This thesis investigates child survival in Malawi using a modified Cox proportional hazards model which includes family and community random effects in addition to the fixed effect of the covariates. The parameters of the model are estimated using the Gibbs sampler, a Bayesian Markov Chain Monte Carlo (MCMC) method. We believe this to be the first time the method has been used for this type of data. The parameters are also estimated using the expectation-maximisation (EM) algorithm, a method that has previously been used to analyse this type of data. Both approaches were implemented in Fortran using the NAG routines. The child survival data used in this study were collected as part of the 1992 Demographic and Health Survey (DHS) of Malawi. The women respondents were obtained through a two-stage cluster sampling procedure. The respondents were systematically interviewed about biological, social and demographic factors relating to all live births that had occurred during the previous five years. The results show that the covariates first birth, birth spacing, breastfeeding duration, maternal age, hospital birth and father’s education are important determinants of infant and early childhood survival in Malawi. The family and community frailty effect variances are modest in magnitude. The study also shows that child mortality varies more considerably across families than over communities after controlling for the observed covariates. Neglecting frailty biases estimates of the observed covariates slightly downwards, although the subsequent substantive findings are not markedly affected. The strength of the family random effect is grossly overestimated when community random frailty is ignored while that of the community random effect shows stability whether family random frailty is controlled or not. The Gibbs sampler is shown to be an important alternative to the EM algorithm and other existing methods for estimating parameters in a multilevel hazards model. It allows the full Bayesian inference without the need to evaluate high-dimensional integrals. We obtain a random sample from the complete posterior distribution of all the parameters and hyperparameters whose behaviour can be studied over their range rather than only around the mode. The Gibbs sampler is computational intensive, but this fact has decreasing relevance due to the availability of very powerful computing equipment.