Introductory Biostatistics

11.1 SURVIVAL DATA

In prospective studies, the important feature is not only the outcome event,
such as death, but the time to that event, thesurvival time. To determine the
survival timeT, three basic elements are needed:

1. A time origin or starting point


2. An ending event of interest


3. A measurement scale for the passage of time

These may be, for example, the life spanTfrom birth (starting point) to death
(ending event) in years (measurement scale) (Figure 11.1). The time origin or
starting point should be defined precisely, but it need not be birth; it could be
the start of a new treatment (randomization date in a clinical trial) or admis-
sion to a hospital or nursing home. The ending event should also be defined
precisely, but it need not be death; a nonfatal event such as the relapse of a
disease (e.g., leukemia) or a relapse from smoking cessation or discharge to the
community from a hospital or nursing home satisfy the definition and are
acceptable choices. The use of calendar time in health studies is common and
meaningful; however, other choices for a time scale are justified—for example,
hospital cost (in dollars) from admission (starting point) to discharge (ending
event).
Distribution of the survival timeTfrom enrollment or starting point to the
event of interest, considered as a random variable, is characterized by either
one of two equivalent functions: the survival function and the hazard function.
Thesurvival function, denotedSðtÞ, is defined as the probability that a person
survives longer thantunits of time:

SðtÞ¼PrðT>tÞ

SðtÞis also known as thesurvival rate; for example, if times are in years,Sð 2 Þis
the two-year survival rate,Sð 5 Þis the five-year survival rate, and so on. A
graph ofSðtÞversustis called asurvival curve(Figure 11.2).
Thehazardorrisk functionlðtÞgives theinstantaneousfailure rate and is
defined as follows. Assuming that a typical patient has survived to timet, and
for a small time incrementd, the probability of an event occurring during time
intervalðt;tþdÞto that person is given approximately by

lðtÞdFPrðtaTatþdjtaTÞ

Figure 11.1 Survival time.

SURVIVAL DATA 381