses, such as detoxification of surfaces etc. The modification of surfaces with thin polymer films is widely used to tailor surface properties such as wettability, biocompatibility, corrosion resistance, and friction [142-144]. The advantage of polymer brushes over other surface modification methods like self-assembled monolayers is their mechanical and chemical robustness, coupled with a high degree of synthetic flexibility towards the introduction of a variety of functional groups. 

This general introduction will continue with a summary of application areas covered in the following chapters and the related robustness questions which have to be solved. Then the different statistical methods that play a role in solving the questions and which are discussed in the following chapters will be put in a general framework. 

A brief introduction into the Taguchi method is given in Chapter 4, whereas in Chapters 2 and 6 some of Taguchi s ideas are touched upon and discussed. The philosophy of Taguchi (build-in robustness) is present in Chapters 2,4, 6, 7 and 8. 

Introduction of the compressible-flow formulation, together with numerical implementation, leads to robust simulations for extremely fast transients. The time steps reduce appropriately to capture high-frequency details of the solution. Moreover there are essentially no convergence failures, indicating that the numerical method remains well conditioned even for extremely small time steps. This behavior demonstrates in practical terms that the system has been successfully reduced to index-one, confirming the analytical result. 

In summary, we were able to develop a chemically robust synthetic route to lipid I, the penultimate intermediate utilized in bacterial cell wall biosynthesis. The identification of a method for stereoselective introduction of the anomeric phosphate and a protocol to enable diphosphate coupling were pivotal to our success and ultimately provided the precedent for our chemical synthesis of lipid II detailed in the sections that follow. 

This top down method is quite reasonable and fairly robust but, as can be seen from the example, it is very simplistic in its assumptions. It would only take the introduction of new products from another class of compounds (which also affect high blood pressure or congestive heart failure in a similar way) that produce better effects to reduce the market opportunity for ACE inhibitors. This would alter the value of the ACE inhibitor market dramatically and even a 25 per cent share might be worth much less than anticipated. As a result of the need to take this into account the next level of model must include all the other players in the market. 

Applications of continuum solvation approaches to MCSCF wavefunctions have required a more developed formulation with respect to the HF or DFT level. Even for an isolated molecule, the optimization of MCSFCF wavefunctions represents a difficult computational problem, owing to the marked nonlinearity of the MCSCF energy with respect to the orbital and configurational variational parameters. Only with the introduction of second-order optimization methods and of the variational parameters expressed in an exponential form, has the calculation of MCSCF wavefunction became routine. Thus, the requirements of the development of a second-order optimization method has been mandatory for any successful extension of the MCSCF approach to continuum solvation methods. In 1988 Mikkelsen el ol. [10] pioneered the second-order MCSCF within a multipole continuum model approach in a spherical cavity. Aguilar et al. [11] proposed the first implementation of the MCSCF method for the DPCM solvation model in 1991, and their PCM-MCSCF method has been the basis of many extensions to more robust second-order MCSCF optimization algorithms [12],... 

Mass spectrometric methods have a huge potential to overcome these limitations. While GC-MS has found a fundamental role in toxicology and occupational medicine it did not find a place in the routine clinical chemistry laboratory due to its very demanding handling not suitable for a 24 h/7 day service laboratory setup. The introduction of LC-MS has improved the practicability and robustness of highly specific and highly multiplexed mass spectrometric analyses very substantially. 

---