Censored Survival Times Research Articles

Further results on the explained variation in proportional hazards regression

SUMMARY Two competing measures of the proportion of variation of possibly censored survival times explained by a given proportional hazards model are compared in a Monte Carlo study. It is shown that the validity of a very quick likelihood based measure (Magee, 1990) depends on a proportionality assumption. The computationally more involved measure V2 of Schemper (1990) is robust in this respect. Both measures produce very close results if the Cox model assumptions are perfectly met.

The Explained Variation in Proportional Hazards Regression

Two new measures of the proportion of the variation of possibly censored survival times explained by a given proportional hazards model are presented. It is shown that the previously suggested proportion of explained log likelihood is not a useful measure of predictive precision. The new measures are intuitively appealing and sufficiently simple for routine application.

The explained variation in proportional hazards regression

Two new measures of the proportion of the variation of possibly censored survival times explained by a given proportional hazards model are presented. It is shown that the previously suggested proportion of explained log likelihood is not a useful measure of predictive precision. The new measures are intuitively appealing and sufficiently simple for routine application.

Efficiency of observation schemes for proportional hazard regression models with censoring

Proportional hazard regression models are considered. The scheme of observation in which observed and censored survival times are available for analysis is compared with the simpler binary scheme in which it is only known whether or not individuals survived the censoring times. A bound on the expected information matrix for the second scheme is given in terms of the information matrix for the first scheme. When the censoring variables are of degenerate type giving fixed censoring times, it is shown that the loss of efficiency from using the binary scheme of observation will be unimportant provided that there is at least a moderate degree of censoring. Power comparisons for likelihood ratio tests of hypotheses for the regression coefficients in an exponential regression model with a logarithmic link for the failure rate function are made for the two schemes of observation.

Stratification by stepwise regression, correspondence analysis and recursive partition: a comparison of three methods of analysis for survival data with covariates

The problem of defining prognostic groups on the basis of censored survival times and covariates is central in medical biostatistics. Several methods have been proposed, but little is known about their relative advantages. Here three methods are discussed: Stepwise Regression, Correspondence Analysis and Recursive Partition. The approach is empirical in that the focus is on the performance on real data sets. Our example is discussed at length. We find that Stepwise Regression has the advantage of flexibility and economy of description, but is limited in discovering interactions and other complex features of the data. Correspondence Analysis is equally flexible, though less economical, and is very powerful in revealing unexpected features of the data: it is recommended as an exploratory tool. Recursive Partition is efficient in discovering interactions within large data sets and has the advantage of being the only method that produces clear descriptions in direct clinical terms; its flexibility, however, is limited, especially when the number of covariates is large relative to the number of individuals. Since no method is universally preferable, their joint use is recommended. A variety of criteria for ranking stratifications are proposed when a choice to be made.

Sequential Analysis of Survival Times in Clinical Trials

AbstractThis paper provides a synopsis of statistical methods which can be used for the sequential analysis of possibly censored survival times in clinical trials. Especially, results on the asymptotic behaviour of the Breslow‐Haug statistic and on the sequential version of the logrank statistic are presented in a standardized terminology. In addition, formulae for the explicit calculation of linear and square‐root boundaries for sequential plans are given and illustrated by an example. Practical problems of applying these methods when monitoring a fixed‐sample clinical trial as well as group sequential methods and calculation of P‐values are also discussed.

Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators

Derivatives of statistical functionals have been used to derive the asymptotic distributions ofL-,M- andR-estimators. This approach is often heuristic because the types of derivatives chosen have serious limitations. The Gâteaux derivative is too weak and the Frechet derivative is too strong. In between lies the compact derivative. This paper obtains strong results in a rigorous manner using the compact derivative onC0(R). This choice of space allows results for a broader class of functionals than previous choices, and the fact that\(\left\{ {\sqrt n \left( {\tilde F_n - F} \right)} \right\}\) is often tight provides the compact set required. A major result is the derivation of the compact derivative of the inverse c.d.f. when the range space is endowed with the uniform norm. It has applications to the asymptotic theory ofL-,M- andR-estimators. We illustrate the power of this result by applications toL-estimators in settings including the one sample problem, data grouped by quantiles, and censored survival time data.

GFREG: A computer program for maximum likelihood regression using the generalized F distribution

A FORTRAN program is described for maximum likelihood estimation within the Generalized F family of distributions. It can be used to estimate regression parameters in a log-linear model for censored survival times with covariates, for which the error distribution may have a great variety of shapes, including most distributions of current use in biostatistics. The optimization is performed by an algorithm based on the generalized reduced gradient method. A stepwise variable search algorithm for covariate selection is included in the program. Output features include: model selection criteria, standard errors of parameter estimates, quantile and survival rates with their standard errors, residuals and several plots. An example based on data from Princess Margaret Hospital, Toronto, is discussed to illustrate the program's capabilities.

Nonparametric Estimation of the Survival Function When Cause of Death is Uncertain

A nonparametric estimator for the survival function, accommodating censored survival times and uncertainty in the assignment of cause of death, is proposed. For example, in a carcinogenicity experiment the data on each animal may consist of an observed age-at-death and some indication of the probability that the tumor type under study caused death. An estimator of the net survival function, for time-to-death due to the cause of interest, is developed. Under certain assumptions, the proposed estimator is consistent and asymptotically normally distributed. Monte Carlo simulations were used to compare this estimator with the Kaplan-Meier estimator. Forcing the cause of death to be specified with certainty, as required by the Kaplan-Meier estimator, may result in substantial biases.

A Concordance Test for Independence in the Presence of Censoring

An extension of the definition of Kendall's coefficient of concordance is proposed for testing independence in bivariate censored survival times. The test statistic counts the number of known concordances minus the number of known discordances. The variance of the statistic is calculated for an random-censorship model which, however, does not require independence of the components of the censoring distribution. Two examples are given.

Rank tests for association with right censored data

SUMMARY Rank tests for association between a time variable and a continuous covariate, and between two time variables are given in the presence of censoring. In the first case, these tests are related to the semiparametric tests of Cox (1972) and Prentice (1978) but differ in that the covariate is replaced by a score computed from its rank. In the second case, the test statistic is again a simple inner product, which in the proportional hazards model reduces to the correlation between the two vectors of log rank scores. The loss of efficiency arising from incorrect score functions is also explored when there is no censoring. In this paper we consider a general model for association between pairs of observations in which either variable or both variables may be right censored. The aim is to develop tests based on the generalized ranks of both variables which have maximal power at alternatives which are near to the null hypothesis of independence. Attention will be concentrated on two special cases of interest in survival analysis: tests for association between a censored survival time and a possibly censored continuous covariate, and tests for association between the times to different events. Beginning with Cox (1972), the first situation has been studied extensively in a semiparametric set-up in which information based on ranks is used for the time variable but exact values are used for the covariate. Procedures which use only the ranks of both variables have been developed by Brown, Hollander & Korwar (1974) and more recently by O'Brien (1978). However, neither of these procedures have been developed with a particular model in mind and one of the achievements of the present paper is to derive rank tests with good local power when a particular model such as the proportional hazards model holds. In the case of a proportional hazards model with a continuous covariate, the present development may be preferable to the semiparametric methods when the functional form of the relation- ship between the hazard and the covariate is not known in advance, or when errors or outliers may exist in the observed covariate values. Thus, significance levels may be obtained which are not inflated by multiple regroupings or transformations of the covariate data, although it would be expected that parametric model building with variables found to be associated with survival would often be carried out in further analyses.

Statistical methods for censored survival data.

Methods of statistical analysis of censored survival times are briefly reviewed and illustrated by application to clinical trials data. These include estimation of the survival curce, nonparametric tests to compare several survival curves, tests for trend, and regression analysis. Extensions of the methodology are made for application to epidemiologic case-control studies. These are used to estimate relative risks for leukemia associated with radiation exposures. A final section provides some annotated references to the recent literature.

Statistical Methods for Censored Survival Data

Methods of statistical analysis of censored survival times are briefly reviewed and illustrated by application to clinical trials data. These include estimation of the survival curce, nonparametric tests to compare several survival curves, tests for trend, and regression analysis. Extensions of the methodology are made for application to epidemiologic case-control studies. These are used to estimate relative risks for leukemia associated with radiation exposures. A final section provides some annotated references to the recent literature.

A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence

SUMMARY The application of Cox's (1972) regression model for censored survival data to epidemiological studies of chronic disease incidence is discussed. A related model for association in bivariate survivorship time distributions is proposed for the analysis of familial tendency in disease incidence. The possible extension of the model to general multivariate survivorship distributions is indicated. This paper is concerned with a problem in the analysis of epidemiological studies of chronic disease incidence. In contrast with problems in the epidemiology of infectious disease, such analysis usually assumes that incidence of disease in different individuals represents independent events, the occurrence of which is influenced by measurable factors describing individuals and their environment. However, in the study of familial tendency in chronic disease incidence, this assumption is called into question. Comparisons of parents and offspring and sibling comparisons investigate possible relationships between disease incidence in related individuals and such studies provide interesting analytical difficulties. Here, this problem is treated as one of estimating association in multivariate life tables. In ? 2 it is shown that epidemiological incidence studies may be regarded as being concerned primarily with the study of the distribution of the age at incidence and that Cox's (1972) regression model for the analysis of censored survival time data may readily be applied to incidence data and is closely related to more specifically epidemiological models. In later sections, a related model for bivariate life tables is developed and applied to the problem of demonstrating association in disease incidence in ordered pairs of individuals. The possible extension of the model into more dimensions is indicated.