Survival analysis is concerned with finding models to predict the survival of patients or to assess the efficacy of a clinical treatment. A key part of the model-building process is the selection of the predictor variables. It is standard to use a stepwise procedure guided by a series of significance tests to select a single model, and then to make inference conditionally on the selected model. However, this ignores model uncertainty, which can be substantial. We review the standard Bayesian model averaging solution to this problem and extend it to survival analysis, introducing partial Bayes factors to do so for the Cox proportional hazards model. In two examples, taking account of model uncertainty enhances predictive performance, to an extent that could be clinically useful. 1 Introduction From 1974 to 1984 the Mayo Clinic conducted a double-blinded randomized clinical trial involving 312 patients to compare the drug DPCA with a placebo in the treatment of primary biliary cirrhosis...

We investigate the Bayesian Information Criterion (BIC) for variable selection in models for censored survival data. Kass and Wasserman (1995) showed that BIC provides a close approximation to the Bayes factor when a unit-information prior on the parameter space is used. We propose a revision of the penalty term in BIC so that it is de ned in terms of the number of uncensored events instead of the number of observations. For the simplest censored data model, that of exponential distributions of survival times (i.e. a constant hazard rate), this revision results in a better approximation to the exact Bayes factor based on a conjugate unit-information prior. In the Cox proportional hazards regression model, we propose de ning BIC in terms of the maximized partial likelihood. Using the number of deaths rather than the number of individuals in the BIC penalty term corresponds to a more realistic prior on the parameter space, and is shown to improve predictive performance for assessing stroke risk in the Cardiovascular Health Study.

This research is concerned with developing improved methods for analyzing survival data and determining appropriate sample sizes for cohort studies. The model proposed for the hazard function incorporating covariables is a polynomial with different functions of the covariables as coefficients of the various powers of time. This model does not require the assumption that the hazards for different individuals be in constant ratio over time, and it allows for testing whether this assumption is reasonable. The model is parametric, which allows for easy specification of the survival curve and interpretation of results. At the same time, it is general enough so that the form of the hazard is not unduly restricted. Methods for fitting the model to data, testing hypotheses about

of possible prognostic importance, were related to the survival of the patients, using regression models (Kay, 1977). For most tumour sites except the testis, the distributions of modal DNA values were bimodal, with peaks at the diploid level and in the triploid-tetraploid range. For all tumour sites except the cervix uteri, patients in the low (near-diploid) range showed better survival; the reverse was true for squamouscell carcinoma of the cervix uteri. Other variables showed the following effects: for all sites except the testis, younger patients showed a better survival; for the cervix and corpus uteri, breast and ovary, increasing clinical stage was associated with poorer survival. Where evaluated, histological grade appeared to be associated with survival rate, the less well differentiated tumours having a worse prognosis, except for the breast, where the reverse correlation was noted. For carcinoma of the bladder, females had a poorer survival rate than males. MEASUREMENTS of nuclear DNA content have been widely used to estimate the approximate modal chromosome numbers

We consider maximum

This paper is concerned with the Bayesian analysis of failure time data in the presence of covariate information. In this paper, the focus will be on the semiparametric Bayesian model. The semiparametric nature of the models allows considerable generality and applicability but enough structure for useful physical interpretation and understanding particular applications in the medical research. The popularity of semiparametric approches for analyzing univariate survival data begins with the seminal paper of Cox(1972) on the proportional hazards model. This model assumes a constant relative risk compared to the baseline hazard function for all values of the failure time given the covariate values. When the assumption is violated, the proportional odds model provides an alternative. In this model the log-odds of the failure time distribution depends linearly on the covariates. There is a temptaion to use proportional hazards model to analyse failure time observations without any formal model checking, even when the model doesnot fit the data well due to its large sample inference properties (Andersen ans Gill, 1982) and easy access to statistical software for this model. In this paper we introduce a general class of models which contains both proportional hazards and odds model. The semiparametric nature of this model creates the flexibility to fit the data well. Also we will propose a criterion to choose between proportional hazards and proportional odds model. Furthermore, a simple extension to the model allows us to include frailties (Clayton and Cuzick, 1985) for multivariate survival data. For nice discussion of Bayesian semiparametric models see Gelfand, 1996 and for Bayesian semiparametric analysis of survival data see Sinha and Dey, 1998. Let T 1 , . . . , T n be f...

The mean residual life function (mrlf) of a subject is defined as the expected remaining lifetime of the subject given that the subject has survived up to a given time. The commonly used regression models as proportional mean residual life (PMRL) and linear mean residual life (LMRL) have limited applications due to adhoc restriction on the parameter space. The regression model we propose does not have any constraints. It turns out that the proposed proportional scaled mean residual life (PSMRL) model is equivalent to the accelerated failure time (AFT) model, which provides an alternative way to estimate the regression parameters of the AFT model and to interpret the regression parameters estimated from the AFT model in terms of the mrlf instead of the survival function. We use full likelihood by nonparametrically estimating the baseline mrlf using the smooth scale mixture estimator of the mrlf based on a single sample of iid observations to develop the statistical inference for the regression parameters, which are estimated using an iterative procedure. A simulation study is carried out to assess the properties of the estimators of the regression coefficients. We illustrate our regression model by applying it to the well-known Veteran’s Administration lung cancer data.

Methods introduced by Lagakos for incorporating infonnation from a time-dependent covariable (an intervening event) into the analysis of failure times are generalized. The research was motivated by two follow-up studies--one involving industrial workers where dis-I ability retirement is the intervening event and the other, patients with coronary artery disease where the first non-fatal myocardial infarctiOn after diagnosis for the disease is the intervening event. The generalized model

this paper, we propose an alternative approach. We condition the likelihood function on the estimated probability of right-censoring by time t, and obtain regression parameter estimates by maximizing this conditional likelihood function. We show that maximum conditional likelihood estimates (MCLE) are consistent and asymptotically normal. Simulation studies show that the MCLE method is favorably compared to the Jung's modification. As an application of the MCLE method, we propose a graphical method of testing the proportional hazard assumption. We also apply the MCLE and the Cox regression method to predicting survival probabilities for a given set of prognostic factors, and compare predicted survival curves. In one data set where the proportional hazards model is reasonable, the two methods produce almost identical curves. But in another survival data where the data do not follow the proportional hazard assumption, two predicted curves are not the same. A simulation 1 study demonstrates a case where the MCLE method is more powerful than the Cox regression method. KEY WORDS: Logistic regression, survival probability, maximum conditional likelihood estimation, test of proportional hazard assumption 2 1. INTRODUCTION In survival data analysis, suppose we are interested in relating the survival probability at time t to a given set of covariates. If there is no censored observation before time t, then we can apply a logistic regression method. But when some observations are censored before time t, we cannot apply the logistic regression method to the survival data. The censoring mechanism is usually assumed to be independent of the survival time itself. Such an assumption is reasonable, for example, when the survival time is administratively-censored. Under the independ...

V ALCINDA WANGSNESS LEWIS. The Burr Distribution as a General Parametric Family in Survivorship and Reliability Theory Applications. (Under the direction of MICHAEL J. SYMONS) In the analysis of survival time studies a parametric form of the survival distribution is commonly assumed. The choice of the parametric form of the survival distribution is an important issue which should be given careful consideration. The Burr Type XII distribution has as special limiting cases of its parameter values the Weibull and the exponential distributions, two commonly used survival distributions. A generalized likelihood ratio test, based upon maximum likelihood procedures, is applied to assess the goodness-of-fit of the two special