Reference values statistical treatment

Thermodynamic principles arise from a statistical treatment of matter by studying different idealized ensembles of particles that represent different thermodynamic systems. The first ensemble that we study is that of an isolated system a collection of N particles confined to a volume V, with total internal energy E. A system of this sort is referred to as an NVE system or ensemble, as N, V, and E are the three thermodynamic variables that are held constant. N, V, and E are extensive variables. That is, their values are proportional to the size of the system. If we combine NVE subsystems into a larger system, then the total N, V, and E are computed as the sums of N, V, and E of the subsystems. Temperature, pressure, and chemical potential are intensive variables, for which values do not depend on the size of the system. 

International Federation of Clinical Chemistry. Approved recommendation (1987) on the theory of reference values. Part 5. Statistical treatment of collected reference values. Determination of reference limits. J Clin Chem Clin Biochem 1987 25 645-56. 

Figure 16-2 The statistical treatment of reference values.The boxes in the flow chart refer to sections in the text. NB = Nota bene The order of the three first actions (partitioning, inspection, and detection and/or handling of outliers) may vary, dependent on the distribution and the statistical methods applied. Y, Yes N, no.

International Federation of Clinical Chemistry, Expert Panel on Theory of Reference Values. Approved recommendation on the theory of reference values. Part 1. The concept of reference values. J CUn Chem Clin Biochem 1987 25 337-42 Part 2. Selection of individuals for the production of reference values. J Clin Chem Clin Biochem 1987 25 639-44 Part 3. Preparation of individuals and collection of specimens for the production of reference values. J Clin Chem Clin Biochem 1988 26 593-8 Part 4. Control of analytical variation in the production transfer and application of reference values. Eur J CUn Chem Clin. Biochem 1991 29 531-5 Part 5. Statistical treatment of collected reference values Determination of reference limits. J Clin Chem... 

Solberg HE. Statistical treatment of reference values in laboratory medicine Testing the goodness-of-fit of an observed distribution to the Gaussian distribution. Scand J Clin Lab Invest 1986 46 (Suppl. 184) 125-32. 

S is taken as the reference monomer with =1.0 and e - -0.8. Values for other monomers are derived by regression analysis based on literature or measured reactivity ratios. The Q-e values for some common monomers as presented in the Polymer Ihwdbook are given in Table 7.7. The accuracy of Q-e parameters is limited by the quality of the reactivity ratio data and can also suffer from inappropriate statistical treatment employed in their derivation. A further problem is that the data analysis makes no allowance for the dependence of reactivity ratios on reaction conditions. Reactivity ratios can he dependent on solvent (Section 7.3.1.2), reaction temperature, pll, etc. It follows that values of e and perhaps Q for a given monomer should depend on the medium, the monomer ratio and the particular comonomer. This is especially true for monomers which contain ionizablc groups e.g. MAA, A A, vinyl pyridine) or arc capable of fomiing hydrogen bonds e.g. HEMA, HEA). 

No one has yet actually applied the statistical method to the evaluation of substituent constants. The closest to it is the work of Wold and Sjostrom (35,36) who have used a large number of sets in the statistical evaluation of what they refer to as the inductive substituent constants. These constants are actually equivalent to the cfi or a" values. Wold and Sjostrom have argued that the statistical treatment is the best approach to the evaluation of substituent constants. In the course of their work, they have used data for ionization of benzoic acids in alcohol-water, acetone-water, and dioxane-water mixtures on the assumption that the competition of the electrical effect was the same in all of these sets. The results of Ehrenson, Brownlee, and Taft (1) and our own results show that the composition of the electrical effect varies with solvent. This would seem to throw some doubt on the results of Wold and Sjostrom and to suggest that the third objection to the statistical evaluation of substituent constants may indeed be valid. 

Braband, J. Schabe, 2013. Assessment of national reference values for railway safety a statistical treatment, Journal of Risk and Reliability, 227(4) 405-410. 

Software sensors and related methods - This last group is considered because of the complexity of wastewater composition and of treatment process control. As all relevant parameters are not directly measurable, as will be seen hereafter, the use of more or less complex mathematical models for the calculation (estimation) of some of them is sometimes proposed. Software sensing is thus based on methods that allow calculation of the value of a parameter from the measurement of one or more other parameters, the measurement principle of which is completely different from an existing standard/reference method, or has no direct relation. Statistical correlative methods can also be considered in this group. Some examples will be presented in the following section. 

This calculation of the p-value comes from the probabilities associated with these signal-to-noise ratios and this forms a common theme across many statistical test procedures. In general, the signal-to-noise ratio is again referred to as the test statistic. The distribution of the test statistic when the null hypothesis is true (equal treatments) is termed the null distribution. 

The formula for the test statistic is somewhat complex, but again this statistic provides the combined evidence in favour of treatment differences. When Mantel and Haenszel developed this procedure they calculated that when the treatments are identical the probabilities associated with its values follow a x i distribution. This is irrespective of the number of outcome categories, and the test is sometimes referred to as the chi-square one degree of freedom test for trend. 

Suppose the truth is that = p,2> the treatments are the same. We would hope that the data would give a non-significant p-value and our conclusion would be correct, we are unable to conclude that differences exist. Unfortunately that does not always occur and on some occasions we will be hoodwinked by the data and get p < 0.05. On that basis we will declare statistical significance and draw the conclusion that the treatments are different. This mistake is called the type I error. It is the false positive and is sometimes referred to as the a error. 

From the data listed in Tables I-V, we conclude that most authors would probably accept that there is evidence for the existence of a compensation relation when ae < O.le in measurements extending over AE 100 and when isokinetic temperature / , would appear to be the most useful criterion for assessing the excellence of fit of Arrhenius values to Eq. (2). The value of oL, a measure of the scatter of data about the line, must always be considered with reference to the distribution of data about that line and the range AE. As the scatter of results is reduced and the range AE is extended, the values of a dimin i, and for the most satisfactory examples of compensation behavior that we have found ae < 0.03e. There remains, however, the basic requirement for the advancement of the subject that a more rigorous method of statistical analysis must be developed for treatment of kinetic data. In addition, uniform and accepted criteria are required to judge quantitatively the accuracy of obedience of results to Eq. (2) or, indeed, any other relationship. 

Table 6.2. UNIVARIATE SUMMARY STATISTICS (MEAN + S.E.M.) FROM THE SIMULATED DATA ILLUSTRATED IN FIGURE 6.5 CONSISTING OF A CONTROL GROUP AND A TREATMENT GROUP The p values refer to a two-sided r-test with 20 degrees of freedom.

The optical excitation of electron-hole pairs represents a non-equilibrium state. The subsequent relaxation processes from the initial state includes both carrier-carrier interactions and coupling to the bath phonons. In some treatments, there is a distinction made between carrier-carrier and carrier-phonon interactions in which the latter is referred to as thermalisation. A two-temperature model is invoked in that the carrier-carrier scattering leads to a statistical distribution that can be described by an elevated electronic temperature, relative to the temperature characterising the lattice phonons (Schoenlein et al, 1987 Schmuttenmaer et al, 1996). This two-temperature model is valid only if the carrier-carrier energy redistribution occurs on time scales much faster (>10 times) than relaxation into phonons. This distinction has limited value when there is not a sufficient separation in time scale to make a two-temperature model applicable. The main emphasis in this section is on the dynamics of the energy distribution of the carriers as this is most relevant to energy storage applications. 

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