Bayes linear bayes network models for medical diagnosis and prognosis

Abstract

In this thesis, we develop the application of Bayes linear kinematics and Bayes linear Bayes graphical models to problems in medical diagnosis and prognosis. In medical diagnosis or prognosis, we might use information from a number of covariates to make inferences about the underlying condition, prediction about survival or simply a prognostic index. The covariates may be of different types, such as binary, ordinal, continuous, interval censored and so on. The covariates and the variable of interest may be related in various ways. We may wish to be able to make inferences when only a subset of the covariates is observed so relationships between covariates must be modelled. In the standard Bayesian framework, such a case might suggest the use of Markov chain Monte Carlo (MCMC) methods to integrate over the distribution of the missing covariate values but this may be impractical in routine use. We propose an alternative method, using Bayes linear kinematics within a Bayes linear Bayes model in which relationships between the variables are specified through a Bayes linear structure rather than a fully specified joint probability distribution. This is much less computationally demanding, easily allows the use of subsets of covariates and does not require convergence of a MCMC sampler. In earlier work on Bayes linear Bayes models, a conjugate marginal prior has been associated with each covariate. We relax this requirement and allow non-conjugate marginal priors by using one-dimensional numerical integration. We compare this approach with one using conjugate marginal priors and with a Bayesian analysis using MCMC and a fully specified joint prior distribution. We illustrate our methods with an application to prognosis for patients with non-Hodgkin’s lymphoma in which we treat the linear predictor of the lifetime distribution as a latent variable and use its expectation, given whatever covariates are available, as a prognostic index.