Keyfitz N., Caswell H. Applied Mathematical Demography

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19.3. Application to Mortality 485

and this last is bound to be true unless µ

= µ

.Wecanproveitifµ

= µ

by trying ﬁrst µ

>µ

,thenµ

<µ

. Thus the mixed population in

question with ﬁxed death rates in each of its two subpopulations will show

a spurious fall in its death rate over time.

19.3.3 Numerical Eﬀect on Mortality

Since in its very nature unobserved heterogeneity cannot be measured sta-

tistically, it is not clear how we are to put to practical use the unquestioned

fact that individuals vary in their probability of dying (or of becoming sick,

divorcing, etc.). A person being either alive or dead, neither before nor after

he dies is there any way of ascertaining his individual probability. All the

facts on numbers exposed and dying contribute no information on individ-

ual variability and we are forever unable to make a life table for a single

individual, except in the trivial ex post sense that his probability was unity

of living to any age up to his death, and from then on was zero. Selection

enters whenever we make a table for a group.

Yet we can consider the individual as the limiting case of a group. Sup-

pose we move from the unselected population to those who are active and

at work, to those who jog every day, etc. By projecting the sequence we

obtain at least the order of magnitude of the probability for the individual

least likely to die. And in the other direction, we can move from the uns-

elected to the disabled, sick, in hospital, etc., to ﬁnd the individual most

likely to die.

Without empirical materials we can at least try out a hypothetical degree

of heterogeneity. Suppose the population in three homogeneous groups, one

with standard mortality, one with mortality of 20 percent of the standard,

and the third with 180 percent of the standard, all taken in relation to a

current life table. This may seem like a wide range, but it corresponds to

a range of expectations of 65.3 to 81.8 years; that is not unbelievable.

Consider then that there is within a population a group whose life expec-

tation is 81.8 years, another whose expectation is 72.10 years, a third whose

expectation is 65.3 years, and suppose that these three groups are initially

numerically equal. Under these conditions, the expectation as observed is

73.09, against 72.10 for the true expectation, i.e., the expectation of the

average individual. The life table as calculated in disregard of heterogeneity

exaggerates expectation by 0.99 years.

Thus a moderate conclusion from what we know now is that the ob-

served expectation for ordinary populations calculated by the usual life

table methods (Chapter 2) is high by about a year in application to the in-

dividual of average frailty at birth. We expect the eﬀect to be much greater

for mobility, divorce, or morbidity, since in these the individual is not re-

stricted to the outcomes of 0 or 1 as for living or dead, but can move, or

divorce, or become sick 0, 1, 2, . . . times.

486 19. Heterogeneity and Selection in Population Analysis

19.4 Modelling Heterogeneity

While measured heterogeneity can be incorporated into the life cycle

graph and made part of a demographic model, unmeasured heterogeneity

(sometimes called unobserved or unmeasurable) confronts the analyst with

including the eﬀects of something that he or she hasn’t seen. Not surpris-

ingly, this can be diﬃcult. This section summarizes some of the approaches

used and some of the problems that arise. For more details and extensive

references, see Trussel and Rodr´ıguez (1990), Vaupel (1990), Yashin et al.

(1994), Yashin and Iachine (1997), Vaupel and Yashin (1999), and Link et

al. (2002). We will speak in terms of mortality, but the principles apply to

any kind of transition from one state to another.

19.4.1 Continuous Distribution of Frailty

Think of a standard individual, whose frailty at age zero is the mean of that

of the population at age zero, and in the ﬁrst round disregard any changes

of frailty over the lifetime of a given person (Vaupel and Yashin, 1985). Call

the frailty of that standard individual unity, and the frailty of any other

person z,sothatwithµ(x) dx the conditional probability of dying between

age x and x+dx for the standard individual, the chance for the individual of

frailty z is zµ(x). We express this assumption as µ(x|z)=zµ(x). To suppose

the frailty ﬁxed through life for any individual simpliﬁes the mathematics

without committing us to any stand on the relative eﬀects of heredity and

environment in the action of mortality.

If for individuals the mortality is zµ(x) then the observed rate at birth

¯µ(0) depends on the distribution of frailty, say f

(z):

¯µ(0) =



(z)zµ(0) dz



(z) dz

=¯z(0)µ(0). (19.4.1)

Following Vaupel and Yashin (1985) we will henceforth use a bar for the

observed mortality, and the unbarred µ for the standard or initially average

individual.

For a later age x the distribution of frailty is no longer f

(z) but

(z)l(x|z)whenl(x|z) is the probability that an individual of frailty z

survives to age x. We have for the observed mortality at age x

¯µ(x)=



(z)l(x|z)zµ(x) dz



(z)l(x|z) dz

=¯z(x)µ(x). (19.4.2)

In words: the observed death rate is the mean frailty multiplied by the

death rate for the standard individual. This simple but fundamental result

factors ¯µ(x) into a part involving z and a part involving µ.

Calculating the derivative of ¯z(x) will show that mean frailty steadily

declines through life under all circumstances where mortality is positive.

19.4. Modelling Heterogeneity 487

In the ﬁrst place we know that

l(x|z)=exp



−



µ(α|z) dα



=exp



−



zµ(α) dα



=exp



−





µ(α) dα



(z)



l(x)

Entering this in an expression for the mean frailty gives

¯z(x)=



(z)

l(x)

zdz



(z)

l(x)

. (19.4.3)

To ﬁnd how ¯z(x) changes over successive ages we take the logarithm of

¯z(x) and diﬀerentiate to obtain

¯z(x)

d¯z(x)

[z(x)]

µ(x)

¯z(x)

+¯z(x)µ(x),

wherewewrite

(z(x))

for the mean square of z(x). The algebra is a further

variant on that employing the arithmetic–geometry inequality. We ﬁnally

have

d¯z(x)

= −σ

(z|x)µ(x), (19.4.4)

which is necessarily a negative quantity. We have thus shown that with

positive mortality and positive variance of z mean frailty can only de-

cline. Expression 19.4.4 describes the selection eﬀect as it diminishes frailty

through the life of a cohort. Together with (19.4.1) it shows that the ob-

served ¯µ(x) is increasingly biased (relative to the true frailty µ(x)atthe

individual level) as the cohort ages.

Note that the results here obtained are applicable starting from any

age, and not only from age zero. The selection that starts to operate in a

cohort just born operates equally in a cohort starting at an arbitrary age.

Even for an advanced age, say 70, whatever distribution of frailties is to

be found will gradually change with diﬀerential survival through ages 75,

80, etc. Age zero is a perfectly arbitrary starting point for this as for other

purposes; one could develop a demography for age 15, age 70, or any other

age, disregarding the individuals below that age, and consider a birth to be

the entry into that starting age. One could go the other way and measure

from conception as was done in Chapter 18.

The results in (19.4.1) to (19.4.4) can be extended if we parameterize the

distribution f

(z). A convenient choice is the gamma distribution:

(z)=

k−1

−kz

Γ(k)

488 19. Heterogeneity and Selection in Population Analysis

where k is the reciprocal of the variance σ

. Without assuming anything

about the value of k, Vaupel (1979) shows that with this f

(z) mean frailty

at age x is

¯z(x)=

1+σ





µ(α) dα



. (19.4.5)

This last tells us that the course of frailty with age is a simple function of

death rates below each age.

As an example of the use of (19.4.5), if mortality is age-invariant, so that

µ(x)=µ, then the course of observed mortality, calculated from (19.4.2),

will be hyperbolic

¯µ(x)=

1+σ

µx

This expression permits calculating the eﬀect numerically knowing only the

variance of the z, assuming a gamma distribution of frailty.

Alternatively, if the baseline mortality follows the Gompertz–Makeham

model

µ(x)=ae

+ c (19.4.6)

with a, b,andc constants, then the true mortality of each individual

increases exponentially with age. But the observed mortality will be

¯µ(x)=

1+σ

− 1)

+ c, (19.4.7)

which is a logistic increase with age, reaching an asymptote of b/aσ

(Yashin et al. 1994). The eﬀect of selection completely obscures the pattern

of mortality increase at the individual level.

19.4.1.1 Parameter Estimation

Suppose that data are available on a cohort of individuals, followed until

the last one dies. We want to estimate, via maximum likelihood, a set of

parameters deﬁning f

(z)andµ(x). The likelihood of a set of parameter

values is proportional to the probability of observing the data given those

values. The fate of an individual depends on a frailty z speciﬁc to that

individual; in a continuous-frailty model, each individual has a unique value

of z. Thus the number of parameters to be estimated appears to exceed

the number of individuals in the sample; this is not promising.

A solution is to focus on the distribution f

(z) rather than on the val-

ues of z realized by the individuals. This treats the individual frailties as

random variables rather than as ﬁxed but unknown constants. This view

is familiar in the Bayesian framework, where all parameters are viewed as

random variables. If we let Y denote the data and θ a vector of parameters,

19.4. Modelling Heterogeneity 489

then a Bayesian analyst writes

f(θ|Y )=

P (Y |θ)p(θ)



P (Y |θ)p(θ)dθ

, (19.4.8)

where f(θ|Y ) is the distribution of θ after observing the data Y (called

by Bayesians the posterior distribution), P (Y |θ) is the probability of the

data given θ (the likelihood ), and p(θ) is the unconditional distribution

of θ before the data Y are obtained (the prior distribution). That is, the

prior distribution of θ is updated, after data are obtained, via the likelihood

function. The posterior distribution shows which values of the parameters

are probable and which are not. The mean or the mode of the posterior

distribution can be used as a point estimate of the parameter if desired.

Bayesian statistics inspires philosophical arguments about the inﬂuence

of the prior distribution on the conclusions. Depending on your point of

view, this inﬂuence is either good, representing the growth of scientiﬁc

knowledge by updating prior beliefs in the light of new data, or bad, cor-

rupting the scientiﬁc process by permitting a subjective prior distribution

to inﬂuence the conclusions. We can leave the philosophers to their own

devices, however, by taking what is sometimes called an objective Bayesian

approach (e.g., Link et al. 2002). If the prior distribution in (19.4.8) is ﬂat

(or very nearly so), it provides no (or nearly no) information about θ prior

to obtaining the data. The posterior distribution is then proportional (or

nearly so) to the likelihood function. The mode of the posterior distribution

is then the maximum likelihood estimate of θ. Thus Bayesian methods can

be used to arrive at maximum likelihood estimates uninﬂuenced by prior

beliefs. The advantage is that this includes models where some parameters,

such as individual frailty, are treated as random variables characterized by

their distributions.

The integral in the denominator of (19.4.8) can be extremely diﬃcult to

evaluate, but the so-called Markov chain Monte Carlo (MCMC) algorithm

solves this problem, and powerful software for this purpose is now available

(Spiegelhalter et al. 1999; see Link et al. 2002 for a description in the

context of population biology).

MCMC methods are attracting attention from both human demogra-

phers (e.g., Bolstad and Manda 2001) and population ecologists. As an

example, Cam et al. (2002; see also Link et al. 2002) studied survival and

fertility in the black-legged kittiwake (Rissa tridactyla, a gull of temper-

ate and arctic waters in the Atlantic and Paciﬁc oceans). Over a period of

22 years, some 845 individuals were followed from ﬁrst reproduction until

death or the end of the study. For each bird, the time it entered the study

and the time of its death are known (unless it survived to the end of the

study), and for each year that it was in the study it is known whether or

not it reproduced. Cam et al. (2002) modelled survival and fertility as

logit

φ(i, x, t)

= a(t)+b(x)+z

(φ)

(19.4.9)

490 19. Heterogeneity and Selection in Population Analysis

logit

β(i, x, t)

= c(t)+d(x)+z

(β)

, (19.4.10)

where x denotes age and t denotes time, and i refers to a particular individ-

ual; the frailties z

(φ)

and z

(β)

are random eﬀects on survival and fertility

speciﬁc to individual i.

Assuming that the frailties z

(φ)

and z

(β)

were ﬁxed, individual-speciﬁc

eﬀects would require estimating 845 × 2 = 1690 parameters to account for

inter-individual heterogeneity. However, assuming that the individual frail-

ties were drawn from a bivariate normal distribution with a mean of zero

reduces the number of parameters to 3 (two variances and a correlation).

Flat priors were used for these 3 parameters, as well as the eﬀects of age

and time, and MCMC methods used to compute the posterior distributions

of all the parameters. A variety of models for the time and age eﬀects were

explored; the best-ﬁtting models all included signiﬁcant individual hetero-

geneity. At the individual level (i.e., the age eﬀect b(x)), survival declined

linearly with age. Because of selection, however, survival at the population

level was nearly age-invariant. The estimated correlation between the in-

dividual eﬀects on survival and reproduction was unambiguously positive,

indicating that, at the level of individual heterogeneity, “frailty” aﬀects

both survival and reproduction in the same way.

19.4.2 Finite Mixture Models

Often there is little theoretical basis for choosing a parametric distribution

of frailty. It is tempting to choose a function on the basis of convenience

(e.g., the gamma distribution, which has support on the non-negative num-

bers), but in at least some circumstances diﬀerent choices of the distribution

lead to diﬀerent estimates of the functional relationship between survival

and other (measured) variables (Heckman and Singer 1982). A nonpara-

metric approach based on a mixture of a ﬁnite number of frailty classes

may be preferable; it is certainly appropriate when there are really classes

within the population (those with and without some medical condition, for

example). In any case, many of the eﬀects of heterogeneity become apparent

by considering only two groups (Vaupel and Yashin 1985).

Suppose there are N types, with frailties z

, i =1,...,N, and that

)=P (z = z

). Then we replace (19.4.1) with

¯µ(0) =



µ(0) = ¯z(0)µ(0). (19.4.11)

The theory proceeds as in the continuous case, but with summations re-

placing integrations (Vaupel and Yashin 1999). Maximum likelihood can

be used to estimate the parameters in the mortality schedule µ(x) and the

probabilities p

). In cases where the follow-up of individuals is less than

perfect, so that an individual may disappear either because of death or be-

cause of imperfect sampling, the heterogeneity can aﬀect not only mortality

19.4. Modelling Heterogeneity 491

but the likelihood of detection (e.g., Pledger 2000, Pledger et al. 2003). The

choice of the number of groups to include (N) can be based on prior knowl-

edge or on a comparison of models with diﬀerent numbers of classes. Hoem

(1990), however, reports on a failed attempt to use this approach to study

heterogeneity in the breakup of conjugal unions (married or otherwise) in

Sweden; he found the number of parameters too large and the likelihood

too diﬃcult to maximize. Whether Bayesian MCMC methods would help

is an open question.

19.4.3 Age-Dependent Frailty

The models discussed so far assume that each individual has a ﬁxed frailty,

assigned at birth and kept until death. An alternative conceptualization

assumes that all individuals are born with the same frailty, and that frailty

then changes with age (e.g., Yashin et al. 1994). Suppose there are discrete

frailty states 0, 1,...,n, representing increasing levels of frailty, and that

frailty aﬀects mortality as

µ(z,x)=µ

(x)+zµ. (19.4.12)

Here, µ is the rate at which mortality increases with increased frailty. Frailty

changes over time, either increasing according to

P (z(t + dt)=z +1)=(γ

+ zγ) dt (19.4.13)

P (z(t + dt)=z)=1− (γ

+ zγ) dt. (19.4.14)

Here, γ is the rate of increase in frailty.

This model leads to a system of diﬀerential equations that can be solved

for the probability distribution of individuals among frailty classes at any

age. From this, as n →∞, the observed mortality is

¯µ(x)=µ

µγ

1 − e

(γ+µ)x

µ + γe

−(γ+µ)x

, (19.4.15)

which produces a logistic pattern of observed mortality (Yashin et al. 1994).

19.4.4 Parameter Identiﬁcation and Model Discrimination

Not surprisingly, any attempt to estimate the eﬀect of invisible variables

leads to problems with identiﬁcation and model discrimination.

First, as emphasized by Trussell and Rodr´ıguez (1990), there are two

ways to write the mortality rate in the presence of heterogeneity. One is to

specify a distribution of frailty and incorporate it in the formula (19.4.2).

The other is to carry out the integration over levels of frailty in (19.4.2)

and obtain a resulting mortality curve in which the distribution of frailty

is no longer apparent. There is no way to tell, from a mortality schedule

alone, whether it is the result of heterogeneity. Thus observed mortality

492 19. Heterogeneity and Selection in Population Analysis

data could follow the logistic age trajectory in (19.4.7) for two reasons.

Each individual in the population might experience the same logistic tra-

jectory of mortality, or each individual might experience the exponentially

increasing Gompertz–Makeham mortality trajectory (19.4.6) modiﬁed by

its unique, gamma-distributed, frailty. No observations on mortality alone

can distinguish the two hypotheses, because they lead to exactly the same

mortality trajectory.

Theemphasisheremustbeon“mortalityalone,”becauseitiscertainly

possible that additional information could help distinguish the models.

One can reduce heterogeneity (e.g., by conducting experiments on geneti-

cally homogeneous cohorts) and see if it changes the form of the mortality

function. Or one can use information from relatives, whose frailty may be

expected to exhibit some correlation, and see how much heterogeneity is

reduced (e.g., Yashin and Iachine 1997).

It is also impossible, under at least some hypotheses, to distinguish ﬁxed

from age-dependent frailty. Yashin et al. (1994) showed that the parameters

of the age-dependent frailty model (19.4.15) can be put in one-to-one cor-

respondence with the parameters of the gamma–Makeham model (19.4.7).

Thus observations of mortality alone cannot distinguish between a frailty

model where each individual has its own ﬁxed level of frailty assigned at

birth and one where all individuals are born with the same frailty, which

increases stochastically through their lives.

19.5 Experimentation

As an example of the way heterogeneity can distort even a careful medical

trial, consider a population that is heterogeneous, containing two equal

subpopulations, both initially stable, equal in number and increasing at

the same rate r, but with diﬀerent mortality (and correspondingly diﬀerent

birth) rates. A procedure is applied that lowers mortality by amount ε at

all ages for one of the subpopulations. Under what circumstances will the

use of this procedure raise the overall crude death rate of the population?

If the initial common rate of increase is r, the two birth rates b

∗

and b,

then the initial crude death rate is

(cdr)

=((b

∗

− r)+(b − r))/2.

Suppose the procedure lowers mortality by ε uniformly at all ages for the

starred subpopulation, increasing its r to r + ε, and having no eﬀect on its

crude birth rate b

∗

. The combined population that results at time t will

have a weight of e

εt

for the starred subpopulation and unity for the other,

so its crude death rate will be

(cdr)

=(e

εt

∗

− r − ε)+(b − r))/(e

εt

+1).

19.5. Experimentation 493

Then (cdr)

will be greater than (cdr)

if and only if

εt

∗

− r − ε)+(b − r)

εt

∗

− r +(b − r)

Ultimately as t becomes large, the condition becomes ε<(b

∗

−b)/2which

is the same as

ε<

∗

− r) − (b − r)

. (19.5.1)

This shows that if the decrease in mortality rates aﬀects the subpopu-

lation of higher mortality only, and if the improvement is less than half

the diﬀerence between the preceding death rates of the two subpopula-

tions, then the eﬀect of the improvement will be an increase in the overall

mortality rate.

Epilogue: How Do We Know the Facts

of Demography?

Demographers know that a slowly increasing population has a higher

proportion of old people than one that is increasing rapidly, and that dif-

ferences in birth rates have a larger inﬂuence on the age distribution than

do diﬀerences in death rates. They often claim that a poor country whose

population is growing rapidly will increase its per capita income faster if it

lowers its birth rate rather than maintaining it at a high level.

How do demographers know these things? Many readers will be surprised

to learn that in a science thought of as empirical and often criticized for its

lack of theory the most important relations cannot be established by direct

observation, which tends to provide enigmatic and inconsistent reports.

Confrontation of data with theory is essential for correct interpretation of

such relationships, even though on a particular issue it more often generates

an agenda for further investigation than it yields useful knowledge. Much of

this book is devoted to examining the ways in which demographers distill

knowledge from observation and from theory. The present summing up

shows a relatively heavy weight of evidence for theory, illustrated brieﬂy

with an application to economic demography. We thank Paul Demeny for

many improvements and clariﬁcations in this account.

Let no one think that the questions of demography, and the issues of

method for ﬁnding answers to them, are remote or purely abstract. The

resolution of major policy issues of our time depends on the answers. How

much of its development eﬀort should a poor country put into birth control

if it deems its rate of population growth excessive? Some would put nothing,

in the expectation that rapid increase of income will by itself bring popula-

tion under control. Once people have automobiles, once their countryside is