Keyfitz N., Caswell H. Applied Mathematical Demography

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12.2. The Technique of Projection 275

Table 12.2. Female-dominant projection matrix A with 36 rows and 36 columns

in 5-year age intervals; l

= l

∗

= 1; male life table distinguished by asterisk

⎛

⎜

⎝









···

00···

00 0 ···

00···

0 ···

00···

. ···

00s

∗





∗





···

00···

00 0 0 ···

∗

0 ···

00 0 0 ···

∗

···

00 0 0 ···

. ···

⎞

⎟

⎠

depending on the relative frequencies of males and females of the various

ages or stages. For a review of such models and the dynamics they can

produce, see MPM Chapter 17).

12.2.1 Age Versus Other Variables

As we saw in Chapter 3, the concentration on age in the projection

described above, though it follows a long and honorable tradition, may

be considered arbitrary from a wider viewpoint. We know that mortality

and fertility rates depend on age, but they also depend (though less de-

cisively) on rural and urban residence, education, social class, and many

other variables.

A projection matrix that recognizes rural and urban residence but

disregards age can easily be devised:

An(0) =







(0), (12.2.4)

where a

is the fraction moving from rural to urban, and a

that from

urban to rural. If a

and a

were zero, a

and a

would be the change

ratios of rural and urban parts. By repeated application of (12.2.4), the

population at the end of t periodsof5yearsisA

n(0). The procedure

would be preferred if a move to an urban area caused a change in natural

increase greater than the change in going from one age group to another.

276 12. Projection and Forecasting

Recognition of rural and urban residence does not preclude recognition of

age. We can make area the primary classiﬁcation and show the breakdown

by age within each area, as does Rogers (1968, p. 13), or else we can take

age ﬁrst and show within each age the rural and urban areas, as does Feeney

(1970). Both of these approaches have been used in recent multiregional

matrix models (Rogers 1995, Hastings 1992, Bravo et al. 1995, 1997, Sanz

and Bravo 1999, 2000, Lebreton 1996, Lebreton and Gonzalez-Davila 1993).

Hunter and Caswell (2004) have presented an approach that makes it easy

to move from one to the other.

The usual model of projection recognizes age and sex, sometimes region,

less often rural and urban parts, and in a few countries race. But age and

sex have been the universal dimensions of population projection since the

1940s; no other is accepted as indispensable.

The concentration on age and sex could represent a cultural lag. It might

be that at certain times this breakdown is important, at other times less so.

To see just how much improvement results from recognizing age and sex, as

against not recognizing them, the experiment of Table 12.3 was carried out.

For each of 20 populations a 15-year projection was made by age and sex,

using the age-sex-speciﬁc rates of birth and death for the initial moment—

1 year, 3 years, or 5 years, but always symmetrically arranged around the

midpoint of the stated jumping-oﬀ year. Except for the constancy of the

rates assumed and the omission of migration, this is the usual projection

by periods and is designated as (b) in the table.

Contrasted with this is the simpler projection in which no account is

taken of age and sex, but the crude or observed rates of birth and death

are made to provide a rate of increase, and that rate is applied to the total

population N

= n(t) of the jumping-oﬀ moment. This method, which

has not been used by demographers in good standing for the last 70 years

or more, is shown as (a) in the table.

The observed population 15 years later is the last column of the table,

and projections (a)and(b) may be compared with it. Our intention, before

seeing the outcome, was to estimate quantitatively how much improvement

is due to the incorporation of age and sex, by seeing how much closer column

(b) was to the subsequent performance. That column (a) would turn out

equally well was never anticipated. In fact, in the slight majority of the

instances (marked with an asterisk) the simple projection (a) comes closer

to the performance than (b), which recognizes age and sex.

There are obvious qualiﬁcations in regard to the result. An experiment

involving 20 cases is too small, especially since they are not entirely in-

dependent of one another; when the European birth rate was falling,

somewhat the same inﬂuences acted on age distribution in all countries.

(However, a similar experiment on an additional 20 cases, using a 10-year

prediction span, gave a similar result.) The use of ﬁxed rather than chang-

ing rates of birth and death is a handicap equally to methods (a)and(b).

The same is true for the absence of migration, omitted because data are

12.2. The Technique of Projection 277

Table 12.3. Projections for 20 populations over a 15-year time period, (a) neglect-

ing age and sex, and (b) taking account of age and sex, in both cases neglecting

migration. The projections are compared with observed populations 15 years

later; the closer of (a) and (b) is starred

Rate of

Projected 15 Years

natural By age and

increase Without sex

Population per 1000 age and sex Total of Observed

(000s) population N

15r

/1000 A

n(t)/1000 time t +1

Time (t) N

/1000 1000r (a)(b) N

t+15

/1000

United States

1920 106,630 10.93 125,627

∗

123,665 127,252

1945 131,976 11.64 157,153

∗

150,910 179,990

Japan, 1951 84,260 15.71 106,651 106,607

∗

98,859

Belgium, 1935 8,279 3.01 8,661

∗

8,346 8,646

France, 1876 36,824 3.54 38,832 38,447

∗

38,139

Hungary, 1951 9,423 8.76 10,746 10,427

∗

10,179

Netherlands, 1935 8,387 11.70 9,996

∗

9,889 10,114

Norway, 1951 3,296 10.08 3,834 3,674

∗

3,753

Sweden

1780 2,104 8.59 2,393 2,381

∗

2,274

1805 2,418 6.39 2,661

∗

2,666 2,573

1830 2,876 6.70 3,180 3,216

∗

3,296

1855 3,625 9.23 4,163

∗

4,144 4,164

1880 4,572 11.96 5,470

∗

5,485 4,896

1905 5,278 10.68 6,195

∗

6,265 5,876

1930 6,131 3.18 6,431

∗

6,409 6,630

1945 6,636 9.12 7,609

∗

7,287 7,499

Switzerland, 1951 4,749 7.39 5,306

∗

5,118 5,917

England and Wales

1941 38,743 1.41 39,571

∗

38,259 44,667

1946 40,595 7.36 45,333

∗

43,273 46,166

Australia, 1951 8,422 13.61 10,329

∗

9,887 11,550

more diﬃcult to come by, and also because it presents special problems:

should one use the absolute numbers of the jumping-oﬀ year or the rates,

and how would they be applied by age and sex?

The conclusion from the experiment is that in some circumstances the

eﬀect of age is important, and in others it is not. The experiments summa-

rized in Table 12.3 are not the only ones in which simple models seem to

outperform models including more demographic detail (Keyﬁtz 1981, 1982;

see the review by Rogers 1995b). Such results hardly constitute justiﬁca-

tion for abandoning age, especially since we usually need the forecast by

age, but we should not be under the illusion that projection by age and sex

is a powerful technique for discerning the future.

To ﬁnd which of the other dimensions can provide real gains in accuracy

of forecast will require additional experimenting similar to that of Table

12.3. We ought to be trying to project marriages, and then applying rates of

nuptial fertility; perhaps not age within marriage but duration of marriage

is the superior variable; perhaps ﬁrst births, second births, and so on are

278 12. Projection and Forecasting

more easily extrapolated separately than are births of all orders together,

and so the model ought to be order-speciﬁc.

Whatever the variables recognized, their change from period to period

is easily provided for. Each 5-year period would have its own projection

matrix, so that after t periodswewouldhavenotA

n(0), but instead, if

i−1

is the matrix for the ith period,

n(t)=A

t−1

···A

n(0). (12.2.5)

Several diﬀerent cases of time variation must be distinguished, because dif-

ferent analytical methods are available for each: periodic variation (MPM

Chapter 13), stochastic variation (MPM Chapter 14), and variation due to

nonlinear feedback between the population and its own vital rates (MPM

Chapter 16). Whereas the changing matrix can be symbolically incorpo-

rated into a formula without diﬃculty, and analyzed with some eﬀort, the

attempt to determine the elements of the matrix, that is, to say what future

mortality and fertility will be, is the most diﬃcult problem of demography,

and will be the subject of Section 12.4.

12.2.2 Projection in a Heterogeneous Population

When a heterogeneous population is projected without recognition of its

subpopulations, the result is always lower than what would have been found

by applying the same projection process to the subpopulations and adding.

This was shown to be true in Section 1.5, without recognizing age. It is true

also for projections by age, but to prove it in general for an arbitrary initial

age vector is diﬃcult.

As an example that will suggest the quantitative eﬀect, the United States

population in 1966 without breakdown by color, as projected with a life

table made from the deaths in that year and the fertility rates implied by

the 1966 births, results in a total population for 1981 of 230,477,000. Rec-

ognizing the two separate groups of Whites and Nonwhites, whose 1966

populations, births, and deaths add exactly to the totals for the United

States, constructing life tables and age-speciﬁc birth rates for the two

groups separately, and then projecting each by means of its own life ta-

ble and birth rates, gives 1981 Whites as 199,287,000 and Nonwhites as

31,441,000, which add to 230,728,000. The deﬁciency of the projection not

recognizing color is 251,000 or, about 1 part in 1000, after 15 years. After

100 years the two separate projections add to 8 percent more than the

projection without breakdown by color.

12.3 Applications of Projection

Projection leads itself to making “if ...then ...” statements; indeed it con-

sists of such statements. If England and Wales have the same age-speciﬁc

12.3. Applications of Projection 279

birth and death rates as in 1968 for the following 25 years, the total pop-

ulation will grow from the 48,593,000 estimated for 1968 to 54,869,000 in

1993. The increase will average 0.0049 or 4.9 per 1000 per year, less than

the intrinsic rate r of 1968, which was 6.9 per 1000. The slower rate of the

projection is due to an unfavorable 1968 age distribution.

The United States 1967 population, projected at 1967 rates, amounts to

230,109,000 by 1982, an average rate of increase of slightly over 1 percent

per year. In this case the rate of increase is raised by an exceptionally large

proportion of women in childbearing ages; the intrinsic rate r of 1967 was

only 0.74 percent.

12.3.1 Population Dynamics with One Cause of Death

Eliminated

A transparently simple application of projection involves the eﬀect on pop-

ulation growth of eliminating a particular cause of death, say cancer. We

want the eﬀect of this change separate from the eﬀects of all other changes

that may take place. Excluding the eﬀect of the birth rate or of income

incurs no diﬃculties, but excluding changes in other causes of death raises

a problem if these causes would become more (or less) serious as a direct

result of eliminating cancer. Since we know little about the interrelations of

the several causes, we assume that age by age the others remain unchanged

when cancer is eliminated. (Section 2.6)

A life table was calculated omitting all mortality from cancer, and an-

other omitting all mortality from heart disease, using the technique of

Section 2.6 and data for the United States, 1964. The unaltered table and

each of these two modiﬁcations were applied to the 1964 female population

of the United States to separate out the eﬀects of the two causes on the rate

of increase and on the age distribution of the ultimate stable population.

The resulting rates and age distributions are shown in Table 12.4, always

supposing that the birth rates of 1964 continue to prevail.

The intrinsic rate of natural increase is aﬀected by less than 1 part in

200 by omission of heart disease: it rises from 0.01573 to 0.01579. Not

many women are struck down by heart disease at ages before the end of

childbearing. Cancer deaths are fewer in total, but they aﬀect younger ages

in somewhat more cases and so their omission produces a slightly greater

rise in the rate of increase.

To represent the eﬀect on age distribution we have used the stable equiv-

alent q, corresponding to an age distribution, or an age-sex distribution

n(0), obtained in (8.8.3) as

q =

n(0)

5rt

, (12.3.1)

where t is large, say 100 cycles of 5 years, and e

is the ratio of increase per

cycle as t becomes large. The stable equivalent is the number of persons in

280 12. Projection and Forecasting

Table 12.4. Ultimate age distribution and intrinsic rates corresponding to United

States 1964 population, with (a) all causes of death, (b) all causes except heart

disease, and (c) all causes except cancer (thousands of persons)

(a)(b)(c)

All causes Death from heart Death from

of death disease removed cancer removed

Stable equivalent Stable equivalent Stable equivalent

Age Males Females Males Females Males Females

0–4 10,535 10,125 10,537 10,127 10,537 10,126

5–9 9,698 9,329 9,699 9,330 9,700 9,330

10–14 8,943 8,609 8,942 8,608 8,944 8,609

15–19 8,229 7,941 8,227 7,939 8,230 7,940

20–24 7,545 7,317 7,543 7,315 7,546 7,315

25–29 6,910 6,737 6,911 6,737 6,914 6,736

30–34 6,326 6,195 6,332 6,198 6,332 6,197

35–39 5,774 5,682 5,795 5,693 5,786 5,693

40–44 5,238 5,193 5,290 5,216 5,260 5,219

45–49 4,700 4,719 4,808 4,762 4,740 4,768

50–54 4,141 4,249 4,335 4,326 4,212 4,332

55–59 3,547 3,780 3,864 3,906 3,662 3,902

60–64 2,921 3,299 3,390 3,503 3,088 3,463

65–69 2,268 2,788 2,907 3,109 2,485 2,992

70–74 1,629 2,242 2,429 2,721 1,872 2,474

75+ 2,003 3,491 7,833 11,856 2,596 4,168

Total 90,407 91,696 98,842 101,346 91,904 93,264

Total both sexes 182,103 200,188 185,168

Females

15–44

Total

× 100 42.60 38.57 41.92

Female mean age 31.32 36.79 32.03

Intrinsic rates

Birth 0.02348 0.02124 0.02308

Death 0.00774 0.00546 0.00726

Natural increase 0.01573 0.01579 0.01582

Sex ratio

All ages 0.986 0.975 0.985

Ages 75+ 0.574 0.661 0.623

each category of age and sex that could be projected to (distant) time t by

multiplication by e

at all ages, with assurance that the same numbers at

time t would result as from projection of the observed population by the

usual components method described in Section 12.2.

The stable equivalent shown in Table 12.4 is little aﬀected at younger

ages by the omission of heart disease or cancer, and even by age 50 the

increase is less than 5 percent for males and 2 percent for females. At the

very oldest ages of 75 and over, however, the eﬀect of eliminating heart

disease is dramatic: a nearly fourfold rise for men and well over a threefold

increase for women.

12.3. Applications of Projection 281

The net result is a rise in the average age of females from 31.32 to 36.79

years with the elimination of heart disease, and to 32.03 with the elimina-

tion of cancer. The percentage of females in the childbearing ages 15 to 44

falls from 43.21 to 39.56 with the elimination of heart disease, and to 42.54

with the elimination of cancer.

The main limitation of the result given in Table 12.4 is the certainty

that if heart disease were eliminated other causes would also be aﬀected.

They would be increased insofar as heart disease selects weaker individuals,

more subject to other ailments than randomly selected members of the

population, an issue raised in Section 4.4.

12.3.2 Eﬀect of Immediate Drop to Replacement Fertility

Continuing with the eﬀort to make population projection perform exper-

iments, we ask another question: “What would be the ultimate stationary

population if birth rates were to drop immediately and permanently to

bare replacement level?” Though a drop to bare replacement would sooner

or later terminate growth, a large concentration of women at childbearing

ages exists as a result of growth in the past, and the ultimate level would

be considerably higher than the present population.

The question may be answered by a population projection, applying the

observed life table and observed age-speciﬁc birth rates divided by R

,the

net reproduction rate. We refrain from trying to guess what mortality is

going to do, nor do we suppose changes in the age distribution of child-

bearing. The object is to see the outcome of a drop to replacement fertility

if everything else remains unchanged.

Taking the birth and death rates for the year listed, making a life table

from the death rates, dividing the birth rates by R

, and projecting some

500 years into the future gives the results shown in Table 12.5 for six coun-

tries. Thus Colombia showed 17.993 million people in 1965; the projection

climbed to 29.786 million, a rise of 65.5 percent. Such a projection takes

full account of the initial age distribution.

The results can be compared with the corresponding calculation on the

stable age distribution. We saw in Section 8.6 that the ratio of the ultimate

stationary to the present population, on the assumption of an immediate

drop to bare replacement fertility, is

rµ



− 1



, (12.3.2)

where the birth rate b, the rate of natural increase r, the net reproduction

rate R

, and the mean age µ of childbearing in the stationary population

are all calculated before the fall. Applying (12.3.2) to Colombian females

of 1965 gives 1.59, that is, an ultimate rise of 59 percent over the 1965

population. This is somewhat less than the 66 percent rise on the projection,

so presumably the observed 1965 Colombian sex–age distribution departed

282 12. Projection and Forecasting

Table 12.5. Current and ultimate stationary populations, on assumption that

birth rates drop immediately to stationary level, for six countries

Percent

Number (000s)

increase

Population Current Ultimate to ultimate

Chile, 1965 8,584 12,916 50.5

Colombia, 1965 17,993 29,786 65.5

Ecuador, 1965 5,109 8,518 66.7

Italy, 1966 53,128 62,189 17.1

Peru, 1963 14,713 23,080 56.9

United States

1966 195,857 259,490 32.5

1967 197,863 267,096 35.0

from the stable one in a direction favorable to reproduction. However, the

departure could be an artifact due to defects in the statistics. In the face

of inaccurate primary data it is an advantage to have available both the

projection and the stable formula (12.3.2).

A number of such experiments are described by Frejka (1973). He sees

what the ultimate stationary population will be with an immediate drop

and with a drop spread over 25 years. He does not keep other factors

constant at their base values, but supposes that mortality will fall according

to a certain curve. This has the advantage of greater realism if indeed the

projected fall of mortality occurs. On the other hand, it has the drawback

of requiring us to keep in mind a particular fall in mortality rather than

just constancy. Such a conﬂict between realism and simplicity is typical of

demographic projection.

We have all thought how convenient it would be if we knew the condition

at some distant future and could interpolate between the jumping-oﬀ point

and then. Sooner or later the population must cease to grow; and if we

could only guess the date at which it would become stationary, most of

the problem of forecasting would be solved, for the interpolation could not

involve the gross errors to which extrapolation is subject. Frejka’s treatment

is an attempt along this line.

12.4 The Search for Constancies

Any forecasting method, whether naive or sophisticated, depends on some

function being approximately constant. For the short term it may be suf-

ﬁcient to suppose that the absolute number of births of the current year

will be repeated over the next few years, and the same for deaths and

migration. More convincing is to suppose that the crude rates will be con-

stant for the next few years. Insofar as age distribution is changing, the

12.4. The Search for Constancies 283

rates that are supposed ﬁxed had better be age speciﬁc. Insofar as a visible

trend appears in the age-speciﬁc rates, one may wish to take year-to-year

diﬀerences of the age-speciﬁc rates as ﬁxed; if

is the death rate at time

t for the age interval x to x +4, then

−

t−1

being constant will

give

t+1

−

t−1

. If the past trend looks to be exponential,

the logarithm of the numbers in previous years can be supposed to have

ﬁxed ﬁrst diﬀerences.

If the trend in total population can be assumed to be logistic, what is

ﬁxed is the ratio of the diﬀerence of reciprocals in successive years; with

standing for the population in year t,

1/N

− 1/N

t−1

1/N

t−1

− 1/N

t−2

is constant.

Better adapted ways of choosing the entities that are to be held con-

stant are provided by relational methods, which deserve a more detailed

treatment.

12.4.1 Relational Methods

To project mortality or fertility or nuptiality—indeed, any component

that varies with age—we need to determine not only overall trends but

also what is going to happen to the several ages. We could suppose that

all ages are aﬀected in the same way, either adding to all age-speciﬁc rates

some constant for each year that goes by, or multiplying them by some

constant. But actual changes are far from being either constant increments

or constant multiples applicable to all ages. We could meet this objection

by extrapolating each age separately, but that would produce gratuitous

errors, as well as inconvenient discontinuities; with most data age distri-

butions would become less and less smooth as we went forward in time on

separate projections for the several ages.

We could observe how past changes have occurred, for instance, how

successive mortality tables for Sweden diﬀer, and use the Swedish trend

as showing the future for countries that are not yet down to the Swedish

level of mortality. Similar possibilities exist for fertility and nuptiality. This

method has been used, but it presents diﬃculties. The problem is to ﬁnd

some way of relating the future to the past that involves a small number

of parameters, say two, and that is a suﬃciently good ﬁt to all times for

the two parameters extrapolated to produce acceptable curves for future

dates.

Mathematical curves would serve this purpose if any could be found

that ﬁtted closely enough. If the age distribution of childbearing could be

satisfactorily ﬁtted by a normal curve, for example, we could extrapolate

the total, mean age, and variance of ages to ﬁnd the age distribution of

childbearing at future times. But Lotka’s (1939, p. 70) normal curve does

284 12. Projection and Forecasting

not ﬁt past data well enough to justify its use for prediction, nor does

the incomplete gamma function applied by Wicksell (1931) or Hadwiger’s

(1940) more elaborate exponential. (Keyﬁtz 1968, Chapter 6, is a secondary

source on these.) Murphy and Nagnur (1972) tried a Gompertz curve, long

used for mortality, with some success.

Success in specifying algebraic or transcendental curves has been incom-

plete enough that most workers resort to sets of model tables, for fertility

as well as for mortality. A set of useful model fertility tables has been pub-

lished by Coale and Trussell (1974), which do for this component what the

Coale and Demeny (1966) tables did for mortality.

An alternative is the relational method evolved by Brass (1974) over the

course of some years of experimenting. Instead of aiming at the desirable

but unavailable mathematical curve, or attempting to sum up the entire

range of empirical materials in a set of model tables, the relational method

embraces changes through time in a ﬂexible description involving two (or

at most three) parameters. Let us see how the Brass method ﬁts mortality;

fertility and nuptiality may be treated in a similar way.

Mortality. In the ﬁrst application, to mortality, Brass found that to go

from one life table to another was easy if both were translated into logits,

that is, their l(x) columns were transformed by

Y (x)=

log



1 − l(x)

l(x)



If the early table is distinguished by a subscript s, the relation is the linear

Y (x)=α + βY

(x),

or, written out in full,

log



1 − l(x)

l(x)



= α +

log



1 − l

(x)



In forecasting one would take a life table for time t

as l

(x) and ﬁt to it

the table for later time t

, ﬁnding the constants α

and β

,say.Anewα

and β

would be found on ﬁtting the table for t

to t

,andα

and β

ﬁtting the table for t

to t

. The several ﬁttings would indicate a trend in α

and in β that could then be extrapolated into the future. The extrapolated

α and β for future dates, along with the base life table l

(x), would carry

the life table to the dates in question. Studying mortality trends of the past

half-century in England and Wales, Brass considers it possible that in the

future β maywellcomeclosetounity,andα decrease by 0.5 every 40 years.

The trend in α and in β must be worked out for each population on which

the method is tried. The method imports nothing—neither a mathematical

curve nor model tables—from the experience of other populations than the

one under consideration: for England and Wales β and the ﬁrst diﬀerence

of α are constant, but other functions of α and β would be used in other

populations.