Mixture model submodel

First, consider a two subpopulation mixture model. The pk block of the control stream needs to be modified to communicate to NONMEM that there may be two subpopulations with respect to the elimination rate constant, k. A variable called MiXNUM indexes the subpopulation, and hence the submodel, for which variables are computed, mixnum may be used as a right-hand quantity in the abbreviated code, as long as the control stream contains a special block of abbreviated code called Mix (see Section 28.4). For the example, the code for k is modified in the PK block as follows ... 

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... 

Next, a three subpopulation mixture model is tried, whereby subjects could remain stable, gain, or lose weight. For this first mixture attempt, drug exposure is not included as a covariate. The three submodels and probability models are as follows (see C9. txt r9. txt). 

Some other kinds of models have shown parameters that seem to follow useful correlation relationships. Among these are the virial coefficient model of Bums (2), the interaction coefficient model of Helgeson, Kirkham, and Flowers (4), and the hydration theory model of Stokes and Robinson (1). The problem shared by all three of these models is that they employ individual ion size parameters in the Debye-Hiickel submodel. This led to restricted applicability to solutions of pure aqueous electrolytes, or thermodynamic inconsistencies in applications to electrolyte mixtures. Wolery and Jackson (in prep.) discuss empirical modification of the Debye-Huckel model to allow ion-size mixing without introducing thermodynamic inconsistencies. It appears worthwhile to examine what might be gained by modifying these other models. This paper looks at the hydration tlieory approach. 

Klein s group developed a mechanism-based lumping scheme for hydrocarbon pyrolysis involving free radicals. The model has two submodels. One is a five-component training set mixture (5CM) that calculates radical concentrations in terms of 42 representative free radical intermediates. The other is a module in which a feed mixture of many components reacts with the 5CM kemal to provide detailed rates and selectivities. The model retains the essence of pyrolysis chemistry with reasonable CPU demand. 

We can divide the literature on FCC modeling into two categories kinetic and unit-level models. Kinetic models focus on chemical reactions taking place within the riser or reactor section of the FCC unit, and attempt to quantify the feed as a mixture of chemical entities to describe the rate of reaction from one chemical entity to another. In contrast, unit-level models contain several submodels to take into account the integrated nature of modem FCC units. A basic unit-level model contains submodels for the riser/reactor, regenerator and catalyst transfer sections. 

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