The regression analysis of group truncated data

Abstract

This thesis considers the regression modelling of grouped binary data that is subject to truncation, and explores some general issues relating to truncation. The likelihood for simple binary and ordinal models is developed and the statistical behaviour of these models is explored. The models are found to be well behaved. The efficiency of the truncated model is compared with that of conditional logistic regression, a competing technique. It is found that the truncated model is always more efficient but requires additional assumptions about the data generation process to be applicable. The estimation of the full sample size, N , before truncation occurs is considered, in quite general regression models. The case where the covariate distribution is discrete is first considered. This is extended to allow continuous covariates, and the additional difficulties involved are explored. The issue of setting confidence intervals for N is discussed. A simulation study is used to explore the methods behaviour. Next, the Bayesian analysis of truncated regression models is considered. The use of the empirical distribution of the observed covariates to facilitate the analysis is explored. The posterior distribution of the models parameters under this approach is derived and a Gibbs sampling algorithm implemented to explore the posterior. The convergence properties of the algorithm is considered, and the techniques behaviour assessed in a small simulation study. The effect of over-dispersion on the analysis of group truncated binary data is considered. The available methods of introducing over-dispersion in clustered binary data are discussed and it is argued that only random effects models provide a viable approach. Parameter estimation in these models is derived via a marginal likelihood. In addition a score test is constructed to test for the presence of random effects in group truncated binary data. The methods performance is demonstrated using a simulation study. Finally, the use of the bootstrap to estimate the sampling distribution of parameter estimates from truncated data is considered in an appendix. The inherent limitations of using resampling methodologies to investigate truncated data is demonstrated. It is shown that the nonparametric advantages of the bootstrap are not realised with truncated data due to the lack of observations on the truncated class.