A multiple regression method for analysing stage-frequency data

Summary A linear regression method that allows survival rates to vary from stage to stage is described for the analysis of stage-frequency data. It has advantages over previously suggested methods since the calculations are not iterative, and it is not necessary to have independent estimates of stage durations, numbers entering stages, or the rate of entry to stage 1. Simulation is proposed to determine standard errors for estimates of population parameters, and to assess the goodness of fit of models.     

Examination of the generalized age distribution

The formula for the age distribution and other relationships that follow from it for any (non-stable) population presented by Preston and Coale are significant contributions to demography. The formulas summarize the relationships among various demographic measures precisely, and are formally analogous to the relationships that hold for stable populations. The significance of these formulas cannot be overstated; they allow us to understand clearly the relationships among demographic measures in any arbitrary population. However, when it comes to using them for estimating demographic measures when census data are defective, the method of estimation is still affected by defective data. The reason is that the series of age-specific growth rates reflects the observed census age distributions exactly so that any defects in the census data are summarized in the growth rates. This paper begins with the formulation of the discrete version of the "new synthesis" developed by Preston and Coale. With the discrete formulation, the three kinds of errors introduced when the continuous time formulas are applied to real data can be avoided. Then it is pointed out that when two accurate census data are available, the Preston-Coale procedure of "estimating" the age distribution at the second census is equivalent to checking the identity of the age distribution formula. Also "estimating" mortality by the procedure of Preston-Coale is shown to be equivalent to obtaining mortality directly from intercensal survival rates. That the procedure which involves the age-specific growth rates is equivalent to those that involve the intercensal survival rates may have escaped notice because there are no a priori constraints for patterns of age-specific growth rates to follow. The irregularity in growth rates due to defective data are not distinguishable from true irregularity that exists in the population, contrary to the well-known regularity in the pattern of survival rates in human populations.     

On the nature of errors involved in estimating stage-specific survival rates by Southwood's method for a population with overlapping stages

Summary This paper has examined the effect of within-stage mortality on the estimation of stage-specific survival rates bySouthwood's (1978, p. 358) method. As pointed out bySouthwood, both the severity and timing of mortality affect the mean duration of a life stage, and consequently the estimate of the number of individuals entering that stage. Knowledge of the form of the survivorship curve permits correction of the estimate under certain circumstances. The use ofSouthwood's method with two overlapping stages having different rates and patterns of mortality leads to complex errors in the estimation of survival for the first stage. The nature of these errors is examined analytically and via a simulation model.Southwood's method is fairly robust, with moderate differences in mortality rates leading to acceptable errors in estimating survival for the first stage. When both the rate and pattern of mortality in both life stages are the same, then the survival estimate is made without error. Precise estimates of stage-specific survival will not usually be possible withSouthwood's method because of the errors introduced by the very parameters being measured. Direct measurement of mortality rates and survivorship patterns (seeSouthwood, 1978, p. 309) is strongly advised, at least in preliminary work.     

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.     

Further improvements to a method for analysing stage-frequency data

Summary An iterative procedure for correcting stage-frequency data is described to allow for situations where the period during which a population is sampled begins after some individuals have entered stage 2 or ends before all individuals are dead. The reason for correcting data in this way is to enableKiritani andNakasuji's method for estimating stage-specific survival rates, with extensions proposed byManly (1976, 1977), to be used to analyse the data. The proposed procedure is illustrated on data obtained by sampling a population of the grasshopperChorthippus brunneus passing through four instar stages to reach the adult stage.     

Estimates of Survival of Diabetics from Repeated, Independent Sample Surveys

Little is known about death rates among diabetic populations. The few prior estimates have used two data systems, usually a registry or a survey to identify diabetics and death certificates to identify deaths. In this research, the diabetic population aged 18–94 in 1996–1998 and those surviving in 2001–2003 were estimated from repeated cross-sectional surveys, the Behavioral Risk Factor Surveillance System of the Centers for Disease Control and Prevention. Forward survival ratios were computed using a method developed for successive censuses and these were used to compute death rates. Nonlinear regression models for age-sex specific survival ratios were used to estimate parametric rates and thereby increase the accuracy of estimates. About 81.4 % (SE = 1.3 %) of diabetics survived 5 years, for an annual death rate of 41.1 per thousand (SE = 3.2). Among men survival was 84.7 % (SE = 2.1 %) with an annual death rate of 33.8 (SE = 4.9) per thousand; among women survival was 78.5 % (SE = 2.2 %) with an annual death rate of 48.1 (SE = 4.1) per thousand. Model estimates of mortality rates showed an odds ratio of 3.17 (95 % CI 2.64, 3.82) for each 10 year age interval and of 1.35 (95 % CI 1.02, 1.79) for women compared with men. Pooled annual samples, longer time intervals for survival, and parametric estimates of rates all help overcome the small numbers and large sampling variation of survey estimates of survival and mortality. Useful estimates of survival rates can be made from a single data system, a sample survey of the general population. This can be done for any condition where a respondent’s status at the earlier survey time is obtained at the later survey time. It could also be used to make estimates from periodic surveys for nations with limited information systems.     

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.     

Home ownership,life cycle stage,and residential mobility

Previous research has shown that mobility rates decline with increasing age and duration of residence. These relationships are investigated further for the case of residential mobility using residence histories obtained in interviews with 2264 Rhode Island residents. Three methods of classifying segments of a person’s life into life cycle stages are compared: age, life cycle stages based on marital status and child-rearing periods, and a combined age-marital status classification. These classifications were not found to be equivalent in that there was considerable variation in mobility rates by life cycle stage within age categories and by age within life cycle categories. The age-marital status classification was selected for use in the remainder of the analysis because it had the least variation in mobility rates within categories and required far less data for computation than the life cycle stages. When mobility rates were examined by home ownership, age-marital status, and duration of previous residence, it was observed that there was little variation in mobility rates by duration for home owners while the mobility rates for renters declined with duration.     

General growth balance method: A reformulation for populations open to migration

This paper proposes a reformulation of the general growth balance method for estimating census and registration completeness so as to make it applicable even to populations that are affected by migration. It also discusses a new procedure of line fitting that could be useful in countries where the input data are severely affected by age misreporting. The method is applicable to countries where data on age distribution of the population are available for two points in time from either censuses or surveys. Following closely the original proposal of Brass, it involves adjusting the ‘partial’ birth rates for age-specific disturbances from growth and migration rates. Beyond correcting the death rates, the method is useful in inferring the relative completeness of the censuses, and in deriving a robust estimate of birth rate under certain conditions. The application of the method is illustrated using the example of the male population of the Indian state of Uttar Pradesh for the period 1981 to 1991.     

General growth balance method: A reformulation for populations open to migration

This paper proposes a reformulation of the general growth balance method for estimating census and registration completeness so as to make it applicable even to populations that are affected by migration. It also discusses a new procedure of line fitting that could be useful in countries where the input data are severely affected by age misreporting. The method is applicable to countries where data on age distribution of the population are available for two points in time from either censuses or surveys. Following closely the original proposal of Brass, it involves adjusting the 'partial' birth rates for age-specific disturbances from growth and migration rates. Beyond correcting the death rates, the method is useful in inferring the relative completeness of the censuses, and in deriving a robust estimate of birth rate under certain conditions. The application of the method is illustrated using the example of the male population of the Indian state of Uttar Pradesh for the period 1981 to 1991.     

The multiregional matrix growth operator and the stable interregional age structure

Current population-forecasting efforts generally adopt minor variants of the cohort-survival projection method. This technique focuses on a population disaggregated into cohorts, a group of people having one or more common characteristics at a point in time and, by subjecting each cohort to class-specific rates of fertility, mortality, and net migration, generates a distribution of survivors and descendants of the original population, at successive intervals of time.Although cohort-survival methods take on a large number of variations, they all are essentially trend-based, dynamic, aspatial models of growth. The temporal element is introduced by a recursive structure which operates over a sequence of unit time intervals. The spatial dimension, when it is included at all, typically is accommodated by replicating the analysis over as many areal units as comprise the study area. Realistically, however, time and space need to be considered jointly in population-forecasting models. The need for interregional models which systematically introduce place-to-place movements and simultaneously consider the spatial as well as the temporal character of interrelated population processes is becoming increasingly apparent.Recently several demographers have taken advantage of the conceptual elegance and computational simplicity of matrix methods of population analysis. Their models, however, assume a "closed" population which is subject only to the processes of fertility and mortality. These, therefore, are not directly applicable to interregional "open" systems in which migration is frequently a much more variable and important contributor to population change than births or deaths. However, a natural extension of the demographer's matrix model allows one to incorporate place-to-place migration and provides an integrated interregional population-forecasting model which easily may be programmed for any of the current generation of digital computers. Such a model is outlined in this paper.     

Species diversity preserved in different numbers of nature reserves of the same total area

Conclusion and Summary The expected number of species occurring in different numbers of reserves of the same total area is examined on different assumptions of the spatial distribution and the probability of extinction. The advantage of one large reserve or several smaller ones of equal total area depends on the spatial distributions of species and the stage after the establishement of reserves. In general, several smaller reserves maintain more species immediately after the establishments unless the spatial distribution are uniform or random, whereas one large reserve excels several smaller ones after some rare species have gone extinct unless the spatial distributions are strongly contagious. Since the extinction of rare species must be facilitated as the size of each reserve reduces, the area of a reserve should be larger than the critical area that ensures the persistence of the species. Hence it is concluded that one or a few large reserves are a better strategy in order to maintain the species diversity.     

Multivariate survivorship analysis using two cross-sectional samples

As an alternative to survival analysis with longitudinal data, I introduce a method that can be applied when one observes the same cohort in two cross-sectional samples collected at different points in time. The method allows for the estimation of log-probability survivorship models that estimate the influence of multiple time-invariant factors on survival over a time interval separating two samples. This approach can be used whenever the survival process can be adequately conceptualized as an irreversible single-decrement process (e.g., mortality, the transition to first marriage among a cohort of never-married individuals). Using data from the Integrated Public Use Microdata Series (Ruggles and Sobek 1997), I illustrate the multivariate method through an investigation of the effects of race, parity, and educational attainment on the survival of older women in the United States.     

Household structure and housing needs

This paper is mainly derived from the material presented in the preceding article by S. P. Brown. Indeed, while the previous analysis is of considerable intrinsic interest, the hypothetical population was constructed and its family distribution was shown for the purpose of providing a basis for estimates of housing needs. For several reasons it appeared to be essential to have such a basis. First, any housing programme has to take the future, as well as the present, distribution of households by type and size into account. Secondly, such a programme has to be designed so as not to prevent household formation—there should be dwellings for all potential households, so that involuntary doubling-up need not occur. Thirdly, most residential areas should have dwellings for an eventually stable population, that is, for one which has variety of age groups and of household types, and also fair stability of housing demand. Estimates of the distribution of potential ‘households’ could be derived from the ‘family’ distribution of the hypothetical population which reflects current demographic trends. Thus although this population is a ‘hypothetical’ one, it provides a realistic premise for considering housing needs, and because it is a ‘stationary’ one, it provides an especially suitable premise. Moreover, since the demographic characteristics of its ‘families’ and therefore of its potential households were established in far greater detail than has ever been the case in sample surveys of existing households, it was possible to classify households in the terms which appear to be most appropriate for the first draft of a housing programme, irrespective of social and economic variations in demand.

The first stage in following up Mr Brown's analysis was the conversion of ‘families’ into ‘households’. Two examples of the possible household distribution of the hypothetical population are presented. Example A, which gives a realistic, but not extreme, picture of the conversion of families into households, is used for the subsequent detailed analysis, while broader figures for distribution B are also included.

In the second stage the various types of household had to be distinguished. For estimating housing needs, two interrelated criteria of household classification are relevant—first, the stage in the life of a household, especially appropriate in considering space requirements; secondly, the age composition of households, which largely determines the type of dwelling needed.

The detailed distribution of households by size and type, based on this classification, is further translated into a distribution of dwellings by type and size. For this purpose, additional assumptions about the number of rooms and the type of dwelling needed by households of various types are introduced and applied to the hypothetical population, both to household distributions A and B. These assumptions are not based on accepted standards, nor do they suggest standards. They are merely used for the purpose of illustrating a possible method of estimating housing needs on the basis of a detailed picture of household structure. They are further designed to represent one possible compromise between economy in dwelling distribution, on the one hand, and flexibility of space for individual households, on the other.

In the final sections of the paper, the implications of the dwelling distributions here presented are discussed in relation to household mobility, and also with reference to the necessity for reconciling short-term and long-term housing needs in any housing programme.     

Estimation of demographic measures for India, 1881-1961, based on census age distributions

Abstract India is one of the very few developing countries which have a relatively long history of population censuses. The first census was taken in 1872, the second in 1881 and since then there has been a census every ten years, the latest in 1971. Yet the registration of births and deaths in India, even at the present time, is too inadequate to be of much help in estimating fertility and mortality conditions in the country. From time to time Indian census actuaries have indirectly constructed life tables by comparing one census age distribution with the preceding one. Official life tables are available for all the decades from 1872-1881 to 1951-1961, except for 1911-1921 and 1931-1941. Kingsley Davis(1) filled in the gap by constructing life tables for the latter two decades. He also estimated the birth and death rates ofIndia for the decades from 1881-1891 to 1931-1941. Estimates of these rates for the following two decades, 1941-1951 and 1951-1961, were made by Indian census actuaries. The birth rates of Davis and the Indian actuaries were obtained basically by the reverse survival method from the age distribution and the computed life table of the population. Coale and Hoover(2), however, estimated the birth and death rates and the life table of the Indian population in 1951 by applying stable population theory. The most recent estimates of the birth rate and death rate for 1963-1964 are based on the results of the National Sample Survey. All these estimates are presented in summary form in Table 1.     

The Network Survival Method for Estimating Adult Mortality: Evidence From a Survey Experiment in Rwanda

Adult death rates are a critical indicator of population health and well-being. Wealthy countries have high-quality vital registration systems, but poor countries lack this infrastructure and must rely on estimates that are often problematic. In this article, we introduce the network survival method, a new approach for estimating adult death rates. We derive the precise conditions under which it produces consistent and unbiased estimates. Further, we develop an analytical framework for sensitivity analysis. To assess the performance of the network survival method in a realistic setting, we conducted a nationally representative survey experiment in Rwanda (n = 4,669). Network survival estimates were similar to estimates from other methods, even though the network survival estimates were made with substantially smaller samples and are based entirely on data from Rwanda, with no need for model life tables or pooling of data from other countries. Our analytic results demonstrate that the network survival method has attractive properties, and our empirical results show that this method can be used in countries where reliable estimates of adult death rates are sorely needed.     

The female population of Chile, 1855-1964: a microcomputer balance sheet method

This article summarizes the essential features of the inverse projection method and applies it to data on the female population of Chile for the period 1855-1964. Changes in age distribution, vital rates, life expectancy, fertility, and gross and net reproduction rates over time are described.     

Uncertain survival and time discounting: intertemporal consumption plans for family trusts

We derive expressions for optimal consumption for family trusts with random wealth and uncertain survival. Using UK birth statistics and the theory of branching processes, we compute size and survival probabilities for single- and multiple-branch families. Survival for a single-branch family is approximated by a Pareto distribution and consumption policies exhibit decreasing discount rates, but multiple-branch families use non-monotonic discount rates. When trust distributions depend on the number of beneficiaries rather than the survival of the whole family unit, spending paths depend on expected membership and the elasticity of intertemporal substitution. We report examples of consumption paths for a range of family trusts with constant relative risk aversion preferences.     

From new entries to retirement: The changing age composition of the U.S. male labor force by industry

The process whereby the age composition of an industry is formed appears to be largely a function of past rates of growth in employment; the social (or institutional) framework sets limits and affects the ensuing age composition but relatively little. The following types will illustrate this process. 1. Consider an industry which has increased considerably more rapidly in employment over several decades than has the total labor force. The rapid growth brings in a disproportionately large share of youth who are first entering the labor market; other younger workers move from slowly growing (or declining) industries. These movements add many more younger workers. On the other hand, there is little, if any, unemployment in the industry so that there are few pressures being exerted on the older workers to retire, and relatively few will retire. Under these circumstances the age composition will be younger than tliat of the entire male working force. 2. Consider an industry which has grown slowly, if at all, for some time. There will be comparatively fewer (in comparison with the first example) new entries and less mobility from other industries. The men already engaged in this industry will continue to work there; they gradually become older and are not counterbalanced by increasing numbers of young workers. Unemployment is likely to be higher, leading to a higher retirement rate. There are also likely to be large numbers of men a decade or two under the retirement age-the heritage of an "ancient" period when the industry had experienced significant increases in employment; these add pressure on the older men and more retire. The age composition of such an industry gradually veers toward the older side; it is considerably older than that of the entire male working force. At any given moment of time most industries will reflect variable past growth rates. For example, one industry may have a very large proportion of young workers because it grew very rapidly in employment only during the decade prior to the time of study (i.e., the time of a decennial census); another may have a large proportion in the middle ages reflecting very rapid growth two or three decades earlier, followed by very slow growth in the decade prior to the time of study; and so forth.In light of the foregoing analysis, it appears that technological change, as measured by average annual changes in output per worker, has little bearing on the age composition of an industry. Conversely, the latter probably does not affect changes in output per worker.     

Sensitivity analysis of discrete-time interregional population systems

This paper shows analytically how (a) the long-run growth rate and (b) the long-run proportional distribution of an interregional population system with a time-homogeneous structural matrix are affected by small changes or errors in (a) the natural growth rates of individual regions and (b) the interregional migration rates. Furthermore, the analytic results are applied to an eight-region Canadian population system. Finally, it is claimed that the method introduced here can be easily applied to sensitivity analysis of both the intrinsic growth rate and the “stable” age-composition of the Leslie model with respect to changes in age-specific birth and survival rates.