Multilevel modelling of clustered grouped survival data using Cox regression model: an application to ART dental restorations

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Abstract
In some controlled clinical trials in dental research, multiple failure time data from the same patient are frequently observed that result in clustered multiple failure time. Moreover, the treatments are often delivered by more than one operator and thus the multiple failure times are clustered according to a multilevel structure when the operator effects are assumed to be random. In practice, it is often too expensive or even impossible to monitor the study subjects continuously, but they are examined periodically at some regular pre-scheduled visits. Hence, discrete or grouped clustered failure time data are collected. The aim of this paper is to illustrate the use of the Monte Carlo Markov chain (MCMC) approach and non-informative prior in a Bayesian framework to mimic the maximum likelihood (ML) estimation in a frequentist approach in multilevel modelling of clustered grouped survival data. A three-level model with additive variance components model for the random effects is considered in this paper. Both the grouped proportional hazards model and the dynamic logistic regression model are used. The approximate intra-cluster correlation of the log failure times can be estimated when the grouped proportional hazards model is used. The statistical package WinBUGS is adopted to estimate the parameter of interest based on the MCMC method. The models and method are applied to a data set obtained from a prospective clinical study on a cohort of Chinese school children that atraumatic restorative treatment (ART) restorations were placed on permanent teeth with carious lesions. Altogether 284 ART restorations were placed by five dentists and clinical status of the ART restorations was evaluated annually for 6 years after placement, thus clustered grouped failure times of the restorations were recorded. Results based on the grouped proportional hazards model revealed that clustering effect among the log failure times of the different restorations from the same child was fairly strong (corrchild=0.55) but the effects attributed to the dentists could be regarded as negligible (corrdentist=0.03). Gender and the location of the restoration were found to have no effects on the failure times and no difference in failure times was found between small restorations placed on molars and non-molars. Large restorations placed on molars were found to have shorter failure times compared to small restorations. The estimates of the baseline parameters were increasing indicating increasing hazard rates from interval 1 to 6. Results based on the logistic regression models were similar. In conclusion, the use of the MCMC approach and non-informative prior in a Bayesian framework to mimic the ML estimation in a frequentist approach in multilevel modelling of clustered grouped survival data can be easily applied with the use of the software WinBUGS. Copyright © 2005 John Wiley & Sons, Ltd.