# How to interpret the output of survival analysis?

I am beginner of survival analysis, so I am confused about some basic ideas in this area.

I think the basic assumption in survival analysis is that time of event for a subject can be estimated given all relevant time-invariable attributes of subjects. Therefore, ideally, if we can regress over the failure time correctly, we can then predict the expected failure time of subjects.

Since regression is hard, researchers transformed it to ranking problem. That is, they want to rank subjects according to their failure time. Furthermore, the performance of ranking is measured by concordance index (c-index).

My question is, "Even with a perfect ranking list (larger score for longer survival time), how can we determine the failure risk of each subject at time T?" @mbq, @ocram :)

• 1) Why is regression "hard"? 2) The ability to determine failure risk at time T depends on the type of survival analysis you did. The most common type is Cox proportional hazards, where the goal is usually to estimate the hazard $ratio$ rather than the risk at a particular time. What model did you use? 3. One big issue in survival analysis (maybe the biggest) is censoring. What did you do about that? – Peter Flom Jun 1 '12 at 10:58
• In addition to what Peter said I don't know where you get the idea that survival analysis is based on ranking the failure times. The focus can be on the hazard rate but usually it is on the survival function. So what you estimate is S(t)= P[X>t]. The probability that a randomly chosen member of the population the sample was taken from X is greater than t for all t in a reasonable range of times. Time is measured from an entry point which can be different in clinical trials for example because the enrollment time varies among patients. – Michael R. Chernick Jun 1 '12 at 15:23
• The survival or more generally speaking the time to event is measure from that entry pont and not from an absolute time scale. Now the data consists of the time to event. This is nice and clean if we can observe patients long enough for everyone to have an event. But often we can't. So the cases that don't have an event over the time we observe them are said to be right censored and the time until we stop observing them is useful information and is called their censoring time. – Michael R. Chernick Jun 1 '12 at 15:28
• Given this information the product limit estimator (also called the Kaplan-Meier estimator) can be used to get a nonparametric estimate of the curve. Methods are also available to get confidence intervals for this estimate at fixed time points. Ranking tests such as the logrank test and the Wilcoxon test can be used to compare two survival curves to determine if they differ. This may be where you are getting the notion of ranking. Also as Peter mentioned there is a model called the Cox proportional hazards model that introduces covariates as factors affecting the survival curve. – Michael R. Chernick Jun 1 '12 at 15:32
• The Cox model uses the same data and the covariate values for each subject to fit a survival curve that is obtained by estimating regression parameters using what Cox called the partial likelihood function. There are also fully parametric models such as the negative exponential and the Weibull family that can be fit to such data. That is survival analysis in a nutshell. Go to the literature and textbooks for more detailed information. – Michael R. Chernick Jun 1 '12 at 15:36

## 1 Answer

To summarize our comments which answer your question, The best thing to do is to either fit survival curves non-parametrically and compare them using log rank or a similar test or fit a parametric or semiparametric regression model such as the Cox model if hazard are proportional. From the models you can obtain hazard ratios (relative risk) or odds ratios and get confidence intervals for them. Good references for this are the book by Kalbfleisch and Prentice (Wiley) and Therneau and Grambsch (Springer).