Introductory Biostatistics

In other words, the hazard or risk functionlðtÞapproximates the proportion of
subjects dying or having events per unit time around timet. Note that this dif-
fers from the density function represented by the usual histogram; in the case of
lðtÞ, the numerator is aconditionalprobability.lðtÞis also known as theforce
of mortalityand is a measure of the proneness to failure as a function of the
person’s age. When a population is subdivided into two subpopulations,E
(exposed) andE^0 (nonexposed), by the presence or absence of a certain char-
acteristic (an exposure such as smoking), each subpopulation corresponds to a
hazard or risk function, and the ratio of two such functions,

RRðtÞ¼

lðt;EÞ

lðt;E^0 Þ

is called therelative riskassociated with exposure to factor E, risk of the
exposured subjectsrelativeto the risk of nonexposed subjects. In general, the
relative risk RRðtÞis a function of time and measures the magnitude of an
e¤ect; when it remains constant, RRðtÞ¼r, we have aproportional hazards
model(PHM):

lðt;EÞ¼rlðt;E^0 Þ

with the risk of the nonexposed subpopulation served as the baseline. This is a
multiplicative model; that is, exposure raises the risk by a multiplicative con-
stant. Another way to express this model is

lðtÞ¼l 0 ðtÞebx

Figure 11.2 General form of a survival curve.

382 ANALYSIS OF SURVIVAL DATA