Page 151 - 20dynamics of cancer

136 CHAPTER 7

7.4 Weibull and Gompertz Models

PR ´ ECIS
Demographers and engineers use Weibull and Gompertz models to
describe age-specific mortality and failure rates. A simple form of the
Weibull model assumes that failure rates versus age fit a straight line on
log-log scales. This matches the simplest multistage model of progres-
sion dynamics under the assumption that log-log acceleration remains
constant over all ages.
The advantage of the Weibull model is that it makes no assumptions
about underlying process, and allows one to reduce data description to
the two parameters of slope and intercept that describe a line. Compari-
son between data sets can be made by comparing the slope and intercept
estimates.
The disadvantage of the Weibull model is that, because it is a de-
scriptive model that makes no assumptions about underlying process,
one cannot easily test hypotheses about how particular factors affect
the processes of progression. I prefer an explicit underlying model of
progression dynamics. In some cases, such as the simplest multistage
model, the solution based on explicit assumptions about progression
leads to an approximate Weibull model.
The common form of the Gompertz model arises by assuming a con-
stant value for the slope of incidence versus age on log-linear scales:
that is, logarithmic in incidence and linear in age. The advantages and
disadvantages for the Weibull model also apply to the Gompertz model.

DETAILS
The Weibull model describes age-specific failure rates. Engineers use
the Weibull model to analyze time to failure for complex control sys-
tems, particularly where system reliability depends on multiple sub-
components. Multicomponent failure models have a close affinity to
multistage models of disease progression. Demographers also use the
Weibull model to describe the rise in age-specific mortality rates with
increasing age.
Both engineers and demographers have observed that the Weibull
model provides a good description of age-specific failure rates in many