Random mixture model

This model is appropriate for random mixtures of elements in which tire pairwise bonding energies remain constant. In most solutions it is found that these are dependent on composition, leading to departures from regular solution behaviour, and therefore the above equations must be conhned in use to solute concentrations up to about 10 mole per cent. 

Note that large density fluctuations are suppressed by construction in a random lattice model. In order to include them, one could simply simulate a mixture of hard disks with internal conformational degrees of freedom. Very simple models of this kind, where the conformational degrees of freedom affect only the size or the shape of the disks, have been studied by Fraser et al. [206]. They are found to exhibit a broad spectrum of possible phase transitions. 

The common disadvantage of both the free volume and configuration entropy models is their quasi-thermodynamic approach. The ion transport is better described on a microscopic level in terms of ion size, charge, and interactions with other ions and the host matrix. This makes a basis of the percolation theory, which describes formally the ion conductor as a random mixture of conductive islands (concentration c) interconnected by an essentially non-conductive matrix. (The mentioned formalism is applicable not only for ion conductors, but also for any insulator/conductor mixtures.)... 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

It is interesting to note that various QSAR/QSPR models from an array of methods can be very different in both complexity and predictivity. For example, a simple QSPR equation with three parameters can predict logP within one unit of measured values (43) while a complex hybrid mixture discriminant analysis-random forest model with 31 computed descriptors can only predict the volume of distribution of drugs in humans within about twofolds of experimental values (44). The volume of distribution is a more complex property than partition coefficient. The former is a physiological property and has a much higher uncertainty in its experimental measurements while logP is a much simpler physicochemical property and can be measured more accurately. These and other factors can dictate whether a good predictive model can be built. 

We now consider the random copolymer model in the presence of solvent— that is, for a copolymer volume fraction p0 < 1. We are not aware of previous work on this model in the literature, but will briefly discuss below the link to models of homopolymer/copolymer mixtures [57]. The excess free energy (86) then depends on two moment densities, rather than just one as in all previous examples. For simplicity, we restrict ourselves to the case of a neutral solvent that does not in itself induce phase separation this corresponds to X = 0, making the excess free energy... 

The symmetry of Figure 6 indicates that two possible models exist an intramolecular bond on chain 1 with an intermolecular bond on chain 2, or vice versa. On looking at molecular models, selection of one option for a particular site does not appear to dictate the selection at neighboring sites. Thus a random mixture of the two options was considered by placing half oxygens at the two positions for each residue. This statistical model is shown in Figure 7, and had a residual of R"=0.188, not significantly different from the non-statistical model. 

A companion study52,53 tested the ability of this model to simulate the partitioning of solute molecules between the two phases, governed by their relative lipophilicity. The addition of a small number of solute molecules was made to the initial, random mixture. As the dynamics proceeded, it was... 

Le Grand (36) has developed a model to account for domain formation and stability based on the change in free energy which occurs between a random mixture of block copolymer molecules and a micellar domain structure. The model also considers contributions to the free energy of the domain morphology resulting from the interfacial boundary between phases and elastic deformation of the domains. 

In order to justify this approach, it was necessary to evaluate if the patients in these studies represented a random sample from a population composed of at least two subpopulations, one with one set of typical values for response and a second with another set of typical values for response. A mixture model describing such a population can be represented by the following equations ... 

A mixture model implicitly assumes that some fraction. (p) of the population has one set of typical values of response, and that the remaining fraction (1 — p) has another set of typical values. In this model, the only difference initially allowed in the typical values between the two groups was the maximal fractional reduction in seizure frequency after treatment with pregabalin, that is, EmaxA EmaxB- Values for these two parameters and the mixing fraction p were estimated. Random interindividual variability effects r i and r 2 were assumed to be normally distributed with zero means and common variance co. The estimation... 

Equation (21-66) estimates the variance of a random mixture, even if the components have different particle-size distributions. If the components have a small size (i.e., small mean particle mass) or a narrow particle-size distribution, that is, and c are low, the random mix s variance falls. Sommer has presented mathematical models for calculating the variance of random mixtures for particulate systems with a particle-size distribution (Karl Sommer, Sampling of Powders and Bulk Materials, Springer-Verlag, Berlin, 1986, p. 164). This model has been used for deriving Fig. 21-46. 

Contrary to the concept of the random mixing of ions, Fellner (1984), Fellner and Chrenkova (1987) proposed the molecular model for molten salt mixtures in which it is assumed that in an ideal molten mixture, molecules (ionic pairs) mix randomly. The model composition of the melt, i.e. the molar fractions of ionic pairs in the molten mixture, is calculated on the basis of simultaneous chemical equilibrium among the components of the mixture. For instance, in the melt of the system M1X-M2X-M2Y one can assume random mixing of the ionic pairs -X , mJ-X , mJ-Y , -X , and 2MJ-XY . 

Ideally, the skewness coefficient is zero. The presence of skewness in this random effect plot is undesirable and data simulated with the model might not statistically reflect the observed data. The absence of bimodality in the plot does not imply the absence of a mixture. Indeed, when a mixture is present, and the means of two subpopulations are close together, or their variances are large, or their relative proportions are unbalanced (i.e., a 90%/10% partition versus a 50%/50% partition), one may not see bimodality in these types of plots. Two possible approaches to this skewness problem are mixture modeling and random effect transformation. 

Whether one is able to fit mixture models with distinct random effects parameters for each subpopulation is dependent on the nature of the underlying mixture. Are the subpopulations close together in mean, how much data is available (per subject and total), and which type of estimation is being used (first order, hybrid, Laplacian) Now to complete the attempt at applying a two subpopulation mixture model to this data, the probability model and number of subpopulations must be communicated to NONMEM via the mix block. Within the mix abbreviated code the number of subpopulations are communicated with the variable nspop and the probabilities associated with the subpopulations with the variable p (i) (or its alias Mixp(i)), where i indexes the subpopulation. Thus, the code would be... 

The example above used two ways of specifying the random effects on K, paving the way to a discussion about how NONMEM calculates the eta-bar statistic when conditional estimation is used. Notice that both control streams (c2.txt and C3. TXT) contain the statement est=mixest in their pk blocks. At finalization of the NONMEM run, est will contain the number (1 or 2) corresponding to each subject s most likely subpopulation. This allows the tabulation of est and the modal ETA values for each patient using the table record with the firstonly option. Each patient will have an eta estimate, only for his/her most likely subpopulation. His/her eta estimate for the other subpopulation will be zero. For the first mixture model report (r2. txt), the output for the eta-bar section is as follows ... 

One must be careful not to become a lazy modeler and rely on mixtures when covariates, structural model modifications, or changes in random effects structure can be used in lieu of a mixture. Before one accepts a mixture model, the covariates should be examined closely, across the subpopulations, with the idea of seeing a way to include them in a nonmixture model. 

Considerable work has been focused on determining the asymptotic null distribution of -2 log-likelihood -ILL) when the alternative hypothesis is the presence of two subpopulations. In the case of two univariate densities mixed in an unknown proportion, the distribution of -ILL has been shown to be the same as the distribution of [max(0, Y)f, where Y is a standard normal random variable (28). Work with stochastic simulations resulted in the proposal that -2LL-c is distributed with d degrees of freedom, where d is equal to two times the difference in the number of parameters between the nonmixture and mixture model (not including parameters used for the probability models) and c=(n-l-p- gl2)ln (31). In the expression for c, n is the number of observations, p is the dimensionality of the observation, and g is the number of subpopulations. So for the case of univariate observations (p = 1), two subpopulations (g = 2), and one parameter distinguishing the mixture submodels (not including the mixing parameter), -2LL-(n - 3)/n with two... 

This section explores various diagnostic plots that might be used to support a mixture model. The issue of skewed random effects distributions as they relate to how well one can correctly classify patients into their correct subpopulations, assuming a mixed population, is also revisited. 

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