Survival function

Abstract: Competing risks occur frequently in the analysis of survival data. A competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. In a study examining time to death attributable to cardiovascular causes, death attributable to noncardiovascular causes is a competing risk. When estimating the crude incidence of outcomes, analysts should use the cumulative incidence function, rather than the complement of the Kaplan-Meier survival function. The use of the Kaplan-Meier survival function results in estimates of incidence that are biased upward, regardless of whether the competing events are independent of one another. When fitting regression models in the presence of competing risks, researchers can choose from 2 different families of models: modeling the effect of covariates on the cause-specific hazard of the outcome or modeling the effect of covariates on the cumulative incidence function. The former allows one to estimate the effect of the covariates on the rate of occurrence of the outcome in those subjects who are currently event free. The latter allows one to estimate the effect of covariates on the absolute risk of the outcome over time. The former family of models may be better suited for addressing etiologic questions, whereas the latter model may be better suited for estimating a patient's clinical prognosis. We illustrate the application of these methods by examining cause-specific mortality in patients hospitalized with heart failure. Statistical software code in both R and SAS is provided.

Abstract: Use of the proportional hazards regression model (Cox 1972) substantially liberalized the analysis of censored survival data with covariates. Available procedures for estimation of the relative risk parameter, however, do not adequately handle grouped survival data, or large data sets with many tied failure times. The grouped data version of the proportional hazards model is proposed here for such estimation. Asymptotic likelihood results are given, both for the estimation of the regression coefficient and the survivor function. Some special results are given for testing the hypothesis of a zero regression coefficient which leads, for example, to a generalization of the log-rank test for the comparison of several survival curves. Application to breast cancer data, from the National Cancer Institute-sponsored End Results Group, indicates that previously noted race differences in breast cancer survival times are explained to a large extent by differences in disease extent and other demographic characteristics at diagnosis.

Abstract: Since John Snow first conducted a modern epidemiological study in 1854 during a cholera epidemic in London, statistics has been associated with medical research. After Austin Bradford Hill published a series of articles on the use of statistical methodology in medical research in 1937, statistical considerations and computational tools have been paramount in conductingmedical research [1]. For the past century, statistics has played an important role in the advancement of medical research and medical research has stimulated rapid development of statistical methods. For example, the development of modern survival analysis-an important branch of statistics has aimed to solve problems encountered in clinical trials and large-scale epidemiological studies. In this era of evidence-based medicine, the development of novel statistical methods will continue to be crucial in medical research. With the expansion of computer capacity and advancement of computational techniques, it is inevitable that modern statistical methods will likely incorporate, to a greater degree, complex computational procedures. This issue focuses on statistical methods in medical research. Several novel methods aiming on solving different medical research questions are introduced. Some unique approaches of statistical analysis are also present. Hanagal and Sharma contribute two papers. The first one deals with a bivariate survival model. They examine a parameter estimation issue when the samples are taken from a bivariate log-logistic distribution with shared gamma frailty. They propose to use a Bayesian approach along with theMarkov ChainMonte Carlo computational technique for implementation. The computer simulation is conducted for performance evaluation. Two well-known datasets, one about acute leukemia and the other about kidney infection are applied as examples. The second paper contributed by Hanagal and Sharma examines the shared inverse Gaussian frailty model with the bivariate exponential baseline hazard. They first derive the likelihood of the joint survival function. In their Bayesian approach, the parameters of the baseline hazard are assumed to follow a gamma distribution while the coefficients of the regression relationship are assumed to follow an independent normal distribution. The dependence of two components of the survival function is tested. Three information criteria are used for model comparisons. The proposed method is applied to analyze diabetic retinopathy data. The paper by Chang, Lyer, Bullitt and Wang provides a method to find determinants of the brain arterial system. They represent the brain arterial system as a binary tree and apply the mixed logistic regression model to find significant covariates. The authors also demonstrate model selection methods for both fixed and random effects. A case study is presented using the method. This paper provides a rigorous approach for analyzing the binary branching structure data. It is potentially applicable to other tree structure data. Chakraborty proposes two probabilistic models to estimate male-to-female HIV-1 transmission rate in one sexual contact. One model is applicable when the transmitter cell counts are known and the other model is applicable when the receptor cell counts are known. By first uniformizing each transmitter (or receptor) cell count and assuming as a beta distribution, this paper algebraically derives the transition probability by imposing some boundary conditions based on scientific phenomena related to HIV infection. The paper by Yeh, Jiang, Garrard, Lei and Gajewski proposes to use a zero-truncated Poisson model to analyze human cancer tissues transplanted to mice when the positive counts of affected ducts is subject to right censoring. A Bayesian approach choosing a Gamma distribution as the prior is adopted. After implementing through complex computational procedures, this paper obtains the estimates of the coefficients and demonstrates model fitting through

Abstract: The estimation of survival distributions for radio-tagged animals is important to wildlife ecologists. Allowance must be made for animals being lost (or censored) due to radio failure, radio loss, or emigration of the animal from the study area. The Kaplan-Meier procedure (Kaplan and Meier 1958), widely used in medical studies subject to censoring, can be applied to this problem. We developed a simple modification of the Kaplan-Meier procedure that allows for new animals to be added after the study has begun. We present 2 examples using telemetry data collected from northern bobwhite quail (Colinus virginianus) to show the simplicity and utility of the Kaplan-Meier procedure and its modifications. The log rank test used to compare 2 survival distributions can also be modified to allow for additions during the study. Simple computer programs that can be run on a personal computer are available from the authors. J. WILDL. MANAGE. 53(1):7-15 Radio-tagged animals are used to study survival. Present techniques for analyzing data from these studies assume that each survival event (typically an animal surviving a day) is independent and has a constant probability over all animals and all periods (Trent and Rongstad 1974, Bart and Robson 1982, Heisey and Fuller 1985). We believe these assumptions are often unrealistic and restrictive. White (1983) generalized discrete approaches using the same framework as that of band return models (Brownie et al. 1985) and he developed a flexible computer program (SURVIV) for use with his approach. Heisey and Fuller (1985) generalized the Trent and Rongstad (1974) approach to allow mortality from different causes (e.g., predation, starvation) and developed a microcomputer program called MICROMORT. Typically an animal's exact survival time (at least to within 1-2 days) is known unless that survival time is right censored (i.e., only known to be greater than some value). Pollock (1984) and Pollock et al. (1989) suggested a useful approach based on continuous survival models allowing right censoring that is widely used in medicine and engineering (Kalbfleisch and Prentice 1980, Cox and Oakes 1984) and provided examples of the Kaplan-Meier procedure. The Kaplan-Meier procedure does not require specification of a particular parametric continuous distribution; e.g., the exponential or Weibull. Related ecological papers using survival methods include Muenchow (1986), Pyke and Thompson (1986), Kurzejeski et al. (1987), and White et al. (1987). We present a simple description of the Kaplan-Meier procedure with an example using northern bobwhite quail survival data collected by PDC. We then generalize the Kaplan-Meier procedure to allow gradual (or staggered) entry of animals into the study. The calculations are illustrated with an example from the quail data. Finally, we present the log-rank test for comparison of survival distributions (modified for staggered entry of animals) with an example. We also present a discussion of model assumptions and directions for future research. We thank J. D. Nichols and W. L. Link for helpful comments on an earlier draft of this paper. We acknowledge G. C. White and D. M. Heisey for their helpful reviews that improved the final version. THE KAPLAN-MEIER OR PRODUCT LIMIT PROCEDURE The Kaplan-Meier or product limit estimator was developed by Kaplan and Meier (1958) and is d scussed by Cox and Oakes (1984:48) and Kalbfleisch and Prentice (1980:13). The survival function (S[t]) is the probability of an arbitrary animal in a population surviving t units of time from the beginning of the study. A nonparametric estimator of the survival function can be obtained by restricting ourselves to the discrete time points when deaths occur a1, a2, ..., ag. We define r, . . . , rg to be the numbers of an-