# Survival Analysis

### Survival Analysis

This is a set of statistical approaches used to investigate the time it takes for an event we are interested in to occur. This concept  is  used in a variety of fields like:

• Patients survival time analyses in cancer studies
• Event history analysis in sociology
• Failure-time analysis in engineering

In cancer studies, survival analysis is used to provide answers to questions like:

• What are the chances that a patient survives 5 years?
• What is the difference in survival between groups of patients?
• What impact do certain clinical characteristics have on a patient’s survival?

Most survival analyses in cancer studies use the following methods:

• Kaplan-Meier plots – This method is used to visualize survival curves
• Log-rank test – it compares the survival curves of two or more groups
• Cox proportional hazards regression – describes the effect variables have on survival.

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The basics concepts and fundamental terms in survival analysis

• Survival time and type of events

Cancer studies have different types of events. Some of them include:

• Relapse
• Progression
• Death

Survival time can be defined as the time from response to treatment to the occurrence of the event of interest. The time of death and the relapse-free survival time are the two most important measures in cancer studies. Relapse free survival time corresponds to the time between response to treatment and recurrence of the disease. It is sometimes also referred to as event-free survival time or disease-free survival time.

Characteristics of survival time

• It is usually continuous
• It is incompletely determined for some subjects. This means that we can know that for some subjects, the survival time was at least equal to time t, whereas, for others, we can know the exact time of the event.
• Survival time is always greater than or equal to zero.

Issues in analysis

Standard regression procedures could be used if there is no censoring. However, the procedures may be inadequate because:

• Time of an event has a skewed distribution and is restricted to be positive
• Sometimes, the scientists are more interested in surviving past a certain point than the expected time of the event
• The hazard function can be more effective than linear regression in lending more insight into the failure of the mechanism
• Censoring

As we stated earlier, survival analysis gives attention to the expected duration of time until the event of interest occurs (relapse of death). The event, however, may not be observed for some patients within the study period. This will produce censored observations.

Mentioned below are the ways censoring may arise:

• The event of interest  such as death or relapse has not yet been experienced by the patient within the study period
• Follow-up is not done on the patient during the study period
• Follow up is made impossible because the patient experiences a different event

The examples above are all right censoring. They are handled in survival analysis.

Types of right censoring

• Fixed type I censoring
• Random type I censoring
• Type II censoring

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• Survival and Hazard Functions

Hazard probability and survival probability are used to describe survival data.

• Survival probability – It is also known as the survivor function S(t). It is the probability that a patient survives from the time of diagnosis of cancer to a specified future time t.
• Hazard probability – It is usually denoted by h(t). It is the probability that a patient under observation at a specified time t has an event at that time. Unlike the survival probability that focusses on not having the event, hazard probability focusses on the event occurring.

The Kaplan-Meier Survival Estimate

(Kaplan and Meier, 1958), defines the Kaplan Meier survival estimate as a non-parametric method that is used to approximate the survival probability from observed survival times. It is also known as the Kaplan-Meier (KM) method and is calculated in the following way:

S(ti) is the survival probability at  time ti

S(ti) = S(ti-1)( 1 – di/ni)

In the formula:

• S(ti) –The probability of being alive at ti-1
• ni – The number of patients alive before ti
• di – This is the number of events at ti
• t0 = 0, S(0) = 1

S(ti) is the estimated probability. It is a step function that changes value only at the time of each event. We can also compute the confidence interval for the survival probability. A plot of the KM survival curve against time is the KM survival curve. It provides a useful summary of data that can be used to estimate measures such as the median survival time.