Inferences in the Interval Censored Exponential Regression Model

Abstract

<p>The problem of estimation when the data are interval censored has been investigated by several authors. Lindsey and Ryan (1998) considered the application of conventional methods to interval mid (or end) points and showed that they tended to underestimate the standard errors of the estimated parameters and give potentially misleading results. MacKenzie (1999) and Blagojevic (2002) conjectured that the estimator of the parameter was artificially precise when analyzing inspection times as if they were exact when the "time to event" data followed an exponential distribution. In this thesis, we derive formulae for pseudo and true (or exact) likelihoods in the exponential regression model in order to examine the consequences for inference on parameters when the pseudo-likelihood is used in place of the true likelihood. We pay particular attention to the approximate bias of the maximum likelihood estimates in the case of the true likelihood. In particular we present analytical work which proves that the conjectures of Lindsey and Ryan (1998), MacKenzie (1999) and Blagojevic (2002) hold, at least for the exponential distribution with categorical or continuous covariates.<br /><br />We undertake a simulation study in order to quantify and analyze the relative performances of maximum likelihood estimation from both likelihoods. The numerical evidence suggests that the estimates from true likelihood are more accurate. We apply the proposed method to a set of real interval-censored data collected in a Medical Research Council (MRC, UK) multi-centre randomized controlled trial of teletherapy in the age-related macular disease (the ARMD) study.</p>