Population modeling estimation methods

In addition to these three major methods mentioned, several other computational approaches can also be used to deal with population stratification. For example, ADMIXMAP (22-26) is a model-based method that estimates the individual history of admixture. It can be applied to an admixed population with two or more ancestral populations. It also tests the association of a trait with ancestry at a marker locus with control for population structure. Wu et al. developed a software package in R (PSMIX) for the inference of population stratification and admixture (27). PSMIX is based on the maximum likelihood method. It performs as well as model-based methods such as STRUCTURE and is more computationally efficient. 

Using a variety of model specifications and estimation methods, lizuka consistently found that while the current treatment population size does not... 

Estimation of nonlinear mixed effects models has been implemented in a number of software packages and includes different estimation methods [12]. As NONMEM is the most commonly used software to estimate population parameters this program is base for the following description. 

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

If the experimental runs are completely randomized, then randomization theory (see Hinkelmann and Kempthorne, 1994) tells us that least squares gives us unbiased estimators of any pre-chosen set of n — 1 linearly independent contrasts among the n combinations of factor levels (treatments). In most factorial experiments the pre-chosen treatment contrasts would be main effects and, perhaps, interactions. However, in supersaturated designs there is no rational basis for choosing a set of n — 1 contrasts before the analysis. Any model selection method will lead to selection biases, perhaps large biases, in the estimators of effects. If a2 is assumed known, then we can test the null hypothesis that all n treatment populations have equal means. This would not be of great interest, because even if this null hypothesis were true it would not imply that all main effects are zero, only that a particular set of n - 1 linear combinations of treatment means are zero. Of course, in practice, a2 is not known. 

External validation is the most stringent type of validation. This type of validation can be executed when both input data (index population) to estimate and develop the model and output data (test population) on which the model can be tested exist. It is the application of the developed model to a new data set (validation data set) from another study (28). When a model is validated externally, it provides the strongest evidence for transportability. There are several approaches to internal validation, some of which have been proved to have excessive type I error (6) and various methods are reviewed below. 

Over the past 25 years a variety of methods have been proposed for the characterization of the population pharmacokinetics of drugs. In this chapter, the statistical framework for estimating population pharmacokinetics in terms of individual and population models is discussed as a prelude to discussing some of the methods used in estimating population pharmacokinetics. In doing so we have adopted a user-friendly approach described previously (14). The goals of a PPK analysis and the data type (1) will determine the method selected for the analysis. 

Note that confidence interval construction for the Cmax ratio represents a challenge because of the difficulty of formulating Cmax as a model parameter. Bootstrap (10) allows this construction, though, because in each bootstrap run, the predicted Cmax for the test and reference formulation, and thus their ratio, can be calculated from the population model parameters. The percentile bootstrap method then uses the 5% and 95% percentiles of the bootstrap runs to form the 90% confidence interval. Specifically, in each bootstrap run, a bootstrap data set can be generated where the subjects were resampled with replacement. Parameter estimates can be obtained for the bootstrap data set, and thus a ratio of ACC and Cmax- Results of all bootstrap data sets can be assembled and the 5% and 95% percentiles used to construct the 90% bootstrap confidence intervals. 

All the objective functions shown in Table 15.1 are derived from a least-squares regression approach as previously described, whereas the estimation method more commonly used in population pharmacokinetics and nonlinear mixed effect modeling in general is based on a maximum likelihood (ML) approach. ML is an alternative to the least-squares objective function it seeks to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function). In general terms, the likelihood function is defined as... 

In conditions of chronic (prolonged) exposure, there is usually no danger of deterministic health effects among the population, and therefore methods of dose assessment based on best parameter estimates should be employed, rather than conservative models as used in emergencies. 

The model presented in this paper represents an extension of the Bayesian population variability assessment method in accidents analysis. A Markov-based model is used for estimation of the expected work time loss distributions due to occupational accidents. The Bayesian population variability assessment method allows the evaluating of population variability of accidents and recovery rates based on exposure data of workers submitted to same occupational risks and the Markov-based model is used to derive the worker unavailability... 

The non-parametric method provides a graphical estimate of the number of recurrences (repairs/ failures) per unit and per the whole population, versus utilization/age. The method is non-parametric in the sense that it does not use a parametric model for the population. This estimation involves no assumptions about the form of the mean function or the process generating the system histories. 

Chapter 5 describes simplified methods of estimating airborne pollutant concentration distributions associated with stationary emission sources. There are sophisticated models available to predict and to assist in evaluating the impact of pollutants on the environment and to sensitive receptors such as populated areas. In this chapter we will explore the basic principles behind dispersion models and then apply a simplified model that has been developed by EPA to analyzing air dispersion problems. There are practice and study problems at the end of this chapter. A screening model for air dispersion impact assessments called SCREEN, developed by USEPA is highlighted in this chapter, and the reader is provided with details on how to download the software and apply it. 

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