Censored observations, repeated measures and mixed effects models: An approach using the EM algorithm and normal errors

SUMMARY Maximum likelihood estimation of a vector regression parameter and variance com- ponents is considered for the mixed effects model when observations are right censored. A general scheme of estimation is given using the EM algorithm and detailed results found for the model with between and within block variation. This model is applied to the logarithms of survival times from a repeated measures design. In this paper we consider the estimation of the parameters when some of the observa- tions are right-censored. This can occur with repeated measurements on subjects where the response is a waiting time and the variability in log times is modelled by a normal distribution. Alternative models for this situation are given by Clayton & Cuzick (1985) and their methods involve partially parametric techniques using ranks and are based on models involving extreme value, log gamma and logistic distributions. We are concerned with use of the EM algorithm (Dempster, Laird & Rubin, 1977) to estimate the parameters of the mixed model. For uncensored data, Hartley & Rao (1967) have considered this solution. The analysis leads to a method which is straightforward to implement for the repeated measures model. Dempster et al. (1984) give an account of the method with uncensored observations. In ? 2 we consider a general implementation of the EM algorithm for the pure random effects model with censored data and then, in ? 3, we give explicit results for the simple random effects model. Section 4 uses these results for the analysis of repeated measures and ? 5 discusses some miscellaneous points arising. Finally, in ? 6, we give an example involving skin graft data, where there has been extensive analysis using matched pairs techniques.