Joining and splitting models with Markov melding.

Analysing multiple evidence sources is often feasible only via a modular approach, with separate submodels specified for smaller components of the available evidence. Here we introduce a generic framework that enables fully Bayesian analysis in this setting. We propose a generic method for forming a suitable joint model when joining submodels, and a convenient computational algorithm for fitting this joint model in stages, rather than as a single, monolithic model. The approach also enables splitting of large joint models into smaller submodels, allowing inference for the original joint model to be conducted via our multi-stage algorithm. We motivate and demonstrate our approach through two examples: joining components of an evidence synthesis of A/H1N1 influenza, and splitting a large ecology model.

Modelling life course blood pressure trajectories using Bayesian adaptive splines.

No single study has collected data over individuals' entire lifespans. To understand changes over the entire life course, it is necessary to combine data from various studies that cover the whole life course. Such combination may be methodologically challenging due to potential differences in study protocols, information available and instruments used to measure the outcome of interest. Motivated by our interest in modelling blood pressure changes over the life course, we propose the use of Bayesian adaptive splines within a hierarchical setting to combine data from several UK-based longitudinal studies where blood pressure measures were taken in different stages of life. Our method allowed us to obtain a realistic estimate of the mean life course trajectory, quantify the variability both within and between studies, and examine overall and study specific effects of relevant risk factors on life course blood pressure changes.

Rapid model exploration for complex hierarchical data: application to pharmacokinetics of insulin aspart.

We consider situations, which are common in medical statistics, where we have a number of sets of response data, from different individuals, say, potentially under different conditions. A parametric model is defined for each set of data, giving rise to a set of random effects. Our goal here is to efficiently explore a range of possible 'population' models for the random effects, to select the most appropriate model. The range of possible models is potentially vast, because the random effects may depend on observed covariates, and there may be multiple credible ways of partitioning their variability. Here, we consider pharmacokinetic (PK) data on insulin aspart, a fast acting insulin analogue used in the treatment of diabetes. PK models are typically nonlinear (in their parameters), often complex and sometimes only available as a set of differential equations, with no closed-form solution. Fitting such a model for just a single individual can be a challenging task. Fitting a joint model for all individuals can be even harder, even without the complication of an overarching model selection objective. We describe a two-stage approach that decouples the population model for the random effects from the PK model applied to the response data but nevertheless fits the full, joint, hierarchical model, accounting fully for uncertainty. This allows us to repeatedly reuse results from a single analysis of the response data to explore various population models for the random effects. This greatly expedites not only model exploration but also cross-validation for the purposes of model criticism. © 2015 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.

Survival extrapolation using the poly-Weibull model.

Recent studies of (cost-) effectiveness in cardiothoracic transplantation have required estimation of mean survival over the lifetime of the recipients. In order to calculate mean survival, the complete survivor curve is required but is often not fully observed, so that survival extrapolation is necessary. After transplantation, the hazard function is bathtub-shaped, reflecting latent competing risks which operate additively in overlapping time periods. The poly-Weibull distribution is a flexible parametric model that may be used to extrapolate survival and has a natural competing risks interpretation. In addition, treatment effects and subgroups can be modelled separately for each component of risk. We describe the model and develop inference procedures using freely available software. The methods are applied to two problems from cardiothoracic transplantation.

Combining MCMC with 'sequential' PKPD modelling.

We introduce a method for preventing unwanted feedback in Bayesian PKPD link models. We illustrate the approach using a simple example on a single individual, and subsequently demonstrate the ease with which it can be applied to more general settings. In particular, we look at the three 'sequential' population PKPD models examined by Zhang et al. (J Pharmacokinet Pharmacodyn 30:387-404, 2003; J Pharmacokinet Pharmacodyn 30:405-416, 2003), and provide graphical representations of these models to elucidate their structure. An important feature of our approach is that it allows uncertainty regarding the PK parameters to propagate through to inferences on the PD parameters. This is in contrast to standard two-stage approaches whereby 'plug-in' point estimates for either the population or the individual-specific PK parameters are required.

Automated covariate selection and Bayesian model averaging in population PK/PD models.

We illustrate the use of 'reversible jump' MCMC to automate the process of covariate selection in population PK/PD analyses. The output from such an approach can be used not only to determine the 'best' covariate model for each parameter, but also to formally measure the spread of uncertainty across all possible models, and to average inferences across a range of 'good' models. We examine the substantive impact of such model averaging compared to conditioning inferences on the 'best' model alone, and conclude that clinically significant differences between the two approaches can arise. The illustrative data that we consider pertain to the drug vancomycin in 59 neonates and infants, and all analyses are conducted using the WinBUGS software with newly developed 'Jump' interface installed.

A Bayesian toolkit for genetic association studies.

We present a range of modelling components designed to facilitate Bayesian analysis of genetic-association-study data. A key feature of our approach is the ability to combine different submodels together, almost arbitrarily, for dealing with the complexities of real data. In particular, we propose various techniques for selecting the "best" subset of genetic predictors for a specific phenotype (or set of phenotypes). At the same time, we may control for complex, non-linear relationships between phenotypes and additional (non-genetic) covariates as well as accounting for any residual correlation that exists among multiple phenotypes. Both of these additional modelling components are shown to potentially aid in detecting the underlying genetic signal. We may also account for uncertainty regarding missing genotype data. Indeed, at the heart of our approach is a novel method for reconstructing unobserved haplotypes and/or inferring the values of missing genotypes. This can be deployed independently or, alternatively, it can be fully integrated into arbitrary genotype- or haplotype-based association models such that the missing data and the association model are "estimated" simultaneously. The impact of such simultaneous analysis on inferences drawn from the association model is shown to be potentially significant. Our modelling components are packaged as an "add-on" interface to the widely used WinBUGS software, which allows Markov chain Monte Carlo analysis of a wide range of statistical models. We illustrate their use with a series of increasingly complex analyses conducted on simulated data based on a real pharmacogenetic example.