Encyclopedia of Sociology

DEMOGRAPHIC METHODS

different from that within the corresponding in-
terval of the life table population, then implicitly
by using this projection device we are improperly
weighing the component parts of the broad inter-
val with respect to survival probabilities.

Parenthetically, if we desired to determine the
size of any component of the population under t
years old—in this particular example, ten years
old—we would have to draw upon fertility as well
as mortality information, because at time τ these
individuals had not yet been born.

HAZARDS MODELS

Suppose we were to examine the correlates of
marital dissolution. In a life table analysis, the
break-up of the marriage (as measured, e.g., by
separation or divorce) would serve as the analogue
to death, which is the means of exit in the standard
life table analysis.

In the study of many duration-dependent phe-
nomena, it is clear that several factors may affect
whether an individual exits from a life table. Cer-
tainly, it is well-established that a large number of
socioeconomic variables simultaneously impinge
on the marital dissolution process. In many popu-
lations, whether one has given birth premaritally,
cohabited premaritally, married at a young age, or
had little in the way of formal education, among a
whole host of other factors, have been found to be
strongly associated with marital instability. In such
studies, in which one attempts to disentangle the
intricately related influences of several variables
on survivorship in a given state, we invoke a haz-
ards model approach. Such an approach may be
thought of as a multivariate statistical extension of
the simple life table analysis presented above (for
theoretical underpinnings, see, e.g., Cox and Oakes
1984 and Allison 1984; for applications to marital
stability, see, e.g., Menken, Trussell, Stempel, and
Babakol 1981 and Bennett, Blanc, and Bloom 1988).

In the marital dissolution example, we would
assume that there is a hazard, or risk, of dissolu-
tion at each marital duration, d, and we allow this
duration-specific risk to depend on individual char-
acteristics (such as age at marriage, education,
etc.). In the proportional hazards model, a set of
individual characteristics represented by a vector

of covariates shifts the hazard by the same propor-

tional amount at all durations. Thus, for an indi-

vidual i at duration d, with an observed set of

characteristics represented by a vector of covariates,

Zi, the hazard function, μi(d), is given by:

μi(d) = exp [ λ (d) ] exp [Zi β] (^12 )

where ß is a vector of parameters and λ(d) is the

underlying duration pattern of risk. In this model,

then, the underlying risk of dissolution for an

individual i with characteristics Zi is multiplied by a

factor equal to exp[Ziß].

We may also implement a more general set of

models to test for departures from some of the

restrictive assumptions built into the proportional

hazards framework. More specifically, we allow for

time-varying covariates (for instance, in this exam-

ple, the occurrence of a first marital birth) as well

as allow for the effects of individual characteristics

to vary with duration of first marriage. This model

may be written as:

μi(d) = exp [ λ (d) ] exp [Zi (d) β (d) ] (^13 )

where λ(d) is defined as in the proportional haz-

ards model, Zi(d) is the vector of covariates, some

of which may be time-varying, and ß(d) represents

a vector of parameters, some of which may give

rise to nonproportional effects. The model pa-

rameters can be estimated using the method of

maximum likelihood. The estimation procedure

assumes that the hazard, μi(d), is constant within

duration intervals. The interval width chosen by

the analyst, of course, should be supported on

both substantive and statistical grounds.

INDIRECT DEMOGRAPHIC ESTIMATION

Unfortunately, many countries around the world

have poor or nonexistent data pertaining to a wide

array of demographic variables. In the industrial-

ized nations, we typically have access to data from

rigorous registration systems that collect data on

mortality, marriage, fertility, and other demograph-

ic processes. However, when analyzing the demo-

graphic situation of less developed nations, we are

often confronted with a paucity of available data

on these fundamental processes. When such data

are in fact collected, they are often sufficiently