Specifications statistical considerations

PHARMACOPEIAS AND SPECIFICATIONS STATISTICAL CONSIDERATIONS IN SETTING SPECIFICATIONS... 

Statistical considerations make it possible to test the assumption of independent additions. Let us approach this topic by considering an easier problem coin tossing. Under conditions where two events are purely random-as in tossing a fair coin-the probability of a specific sequence of outcomes is given by the product of the probabilities of the individual events. The probability of tossing a head followed by a head-indicated HH-is given by... 

Finally most therapeutic specific guidelines contain recommendations that directly impact on statistical considerations, for example in terms of the definition of endpoints, the requirement for more than one primary endpoint, the definition of analysis sets and the choice of A. In a particular therapeutic setting it is self-evident that the requisite guidelines should be studied carefully in order to extract relevant information for statistical aspects of design and analysis. 

Once the regulatory submission has been made there will inevitably be questions and issues coming back from the regulators. There may be concerns about the way the data has been handled from a statistical point of view. There may be requests for additional specific analyses to resolve uncertainty. There may be more open issues that the regulators are unhappy with which may require a substantial amount of further analysis. At the extreme end there may be outright rejection and the company may then be back to the drawing board in terms of the product. In all of these cases there will usually be a need for further statistical considerations. 

In the present state of the art of polymer physics, an exhaustive solution of the first of these two problems has not been found so far, although some attempts in this direction have been undertaken. At the same time, an intensive development of production of polymers puts forward demands of the prediction of their properties. To cope with this task chemical engineers generally use in practice some simple semi-empirical correlations. In doing so, they resort to certain qualitative theoretical approaches to treat the available experimental data. According to the most reputable adherent of this method, van Krevelen, such semi-empiri-cal correlations are highly effective and provide rather reliable results in most practically important cases (van Krevelen and te Nijenhuis, 2009). However, even in the framework of the above approach, considerable difficulties are encountered, because often there is no clear idea about which specific statistical characteristics of a polymer are responsible for a particular mechanical and physicochemical property. It especially concerns copolymers because the number of their characteristics of such a kind is larger than that for homopolymers. 

In Section 8B-4, we presented the statistical considerations involved in reducing the particle size of the gross sample so as to obtain a laboratory sample. Here, some of the specific techniques are described. 

Statistical reports are always welcome, providing analytical data focused on specific subjects of interest. A statistical consideration of the Socrates/Erasmus CD projects shows the evolution of the large number of partner institutions that put their efforts, experience, and expertise in order to develop joint new study programs. Statistical data, as well as an ex-post evaluation report of the Erasmus CD projects is available on the Socrates website (http //www.socleoyouth.be) for the period of 1997-2001 (Sahlin et al., 2005a). A less detailed presentation, but still containing an analytical... 

The reaction coordinate for dissociation must be some linear combination of stretch modes. If the energy remains localized in the stretch modes then the actual rate of dissociation will be much faster than that expected from statistical considerations. The latter assume that energy is fully shared over all modes prior to dissociation. Specifically, the RRK (115) unimolecular reaction rate at the energy E is given by... 

Thus, given the requisite total energy in the various normal modes of a molecule, fleeting accumulations in particular bonds are inevitable provided that there is no disturbance by further collision. The time required for the process is of the order of magnitude of the period of the complex motion. If the amplitude reinforcement had to be very exact, and if the periods of the individual normal modes were incommensurable, the time could be very long indeed, but the first condition is not likely to be at all rigorous. Even so, the time clearly depends upon the arithmetical relations of the normal frequencies and is thus specific and not calculable from statistical considerations alone. 

Protocol sampling. These are specified plans used for decision purposes in a given situation. Regulations (of the protocol) usually specify the type, size, frequency, and period of sampling in addition to the test methods to be used and other important test issues. A combination of intuitive and statistical considerations may be used, and the above population estimates may be obtained depending on the protocol. Testing for conformance with producer-user specifications for commercial transactions is typical for this approach. 

From the kinetic theory of gases, an expression for the net-mass flux at the interphase can be derived based on the works of Hertz [223] and Knudsen [224]. From a statistical consideration under the assumption of a Maxwell-Boltzmann distribution for the velocity of the gas molecules, the maximum condensation mass flux can be calculated. The evaporation mass flux has to equal the condensation mass flux at equilibrium. The resulting Hertz-Knudsen equation for calculating the area specific net-mass flux is given below ... 

In Chap. 8 we discuss the thermodynamics of polymer solutions, specifically with respect to phase separation and osmotic pressure. We shall devote considerable attention to statistical models to describe both the entropy and the enthalpy of mixtures. Of particular interest is the idea that the thermodynamic... 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

There are a variety of ways to express absolute QRA results. Absolute frequency results are estimates of the statistical likelihood of an accident occurring. Table 3 contains examples of typical statements of absolute frequency estimates. These estimates for complex system failures are usually synthesized using basic equipment failure and operator error data. Depending upon the availability, specificity, and quality of failure data, the estimates may have considerable statistical uncertainty (e.g., factors of 10 or more because of uncertainties in the input data alone). When reporting single-point estimates or best estimates of the expected frequency of rare events (i.e., events not expected to occur within the operating life of a plant), analysts sometimes provide a measure of the sensitivity of the results arising from data uncertainties. 

The information obtained during the background search and from the source inspection will enable selection of the test procedure to be used. The choice will be based on the answers to several questions (1) What are the legal requirements For specific sources there may be only one acceptable method. (2) What range of accuracy is desirable Should the sample be collected by a procedure that is 5% accurate, or should a statistical technique be used on data from eight tests at 10% accuracy Costs of different test methods will certainly be a consideration here. (3) Which sampling and analytical methods are available that will give the required accuracy for the estimated concentration An Orsat gas analyzer with a sensitivity limit of 0.02% would not be chosen to sample carbon monoxide... 

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