Inverse Projection and Back Projection: A Critical Appraisal,and Comparative Results for England, 1539 to 1871

Inverse projection and back projection are two methods for exploiting long historical series of births and deaths to produce estimates of population size and age structure, net migration, and vital rates. While inverse projection requires extraneous information on population size at scattered dates, back projection does not. In this paper I argue that back projection attempts an impossible task, and can only arbitrarily select one demographic past from among an infinite set of equally plausible and acceptable ones, which are also consistent with the input data. Inverse projection, on the other hand, is more modest in its goal, but is robust and straightforward. In an important and outstanding book, Wrigley and Schofield use back projection to reconstruct English demographic history from 1539 to 1871. In this paper, inverse projection is used to replicate their reconstruction under assumptions that are in important respects weaker, although these estimates are contingent on independent population size estimates for 1541 and 1696. The results buttress Wrigley and Schofield's reconstruction. However, it is argued that their data and reconstruction cannot offer independent evidence for the general levels of population before the mid-eighteenth century; rather, they help us to interpolate among benchmarks for which we have extraneous evidence, and contingent on these benchmarks, fill in the rich details of the demographic past.     

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.     

Back Projection and Inverse Projection: Members of a Wider Class of Constrained Projection Models

Constrained by estimated population totals and observed totals of births and deaths, estimates of age-structures from inverse projection have been widely used in historical demography. Back Projection attempted, by using hypothetical constraints on net-migration, to estimate the population totals as well, and has been used to derive ‘censuses’ for England from 1541 to 1871. A wider formulation, called Generalised Inverse Projection, is proposed in this paper, which replaces back-projection and establishes its relation to Inverse Projection more clearly. Under the same assumptions as back projection, but using the new method, the estimates for England are virtually unchanged. This new method is capable of performing population projections, subject to a wide variety of hypothetical and empirical constraints, for contemporary as well as historical data.     

Combining forward and backward mortality estimation

Demographers often form estimates by combining information from two data sources—a challenging problem when one or both data sources are incomplete. A classic example entails the construction of death probabilities, which requires death counts for the subpopulations under study and corresponding base population estimates. Approaches typically entail ‘back projection', as in Wrigley and Schofield's seminal analysis of historical English data, or ‘inverse’ or ‘forward projection’ as used by Lee in his important reanalysis of that work, both published in the 1980s. Our paper shows how forward and backward approaches can be optimally combined, using a generalized method of moments (GMM) framework. We apply the method to the estimation of death probabilities for relatively small subpopulations within the United States (men born 1930–39 by state of birth by birth cohort by race), combining data from vital statistics records and census samples.     

The abs population projections: Overview and evaluation

The paper provides a brief history of the ABS projection services in the context of its role as the central statistical agency in Australia. It also presents an overview of the methodology used for generating the projections and details the procedures used for compiling the various components of change incorporated in the projections. The accuracy of the ABS past projections is assessed by comparing their results, both at the national and state/territory levels, with corresponding estimated resident population figures. The paper also describes the range of projection services currently provided by the ABS, lists present unmet needs and indicates some of the developments likely to occur in the future in satisfying unmet demand, consolidation of existing services and dissemination of output.     

Long-range trends in adult mortality: Models and projection methods

In the study reported here, I had two objectives: (1) to test a new version of the logistic model for the pattern of change over time in age-specific adult mortality rates and (2) to develop a new method for projecting future trends in adult mortality. A test of the goodness of fit of the logistic model for the force of mortality indicated that its slope parameter is nearly constant over time. This finding suggests a variant of the model that is called the shifting logistic model. A new projection method, based on the shifting mortality model, is proposed and compared with the widely used Lee-Carter procedure.     

Probabilistic Projections of the Total Fertility Rate for All Countries

We describe a Bayesian projection model to produce country-specific projections of the total fertility rate (TFR) for all countries. The model decomposes the evolution of TFR into three phases: pre-transition high fertility, the fertility transition, and post-transition low fertility. The model for the fertility decline builds on the United Nations Population Division’s current deterministic projection methodology, which assumes that fertility will eventually fall below replacement level. It models the decline in TFR as the sum of two logistic functions that depend on the current TFR level, and a random term. A Bayesian hierarchical model is used to project future TFR based on both the country’s TFR history and the pattern of all countries. It is estimated from United Nations estimates of past TFR in all countries using a Markov chain Monte Carlo algorithm. The post-transition low fertility phase is modeled using an autoregressive model, in which long-term TFR projections converge toward and oscillate around replacement level. The method is evaluated using out-of-sample projections for the period since 1980 and the period since 1995, and is found to be well calibrated.     

Procedures in the biochemical estimation of age of vertebrates

Summary The most accurate method known to data for estimating age of wild vertebrates involves biochemical assay of eye lens proteins. Laboratory procedures that may be unfamiliar to many ecologists are described in detail. The method is based on precise changes in the amount of insoluble lens protein, and consists of two procedures: obtaining the appropriate lens fraction, and quantitatively analyzing its protein composition. Lenses are homogenized, and the insoluble fraction isolated by centrifugation. Protein content is measured colorimetrically by using the Lowry test. Instructions are given for single and double-beam spectrophotometers.     

开放区域的人口分组要素组合预测

随着经济发展和区域开放程度的提高,传统的区域人口预测方法已不合时宜。本文依据人口平衡方程,运用人口分要素组合预测法,阐述了开放区域人口预测的基本方法和步骤,从而为具体的人口预测工作提供思路借鉴。     

含迁移要素的区域人口随机预测方法及其应用--以浙江省宁波市为例

以宁波市为例,基于宁波市“六普”与“五普”数据,首先利用年龄别死亡率对同批次队列人口进行年龄移算,即人口留存分析,对比2010年的实际人口数和无迁移模式下的封闭人口数之差,进一步测算两次普查期间分性别、年龄别人口的净迁移量和净迁移率。然后,将迁移率与出生率和死亡率整合,构建多要素随机人口预测模型,预测宁波市未来人口变化趋势,并评估迁移、死亡、出生三要素对人口变动的弹性影响效果。结果表明：人口净迁入对宁波市人口总量变化影响最大,而低死亡率和低出生率是未来50年加剧宁波市人口老龄化的主要原因。未来50年内,劳动力人口净迁入对降低宁波市人口老年负担系数的作用有限。鼓励夫妇生育二孩,优化人才引进与落户机制,尽快建立起应对高龄社会的公共服务体系尤其是长期照护体系,应是宁波市人口政策改革的当务之急。     

小区域家庭户预测的理论、方法与应用

小区域人口和家庭户预测越来越受到学界和社会各界的重视 ,但该领域的研究受方法论和数据可得性等的限制 ,还存在许多没有解决的问题。而且发展中国家的综合家庭户和人口预测更是少见。本文应用近年来有关家庭人口学、区域人口学、小区域人口估计的理论和方法的最新成果 ,采用多维动态宏观家庭户预测模型ProFamy,对中国两个区域泰和县和深圳特区的人口和家庭户进行了同时、一致的预测。通过实际应用 ,提出了小区域家庭户预测的理论、方法和应用的一些思考     

Probabilistic population projections for countries with generalized HIV/AIDS epidemics

In 2015, the United Nations (UN) issued probabilistic population projections for all countries up to 2100, by simulating future levels of total fertility and life expectancy and combining the results using a standard cohort component projection method. For the 40 countries with generalized HIV/AIDS epidemics, the mortality projections used the Spectrum/Estimation and Projection Package (EPP) model, a complex, multistate model designed for short-term projections of policy-relevant quantities for the epidemic. We propose a simpler approach that is more compatible with existing UN projection methods for other countries. Changes in life expectancy are projected probabilistically using a simple time series regression and then converted to age- and sex-specific mortality rates using model life tables designed for countries with HIV/AIDS epidemics. These are then input to the cohort component method, as for other countries. The method performed well in an out-of-sample cross-validation experiment. It gives similar short-run projections to Spectrum/EPP, while being simpler and avoiding multistate modelling.     

Ethnic population projections for the UK, 2001–2051

This paper reports on projections of the United Kingdom’s ethnic group populations for 2001–2051. For the years 2001–2007 we estimate fertility rates, survival probabilities, internal migration probabilities and international migration flows for 16 ethnic groups and 355 UK areas. We make assumptions about future component rates, probabilities and flows and feed these into our projection model. This model is a cohort-component model specified for single years of age to 100+. To handle this large state space, we employed a bi-regional model. We implement four projections: (1) a benchmark projection that uses the component inputs for 2001; (2) a trend projection where assumptions beyond 2007 are adjusted to those in the UK 2008-based National Population Projections (NPP); (3) a projection that modifies the NPP assumptions and (4) a projection that uses a different emigration assumption. The projected UK population ranges between a low of 63 millions in 2051 under the first projection to a high of 79 million in the third projection. Under all projections ethnic composition continues to change: the White British, White Irish and Black Caribbean groups experience the slowest growth and lose population share; the Other White and Mixed groups to experience relative increases in share; South Asian groups grow strongly as do the Chinese and Other Ethnic groups. The ethnic minority share of the population increases from 13% (2001) to 25% in the trend projection but to only 20% under our modified emigration projection. However, what is certain is that the UK can look forward to be becoming a more diverse nation by mid-century.     

Estimation of local demographic variation in a flexible framework for population projections

A cohort component projection of local populations based on sex and single year of age offers great value for planning local services, but demands data beyond the detail available. Local fertility, mortality and migration schedules by age and sex must be estimated sensitively to local variation if the results are to be of greater value than simpler methods of projection. Two approaches are compared, using data for the recent past: (a) direct estimation of local area age-specific schedules of fertility, mortality and migration based on data available to the national statistical agency; (b) graduation of national schedules using only local area population estimates by age, total numbers of births, and total numbers of deaths; age-specific migration is indirectly estimated from successive population estimates. These two approaches are compared with a projection using the same rates for each area. The three projections have been implemented for electoral wards in the Fife local government area of Scotland, using the flexible framework provided by POPGROUP software. Persuasive local population projections based on standard data for standard areas are feasible without the regular publication of migration flows.     

Sensitivity analysis in a multidimensional demographic projection model with a two-sex algorithm

"Formulas are derived for the effect that a slight change in the occurrence/exposure rate of the multidimensional projection model has on the elements of the population vector. The projection model classifies the population by sex, age, and marital status. The model includes a two-sex algorithm in order to ensure consistency between numbers of male and female marriages, number of divorces for the two sexes, and new widows (widowers) and deceased spouses. The sensitivity functions and elasticities are applied to data from the Netherlands for the period 1980-1984. The results indicate that marriage market mechanisms, in particular competition and substitution effects, are reasonably well modelled."     

On the decomposition of changes in expectation of life and differentials in life expectancy

The projection of mortality rates requires inter alia close examination of the mortality experience of a population over a long period of time and will usually also involve the analysis of mortality trends by cause of death. In two of the more important recent contributions, techniques were devised for explaining change in life expectancy in terms of mortality changes in particular age groups and by different causes of death. The approaches adopted by the authors differ, and the purpose of this article is to reconcile the two and tie the results in with those obtained by earlier writers. A new method for explaining the change in a life expectancy differential in terms of the observed changes in the mortality differentials and the observed change in overall mortality level is also described.     

The fertility of two-generation urbanites

In this paper it is argued that in studies of urban fertility, the relationship between socio-economic variables and fertility has been obscured by the presence of rural migrants in the populations under investigation. Accordingly, data obtained from families of completed fertility in six probability samples of metropolitan Detroit are divided into two groups, farm migrants and two-generation urbanites.

In general, the socio-economic differences in fertility observed among the “pure” urban types in Detroit are found to be small and inconsistent, most of them being statistically insignificant. The inverse fertility pattern found in the total Detroit population is attributed to : (a) the overrepresentation of farm migrants (who have high fertility) in the lowest social and economic positions in the city, and (b) the pronounced inverse pattern of fertility among the farm migrants.

It is suggested that the absence of an inverse fertility pattern among twogeneration urbanites and its presence among the farm migrants can be attributed to differences in family organization.     

Lessons from stochastic small-area population projections: the case of Waikato subregions in New Zealand

Subnational population projections in New Zealand by means of the conventional deterministic cohort-component method have had a tendency to be conservative: underprojecting fast-growing populations and overprojecting slow-growing ones. In this paper we use a stochastic population projection method as an alternative. We generate population projections for five demographically distinct administrative areas within the Waikato region of New Zealand: Hamilton City, Franklin District, Thames-Coromandel District, Otorohanga District and South Waikato District. The results are compared to official subnational deterministic projections. The accuracy of subnational population projections in New Zealand is strongly affected by the instability of migration as a component of population change. Differently from the standard cohort-component method, in which net migration levels are projected, the key parameters of our method are age-gender-area specific probabilistic net migration rates. Generally, the identified and modelled uncertainty makes the traditional ‘mid-range’ scenario of subnational deterministic projections of limited use for policy analysis or planning beyond a relatively short projection horizon. We find that the projected range of rates of population growth is wider for smaller regions and/or regions more strongly affected by net migration. Directions for further development of the methodology are suggested.     

A Comparative Evaluation of Error and Bias in Census Tract-Level Age/Sex-Specific Population Estimates: Component I (Net-Migration) vs Component III (Hamilton–Perry)

While the housing-unit method continues to be the preferred method nationwide for producing small-area population estimates, this procedures lacks a method for making age/sex-specific estimates. This paper reports evaluation research on implementation of component-based methods for estimating census tract populations with age/sex detail. Two alternatives are explored: (1) the Component I method relying upon estimates of births, deaths, and net-migration and (2) the Component III method relying solely upon 1990 and 2000 Census counts. From an April 1, 2000 base, each method is used to make estimates moving forward to an April 1, 2010 estimate that is compared to the results of the 2010 Census. The two methods are compared in terms of accuracy and bias using both absolute and algebraic mean and median percentage errors. Results are reviewed and discussed in light of their implications for applied demographers tasked with making small-area demographic estimates.     

Some aspects of the use of birth expectations data from sample surveys for population projections

The value of national sample survey data relating to birth expectations for projecting births is reassessed in the light of data limitations pointed out by Ryder and Westoff among others and of the methods of projection used by the United States Bureau of the Census. The annual level of fertility under the cohort-fertility projection method depends on the assumptions regarding completed fertility, about which the available survey data are fairly informative, and on the assumptions regarding the timing of births, about which the survey data tell us very little. Test calculations suggest that Ryder and Westoff have overstated the significance of timing relative to completed family size for the level of future births. We believe that the fall in the annual total fertility rate in the first half of the sixties is to be explained only in part by a general delay in childbearing; a moderate to substantial decrease in completed family size has also occurred. Analysis of the latest set of fertility projections of the Census Bureau also suggests that the assumptions about completed fertility are a much more important determinant of the level of future births than timing, both in the short and long term. Although the available expectations data cannot help in predicting short-term annual changes in fertility, they appear useful for making long-term projections of annual fertility. Expansion of the size, frequency, and content of the sample surveys and incorporation of parity and birth interval into the projection method may improve projections.