Diagnosis and Prediction pp 75-90 | Cite as

# Survival Analysis for Interval Data

## Abstract

This article develops Bayesian methods for the analysis of a particular kind of interval and right censored data when no parametric structure for the distribution of the data is assumed. It is often not feasible or practical to continuously observe a group of individuals under study. For example, suppose the response variable is the time under experimental conditions until infection with a particular disease, and that the goal is to (i) obtain the predictive survival curve (PSC), (ii) predict a future observation, or (iii) estimate the mean or median survival time. If the number of individuals is large and/or if the method of detection is expensive, it would be sensible to check for the disease at times which have been spaced out; and furthermore, it may be sensible to not check all individuals at all times. In this paper we consider sampling situations where come individuals may be checked for a response more frequently than others.

We assume a prior guess for the entire distribution is available and in conjunction with this, we assume a Dirichlet process prior for the underlying survival distribution. The parameter *α*(⋅) for the prior would often be available from previous parametric analyses of the same type of data. It is also possible to simply place a Dirichlet prior on the vector of probabilities corresponding to the partition of intervals induced by the checking times; the resulting survival curve estimate would be a special case of the one we develop here.

Explicit formulas for Bayesian survival probabilities and curves are obtained. When the prior measure *α*(⋅) for the Dirichlet Process is absolutely continuous, so is the PSC. If the weight *w* attached to the prior measure *α*(⋅) tends to zero, the PSC tends to the nonparametric maximum likelihood estimate, which is explicitly obtained. Under some conditions, the Bayesian curves are shown to be consistent. We also address the issues of estimating the mean and residual mean times until response, and of obtaining the predictive probability that a fraction of future individuals will respond in an interval.

## Key words

Predictive Distribution Nonparametric Maximum Likelihood Estimate Dirichlet Process Dirichlet Prior Censored Data## Preview

Unable to display preview. Download preview PDF.

## References

- Breslow, N. And Crowley, J. (1974). A large sample study of the life table and product limit estimates under random censorship.
*Ann. Statist.***2**, 437–53.MathSciNetzbMATHCrossRefGoogle Scholar - Campbell, G. And Hollander, M. (1982). Prediction intervals with a Dirichlet process prior distribution.
*Can. J. Statist.***10**, 103–111.MathSciNetzbMATHCrossRefGoogle Scholar - Cornfield, J. And Detre, K. (1977). Bayesian life table analysis.
*J. Roy Statist. Soc. Ser. B***39**, 86–94.MathSciNetzbMATHGoogle Scholar - Di Finetti, B. (1937). La prevision: ses lois logiques, ses sources subjectives.
*Annales de l’Institut*Henri poincaire**7**, 1–68.Google Scholar - Dickey, J. (1983). Multiple hypergeometric functions: probabilistic interpretations and statistical uses.
*JASA***78**, 628–637.MathSciNetzbMATHGoogle Scholar - Dickey, J., Jiang, J., And Kadane, J. (1987). Bayesian methods for censored categorical data.
*JASA***82**, 773–781.MathSciNetzbMATHGoogle Scholar - Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions.
*Ann. Probab.***2**, 183–201.MathSciNetzbMATHCrossRefGoogle Scholar - Finkelstein, D.M. (1986). A proportional hazards model for interval-censored failure time data.
*Biometrics***42**, 845–54.MathSciNetzbMATHCrossRefGoogle Scholar - Finkelstein, D.M. And Wolfe, R.A. (1985). A semiparimetric model for regression analysis of interval-censored failure time data.
*Biometrics***41**, 933–45.MathSciNetzbMATHGoogle Scholar - Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems.
*Ann. Statist.***1**, 209–230.MathSciNetzbMATHCrossRefGoogle Scholar - Geisser, S. (1982). Aspects of the predictive and estimative approaches in the determination of probabilities.
*Biometrics Supplement*, 75–85.Google Scholar - Geisser, S. (1984). Predicting Pareto and exponential observables.
*Can. J. Statist.***12**, 143–152.MathSciNetzbMATHCrossRefGoogle Scholar - Geisser, S. (1985). Interval prediction for Pareto and exponential observables.
*J. Econometrics***29**, 173–185.MathSciNetzbMATHCrossRefGoogle Scholar - Gentleman, R. And Geyer, C.J. (1994). Maximum likelihood for interval censored data: Consistency and computation.
*Biometrika***81**, 618–23.MathSciNetzbMATHCrossRefGoogle Scholar - Johnson, W. And Christensen, R. (1986). Bayesian nonparametric survival analysis for grouped data.
*Can. J. Statist.***14**, 307–314.MathSciNetzbMATHCrossRefGoogle Scholar - Kalbfleisch, J. And Mac Kay, R.J. (1978). Remarks on a paper by Cornfield and Detre.
*J. Roy. Statist. Soc. Series B***40**, 175–177.MathSciNetzbMATHGoogle Scholar - Kuo, Lynn (1991). Sampling based approach to computing nonparametric Bayesian estimators with doubly censored data. Cmp. Sc. St.
**23**, 612–15.MathSciNetGoogle Scholar - Lassauzet, M., Johnson, W. And Thurmond, M. (1987). Regression models for timeto-seroconversion following experimental Bovine leukemia virus infection.
*Statistics in Medicine***8**, 725–41.CrossRefGoogle Scholar - Morales, D., Pardo, L. And Quesada, V. (1990). Estimation of a survival function with doubly censored data and Dirichlet Process Prior knowledge on the observable variable.
*Comm. Statist.-Simula.***19**(1), 349–61.MathSciNetzbMATHCrossRefGoogle Scholar - Susarla, V. And Van Ryzin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete observations.
*J. Amer. Statist. Assoc.***71**, 897–902.MathSciNetzbMATHCrossRefGoogle Scholar - Turnbull, B.W. (1976). The empirical distribution function for arbitrarily grouped, censored and truncated data.
*J. Roy. Statist. Soc. Series B***38**, 290–295.MathSciNetzbMATHGoogle Scholar