Statistica Sinica

Ross A. Maller and Xian Zhou

Abstract:``Competing risk'' or ``multiple cause'' survival data arise in medical, criminological, financial, engineering, and many other contexts when death or failure of an individual or unit is classified into one of a variety of types or causes. Important issues in the analysis of such data range from basic properties, such as consistency of estimation of parameters, through more complex boundary hypothesis-testing problems, such as whether a specified list of causes is ``exhaustive'' as opposed to the possibility that some individuals may be ``immune'' to all of these causes. We give a carefully formulated parametric mixture model for competing risk data which allows for censoring and immune individuals, and for which a large-sample analysis can be developed. Under some mild assumptions, we are able to show the existence, uniqueness (local to the true parameter values with probability approaching 1), consistency and asymptotic normality of the maximum likelihood estimators when the parameters are interior to the parameter space. A formulation using ``cause-specific hazards'' can be treated in the same way.

Consistent estimators also exist when the parameters are on the boundary of the parameter space, as is the case for example when testing for exhaustiveness of causes. The ``deviance'' statistic for testing this hypothesis is shown to have as its large-sample distribution a 50-50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. Competing risks data with no censoring can be analyzed similarly.

The large-sample results we give allow many of the data-analytic questions for competing risks data to be formulated and answered in a satisfying way. The methods and approaches are illustrated on a set of criminological (re-arrest) data from Western Australia.

Key words and phrases:Competing risks, survival analysis, mixture models, censored data, causes of death, maximum likelihood estimation, likelihood ratio test.