# Survival analysis, part 1: Introduction

When time to an event (eg, how long until an event occurs?) is the main outcome of interest in a trial, the relevant data are often called survival data. The survival time for each participant is the interval from a predetermined start point, such as entry into the study, until the occurrence of the event of interest (eg, lingual retainer failure). Participants can have the same survival time, but they may have been enrolled at different calendar times. Survival analysis can be applied to various situations such as time to death from any cause after treatment for cancer, time to finish an orthodontic treatment, time to achieve initial orthodontic alignment, time to bracket failures, time to eruption of impacted canines, and so on. Survival analysis is suitable when the risk of an event varies with time: ie, lingual retainer or bracket failures may be higher early in posttreatment rather than later.

## Why use survival analysis?

• Recruitment at different dates, but the study end is the same for all patients

• Patients may be lost to follow-up, but they still contribute information for the time before they were lost

• Risk of an event varies with time

• Patients might experience the event from other causes (competing risk)

## Rates

As previously discussed, the rate quantifies the number of new events per person per unit of time (person-time) :

rate = number of ( new ) events total person-years of observation

It is interpreted as the probability that a participant will experience the event during a specified period of time. Usually, it applies to nonrecurrent events or to the first episode of an event. The rate indicates the average rate (or speed) at which a participant experiences the event in a given population at risk. The rate has units such as the number of events per person-time; it can be measured in months, years, and so on, and persons can be counted per 100 or 1000. For example, if the number of lingual retainer failures in adolescent patients is on average 20 per year in a population of 100 persons, then:

rate = 20 100 * 1 = 0.2 events per person-years = 1 event per 5 person-years

In other words, for every 5 persons followed over a 1-year period, we are expecting 2 retainer failures.

Similarly, if we follow 100 patients with fixed lingual retainers for 5 years, and 40 patients experience the event (any type of first-time failure) in that period, we can say that the proportion of broken retainers (risk of retainer failure) for this cohort is 40/100 = 0.4 (proportion).

If our interest is in events over particular time periods (rates), we need to know the times of the retainer failure so that we can calculate the follow-up time from the beginning until the occurrence of the event. The rate for the previous example would be:

λ = 40 100 · sum of follow up

That is:

rate = 40 100 * 5 = 0.08 events per person-years = 8 events per 100 person-years