Model statistical

Statistical models can be used to predict the risk, or probability, of a disease outbreak based on data from previous outbreaks. The most common statistical model used in this context is the logistic model 

2 Statistical models. A number of statistical dose-response extrapolation models have been discussed in the literature (Krewski et al., 1989 Moolgavkar et al., 1999). Most of these models are based on the notion that each individual has his or her own tolerance (absorbed dose that produces no response in an individual), while any dose that exceeds the tolerance will result in a positive response. These tolerances are presumed to vary among individuals in the population, and the assumed absence of a threshold in the dose-response relationship is represented by allowing the minimum tolerance to be zero. Specification of a functional form of the distribution of tolerances in a population determines the shape of the dose-response relationship and, thus, defines a particular statistical model. Several mathematical models have been developed to estimate low-dose responses from data observed at high doses (e.g., Weibull, multi-stage, one-hit). The accuracy of the response estimated by extrapolation at the dose of interest is a function of how accurately the mathematical model describes the true, but unmeasurable, relationship between dose and response at low doses. 

For the most frequently used low-dose models, the multi-stage and one-hit, there is an inherent mathematical uncertainty in the extrapolation from high to low doses that arises from the limited number of data points and the limited number of animals tested at each dose (Crump et al., 1976). The statistical term confidence limits is used to describe the degree of confidence that the estimated response from a particular dose is not likely to differ by more than a specified amount from the response that would be predicted by the model if much more data were available. EPA and other agencies generally use the 95 percent upper confidence limit (UCL) of the dose-response data to estimate stochastic responses at low doses. 

By using UCL and assuming that the model accurately reflects the dose-response relationship at low doses, there is only a five percent chance that the true response is higher than the response predicted by the model. 

UCL takes into account measurement uncertainty in the study used to estimate the dose-response relationship, such as the statistical uncertainty in the number of tumors at each administered dose, but it does not take into account other uncertainties, such as the relevance of animal data to humans. It is important to emphasize that UCL gives an indication of how well the model fits the data at the high doses where data are available, but it does not indicate how well the model reflects the true response at low doses. The reason for this is that the bounding procedure used is highly conservative. Use of UCL has become a routine practice in dose-response assessments for chemicals that cause stochastic effects even though a best estimate (MLE) also is available (Crump, 1996 Crump et al., 1976). Occasionally, EPA will use MLE of the dose-response relationship obtained from the model if human epidemiologic data, rather than animal data, are used to estimate risks at low doses. MLEs have been used nearly universally in estimating stochastic responses due to radiation exposure. 

In order to derive further information from NMR sequence data, it is necessary to relate the sequence distribution to some form of statistical model describing copolymerisation. The basis of these models is the concept of monomer addition probabilities. It will be apparent that some of these models are closely related to those used to describe configurational effects in homopolymers (e.g. see chapter 1). 

The Bernoullian model The Bernoullian model is the simplest of the statistical models used to describe copolymerisation, whereby the addition of each comonomer to a growing polymer chain is regarded as a random process. Thus, the framework for the model is that the probability of addition of a given monomer unit to a growing polymer chain is only dependent upon the mole fraction of the monomer in the feed. 

The Bernoullian process is therefore defined for a copolymer by two transition probabilities, and Pg, which reflect the mole fractions of monomers A and B within the resulting copolymer. Given the two addition probabilities, the mole fraction of any given sequence can be calculated straightforwardly. Thus, for example, the abundance of an A A dyad is given by P, whilst that of an AB dyad is equal to 2P P (the factor of two arises because the AB dyad represents both AB and BA sequences). Table 2.2 shows the set of Bernoullian expressions for the three dyad and six triad sequences in an A/B copolymer. 

Given a set of sequence distribution data from NMR measurements on a copolymer, it is normal practice to test how well that set matches the data 

Only one independent variable is necessary to describe the Bernoullian process in full since the sum of and is equal to unity. Dyad distribution data contain two independent observations which is sufficient to check for conformity to Bernoullian statistics, (Although there are three dyad types, the mole fraction of one of them is always defined by the mole fractions of the other two since the mole fraction of the total dyad is, of course, equal to one, i.e. there are only two independent observations.) However, a better test for conformity is found in a triad distribution since this contains five independent observations (there are six triads). 

It is a truism that reactive collisions are exceedingly complicated and their outcome is determined by a variety of initial parameters— impact parameter, relative velocity, internal energy, etc.—whose specific individual roles are difficult to disentangle. Any justifiable simplification 

We consider two such approaches here, one based on the quasiequilibrium theory of mass spectra and the other on the phase-space theory. In neither case is the principal utility of the model the ability to predict absolute rate constants or cross sections for particular channels. Such absolute rate parameters would require knowledge of the same for the formation of the ion-molecule intermediate and is not available. Rather, the strength of these two theories is in the prediction of relative rate parameters or branching ratios for the various channels. 

Qualitative applications date back to the very early days, to the first major paper of Field et quantitative applications are a recent 

Since reference is still made to the qualitative applications, it is relevant to review their significance. To illustrate such applications. Field et noted a striking similarity between the spectrum of ionic 

A strict comparison between the product spectra of the ion-molecule reaction and the electron-impact mass spectra may only be made if the 

The number of scattering channels that partake in molecular reactions may increase very rapidly when chemical species or kinematical parameters are changed. For example rotational channels multiply as the moments of inertia of reactants or products decrease vibrational modes become numerous in polyatomic molecules and generally the number of accessible channels increases with increasing collision energies. In these cases it may not be possible, and sometimes it is not needed, to develop a detailed treatment of cross sections. Rather, one may want to know only averages of cross 

A simple and popular approximation has been the phase-space model (see Light, 1967, for a review and previous references also Nikitin, 1968, op. cit.). Within the foregoing scheme, this model may be obtained by assuming 

The criterion for choosing la max is crucial in this model, because / max is the only parameter that depends on the physical nature of the intermediate complex. It has usually been taken as 

Considering the assumptions in the phase-space model, this appears most appropriate for calculation of magnitudes of cross sections, and particularly of branching ratios. But it may be expected to fail with regard to product energy distributions, because these are fixed a priori by the process of state counting. 

The phase-space model oversimplifies matters when it assumes randomness and decoupling in S 7 irrespective of the values of J. In actuality, one expects different coupling regimes for different ranges of J (Lester and Bernstein, 1970). The strong coupling required by the phase-space model is not likely to exist for very small or very large J. 

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Sarle, W.S., (1994), Neural Networks and Statistical Models , Proceedings of the Nineteenth Annual SAS Users Group International Conference, April, 1994. 

In a regression approach to material characterization, a statistical model which describes the relation between measurements and the material property is formulated and unknown model parameters are estimated from experimental data. This approach is attractive because it does not require a detailed physical model, and because it automatically extracts and optimally combines important features. Moreover, it can exploit the large amounts of data available. 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

Miller W H 1976 Unified statistical model for complex and direct reaction mechanisms J. Chem. Rhys. 65 2216-23... 

Quack M 1981 Faraday Discuss. Chem. Soc. 71 309-11, 325-6, 359-64 (Discussion contributions on flexible transition states and vibrationally adiabatic models statistical models in laser chemistry and spectroscopy normal, local, and global vibrational states)... 

The best recipe we found so far is based on the statistical model... 

M. Smith, Neural Networks in Statistical Modelling. Van Nostrand Reinhold, New York, 1993. 

The statistical model of the random coil discussed in Chap. 1 illustrates many of these items. 

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... 

A second way of dealing with the relationship between aj and the experimental concentration requires the use of a statistical model. We assume that the system consists of Nj molecules of type 1 and N2 molecules of type 2. In addition, it is assumed that the molecules, while distinguishable, are identical to one another in size and interaction energy. That is, we can replace a molecule of type 1 in the mixture by one of type 2 and both AV and AH are zero for the process. Now we consider the placement of these molecules in the Nj + N2 = N sites of a three-dimensional lattice. The total number of arrangements of the N molecules is given by N , but since interchanging any of the I s or 2 s makes no difference, we divide by the number of ways of doing the latter—Ni and N2 , respectively—to obtain the total number of different ways the system can come about. This is called the thermodynamic probabilty 2 of the system, and we saw in Sec. 3.3 that 2 is the basis for the statistical calculation of entropy. For this specific model... 

The next part of the procedure involves risk assessment. This includes a deterrnination of the accident probabiUty and the consequence of the accident and is done for each of the scenarios identified in the previous step. The probabiUty is deterrnined using a number of statistical models generally used to represent failures. The consequence is deterrnined using mostiy fundamentally based models, called source models, to describe how material is ejected from process equipment. These source models are coupled with a suitable dispersion model and/or an explosion model to estimate the area affected and predict the damage. The consequence is thus determined. 

Empirical—statistical models ate based on estabUshing a relationship between historically observed air quaUty and the corresponding emissions. The linear rollback model is simple to use and requites few data, and for these reasons has been widely appHed (3,4). Linear rollback models assume that the highest measured pollutant concentration is proportional to the basinwide emission rate, plus the background value that is,... 

A unified statistical model for premixed turbulent combustion and its subsequent application to predict the speed of propagation and the stmcture of plane turbulent combustion waves is available (29—32). 

What is a reasonable statistical model, or equation, to approximate the relationship between the independent variables and each response variable ... 

Can the relationship be approximated by an equation involving linear terms for the quantitative independent variables and two-factor interaction terms only or is a more complex model, involving quadratic and perhaps even multifactor interaction terms, necessary As indicated, a more sophisticated statistical model may be required to describe relationships adequately over a relatively large experimental range than over a limited range. A linear relationship may thus be appropriate over a narrow range, but not over a wide one. The more complex the assumed model, the more mns are usually required to estimate model terms. 

SH Bryant, CE Lawrence. The frequency of lon-pair substructures m proteins is quantitatively related to electrostatic potential A statistical model for nonbonded interactions. Proteins 9 108-119, 1991. 

The statistical model makes use of the fact that the probability of all the components being at the extremes of their tolerance range is very low (see equation 3.2). The statistical model is given by ... 

Data that is not evenly distributed is better represented by a skewed distribution such as the Lognormal or Weibull distribution. The empirically based Weibull distribution is frequently used to model engineering distributions because it is flexible (Rice, 1997). For example, the Weibull distribution can be used to replace the Normal distribution. Like the Lognormal, the 2-parameter Weibull distribution also has a zero threshold. But with increasing numbers of parameters, statistical models are more flexible as to the distributions that they may represent, and so the 3-parameter Weibull, which includes a minimum expected value, is very adaptable in modelling many types of data. A 3-parameter Lognormal is also available as discussed in Bury (1999). 

Bury, K. V. 1975 Statistical Models in Applied Science. NY Wiley. 

Leitch, R. D. 1990 A Statistical Model of Rough Loading. In Proceedings 7th International Conference on Reliability and Maintainability, Brest, France, 8-12. 

The limitations of mathematical modeling described above increase the importance of statistical analysis of accidental explosions. However, gathering all needed data to perform a statistical analysis is often very complicated, so results are often incomplete. Wherever possible, both theoretical and statistical models should both be applied in estimating effects. 

These are the six unknowns in a statistical model of the concentrations. They satisfy the following three constraints ... 

By a statistical model of a solution we mean a model which does not attempt to describe explicitly the nature of the interaction between solvent and solute species, but simply assumes some general characteristic for the interaction, and presents expressions for the thermodynamic functions of the solution in terms of an assumed interaction parameter. The quasi-chemical theory is of this type, and we have noted that a serious deficiency is its failure to consider the vibrational effects in the solution. It is of interest, therefore, to consider briefly the average-potential model which does include the effect of vibrations. 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

Spin orbitals, 258, 277, 279 Square well potential, in calculation of thermodynamic quantities of clathrates, 33 Stability of clathrates, 18 Stark effect, 378 Stark patterns, 377 Statistical mechanics base, clathrates, 5 Statistical model of solutions, 134 Statistical theory for clathrates, 10 Steam + quartz system, 99 Stereoregular polymers, 165 Stereospecificity, 166, 169 Steric hindrance, 376, 391 Steric repulsion, 75, 389, 390 Styrene methyl methacrylate polymer, 150... 

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