Sankhya: The Indian Journal of Statistics

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

POPULATION MODELS WITH A NONPARAMETRIC RANDOM COEFFICIENT DISTRIBUTION

By

STEPHEN WALKER, *Imperial College of Science, Technology and Medicine, London*

and

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.