234
14 New Zealand Palaeodemography
to determine the annual rate of change r of a population, as this is a second-order
parameter that is found indirectly. One way to combat such a problem is to make
use of prior information.
Let S(x) denote the probability that a woman lives at least to age x years and F(x)
the mean number of children that she has had by such an age if she lives that long.
The value of x need not be an integer. There is a considerable literature indicating
that suitable functions involving only two (sometimes three) parameters can give
good fits to empirical data for S(x) (the survival function) and F(x) (the cumulative
fertility function).
Our approach utilizes such a pair of functions. This makes the use of information
about the general form of S, F in human populations and entails estimation from the
data of only five parameters. At the most basic level, our knowledge of the onset
of menarche means that for practical purposes we can take F(x) = 0 for x < 15.
This procedure has such further advantages as automatically smoothing data (which
would otherwise usually need to be performed as a further step) and obviating issues
of quadrature errors involved in discrete calculations with lifetime tables.
Since the parameters are in general correlated, there are in principle problems
in determining standard errors that reflect their joint values. However, the theory
of Fisher information supplies (for a given probability p of a woman surviving to
age 15) a joint approximately bivariate normal distribution for the two parameters
associated with S and a joint approximately trivariate normal distribution for the
three parameters associated with F.
Because distinguishing gender from a skeleton is difficult when death occurs
before age 15, the estimation of p is made from a pooled sample of males and
females.
An integral formula of Lotka linking S, F, p and q may be used to estimate r.
A standard error for the estimate is induced by the approximate bi- and trivariate
normal distributions associated with S, F and estimated from those distributions
using Monte Carlo methods.
Full details of the statistical analysis would be out of place in this book and are
published separately (Pearce et al. [21]).
Table 14.2 shows a spectrum of estimates for the population growth rate r associated with a range of infant mortality rates. All figures are presented as percentages.
We note that while our point estimates differ somewhat from those of Brewis et al.,
their estimates lie in the relevant 95% confidence intervals for r corresponding to
the relevant infant mortality level. We have in fact confirmed their skeletal-based
result of a decreasing population. Both our results, given in Table 14.2, and those
of Brewis et al. challenge Anderson’s late first settlement claim which depends on
consistently high population growth rates for the 600-year period covered by the
skeletal data. It is clear from Table 14.2 that there is a very low probability for even
a modest positive population growth rate occurring. To obtain such a growth rate
would further require an implausibly low level for infant mortality.
The point estimates in Table 14.2 belong to a larger possible range of decline
rates corresponding to various levels of infant mortality. They can be thought of as
representing eight options which create a context in which plausible bounds for a
14 New Zealand Palaeodemography
to determine the annual rate of change r of a population, as this is a second-order
parameter that is found indirectly. One way to combat such a problem is to make
use of prior information.
Let S(x) denote the probability that a woman lives at least to age x years and F(x)
the mean number of children that she has had by such an age if she lives that long.
The value of x need not be an integer. There is a considerable literature indicating
that suitable functions involving only two (sometimes three) parameters can give
good fits to empirical data for S(x) (the survival function) and F(x) (the cumulative
fertility function).
Our approach utilizes such a pair of functions. This makes the use of information
about the general form of S, F in human populations and entails estimation from the
data of only five parameters. At the most basic level, our knowledge of the onset
of menarche means that for practical purposes we can take F(x) = 0 for x < 15.
This procedure has such further advantages as automatically smoothing data (which
would otherwise usually need to be performed as a further step) and obviating issues
of quadrature errors involved in discrete calculations with lifetime tables.
Since the parameters are in general correlated, there are in principle problems
in determining standard errors that reflect their joint values. However, the theory
of Fisher information supplies (for a given probability p of a woman surviving to
age 15) a joint approximately bivariate normal distribution for the two parameters
associated with S and a joint approximately trivariate normal distribution for the
three parameters associated with F.
Because distinguishing gender from a skeleton is difficult when death occurs
before age 15, the estimation of p is made from a pooled sample of males and
females.
An integral formula of Lotka linking S, F, p and q may be used to estimate r.
A standard error for the estimate is induced by the approximate bi- and trivariate
normal distributions associated with S, F and estimated from those distributions
using Monte Carlo methods.
Full details of the statistical analysis would be out of place in this book and are
published separately (Pearce et al. [21]).
Table 14.2 shows a spectrum of estimates for the population growth rate r associated with a range of infant mortality rates. All figures are presented as percentages.
We note that while our point estimates differ somewhat from those of Brewis et al.,
their estimates lie in the relevant 95% confidence intervals for r corresponding to
the relevant infant mortality level. We have in fact confirmed their skeletal-based
result of a decreasing population. Both our results, given in Table 14.2, and those
of Brewis et al. challenge Anderson’s late first settlement claim which depends on
consistently high population growth rates for the 600-year period covered by the
skeletal data. It is clear from Table 14.2 that there is a very low probability for even
a modest positive population growth rate occurring. To obtain such a growth rate
would further require an implausibly low level for infant mortality.
The point estimates in Table 14.2 belong to a larger possible range of decline
rates corresponding to various levels of infant mortality. They can be thought of as
representing eight options which create a context in which plausible bounds for a