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\title{Demographic Analysis: Stochastic Approach}
\author{Krishnan Namboodiri\\
Professor Emeritus\\
Ohio State University, Columbus, Ohio, USA}

\maketitle

\section{Introduction}

Demographers study population dynamics: changes in population size and
structure resulting from fertility (reproduction), mortality (deaths),
and spatial and social mobility. The focus may be the world
population or a part of it, such as the residents of a country or the
patients of a hospital. Giving birth, dying, shifting usual place of
residence, and trait changes (e.g., getting married) are called
events. Each event involves transition from one ``state'' to another
(e.g., from never-married state to married state). A person is said to
be ``at risk'' or ``exposed to the risk'' of experiencing an event, if for
that person the probability of that experience is greater than zero.
The traits influencing the probability of experiencing an event are
called the risk factors of that event (e.g., high blood pressure, in
the case of ischemic heart disease). Demographic data are based on
censuses, sample surveys, and information reported to offices set up
for continuously recording demographic events. Some observational
studies can be viewed as random experiments. For an individual
selected at random from a population at time $t$, the value of the
variable $y_{t+\theta}$, denoting whether that individual will be
alive as of a subsequent moment $t+\theta$ is unpredictable. This
unpredictability of the value of $y_{t+\theta}$ qualifies the
observational study involving observations at times $t$ and $t+\theta$
to be considered as a random experiment and $y_{t+\theta}$,
as a random variable, defined by a set of possible values it may take
(e.g., $1$ if alive at time $t+\theta$, and $0$, otherwise), with
a probability function associated therewith (Kendall and Buckland
1971). The interval between a fixed date and a subsequent event
is a random variable, in the above-mentioned sense.

The term rate is used in demography for the number of events (e.g.,
deaths) expressed per unit of some other quantity, such as
person-years at risk (often expressed per $1000$). For example, the
\textit{crude death rate} (annual number of deaths expressed
per $1000$ mid-year
population) in Japan in 2007 was $9$. The mid-year population in such
calculations is an approximation to the sum of the person-years lived
by the members of the population involved during the specified
year. Death rates calculated for sub- populations, homogeneous, to
some degree, with respect to one or more relevant risk factors are
called specific death rates. Examples are age-specific and age-sex
specific death rates.

A \textit{life-table} shows the life-and-death history of a group of persons,
called a \textit{cohort}, born at the same time (e.g., a year), as the cohort
members survive to successive ages or die in the intervals, subject to
the mortality conditions portrayed in a schedule of age-specific death
rates. An account of the origin, nature, and uses of life tables is
available in P. R. Cox (1975). Life tables have become powerful tools
for the analysis of non-renewable (non-repeatable) processes. If a
repeatable process, such as giving births, can be split into its
non-renewable components (e.g., births by birth order) then each
component can be studied, using the life-table method. The term:
\textit{survival analysis} is applied to the study of non-renewable processes,
in general. Associated with the survival rate is the hazard rate,
representing the instantaneous rate of failure (to survive). Hazard
rate corresponds to the instantaneous death rate or force of
mortality, as used in connection with life tables.

\section{Macro-Level Focus}

A great deal of demographic research is linked directly or indirectly
to model construction and validation, viewing observations as outcomes
of random experiments. Birth-and-death process (see Kendall, 1948;
Bhat, 1984) is a continuous time, integer valued, counting process, in
which population size at time $t$, remains constant, increases by one
unit (a birth), or decreases by one unit (a death), over the period: $t$
to $t+\Delta t$. Time-trend in population size is studied using branching
processes, in a simple version of which, each member of each
generation produces offspring, in accordance with a fixed probability
law common to all members (see, e.g., Grimmett and Stirzaker, 1992 for
a discussion of simple as well as complex models of branching
processes). The logistic process for population growth of the
``birth-and-death'' type views the instantaneous rates of birth and
death per individual alive at a given moment as linear functions of
population size (see Brillinger, 1981; Goel and Dyn, 1979; Mollison,
1995). For compositional analysis, one may apply an appropriate
log-ratio transformation to the composition of interest, and treat the
resulting values as a random vector from a multivmultivariate normal
distribution (see Aitchison, 1986; Namboodiri, 1991).

Using the component model (see Keyfitz, 1971) of population projection,
one obtains internally consistent estimates of the size and age-sex
composition of populations as of future years by combining
hypothesized patterns of change in fertility, mortality, and
migration. On the basis of such projections, issues such as the
following can be examined: (1) Reduction in population growth rate
resulting from the elimination of deaths due to a specific cause,
e.g. heart disease; (2) Relative impact on the age-composition, in the
long-run, of different combinations of population- change components
(e.g., fertility and mortality); and (3) tendency of populations to
``forget'' the past features (e.g., age composition) if the components
of population dynamics were to continue to operate without
change over a sufficiently long time.

To estimate and communicate the uncertainty of population projections,
the practitioners have been combining ``high,'' ``medium,'' and
``low'' scenarios for the components of population change in various
ways (e.g., ``high'' fertility combined with ``low'' mortality to produce
``high'' population projection) to show different possibilities
regarding future population size and composition. Since such
demonstrations of uncertainties have no probabilistic interpretations,
Lee and Tuljapurkar, among others, have pioneered efforts to develop
and popularize the use of stochastic population projections (see Lee,
2004). Lee and Tuljapurkar (1994) demonstrated, for example, how to
forecast births and deaths, from time-series analyses of fertility and
mortality data for the United States, and then combine the results
with deterministically estimated migration to forecast population size
and composition. They used in the demonstration, products of
stochastic matrices.

Comparison of the simple non-stochastic trend model:
$y_t=\beta_0+\beta_t (t)+e_t$, with the stochastic (random-walk with a
drift) model: $y_t=\alpha_0+y_{t-1}+e_t$, where $e_t$'s are $NID
(0,\sigma_e)$ for all $t$, shows that even when the error terms have
equal variance $(\sigma^2_e)$ in the two models, the prediction
intervals for the latter are wider than those of the former: For a
forecast horizon $H$, the variance of the forecast error (the
departure of the forecast from the actual) in the case of
$y_t=\beta_0+\beta_1 (t)+e_t$ is $\sigma^2_e$, while the corresponding
quantity is $H\sigma^2_e$, in the case of $y_t=\alpha_0+y_{t-1}+e_t$.

\section{Micro-Level Processes}

At the micro level, one focuses on events (such as giving birth to the
first child, dying, recovering from illness, and so on) experienced by
individuals. In event histories, points of time at which transitions
occur (e.g., from not in labor force to employed) are represented by a
sequence of non-negative random variables: $(T_1,T_2,\dots )$, and the
differences: $V_k=T_k - T_{k-1}$, $k=2,3,\dots$, are commonly referred to
as waiting times. Comprehensive discussions of waiting times are
available, for example, in: Cleves et al.\ (2004); Collett (2003);
Elandt-Johnson and Johnson (1980/1999); and Lawless (1982/2003).

D. R. Cox (1972) introduced, what has come to be known as, the
proportional hazards model: $\lambda (t)=\lambda_0(t)\psi (z_1,z_2,
\dots ,z_k)$, where ``$t$'' represents time, and the multiplier,
$\psi (z_1,z_2,\dots ,z_k)$, is
positive and time-independent. A special form of the model is:
$\lambda (t) = \lambda_0(t)\exp (\Sigma\beta_jz_j)$, in which
$\{\beta_j\}$ are unknown regression coefficients.

An important feature of waiting time is heterogeneity (variation among
individuals) in the hazard rate (see Sheps and Menken, 1973; Vaupel et
al., 1979; Heckman and Singer, 1982). Heterogeneity is incorporated
often as a multiplier in the Cox proportional hazards model. For
example, the hazard function for the $i$th individual may be specified
as: $\lambda (t) =\lambda_0(t)\nu_i\exp (\Sigma\beta_jZ_{ij})$,
representing an individual-specific,
unobserved heterogeneity factor by $\nu_i$. Vaupel et al.\ (1979) called
such models: ``frailty'' models.

Heckman and Singer (1982) suggested the specification of the
unobserved heterogeneity factor in $\lambda (t) =
\lambda_0(t)\nu_i\exp (\Sigma\beta_jZ_{ij})$, as a
$K$-category discrete random variable. Thus the $i$th individual is
presumed to belong to one of $K$ groups. The value of $K$ is determined
empirically so as to maximize the likelihood of the sample on hand,
under a specified (e.g., the exponential or Weibull) form for
$\lambda_0(t)$. In the presence of heterogeneity, inference becomes sensitive
to the form assumed for the hazard function (see, e.g., Trussell and
Richards, 1985).

As Sheps and Perin (1963) and Menken (1975), among others, have
pointed out, simplified models, unrealistic though they may be, have
proved useful in gaining insights such as that a highly effective
contraceptive used by a rather small proportion of a population
reduces birth rates more than does a less effective contraceptive used
by a large proportion of the population.

Some fertility researchers have been modeling parts rather than the
whole of the reproductive process. The components of birth intervals
have been examined, with emphasis on the physiological and behavioral
determinants of fertility (see Leridon, 1977). Another focus has been
abortions, induced and spontaneous (see: Abramson, 1973; Potter et
al., 1975; Michels and Willett, 1996). Fecundability investigations
have been yet another focus (see Menken, 1975; Wood et al., 1994).
Menken (1975) alerts researchers to the impossibility of reliably
estimating fecundability from survey data. The North Carolina
Fertility Study referred to in Dunson and Zhou (2000) is of interest
in this connection: In that study couples were followed up from the
time they discontinued birth control in order to attempt
pregnancy. The enrolled couples provided base-line data and then
information regarding ovulation in each menstrual cycle, day-by-day
reports on intercourse, first morning urine samples, and the
like. Dunson and Zhou present a Bayesian Model and Wood et al.\ (1994)
present a multistate model for the analysis of fecundability and
sterility.

To deal with problems too complex to be addressed using analytic
models, researchers have frequently been adopting the simulation
strategy, involving computer-based sampling and analysis at the
disaggregated (e.g., individual) level. See, for example, the study of
(1) kinship-resources for the elderly (Murphy, 2004; Wachter, 1997); (2)
female family-headship (Moffit and Rendall, 1995); (3) AIDs and the
elderly (Wachter et al., 2002); and (4) the impact of heterogeneity on
the dynamics of mortality (Vaupel and Yashin, 1985; Vaupel et
al., 1979). Questions such as the following arise: Is it possible to
reproduce by simulation the world-population dynamics, detailing the
changes in the demographic-economic-spatial-social DESS) complex, over
the period, say: 1900-2000? Obviously, in order to accomplish such a
feat, one has to have a detailed causal model of the observed changes
to be simulated. As of now no satisfactory model of that kind is
available. Thinking along such lines, demographers might begin to view
micro-simulation as a challenge and an opportunity to delve into the
details of population dynamics.

Based on an article from Lovric, Miodrag (2011),
\textit{International Encyclopedia of Statistical Science}.
Heidelberg: Springer Science+Business Media, LLC.

Dr.\ Krishnan Namboodiri was Robert Lazarus Professor of Population
Studies at The Ohio State University, Columbus, Ohio, USA, (1984-2000)
and has been Professor Emeritus at the same institution since 2000.
Before joining The Ohio State University, he was Assistant Professor,
Associate Professor, Professor, and Chairman, Department of Sociology,
University of North Carolina at Chapel Hill, USA, (1966-1984); Reader
in Demography, University of Kerala, India,
(1963-1966). Dr. Namboodiri was Editor of Demography (1976-1979), and
Associate Editor of a number of professional journals such as
Mathematical Population Studies (1985-1989). He has authored or
co-authored over 80 publications including 12 books. He is a Fellow of
the American Statistical Association, and is a recipient of honors
such as Lifetime Achievement Award from Kerala University, and has
been consultant from time to time to Ford Foundation, World Bank,
United Nations, and other organizations.

\section*{References and Further Readings}

\hangindent=2em

\hangindent=2em Abramson, F.D. (1973) High foetal mortality and birth
intervals. \textit{Popul. Stud.} 27:235-242.

\hangindent=2em Aitchison, J. (1986) \textit{The Statistical Analysis
of Compositional Data}. Chapman and Hall, London.

\hangindent=2em Bhat, U.N. (1984) \textit{Elements of Applied
Stochastic Processes, 2nd edn.} Wiley, New York.

\hangindent=2em Brillinger, D.R. (1981) Some aspects of modern population
mathematics. \textit{Can. J. Stat.} 9:173-194.

\hangindent=2em Cleves, M., Gould, W.G., and Gutierrez, R.G. (2004)
\textit{An Introduction to Survival Analysis Using Stata, Revised
edn.} Stata Press, College Station.

\hangindent=2em Collett, D. (2003) \textit{Modeling Survival Data in
Medical Research, 2nd edn.} Chapman and Hall/CRC, Boca Raton.

\hangindent=2em Cox, D.R. (1972) Regression models and life tables
(with discussion). \textit{J. Roy. Stat. Soc., Series B} 34:187-202.

\hangindent=2em Cox, P.R. (1975) \textit{Population Trends, Vols
I.II.} Her Majesty's Stationary Office, London.

\hangindent=2em Dunson, D.B. and Zhou, H. (2000) A Bayesian model for
fecundability and sterility. \textit{J. Am. Stat. Assoc.}
95(452):1054-1062.

\hangindent=2em Elandt-Johnson, R. and Johnson, N. (1980/1999)
\textit{Survival Models and Data Analysis}. Wiley, New York.

\hangindent=2em Goel, N.S. and Dyn, N.R. (1979) \textit{Stochastic Models
in Biology}. Academic Press, New York.

\hangindent=2em Grimmett, G.R. and Stirzaker, D.R. (1992)
\textit{Probability and Random Processes, 2nd edn.} Clarendon
Press, Oxford.

\hangindent=2em Heckman, J.J. and Singer, B. (1982) Population
heterogeneity in demographic models. In: \textit{Multidimensional
Mathematical Demography}, Land, K.C., Rogers, A. (eds). Academic
Press, New York, pp.\ 567-599.

\hangindent=2em Kendall, D.G. (1948) A generalized birth and death
process. \textit{Ann. Math. Stat.} 19:1-15.

\hangindent=2em Kendall, M.G. and Buckland, W.R. (1971)
\textit{Dictionary of Statistical Terms, 3rd edn.} Hafner, New York.

\hangindent=2em Keytz, N. (1971) Models. \textit{Demography} 8:329-352.

\hangindent=2em Lawless, J.F. (1982/2003) \textit{Statistical Models
and Methods for Lifetime Data, 2nd edn.} Wiley, New York.

\hangindent=2em Lee, R.D. (2004) Quantifying our ignorance:
Stochastic forecasts of population and public budgets.
In: \textit{Aging, Health, and Public Policy: Demographic and
Economic Perspectives}, Waite LJ (ed). \textit{Population and Development
Review (A Special Supplement)} 30:153-175.

\hangindent=2em Lee, R.D. and Tuljapurkar, S. (1994) Stochastic
population projections for the United States beyond high, medium,
and low. \textit{J. Am. Stat. Assoc.} 89(438):1175-1189.

\hangindent=2em Leridon, H. (1977) \textit{Human Fertility: The
Basic Components} (trans: Helzner, J.F.). University of Chicago
Press, Chicago.

\hangindent=2em Menken, J. (1975) Biometric models of fertility.
\textit{Soc. Forces} 54:52-65.

\hangindent=2em Michels, K.B. and Willett, W.C. (1996) Does induced or
spontaneous abortion affect the risk of cancer? \textit{Epidemiology}
7:521-528.

\hangindent=2em Moffit, R.A. and Rendall, M.S. (1995) Cohort trends in
the lifetime distribution of family headship in the United States,
1968-1985. \textit{Demography} 32:407-424.

\hangindent=2em Mollison, D. (ed) (1995) \textit{Epidemic Models:
Their Structure and Relation to Data}. Cambridge University Press,
London.

\hangindent=2em Murphy, M. (2004) Tracing very long-term kinship
networks using SOCSIM. \textit{Demogr. Res.} 10:171-196.

\hangindent=2em Namboodiri, K. (1991) \textit{Demographic Analysis:
A Stochastic Approach}. Academic Press, San Diego/New York.

\hangindent=2em Potter, R.G., Ford, K., and Boots, B. (1975)
Competition between spontaneous and induced abortions.
\textit{Demography} 12:129-141.

\hangindent=2em Shepsm, M.C. and Menken, J. (1973)
\textit{Mathematical Models of Conception and Birth}. University
of Chicago Press, Chicago.

\hangindent=2em Sheps, M.C. and Perin, E.B. (1963) Changes in birth
rates as a function of contraceptive effectiveness: Some applications
of a stochastic model. \textit{Am. J. Public Health} 53:1031-1046.

\hangindent=2em Trussell, T.J. and Richards, T. (1985)
Correcting for unobserved heterogeneity in hazard models:
An application of the Heckman-Singer model for demographic data. In:
\textit{Sociological Methodology}, Tuma, N.B. (ed). Jossey-Bass, San
Francisco, pp.\ 242-276.

\hangindent=2em Vaupel, J.W., Manton, K.G., and Stallard, E.
(1979) The impact of hetero geneity in individual frailty on
the dynamics of mortality. \textit{Demography} 16:439-454.

\hangindent=2em Vaupel, J.W. and Yashin, A.J. (1985) Heterogeneity
ruses: Some surprising effects of selection in population dynamics.
\textit{Am. Stat.} 39:176-185.

\hangindent=2em Wachter, K.W. (1997) Kinship resources for the
elderly. \textit{Philos. T. Roy. Soc. B} 352:1811-1817.

\hangindent=2em Wachter, K.W., Knodel, J.E., and Vanlandingham, M.
(2002) AIDs and the elderly of Thailand: Projecting familial
impacts. \textit{Demography} 39:25-41.

\hangindent=2em Wood, J.W., Holman, D.J., Yashin, A.I., Peterson,
R.J., Weinstein, M., and Chang, M.C. (1994) A multistate model of
fecundability and sterility. \textit{Demography} 31:403-426.
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