Time to Event

by Steve Simon, PhD

There are two features of survival models.

First is the process of measuring the time in a sample of people, animals, or machines until a specific event occurs. In fact, many people use the term “time to event analysis” or “event history analysis” instead of “survival analysis” to emphasize the broad range of areas where you can apply these techniques.

Second is the recognition that not everyone/everything in your sample will experience the event. Those not experiencing the event, either because the study ended before they had the event or because they were lost to follow-up, are classified as censored observations.

[Read more…] about What is Survival Analysis and When Can It Be Used?

Time to event analyses (aka, Survival Analysis and Event History Analysis) are used often within medical, sales and epidemiological research. Some examples of time-to-event analysis are measuring the median time to death after being diagnosed with a heart condition, comparing male and female time to purchase after being given a coupon and estimating time to infection after exposure to a disease.

Survival time has two components that must be clearly defined: a beginning point and an endpoint that is reached either when the event occurs or when the follow-up time has ended.

One basic concept needed to understand time-to-event (TTE) analysis is censoring.

In simple TTE, you should have two types of observations:

1. The event occurred, and we are able to measure when it occurred OR

2. The event did NOT occur during the time we observed the individual, and we only know the total number of days in which it didn’t occur. (CENSORED).

Again you have two groups, one where the time-to-event is known exactly and one where it is not. The latter group is only known to have a certain amount of time where the event of interest did not occur. We don’t know if it would have occurred had we observed the individual longer. But knowing that it didn’t occur for so long tells us something about the risk of the envent for that person.

For example, let the time-to-event be a person’s age at onset of cancer. If you stop following someone after age 65, you may know that the person did NOT have cancer at age 65, but you do not have any information after that age.

You know that their age of getting cancer is greater than 65. But you do not know if they will never get cancer or if they’ll get it at age 66, only that they have a “survival” time greater than 65 years. They are censored because we did not gather information on that subject after age 65.

So one cause of censoring is merely that we can’t follow people forever.  At some point you have to end your study, and not all people will have experienced the event.

But another common cause is that people are lost to follow-up during a study. This is called random censoring. It occurs when follow-up ends for reasons that are not under control of the investigator.

In survival analysis, censored observations contribute to the total number at risk up to the time that they ceased to be followed.  One advantage here is that the length of time that an individual is followed does not have to be equal for everyone. All observations could have different amounts of follow-up time, and the analysis can take that into account.

Allison, P. D. (1995). Survival Analysis Using SAS. Cary, NC: SAS Institute Inc.

Hosmer, D. W. (2008). Applied Survival Analysis (2nd ed.). Hoboken, NJ: John Wiley & Sons, Inc.