Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory, reliability analysis or reliability engineering in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival? To answer such questions, it is necessary to define "lifetime". In the case of biological survival, death is unambiguous, but for mechanical reliability, failure may not be well-defined, for there may well be mechanical systems in which failure is partial, a matter of degree, or not otherwise localized in time. Even in biological problems, some events (for example, heart attack or other organ failure) may have the same ambiguity. The theory outlined below assumes well-defined events at specific times; other cases may be better treated by models which explicitly account for ambiguous events. More generally, survival analysis involves the modelling of time to event data; in this context, death or failure is considered an "event" in the survival analysis literature – traditionally only a single event occurs for each subject, after which the organism or mechanism is dead or broken. Recurring event or repeated event models relax that assumption. The study of recurring events is relevant in systems reliability, and in many areas of social sciences and medical research.
Introduction to survival analysis Survival analysis is used in several ways:
To describe the survival times of members of a group Life tables Kaplan–Meier curves Survival function Hazard function To compare the survival times of two or more groups Log-rank test To describe the effect of categorical or quantitative variables on survival Cox proportional hazards regression Parametric survival models Survival trees Survival random forests
Definitions of common terms in survival analysis The following terms are commonly used in survival analyses:
Event: Death, disease occurrence, disease recurrence, recovery, or other experience of interest Time: The time from the beginning of an observation period (such as surgery or beginning treatment) to (i) an event, or (ii) end of the study, or (iii) loss of contact or withdrawal from the study. Censoring / Censored observation: Censoring occurs when we have some information about individual survival time, but we do not know the survival time exactly. The subject is censored in the sense that nothing is observed or known about that subject after the time of censoring. A censored subject may or may not have an event after the end of observation time. Survival function S(t): The probability that a subject survives longer than time t.
Example: Acute myelogenous leukemia survival data This example uses the Acute Myelogenous Leukemia survival data set "aml" from the "survival" package in R. The data set is from Miller (1997) and the question is whether the standard course of chemotherapy should be extended ('maintained') for additional cycles. The aml data set sorted by survival time is shown in the box.
Time is indicated by the variable "time", which is the survival or censoring time Event (recurrence of aml cancer) is indicated by the variable "status". 0 = no event (censored), 1 = event (recurrence) Treatment group: the variable "x" indicates if maintenance chemotherapy was given The last observation (11), at 161 weeks, is censored. Censoring indicates that the patient did not have an event (no recurrence of aml cancer). Another subject, observation 3, was censored at 13 weeks (indicated by status=0). This subject was in the study for only 13 weeks, and the aml cancer did not recur during those 13 weeks. It is possible that this patient was enrolled near the end of the study, so that they could be observed for only 13 weeks. It is also possible that the patient was enrolled early in the study, but was lost to follow up or withdrew from the study. The table shows that other subjects were censored at 16, 28, and 45 weeks (observations 17, 6, and 9 with status=0). The remaining subjects all experienced events (recurrence of aml cancer) while in the study. The question of interest is whether recurrence occurs later in maintained patients than in non-maintained patients.
Kaplan–Meier plot for the aml data The survival function S(t), is the probability that a subject survives longer than time t. S(t) is theoretically a smooth curve, but it is usually estimated using the Kaplan–Meier (KM) curve. The graph shows the KM plot for the aml data and can be interpreted as follows:
The x axis is time, from zero (when observation began) to the last observed time point. The y axis is the proportion of subjects surviving. At time zero, 100% of the subjects are alive without an event. The solid line (similar to a staircase) shows the progression of event occurrences. A vertical drop indicates an event. In the aml table shown above, two subjects had events at five weeks, two had events at eight weeks, one had an event at nine weeks, and so on. These events at five weeks, eight weeks and so on are indicated by the vertical drops in the KM plot at those time points. At the far right end of the KM plot there is a tick mark at 161 weeks. The vertical tick mark indicates that a patient was censored at this time. In the aml data table five subjects were censored, at 13, 16, 28, 45 and 161 weeks. There are five tick marks in the KM plot, corresponding to these censored observations.
Life table for the aml data A life table summarizes survival data in terms of the number of events and the proportion surviving at each event time point. The life table for the aml data, created using the R software, is shown.
The life table summarizes the events and the proportion surviving at each event time point. The columns in the life table have the following interpretation:
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