Applications of Hierarchical Statistical Models to Interpret Biological Data

Abstract

This thesis looks at two biological applications of hierarchical models. The first application considers the non-linear decay of a contaminant in beef cattle, while the other quite different application considers the transmission probability of genetic markers in the mouse. In the first application, a contaminant was measured over time on 11 animals. Each animal was then a random effect in a small data set. Application of a non-linear random effects model was the only way to analyse these data. This required integrating over animal effects to obtain the marginal model for the small animal population. Lack of a closed form solution for the integration, necessitated use of LaPlace's approximation. A better analysis was demonstrated using MCMC, which provided confidence intervals for parameter estimates, and was not limited by the number of parameters in the model. The second application was seemingly very different. The data consisted of counts of genetic markers in mouse progeny, and so was binary with many elements in common to a time series. Of interest was change in transmission probability of markers from parents, as a function of marker location down each chromosome. An empirical (unconstrained) regression was found to potentially confound transmission effects with local error effects, and constrained the models sensitivity to very local effects. A better approach was to constrain transmission probabilities to specific states, which allowed nearby markers to have either abrupt changes in state, or no change. Hidden Markov Models confer this structure to data, having some similarity to step functions. The best method of fitting these models to data was shown to be the Baum-Welch algorithm, which allows model flexibility and expansion. It was shown that Hidden Markov Models provided a good fit to the mouse data. The ability to pool information within states provided a better estimate of transmission means and standard error, than is incurred by treating each locus in isolation and attaching a global error. In common to both hierarchical models, was the bringing of some sort of pre-existing information to models. For the random effects model, this was the constraint that random effects were normally distributed, and for the mouse data, that nearby markers may have the same transmission probability. This had the effect of adding to the information in the data, and so improved parameter estimation, and gave models stability. This latter benefit is particularly beneficial to small data sets. The bringing of pre-existing information to a model is implicitly a Bayesian approach to analysis.