Distributions, selection random-effects models

Data used to describe variation are ideally representative of some population of risk assessment interest. Representativeness was a focus of an earlier workshop on selection of distributions (USEPA 1998). The role of problem formulation is emphasized. In case of representativeness issues, some adjustment of the data may be possible, perhaps based on a mechanistic or statistical model. Statistical random-effects models may be useful in situations where the model includes distributions among as well as within populations. However, simple approaches may be adequate, depending on the assessment tier, such as an attempt to characterize quantitatively the consequences of assuming the data to be representative. 

In addition it is now time to think about the two assumption models, or types of analysis of variance. ANOVA type 1 assumes that all levels of the factors are included in the analysis and are fixed (fixed effect model). Then the analysis is essentially interested in comparing mean values, i.e. to test the significance of an effect. ANOVA type 2 assumes that the included levels of the factors are selected at random from the distribution of levels (random effect model). Here the final aim is to estimate the variance components, i.e. the variance fractions with respect to total variance caused by the samples taken or the measurements made. In that case one is well advised to ensure balanced designs, i.e. equally occupied cells in the above scheme, because only then is the estimation process straightforward. 

Several simulation approaches have been employed to model the effect of uniaxial orientation of rodlike fillers on the percolation behavior in 2D and 3D. In 2D, simple anisotropic stick orientation distributions can be obtained by allowing the angle 0, between stick i and the longitudinal axis to be selected randomly in the interval where... 

Many stochastic models are used in population genetics to describe the effects such as random drift selective force, and mutation pressure (e.g. Karlin McGregor, 1964 Crow Kimura, 1970 Maruyama, 1977). Diffusion models are mostly used, and some of them can be interpreted in terms of the CCS model of chemical reactions. The models are in terms of the probability distribution of gene frequencies. 

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