## Bayesian information criterion for censored survival models

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Venue: | Biometrics |

Citations: | 18 - 2 self |

### BibTeX

@ARTICLE{Volinsky_bayesianinformation,

author = {Chris T. Volinsky and Adrian E. Raftery},

title = {Bayesian information criterion for censored survival models},

journal = {Biometrics},

year = {}

}

### Years of Citing Articles

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### Abstract

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.

### Citations

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1043 | Bayes factors
- Kass, Raftery
- 1995
(Show Context)
Citation Context ...from 100 Splits . . . . . . . . . . . . . . . 11 i 1 Introduction The Bayesian framework for hypothesis testing uses Bayes factors to quantify the evidence for one hypothesized model against another (=-=Kass and Raftery 1995-=-). Schwarz (1978) derived the Bayesian Information Criterion (or BIC) as a large sample approximation to twice the logarithm of the Bayes factor. For a model M j parameterized by an m j -dimensional v... |

531 | The Statistical Analysis of Failure Time Data - Kalbfleisch, Prentice - 2002 |

352 |
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Citation Context ...of censored survival models. When censoring is present it is unclear whether the penalty in BIC should be n, the number of observations, or d, the number of events. When using the partial likelihood (=-=Cox 1972-=-) there are only as many terms in the partial likelihood as there are events d. Kass and Wasserman (1995) indicate that the term used in the penalty should be the rate at which the Hessian matrix of t... |

277 | Model selection and accounting for model uncertainty in graphical models using Occam's window
- Madigan, Raftery
- 1994
(Show Context)
Citation Context ...of any quantity of interest is a weighted average over the models considered. In several reported experiments with real data, BMA has yielded improved predictive performance and parameter estimation (=-=Madigan and Raftery 1994-=-; Madigan et al. 1995; Volinsky et al. 1997; Raftery et al. 1997). An S-PLUS function to do Bayesian model averaging for Cox regression models, bic.surv, is available from Statlib (lib.stat.cmu.edu/S/... |

277 | Bayesian model selection in social research
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(Show Context)
Citation Context ...Splits . . . . . . . . . . . . . . . 11 i 1 Introduction The Bayesian framework for hypothesis testing uses Bayes factors to quantify the evidence for one hypothesized model against another (Kass and =-=Raftery 1995-=-). Schwarz (1978) derived the Bayesian Information Criterion (or BIC) as a large sample approximation to twice the logarithm of the Bayes factor. For a model M j parameterized by an m j -dimensional v... |

197 | Bayesian Model Averaging for Linear Regression Model
- Raftery, Madigan, et al.
- 1997
(Show Context)
Citation Context ...sidered. In several reported experiments with real data, BMA has yielded improved predictive performance and parameter estimation (Madigan and Raftery 1994; Madigan et al. 1995; Volinsky et al. 1997; =-=Raftery et al. 1997-=-). An S-PLUS function to do Bayesian model averaging for Cox regression models, bic.surv, is available from Statlib (lib.stat.cmu.edu/S/bic.surv), and can also be obtained by sending the email message... |

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Citation Context ...odels can be compared via their predictive performance. The data were split in half to create a model building set and a validation set. The models are assessed using an analogue of Good's log score (=-=Good 1952-=-) called the partial predictive score (PPS) (Volinsky et al. 1997). For 50 different splits of the data, we calculated the difference in the PPS. Figure 2 (a) shows the histogram of these 50 differenc... |

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135 |
Partial likelihood
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Citation Context ...ed on the partial likelihood, namely PL(fi) = n Y i=1 / exp(x T i fi) P `2R i exp(x T ` fi) ! w i ; (11) where R i is the set of individuals at risk at time t i (often called the risk set) (Cox 1972; =-=Cox 1975-=-). Equation (11) assumes that there are no ties between the times at which deaths occur; when there are ties modifications are necessary, but for simplicity we do not consider this here. The parameter... |

126 |
Counting processes and survival analysis
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(Show Context)
Citation Context ...V.A. Lung Cancer trial (Prentice 1973; Kalbfleisch and Prentice 1980): a randomized clinical trial designed to assess a test chemotherapy treatment, ffl the Mayo Clinic PBC Data (Dickson et al. 1985; =-=Fleming and Harrington 1991-=-): a double blind randomized clinical trial studying the effect of DPCA on primary biliary cirrhosis of the liver, and ffl the Cardiovascular Health Study (Fried et al. 1991; Manolio et al. 1996): a l... |

70 |
A statistical paradox
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Citation Context ...ormation in one observation. The prior is vague enough since it is based on only one observation, yet it is proper and not too spread out, and so avoids well documented problems with improper priors (=-=Lindley 1957-=-; Spiegelhalter and Smith 1982). For a review of this and other reference priors, see Kass and Wasserman (1996). 3 Model Selection in Censored Survival Models 3.1 Model Selection Model selection crite... |

54 |
Nonparametric Bayesian analysis of survival time data
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(Show Context)
Citation Context ... for the approximation (12). The first is that if the prior forsis a diffuse gamma process, then, to a first order approximation, the partial likelihood is indeed the integral of the likelihood overs(=-=Kalbfleisch 1978-=-). The second justification is that the partial likelihood (11) actually becomes a full likelihood if a part of the data is discarded, namely the times at which deaths occur. It is a full likelihood f... |

42 |
Bayes Factors for linear and log-linear models with vague prior information
- Smith, Spiegelhalter
- 1982
(Show Context)
Citation Context ...e observation. The prior is vague enough since it is based on only one observation, yet it is proper and not too spread out, and so avoids well documented problems with improper priors (Lindley 1957; =-=Spiegelhalter and Smith 1982-=-). For a review of this and other reference priors, see Kass and Wasserman (1996). 3 Model Selection in Censored Survival Models 3.1 Model Selection Model selection criteria such as BIC are often used... |

41 | Formal Rules for Selecting Prior Distributions: A Review and Annotated Bibliography - Kass, Wasserman - 1996 |

36 | Marginal likelihood based on Cox’s regression and life model - Kalbfleisch, Prentice - 1973 |

30 | Bayesian model averaging in proportional hazard models: assessing the risk of a stroke - CT, Madigan, et al. - 1997 |

27 |
Eliciting prior information to enhance the predictive performance of Bayesian graphical models
- Madigan, Gavrin, et al.
- 1995
(Show Context)
Citation Context ...t is a weighted average over the models considered. In several reported experiments with real data, BMA has yielded improved predictive performance and parameter estimation (Madigan and Raftery 1994; =-=Madigan et al. 1995-=-; Volinsky et al. 1997; Raftery et al. 1997). An S-PLUS function to do Bayesian model averaging for Cox regression models, bic.surv, is available from Statlib (lib.stat.cmu.edu/S/bic.surv), and can al... |

22 |
Contributions to the theory of rank order statistics: Computation rules for probabilities of rank orders
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(Show Context)
Citation Context ...carded, namely the times at which deaths occur. It is a full likelihood for the part of the data consisting of the order in which individuals die and the risk sets, R i , corresponding to each death (=-=Savage 1957-=-; Kalbfleisch and Prentice 1973). When estimation for the Cox model is based on the partial likelihood, as is usually the case, standard likelihood theory is not directly applicable. However, Andersen... |

18 |
Exponential survivals with censoring and explanatory variables
- PRENTICE
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(Show Context)
Citation Context ...hich deaths occur. It is a full likelihood for the part of the data consisting of the order in which individuals die and the risk sets, R i , corresponding to each death (Savage 1957; Kalbfleisch and =-=Prentice 1973-=-). When estimation for the Cox model is based on the partial likelihood, as is usually the case, standard likelihood theory is not directly applicable. However, Andersen and Gill (1982) proved asympto... |

13 | A reference Bayesian test for nested hypotheses with large samples - Kass, Wasserman - 1995 |

13 | The Statistical Analysis of Failure Time Data - eisch, John, et al. - 1980 |

10 | The cardiovascular health study: Design and rationale - Fried, Borhani, et al. - 1991 |

9 | BE: Aspirin use and chronic diseases: a cohort study of the elderly. Br Med J - Paganini-Hill, Chao, et al. - 1989 |

3 | Nonparametric Bayesian analysis of survival time data - eisch, D - 1978 |

3 | Marginal likelihoods based on Cox's regression and life model - eisch, D, et al. - 1973 |

2 |
Trial of penicillamine in advanced primary biliary cirrhosis
- Dickson, Fleming, et al.
- 1985
(Show Context)
Citation Context ...s. They are: ffl the V.A. Lung Cancer trial (Prentice 1973; Kalbfleisch and Prentice 1980): a randomized clinical trial designed to assess a test chemotherapy treatment, ffl the Mayo Clinic PBC Data (=-=Dickson et al. 1985-=-; Fleming and Harrington 1991): a double blind randomized clinical trial studying the effect of DPCA on primary biliary cirrhosis of the liver, and ffl the Cardiovascular Health Study (Fried et al. 19... |

2 |
Short-term predictors of incident stroke in older adults: The Cardiovascular Health Study
- Manolio, Kronmal, et al.
- 1996
(Show Context)
Citation Context ...eming and Harrington 1991): a double blind randomized clinical trial studying the effect of DPCA on primary biliary cirrhosis of the liver, and ffl the Cardiovascular Health Study (Fried et al. 1991; =-=Manolio et al. 1996-=-): a longitudinal observational study on risk factors for cardiovascular health in the elderly U.S. population. For each of the studies, we found both the overall and the uncensored unit information p... |