# Multistate Demography

# MULTISTATE DEMOGRAPHY

Multistate demography is the study of populations stratified by age, sex, and one or several attributes, such as region of residence, marital status, number of children, living arrangement, employment status, occupation, and health status. A population that is stratified is a multistate population, and people who occupy the same state constitute a subpopulation. The dynamics of multistate populations are governed by differential fertility and mortality, the transfer of individuals between subpopulations.

## A Brief History

Multistate demography was pioneered by Andrei Rogers in the 1960s and 1970s. Rogers's aim was to generalize classical demographic models–the life table, population projection models, the stable population model–from two states (alive and dead) to multiple states of existence. As a specialist in urban and regional planning, Rogers's interest was mainly in regional population dynamics and migration, and the generalization was to a system of regions: a multiregional system. His first results appeared in *Demography* in 1966 and in book form in 1975. The book covers the multiregional life table, the continuous and discrete models of multiregional demographic growth (the Lotka and Leslie models), and the estimation of multiregional demographic measures from incomplete data. Rogers demonstrated that the generalization of demographic techniques to multiple states is relatively straightforward. Changes in multiregional populations are described by systems of simultaneous linear equations, conveniently represented in matrix notation. The broadening of multiregional demography into multistate demography was triggered by Robert Schoen's treatment of a population stratified by marital status (Schoen 1975).

In England, the geographer Philip Rees developed an accounting system for multiregional populations, pursuing some ideas from Rogers and having been influenced by earlier work of the economist Richard Stone (1913–1991), who initiated economic and social accounting in the early 1960s (Rees and Wilson, 1977). Accounts–in this case comprising population stocks and flows–have a great advantage: They must balance. Differences in data type, inconsistencies, and other data problems are easily identified.

From their beginning, multistate models followed the accounting tradition prevailing in demography and the actuarial sciences. Multistate models, and in particular the multistate life table, however, could also be viewed as applications of mathematical statistics, based in probability theory. Supporters of this perspective–Jan M. Hoem, Michael T. Hannan, and others–identified common features of the questions demographers try to answer using the life table and those addressed in the fields of survival analysis and event-history analysis with their focus on models of duration dependence. Age is viewed as a duration variable. The two distinct traditions persist (see Bogue et al. 1993, Chapters 21-22 for an accessible introduction).

## Multistate Models

At any point in time, an individual occupies a state, and the distribution of people over the various states determines the population structure. State occupancies change over time as a result of (1) interstate transitions people experience–for example, from being single to being married, from being diseased to being healthy, or from being a resident of one region to being a resident of another region, and (2) differential entries from and exits to the rest of the world. The multistate life table describes how the size and composition of a (synthetic) cohort change over time. Multistate projection models describe how the population structure (stock) at a given time depends on the initial population and the transitions people make (flows).

The dynamics of a multistate population–a cohort or an age-graded population–are based on transition rates and transition probabilities. Rates relate the number of transitions people make to the *duration* at risk of a transition. Probabilities relate transitions to the *population* at risk at the beginning of an interval.

Transition rates and transition probabilities are estimated from the data. The estimation of probabilities directly from the data is complicated in the presence of censoring (i.e., if individuals enter or leave the population during the period of observation for a reason unrelated to the transitions being studied). In survival analysis, the concept of *risk set* has been introduced to distinguish the population at risk of experiencing an event during an interval from the population present at the beginning of that interval. The estimation of rates does not present that problem since the transitions are related to the time spent in the origin state during the interval. In this approach, people may enter and/or leave a state during an interval. Transition rates must be converted into probabilities. The task is straightforward if the rates vary between age intervals but not within age intervals, or when the transitions that occur during an interval are uniformly distributed.

## Applications

Early applications considered populations stratified by age and region of residence. The life table was used to estimate the regional distribution of members of a synthetic cohort and the number of years spent in the different regions. For example, using data from the 1980 census and vital statistics, Rogers (1995, p. 91) found that a person born in New York and subjected to migration and mortality patterns of the late 1970s, may expect to live 74 years, of which 18 years are in the South including 3 years in Florida. The period in the South is concentrated at higher ages. The life table also gives the number of migrations cohort members may experience in a lifetime. Rogers and Frans Willekens (1986) present multiregional life tables for several countries. Multiregional population projections are used widely because regional populations change more in response to migration than to fertility or mortality. A multiregional model is the only one that considers migration by origin and destination.

Another popular area of application is family and household demography. The life table produces indicators such as the probability that a marriage ends in a divorce, the mean age at divorce, the expected duration of marriage at divorce, and the expected number of divorces in a lifetime. It also reports the probability that a married woman at a given age, 32 (for example), will experience a divorce within 10 years and the probability that she will be divorced at age 60 (for example). The multistate life tables follow women and men through their marital careers. The text *Family Demography* (1987), edited by John Bongaarts, Thomas Burch, and Kenneth Wachter, which includes descriptions of the marital careers of American women, children's experiences with different types of families, and the family types generated over the life course of a cohort, has stimulated the development and application of multistate models. Multistate projection models allow researchers to move beyond the widely used headship rate method and to consider changes in the number and types of families and households in terms of the demographic events people experience and the transitions they make to new family or household types.

Epidemiology and public health are other important areas of application of the multistate models. The state space distinguishes states of health and may consider specific diseases, impairments, disability or handicaps. The life table estimates the probability that a person of a given age develops a disease over a period of 5 years, 10 years, or a lifetime, and adds the probability of recovery if data permit. It also yields estimates of expected duration of the disease. Kenneth Manton and Eric Stallard (1988) developed multistate life tables for chronic diseases. Historically, multistate models have been applied and further developed in epidemiology and public health, in particular to assess the length of healthy life and the effect of risk factors on morbidity and mortality. A review by D. Commenges (1999) reveals, however, that many studies using multistate models do not stratify the population by age and do not use a multistate life table. Commenges concludes that "the strong effect of age is not very well taken into account" (1999, p. 332). The situation is changing, however, age is becoming a significant time scale in epidemiology and is leading to the new subfield of life-course epidemiology. A paper on the cardiovascular life course by Anna Peeters and others (2002) illustrates how the multistate life table may be used to describe a particular disease history of a cohort and how the life table may be used to improve the estimates of lifetime risk of the disease and years with the disease attributable to risk factors.

As the area of application is extending into new fields of research, the multistate life table is developing into a technique that moves beyond the description of a (synthetic) cohort into a method that accounts for intra-cohort variation. Two developments are currently under way. The first considers the effect of covariates on transition rates and probabilities. The multistate life table with covariates generalizes the semiparametric Cox model and parametric duration models to multiple and transient states. This change is a movement toward the construction of synthetic individual biographies rather than cohort biographies. That action requires techniques of microsimulation to produce samples of individual life histories on the computer that are consistent with the empirical evidence on life histories. The second development results in probabilistic multistate life tables that account for the effects of sampling variation. The most modern procedure is to produce probabilistic life tables using bootstrapping.

The prediction of individual life histories combining data on the individual and on people with similar characteristics, and accounting for the uncertainties involved, may initiate a new era in which the multistate life table becomes an instrument for life planning and contingency analysis.

**See also:** *Event-History Analysis*; *Life Tables*; *Migration Models*; *Renewal Theory and the Stable Population Model*; *Stochastic Population Theory*.

## bibliography

Bogue, Donald J., Eduardo E. Arriaga, and Douglas L. Anderton, eds. 1993. *Readings in Population Research Methodology.* Chapter 21: "Survival and Event History Methods"; Chapter 22: "Multistate Methods." Chicago: Published for the United Nations Population Fund by Social Development Center.

Bongaarts, J., T. Burch, and K. Wachter, eds. 1987. *Family Demography.* Oxford: Clarendon Press.

Commenges, D. 1999. "Multi-state Models in Epidemiology." *Lifetime Data Analysis* 5: 315–327.

Hannan, Michael T. 1984. "Multistate Demography and Event History Analysis." In *Stochastic Modeling of Social Processes,* ed. A. Diekmann and P. Mitter. Orlando, FL: Academic Press.

Hoem, Jan M., and U. F. Jensen. 1982. "Multistate Life Table Methodology: A Probabilist Critique." In *Multidimensional Mathematical Demography,* ed. K. C. Land and A. Rogers. New York: Academic Press.

Manton, K. G., and E. Stallard. 1988. *Chronic Disease Modelling: Measurement and Evaluation of the Risks of Chronic Disease Processes.* London: Charles Griffin.

Peeters, A., A. A. Mamun, F. Willenkens, and L. Bonneux. 2002. "A Cardiovascular Life History: A Life Course Analysis of the Original Framingham Heart Study Cohort." *European Heart Journal* 23: 458–466.

Rees, Philip, and A. Wilson. 1977. *Spatial Population Analysis.* London: Edward Arnold.

Rogers, Andrei. 1975. *Introduction to Multiregional Mathematical Demography.* New York: Wiley.

——. 1995. *Multiregional Demography: Principles, Methods, and Extensions.* New York: Wiley.

Rogers, Andrei, and F. Willekens, eds. 1986. *Migration and Settlement: A Multiregional Comparative Study.* Dordrecht, Netherlands: Reidel.

Schoen, Robert. 1975. "Constructing Increment-Decrement Life Tables." *Demography* 12: 313–324.

——. 1988. *Modeling Multigroup Populations.* New York: Plenum Press.

Frans Willekens

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