Aust.N.Z.J.Stat. 43(1), 2001, 7–16 A COMPARISON OF METHODS FOR ANALYSING A NESTED FRAILTY MODEL TO CHILD SURVIVAL IN MALAWI SAMUEL O.M. MANDA 1 University of Waikato Summary Demographic and Health Surveys collect child survival times that are clustered at the fam- ily and community levels. It is assumed that each cluster has a specific, unobservable, random frailty that induces an association in the survival times within the cluster. The Cox proportional hazards model, with family and community random frailties acting mul- tiplicatively on the hazard rate, is presented. The estimation of the fixed effect and the association parameters of the modified model is then examined using the Gibbs sampler and the expectation–maximization (EM) algorithm. The methods are compared using child survival data collected in the 1992 Demographic and Health Survey of Malawi. The two methods lead to very similar estimates of fixed effect parameters. However, the estimates of random effect variances from the EM algorithm are smaller than those of the Gibbs sampler. Both estimation methods reveal considerable family variation in the survival of children, and very little variability over the communities. Key words: EM algorithm; Gibbs sampling; nested frailty; proportional hazards model. 1. Introduction Child survival data collected as part of the Demographic and Health Survey program (DHS) are clustered at the family and community levels. The mortality risks of children from the same cluster tend to be more alike than the risks of children chosen at random from the whole population. This might result from the fact that children in the same family share similar genetic factors and those in a community share the same environmental conditions that affect their survival. Thus, the usual assumption of independence between event times may not be valid. As a result, analyses that fail to account for the association in the survival times are more likely to underestimate the variances of parameters. Research on the statistical analysis of correlated survival data began primarily with the work of Holt & Prentice (1974) who studied survival experiences in twins. Clayton (1978) proposed a bivariate survival model that can be interpreted in terms of the proportional hazards model with a gamma distributed random effect. Subsequently, Oakes (1982) and Clayton & Cuzick (1985) worked on non-parametric estimation of the association parameter in correlated survival times. On the other hand, Vaupel, Manton & Stallard (1979) introduced the notion of frailty in demographic survival models. Heckman & Singer (1984) studied the sensitiv- ity of covariate effects to the choice of frailty distribution in employment duration models Received October 1998; revised January 2000; accepted March 2000. 1 Dept of Statistics, University of Waikato, Private Bag 3105, Hamilton, New Zealand. Present address: Dept of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand. e-mail: samuelm@stat.auckland.ac.nz Acknowledgments. The author thanks William M. Bolstad and two anonymous referees for their helpful and constructive comments that greatly improved the quality of the paper. The author also thanks the National Statistics Office of Malawi for allowing the use of their data. c  Australian Statistical Publishing Association Inc. 2001. Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden MA 02148, USA