Keyfitz N., Caswell H. Applied Mathematical Demography

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3.6. Age as a State Variable: When Does it Fail? 65

2. The eﬀect of the population on the environment can be written as

the sum of the contributions of the individuals.

This is an extremely important conclusion. It justiﬁes the practice of

writing p-states as vectors of abundances of individuals in various i-

state categories, which is fundamental to all types of demographic models

(Tuljapurkar and Caswell 1997).

The Metz–Diekmann conditions guarantee that we can test a would-be p-

state variable by testing the adequacy of the corresponding i-state variable.

This can be done by measuring the i-states for a set of individuals, doc-

umenting their responses to a common environment, and quantifying the

information about individual fate provided by the i-state. Loglinear mod-

els (Section 17.6) are now widely used in ecology for this purpose (Caswell

1988, MPM Chapter 3).

The Metz–Diekmann conditions are often satisﬁed, at least approx-

imately. For instance, in many demographic models, the eﬀect of the

population on the environment is not represented explicitly. Instead, the

vital rates are written as functions of density, assuming some mechanism

by which density is translated into an eﬀect on the environment (e.g., con-

sumption of resources), and hence back to the population. In such models,

the second condition is satisﬁed automatically.

When the Metz–Diekmann conditions are violated, the state of the pop-

ulation cannot be described by the distribution of individuals among the

i-states. Instead, the state of each individual must be accounted for explic-

itly. Such models are called i-state conﬁguration models (Caswell and John

1992), individual-based models (DeAngelis and Gross 1992, Grimm et al.

1999), or, in the human demographic literature, microsimulation models

(e.g., Ruggles 1993, Wachter 1997).

The conditions are violated when the vital rates are determined by inter-

actions among speciﬁc individuals. The most common case involves sessile

organisms where individuals interact only with their immediate neighbors.

In such populations, each individual may experience a diﬀerent environ-

ment, and the eﬀects of an individual on the environment depend on where

it is located and cannot be written as a sum of individual eﬀects.

3.6 Age as a State Variable: When Does it Fail?

The adequacy of age as a state variable depends on how much information

it provides about the demographically relevant aspects of individual devel-

opment. Several circumstances combine to limit this information and make

other state variables more suitable than age.

66 3. The Matrix Model Framework

3.6.1 Size-Dependent Vital Rates and Plastic Growth

If the vital rates depend on body size and if growth is suﬃciently plastic

that individuals of the same age may diﬀer appreciably in size, then age

will provide little information about the fate of an individual. If the vital

rates depend on developmental stage (e.g., instar) and stage duration varies

among individuals, age will be a poor state variable. Such plasticity is

widespread in plants, ﬁsh, and arthropods.

Size-dependent demography is not limited to species with indetermi-

nate growth, however. Sauer and Slade (1987) have documented eﬀects

of body size on survival and reproduction in vertebrates, and they have

used size-based demographic models for small mammals (Sauer and Slade

1985, 1986). Weight and body composition are known to aﬀect the age at

reproductive maturity in humans (Frisch 1984).

Size- or stage-dependent vital rates by themselves are not enough to ren-

der age inadequate as a state variable. If growth is so tightly regulated that

age is a good predictor of size, then even if fecundity and mortality depend

on size, age will work as an i-state variable. Some authors (e.g., Stearns

and Koella 1986) have used this approach to develop models phrased in

terms of size, but using age-classiﬁed demography.

3.6.2 Multiple Modes of Reproduction

Many organisms exhibit both sexual and vegetative or clonal reproduction

(Jackson et al. 1985). Sexual and vegetative oﬀspring of the same age may

diﬀer markedly in their demographic properties. Cook (1985), for example,

summarizes data on several clonal plant species showing that the probabil-

ity of successful establishment for vegetative oﬀspring is from 3 to 30 times

higher than the corresponding probability for oﬀspring produced from seed.

To the extent that individuals of the same age have diﬀerent vital rates,

age is an inadequate i-state variable for such species.

3.6.3 Population Subdivision and Multistate Demography

A third situation leading to inadequacy of age as an i-state variable re-

sults from population subdivision, when the subpopulations are exposed

to diﬀerent environments. Speciﬁcation of an individual’s state in such a

population requires speciﬁcation of its age and its environment. Spatial

subdivision is an obvious example; if individuals migrate between regions

characterized by diﬀerent vital rates, the age × region distribution is re-

quired to specify the state of the population. There is a large literature on

such “multiregional” models (e.g., Rogers 1975, 1985, 1995). These models

also apply to populations with other kinds of heterogeneity (sex, marital

status, parity, employment). The term “multistate demography” is used

3.7. The Life Cycle Graph 67

to describe models in which individuals are classiﬁed by multiple i-state

variables (Land and Rogers 1982, Schoen 1988; see Chapter 17).

The choice of an appropriate state variable is a critical step in model

construction. The concepts of i-state and p-state clarify the biological and

statistical issues involved in that choice. They provide criteria for decid-

ing, from studies of individuals, what variables need to be included in a

population model.

3.7 The Life Cycle Graph

We turn now to constructing matrix population models for any discrete

i-state variable; we will refer to these models as “stage-classiﬁed.” Age-

classiﬁed models are a special case of stage-classiﬁed models. We will see

in Chapter 7 that one of the advantages of the matrix formulation is

that results on asymptotic dynamics, ergodicity, transient analysis, and

perturbation analysis apply equally to age- and stage-classiﬁed models.

We begin with a simple graphical description of the life cycle, called the

life cycle graph. Figure 3.9 shows two examples. The construction proceeds

as follows.

1. Choose a set of stages (i.e., i-states) in terms of which to describe the

life cycle.

2. Choose the projection interval, deﬁning the time step in the model.

3. Create a node for each stage; number the nodes from 1 through s.

The order of the numbering is irrelevant, but it is often convenient to

assign the number 1 to a stage representing “newborn” individuals.

The symbol N

denotes node i.

4. Put a directed line or arc from N

to N

if an individual in stage i at

time t can contribute individuals (by development or reproduction)

to stage j at time t + 1. If an individual in stage i at time t can

contribute to stage i at t + 1 (e.g., by remaining in the same stage

from one time to the next), put an arc from N

to itself; such an arc

is called a self-loop.

5. Label each arc by a coeﬃcient; the coeﬃcient a

on the arc from N

to N

gives the number of individuals in stage i at time t +1 per

individual in stage j at time t.Thus

(t +1)=



j=1

(t).

68 3. The Matrix Model Framework

(a)

421

(b)

421

Figure 3.9. (a) A life cycle graph for an age-classiﬁed life cycle, in which the

width of the age classes equals the projection interval. (b) A life cycle graph for

the standard size-classiﬁed model. Nodes represent size classes, and an individual

can grow no more than a single size class in the interval (t, t +1).

These coeﬃcients may be transition probabilities or reproductive

outputs.

†

Note the order of the subscripts!

Figure 3.9a shows the graph for an age-classiﬁed life cycle in which the

age interval and projection interval are identical. Individuals survive, with

probability P

, to become one unit older, and reproduce, with fertility F

producing new individuals in age class 1. Figure 3.9b shows the graph for a

size-classiﬁed life cycle. An individual in size class i may survive and grow

to size class i + 1 with probability G

, or may survive and remain in size

class i with probability P

. Reproduction produces new individuals in the

smallest size class. As drawn, this graph asserts that it is impossible for an

individual to shrink, or to grow two or more size classes in a single time

interval.

Figure 3.10 shows a life cycle graph for the killer whale Orcinus orca

(Brault and Caswell 1993). The projection interval was one year, and indi-

viduals were classiﬁed as yearlings, juveniles (past their ﬁrst year but not

mature), mature females, and postreproductive females.

†

The life cycle graph as deﬁned here is known technically as the Coates graph of the

matrix A (Chen 1976); its use in demographic analysis is due to Lewis (1972, 1976),

Hubbell and Werner (1979), and Caswell (1982a,b). Some of the results that can be

obtained from the graph will be seen in Chapter 9; see MPM Chapter 7 for details.

3.8. The Matrix Model 69

421

Figure 3.10. A life cycle graph for the killer whale (Brault and Caswell 1993).

Nodes represent stages: N

= yearlings, N

= juveniles, N

= mature females,

and N

= postreproductive females.

Three things about Figure 3.10 are worth noting. First, because stage

1 is an age class of the same length as the projection interval, individuals

cannot remain in stage 1 from one time to the next. Thus P

= 0 and

no self-loop has been drawn on N

. The loop could have been drawn and

its coeﬃcient set to zero without changing anything. Second, there is a

postreproductive stage, which does not contribute individuals to any of the

other stages. Third, it includes positive fertility for juvenile females. This

arises because in a birth-ﬂow model it is assumed that some juveniles may

mature during the interval from t to t + 1 and produce oﬀspring at t +1.

3.8 The Matrix Model

The life cycle graph is more than a handy pictorial description of the life

cycle. It is isomorphic to the population projection matrix A in the equation

n(t +1)=An(t),

where n(t) is now a vector of stage abundances. The rule for generating

the matrix is simple: the matrix entry a

is the coeﬃcient on the arc from

to N

in the life cycle graph. Note the order of the subscripts.

The projection matrices corresponding to the two graphs in Figure 3.9

are

⎛

⎜

⎝

0 F

000

0 P

00P

⎞

⎟

⎠

(3.8.1)

⎛

⎜

⎝

0 G

00G

⎞

⎟

⎠

. (3.8.2)

70 3. The Matrix Model Framework

The age-classiﬁed graph (Figure 3.9a) yields a Leslie matrix. The matrix

resulting from the size-classiﬁed graph (Figure 3.9b) includes positive

elements on the diagonal, corresponding to individuals remaining in the

same size class, as well as on the subdiagonal and the ﬁrst row.

‡

The matrix

for the killer whale from the life cycle in Figure 3.10 is

A =

⎛

⎜

⎝

0 F

0 G

00G

⎞

⎟

⎠

. (3.8.3)

Deciding which properties of individuals to include in the life cycle calls

on all the biological and sociological information available, and is always

a compromise between recognizing the potential importance of even the

tiniest diﬀerences and the acknowledging the limitations of the available

data. But once the decision has been made and the projection matrix con-

structed, all manner of population analyses are possible. We return to these

in Chapter 7.

‡

This model, called the standard size-classiﬁed model, is widely used in ecology [e.g.,

Usher (1966), Pinero et al. (1984) for trees, Hughes and Jackson (1985) for corals, Heppel

et al. (1996) for sea turtles, Doak et al. (1994) for tortoises, Warner and Hughes (1988)

for ﬁsh].

Mortality Comparisons;

The Male-Female Ratio

The United States in 1975 showed an expectation of life at birth for males

of 68.5 years and for females of 76.4 years, a diﬀerence of 11.5 percent. But

male death rates at most ages are at least 50 percent higher than female

rates. The ratio of male to female rates, simply averaged over the ages,

may show males 80 percent higher; the average with living population as

weights may show males 70 percent higher; with deaths as weights males

may be 50 percent higher. Our question is whether male mortality is 10,

50, 80, or 70 percent higher than female. The issue is raised in Sheps (1959)

and Golini (1967).

The same diﬃculty arises when we compare two countries—the United

States and Italy, or the United States and India—or two parts of the same

country—South and North in Italy, or city and countryside. It arises also

in comparisons through time: has mortality been improving in Europe and

America during recent years, and if so by how much? Essentially the same

issue can be expressed in terms of the conditional future: if mortality is

reduced on an average of 5 percent at each age, what diﬀerence will this

make to the expectation of life? The question is of special interest with

respect to cause of death: if deaths from cancer are reduced by 10 percent

on the average of the several ages, what will the eﬀect be on expectation

of life, provided that all other causes are unchanged? A more sophisticated

form of the question is to suppose that deaths from cancer are reduced by

10 percent and to note the eﬀect if cancer mortality is positively correlated

to a given degree with other causes, that is, if those subject to cancer are

somehow more susceptible also to other causes, but this is out of our scope.

72 4. Mortality Comparisons; The Male-Female Ratio

Table 4.1. Ratio of male to female age-speciﬁc death rates for four countries

United

States, France, Greece, Mexico,

Age 1967 1967 1968 1966

–1 1.30 1.31 1.11 1.20

1–4 1.25 1.21 1.10 0.95

5–9 1.38 1.50 1.37 1.06

10–14 1.72 1.63 1.55 1.17

15–19 2.51 2.14 1.92 1.25

20–24 2.80 2.31 2.10 1.31

25–29 2.26 2.08 1.78 1.31

30–34 1.81 1.95 1.56 1.40

35–39 1.68 1.89 1.34 1.41

40–44 1.72 2.02 1.51 1.56

45–49 1.79 2.07 1.59 1.50

50–54 1.95 2.09 1.73 1.45

55–59 2.07 2.37 1.69 1.41

60–64 2.10 2.35 1.68 1.21

65–69 1.91 2.19 1.62 1.23

70–74 1.82 1.80 1.32 1.12

75–79 1.56 1.59 1.17 1.08

80–84 1.33 1.41 1.06 0.88

85+ 1.08 1.24 1.15 0.75

Source: Based on Keyﬁtz and Flieger (1971).

In the simplest aspect of the problem a person saved from death through

one cause at a particular time is rescued only to be exposed to the same

cause and others in the subsequent period. To avoid death from heart

disease at an age typical of the ages at which heart disease occurs can

oﬀer only a brief respite, since overall death rates increase rapidly in later

life. Lower mortality of young and middle-aged females allows them to

survive into ages at which mortality is very high for everyone. That the

curve of death rates by age is concave upward and that populations are

heterogeneous in risk are what give rise to the questions with which the

present chapter deals.

4.0.1 Variation by Age in the Sex Ratio of Mortality

To bring the theory into contact with concrete data we show the ratios of

male to female age-speciﬁc death rates for four countries in Table 4.1. The

tendency is toward a characteristic pattern, with peaks around 0, 20, and

60, and troughs around 1, 35, and 85, the pattern being shown clearly by

the United States and less clearly by Mexico.

4.1. The Multiplicity of Index Numbers 73

4.1 The Multiplicity of Index Numbers

The most familiar comparison of mortality is by direct standardization: how

many deaths would occur in a given population if it had the age distribution

of the (standard) population with which it is being compared, and what is

the ratio of this number to the deaths in the standard population? Kitagawa

(1964) widened the perspective by assimilating the issues to those of price

index numbers. Her formulae can be applied readily to the excess mortality

of males. If the male population is p

, the female population p

,themale

death rate µ

, and the female death rate µ

, the relative mortality at age

x is µ

/µ

, and the index in which such relatives are weighted by male

population is



(µ

/µ

)



, (4.1.1)

and by female population is



(µ

/µ

)



. (4.1.2)

The index weighted by male deaths is



(µ

/µ

)



, (4.1.3)

and by female deaths is



(µ

/µ

)



. (4.1.4)

To one of these four indices preferred in demography is I

, the directly

standardized rate (4.1.4). In economics I

is called an aggregative index,

in which the µ

might be prices and the p

quantities; f might be the base

period and m the current period in a time comparison.

In addition to the four index numbers of (4.1.1) to (4.1.4) we can ob-

tain four further numbers by using the harmonic means. For the male-

population-weighted harmonic mean H

we have



/(µ

/µ

)

and similarly for the other three indices obtained by weighting by females

and by deaths.

One might have thought that taking the male death rates as the denomi-

nator and then inverting the result would give additional indices. However,

this merely reproduces in a diﬀerent order the eight indices described above.

74 4. Mortality Comparisons; The Male-Female Ratio

Geometric averaging does give diﬀerent results (Schoen, 1970). The male-

population-weighted index would be



(µ

/µ

)





and similarly for G

and others.

A further variant is the exponential of the harmonic mean of the

logarithms of the ratios,

=exp





/ log(µ

/µ

)



which gives another four ways of expressing the relation of the given

population to the standard.

We could obtain additional indices by cross-weighting and averaging.

One instance is I =

)(I

), which may well give about as reﬁned a

comparison as any. England and Wales show considerably higher death

rates than the United States at older ages and lower rates at younger ages.

To ﬁnd how the two countries stand on the whole we can ﬁrst standardize on

the United States 1960 census age distribution taking both sexes together,

and calculate the intercountry value of (4.1.4), which turns out to be 1.218.

Using the same formula but standardizing on the 1961 census of England

and Wales, we ﬁnd 1.231. The geometric mean of these quantities is 1.224.

Other compromises between the given and the standard age distributions

are possible, as well as other kinds of averages. However, the conceptual

points raised later in this section will lead to some skepticism about the

usefulness of extreme reﬁnement.

4.1.1 Weighted Index of Male to Female Mortality

The ratios of Table 4.1 (but to more decimal places) were combined in

the four ways designated above as I

, I

, that is, weighting by

male population, female population, male deaths, and female deaths, as in

expressions 4.1.1 and 4.1.4. They were also combined to give the harmonic

mean H, the geometric mean G, and the harmonic mean of the logarithm

HL. The results are shown in Table 4.2 for six countries. Especially for

low-mortality countries weighting by population gives a much higher excess

mortality of males than weighting by deaths. The reason is that deaths are

relatively heavier than population at the oldest ages, where the age-speciﬁc

rates of females approach those of males.

Note that the arithmetic ratios are higher than the geometric, which

in turn are higher than the harmonic. Moreover, the geometric ratios are

equal to the means of the corresponding arithmetic and harmonic ratios:

G =



(I)(H).