Some extensions of the keyfitz momentum relationship

A stable population, such that the total birthrateB(t) =B _o e ^ro^t, is abruptly altered by modifying the age-specific birth rate,m(x). The survivor function remains unaltered. The modified population ultimately settles down to a stable behavior, such thatB(t) =B ₁ e ^r ₁ ^t . It is shown thatB ₁/B ₀ = (R ₀ ?R ₁)/[(r ₀ ?r ₁)R ₀ Z ₁], whereR ₀,R ₁ are the net reproduction rates before and after the change, and \(\bar Z_1 \) expected age giving birth for the stable population after the change. The age structure and transients resulting from the change are also described. The effect of an abrupt change in the survivor functionl(x) is also investigated for the simple case where the change is caused by alteringl(x) toe ^?λx l(x). It is shown that the above ratio becomes \(B_1 /B_0 = N_1 /N_0 = [1 - \smallint _0^\infty e^{ - kx} g(x)dx]/\bar Z_1 \lambda \) , whereN refers to the numbers in the population,k =r ₀ + λ, andg(x) =m(x)l(x), the value before the change. A measure for the reproductive worth of the population is also established.     

Life table technique for the working ages

Some results of efforts to design an integrated formula system for age-specific death rates, survivals and expectation of life are presented. The report deals only with the active ages 15–80. The system is based on the assumption that the age-specific central mortality rate (m _x ) can be expressed satisfactorily by means of a polynomial m = a ₀ + a ₁ x + a ₂ x ²+ ….     

On the momentum of population growth

If age-specific birth rates drop immediately to the level of bare replacement the ultimate stationary number of a population will be given by (9): $$\left( {{\textstyle{{b\mathop e\limits^ \bullet {}_0} \over {r\mu }}}} \right)\left( {\frac{{R_0 - 1}}{{R_0 }}} \right)$$ multiplied by the present number, where b is the birth rate, r the rate of increase, \(\mathop e\limits^ \bullet _0 \) the expectation of life, and R ₀ the Net Reproduction Rate, all before the drop in fertility, and μ the mean age of childbearing afterwards. This expression is derived in the first place for females on the stable assumption; extension to both sexes is provided, and comparison with real populations shows the numerical error to be small where fertility has not yet started to drop. The result (9) tells how the lower limit of the ultimate population depends on parameters of the existing population, and for values typical of underdeveloped countries works out to about 1. 6. If a delay of 15 years occurs before the drop of the birth rate to replacement the population will multiply by over 2. 5 before attaining stationarity. The ultimate population actually reached will be higher insofar as death rates continue to improve. If stability cannot be assumed the ultimate stationary population is provided by the more general expression (7), which is still easier to calculate than a detailed projection.     

On long-term mortality trends in the United States, 1850–1968

This study of United States life tables analyzes the process of mortality transition during 1850–1968. Special features of the study are (1) a phase-specific, rather than an age-specific, analysis of mortality and (2) use of measures based on person-years of life (nL _x ) in phase-intervals, rather than survival rates (nPx) or expectation of life at given ages (e _x ^o). The analysis suggests that the historical transition of mortality in the United States can be described as a three-stage process: an initial stage of slow improvement in life expectancy during 1850–1900, a second stage of rapid improvement during 1900–1950, and a third stage of slower improvement since 1950. Quantitative measures of rapidity of mortality decline in the several phases indicate that they are not identical for all phases and in all stages. The analysis also suggests that there have been rapid changes in the components of overall mortality differentials by sex and race in the United States. The paper draws attention to the need for studies of factors in variations of mortality at ages beyond 50 in the United States population subgroups.     

Effect of mortality change on stable population parameters

The stable population model is used to establish formulas expressing the effects of mortality change on population growth rates, birth rates, and age composition. The change in the intrinsic growth rate is shown to be quite accurately approximated by the average decline in age-specific death rates between age zero and the mean age at childbearing in the stable population. This change is essentially independent of the initial level of fertility in the population. Changes in birth rates and age composition are shown to be simple functions of the age pattern of cumulative changes in mortality rates relative to an appropriately defined “neutral” standard.     

A short note on the taeuber paradox

When the force of mortality is reduced by a constant fraction 0 at every age, the relative increase in life expectancy e(0) can be measured by δH, where H is determined by the l(a) values of the life table. Although H is not easily reducible in terms of the well-known life table parameters, it has been shown that it can be approximately estimated by 2 – e(0)/à in which d is the average age of the stationary population. It has been found that, for a given value of 0, the relative gain in life expectancy is less appreciable in countries with larger values of e(0).     

The Eventual Frequencies of Kin in a Stable Population

Associated with every real birth cohort of women is a set of probabilities {f _k} of eventually having _k daughters. With a variant of stable population theory, these probabilities are used to generate the entire probability distributions, as well as all moments, for all categories of kin who are female and female-related. With additional assumptions, a full two-sex model for all kin also is given. The two-sex model is applied to a cohort of U.S. women born in the mid-twentieth century, suggesting plausible frequencies of kin in a stationary population.     

Social components of metropolitan population densities

The decline of population density from the center of metropolitan areas can be expressed mathematically as: d _r = d _o e ^gr where d _r is the population density of a subarea at distance r from the center, d _o is the hypothetical density at the center, and g is the population-density gradient, empirically always negative. Expanding this exponential model permits examining systematically the relationship between distance from center and various components of population density—housing-unit density, vacant units, household size, and group-quarters population-and the change over time in these components. For the metropolitan areas of Columbus, Dayton, Hartford, Miami, and Syracuse in 1950 and 1960, housing-unit density decreased from the center more sharply than population density. Vacancies, which increased slightly at the center, were proportionately low in the stable middle zones but somewhat higher in the rapidly growing outer zones. While household size decreased around the center between 1950 and 1960, on the periphery it remained constant or increased slightly because of increased family size. During the same decade, the group-quarters population, relative to total population, shifted outward from the center to the periphery to a small extent.     

The precision of population projections studied by multiple prediction methods

A population projection is a prediction of a random vector variable X _T . which represents the size and age/sex distribution of the population in year T. The population is assumed to be closed and to develop according to fixed and known schedules of birth and death probabilities as a multitype branching process. The precision of the usual projection e _T (= EX _T) is studied by a family of prediction intervals of linear functions of the vector of deviations X _T — e _T , which has a preassigned probability level. This family is obtained by a multi-normal approximation and an argument similar to the one leading to Scheffé's method of multiple comparison. From the family of prediction intervals, an upper limit of the total absolute deviation Σ |X _iT ? e _it | is obtained, and the ratio of this limit to the projected total population is proposed as a measure of the relative precision of the projection. For a numerical study, Norwegian population data is used.     

Including the Smoking Epidemic in Internationally Coherent Mortality Projections

We present a new mortality projection methodology that distinguishes smoking- and non-smoking-related mortality and takes into account mortality trends of the opposite sex and in other countries. We evaluate to what extent future projections of life expectancy at birth (e ₀) for the Netherlands up to 2040 are affected by the application of these components. All-cause mortality and non-smoking-related mortality for the years 1970–2006 are projected by the Lee-Carter and Li-Lee methodologies. Smoking-related mortality is projected according to assumptions on future smoking-attributable mortality. Projecting all-cause mortality in the Netherlands, using the Lee-Carter model, leads to high gains in e ₀ (4.1 for males; 4.4 for females) and divergence between the sexes. Coherent projections, which include the mortality experience of the other 21 sex- and country-specific populations, result in much higher gains for males (6.4) and females (5.7), and convergence. The separate projection of smoking and non-smoking-related mortality produces a steady increase in e ₀ for males (4.8) and a nonlinear trend for females, with lower gains in e ₀ in the short run, resulting in temporary sex convergence. The latter effect is also found in coherent projections. Our methodology provides more robust projections, especially thanks to the distinction between smoking- and non-smoking-related mortality.     

Cyclically stable populations∗

The cyclically stable population relaxes the stable population assumption of fixed vital rates and replaces it with the assumption of a recurring sequence of schedules of vital rates. From any point (or stage) in one cycle of the sequence to the same stage in the next cycle, the cyclically stable population grows at a constant rate (λ). While the age composition of the cyclically stable population is different at different stages of the same cycle, it always has the same age composition at the same stage of every cycle. The essential dynamics of the cyclically stable model are captured by its birth projection matrix (BPM). The dominant eigenvalue of the BPM is growth rate A, and the right eigenvector associated with λ gives the within cycle‐birth sequence.

An important special case occurs when λ = 1, and a cyclically stationary population arises. Such populations challenge simplistic ideas about “Zero Population Growth.”; A population projection based on the sets of rates observed in the United States, 1970–90, shows a cyclically stationary population arising in less than 100 years. While it experiences no long term growth, that cyclically stationary population exhibits fluctuations in total size and considerable variability in age structure.     

The analysis of linkages in demographic theory

Many seemingly different questions that arise in the analysis of population change can be phrased as the same technical question: How, within a given demographic model, would variable y change if the age- or time-specific function f were to change arbitrarily in shape and intensity? At present demography lacks the machinery to answer this question in analytical and general form. This paper suggests a method based on modern functional calculus for deriving closed-form expressions for the sensitivity of demographic variables to changes in input functions or schedules. It uses this “linkage method” to obtain closed-form expressions for the response of the intrinsic growth rate, birth rate, and age composition of a stable population to arbitrary marginal changes in its age patterns of fertility and mortality. It uses it also to obtain expressions for the transient response of the age composition of a nonstable population to time-varying changes in the birth sequence, and to age-specific fertility and mortality patterns that change over time. The problem of “bias” in period vital rates is also looked at.     

Dynamics of Mixed Coalitions Under Social Cohesion Constraints

ABSTRACT

Parameters for the birth and death diffusion life table model subject to downward jumps randomly occurring at a constant rate are estimated. The jump magnitudes have a beta distribution with support [0, l_x ], where l_x is the total number of survivors prior to the jump. The estimation method is maximum likelihood. The Cramer–Rao Lower bound and the asymptotic distribution for the MLE are derived. The model is applied to the U.S. men′s population from 1900 to 1999.     

Counter-Examples about Lower- and Upper-Bounded Population Growth

ABSTRACT

For a unimodal growth function f having its maximum at a critical state x _c , the interval bounding the population size asymptotically is usually presented as being equal to [f ^○2(x _c ), f(x _c )]. This interval however does not represent the maximum range within which the population size can vary, even asymptotically. The actual invariant interval containing the population size is equal to: [min(x*, f ^○2(x _c )), f(x _c )], where x* denotes the non-zero fixed point, assumed to be unique, of the iteration of f.     

Tobacco smoking and the sex mortality differential

This paper examines the effects of tobacco smoking on the sex mortality differential in the United States. It is found that all forms of smoking combined account for about 47 percent of the female-male difference in ₅₀ e ₃₇ (life expectancy between ages 37 and 87) in 1962,and about 75 percent of the increase in the female-male difference in ₅₀ e ₃₇over the period 1910–62. When these percentage effects of smoking are decomposed each into a sum of contributions by age and immediate medical cause of death, the degenerative diseases acting at the older ages are found to be of primary importance. The above results appear in large part to explain why the degenerative diseases also account for most of the 1910–65 increase in the female-male difference in life expectancy at birth. The analysis assumes that spurious effects due to the correlation of tobacco consumption with other mortality-related factors are small compared to the causal effects of tobacco consumption itself.     

Expectancy of life at birth in 36 nationalities of the Soviet Union: 1958-60

In the 36 nationalities of the Soviet Union the estimated expectancy of life at birth ranged from 50·0 years for Chechens to 71·1 years for Latvians with a median of about 67·5 years for Russians.

In essence, the life table function e₀ was generated from the child-woman ratios with the use of intricate equations based on empirical data obtained from official Soviet publications. A modified version of Bourgeois-Pichat's model was used to estimate life expectancies at birth among the 36 nationalities on the basis of their crude death rates and the percentage of population aged 65 years and over. The 1959 U.S.S.R. Census of Population provided information pertaining to the older age groups. The crude death rates were estimated separately with the aid of second-degree polynomials fitted to the crude demographic measures for 109 administrative areas of the Soviet Union for 1960.

Information about recent improvements in public health, as well as conjectural evaluations of economic advancement in recent years were examined and related to the past and present level of mortality among the Russian people and the remaining population of minorities.     

Measurement of non-randomness in spatial distributions

Summary The measurement of departure from randomness in spatial distributions has widespread application in ecological work. Several “indices of non-randomness” are compared with regard to their dependence on sample number, sample size and density. Criteria for the best choice of index for specific situations are discussed. A new coefficientC _x is proposed for use with positively contagious distributions and tests of significance are given. WhenC _x and another index (S ²/m−1) are used for positive and negative contagion respectively, values ranging from −1 through 0 (random) to +1 are obtained, regardless of sample number, sample size or density.     

The momentum of mortality change

Mortality change is not usually assigned much importance as a source of population growth when future population trends are discussed. Yet it can make a significant contribution to population momentum. In populations that have experienced mortality change, cohort survivorship will continue varying for some time even if period mortality rates become constant. This continuing change in cohort survivorship can create a significant degree of mortality-induced population change, a process we call the ‘momentum of mortality change’. The momentum of mortality change can be estimated by taking the ratio of e ₀ (the period life expectancy at birth) to CAL (the cross-sectional average length of life) for a given year. In industrialized nations, the momentum of mortality change can attenuate the negative effect on population growth of declining fertility or sustained below-replacement fertility. In India, where population momentum has a value of 1.436, the momentum of mortality change is the greatest contributor to its value.     

A stochastic computer model for simulating population growth

Summary A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model’s behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka’s Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values orr _m andK. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetleTribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Age structure,growth, attrition and accession: A new synthesis

This paper shows that each equation describing relationships among demographic parameters in a stable population is a special case of a similar and equally simple equation that applies to any closed population and demonstrates some implications of these new equations for demographic theory and practice. Much of formal demography deals with functions that pertain to individuals passing through life, or to a stationary population in which births of individuals are evenly distributed over time. These functions include life expectancy, probabilities of survival, net and gross reproduction rates, expected years spent in various states and the probability that certain events will occur in the course of life. The stable population model permits the translation of population structure or processes in a more general type of population, with constant growth rates, back into equivalent populations for a stationary population. The method for translation developed in this paper, requiring only a set of age-specific growth rates is even more general, applying to any population. Age specific growth rates may also be useful for performing reverse translations, between a population's life table and its birth rate or its age distribution. Tables of numbers of females by single years of age in Sweden are used to illustrate applications. Tables summarize the basic relations among certain functions in a stationary population, a stable population and any population. Applications of new equations, particularly to demographic estimation of mortality, fertility and migration, from incomplete data, are described. Some other applications include; the 2 sex problem, increment decrement tables, convergence of population to its stable form, and cyclical changes in vital rates. Stable population models will continue to demonstrate long term implications of changes in mortality and fertility. However, in demographic estimation and measurement, new procedures will support most of those based on stable assumptions.