Bayes estimation

C, microlayer, subsurface natural water of salinity 17% and TOC 0.4-1.0 ppm, from Pt. Lookout, Chesapeake Bay estimated value adjusted to salinity, Rice et al. 1997b)... 

Lookout, Chesapeake Bay estimated value adjusted to salinity, Rice et al. 1997b)... 

Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

The maximum a posteriori estimation method (the Bayes estimation)... 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

One should note that test independence is important for accurate probability estimation, but not necessarily for classification. In classification, one has to determine only which disease is the more likely, not its exact probability. Studies have shown that using Bayes estimation gives quite good diagnostic accuracy even when the tests are not independent. Tests with correlations of up to 0.7 can stiU be used together to give an idea of the most likely disease. 

Individual estimates of the parameters (Eq. (7.2)), P can be expressed as a function q ) based on the typical individual parameter estimate and the zero mean, symmetrically distributed variable r, whose standard deviation is (o. The values of ], are obtained as posterior Bayes estimates conditioned on a set of typical individual parameter estimates, (o and the data for the individual. The important property of the individual parameter estimates is that they will be shrunken toward the typical individual parameter estimate, the degree of shrinkage determined by the amount of information in the data from the individual relative to the size of (o. In the extreme case with no data at all, the individual estimate will be exactly the same as the typical individual estimate. In the other extreme, with an infinite amount of information, the individual parameter estimate will be independent of the typical individual estimates and (o. 

In Figures 7.13 and 7.14, the unexplained variability in clearance was expressed as the corresponding r value (obtained by posterior Bayes estimation). There are other alternatives, as shown in Figure 7.15. 

Mixed models and empirical Bayes estimates of exposure... 

Friesen, M. C., Macnab, Y. C., Marion, S. A., Demers, P. A., Davies, H. W., and Teschke, K. (2006). Mixed models and empirical Bayes estimation for retrospective exposure assessment of dust exposures in Canadian sawmills. Ann Occup Hyg 50, 281-288. 

ESTIMATION OF THE RANDOM EFFECTS AND EMPIRICAL BAYES ESTIMATES (EBEs)... 

The linear mixed effect model assumes that the random effects are normally distributed and that the residuals are normally distributed. Butler and Louis (1992) showed that estimation of the fixed effects and covariance parameters, as well as residual variance terms, were very robust to deviations from normality. However, the standard errors of the estimates can be affected by deviations from normality, as much as five times too large or three times too small (Verbeke and Lesaffre, 1997). In contrast to the estimation of the mean model, the estimation of the random effects (and hence, variance components) are very sensitive to the normality assumption. Verbeke and Lesaffre (1996) studied the effects of deviation from normality on the empirical Bayes estimates of the random effects. Using computer simulation they simulated 1000 subjects with five measurements per subject, where each subject had a random intercept coming from a 50 50 mixture of normal distributions, which may arise if two subpopulations were examined each having equal variability and size. By assuming a unimodal normal distribution of the random effects, a histogram of the empirical Bayes estimates revealed a unimodal distribution, not a bimodal distribution as would be expected. They showed that the correct distributional shape of the random effects may not be observed if the error variability is large compared to the between-subject variability. 

Covariate screening methods are used when there are a large number of covariates, such that evaluating every possible combination in a model is prohibitive. With this methodology, EBEs of the random effects are treated as data and then exploratory methods are used to assess the relationship between the random effects and the covariate of interest. In other words, each individual s pharmacokinetic parameter, clearance for example, is estimated and treated as a being without measurement error. These Bayes estimates are then compared against subject-specific covariates for a relationship using either manual or automated methods. 

Figure 9.8 Scatter plot of creatinine clearance against the empirical Bayes estimate for tobramycin systemic clearance under the 2-compartment model with reduced unstructured covariance matrix. Solid line is the LOESS smoother with 0.3 sampling proportion.

A third sort of measure is the empirical Bayes estimate. Here we assume that each Nij is drawn from Poisson distribution with mean i ij. We are really interested in... 

Examine distribution of empirical Bayes estimates and determine extremes/outliers. Determine if lOV or mixture model improves results. 

Davis CE, Leffingwell DP. Empirical Bayes estimates of subgroup effects in clinical trials. Controlled Clinical Trials, 11 37-42,1990. 

Liang, T. 2004. Empirical Bayes estimation with random right censoring. International J. Information and Management Sciences 15(4) 1-12. 

Susarla, V and Van Ryzin, J. 1978. Empirical Bayes estimation of a distribution (survival) function from right censored observations.Tnn. Statist. 6 740-754. 

To conclude, two important limitations of this work deserve attention. Firstly discussing semi-Markov processes in general, we have the well-known and already cited difficulty in obtaining the requisite data to analyze semi-Markov processes on the non-homogeneous environment. On that, El-Gohary (2004) presents max-iminn likelihood and Bayes estimates of the parameters included in a semi-Markov reliability model of three states. 

The empirical Bayes estimate of the trigger rate is given by the expectation of the posterior distribution, which under the Gamma-Pois son model is Gamma (a -I- i, p + ti). The expectation of this distribution is ... 

Elvik, R. (2008). The predictive validity of empirical Bayes estimates of road safety. Accident Analysis and Prevention 40, 1964-1969. 

Sarhan, A.M. (2003). Empirical Bayes estimates in exponential reliability model. Applied Mathematics and Computation 135,319-332. 

Estimates of 9 can be determined by several methods. These methods can be divided in two major categories. In the so-called Fisher estimation approach, only the data vector z of Equation 9.16 is supplied to the estimator in order to estimate the unknown model parameters 0. The second approach, known as the Bayes estimation approach, takes into account not only z but also some statistical information that is a priori available on the unknown parameter vector 0. 

Bayes estimators assume that the parameter vector is the realization of a random vector 0, the a priori probability distribution of which p0(0) is available, for example, from preliminary population studies. Starting from the knowledge of both the model of Equation 9.16 and the probability distribution of the noise vector v, one can calculate the Hkelihood function p 0(z 0) (i.e., the probability distribution of the measurement vector in dependence of the parameter vector). From p0(0) and p 0(z 0), the a posteriori probability distribution pe z (0 z) (i.e., the probability distribution of the parameter vector given the data vector) can be determined by exploiting the Bayes theorem ... 

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