Keyfitz N., Caswell H. Applied Mathematical Demography

Подождите немного. Документ загружается.

18.6. Family Policy 475

The utilities in the Luce and Raiﬀa payoﬀ matrix are

(A) Husband’s choice

I II

(B) Wife’s

a, b



c, c



a>b>c,

choice

c, c



b, a



(18.5.1)

If a



,andc



>cthen the man has power. If c and c



are more

nearly equal, then the sexes are more equal and the woman has a chance

of attaining her goal. The model applies to a matter as trivial as going out

to dinner versus staying home, as well as to the most solemn decisions that

married couples make.

The implications of a utilitarian or contractual relation have been worked

out in a tradition of thought about the family that runs through Durkheim,

Schumpeter (1950, p. 157), and more recently William Goode. Thus Schum-

peter in a characteristically farsighted phrase speaks of “the heavy personal

sacriﬁces that family ties and especially parenthood entail under modern

conditions,” and Goode, “If larger part of one’s life beneﬁts are to be de-

rived from job holding, in a social setting where emotional relations can be

ﬂeeting and superﬁcial without incurring social disapproval, then it seems

likely that future investments in the family . . . may be lessened” (p. 79, in

Toqueville Review). The stability of the family was all along based on the

division of labor within it, but this could only be maintained with a con-

centration of authority, almost invariably in the male. Such a view, with

its pessimistic conclusions, is supported by statistics of divorce, of later

marriage, of nonmarriage, of increasing illegitimacy.

More upbeat is the argument that democracy within the family of course

changes its nature, and of course no one can deny that divorce has increased,

but the fact is that most people still marry, and the usual purpose of divorce

is to escape one marriage in order to engage in another. It is almost as

though the very concern about the quality of marriage and family life is

what leads to divorce and trying again. No empirical evidence is likely to

disprove this.

18.6 Family Policy

One test of theories of the family is whether policies based on such theories

work. There has been a good deal of such testing in the United States by

policy analysis and supplementation in recent years, especially in the ﬁeld

of welfare, and the results are not encouraging (Glazer 1984): “A program

meant to reduce the distress of widows and unmarried mothers and their

children, and designed to maintain them at a minimal but decent level,

476 18. Family Demography

seemed to be accompanied by a rising number of such women. Welfare

assistance was an incentive for mothers to push fathers out of the home.”

The hope had been that mothers and children would have husbands and

fathers, and that with the support of unemployment insurance, old-age

pensions, and full-employment policies the family would be held together,

with the husband-father the principal wage earner and Aid to Families with

Dependent Children (AFDC) a transitory necessity only. But in fact AFDC

became a program for mothers of illegitimate children, and the number of

these increased rapidly, with half of the number being black.

If that incentive system was wrong, then could not ingenuity devise a

better one? Just as the income tax does not deprive people of the incentive

to work, so support should not prevent them from working, provided they

were able to keep some fraction—say half—of their earnings. This applica-

tion of the income tax principle did operate as theory said it should, but it

too had a perverse eﬀect: in repeated trials it raised the divorce rate. Ev-

idently the grant made wives independent and a larger number of couples

took advantage of this to separate or divorce. The Negative Income Tax

could maintain the incentive to work, but only at the cost of increasing

divorce (Hannan et al. 1977).

An example of the perverse way in which policy measures can operate

is shown by the Swedish experience. In the postwar period Sweden experi-

enced a labor shortage, and its response was various measures to encourage

women to enter the labor force. The measures were successful. As among

women with children, 44.6 percent were in the labor force in 1965 and 69.0

percent in 1975. But the longer-run eﬀect was delay of childbearing and a

lower birth rate. As the resultant small cohorts reach maturity the entrants

into the labor force will diminish.

Thus measures to increase the labor force that were successful in the

short run can have the eﬀect of diminishing the labor force in the longer

run.

Boudon (1977) gives other examples of the perversity of social life in the

face of measures to inﬂuence it. More speciﬁcally in relation to our ﬁeld,

Henripin (1977) shows how diﬃcult it is to alter demographic trends. Yet

in certain less-developed countries genuine examples of the success of the

policy are apparently to be found.

Heterogeneity and Selection in

Population Analysis

Heterogeneity in the underlying population places diﬃculties in the way of

interpretation of all statistical data based on averages. No two persons are

equally likely to die in the next year; no two marriages are equally likely to

be broken up by divorce; no two businesses are equally likely to fail; no two

automobiles are equally likely to break down. That on the average a given

make of automobile will travel 50,000 miles without major repairs oﬀers

little assurance for any particular automobile. Averages can be applied to

individual cases only at great risk. This gross aspect of heterogeneity is not

the subject of the present chapter.

People vary in respect of age, and mortality comparisons between pop-

ulations have attempted to take account of age at least since the time of

John Graunt. More recently, matrix and multi-state methods have made it

straightforward to incorporate measurable sources of heterogeneity other

than, or in addition to, age. Nothing more need be said here beyond the cus-

tomary exhortation to break down aggregates into homogeneous subgroups

for purposes of analysis.

But what about diﬀerences among individuals not ordinarily tabulated

in mortality statistics? Mr. A has a heart murmur, or is a heavy drinker,

and these aﬀect his chances of survival. Beyond such diﬀerences, that

could be distinguished in the data collection but are usually neglected,

are diﬀerences that could not possibly be described. One cannot provide

examples of the indescribable, but one can be certain that two individuals,

even though alike in every possible statistical categorization, do not have

identical probabilities of dying in the next year.

478 19. Heterogeneity and Selection in Population Analysis

People thought of heterogeneity as at most aﬀecting the precision of a

mean; now we know that under many conditions it introduces a clear bias in

the mean as customarily estimated. That bias arises because of a selection

eﬀect: in the random process under consideration those persons likely to

die ﬁrst disappear from the exposed group, leaving the group on average

more robust. This selective eﬀect acts always in the same direction—in the

case of mortality to underestimate the rates to which the average individual

will be subject.

19.0.1 Historical Note

The selection within a heterogeneous population that is our main con-

cern has turned up in various areas of statistics and demography. An early

recognition of the eﬀect is that of Gini (1924) in a two-page paper. In the

1920s, Raymond Pearl followed a group of women from marriage to ﬁrst

pregnancy, counting the number of pregnancies over a considerable num-

ber of months, and dividing by the total woman-months of exposure. Gini

showed that this gives too low a ratio to represent the pregnancy rate for

an unselected group, because the less fecund women are steadily increasing

their weight among the observations. Only the ﬁrst month’s observations

provide a meaningful ratio, and one must throw away all the data for later

months as having no reference to an unselected population. Potter and

Parker (1964) tried by means of graduation to rescue these rejected data,

but they were compelled to use strong assumptions on the distribution of

fecundability among women (see also Wood and Weinstein 1990, Wood

1994)

In respect to migration it was observed (Blumen, Kogan, and McCarthy

1955) that to project a population forward through time as though everyone

were subject to the rates of a given period exaggerates the subsequent

movement. It is an improvement to suppose that the population consists of

two kinds of people: those who never move, and those who have the same

chance of moving in each period.

The Gini problem came to life again in the work of Sheps and Menken

(1973). In respect of mortality it was taken up by Vaupel, Manton, and Stal-

lard (1979), Manton, Stallard, and Vaupel (1981), and Vaupel and Yashin

(1985). Shepard and Zeckhauser (1980) showed the issue to be important

for the interpretation of medical trials. Reliability engineers have studied

the question under the heading of mixed distributions (Mann, Schafer, and

Singpurwalla 1974). Automobiles show higher failure rates in the ﬁrst weeks

of use than later. This is not likely to be due to individual items of equip-

ment becoming better as they are used, the way an organism can gain

strength; most of it must be due to defective machines being repaired or

eliminated.

19.1. Conditioning and the Interpretation of Statistical Data 479

19.1 Conditioning and the Interpretation

of Statistical Data

19.1.1 Simpson’s Paradox

Some numbers will illustrate how unrecognized heterogeneity can make

a comparison uncertain, in a situation more general than those discussed

above in that it does not depend on selection over time (Cohen 1986, Simp-

son 1951). This pervasive aspect of heterogeneity deserves more attention

than it has received. It can be summarized in the assertion that every

comparison depends on the variables on which it is conditioned.

There are two regions, A and B, each with a 1,000,000 labor force. In A

there are 100,000 unemployed, in B 90,000. That is, we have

A B

Labor Percent Labor Percent

force Unemployed unemployed

1,000,000 100,000 10

1,000,000 90,000 9

Apparently, A’s unemployment rate at 10 percent is greater than B’s at

9 percent. Abstracting from sampling error as we do throughout, nothing

would seem more incontestable.

Yet, the obvious conclusion can be wrong. Suppose each of A and B to

break down into two groups, young and old, with the following proportions:

A B

Labor Percent

force Unemployed unemployed

Young 500,000 75,000 15 300,000 48,000 16

Old 500,000 25,000 5

700,000 42,000 6

For the young, B has the higher unemployment, and likewise for the old.

We have to reverse the conclusion based on the aggregate data; now B’s

unemployment is higher. Might a further breakdown make another reversal?

There is no way of knowing, and the eﬀect of underlying heterogeneity can

only induce modesty in the presentation of the comparison. We have to

visualize a penumbra of uncertainty based on the ever-present possibility

that some unsuspected source of heterogeneity will show up, or exists but

will never show up. It would be useful to develop some theory showing the

bounds of that penumbra.

An actual occurrence of this paradox was observed (Cohen and

Nagel 1934, p. 449) in a comparison of tuberculosis deaths

in New York City and Richmond, Virginia, during the year

480 19. Heterogeneity and Selection in Population Analysis

1910. Although the overall tuberculosis mortality rate was lower

in New York, the opposite was observed when the data were

separated into two racial categories; in both the white and non-

white categories, Richmond had a lower mortality rate. (Wagner

1982.)

A more recent real instance appears in the Canadian census of 1976:

Mean number of children

1971–76

French English

Canada 1.85 1.95

Quebec 1.80 1.64

Other provinces 2.14 1.97

Source:R´ejean Lachapelle (1980), La situation demolinguis-

tique au Canada: Evolution passee et prospective. L’Institut

de Recherches Politiques.

Average children for the period between 1971 and 1976 were 1.85 for the

population of French mother tongue and 1.95 for the population of English

mother tongue. Should we conclude, then, that English fertility is higher

than French? Not at all: in a breakdown of rates between Quebec and the

rest of Canada, Quebec showed 1.80 for the French and 1.64 for the English.

The same relation held for other provinces, with the French higher at 2.14

as compared with 1.97 for the English. The initial conclusion is vulnerable

to challenge on the basis of heterogeneity.

The paradox can be grasped intuitively as an aspect of weighting. French

speakers in Canada are mostly in Quebec; English speakers are mostly in

the other provinces. Thus the Canada ﬁgure for the French is mostly a

reﬂection of Quebec fertility; for the English it mostly reﬂects the other

provinces. For the statistician there is no paradox here; he understands

that every comparison depends on the variables on which it is conditioned.

Yet others are likely to feel uncomfortable. The man in the street might

well say, “I am asking the simple question and looking for a yes or no

answer: after all these years of legendary high fertility for the French has

English fertility now come to be higher?” To say that it is literally higher

but that we do not know whether to attribute this to people’s being English

or to their living in a certain part of the country sounds like an evasion. Yet

we have to insist that the simple question is more diﬃcult than it looks.

Extending somewhat an argument due to Colin Blyth (1972), suppose

that we are trying out a new medical treatment in two hospitals and

comparing it with the standard treatment. Suppose also that one of the

hospitals is very famous and attracts the more diﬃcult cases; the other

gets easy cases most of whom would recover however they were handled.

If the new treatment is tried mostly in the hospital with the more diﬃ-

19.2. Heterogeneity and Selection 481

cult cases, and the standard one in the hospital with the easy cases, then

the new treatment will show a higher failure rate. What spoils any such

comparison is the correlation between treatments and hospitals; the only

way to escape the diﬃculty and ﬁnd out which treatment is really better

is to assign cases to the treatments at random in both hospitals. That is

a recourse we do not have in the French–English fertility question. Ran-

domization enables us to avoid the error that can be called misnaming:

attributing to treatment what is really due to hospital.

Note that the diﬃculty is not removed when the whole experiment is

done in a single hospital. If the assignment is not made at random there is

always the possibility that some other variable—unobserved and perhaps

unobservable—is inﬂuencing the comparison of the way the two hospitals

were doing.

In Colin Blyth’s (1972) notation that expresses the principle in its most

general form, where it is known as Simpson’s paradox, it is possible to have

P (A|B) <P(A|B



) (19.1.1)

and yet both

P (A|BC) >P(A|B



C) (19.1.2)

and

P (A|BC



) >P(A|B



). (19.1.3)

Here B = B



and C = C



. In words, the probability of A in the Group B

is less than that in B



, even though within each of two subgroups, say C

and C



, the probability of A is greater for B than for B



Some deep and very general questions of interpretation of statistics are

raised by Simpson’s paradox. The eﬀect of heterogeneity in the unobserv-

ables applies in all comparisons, and so aﬀects nearly any interpretation

of statistical data. The apparent solution, adding additional dimensions

of cross-classiﬁcation, creates its own problems, because as the number of

categories increases, the data become more and more thinly spread over

the possible combinations. Cohen (1986) refers to this as a demographic

uncertainty principle.

19.2 Heterogeneity and Selection

If we take an unbiased sample of the unemployed and follow the individuals

in it month by month to establish the duration of unemployment, we come

face to face with the selective eﬀect of heterogeneity. For those who are more

capable or more active will ﬁnd jobs sooner and drop out of observation,

leaving in the observations those less likely to ﬁnd jobs, and ultimately only

those most diﬃcult to place. Statistics based on this group will exaggerate

the duration of unemployment, as well as the rate of unemployment.

482 19. Heterogeneity and Selection in Population Analysis

If everyone had the same chance of ﬁnding a job the selective eﬀect

could not exist; the bias is the consequence of variability in that chance,

i.e., of heterogeneity. This eﬀect cannot occur instantaneously; it acts when

a group is followed over time. If the heterogeneity is between observable

categories it can be removed; the unobservable heterogeneity is unremov-

able, though indirect ways of estimating its eﬀect have been suggested.

Matters such as unemployment where the event can occur to an individual

more than once are more tractable than mortality, and indeed one could in

principle work out the “within person” variability of the chance of falling

unemployed. Yet even in this case it is highly uncertain that heterogeneity

as measured for one epoch is the same as heterogeneity for another.

Similar statements can be made about divorce. In most populations the

probability of divorce is low in the ﬁrst years of marriage, then rises to

a peak (“the seven-year itch”), then falls oﬀ. Does that mean that the

chance of a particular marriage breaking up rises to a peak, then declines?

Not necessarily: if, for example, there are two kinds of couples—one with

a low and constant probability of divorce, and one with a steadily rising

probability, then the observations would be accounted for by an argument

similar to that applying to death (Vaupel and Yashin 1985, Hoem 1990).

The subject matter ﬁelds aﬀected are numerous: mortality, unemploy-

ment, divorce, risk of conception, migration, mechanical failure, in short

any ﬁeld where individuals drop out of the observed category when the

contingency in question materializes.

19.3 Application to Mortality

In the usual deterministic life table model, for a person of given age the

probability of surviving a year, say 0.99, is based on the collection of

observed deaths and the corresponding exposed population. We may care-

lessly argue that diﬀerent probabilities for individuals, being unmeasurable,

have no meaning; after all the person will be alive or dead in the succeed-

ing period—there is no middle possibility. But that argument falls to the

ground when we think of one person in the hospital with a diagnosis of

incurable cancer, and another of the same age, going about his business in

evident good health. The unrealistic assumption of homogeneity has been

thought to be innocent, in that it would not aﬀect the overall conclusions

drawn from the numbers. Diﬀerences between individuals, therefore, being

both unmeasurable and inconsequential for averages, can be disregarded.

This is the viewpoint that recent research has shown to be unacceptable.

A person of average health or frailty

∗

has something of the order of 1 year

∗

The “frailty” of an individual is supposed to measure the susceptibility of the

individual to risks of death, beyond what is determined by age or other measured co-

19.3. Application to Mortality 483

less expectation of life than is given by the life tables for the population of

whichheisamember.

If an insurance company were to set its rates from population statistics,

and then insure a random subset of the population, it would break even

(abstracting from expenses, etc.). But if its underwriters were to accept as

risks only persons of average robustness, then it would lose.

A medical improvement can save many lives—i.e., it can prevent the

death of persons who without it would have died—and yet raise the overall

death rate in the population. It does this if it changes suﬃciently the mix

of frail and robust individuals in the population.

A mix of populations, in both of which mortality is rising steadily and in-

deﬁnitely, can show a rise to a peak in the death rate, followed by a decline,

then a further rise. That is to say there would be a period when mortality

at the individual level is actually rising, while the overall statistics, the

“observations” at the population level, show a fall.

This issue has become inescapable in studies of mortality among very

old individuals. Studies of humans, as well as of laboratory populations of

fruit ﬂies, nematodes, and yeast, show mortality curves that decline, or at

least decelerate, at the oldest ages (Vaupel et al. 1998). These data have

been obtained by following huge cohorts (millions or even billions of indi-

viduals) to increase sample sizes among the rare extremely old individuals.

Evolutionary theory predicts that the selective pressure against mortality

should decline with age, so that traits with beneﬁcial eﬀects early in life

and detrimental eﬀects late in life should be favored (e.g., Hamilton 1966).

This should lead to the accumulation of genes that increase mortality late

in life, and to mortality rates that increase with age. The observations of

declining mortality with age could mean that the evolutionary theory has

left something out (e.g., Lee 2003), or that the predicted increase is masked

by heterogeneity. The problem becomes only more acute (and interesting)

as more is learned about the biological determinants of longevity (e.g.,

Wachter and Finch 1997).

19.3.1 A Mixture of Populations Having Diﬀerent Rates of

Increase

In one sense the mortality application is a special case of changing pro-

portions of subpopulations that applies to overall population increase or

decrease.

A population of initial size Q growing at rate r numbers Qe

at time

t, r being taken as ﬁxed and the population as homogeneous. Now suppose

heterogeneity—a number of subpopulations, of which the ith is initially Q

variates. It could be determined by genetics, physiology, behavior, or the current or past

environment.

484 19. Heterogeneity and Selection in Population Analysis

growing at rate r

,sothatattimet the total number is P (t)=



We showed (Section 1.5) that the total never stabilizes, that its rate of

increase forever increases, and that the composition constantly changes.

The argument of Section 1.5 leads to the conclusion that the change in

the mean rate of change is σ

(t), the variance among the rates r

, each

weighted according to its current subpopulation Q

. That proves that

with all r

ﬁxed the rate of increase steadily increases, unless the r

are

equal.

For two subpopulations that start out equal with unequal rates of in-

crease the fact that the sum of the separate projections will always be

greater than the combined population projection may be shown as a con-

sequence of the arithmetic–geometric inequality. The sum of the separate

projections from initial values of unity is e

and the single projection

is 2e

)t/2

. The former is greater if

+ e

> 2e

)t/2

i.e., if



√

−

√



> 0,

which is always the case if r

= r

. The central proposition is that when one

projects a heterogeneous population in disregard of the heterogeneity, which

is to say using the average rate of increase for the whole, one underestimates

the subsequent population. To project the population of the United States,

for example, with the parameters of the country as a whole necessarily

gives a lower answer than projecting each state with its own parameters,

and then taking the total for the United States.

19.3.2 Two Classes of Frailty

An especially simple application of the foregoing is where a population with

ﬁxed mortality rates for individuals shows a spurious time-dependence. If

at the outset half of the population suﬀers mortality rate µ

, and half µ

and there are no births, then after x years there will be e

−µ

individuals

of the ﬁrst type, and e

−µ

individuals of the second type. The mortality

rate will then be:

−µ

+ e

−µ

+ e

−µ

and this is bound to be less than the initial rate (µ

+ µ

)/2. To show that

we note by multiplying up that

−µ

+ e

−µ

+ e

−µ

+ µ

if and only if

(µ

− µ

)(e

−µ

− e

−µ

) < 0