Biology Reference
In-Depth Information
indicators in a target sample against age categories in a reference sample in order to esti-
mate proportions of deaths by age category for the target, in what we (
Konigsberg and
Frankenberg, 2002
) have called “contingency table paleodemography.” Although there
have been major new developments in these methods (
Bocquet-Appel and Masset, 1996;
Eshed et al., 2004; Bonneuil, 2005; Bocquet-Appel and Bacro, 2008
;
Caussinus and Cour-
geau, 2010
) the explicit use of hazard models (see section below) has increased (
Boldsen
et al., 2002; Hoppa and Vaupel, 2002b
;
Konisgberg and Herrmann, 2002
;
M ¨ ller et al.,
2002; Wood et al., 2002; DeWitte and Wood, 2008; Gage, 2010; Redfern and DeWitte, 2011
)
and is starting to supplant life tables. We focus here on the estimation of age-at-death
and of the parameters of hazard models, leaving coverage of life tables to a previous publi-
cation (
Frankenberg and Konigsberg, 2006
).
Hazard Models
Hazard models are a class of statistical models that specify the time until particular events
occur. In our case, the event of interest is the death of individuals, and so the times until the
events (deaths) are the ages-at-death. Hazard models are simply a tool that can be used to
represent a continuous age-at-death distribution using a relatively small number of param-
eters, where the parameters are numbers that characterize the age-at-death distribution. We
start with the simplest of possible models: an exponential hazard. In the exponential hazard
model the survivorship to age t is
SðtjlÞ¼expðltÞ;
(11.12)
where
is the instantaneous hazard of death (risk of dying), which is constant across age.
This is a completely unreasonable model for most human mortality, but it serves as a good
starting point. Note that at age zero we have the exponentiation of zero, which is one, so
the survivorship at the initial age is 1.0. The distribution of ages-at-death will be
l
f
ðtjlÞ¼l expðltÞ:
(11.13)
l ¼
0.34. (There is no logic behind
choosing this particular value except that 34 is LWK's favorite number.) In the exponential
hazard model the mean age-at-death is
1=l
As the example in this section we will use a hazard of
,so
we have about three years for the mean age-at-death and about two years for the median
age-at-death. Again, this is a completely unreasonable model for human mortality, but if
you persevere we will get to more reasonable models.
Figure 11.3
shows a simulation of 10,000 deaths from the exponential hazard model,
where the histogram represents the simulated ages-at-death and the dashed curve is the
expected distribution given the hazard parameter of 0.34. The exponential hazard model
represents a constant hazard of death, and as such the hazard does not change across
age. This model is sometimes used to represent a constant “baseline” hazard in a five-
parameter model known as the Siler model (
Siler, 1979; Gage and Dyke, 1986
). The Siler
model has three components of mortality: (1) juvenile mortality represented by a negative
Gompertz model, (2) senescent (old-age) mortality represented by a positive Gompertz
model, and (3) a baseline age-independent hazard represented by the exponential model.
while the median age-at-death is
lnð2Þ=l
