Bayesian Analysis of Linear and Non-Linear Population Models by Using the Gibbs Sampler

Abstract

(Fearn, 1975), a linear growth relationship between the dental measurement and age was assumed. This is also assumed in our subsequent analysis, together with homoscedastic normal errors within each separate population (girls, boys). Let x j , Y j denote, respectively, the Jth time point (using age 11 as origin) and associated measurement on the ith individual ( -1,... . 11 for the population of girls, J -12,..., 27 for the population of boys, J -1,...,4). Table 1 Here ------------------- -------------------------- 3For both the girl and boy populations the first stage model of (1. 1) ( 1 ) = p2, so that the individual straight-line growth curves are, n affect, regarded as distributed around a 'mean' population growth curve, p + p 2 x with population variation described by the 2 x 2 covariance matrix Z.(We digress to note that since the 13 are positive In this case it might be more reasonable to assume log 1 to be normally distributed. Such a refinement is not, in fact, important In this example and so we shall not pursue it, In order to keep this Initial exposition as simple as possible.We shall illustrate such a transformation n our second, nonlinear, example.) In what follows, we shall denote p, by aG (ca) for the girl (boy) populations and u2 by 3 G (3B).Since information from Individuals within each population is effectively 'pooled' to give population 'mean' Inferences, it can be important in such studies to guard against an aberrant or 'outlying' Individual unduly influencing the population Inference.Proceeding naively, examining, for example, the pooled population of girls and boys. one might plot the leastsquares estimates of ntercepts and slopes, as in Figure 1. Should one conclude from the plot that the boy labelled 24 Is a 'slope outlier'? Or that the boy labelled 21 is an 'intercept outlier'? We seek a modelling analysis 4 strategy which will .provide both a coherent outlier detection diagnostic and direct inferences which accommodate the effect of any outliers present. The main inference questions in this study relate to differences in growth between the girl and boy populations. We shall provide illustrative analyses of this n Section 4, taking into account the outlier issue discussed above. Table 2 presents pharmacokinetic data on the plasma concentration of the drug Cadralazine in 10 cardiac failure patients at various times after the administration of a single dose of 30mg. where a , Pi (> 0) are, respectively, the volume of distribution and elimination rate for Individual 1. I -1,.., 10. A nonlinear povulation vharmacokinetlc example 5Measurement variance is certainly related to underlying concentration level in studies such as this, so that a simple additive homoscedastic normal error assumption for the first stage distribution is inappropriate. We shall illustrate possible models and their subsequent analyses by considering the following three intra-individual error structures.As a first possibility, lettingPi. i) and denoting the righthand-side of (2.1) by iij(01), we assume that log Yij "...

Statistical analysis of Clewell et al. PBPK model of trichloroethylene kinetics.

“…These iterative sampling procedures are particularly convenient in the case of hierarchical models. They belong to a class of Markov chain Monte Carlo (MCMC) techniques that has recently received much interest (10,(29)(30)(31)(32)(33)(34). Three independent Markov chain Monte Carlo runs were performed for each species.…”

Detecting stage-wise outliers in hierarchical Bayesian linear models of repeated measures data

“…Alternative model elaboration steps to accommodate outliers (e.g., replacement of the normal priors by $ -like priors) could also be entertained as in Wakefield et al (1994) and, more recently, Spiegelhalter and Marshall (1999).…”

“…The Bayesian analysis of repeated measures data is typically based on hierarchical generalizations of mixed effects models. Early references are Berger and Hui (1983) and Wakefield et al (1994). In recent years such models have found increasingly fruitful application in both medicine and epidemiology (see for example Palmer and Müller, 1998;Joseph et al, 1999;Tan et al, 1999;Laird, 2000, 2002;Lambert et al, 2001;Berlin et al, 2002).…”

Applications of population approaches in toxicology

“…Indeed, the scope of population models is not limited to pharmaco-or toxicokinetics. Dose-response studies, for example, can be fruitfully analysed with the same tools (Davidian and Giltinian, 1995;Palmer and Müller, 1998;Patron-Bizet et al, 1998;RacinePoon, 1985;Racine-Poon and Smith, 1990;Wakefield et al, 1994) .…”