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A simple formula for estimating age-specific event rates for a period from "before" and "after" cross sections is developed and demonstrated. The general approach applies to a wide range of estimation problems in demography, the social sciences, and epidemiology. The method arises from the formal mathematics of unstable populations and is similar in spirit to "variable-r" methods. Unlike those methods, however, the new technique does not require specialized computer programming or iterative calculations, and event rates can be calculated directly from cross-sectional data in simple spreadsheets. The article includes a formal mathematical exposition of the method, simulation tests, and several examples.
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A SIMPLE METHOD FOR ESTIMATING AGE-SPECIFIC RATES FROM SEQUENTIAL CROSS SECTIONS*
I develop and demonstrate a simple formula for estimating age-specific event rates for a period from "before" and "after" cross sections. The general approach applies to a wide range of estimation problems in demography, the social sciences, and epidemiology. The method arises from the formal mathematics of unstable populations and is similar in spirit to "variable-r" methods. Unlike those methods, however, the new technique does not require specialized computer programming or iterative calculations, and event rates can be calculated directly from cross-sectional data in simple spreadsheets. The article includes a formal mathematical exposition of the method, simulation tests, and several examples.
Demographers and other scientists frequently wish to learn about event rates by comparing sequential cross sections. For example, we may want to investigate age-specific fertility over a period by comparing "before" and "after" parity schedules (Coale, John, and Richards 1985), to learn about young adults' moves in and out of parental homes by comparing age schedules of the proportion living with parents from consecutive censuses (Zeng et al. 1994), or to study changes in smoking behavior by comparing sequential surveys on the prevalence of smoking by age (Stoto 1988). Table 1 presents several more examples, illustrating that the issue of inferring rates from cross sections arises in many fields.
In Table 1 and throughout this article, V(a,t) represents a population value of interest at exact age a and time t (e.g., the proportion of women sterilized, the proportion agreeing that Communists have a right to free speech, or the average number of previous arrests for drunk driving). In applied work, a demographer usually observes average values for age groups, denoted ^sub n^V^sub x^(t) for age group x to x + n at time t, and must interpolate to find V values at exact ages.1
The function delta(a,t) is the focus of analysis-it represents net, additive, intracohort change in V per unit time.2 In other words, V(a,t) represents a cumulative sum of changes within a cohort between birth and age a, and delta(a,t) represents the rate at which cohort V changes per unit of time. In some applications, the events of interest cause one-way changes in V-for example, fertility "events" can only increase mean cohort parity. In other applications, events could drive V in either direction-for example, changes in individual living arrangements could cause the proportion of cohort members who coreside with children to rise or fall. In the latter situation delta measures the net addition to V per unit time caused by events of interest.
In this article I develop and demonstrate a new method for estimating age- and period-- specific event rates delta from sequential cross sections on V. The method arises from the formal mathematics of unstable populations and is similar in spirit to "variable-r" methods developed by Preston, Coale, and others.3 Unlike existing methods, however, the new approach does not require specialized computer programming or iterative calculations, and event rates can be estimated using elementary, spreadsheet-style calculations.
1. There are many possible interpolation methods, including moving averages and splines. I illustrate several of these approaches in this article.
2. Note that 5 is a rate in the sense that it measures events per year, day, or minute, but not in the sense of events per unit of exposure. I discuss this issue more thoroughly in later sections.
3. The elaboration of precise mathematical expressions for nonstable population identities by Preston and Coale (1982) was pivotal in a series of important articles on estimation methods and nonstable demography. Among these articles were those by Bennett and Horiuchi (1981, 1984), Preston (1983), Coale (1984, 1985), Arthur and Vaupel (1984), Coale et al. (1985), Stoto (1988), and Stupp (1988). In terms of the underlying mathematics, the closest cousins to this article are those by Arthur and Vaupel (1984), which analyzed the geometry of the Lexis surface V(a,t), and Coale (1985), which examined additive relationships between vital events and the number of individuals crossing each edge of a Lexis rectangle.
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CARL P. SCHMERTMANN
*Carl P. Schmertmann, Center for the Study of Population, Florida State University, Tallahassee, FL 323062240; E-mail: [email protected]. Earlier versions of this article were presented at the 2000 Southern Demographic Association meetings in New Orleans and at the 2001 annual meetings of the Population Association of America in Washington, DC. I thank Monica Boyd, Tom Bryan, Joe Potter, Ken Wachter, and an anonymous reviewer for their helpful comments and critiques.
Copyright Population Association of America May 2002