Keyfitz N., Caswell H. Applied Mathematical Demography

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17.4. Projection 455

for those single at age 20 and 53.587 for those already married. The single

can expect 11.687 years single, 28.896 + 7.580 = 36.476 years married.

17.4 Projection

A common demographic activity is population projection. If a given set

of probabilities applies, and we know the vector representing the several

categories of population at a certain point of time, the expected numbers at

a later point of time are calculable. Suppose the age interval as well as the

time interval to be h years as before; then for a population that happened

to be concentrated at ages 0,h,2h,..., the matrix L(x + h)L

−1

(x)would

be appropriate for the projection in all cases where L(x) is nonsingular.

But to approximate the observed population by a series of spikes seems

inferior to approximation by a series of histograms.

We need a ratio corresponding to S

x+h

of the single-region

case. This is obtained by applying the multiplicative property (17.2.2) to

show that L

x+h

= S

, and multiplying on the right by L

−1

to obtain

= L

x+h

−1

, again assuming the inverse of L

exists. (It does not ex-

ist for a marriage or labor force table prior to age 15 or so, and some

other device is needed.) Now suppose the transitions pertaining to each

given age,perhaps transitions among regions, are assembled into a block.

Such blocks for the several ages x may be assembled into a matrix S with

zero blocks everywhere except in the subdiagonal. The arbitrary (that is,

observed) n(t) is projected to time t + h by

n(t + h)=Sn(t). (17.4.1)

This projects the part of the population already alive, but it gives no

attention to births. To allow for them we need a matrix F, whose nonzero

elements are in its ﬁrst row of submatrices, so the complete projection with

time-invariant coeﬃcients is (Rogers 1975, Feeney 1970)

n(t + h)=(S + F) n(t). (17.4.2)

Table 17.4. Multi-state life table: expected number of years in each of ﬁve states

for Canadian males aged 20; vital statistics of 1981

Once Later

Single married Widowed Divorced marriage

Single 11.687 0 0 0 0

Once married 28.896 36.788 0 0 0

Widowed 1.904 2.182 10.669 2.164 2.182

Divorced 2.363 3.063 2.495 6.765 3.063

Later marriage 7.580 11.554 38.197 44.407 48.342

Total 52.430 53.587 51.361 53.337 53.587

Source: Author’s calculation.

456 17. The Multi-State Model

The procedure described by (17.4.2) permits a certain kind of experi-

mentation with the elements of S and F and serves to ascertain the eﬀects

on future population of marriage ages, migration rates from more to less

fertile regions, and so on. For forecasting purposes, some gains can be ob-

tained by deliberately allowing the coeﬃcients to change according to what

one thinks may be the trends over future time. One must be cautious here;

in more than one case varying the coeﬃcients has produced a less accurate

forecast than ﬁxed coeﬃcients would have done.

17.5 Transition Versus Instantaneous Probability

of Moving

Most data on geographical mobility come in the form of transitions over a

ﬁnite period—we know that the person was in state j at one moment of

time and in state i 1 year or 5 years later. The census, for instance, asks

people where they were 5 years ago without inquiring about intermediate

moves. We shall speak of transitions in terms of a P matrix, whose p

the probability that a person is in the jth state this year and in the ith

state 1 year or 5 years later, in contrast to the µ matrix, where the chance

of moving from the jth state to the ith in inﬁnitesimal time dx is µ

(x) dx.

Although our theory started out in terms of movements in time separated

by an inﬁnitesimal interval dx, it then dropped the µ and took no account

of what happens within the ﬁnite interval beyond the probability that a

person in the jth state at the beginning is in the ith state at the end. This

is satisfactory if the time interval is short enough that two moves do not

occur within it—that a person does not move and die in the same speciﬁed

h years, for example. But what if the data cover a wide enough interval

that several moves are possible?

Fortunately, the bias does not apply to most of the quantities calculated

from the model. For projecting population according to the spike represen-

tation of the L(x), the observed transitions are obviously the right ones to

use; one wants to omit intermediate transitions (people going from j to i

and back again, all within one time interval) in the future as they have been

omitted in the past. Slightly less obviously, the same applies to projection

by histograms (equation 17.4.2). On the other hand, the expected number

of moves calculated from the model will be understated if multiple moves

occurring within the unit time interval of the data are neglected.

The Markov assumption of our model assigns the same probabilities to

everyone in a given state at the beginning of each interval; no past history

is allowed to inﬂuence the chance of transition beyond what is implied by

the state that the person is then occupying. If we suppose this also applies

in all subintervals, then the Markov assumption allows us to capture and

add in those jumps that occur within intervals.

17.5. Transition Versus Instantaneous Probability of Moving 457

Table 17.5. Transition probability p, corresponding instantaneous rate µ, and

probability of 0, 1, 2, and 3 moves in two-state model with P

= P

= p

Corresponding

Transition instantaneous

Probability of

probability rate

(p)(µ) 0 Moves 1 Move 2 Moves 3 Moves

0.001 0.001 0.999 0.001 0.000 0.000

0.050 0.053 0.949 0.050 0.001 0.000

0.100 0.112 0.894 0.100 0.006 0.000

0.150 0.178 0.837 0.149 0.013 0.001

0.200 0.255 0.775 0.198 0.025 0.002

0.250 0.347 0.707 0.245 0.042 0.005

0.300 0.458 0.632 0.290 0.066 0.010

0.350 0.602 0.548 0.330 0.099 0.020

Source: Calculated by Frans Willekens.

Consider transitions of one period between two states, in which p

= p, so that the fraction of persons initially in the jth state that are

found in the ith state at the end of the unit period is p. Suppose also that

the instantaneous probability of moving (not known) in the time dt is µdt;

then we can write out in terms of µ what fraction of the individuals initially

in the jth state would be found in the ith state at the end of the period. It

is those who made an odd number of moves, 1 or 3 or 5 or 7, this being as

far as we need to go. If the instantaneous rate is µ, then by integration the

probability of one transition in unit time is µe

−µ

, the probability of three

transitions is (µ

/3!)e

−µ

, and so on, and the sum of these can be equated

to p:

p = µe

−µ

/3!

−µ

+ ···

−µ

− e

−µ

1 − e

−2µ

/2. (17.5.1)

Solving for µ gives

1 − 2p = e

−2µ

µ = −

log(1 − 2p).

(We are grateful to Jan Hoem for this form of the solution.)

Table 17.5 gives values of p and the corresponding values of µ.Itsays,

for example, that if the transition probability from the jth to the ith state

is 0.2, and the transition probability from the ith to the jth is the same

as this, and there are no other states, then an instantaneous rate of 0.255

will produce the 0.200 chance of transition. There will be a probability

−µ

=0.775 of no moves, µe

−µ

=0.198 of one move, (µ

/2!)e

−µ

=0.025

of two moves, and so on.

To generalize, if p

is the probability of a transition from state j to state

i in unit time, we would like to ﬁnd the instantaneous probability µ

(ﬁxed

458 17. The Multi-State Model

over the interval) that would provide gross movement but still show a net

of p

. We have discussed only the simple case where p

= p

= p is the

probability of transition in either direction between the two states.

The generalization that follows from the Kolmogorov equation (Kitsul

1978) is the matrix relation P =exp(−µ), or written out:

⎛

⎜

⎝

··· ···

⎞

⎟

⎠

=exp

⎡

⎢

⎣

−

⎛

⎜

⎝

··· ···

⎞

⎟

⎠

⎤

⎥

⎦

, (17.5.2)

where the diagonals µ

are such as to make the columns sum to zero. If

the matrices have distinct eigenvalues, (17.5.2) may be solved numerically

in either direction.

Let us prove that (17.5.1) is a special case of (17.5.2). Take the matrices

as 2 × 2 and make the µ’s equal to µ or −µ, so that (17.5.2) becomes





=exp



−



µ −µ

−µµ



To evaluate the right-hand side, we need eigenvalues, eigenvectors, and

spectral components. For eigenvalues,

µ − λ −µ

−µµ−λ

=0,

− 2λµ + λ

− µ

=0.

Therefore λ = 0 or 2µ. Left eigenvectors are

and

,up

to constant factors. Right eigenvectors are

1 −1

and

1 −1

Thespectralcomponentsare







1 −1

−11



Hence

exp(−µ)=









−2µ



1 −1

−11



Multiply by e

−µ

to obtain as the P matrix



−µ

'





+ e

−µ



1 −1

−11

(

The lower left element of this is

−µ



− e

−µ



= e

−µ



µ +

+ ···



which is identical to (17.5.1).

17.6. Stable Population 459

Equation 17.5.2 allows us to infer the instantaneous probabilities of move-

ment that are equivalent to a given set of transitions and hence to know

the multiple transitions expected in any ﬁnite period. This is based on the

Poisson process in which the probability of moving from the jth to the ith

state is equal for all moments within the unit time period. In the case of

death, a movement is the same as a transition and no reversal is possible;

for other changes, transition in either direction has to be admitted. Under

the stated assumptions, equation 17.5.2 read as an equation in P tells the

transitions that correspond to a given set of movement probabilities; read

as an equation in µ, it tells the movement probabilities that account for

the given transitions.

In a series of papers Singer and Spilerman (1976) have shown how to ﬁnd

the continuous-time Markov chain of which an observed set of transitions

can be considered the manifestation. Not every series of transitions is em-

beddable in a stationary Markov chain (that is, one with ﬁxed parameters).

They represent the solution of P = e

−µ

formally as µ = −log P and pro-

ceed to create a suitable deﬁnition of log P in terms of the decomposition

of P in spectral components. Because the logarithm has multiple branches,

the result is not unique, and identifying the answer that corresponds to the

problem in hand requires some ingenuity. The reader is referred to Singer

and Spilerman for a highly sophisticated treatment of the relation between

P and µ.

17.6 Stable Population

When we seek probabilities and expected values for individuals, the radix

is taken as L(0) = I. But for stationary populations the radix must be the

number of births into each of the categories. The point does not arise in

tables of the single and married population, nor of the working population

or the school population, because for each of these classiﬁcations everyone

starts out in the same state—single, not in the labor force, and not at

school. But the classiﬁcation by territory is diﬀerent: Portrayal of station-

ary conditions in the several regions requires that we enter as the radix a

diagonal matrix Q showing the number of equivalent births in each region;

here we use not L(x) but L(x)Q.

Like the usual single-region life table, our multi-group table is not only

suited to provide probabilities and expectations but is also a population

model for such groups as regions, occupations, marital statuses, or years of

schooling. It gives the age and group distribution to which the age-group-

speciﬁc rates would lead if they were in operation long enough that the

peculiarities of the original distribution were forgotten.

But for this purpose the stationary model can readily be improved on

by incorporating population increase. In the stable model the people now

460 17. The Multi-State Model

aged x were born x years ago when the population was smaller than it is

now in the ratio e

−rx

, so the number that would be counted as of age x

would be proportional to the survivors from one birth times e

−rx

.Ona

radix of one birth in each region the stable multi-group population would

show the age and region distribution e

−rx

l(x); on a basis of q

births in the

ith region it would show

−rx

l(x)Q, (17.6.1)

where Q is the diagonal matrix containing the q

.Thejth element of the

ith row of e

−rx

l(x)Q is e

−rx

(x)q

, which is the number of persons of age

x out of the q

born in the jth region that will be found at stability in

the ith region. The Q is the multi-region analogue of the stable equivalent

(Keyﬁtz 1968; see Chapter 8).

Such a model mimics observed populations with a closeness that depends

on how nearly constant is the regime of mortality and fertility to which they

have been subject. The model tells the implication of present rates—what

the outcome will be if they continue unchanged for two or three generations.

It tells, for example, where the Northeast would stand with respect to the

rest of the United States if the age-speciﬁc rates of 1967–72 continued to

prevail.

Formulae of this chapter have been applied to census and vital records

that profess to cover the whole population. Another way of saying this

is that the errors of such data are not of a random nature, and so the

guarantees of accuracy that probability sampling can oﬀer are not available

for them.

Demographers increasingly gather their own data by sample survey

methods. The results of such enquiries are usually tabulated in exten-

sive cross-classiﬁcations, and bring the authors face to face with diﬃcult

questions of statistical signiﬁcance. The pioneer on searching through con-

tingency tables to ﬁnd what conclusions can properly be drawn from them

is Leo A. Goodman, whose log-linear methods are now widely used; for

presentations relevant to our applications, see Bishop et al. (1975) and

Fingleton (1984), for applications to the problem of choosing demographic

state variables see Caswell (1988a, MPM Chapter 3).

Family Demography

For most purposes of population study there is no need to consider any

unit intermediate between the individual and the larger group consisting

of all the individuals included within some area—state, province, county,

or nation. Recognizing only two units, population and individual, permits

the construction of models that are readily expounded and understood.

Demographers, following in the footsteps of Lotka and other predecessors,

have worked hard to simplify this much-too-complex material.

The choice of unit for demographic (as for any other) analysis depends

on the problem to be solved. For forecasts of total population, of the future

labor force, or of the pension burden, it has seemed suﬃcient to work with

the individual, and often without characterizing each individual beyond

age and sex. One supposes that individuals give rise to other individuals

over the course of time in a renewal process, irrespective of marriage or

co-residence; individuals are discrete from birth; they live their separate

lives, reproduce, and die.

For some purposes the recognition of an intermediate unit is unavoidable.

Individual demography can tell us little about how the population ﬁts into

the housing stock; it can tell nothing about the kin networks among which

mutual aid and protection take place. It falls short of explaining fertility

change, insofar as the couple rather than the individual is the decision-

making unit. Children are not born to couples independently at random,

but couples take account of the number of children already born to them

and of other aspects of their family situation at any given moment. When

family constitution was stable it attracted less attention from scholars and

462 18. Family Demography

lay people. Enormous changes in family and residential arrangements since

World War II have aroused the current interest in family demography.

18.1 Deﬁnitions

This chapter concerns three kinds of unit intermediate between the in-

dividual and the population: the kin group, the residential family, and

the household. The kin group are parents and children, siblings, nephews,

nieces and cousins, irrespective of where they are living. The family, for

census purposes, is not the totality of relatives but only those that live

together in a given household or dwelling. This is the convenient unit of

census enumeration—since the members know one another, the census in-

formation on all can be obtained from whoever answers the door when

the enumerator calls. The members of the kin groups recognized by the

census taker are in the ﬁrst place the nuclear family—husband, wife, and

children—and then the resident extended family that includes parents and

in-laws of the husband or wife. The household is the persons living in a

given residential unit, however that may be deﬁned—by separate entrance

from the street (i.e., without the need to pass through the premises of some

other household), or full complement of cooking, washing, and other facil-

ities. It is not easy to apply such deﬁnitions across cultures; those cultures

in which privacy is important are likely to have more stringent physical

requirements for a living space to be considered a separate household than

those in which less privacy is desired or can be aﬀorded.

Thus the unit for the analysis with which this chapter deals can be:

1. The kin group, irrespective of its living arrangement, and however dis-

persed. In demographic (though not in genealogical) work the data

are conﬁned to living members. As among kin groups one can recog-

nize: (i) the nuclear family, consisting of father, mother, and children;

(ii) the stem family, including ancestors in direct line and children of

all generations; (iii) the extended family that includes all genera-

tions and all collaterals. Individuals belong in general to more than

one family; even on the narrowest deﬁnition, individuals are usually

members of a family of orientation, into which they are born, and a

family of procreation, where they in their turn become parents.

2. The residential group, people living in a household, which is to say,

sharing kitchen, bathroom, and other facilities, whether or not they

are related.

3. Those members of a family that live in a given household. This cross

between the kin group and the residential group is the natural object

of census inquiries, and it is widely enumerated and tabulated. But it

omits much; the common case where a couple have parents living in an

18.1. Deﬁnitions 463

adjacent apartment is unrecognized; for many purposes propinquity

is important and makes for what is in some respects a joint family,

not captured by censuses.

18.1.1 Classiﬁcations

The problems of family demography do not end with the deﬁnition of the

unit, but go on with the classiﬁcation of types of that unit, whatever it

may be. For the residential family, i.e., kin members within a household,

the classiﬁcations used by censuses are many. The problem is to recognize

the main forms of co-residence without multiplying excessively the number

of types.

One summary solution is to classify by generations. Such a summary

has been used by the Chinese census of 1982 and elsewhere. Comparing

the Chinese material with that for Canada 1981, we ﬁnd that one-person

families were 20.3 percent of the whole in Canada, against 7.9 percent in

China; one couple with no others, 22 percent in Canada against 4.8 percent

in China. In both countries the biggest group was two generations (typically

a couple and their children with no others) numbering 47.4 percent of the

whole in Canada, 64.8 percent in China. Three or more generations, without

unrelated members were 2.3 percent in Canada, 17.2 percent in China which

accords with the common stereotype.

Trends over time are substantial and unidirectional. The United States

proportion of one-person households was 13.3 percent in 1960, 17.5 percent

in 1970. Three or more generations, with or without unrelated members,

dropped from 4.7 percent to 3.5 percent in the United States in the course

of the 1960s. Over longer periods the diﬀerences are dramatic; one-person

households were 4 percent of the whole in 1790 and 23 percent in 1980.

One can speculate on the eﬀect of China’s new marriage law; will requir-

ing children to look after their elderly parents prevent China from following

the evolution of the West toward separate living for old people?

Another equally simple classiﬁcation is given in the Canadian census of

1981. Five categories in all are recognized, and these are tabulated by age

of wife and other characteristics. Thus we have

Husband-wife, no children 851,000

Husband-wife with children 3,267,000

Husband-wife, empty nest 968,000

Male lone parent 83,000

Female lone parent 464,000

out of a total of 5,632,000 families renting or owning their dwellings. By

using the distribution of these by age of wife it is possible to see in cross-

section the various stages in the family life cycle (Canadian Statistical

Review, September 1984, p. viii).

464 18. Family Demography

Once the unit intermediate between the individual and the population

is deﬁned and enumerated in a census, so that classiﬁed counts are avail-

able of the number of households and the number of persons in individual

households, tabulations can be made of the distribution of units by type

(husband and wife and children, widow with child, etc.) according to census

characteristics of individuals (occupation, earnings of senior member) and

of the dwellings that contain them (number of rooms, rent paid).

Beyond these static qualities at the time of the census are family his-

tory materials, portraying longitudinally how individuals circulate through

families over time—arrival of children, their departure 20 or so years later,

the deaths of successive family members. Analysis of typical family histo-

ries and the way a family is transformed from one type to another, i.e.,

following the family through its event history, is the object of extensive

modelling and simulation that will be discussed below. Comparative study

between countries and cultures, as well as between historical epochs, is an

important part of family history.

18.1.2 Theory and Statistical Compilation

The Italian statistician Barberi complained about the shortage of statisti-

cal data on the family, attributing it to the lack of theoretical study of the

family as such (in quanto tale) as against its individual components (Bar-

beri 1972). A statistical agency could not usefully collect data on an entity

whose essential features had not been brought out by theoretical analysis.

On the other hand, the United Nations (1963) says the opposite—that

the lack of theory is due to the lack of statistical information: “The paucity

of demographic studies of families and households is due largely to the lack

of pertinent census and survey data.”

This is the classical problem of the direction of causation that dogs all

social science. Both contentions are right. Theory and data inﬂuence each

other reciprocally, and the absence of one is a handicap to the other. But

an underlying factor operates to hamper both—as John Bongaarts (1983)

tells us, the complexity of the subject makes it diﬃcult both to gather data

and to develop rational understanding.

Thus at the meta-level of the discussion of family demography, i.e., be-

fore we get into the subject proper, there lies a problem: does A cause B

(Barberi), does B cause A (United Nations), or are both A and B caused

by C (Bongaarts)?

18.2 Kinship

To see the eﬀect of the ﬂows of mortality, fertility, and divorce on kinship we

need not census-type data but a model in which cross-sectional numbers of