Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series B, Pt. 1, 196-214



STEPHEN WALKER, Imperial College of Science, Technology and Medicine, London
JON WAKEFIELD , Imperial College School of Medicine at St Mary's, London

SUMMARY. Population data fit very naturally into a hierarchical framework. At the first stage of this hierarchy the data of a particular individual are modelled, typically by a nonlinear regression model, with the same regression model assumed for each individual. Inter-individual variability is accommodated at the second stage by assuming that the parameters of each individual are independently and identically distributed from a population distribution F. Often, interest is in learning about F so that predictions can be made for future individuals from the population. Previous Bayesian work has largely concentrated on F being assigned a specific parametric form, typically the normal. In this paper we propose a Bayesian nonparametric approach using the Dirichlet process (Ferguson, 1973; Antoniak, 1974) as a class of prior distributions for F. We consider the important case where covariate relationships are modelled at the second stage, and allow for errors-in-variables in the measured covariates. Relevant posterior distributions are summarised using Markov chain Monte Carlo methods. A challenging population pharmacokinetic dataset, involving a nonlinear concentration/time relationship and individual-specific covariates, is analysed and the results are compared with those of previous non-Bayesian parametric and non-parametric analyses.

AMS (1991) subject classification. 62C10.

Key words and phrases. Dirichlet process; Errors-in-variables; Markov chain Monte Carlo; Nonlinear random coefficient model; Pharmacokinetics; Random distributions.

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